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Dimensional cross-over of the charge density wave order parameter in thin exfoliated 1T-VSe 2 Árpád Pásztor DQMP Université de Genève 24 quai Ernest AnsermetCH-1211GenevaSwitzerland Alessandro Scarfato DQMP Université de Genève 24 quai Ernest AnsermetCH-1211GenevaSwitzerland Céline Barreteau DQMP Université de Genève 24 quai Ernest AnsermetCH-1211GenevaSwitzerland Enrico Giannini DQMP Université de Genève 24 quai Ernest AnsermetCH-1211GenevaSwitzerland Christoph Renner DQMP Université de Genève 24 quai Ernest AnsermetCH-1211GenevaSwitzerland Dimensional cross-over of the charge density wave order parameter in thin exfoliated 1T-VSe 2 1 The capability to isolate one to few unit-cell thin layers from the bulk matrix of layered compounds 1 opens fascinating prospects to engineer novel electronic phases. However, a comprehensive study of the thickness dependence and of potential extrinsic effects are paramount to harness the electronic properties of such atomic foils. One striking example is the charge density wave (CDW) transition temperature in layered dichalcogenides whose thickness dependence remains unclear in the ultrathin limit 2-5 .Here we present a detailed study of the thickness and temperature dependences of the CDW in VSe 2 by scanning tunnelling microscopy (STM). We show that mapping the real-space CDW periodicity over a broad thickness range unique to STM provides essential insight 6 . We introduce a robust derivation of the local order parameter and transition temperature based on the real space charge modulation amplitude. Both quantities exhibit a striking non-monotonic thickness dependence that we explain in terms of a 3D to 2D dimensional crossover in the FS topology. This finding highlights thickness as a true tuning parameter of the electronic ground state and reconciles seemingly contradicting thickness dependencies determined in independent transport studies. Following the ground-breaking exfoliation of graphite into one atom thin carbon sheets, an increasing number of layered compounds can now be isolated from their bulk matrix in the form of one to few unit-cell thin layers. These sheets often feature unique [7][8][9] or enhanced 2,10 properties in comparison to their parent bulk compounds. They depend on material thickness and can be further tuned through doping, electrostatic gating and assembly of distinct layers into complex heterostructures. Transition metal dichalcogenides (TMDs) are of particular interest in this context. They can be readily exfoliated into thin flakes down to the single unitcell limit 11 and offer a unique playground for studying the thickness dependence of their electronic properties. For example, in MoS 2 , photo active transitions become available in the single layer limit due to the appearance of a direct gap in the band structure. Metallic TMDs host a variety of electronic phases like superconductivity and charge density waves (CDW) [12][13][14][15][16][17] whose transition temperatures can be modified by reducing the thickness of the host crystal 2,4,5,10 . Exfoliation offers a new degree of freedom to engineer these electronic ground states. Unfortunately, thin flakes of metallic TMDs often degrade in air 18 and have thus been much less studied than their semiconducting and insulating counterparts. To overcome this limitation, we developed a mechanism enabling in-situ exfoliation. This study is focused on the thickness dependence of the CDW phase in VSe 2 , a metallic TMD that grows in the 1T polymorph. It consists of van der Waals bonded slabs of triangular vanadium layers sandwiched between two triangular selenium layers (Fig. 1a, left inset). Each vanadium is surrounded by six selenium atoms in an octahedral configuration with in-plane and out of plane lattice constants = = 3.36 Å and = 6.104 Å, respectively 19 A more quantitative analysis of the CDW contrast revealed by STM is required to go beyond the above approximate visual assessment. The amplitude of the gap near associated with the CDW phase transition would be a natural order parameter. However, tunneling spectroscopy does not show any significant reduction in the LDOS at at the phase transition ( Fig. 1b), consistent with the tiny portion of the Fermi surface that is gaped in the CDW phase of 1 -VSe 2 25 . Moreover, the vanadium derived -band just below obscures the CDW gap. A convenient alternative measure to describe the phase transition when the quasi-particle gap is not clearly observable is the CDW modulation amplitude 26 , which depends linearly on the gap in a mean field description 27 . For a quantitative analysis of the CDW phase transition, we thus introduce an order parameter defined as = ∫ ( , )d d ∫ ( , )d d(1) where ( , ) is the amplitude in Fourier space and and are circular shaped integration areas around the CDW and lattice peaks, respectively (Fig. 1d). and were chosen such that the k-space area is the same for all examined micrographs (| | = | | = 0.2 nm −2 ). We normalize to the atomic lattice components to account for possible differences in tunneling conditions, which result in small variations in the appearance of the CDW patterns imaged by STM. Note this variability is the same for bulk crystals and thin exfoliated flakes and does not affect the calculated order parameter . The temperature dependence of near the phase transition of a single crystal (Fig. 3a) can be modeled in a mean field description by a phenomenological BCS gap equation 28 : ( ) = ⋅ ⋅ tanh (1.74 ⋅ √ − 1)(2) where is a scaling factor to be determined. The solid line in Fig. 3a is a fit of equation (2) to the experimental data points, where we have set = T =105 K. The fit has a single adjustable parameter which we find to be = 0.0165 ± 0.0004. The consistent picture emerging from this analysis confirms that equation (1) is an adequate measure of the order parameter enabling a quantitative analysis of the CDW phase transition. Examining a large set of STM images from different terraces and flakes at = 77.6 K, we find a non-monotonic thickness dependence of the CDW order parameter (Fig. 3c). is gradually decreasing from its bulk value when reducing the crystal thickness down to 20 nm. When thinning the crystal further, below 20 nm, the thickness dependence is reversed and is increasing to even significantly exceed the bulk value in the thinnest regions measured here (2.2 nm). The order parameter is not suitable for a direct comparison of our STM data with published transport experiments 4,5 . The latter report as a function of thickness, which is challenging to measure by STM because of thermal drift making it difficult to maintain the tip position over a specific location while changing the temperature above 77 K. As it turns out, we can use equation (2) to calculate the local based on ( ) extracted from STM images at a given temperature < . Assuming the scaling factor A is the same for all thicknesses, equation (2) provides a direct correspondence between and the order parameter ( ). To verify this assumption, we plot equation (2) for the 10 nm, 20 nm, and 50 nm thin terraces, using the transition temperature =122 K, 87 K, and 100 K, respectively, calculated for each thickness from ( = 77.6 ). The agreement with the experimental data points is excellent as shown in Fig. 3b. This demonstrates the validity of this method, providing an unprecedented ability to determine the local CDW transition temperature solely based on the charge modulation amplitude measured by STM. The expected CDW gap amplitudes within this weak coupling model are in the range of 3.8 meV to 5.7 meV depending on crystal thickness, too small to be properly resolved above 77 K. The CDW transition temperature calculated for different thicknesses using the above method are plotted in Fig. 4. They are in remarkable quantitative agreement with independent transport studies 4,5 . The most striking findings of our analysis are a non-monotonic thickness dependence of and a significant increase of above the bulk value in the thinnest terraces measured here. The opposite thickness dependencies of we find in very thin compared to thicker terraces lift the contradicting results reported by Xu et al. 4 and Yang et al. 5 . That discrepancy has been blamed on the distinct liquid and mechanical exfoliation techniques used in these studies, when it is in fact the result of investigating different thickness ranges. The STM data presented in Fig. 4 suffers no such ambiguity; they were obtained with the same tip probe on flakes prepared in an identical mechanical exfoliation process. The two distinct and opposite thickness dependencies of the CDW transition temperature shown in Fig. 4 suggest a crossover from a three-dimensional (3D) to a two-dimensional (2D) regime around 20 nm. has been found to increase with decreasing thickness also for other very thin TMD compounds 2,10 . However, these studies lack important real space information to fully assess the nature of the CDW. STM directly and unambiguously shows no alteration in the CDW symmetry and periodicity with thickness and temperature. The only modification we observe is the charge modulation amplitude associated with the change in T c . We propose that the enhanced T c in the thinnest samples is a consequence of spatial confinement, in analogy to BCS superconductors governed by a gap equation similar to equation (2). In that case, for a confinement potential above a critical value, theory predicts the superconducting transition temperature ( ) to increase with decreasing thickness before vanishing to zero in the zero thickness limit 29 . The characteristic thickness for which is maximum in this model depends on , on the coupling strength, and on the carrier density. It is important to note that does not necessarily correspond to the single layer limit. It can be larger and it is thus necessary to examine a range of thicknesses to draw definite conclusions about the thickness dependence of . In particular, considering only bulk and single layer crystals may lead to contradicting conclusions about the effect of dimensional confinement, even in the same material if different preparation and substrates result in different . The decreasing with decreasing thickness above 20 nm can be understood considering the Fermi surface (FS) topology of VSe 2. It has a significant dispersion of a few eV along 21, 25 , different from the mostly 2D FS of other layered TMDs. Photoemission 21,30 reveals large parallel FS portions centered at the M(L) points of the Brillouin zone. They offer good inplane nesting conditions that persist for all . This nesting is strongest at a particular , resulting in an effective out of plane nesting vector and a 3D CDW 21 in bulk VSe 2 . Upon thinning the crystal to 56 nm and 20 nm (~93 and ~33 layers), the out-of-plane nesting condition becomes weaker due to the discretization of the FS by the reduced number of available points. This drives the system into a weaker 2D charge order that is further suppressed by enhanced fluctuations expected in 2D. In summary, we find a striking non-monotonic thickness dependence of the CDW transition temperature in mechanically exfoliated 1T-VSe 2 from bulk to 2.2 nm thin flakes. On the other hand, the modulation period and alignment with the atomic lattice are entirely independent on thickness. We propose this behavior is a consequence of a 3D to 2D dimensional crossover in the FS topology around 20 nm thickness combined with quantum confinement in thinner flakes. The dimensional crossover weakens the bulk CDW in the thicker flakes and the confinement enhances it in the thinner ones. Unambiguous evidence for this behavior is provided by the local determined from the CDW order parameter measured by STM over an unprecedented broad range of thicknesses in a given experiment. We demonstrate that the charge modulation amplitude provides a suitable measure of the CDW phase transition order parameter. This allows a robust determination of the local critical temperature based solely on STM topographic images of the CDW at a given temperature below . Interestingly, the exact same approximate form of the (weak coupling) BCS equation quantitatively describes the CDW order parameter and critical temperature, independent of sample thickness and temperature. The present study strongly suggests that the thickness dependence reported hereis not a consequence of a varying coupling strength or the signature of a different CDW phase, but indeed due to the Fermi surface topology, dimensional crossover and quantum confinement.Methods1 -VSe 2 single crystals were grown by chemical vapor transport using I 2 as a transport agent and then mechanically exfoliated in-situ (3⋅10 -8 mbar) onto Au(111) single crystal substrates.Prior to the exfoliation, the Au(111) surface was cleaned and reconstructed by repeated cycles of Ar + ion sputtering and annealing at 450°C in ultra-high vacuum (UHV). The exfoliated flakes are hardly visible by optical means in our UHV scanning tunneling microscope (STM) setup. We thus relied on a suitable coverage density produced by our bespoke in-situ exfoliation mechanism to position the STM tip over an exfoliated flake in a systematic scan and search procedure. The STM experiments were done in UHV (base pressure 2⋅10 -10 mbar) using tips electrochemically etched from an annealed tungsten wire. The bias voltage was applied to the sample. Tunneling ( ) and differential conductance / ( ) spectra were acquired simultaneously using a standard lock-in technique with a 7.1 mV rms bias modulation at 337.7 Hz. Figure 1 | 1Transport and STM characterization of bulk 1T-VSe 2 single crystals. (a) Resistivity as a function of temperature with a kink near 105 K associated with the CDW phase transition (left inset: 1T-VSe 2 crystal structure; right inset: ( )/ ). (b) Tunnelling spectra measured at 77.6 K on the surface shown in panel c. (c) 10×10 nm 2 atomic resolution STM image (V bias =-100 meV, =10 pA) of a cleaved surface at 77.6 K and (d) corresponding Fourier transform. Red and green circles indicate the first-order atomic lattice and CDW modulation peaks, respectively. The circle size depicts the integration area around each peak used to calculate the CDW order parameter . Figure 2 | 2STM images of exfoliated 1T-VSe 2 flakes. 5×5 nm 2 atomic resolution micrographs (V bias =-100 meV, =10 pA) measured on different thickness terraces and flakes at 77.6 K and 95.0 K. The atomic lattice is well resolved in all images. The CDW is strongest in the thinnest regions (10 nm) at both temperatures and nearly absent in the 20 nm and 50 nm thin regions at 95.0 K. Figure 3 | 3Temperature and thickness dependencies of the CDW order parameter in 1 -VSe 2 . (a) as a function of temperature in a bulk single crystal near the phase transition. The solid line is a fit to the BCS approximate form ( ) = ⋅ ⋅ tanh(1.74 ⋅ √( / − 1)), where = 105 K and the only fitting parameter is determined to be = 0.0165 ± 0.0004. (b) as a function of temperature for three different thicknesses. The solid lines are calculated with the above BCS interpolation using the bulk scaling factor A and the local T c calculated from ( = 77.6 ) for each thickness. (c) as a function of thickness at 77.6 K. In all panels the error bars correspond to the dispersion of as determined by analyzing many different STM images for a given temperature and thickness. Figure 4 | 4Thickness dependence of the CDW transition temperature in 1 -VSe 2 . Solid squares represent T c calculated from the charge modulation amplitude imaged by STM using equations (1) and (2). They reveal a clear non-monotonic dependence of T c on thickness. The vertical error bars correspond to the dispersion from many different STM images. The green dashed line symbolizes the transition temperature of a bulk sample. Our data are in quantitative agreement with and reconcile data from previous transport experiments covering separate thickness ranges (represented by the solid red circle 4 and the solid blue line 5 ). . Bulk 1T-VSe 2 undergoes a CDW phase transition at ≃ 105 K into a commensurate 4 × 4 superlattice within the layers (ab-plane) and an incommensurate ∼ 3.1 modulation along the c-axis[19][20][21] . The CDW transition temperature in thin flakes (T c ) has been found to deviate up to 30% from these bulk values, with contradicting findings where T c is either increased 4 or reduced5 , seemingly dependent on sample preparation.Before investigating the thickness dependence of the CDW, we have characterized it in bulk single crystals by transport measurements and STM. Resistivity as a function of temperature (Fig. 1a) shows a characteristic kink at the CDW phase transition ≃ 105 K, in agreement with previous studies 22 . Constant current STM images and corresponding Fourier transforms at 77.6 K (Figs. 1c,d) clearly reveal a triangular atomic lattice ( = 3.36 Å) and the in plane 4 × 4 commensurate CDW modulation 23 . Tunneling spectroscopy (Fig. 1b) is consistent with data reported elsewhere 24 , including a characteristic conductance peak associated with the vanadium derived -band below the Fermi level ( ) and an asymmetric -shaped background centered on . To gain insight into the thickness dependence of the CDW, we take advantage of steps and terraces naturally present on the exfoliated flakes. The local thickness is directly quantified from the STM topographic traces as the height of the terrace above the reconstructed Au(111) substrate. Here, we concentrate on STM micrographs measured above 77 K, near where thickness dependent CDW features are most prominent. Topographic and CDW features imaged by STM on different exfoliated thin flakes and terraces with distinct thicknesses (Fig. 2) are very similar to those in bulk crystals. Remarkably, we observe the same 4 × 4 charge order down to the thinnest sample studied (2.2 nm). However, a closer inspection of the 77.6 K micrographs reveals a noticeably weaker CDW amplitude in the 20 nm thin region than in all other thicknesses. At 95.0 K, closer to , the CDW is almost completely suppressed on the 20 nm and 50 nm thin flakes -similar to what we observe in bulk crystals - while it remains surprisingly strong on the thinnest 10 nm flake. AcknowledgementWe acknowledge A. Morpurgo, Ch. Berthod and T. Giamarchi for stimulating discussions, L.Musy for his contributions to the initial attempts of in-situ exfoliation, and G. Manfrini and A.Guipet for their technical assistance. 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Scanning tunneling microscopy of atoms and charge-density waves in 1T-TaS 2 , 1T-TaSe 2 , and 1T-VSe 2. B Giambattista, C G Slough, W W Mcnairy, R V Coleman, Phys. Rev. B Condens Matter. 41Giambattista, B., Slough, C. G., McNairy, W. W. & Coleman, R. V. Scanning tunneling microscopy of atoms and charge-density waves in 1T-TaS 2 , 1T-TaSe 2 , and 1T-VSe 2 . Phys. Rev. B Condens Matter 41, 10082-10103 (1990). Locally modified charge-density waves in Na intercalated VSe 2 studied by scanning tunneling microscopy and spectroscopy. Inger Ekvall, E W Hans, E Brauer &amp; Olin, H , Phys. Rev. B. 59Inger Ekvall, E. W., Hans E. Brauer & Olin, H. Locally modified charge-density waves in Na intercalated VSe 2 studied by scanning tunneling microscopy and spectroscopy. Phys. Rev. B 59, (1999). Three-Dimensional Fermi-Surface Nesting in 1T-VSe 2 Studied by Angle-Resolved Photoemission Spectroscopy. T Sato, J. Phys. Soc. Jpn. 73Sato, T. et al. 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{'abstract': 'The capability to isolate one to few unit-cell thin layers from the bulk matrix of layered compounds 1 opens fascinating prospects to engineer novel electronic phases. However, a comprehensive study of the thickness dependence and of potential extrinsic effects are paramount to harness the electronic properties of such atomic foils. One striking example is the charge density wave (CDW) transition temperature in layered dichalcogenides whose thickness dependence remains unclear in the ultrathin limit 2-5 .Here we present a detailed study of the thickness and temperature dependences of the CDW in VSe 2 by scanning tunnelling microscopy (STM). We show that mapping the real-space CDW periodicity over a broad thickness range unique to STM provides essential insight 6 . We introduce a robust derivation of the local order parameter and transition temperature based on the real space charge modulation amplitude. Both', 'arxivid': '1703.07212', 'author': ['Árpád Pásztor \nDQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland\n', 'Alessandro Scarfato \nDQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland\n', 'Céline Barreteau \nDQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland\n', 'Enrico Giannini \nDQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland\n', 'Christoph Renner \nDQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland\n'], 'authoraffiliation': ['DQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland', 'DQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland', 'DQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland', 'DQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland', 'DQMP\nUniversité de Genève\n24 quai Ernest AnsermetCH-1211GenevaSwitzerland'], 'corpusid': 119493559, 'doi': '10.1088/2053-1583/aa86de', 'github_urls': [], 'n_tokens_mistral': 7976, 'n_tokens_neox': 6706, 'n_words': 4158, 'pdfsha': '03ffacfd0c3c5cefa7875c5e2d3762bcf6ff2003', 'pdfurls': ['https://arxiv.org/pdf/1703.07212v1.pdf'], 'title': ['Dimensional cross-over of the charge density wave order parameter in thin exfoliated 1T-VSe 2', 'Dimensional cross-over of the charge density wave order parameter in thin exfoliated 1T-VSe 2'], 'venue': []}

arxiv

Speedup of Micromagnetic Simulations with C++ AMP On Graphics Processing Units Ru Zhu Graceland University 50140LamoniIowaUSA Speedup of Micromagnetic Simulations with C++ AMP On Graphics Processing Units 1MicromagneticsGPUC++ AMP A finite-difference Micromagnetic solver is presented utilizing the C++ Accelerated Massive Parallelism (C++ AMP). The high speed performance of a single Graphics Processing Unit (GPU) is demonstrated compared to a typical CPU-based solver. The speed-up of GPU to CPU is shown to be greater than 100 for problems with larger sizes. This solver is based on C++ AMP and can run on GPUs from various hardware vendors, such as NVIDIA, AMD and Intel, regardless of whether it is dedicated or integrated graphics processor. Introduction Micromagnetic simulations are important tools to study magnetic dynamics and design new magnetic devices. Micromagnetic solvers running on Central Processing Unit (CPU) such as OOMMF [1] and magpar [2] have been widely used in magnetism research. Micromagnetic simulations of complex magnetic structures require fine geometrical discretization, and are time consuming. There has been growing research work on applying general purpose Graphics Processing Units (GPU) in the fields of Micromagnetics, such as MuMax, FastMag and GPMagnet [3] - [7]. Due to the high computing power of GPU units, these works have achieved considerable speed-ups as compared to previous CPU based implementations. On the other hand, general purpose GPU units are cheap, most costing less than $1000. Therefore complex micromagnetic simulations can be done at much lower cost. However, these implementations are exclusively based on NVIDA's parallel computing platform Compute Unified Device Architecture (CUDA) and their applications are limited to NVIDIA GPUs. In 2012, Microsoft released its own parallel programming library named C++ AMP which is an open specification and is hardware platform independent [8]. Software based on C++ AMP can run on virtually all latest GPUs, including those from NVIDIA, AMD and Intel. The purpose of this work then is to implement a cross-platform micromagnetic solver for solving the Landau-Lifshitz-Gilbert (LLG) equation. Section 2 discusses the formulation of the magnetic dynamics and LLG equation, especially the formulas used to calculate the demagnetization field. Section 3 describes the software implementation of the micromagnetic solver. Section 4 presents the performance of this micromagnetic solver at various problem sizes, and compares it with a popular CPU-based micromagnetic solver. Principle Consider a magnetization vector M  = (M x , M y , M z ) in) ( ) ( 2 1 ) ( ] ) ( ) ( ) extern demag s z y u s x s x s x                  (1) The right hand side of (1) consists of the exchange, anisotropy, demagnetization and Zeeman energy densities, where A is the material exchange constant, K u is the uniaxial anisotropy constant, 0  is the vacuum permeability, H demag is the demagnetization field and H extern is the external field. The anisotropy energy is assumed to be uniaxial with an easy axis on the x direction. The change of magnetization vector is caused by the effective magnetic field H eff derived from the magnetic energy density: According to (1) and (2) , extern demag anis exch eff H H H H M H               (2)x s x exch M M A H 2 2 , 2   .(3) To derive the exchange field H exch we need to discretize the computing region properly and consider the magnetizations of neighboring computational cells. The entire computing region is divided into n x ×n y ×n z cells, each cell with an equal volume of Note that i, j and k are zero-indexed to follow the convention of C++ programming language. According to (3), the Cartesian components of the effective field can be expressed as }. ) 1 , , ( ) , ,( 2 )1 , , ( ) , 1 , ( ) , ,( 2 ), 1 , ( ) , , 1 ( ) , , ( 2 ) , , 1 ( { 2 2 2 2 2 , z k j i M k j i M k j i M y k j i M k j i M k j i M x k j i M k j i M k j i M M A H x x x x x x x x x s x exch                   (4) Other components of exch H  can be obtained by replacing x with y or z in (4). According to (1) and (2), . 2 2 0 , x s u x anis M M K H    (5) The LLG equation in the low damping limit is [9] )] ( [ ) 1 ( ) ( 1 0 2 0 2 eff s eff H M M M H M dt M d                    (6) where α is the damping constant, and γ is the gyromagnetic ratio. To speed up the micromagnetic simulation, it is necessary to decrease per-step simulation time, most of which is consumed by the calculation of the demagnetization field. The brute force calculation of demagnetization field is known to be proportional to the square of the number N of the computational cells [10]. However, this calculation can be accelerated by taking advantage of the discrete convolution theorem and the fast Fourier transform (FFT) [11]. Consider a rectangular magnetic sample with its edges parallel to Cartesian coordinate axes x, y and z. For simplicity, we will start from one-dimensional case. Divide the sample into n x cells on the x direction, and label each cell with an index i, 1 , , 0   x n i  . Then the demagnetization field applied on cell i is      1 0 ) ( ) ( ) ( x n l demag i l K l M i H(7) Where K(l-i) is the demagnetization tensor giving the contribution of source cell M(l) to observation cell i. Since the tensor K is solely determined by the difference of l and i, and it is obvious that H demag is a convolution of K (demagnetization tensor) and M (magnetization). To avoid the effect of cyclic convolution, we have to do zero-padding to original data so that fast Fourier transform can be performed. It is also necessary to achieve correct output in real space after performing the inverse FFT as described in step (5) below. Here is how the zero-padding algorithm can be implemented [11]: (4) According to the DFT theorem, the DFT of right hand side of (7) is member-wise dot product of DFTs of M and K: (1) Set M (i), 1 2 , ,   x x n n i  to zero while keeping the original M(i) 1 , , 0   x n i  intact. 0 1 n x -1 2n x -1 (2) Set K (2n x -1) = 0 while keeping the original K(i), 1 , , 0 , ), 1 (     x x n n i   intact. 0 1 n x -1 2n x -2 2n x -1(3)) ( ) ( ) ( i K i M i H demag  (H FFT  . In three-dimensional space a rectangular magnetic sample can be divided into n x ×n y ×n z cells. After zero padding the input data size increases to 2n x ×2n y ×2n z , as demonstrated by Fig. 1. • • • • 0 0 0 0 • • • • • • • 0                    (12) Other components of H demag can be obtained by replacing x with y or z in (12). By applying DFT theorem to both sides of the equation, we can get ) , ,( ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( , kj i K k j i M k j i K k j i M k j i K k j i M k j i H xz z xy y xx x x demag      (13) Finally, the demagnetization field demag H can be obtained by taking the inverse FFT of demag H , as described by (10). Implementation GPUs have intrinsically different hardware architecture from CPUs, notably for its large number of Arithmetic Logic Units (ALU) that was initially designed for graphics rendering but now also used for general purpose computing. Since GPU is specialized for computing-intensive, highly parallel computation, it is ideally suitable for micromagnetic simulations in which large number of computational cells can be processed in parallel. This idea is schematically illustrated by Fig. 2. C++ AMP was implemented with High Level Shading Language (HLSL), which was initially designed for Microsoft's graphics library DirectX [8]. Compared with popular GPU programming languages such as CUDA, it is fully cross-platform, which means the programs written in C++ AMP can be migrated to another hardware vendor without any modification. Compared with other cross-platform GPU programming language such as Open Computing Language (OpenCL), it has much simplified Application Programming Interface (API), thus reducing the programming effort of programmers. Readers can refer to Append. A for a comparison between APIs of OpenCL and C++ AMP. GPUs usually have their own memory, also known as graphic memory. The data I/O is very fast between its ALUs and its own memory (> 100 GB/s), compared to the I/O between GPU and CPU (about 10 GB/s). Therefore the bottleneck to boost GPU computing performance is the data transfer from CPU to GPU or vice versa. In the micromagnetic solver presented, the only data transfer between CPU and GPU takes place when the initial conditions of the computing region are specified and when the final data is calculated by GPU and transferred back to CPU for display. In this way the simulation speed can be maximized. 3. Time need to carry out one time step at different 3D problem sizes N×N×N. The CPU time data is taken from report by [4]. Summary A GPU-based micromagnetic solver is presented to address the slow speed problem of large simulation problems. The speed boost relative to CPU simulations is significant at problem with large input sizes. This solver can not only run on expensive professional workstations but also economy personal laptops and both achieve considerable speed-ups. . gives the functional derivative of ε with respect to M  In Carry out Discrete Fourier Transform (DFT) to the padded M and K data with FFT algorithm: Fig. 1 A 1cross-sectional view of a rectangular magnetic sample after zero-padding.In the case of a finite three-dimensional computing problem, the demagnetization field can be Fig. 2 A 2comparison between Hardware architectures of CPU and GPU. The GPU has more ALUs dedicated to data processing. Fig. 4 . 4Speed-up of GPU solver on an AMD Radeon HD 7970 GHz Edition compared to CPU solver OOMMF. The speed-up increases with problem size. Table 1 . 1Per-step simulation time needed by CPU and GPU solvers for different 3D problem sizes (N×N×N) with the Euler algorithm. Numbers are in milliseconds.Size CPU (ms) GPU (ms) speedup 8 3 0.8492 1.95 × 0.43 16 3 4.066 2.723 × 1.5 32 3 36.14 3.151 × 11 64 3 489.6 6.558 × 74 128 3 4487 26.34 × 170 Email address: zhu@graceland.edu. AcknowledgementsThis work is supported by Graceland University professional development program. The author wishes to acknowledge Dr. Pieter B. Visscher of the University of Alabama, for helpful discussions of demagnetization field calculation.As mentioned before, the most time-consuming part of micromagnetic simulation is the calculation of demagnetization field. In each time step, the calculation requires three different phases:a. Perform FFTs to magnetization components:).z yz y xz x z demag yz z yy y xy x y demag xz z xy yc. Carry out inverse FFT of result of (15 In three-dimensional space, there are six FFTs to perform for each time step. The FFTs of demagnetization tensor K have been carried out at the beginning of simulation and will not be taken later, since K is constant.A FFT library based on C++ AMP has been implemented before[12]. It is adapted to the calculation of demagnetization field in the micromagnetic solver. At the point of publication the FFT library can only handle single-precision floats so this solver is currently limited to singleprecision computing.ResultsThe micromagnetic standard problem 3[13]was used to test the performance of this solver. A cubic magnetic particle is divided in to grids of N×N×N and the minimum energy state is reach by applying the LLG equation to each computational cell. The relaxation process involves the magnetization dynamics under the influence of demagnetization field, exchange field and uniaxial anisotropy field. To benchmark the solver presented, a hardware system with Intel Xeon E5410 CPU and an AMD Radeon HD 7970 GHz Edition GPU was used. The GPU chipset was among the fastest on the consumer market but still cost less than $500. For comparison, the benchmark of CPU micromagnetic solver OOMMF is also presented, with data from the report of another research group who used an Intel i7-930 CPU[4]. Dimensions with powers of two are benchmarked to demonstrate the performance of solvers varying with problem size, as shown in table 1. However this magnetic solver can solve problems of any size limited by the graphic memory allocable by the GPU.It is noticeable that at smaller problem sizes (N < 20) GPU solver is not significantly faster or even slower than CPU solver. This is caused by two factors. The first factor is that the data I/O overhead. The data transfer between GPU and CPU's main memory takes time. For a smaller problem size the calculation on GPU can be completed very soon, so in this case the computing power of GPU will not be fully utilized. For larger problems the data I/O time can be negligible when compared to the computing time. The second factor is the kernel launching overhead of GPU. This overhead is a constant regardless of the problem size, thus it is significant when the problem size is small.Append. A. Comparison between APIs of OpenCL and C++ AMP.OpenCL OOMMF User's guide. US Department of Commerce, Technology Administration. Michael Donahue, Donald Gene Joseph, Porter, National Institute of Standards and TechnologyDonahue, Michael Joseph, and Donald Gene Porter. OOMMF User's guide. US Department of Commerce, Technology Administration, National Institute of Standards and Technology, 1999. Scalable parallel micromagnetic solvers for magnetic nanostructures. Werner Scholz, Computational Materials Science. 28Scholz, Werner, et al. "Scalable parallel micromagnetic solvers for magnetic nanostructures." Computational Materials Science 28.2 (2003): 366-383. Speedup of FEM micromagnetic simulations with Graphics Processing Units. Attila Kakay, Elmar Westphal, Riccardo Hertel, IEEE Transactions on. 46MagneticsKakay, Attila, Elmar Westphal, and Riccardo Hertel. "Speedup of FEM micromagnetic simulations with Graphics Processing Units." Magnetics, IEEE Transactions on 46.6 (2010): 2303-2306. MuMax: a new high-performance micromagnetic simulation tool. Arne Vansteenkiste, Ben Van De Wiele, Journal of Magnetism and Magnetic Materials. 323Vansteenkiste, Arne, and Ben Van de Wiele. "MuMax: a new high-performance micromagnetic simulation tool." Journal of Magnetism and Magnetic Materials 323.21 (2011): 2585-2591. FastMag: Fast micromagnetic solver for complex magnetic structures. R Chang, Journal of Applied Physics. 109Chang, R., et al. "FastMag: Fast micromagnetic solver for complex magnetic structures." Journal of Applied Physics 109.7 (2011): 07D358. Graphics processing unit accelerated micromagnetic solver. Shaojing Li, Boris Livshitz, Vitaliy Lomakin, IEEE Transactions on. 46MagneticsLi, Shaojing, Boris Livshitz, and Vitaliy Lomakin. "Graphics processing unit accelerated micromagnetic solver." Magnetics, IEEE Transactions on 46.6 (2010): 2373-2375. Micromagnetic simulations using graphics processing units. L Lopez-Diaz, Journal of Physics D: Applied Physics. 45323001Lopez-Diaz, L., et al. "Micromagnetic simulations using graphics processing units." Journal of Physics D: Applied Physics 45.32 (2012): 323001. C++ AMP: Accelerated Massive Parallelism with Microsoft® Visual C++®. Kate Gregory, Ade Miller, O'Reilly Media, IncGregory, Kate, and Ade Miller. C++ AMP: Accelerated Massive Parallelism with Microsoft® Visual C++®. " O'Reilly Media, Inc.", 2012. A phenomenological theory of damping in ferromagnetic materials. Thomas L Gilbert, IEEE Transactions on. 40MagneticsGilbert, Thomas L. "A phenomenological theory of damping in ferromagnetic materials." Magnetics, IEEE Transactions on 40.6 (2004): 3443-3449. Direct solution of the Landau-Lifshitz-Gilbert equation for micromagnetics. Yoshinobu Nakatani, Yasutaro Uesaka, Nobuo Hayashi, Japanese Journal of Applied Physics. 282485Nakatani, Yoshinobu, Yasutaro Uesaka, and Nobuo Hayashi. "Direct solution of the Landau- Lifshitz-Gilbert equation for micromagnetics." Japanese Journal of Applied Physics 28.12R (1989): 2485. Calculation of demagnetizing field distribution based on fast Fourier transform of convolution. Nobuo Hayashi, Koji Saito, Yoshinobu Nakatani, Japanese journal of applied physics. 35Hayashi, Nobuo, Koji Saito, and Yoshinobu Nakatani. "Calculation of demagnetizing field distribution based on fast Fourier transform of convolution." Japanese journal of applied physics 35.12A (1996): 6065-6073. C++ AMP FFT Library. Daniel Moth, CodePlex. Moth, Daniel, et al. "C++ AMP FFT Library." CodePlex, Jan 2013. Web. 23 Jun. 2014 µMAG Standard Problem #3. Michael Donahue, Michael Donahue, et al. "µMAG Standard Problem #3." µMAG organization. Mar 1998. Web. 23 Jun. 2014

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{'abstract': 'A finite-difference Micromagnetic solver is presented utilizing the C++ Accelerated Massive Parallelism (C++ AMP). The high speed performance of a single Graphics Processing Unit (GPU) is demonstrated compared to a typical CPU-based solver. The speed-up of GPU to CPU is shown to be greater than 100 for problems with larger sizes. This solver is based on C++ AMP and can run on GPUs from various hardware vendors, such as NVIDIA, AMD and Intel, regardless of whether it is dedicated or integrated graphics processor.', 'arxivid': '1406.7459', 'author': ['Ru Zhu \nGraceland University\n50140LamoniIowaUSA\n'], 'authoraffiliation': ['Graceland University\n50140LamoniIowaUSA'], 'corpusid': 8923650, 'doi': '10.1109/mcse.2015.132', 'github_urls': [], 'n_tokens_mistral': 5259, 'n_tokens_neox': 4490, 'n_words': 2775, 'pdfsha': '39fbcebe7ef714745391505ce909bc5873054c81', 'pdfurls': ['https://arxiv.org/pdf/1406.7459v1.pdf'], 'title': ['Speedup of Micromagnetic Simulations with C++ AMP On Graphics Processing Units', 'Speedup of Micromagnetic Simulations with C++ AMP On Graphics Processing Units'], 'venue': []}

arxiv

Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 2024 Nouzha Lamdouar Chakir Tajani Mohammed Touati Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse Communications in Mathematics 321202410.46298/cm.1077435 The aim of this paper is to evaluate the train/track induced loads on the substructure by modelling the wheel, at each instant, as a moving sinusoidal pulse applied in a very short period of time. This assumption has the advantage of being more realistic as it reduces the impact of time on the load definition. To that end, mass, stiffness, and dumping matrices of an elementary section of track will be determined. As a result, the equations of motion of a section of track subjected to a sinusoidal pulse and a rectangular pulse respectively is concluded. Two numerical methods of resolution of that equation, depending on the nature of the dumping matrix, will be presented. The computation results will be compared in order to conclude about the relevance of that load model. This approach is used in order to assess the nature and the value of the loads received by the substructure.MSC 2020: 74S05, 37M05, 74-10, 37N30 Introduction Various theoretical and experimental researches have been performed in order to assess train/track induced loads on the substructure. Mohammed Touati and al. [1] determined the loads induced by a non-linear 3D multi-body modelled train on the track with taking into account wheel/rail contact properties and track irregularities. Yang Xinwen and al. [2] concluded, through a vehicle-track-subgrade coupling dynamic theory and finite element method, about the train/track induced loads on each layer of the substructure. As an experimental study, Al Shaer and al. [3] presented the dynamic behavior of a portion of ballasted railway track subjected to cyclic loads in substitution of a moving wheelset. In conclusion, the dynamics behavior of the substructure is widely studied in the literature ( [4], [5], [6], [7], [8]) based on the train/track coupling model. Actually, even if modelling a wheel load as a rectangular pulse is a common assumption, real measurements don't show the same shape. In fact, ONCF (Moroccan railway network manager) has many tools that record wheel pulse like GOTCHA. This system shows that the shape of the load has never been rectangular, but it's more likely compared to a sinusoidal pulse. Then, this paper deals with evaluating train/track induced loads on the substructure by proposing a new approach when it comes to modelling the shape of the wheel impact. Indeed, it's common to consider a moving load as a rectangular impulse applied on the nodes of a mesh structure in each period of time depending on signal sampling. This paper shows that assuming the wheel load as a sinusoidal pulse may reduce the impact of the period of time of its application and, consequently, minimize the loads induced on the substructure oversized by the common assumption. In that matter, a finite element model of the track will be presented and the numerical results will be compared. Track elementary section modeling 2.1 Determination of mass, stiffness et dumping matrices Let's assume a portion of ballasted track composed of two elements of rail considered as a continuous Euler-Bernoulli beam, fixed to two sleepers by a couples of springs/dampers representing the railpads. The ballast is modelled as a couples of springs/dampers under each sleeper ( Figure 1). The displacement vector is written as: U = [u 1 , θ 1 , u 2 , θ 2 , u 3 , θ 3 , u T 1 , u T 2 ] The effective mass and the stiffness matrices of an element of rail [9], are given by: M r = (ρ r A r L/420)         156 22L 54 −13L 0 0 22L 4L 2 13L −3L 2 0 0 54 13L 312 0 54 −13L −13L −3L 2 0 8L 2 13L −3L 2 0 0 54 13L 156 −22L 0 0 −13L −3L 2 −22L 4L 2         K r = E r I r /L 3         12 6L −12 6L 0 0 6L 4L 2 −6L 2L 2 0 0 −12 −6L 24 0 −12 6L 6L 2L 2 0 8L 2 −6L 2L 2 0 0 −12 −6L 12 −6L 0 0 6L 2L 2 −6L 4L 2         Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 37 where ρ r is the density of the rail, A r is the surface of the rail section, E r is Young modulus, and I r is the rail moment of inertia. The dumping matrix of the rail is obtained as a linear combination of mass and stiffness matrices by assuming that the displacements u 1 and u 3 are completely dumped by the effect of railpads. Therefore, the dumping matrix is written as: C * r = a 0 · M * r + a 1 · K * r where, M * r = (ρ r A r L/420)     4L 2 13L −3L 2 0 13L 312 0 −13L −3L 2 0 8L 2 −3L 2 0 13L −3L 2 4L 2     K * r = E r I r /L 3     4L 2 −6L 2L 2 0 −6L 24 0 6L 2L 2 0 8L 2 2L 2 0 6L 2L 2 4L 2     a 0 and a 1 are concluded from the equation: a 0 a 1 = 2ω 1 ω 2 / ω 2 2 − ω 2 1 ω 2 −ω 1 −1/ω 2 1/ω 1 ζ 1 ζ 2 where ω i 2 , (i = 1, 2) are the eigenvalues associated to the vibration of the rail described by the matrices M r * and K r * , and ζ i , (i = 1, 2) are the dumping ratios according to the first and second modes. In one hand, the equation of motion of the rail is written as: where C r is the transformation of the matrix C * r in the base U * , and U * is defined by: M rÜ * + C rU * + K r U * = F(1)U * = [u 1 , θ 1 , u 2 , θ 2 , u 3 , θ 3 ] F is given by: F =         −k s (u 1 − u T 1 ) − c s (u 1 −u T 1 ) 0 0 0 −k s (u 3 − u T 3 ) − c s (u 3 −u T 3 ) 0         In the other hand, the equations of motion of the sleepers are written as: m TüT 1 = k s (u 1 − u T 1 ) + c s (u 1 −u T 1 ) − k b u T 1 − c buT 1 m TüT 2 = k s (u 3 − u T 2 ) + c s (u 3 −u T 2 ) − k b u T 2 − c buT 2(2) From (1) and (2), we may conclude about the equation of motion of the track elementary section as it's modelled. It's written as: MÜ + CU + KU = 0 where M , C and K are the mass, dumping, and the stiffness of the track elementary section respectively. Numerical application Let's assume a track elementary section characterized by the data given in table 1 (we can refer to ( [10], [11], [12]). The figure 2 illustrates the evolution of natural frequencies according to vibration modes. It shows that: Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 39 • The frequency of the 3 rd mode corresponds to a movement in opposition of phase between rail and sleepers. It's equal to 381.1 Hz. 3 Track response to a rectangular and a sinusoidal pulses Description of the studied track Let's assume a section of track composed of N track elementary sections subjected to an external load F as it's shown in figure 3. The number of degrees of freedom is given by: N dof = 8N − 3(N − 1) The displacement vector is written as: U * j = [u j,1 , θ j,1 , u j,2 , θ j,2 , u j,3 , θ j,3 , u j,T 1 , u j,T 2 ] j refers to the element's number. U =    . . . The mass, stiffness and dumping matrices in the base U are obtained by assembling those of a track elementary section determined earlier. The vector of loads is defined by: F =    . . . f j . . .    where,                        N is even            N = 2 f j = P if j = 5 f j = 0 else N = 2 f j = P if j = (5N/2) + 1 f j = 0 else N is uneven f j = P if j = (5(N + 1)/2) − 1 f j = 0 else Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 41 Figure 4: Sinusoidal and rectangular pulses over a period of t d P is a rectangular or a sinusoidal load given as: • Sinusoidal pulse: P = P 0 sin ωt if t ≤ t d P = 0 else • Rectangular pulse: P = P 0 if t ≤ t d P = 0 else Its shape is shown in the figure 4. Description of the methods of resolution The dynamic behavior of the section of track may be analyzed by modal superposition if the dumping matrix verifies orthogonality properties. That method is used in particular for an undumped system. In that case, the equation of motion is reduced to: MÜ + KU = F. Let's assume that ω 2 i are the eigenvalues associated to the track vibration. We note {φ i } the normalized eigenvectors related to ω 2 i . Therefore, the equation of motion is written as: Z + diag(ω 2 i )Z = φ T F(3) where diag(ω 2 i ) is a diagonal matrix of the eigenvalues and: U = Φ.Z The system of equations (3) is uncoupled where each equation is written as: The resolution of that equation is given by DUHAMEL integral: z i + ω 2 i z i = Φ j,i P (t)z i (t) = (1/ω i ) t 0 Φ j,i P (τ ) sin ω i (t − τ )dτ Therefore, the solution for a sinusoidal pulse load is given as: z i (t) = (Φ j,i P 0 /ω 2 i ).(1/(1 − β 2 ))(sin ωt − β sin ω i t) if t ≤ t d (ż i (t d )/ω i ) sin ω i (t − t d ) + z i (t d ) cos ω i (t − t d ) if t ≥ t d where, β = ω/ω i and the solution for a rectangular pulse load is given as: z i (t) = (Φ j,i P 0 /ω 2 i )(1 − cos ω i t) if t ≤ t d (Φ j,i P 0 /ω 2 i )(cos ω i (t − t d ) − cos ω i t if t ≥ t d The figure 5 shows the response z(t) to a sinusoidal and a rectangular pulse. It's obvious that in the forced phase, the maximum rectangular response is higher than the maximum sinusoidal response. In general, the dumping matrix doesn't verify the orthogonality characteristics. Therefore, the modal superposition method is substituted by the following method. The equation of motion can be written as: Z + φ T Cφ.Ż + diag(ω 2 i ).Z = φ T F(4) where diag(ω i 2 ) and φ are defined earlier. Knowing that: Z −Ż = 0(5) Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 43 (4) and (5) could be written as: Ẏ = D.Y + F *(6) where, Y = Ż Z , D = A −1 B , F * = A −1 φ T F 0 and, A = φ T Cφ I I 0 , B = diag(ω 2 i ) 0 0 −I Let's assume that {λ i } are the eigenvalues associated to the matrix D. We note {ψ i } the normalized eigenvectors related to {ω i 2 }. We define X(t) as: Z = ψ.X The equation (6) is written as:Ẋ = diag(λ i ).X + ψ −1 F *(7) The system of equations (7) is uncoupled where each equation is written as: x i (t) = a i .x i (t) + b i .P(8) where, a i = λ i and b i = χ i and, χ = ψ −1 φ T 0 0 0 The resolution of the equation (8) gives: • Sinusoidal pulse: x i (t) = b i P ω a 2 i +ω 2 e a i t − a i b i P a 2 i +ω 2 sin ωt − b i P ω a 2 i +ω 2 cos ωt if t ≤ t d x i (t d )e a i (t−t d ) if t ≥ t d • Rectangular pulse: x i (t) = (bP/a)(e a i t − 1) if t ≤ t d x i (t d )e a i (t−t d ) if t ≥ t d 44 Nouzha Lamdouar, Chakir Tajani and Mohammed Touati Results and discussion The figures presented in this section show the numerical resolution of the system of equations of a dumped track section subjected to a rectangular and sinusoidal loads. The properties of the track are defined in table 1. In figure 6 and figure 7, the sinusoidal pulse is presented in red; however, the rectangular pulse is presented in black. These results have many consequences in the railway field. Actually, we may optimize railway infrastructure components for example (like ballast height). Moreover, the study is made by considering a static load (10 T). This load is mainly amplified by rail/wheel interaction and train speed [1]. Conclusion Based on the results of the model analysis studied in order to determine the loads induced on the substructure, the following conclusions can be drawn: • The common modelling of the load applied on the track due to a moving wheel as a rectangular pulse acting in the time sample of a force signal generates a higher rate of movement in the track and over sizes the loads induced in the substructure than a sinusoidal pulse model; Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse 47 • Dumping matrix has a major influence on reducing the loads induced in the substructure. Therefore, it's necessary to preserve the quality of the track components while maintaining it. As an application, we may evaluate the track behavior according to different characteristics of the track elements that degrade because of maintenance operations. Indeed, the ballast is considered as the most affected element because of operations of damping required for track geometry corrections. Figure 1 : 1Elementary track modelling Figure 2 : 2Natural frequencies of an elementary track section • The frequencies of the 1 st and 2 nd modes correspond to a movement in phase between rail and sleepers. It's equal to 81.62 Hz; Figure 3 : 3Track section modellingwhere,u j,k = u * j,k where k ∈ [1, 8] if j = 1 u j,k = u * j,k where k ∈ [3,4,5,6,8] if j = 1 and, Figure 5 : 5z(t) response to a rectangular and sinusoidal pulse t d /T = 0.75 Figure 6 :Figure 7 :Figure 8 : 678Rail response under sinusoidal, rectangular pulses (N = 4, t d = 0.01s, P = 10T) Sleeper response under sinusoidal and rectangular pulses (N = 4, t d = 0.01s, P = 10T) Loads induced in the substructure (N = 30, t d = 0.01s, P = 10T) 1 . 1Displacements and rotations of the rail 2. Displacements of the sleepers It's clear that the maximum values of rail and sleepers movement under rectangular pulse are higher than those reached under a sinusoidal pulse. The figure 8 shows the maximum loads induced in the substructure. The table 2 shows the repartition of the loads under the sleepers. Table 1 : 1Track properties Table 2 : 2Repartition of the loads under the sleepers (N = 30, t d = 0.01s, P = 10T) Railway vehicle response under random irregularities on a tangent track -nonlinear 3D multi-body modelling. M Touati, N Lamdouar, A Bouyahyaoui, International Journal of Mechanical Engineering and Technology. 97Touati M., Lamdouar N. and Bouyahyaoui A.: Railway vehicle response under random irregularities on a tangent track -nonlinear 3D multi-body modelling. International Journal of Mechanical Engineering and Technology 9 (7) (2018) 944-956. Vertical Vibration Analysis of Vehicle-Track-Subgrade Coupled System in High Speed Railway with Dynamic Flexibility Method. Y Xiwen, G Shaojie, Z Shunhua, S Yao, M Xiaoyun, Transportation Research Procedia. 25Xiwen Y., Shaojie G., Shunhua Z., Yao S. and Xiaoyun M.: Vertical Vibration Analysis of Vehicle-Track-Subgrade Coupled System in High Speed Railway with Dynamic Flexibility Method. Transportation Research Procedia 25 (2017) 291-300. Experimental settlement and dynamic behavior of a portion of ballasted railway track under high speed trains. Al Shaer, A Duhamel, D Sab, K Foret, G Schmitt, L , Journal of Sound and Vibration. 3161-5Al Shaer A., Duhamel D., Sab K., Foret G. and Schmitt L.: Experimental settlement and dynamic behavior of a portion of ballasted railway track under high speed trains. Journal of Sound and Vibration 316 (1-5) (2008) 211-233. Ground Vibration from High-speed Trains: Prediction and Countermeasure. M Kaynia, A Madshus, C Zackrisson, P , Journal of Geotechnical and Geoenvironmental Engineering. 1266M. Kaynia A., Madshus C. and Zackrisson P.: Ground Vibration from High-speed Trains: Prediction and Countermeasure. Journal of Geotechnical and Geoenvironmental Engineering 126 (6) (2000) 531-537. Simulation of Track-ground Vibrations due to High-speed Trains. H Takemiya, Proceedings of the Eighth International Congress on Sound and Vibration. the Eighth International Congress on Sound and VibrationHong Kong, ChinaTakemiya H.: Simulation of Track-ground Vibrations due to High-speed Trains. . Proceedings of the Eighth International Congress on Sound and Vibration. Hong Kong, China (2000) 2875-2882. High-speed Railway Lines on Soft Ground: Dynamic Behaviour at Critical Train Speed. C Madshus, K Kaynia, Journal of Sound and Vibration. 2313Madshus C. and Kaynia K.: High-speed Railway Lines on Soft Ground: Dynamic Behaviour at Critical Train Speed. Journal of Sound and Vibration 231 (3) (2000) 689-701. A Spatial Time-Varying Coupling Model for Dynamic Analysis of High Speed Railway Subgrade. S Qian, C Ying, Journal of Southwest Jiaotong University. 145Qian S. and Ying C.: A Spatial Time-Varying Coupling Model for Dynamic Analysis of High Speed Railway Subgrade. Journal of Southwest Jiaotong University 14 (5) (2001) 509-513. Dynamic response of high-speed ballasted railway tracks: 3D periodic model and in situ measurements. H Chebli, D Clouteau, L Schmitt, Soil Dynamics and Earthquake Engineering. 282Chebli H., Clouteau D. and Schmitt L.: Dynamic response of high-speed ballasted railway tracks: 3D periodic model and in situ measurements. Soil Dynamics and Earthquake Engineering 28 (2) (2008) 118-131. P Mario, L William, Structural dynamics: Theory and Computation. USSpringerMario P. and William L.: Structural dynamics: Theory and Computation. Springer US (2004). DIN-EN-13481-1 : Railway applications.Track. Performance requirements for fastening systems Part 1 : Definitions. DIN-EN-13481-1 : Railway applications.Track. Performance requirements for fastening systems Part 1 : Definitions (2012). . Prediction of vibratory nuisances of rail transport vehicles. G Kouroussis, O Verlinden, C Conti, National Congress on Theoretical and Applied Mechanics -NCTAM. Kouroussis G., Verlinden O. and Conti C.: Prediction of vibratory nuisances of rail transport vehicles. . 6th National Congress on Theoretical and Applied Mechanics -NCTAM (2003) . A detailed model for investigating vertical interaction between railway vehicle and track. W Zhai, X Sun, Vehicle System Dynamics. 23Sup1Zhai W. and Sun X.: A detailed model for investigating vertical interaction between railway vehicle and track. Vehicle System Dynamics 23 (Sup1) (1994) 603-615.

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{'abstract': 'The aim of this paper is to evaluate the train/track induced loads on the substructure by modelling the wheel, at each instant, as a moving sinusoidal pulse applied in a very short period of time. This assumption has the advantage of being more realistic as it reduces the impact of time on the load definition. To that end, mass, stiffness, and dumping matrices of an elementary section of track will be determined. As a result, the equations of motion of a section of track subjected to a sinusoidal pulse and a rectangular pulse respectively is concluded. Two numerical methods of resolution of that equation, depending on the nature of the dumping matrix, will be presented. The computation results will be compared in order to conclude about the relevance of that load model. This approach is used in order to assess the nature and the value of the loads received by the substructure.MSC 2020: 74S05, 37M05, 74-10, 37N30', 'arxivid': '2301.01524', 'author': ['Nouzha Lamdouar ', 'Chakir Tajani ', 'Mohammed Touati '], 'authoraffiliation': [], 'corpusid': 255415975, 'doi': '10.46298/cm.10774', 'github_urls': [], 'n_tokens_mistral': 6098, 'n_tokens_neox': 5143, 'n_words': 3265, 'pdfsha': '889ad43c9d4e8af0db62cb1f5b98c69eca1f4d9f', 'pdfurls': ['https://export.arxiv.org/pdf/2301.01524v2.pdf'], 'title': ['Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse', 'Dynamic behavior of a railway track under a moving wheel load modelled as a sinusoidal pulse'], 'venue': ['Communications in Mathematics']}

arxiv

Regular patterns in the information flow of local dephasing channels Filippo Giraldi School of Chemistry and Physics South Africa Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM University of KwaZulu-Natal and National Institute for Theoretical Physics (NITheP) Westville Campus 4000Durban Istituto Nazionale di Alta Matematica Francesco Severi Cittá Universitaria Piazza Aldo Moro 500185RomaItaly Regular patterns in the information flow of local dephasing channels Consider local dephasing processes of a qubit that interacts with a structured reservoir of frequency modes or a thermal bath, with Ohmic-like spectral density (SD). It is known that non-Markovian evolution appears uniquely above a temperature-dependent critical value of the Ohmicity parameter, and non-Markovianity can be induced by properly engineering the external environment. In the same scenario, we find that the flow of quantum information shows regular patterns: alternate directions appear in correspondence of periodical intervals of the Ohmicity parameter α 0 . The information flows back into the system over long times for 2 + 4n < α 0 < 4 + 4n, at zero temperature, and for 3 + 4n < α 0 < 5 + 4n, at non-vanishing temperatures, where n = 0, 1, 2, . . .. Otherwise, the long time information flows into the environment. In the transition from vanishing to arbitrary non-vanishing temperature, the long time back-flow of information is stable for 3 + 4n < α 0 < 4 + 4n, while it is reverted for 2 + 4n < α 0 < 3 + 4n and 4 + 4n < α 0 < 5 + 4n. The patterns of the information flow are not altered if the low frequency Ohmic-like profiles of the SDs are perturbed with additional factors that consist in arbitrary powers of logarithmic forms. Consequently, the flow of information can be controlled, directed and reverted over long times by engineering a wide variety of reservoirs that includes and continuously departs from the Ohmic-like structure at low frequencies. Non-Markovianity and recoherence appear according to the same rules along with the back-flow of information. Introduction In open quantum systems the loss, revival or maintenance of quantum correlations is deeply related to the structure of the external environment [1,2]. The persistent interactions between system and environment and the appearance of memory effects were usually referred as non-Markovianity. These conditions have been largely investigated and in the last years new definitions and measures of non-Markovianity have been proposed. See Refs. [3,4] for a review. Non-Markovianity can be interpreted in terms of the flow of quantum information. This quantity is defined in various ways, via the Fisher information [5], the fidelity [6] or the mutual information [7], to name a few. The trace-distance measure introduced in Ref. [8] estimates the relative distinguishability of two arbitrary quantum states. In Markovian processes this measure diminishes monotonically in time. This behavior can be seen as a loss of quantum information by the open system, while in non-Markovian dynamics the memory effects can be interpreted as a flow of quantum information from the external environment back into the open system. In this way, the structures of the environment that induce non-Markovian dynamics can be found by studying the direction of the flow of quantum information [9,10,11,12]. The dephasing process of qubit (two-level system) that interacts with a structured reservoir of frequency modes [1,2,13,14,15] is a referential scenario for the study of non-Markovianity. In this system the measures of non-Markovianity mentioned above suggest the same conditions for the appearance of non-Markovian dynamics [16] and are easily evaluated from the dephasing rate and the dephasing factor of the system [17,9,10,11,12]. In fact, persistent negative values of the dephasing rate or, equivalently, a decreasing dephasing factor indicate back-flow of information into the open system and witness non-Markovianity. A remarkable analysis of the dependence of non-Markovianity on the structure of the external environment is performed in Ref. [9]. Convexity properties that involve the general reservoir spectrum provide conditions for the appearance of non-Markovian dynamics. For Ohmic-like SDs with exponential cutoff the transition from Markovian to non-Markovian dynamics appears in correspondence of a critical value of the Ohmicity parameter. In fact, temporary back-flow of information and recoherence are found uniquely for values of the Ohmicity parameters that are larger than such critical value. At zero temperature the critical value is equal to 2, it increases monotonically with temperature and becomes 3 at infinite temperature. See Ref. [9] for details. Great efforts have been made for the experimental observation of these phenomena. Simulations of open system dynamics have been performed with trapped ions [18] and transition from Markovian to non-Markovian dynamics has been obtained in an all-optical experiment [19], to name a few. As a continuation of the scenario described above, here, we consider the local dephasing process of a qubit that interacts either with a structured reservoir of frequency modes or with a thermal bath. In addition to the Ohmic-like structure, the SDs under study include at low frequencies removable logarithmic singularities [20,21]. At higher frequencies the SDs are arbitrarily shaped. We study the decoherence and recoherence processes by evaluating the dephasing factor and we investigate the flow of quantum information by analyzing the dephasing rate, in dependence on the structure of the reservoir or of the thermal bath. We also search for regular patterns in the direction of the flow of information that allow a full manipulation of the flow itself and, consequently, of non-Markovianity and recoherence by engineering the low frequency structure of the external environment. The paper is organized as follows. Section 2 is devoted to the description of the model. In Section 3, asymptotic coherence is described in term of integral properties of the SD. Section 4 is devoted to the description of the Ohmic-like spectral densities with additional removable logarithmic singularities. The decoherence and recoherence processes are studied in Section 5, by analyzing the dephasing factor both at zero and non-vanishing temperature. Patterns in the flow of quantum information are shown in Section 6. Conclusions are drawn in Section 7, and details on the calculations are provided in the Appendix. The model The system of a qubit that interacts locally with a reservoir of frequency modes is described by the following microscopic Hamiltonian [1,2,9,10], H = ω 0 σ z + k ω k b † k b k + k σ z g k a k + g * k a † k ,(1) in units where = 1. The transition frequency of the qubit is ω 0 , while σ z represents the zcomponent Pauli spin operator. The index k runs over the frequency modes. The parameter ω k represents the frequency of the kth mode, while b † k and b k are the rising and lowering operator, respectively, of the same mode. The coefficient g k represents the coupling strength between the qubit and the kth frequency mode. The reduced density matrix ρ(t) represents the mixed state of the qubit at the time t and is obtained by tracing the density matrix of the whole system at the time t over the Hilbert space of the external environment [1]. The model is exactly solvable [13,14,15]. Let the qubit be initially decoupled from the external environment that is represented by a structured reservoir of field modes or by a thermal bath. The reduced time evolution is described in the interaction picture by the following master equation, ρ(t) = γ(t) (σ z ρ(t)σ z − ρ(t)) .(2) The function γ(t) represents the dephasing rate and is related to the temperature of the thermal bath. At zero temperature, T = 0, the dephasing rate is labeled here as γ 0 (t) and reads γ 0 (t) =ˆ∞ 0 J (ω) ω sin (ωt) dω.(3) The function J (ω) represents the SD of the system and is defined in terms of the coupling constants g k via the following form, J (ω) = k |g k | 2 δ (ω − ω k ). If the external environment is initially in a thermal state, T > 0, the dephasing rate is represented here as γ T (t) and reads γ T (t) =ˆ∞ 0 J (ω) ω coth ω 2k B T sin (ωt) dω.(4) The parameter k B is the Boltzmann constant. The quantum coherence between the states |0 and |1 of the qubit is described by the off-diagonal element ρ 0,1 (t) of the density matrix that undergoes the following evolution [13,14,15], ρ 0,1 (t) = ρ * 1,0 (t) = ρ 0,1 (0) exp {−Ξ(t)} .(5) The function Ξ(t) represents the dephasing factor and depends on the temperature T of the thermal bath and on the coupling between the system and the environment. At zero temperature, T = 0, the dephasing factor is indicated here as Ξ 0 (t) and results in the following form, Ξ 0 (t) =ˆ∞ 0 J (ω) 1 − cos (ωt) ω 2 dω.(6) If the external environment is initially in a thermal state, T > 0, the dephasing factor is represented here as Ξ T (t) and reads Ξ T (t) =ˆ∞ 0 J (ω) ω 2 (1 − cos (ωt)) coth ω 2k B T .(7) Both for vanishing and non-vanishing temperature, the dephasing factor is related to the dephasing rate via the time derivative, γ 0 (t) =Ξ 0 (t) and γ T (t) =Ξ T (t). According to Eq. (5), recoherence corresponds to negative values of the dephasing rate. Coherence The loss or persistence of coherence between the two energy eigenstates of the qubit depends on integral and low frequency properties of the SDs. At zero temperature, T = 0, coherence is not entirely lost over long times if the second negative moment of the SD is finite, ∞ 0 J (ω) ω 2 dω < ∞.(8) Under this condition the coherence term shows persistence of residual coherence, ρ 0,1 (∞) = ρ 0,1 (0) exp −ˆ∞ 0 J (ω) ω 2 dω .(9) If the external environment consists in a thermal bath, T > 0, and the following condition holds,ˆ∞ 0 J (ω) ω 2 coth ω 2k B T dω < ∞,(10) the coherence term tends over long times to the non-vanishing asymptotic value ρ 0,1 (∞) = ρ 0,1 (0) exp −ˆ∞ 0 J (ω) ω 2 coth ω 2k B T dω .(11) The maximum modulus of the ratio between asymptotic and initial coherence is obtained at zero temperature, T = 0, from Eq. (9). According to Eq. (5), residual coherence persists over long times if the dephasing factor does not diverge asymptotically, while coherence is fully lost if the dephasing factor diverges. Consequently, the dependence of coherence on the structure of the SD can be analyzed via the dephasing factor itself. We intend to study the short and long time behavior of the dephasing factor for a large variety of SDs. We consider low frequency structures of the SDs such that, for T = 0, the second negative moment is either finite, Eq. (8), or infinite, ∞ 0 J (ω) ω 2 dω = ∞.(12) At non-vanishing temperatures, T > 0, we choose low frequency profiles of the SDs that fulfill either the constraint (10) or the following one, ∞ 0 J (ω) ω 2 coth ω 2k B T dω = ∞.(13) Details on the structure of the SDs under study are provided below. Spectral densities with removable logarithmic singularities The fast development of quantum technologies allows the engineering of the most various environments. According to the remarkable analysis performed in Refs. [22,23], an impurity that is trapped in a double-well potential and is surrounded by a cold gas reproduces, under suitable conditions, a qubit that interacts with an Ohmic-like environment. The Ohmicity parameter increases by enhancing the scattering length that is related to the boson-boson coupling [23]. In case the gas is free and one-dimensional, the SD changes from sub-Ohmic to Ohmic and to super-Ohmic by increasing the scattering length. In the two-dimensional non-interacting condition the spectrum is Ohmic and the super-Ohmic regime is obtained if the magnitude of the interaction decreases. The SD is super-Ohmic in the non-interacting condition if the gas is three-dimensional. We refer to [23] for details. In light of the above observation we focus on SDs that are Ohmic-like at low frequencies and are arbitrarily shaped at higher frequency. We intend to analyze the feature of the open dynamics, the flow of quantum information, non-Markovianity and recoherence of the qubit. We also evaluate the accuracy of the results obtained for the experimentally feasible Ohmic-like SDs by perturbing the power laws of the Ohmic-like profiles with additional factors that are represented by arbitrary powers of logarithmic forms. In this way, we consider a wide variety of SDs that includes and continuously departs from the Ohmic-like condition. Positive (negative) or vanishing values of the first logarithmic power enhance (reduce) or unchange the power law profile, and, especially, provide legitimate SDs [20]. For the sake of convenience, the SDs J (ω) are described via the dimensionless auxiliary function Ω (ν). This function is defined for every ν ≥ 0 by the following scaling property, J (ν∆) /∆ = Ω (ν), in terms of a general scale frequency ∆ of the system. At non-vanishing temperatures it is convenient to define the effective SD J T (ω) as follows, J T (ω) = J (ω) coth ω 2k B T .(14) The corresponding auxiliary function Ω T (ν) reads Ω T (ν) = Ω (ν) coth (( ∆ν) / (2k B T )). In this way, the action of the thermal bath can be represented via a transformed SD. The first class of SDs under study is defined by auxiliary functions Ω (ν) that are continuous for every ν > 0 and exhibit the following asymptotic behavior [24] as ν → 0 + , Ω (ν) ∼ ∞ j=0 n j k=0 c j,k ν α j (− ln ν) k ,(15) where ξ > 0, ∞ > n j ≥ 0, α j+1 > α j , for every j ≥ 0, and α j ↑ +∞ as j → +∞. Furthermore, we consider α 0 ≥ 0, and n 0 = 0 if α 0 = 0. The power α 0 is referred as the Ohmicity parameter [1,2]. In fact, if n 0 = 0, the corresponding SDs are super-Ohmic for α 0 > 1, Ohmic for α 0 = 1 and sub-Ohmic for 1 > α 0 > 0, as ω → 0 + . The singularity in ν = 0 is removable by defining Ω(0) as the finite limit of Ω (ν) as ν → 0 + . Notice that Eq. (15) describes a large variety of asymptotic forms that include exponential and stretched exponential functions, inverse power laws and natural powers of logarithmic forms. The summability of the SD is guaranteed by the constraint Ω (ν) = O ν −1−χ 0 , as ν → +∞, where χ 0 > 0. Additionally, the Mellin transformsΩ (s) andΩ T (s) of the auxiliary functions Ω (ν) and Ω T (ν), and the meromorphic continuations [24,25] are required to decay sufficiently fast as |Im s| → +∞. See Appendix for details. In light of the asymptotic analysis performed in Refs. [26,25], the second class of SDs under study is described by auxiliary functions with the following asymptotic expansion as ν → 0 + , Ω (ν) ∼ ∞ j=0 w j ν α j (− ln ν) β j .(16) The powers β j are real valued, while α 0 > 0. The logarithmic singularity in ν = 0 is removed by setting Ω(0) = 0. Let the parametern be the least natural number such that α k−1 + 1 ≤ n < α k + 1, where the index k is a non-vanishing natural number. The function Ω (n) (ν) is required to be continuous on the interval (0, ∞). The integral´∞ 0 Ω (ν) exp {−ıξν} dν must converge uniformly for all sufficiently large values of the variable ξ and the integraĺ Ω (n) (ν) exp {−ıξν} dν has to converge at ν = +∞ uniformly for all sufficiently large values of the variable ξ. The auxiliary function is required to be differentiable k times and the following asymptotic expansion at ν → 0 + , Ω (k) (ν) ∼ ∞ j=0 w j d k dν k ν α j (− ln ν) β j , is required to hold for every k = 0, 1, . . . ,n. Furthermore, for every k = 0, . . . ,n − 1, the function Ω (k) (ν) has to vanish as ν → +∞. If compared to the first class, the second class of SDs has to fulfill more constraints but includes arbitrary powers of logarithmic forms. In both the classes under study the auxiliary functions Ω (ν) are non-negative, bounded and summable, due to physical grounds, and, apart from the above constraints, arbitrarily shaped at high frequencies [20]. The dephasing factor We start the analysis of the dephasing factor by considering a reservoir of frequency modes at zero temperature, T = 0, as external environment. The SDs under study belong to the first class introduced in Section 4. Over short times, t 1/∆, the dephasing factor increases quadratically in time, Ξ 0 (t) ∼ l 0 2 t 2 ,(17) where l 0 =´∞ 0 J (ω) dω. This behavior is independent of the low or high frequency structure of the SDs under study. Instead, the evolution of the dephasing factor over long times, t 1/∆, is various and is determined by the low frequency structure of the SD, given by Eq. (15). If α 0 = n 0 = 0 the dephasing factor grows linearly for t 1/∆, Ξ 0 (t) ∼ πc 0,0 2 ∆t.(18) If 0 < α 0 < 1 we find for t 1/∆ the following divergent behavior, Ξ 0 (t) ∼ c 0,n 0 r 1 (∆t) 1−α 0 ln n 0 (∆t) ,(19) where r 1 = sin (πα 0 /2) Γ (α 0 ) / (1 − α 0 ) . The above form grows as a power law for n 0 = 0, Ξ 0 (t) ∼ c 0,n 0 r 1 (∆t) 1−α 0 .(20) If α 0 = 1 we obtain over long times, t 1/∆, the divergent logarithmic form as below, Ξ 0 (t) ∼ c 0,n 0 n 0 + 1 ln n 0 +1 (∆t) .(21) If α 0 > 1 and α 0 is not an even natural number, the dephasing factor tends to the following asymptotic value, Ξ 0 (∞) =ˆ∞ 0 J (ω) ω 2 dω,(22) according to relaxations that involve logarithmic forms, Ξ 0 (t) ∼ Ξ 0 (∞) + c 0,n 0 r 1 (∆t) 1−α 0 ln n 0 (∆t) .(23) The above expression turns into inverse power laws for n 0 = 0, Ξ 0 (t) ∼ Ξ 0 (∞) + c 0,n 0 r 1 (∆t) 1−α 0 .(24) If α 0 > 1 and α 0 = 2m 0 , where m 0 and n 0 are non-vanishing natural numbers, we find Ξ 0 (t) ∼ Ξ 0 (∞) + c 0,n 0 r 1 (∆t) 1−2m 0 ln n 0 −1 (∆t) ,(25) where r 1 = π(−1) m 0 n 0 (2m 0 − 2)!/2. The above relaxation provides power laws for n 0 = 1, Ξ 0 (t) ∼ Ξ 0 (∞) + c 0,n 0 r 1 (∆t) 1−2m 0 .(26) If α 0 is an even natural number and n 0 vanishes, consider the least non-vanishing index k 0 such that either α k 0 does not take even natural values or α k 0 = 2m k 0 , where the natural numbers m k 0 and n k 0 do not vanishes. The function Ξ 0 (t) is obtained in the former case from Eqs. (23) and (24) by substituting the power α 0 with α k 0 and n 0 with n k 0 , or in the latter case from Eqs. (25) and (26) by substituting the power m 0 with m k 0 and n 0 with n k 0 . We consider SDs such that the index k 0 exists. At this stage we focus on SDs that belong to the second class introduced in Section 4 with a finite negative second moment, Eq. (8). This conditions requires that the Ohmicity parameter α 0 is larger than unity, α 0 > 1. Over long times, t 1/∆, the dephasing factor relaxes to the asymptotic value Ξ 0 (∞) according to arbitrarily positive or negative, or vanishing powers of logarithmic forms, (a) (b) (c ) (d) (e) (f ) (g) (h) (i) ( j) 1 2 3 4 5 6 ΔtΞ 0 (t) ∼ Ξ 0 (∞) + w 0 (∆t) 1−α 0 r 1 ln β 0 (∆t) +r 1 ln β 0 −1 (∆t) ,(27)wherer 1 = β 0 sin (πα 0 /2) Γ (1) (α 0 − 1) + πΓ (α 0 − 1) /2 . If the Ohmicity parameter α 0 is not an even natural number, the dominant part of the above relaxation is equivalent to the following one, Ξ 0 (t) ∼ Ξ 0 (∞) + w 0 r 1 (∆t) 1−α 0 ln β 0 (∆t), and provides the inverse power laws Ξ 0 (t) ∼ Ξ 0 (∞) + w 0 (∆t) 1−α 0 if β 0 = 0. If the Ohmicity parameter α 0 takes even natural values, Eq. (27) gives Ξ 0 (t) ∼ Ξ 0 (∞) + w 0r1 (∆t) 1−α 0 ln β 0 −1 (∆t), and turns into the inverse power law relaxations Ξ 0 (t) ∼ Ξ 0 (∞) + w 0r1 (∆t) 1−α 0 if β 0 = 1. Thermal bath Let the external environment consist in a thermal bath, T > 0. If the SD belongs to the first class under study with α 0 > 0 the dephasing factor results again in a quadratic function of time for t 1/∆, Ξ T (t) ∼ l T 2 t 2 ,(28) where l T =´∞ 0 J (ω) coth ( ω/ (2k B T )) dω. This behavior is again independent of the low or high frequency structure of the SD. Instead, the evolution of the dephasing factor over long times, t 1/∆, exhibits various behaviors in dependence on the low frequency structure of (a) (b) (c ) (d) (g) (h) (e) (f ) (i) ( j) (k ) (m) (l) (n) -2 2 4 ln(Δt) -10 -8 -6 -4 -2 2 4 ln(ln(ρ 0,1 (0)/ρ 0,1 (t))) the SD. If 0 < α 0 < 2 and α 0 = 1 the dephasing factor diverges for t 1/∆ as below, Ξ T (t) ∼ c 0,n 0 r T (∆t) 2−α 0 ln n 0 (∆t) ,(29) where r T = 2k B T cos (πα 0 /2) Γ (α 0 − 2) / ( ∆). Again, power laws appear from the above conditions for n 0 = 0, Ξ T (t) ∼ c 0,n 0 r T (∆t) 2−α 0 .(30) If α 0 = 1 the dephasing factor diverges for t 1/∆ as below, Ξ T (t) ∼ c 0,n 0 r T (∆t) ln n 0 (∆t) ,(31) where r T = π k B T / ( ∆). The divergence becomes linear in time for α 0 = 1 and n 0 = 0, Ξ T (t) ∼ c 0,n 0 r T (∆t) .(32) If α 0 = 2 the dephasing factor grows for t 1/∆ according to natural powers of logarithmic forms, Ξ T (t) ∼ c 0,n 0 r T ln n 0 +1 (∆t) ,(33) where r T = 2k B T / ( ∆ (n 0 + 1)). If α 0 > 2 and α 0 is not an odd number, the dephasing factor converges for t 1/∆ to the asymptotic value Ξ T (∞) =ˆ∞ 0 J (ω) ω 2 coth ω 2k B T dω,(34) according to the following relaxations, that become inverse power laws for n 0 = 0, Ξ T (t) ∼ Ξ T (∞) + c 0,n 0 r T (∆t) 2−α 0 ln n 0 (∆t) ,(35)(a) (b) (c ) (d) (e) (f ) (g) (h) (i ) ( j ) (k ) (l ) (m) (n) -1 0 1 2 3 4 5 ln(ln(Δt))Ξ T (t) ∼ Ξ T (∞) + c 0,n 0 r T (∆t) 2−α 0 .(36) If α 0 = 2m 1 + 1, where m 1 is a natural number and n 0 does not vanish, we find Ξ T (t) ∼ Ξ T (∞) + c 0,n 0 r T (∆t) 1−2m 1 ln n 0 −1 (∆t) ,(37) where r T = π(−1) m 1 k B T n 0 (2m 1 − 2)!/ ( ∆). The above relaxation becomes a power law for n 0 = 1, Ξ T (t) ∼ Ξ T (∞) + c 0,n 0 r T (∆t) 1−2m 1 .(38) If α 0 is an odd natural number and n 0 vanishes, consider the least non-vanishing index k 1 such that either α k 1 does not take odd natural values or α k 1 = 2m k 1 + 1, where the natural numbers m k 1 and n k 1 do not vanishes. The function Ξ T (t) is obtained in the former case from Eqs. (35) and (36) by substituting the power α 0 with α k 1 and n 0 with n k 1 , or in the latter case from Eqs. (37) and (38) by substituting the power m 1 with m k 1 and n 0 with n k 1 . We consider SDs such that the index k 1 exists. At this stage we focus on SDs such that the auxiliary functions Ω T (ν) belong to the second class under study and that exhibit a finite second negative moment, Eq. (10). This constraint requires the Ohmicity parameter α 0 to be larger than 2. Under this condition, α 0 > 2, a variety of logarithmic relaxations of the dephasing factor to the asymptotic value Ξ T (∞) are obtained for t 1/∆, Ξ T (t) ∼ Ξ T (∞) + w 0 (∆t) 2−α 0 r T ln β 0 (∆t) +r T ln β 0 −1 (∆t) ,(39)wherer T = β 0 k B T ∆ π sin πα 0 2 Γ (α 0 − 2) − 2 cos πα 0 2 Γ (1) (α 0 − 2) . If the Ohmicity parameter α 0 is not an odd natural number, the dominant part of the above relaxation is equivalent to the following one, Ξ T (t) ∼ Ξ T (∞) + w 0 r T (∆t) 2−α 0 ln β 0 (∆t). If the Ohmicity parameter α 0 takes odd natural values, Eq. (39) gives Ξ 0 (t) ∼ Ξ 0 (∞) + w 0r1 (∆t) 2−α 0 ln β 0 −1 (∆t), and provides the power laws Ξ 0 (t) ∼ Ξ 0 (∞) + w 0r1 (∆t) 2−α 0 for β 0 = 1. Notice the expected similarities between the Eqs. (35) and (39). Numerical computations of the coherence term ρ 1,0 (t) are plotted in Figure 1. Numerical analysis of the dephasing factor are displayed In Figures 2 and 3. The short time quadratic growth is confirmed by the parallel asymptotic lines appearing in Figure 2. The long time logarithmic relaxations result in the asymptotic lines plotted in Figure 3. Regular patterns in the long time information flow For the system under study the trace distance measure of non-Markovianity that is defined in Refs. [8,10] takes a simple expression in terms of the dephasing rate and dephasing factor and the non-Markovianity measure results in the following form [17,27,9], N =ˆγ (t)<0 |γ(t)| e −Ξ(t) dt.(40) The open dynamics is Markovian if the dephasing rate is non-negative. On the contrary, persistent negative values of the dephasing rate are source of non-Markovianity and are interpreted as a flow of information from the environment back into the system. At zero temperature, T = 0, the open dynamics is Markovian if the function J (ω) /ω is non-increasing. If the SD is differentiable this condition reads as below, J (ω) ≤ J (ω) ω ,(41) for every ω > 0. At non-vanishing temperatures, T > 0, the open dynamics is Markovian if the function J T (ω) /ω is non-increasing. If the SD is differentiable, this requirement results in the following constraint, J (ω) ≤ 1 ω + k B T cosech ω k B T J (ω) ,(42) for every ω > 0. Consequently, if the open dynamics is non-Markovian, the function J (ω) /ω, for T = 0, or the function J T (ω) /ω, for T > 0, is increasing in an interval of frequencies, at least. Let the SD be differentiable for every ω > 0. If the open dynamics is non-Markovian the constraint (41), for T = 0, or (42), for T > 0, is not fulfilled for one value of the frequency, at least. In general, the asymptotic behavior of the dephasing rate depends on integral properties of the SDs. Over long times, t 1/∆, the dephasing rate vanishes at zero temperature, T = 0, if the following condition is fulfilled, ∞ 0 J (ω) ω dω < ∞.(43) Same behavior is obtained at non-vanishing temperature, T > 0, if ∞ 0 J T (ω) ω dω < ∞.(44) The long time relaxations depend on the low frequency structure of the SD. We start the analysis of the decays by considering a reservoir of frequency modes at zero temperature, T = 0, as external environment, and the first class of SDs introduced in Section 4. Over short times, t 1/∆, the dephasing rate increases linearly, γ 0 (t) ∼ l 0 t.(45) This behavior is independent of the low or high frequency structure of the SDs under study. Over long times, t 1/∆, different forms of relaxations are obtained in dependence on the low frequency structure of the SD, given by Eq. (15). If α 0 = n 0 = 0 the dephasing rate tends to the following non-vanishing asymptotic value for t 1/∆, γ 0 (t) ∼ πc 0,0 ∆ 2 .(46) If α 0 > 0 and α 0 is not an even natural number, the dephasing rate vanishes for t 1/∆ according to the relaxations as below, γ 0 (t) ∼ c 0,n 0 g 1 (∆t) −α 0 ln n 0 (∆t) ,(47) that become inverse power laws for n 0 = 0, γ 0 (t) ∼ c 0,n 0 g 1 (∆t) −α 0 ,(48) where g 1 = ∆ sin (πα 0 /2) Γ (α 0 ). Notice that Eq. (46) is recovered from Eq. (48) as α 0 → 0 + . If α 0 = 2m 2 where m 2 and n 0 are non-vanishing natural numbers, the dephasing rate vanishes for t 1/∆ as follows, γ 0 (t) ∼ c 0,n 0 g 1 (∆t) −2m 2 ln n 0 −1 (∆t) ,(49) where g 1 = π(−1) m 2 +1 n 0 (2m 2 − 1)!∆/2. The above relaxation becomes an inverse power laws for n 0 = 1, γ 0 (t) ∼ c 0,n 0 g 1 (∆t) −2m 2 .(50) If α 0 is an even natural number and n 0 vanishes, consider the least non-vanishing index k 2 such that either α k 2 does not take even natural values or α k 2 = 2m k 2 , where the natural numbers m k 2 and n k 2 do not vanishes. The function γ 0 (t) is obtained in the former case from Eqs. (47) and (48) by substituting the power α 0 with α k 2 and n 0 with n k 2 , or in the latter case from Eqs. (49) and (50) by substituting the power m 2 with m k 2 and n 0 with n k 2 . We consider SDs such that the index k 2 exists. At this stage we focus on SDs that belong to the second class introduced in Section 4. At zero temperature, T = 0, we find various forms of relaxations of the dephasing rate for t 1/∆, γ 0 (t) ∼ w 0 (∆t) α 0 g 1 ln β 0 (∆t) −ḡ 1 ln β 0 −1 (∆t) ,(51)whereḡ 1 = β 0 ∆ π 2 cos πα 0 2 Γ (α 0 ) + sin πα 0 2 Γ (1) (α 0 ) . If the Ohmicity parameter α 0 does not take even natural values, the dominant part of the above asymptotic form is γ 0 (t) ∼ w 0 g 1 (∆t) −α 0 ln β 0 (∆t), and becomes the power law γ 0 (t) ∼ w 0 g 1 (∆t) −α 0 if β 0 = 0. If the Ohmicity parameter α 0 is an even natural number, Eq. (51) gives γ 0 (t) ∼ −w 0ḡ1 (∆t) −α 0 ln β 0 −1 (∆t) and becomes the power law γ 0 (t) ∼ −w 0ḡ1 (∆t) −α 0 if β 0 = 1. Notice the expected similarities between Eqs. (47) and (51). According to the above analysis, at zero temperature, T = 0, for the first class of SDs under study the information is lost into the environment over short times, t 1/∆. Over long times, t 1/∆, the information flows back into the system for the following values of the Ohmicity parameter, 2 + 4n < α 0 < 4 + 4n, where n = 0, 1, 2, . . .. For the second class of SDs under study we observe the same dependence of the long time information back-flow on the Ohmicity parameter. The corresponding long time dynamics is non-Markovian. The modulus of the coherence term increases, along with the back-flow of information, up to the non-vanishing asymptotic value and recoherence is observed over long times for 2 + 4n < α 0 < 4 + 4n, at zero temperature. Otherwise, the long time information is lost into the environment, the long time dynamics is Markovian and the modulus of the coherence term decreases down to the asymptotic value. If compared to the initial condition, coherence is partially lost for α 0 > 1. Coherence is fully lost if 0 ≤ α 0 ≤ 1. Notice that in the whole paper the analysis concerns uniquely the short and long time flow of information. Consequently, the dynamics can still be non-Markovian, due to an intermediate back-flow, even if no information flows from the environment back into the system over long times. (a) (b) (c ) (d) (e) (f ) (g) (h) (i ) ( j ) (k ) (l ) (m) (n) (o) Thermal bath Let the external environment be a thermal bath, T > 0. For SDs that belong to the first class under study and α 0 > 0 the dephasing rate increases linearly over short times, t 1/∆, γ T (t) ∼ l T t.(52) This behavior is independent of the low or high frequency structure of the SDs under study. Over long times, the dephasing rate divergences or vanishes in dependence on the low frequency profile of the SD that is given by Eq. (15). If 0 < α 0 < 1 the dephasing rate diverges for t 1/∆ according to the following form, that describes power laws for n = 0, γ T (t) ∼ c 0,n 0 g T (∆t) 1−α 0 ln n 0 (∆t) ,(53)(a) (b) (c ) (d) (e) (f ) (g) (h) (i ) ( j ) (k ) (l ) (m) (n) (o)γ T (t) ∼ c 0,n 0 g T (∆t) 1−α 0 .(54) The coefficient g T reads g T = 2k B T cos (πα 0 /2) Γ (α 0 ) / ( (1 − α 0 )). If α 0 = 1 the dephasing rate diverges for t 1/∆ as below, γ T (t) ∼ c 0,n 0 πk B T ln n 0 (∆t) .(55) If α 0 = 1 and n 0 = 0 the dephasing rate converges for t 1/∆ to the following non-vanishing value, γ T (t) ∼ c 0,n 0 πk B T .(56) If α 0 > 1 and α 0 is not an odd natural number, the dephasing factor vanishes for t 1/∆ according to Eq. (53). If α 0 = 1 + 2m 3 where m 3 and n 0 are non-vanishing natural numbers, the dephasing rate vanishes for t 1/∆ as follows, γ T (t) ∼ c 0,n 0 g T (∆t) −2m 3 ln n 0 −1 (∆t) ,(57) where g T = π(−1) 1+m 3 k B T n 0 (2m 3 − 1)!/ . The above relaxation provides inverse power laws for n 0 = 1, γ 0 (t) ∼ c 0,n 0 g T (∆t) −2m 2 .(58) If α 0 is an odd natural number and n 0 vanishes, consider the least non-vanishing index k 3 such that either α k 3 does not take odd natural values or α k 3 = 1 + 2m k 3 , where the natural numbers m k 3 and n k 3 do not vanishes. The function γ T (t) is obtained in the former case from Eqs. (53) and (54) by substituting the power α 0 with α k 3 and n 0 with n k 3 , or in the latter case from Eqs. (57) and (58) by substituting the power m 3 with m k 3 and n 0 with n k 3 . We consider SDs such that the index k 3 exists. Let the external environment consist in a thermal bath, T > 0, and the auxiliary functions Ω T (ν) belong to the second class under study. The dephasing rate vanishes for t 1/∆ according to arbitrary powers of logarithmic forms, (a) (b) (c ) (d) (e) (g) (f ) (h) (i ) ( j ) (l )(γ T (t) ∼ w 0 (∆t) 1−α 0 g T ln β 0 (∆t) +ḡ T ln β 0 −1 (∆t) ,(59) whereḡ T = k B T β 0 2 cos πα 0 2 Γ (1) (α 0 − 1) − π sin πα 0 2 Γ (α 0 − 1) . If the Ohmicity parameter α 0 does not take odd natural values, the dominant part of the above relaxation is γ T (t) ∼ w 0 g T (∆t) 1−α 0 ln β 0 (∆t), and becomes the power law γ T (t) ∼ w 0 g T (∆t) 1−α 0 if β 0 = 0. If the Ohmicity parameter α 0 is an odd natural number, Eq. (59) gives γ T (t) ∼ w 0ḡT (∆t) 1−α 0 ln β 0 −1 (∆t) and becomes the power law γ T (t) ∼ w 0ḡT (∆t) 1−α 0 if β 0 = 1. Notice the expected similarities between Eqs. (53) and (59). The above results show that at non-vanishing temperatures, T > 0, and for the first class of SDs under study, the information flows into the environment over short times, t 1/∆. Over long times, t 1/∆, back-flow of information appears for the following values of the Ohmicity parameter, 3 + 4n < α 0 < 5 + 4n, where n = 0, 1, 2, . . .. Same conditions hold for the appearance of the long time back-flow of information by considering the second class of SDs under study. The corresponding long time evolution is non-Markovian. Along with the back-flow of information, the modulus of the coherence term increases up to the non-vanishing asymptotic value and recoherence is observed over long times for 3+4n < α 0 < 5+4n, at nonvanishing temperatures. Otherwise, the long time information is lost into the environment, the long time dynamics is Markovian and the modulus of the coherence term decreases down to the asymptotic value. If compared to the initial condition, coherence is partially lost if α 0 > 2. Coherence is fully lost if 0 < α 0 ≤ 2. Consider the transition from vanishing to an arbitrary non-vanishing temperature. For the SDs under study we observe that the back-flow of information does not change for 3 + 4n < α 0 < 4 + 4n, while it is inverted for 2 + 4n < α 0 < 3 + 4n and 4 + 4n < α 0 < 5 + 4n. In the transition from vanishing to an arbitrary non-vanishing temperature, the long time recoherence results to be unaffected in the former condition and it is destroyed in the latter ones. Numerical computations of the dephasing rate are displayed in Figures 4, 5 and 6. The short time linear growth is shown in Figure 5. The long time logarithmic relaxations are confirmed by the asymptotic lines plotted in Figure 6. Conclusions We have considered the local dephasing process of a qubit that interacts with a structured reservoir of frequency modes or a thermal bath. We have studied the coherence between the two energy eigenstates of the qubit and the flow of quantum information by analyzing the dephasing factor and dephasing rate over short and long times. The SDs under study are Ohmic-like, at low frequencies, with additional logarithmic factors that are represented by arbitrarily positive or negative powers of logarithmic forms. In this way, the SDs are approximately proportional to the form ∆ (ω/∆) α 0 (− ln (ω/∆)) β 0 for ω ∆. The corresponding singularities are removable and provide legitimate SDs that contain, enhance and reduce the low frequency power law profiles of the physically feasible Ohmic-like condition. The SDs are arbitrarily tailored at higher frequencies. In general, the full loss or persistence of coherence, over long times, is determined by integral and low frequency properties of the SD. Over short times, for the SDs under study, the dephasing factor increases quadratically and the dephasing rate grows linearly in time both at zero and at an arbitrary non-vanishing temperature. Over long times, the evolution of the dephasing factor and dephasing rate exhibits various behaviors that are described by logarithmic and power laws, in dependence on the low frequency structure of the SD and on the temperature of the thermal bath. The information flows into the environment over short times both at vanishing and non-vanishing temperature. Over long times, we have found that regular patterns appear in the direction of the flow of information, back into the system or forth into the environment, in dependence on the Ohmicity parameter α 0 of the SD, regardless of the logarithmic form factors. At zero temperature, the long time information flows from the environment back into the system in correspondence of the following periodical intervals, 2 + 4n < α 0 < 4 + 4n, for every n = 0, 1, 2, . . .. At non-vanishing temperatures back-flow of information is obtained over the periodical intervals 3 + 4n < α 0 < 5 + 4n. In the transition from vanishing to an arbitrary non-vanishing temperature, the back-flow of information stably persists over the intervals 3 + 4n < α 0 < 4 + 4n. Instead, the back-flow is inverted over the intervals 2 + 4n < α 0 < 3 + 4n and 4 + 4n < α 0 < 5 + 4n. Non-Markovianity and recoherence of the qubit appear along with the back-flow of information. Consequently, recoherence is observed over long times for 2 + 4n < α 0 < 4 + 4n, at zero temperature, and for 3 + 4n < α 0 < 5 + 4n, at non-vanishing temperature. For 3 + 4n < α 0 < 4 + 4n the transition from vanishing to an arbitrary non-vanishing temperature does not destroy the recoherence process. The presentation of an experimental setting is beyond the purposes of this paper. Still, the reported results apply to the Ohmic-like SDs of trapped impurity atoms that are immersed in a Bose-Einstein condensate environment. Furthermore, if the low frequency power law profiles of the Ohmic-like SDs are enhanced or reduced via arbitrary positive or negative powers of logarithmic form factors, the direction of the information flow is not altered by the logarithmic terms and depends uniquely on the Ohmicity parameter of the Ohmic-like term. Consequently, the patterns in the information flow remain stable with respect to the mentioned logarithmic perturbations of the Ohmic-like SDs. We believe that the present analysis provides further scenarios for the implementation of a stable control of the flow of quantum information and the appearance of non-Markovian dynamics and recoherence via the engineering reservoir approach. A details The evolution of the reduced density matrix ρ(t) is given by the master equation (2). The off-diagonal elements of the reduced density matrix are described by Eq. (5) in terms of the dephasing factor Ξ(t). This function is given by Eq. (6), for T = 0, and Eq. (7), for T > 0. If the second negative moment of the SD is finite, Eq. (8), the expression´∞ 0 J (ω) cos (ωt) /ω 2 dω vanishes over long times due to the Riemann-Lebesgue lemma. In this way, Eq. (9) is obtained. Again, according to the Riemann-Lebesgue lemma, if the second negative moment of the effective SD is finite, Eq. (10), the expression´∞ 0 J T (ω) cos (ωt) /ω 2 dω vanishes over long times. In this way, Eq. (11) is obtained. The asymptotic behavior of the function Ξ 0 (t) is studied in the dimensionless variables ν = ω/∆ and τ = ∆t by considering the function F 0 (τ ), that is defined as F 0 (τ ) = Ξ 0 (τ /∆). According to this definition, the function reads F 0 (τ ) =ˆ∞ 0 Ω (ν) ν 2 sin 2 τ ν 2 dν.(60) The Mellin transform [24,25] of the function F 0 (τ ) is defined as follows,F 0 (s) =´∞ 0 τ s−1 F 0 (τ ) dτ , and readsF 0 (s) = − cos πs 2 Γ (s)Ω (−1 − s) .(61) The fundamental strip depends on the asymptotic behavior of the auxiliary function [24,25]. Consider the first class of SDs introduced in Section 4 via the asymptotic form (15). The fundamental strip of the Mellin transformF 0 (s) is min {0, α 0 − 1} > Re s > −2. The following asymptotic relationship [28], cos πs 2 Γ (s) ∼ sin πs 2 Γ (s) ∼ π 2 1/2 |Im s| Re s−1/2 ,(62) holds for |Im s| → +∞. For max {−4, −2 − χ 0 } < Re s < min {−1/2 − 0 , α 0 − 1} and |Im s| → +∞ the Mellin transform of the function F 0 (t) vanishes as follows,F 0 (s) = o |Im s| −1− 0 , where 0 ∈ (0, 3/2). Consequently, the functionF 0 (s) decreases sufficiently fast in the strip as | Im s| → +∞ and the singularity in s = −2 provides the asymptotic expansion of the dephasing factor at short times, given by Eq. (17). As far as the long time evolution is concerned, let the strip µ 0 ≤ Re s ≤ δ 0 exist such that the functionΩ (−1 − s), or the meromorphic continuation, vanishes as follows, Ω (−1 − s) = O |Im s| −ζ 0 ,(63) for | Im s| → +∞, where ζ 0 > 1/2 + δ 0 . The parameters µ 0 and δ 0 fulfill the constraints as below, µ 0 ∈ (−2, min {0, α 0 − 1}), δ 0 ∈ (α 0 − 1, 0) for 0 ≤ α 0 < 1, or δ 0 ∈ (α k 4 , α k 5 ) if α 0 ≥ 1. The parameter α k 4 coincides with the positive power α 0 if α 0 is not an even natural number, or if α 0 = 2m 0 and n 0 > 0, otherwise α k 4 coincides with the parameter α k 0 that is defined in Section 3. The index k 4 is the least natural number that is larger than k 3 and such that α k 4 is not an even natural number, or such that α k 4 is an even natural number and n k 4 > 0. Under the above conditions, the singularity of the functionF 0 (s) in s = α 0 − 1 and 0 ≤ α 0 ≤ 1, in s = 0 and s = α k 4 − 1 if α 0 > 1, provides the asymptotic forms given by Eqs. (18)- (26). At non-vanishing temperature the asymptotic behavior of the dephasing factor is evaluated via the function F T (τ ), defined as F T (τ ) = Ξ T (τ /∆), and the Mellin transform,F T (s), that readsF T (s) = − cos πs 2 Γ (s)Ω T (−1 − s) .(64) The fundamental strip is min {0, α 0 − 2} > Re s > −2, for α 0 > 0. The relationship (62) implies that for max {−4, −2 − χ 0 } < Re s < min {−1/2 − 1 , α 0 − 2} and | Im s| → +∞ the Mellin transform of the function F T (t) vanishes as follows,F T (s) = o |Im s| −1− 1 , where 1 ∈ (0, 3/2). Consequently, the functionF T (s) decreases sufficiently fast in the strip as | Im s| → +∞ and the singularity in s = −2 provides the asymptotic expansion of the dephasing factor at short times, given by Eq. (28). As far as the long time behavior is concerned, let the strip µ 1 ≤ Re s ≤ δ 1 exist such that the functionΩ T (−1 − s), or the meromorphic continuation, vanishes as follows, Ω T (−1 − s) = O |Im s| −ζ 1 ,(65) for | Im s| → +∞, where ζ 1 > 1/2 + δ 1 . The parameters µ 1 and δ 1 fulfill the following constraints, µ 1 ∈ (−2, α 0 − 2), δ 1 ∈ (α 0 − 2, 0) for 0 < α 0 < 2, or µ 1 ∈ (−2, 0) and δ 1 ∈ (α k 6 , α k 7 ) for α 0 ≥ 2. The parameter α k 6 coincides with the positive power α 0 if α 0 is not an odd natural number, or if α 0 = 1 + 2m 1 and n 0 > 0, otherwise α k 6 coincides with the parameter α k 1 that is defined in Section 3. The index k 7 is the least natural number that is larger than k 6 and such that α k 7 is not an odd natural number, or such that α k 7 is an odd natural number and n k 7 > 0. Under the above conditions, the singularity of the function F T (s) in s = α 0 − 2 for 0 < α 0 ≤ 2, or in s = 0 and s = α k 6 − 2 for α 0 > 2 provides Eqs. (29)-(38). For the second class of SDs introduced in Section 4 the study performed in Refs. [26,25], allows the asymptotic analysis of the expression (6), for T = 0, and (7), for T > 0, in terms of the dimensionless variables ν and τ . In this way, the asymptotic forms (27), for T = 0, and (39), for T > 0, are obtained. The dephasing rate γ(t) is defined by Eq. (3), for T = 0, and by Eq. (4), for T > 0. The constraints (41) and (42) are obtained by observing that the sine transforms of non-increasing functions are non-negative. For the first class of SDs under study the asymptotic behavior of the dephasing rate γ 0 (t) is studied by considering the function G 0 (τ ), that is defined as G 0 (τ ) = γ 0 (τ /∆) and reads G 0 (τ ) = ∆ˆ∞ 0 Ω (ν) ν sin (ντ ) dν. The Mellin transformĜ 0 (s) results as below, G 0 (s) = ∆ sin πs 2 Γ (s)Ω (−s) . The fundamental strip is min {1, α 0 } > Re s > −1. The relationship (62) suggests that for max {−3, −1 − χ 0 } < Re s < −1/2− 2 and | Im s| → +∞ the Mellin transform of the function G 0 (t) vanishes as follows,Ĝ 0 (s) = o |Im s| −1− 2 , where 2 ∈ (0, 1/2). Consequently, the functionĜ 0 (s) decreases sufficiently fast in the strip as | Im s| → +∞ and the singularity in s = −1 provides Eq. (45). As far as the long time evolution is concerned, let the strip µ 2 ≤ Re s ≤ δ 2 exist, such that the functionΩ (−s), or the meromorphic continuation, vanishes as follows,Ω (−s) = O |Im s| −ζ 2 ,(68) for | Im s| → +∞, where ζ 2 > 1/2 + δ 2 . The parameters µ 2 and δ 2 fulfill the constraints as below, µ 2 ∈ (−1, min {1, α 0 }), δ 2 ∈ (α k 8 , α k 9 ). The parameter α k 8 coincides with the positive power α 0 if α 0 is not an even natural number, or if α 0 = 2m 2 and n 0 > 0, otherwise α k 8 coincides with the power α k 2 that is defined in Section 3. The index k 9 is the least natural number that is larger than k 8 and such that α k 9 is not an even natural number, or such that α k 9 is an even natural number and n k 9 > 0. Under the above condition, the singularity of the functionĜ 0 (s) in s = α k 8 provides the asymptotic forms given by Eqs. (46)-(50). For non-vanishing temperatures, T > 0, we study the function G T (τ ), that is defined as G T (τ ) = γ T (τ /∆) and reads G T (τ ) = ∆ˆ∞ 0 Ω (ν) ν coth ∆ν 2k B T sin (ντ ) dν.(69) The Mellin transformĜ T (s) results in the following form, where 3 ∈ (0, 1/2). Consequently, the functionĜ T (s) decreases sufficiently fast in the strip as | Im s| → +∞ and the singularity in s = −1 gives Eq. (52). As far as the long time behavior is concerned, let the strip µ 3 ≤ Re s ≤ δ 3 exist such that the functionΩ (−s), or the meromorphic continuation, vanishes as follows, Ω T (−s) = O |Im s| −ζ 3 ,(71) for | Im s| → +∞, where ζ 3 > 1/2 + δ 3 . The parameters µ 3 and δ 3 fulfill the following constraints, µ 3 ∈ (−1, min {1, α 0 − 1}) for α 0 > 0, δ 3 ∈ (α k 10 , α k 11 ). The parameter α k 10 coincides with the positive power α 0 if α 0 is not an odd natural number, or if α 0 = 1 + 2m 3 and n 0 > 0, otherwise α k 10 coincides with the power α k 3 that is defined in Section 3. The index k 11 is the least natural number that is larger than k 10 and such that α k 11 is not an odd natural number, or such that α k 11 is an odd natural number and n k 11 > 0. Under the above conditions the singularity of the functionĜ T (s) in s = α k 10 − 1 provides the asymptotic forms given by Eqs. (53)-(58). For the second class of SDs introduced in Section 4, the long time behavior of the dephasing rate is obtained from the study performed in Refs. [26,25] in terms of the dimensionless variables ν and τ . In this way, we obtain the expressions (51), for T = 0, and (59), for T > 0. The direction of the flow of information over short and long times is performed by studying the sign of the first term of the asymptotic expansion over short and long times, respectively. Negatives values correspond to back-flow of information. This concludes the demonstration of the present results. Figure 1 : 1(Color online) The quantity ρ 0,1 (t)/ρ 0,1 (0) versus ∆t for 0 ≤ ∆t ≤ 6, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆) and different values of the parameters α and λ. The curve (a) corresponds to α = 1.6, λ = 0.3; (b) corresponds to α = 1.6, λ = 0.4, (c) corresponds to α = 1.6, λ = 0.48; (d) corresponds to α = 1.6, λ = 0.6; (e) corresponds to α = 2, λ = 0.8; (f ) corresponds to α = 2, λ = 1; (g) corresponds to α = 2.5, λ = 1.2; (h) corresponds to α = 5, λ = 2; (i) corresponds to α = 5, λ = 2.2, (j) corresponds to α = 3, λ = 3. Over long times the curves tend to non-vanishing values. Figure 2 : 2(Color online) The quantity ln (ln (ρ 0,1 (0)/ρ 0,1 (t))) versus ln (∆t) for exp {−3} ≤ ∆t ≤ exp {5}, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆) and different values of the parameters α and λ. The curve (a) corresponds to α = 5, λ = 15; (b) corresponds to α = 5, λ = 10, (c) corresponds to α = 5, λ = 7; (d) corresponds to α = 2, λ = 22; (e) corresponds to α = 1.5, λ = 20; (f ) corresponds to α = 1.5, λ = 9; (g) corresponds to α = 10, λ = 4.8; (h) corresponds to α = 10, λ = 4.3; (i) corresponds to α = 1.5, λ = 1, (j) corresponds to α = 10, λ = 3.4; (k) corresponds to α = 1.5, λ = 0.4; (l) corresponds to α = 20, λ = 6.1; (m) corresponds to α = 1.5, λ = 0.2; (n) corresponds to α = 20, λ = 5.55. Over short times each curve tends to an asymptotic line with the slope 2. Figure 3 : 3(Color online) The quantity ln (∆t) α−1 ln (ρ 0,1 (0)/ρ 0,1 (t)) versus ln (ln (∆t))for exp {exp {−1}} ≤ ∆t ≤ exp {exp {5}}, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆)and different values of the parameters α and λ. The curve (a) corresponds to α = 1.5, λ = 10000; (b) corresponds to α = 1.5, λ = 0.01, (c) corresponds to α = 1.5, λ = 0.0001; (d) corresponds to α = 2.5, λ = 10000; (e) corresponds to α = 2.5, λ = 1; (f ) corresponds to α = 2.5, λ = 0.02; (g) corresponds to α = 2.5, λ = 0.0001; (h) corresponds to α = 6, λ = 10; (i) corresponds to α = 6, λ = 0.8, (j) corresponds to α = 6, λ = 0.1; (k) corresponds to α = 6, λ = 0.01; (l) corresponds to α = 6, λ = 0.001; (m) corresponds to α = 6, λ = 0.0002; (n) corresponds to α = 6, λ = 0.00003. Over long times each curve tends to an asymptotic line and the the slope depends on the value of the parameter α. Figure 4 : 4(Color online) The ratio γ 0 (t)/∆ versus ∆t for 0 ≤ ∆t ≤ 9, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆) for different values of the parameters α and λ. The curve (a) corresponds to α = 2, λ = 1.1; (b) corresponds to α = 0.9, λ = 40, (c) corresponds to α = 1.5, λ = 3; (d) corresponds to α = 0.8, λ = 27; (e) corresponds to α = 1.3, λ = 2.9; (f ) corresponds to α = 1.3, λ = 0.7; (g) corresponds to α = 0.8, λ = 17; (h) corresponds to α = 1.1, λ = 2.9; (i) corresponds to α = 1.1, λ = 1.2, (j) corresponds to α = 0.8, λ = 10; (k) corresponds to α = 0.9, λ = 4.8; (l) corresponds to α = 1, λ = 0.7; (m) corresponds to α = 0.7, λ = 9.5; (n) corresponds to α = 0.8, λ = 4.5, (o) corresponds to α = 2, λ = 0.9. Figure 5 : 5(Color online) The ratio γ 0 (t)/∆ versus ∆t for 0 ≤ ∆t ≤ 10, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆) for different values of the parameters α and λ. The curve (a) corresponds to α = 1.5, λ = 12; (b) corresponds to α = 1.5, λ = 5, (c) corresponds to α = 0.8, λ = 12.5; (d) corresponds to α = 0.7, λ = 11.5; (e) corresponds to α = 1.6, λ = 1.9; (f ) corresponds to α = 1.6, λ = 1.7; (g) corresponds to α = 2, λ = 1.6; (h) corresponds to α = 2, λ = 1.5; (i) corresponds to α = 1.4, λ = 1.4, (j) corresponds to α = 1.4, λ = 1.3; (k) corresponds to α = 0.8, λ = 1.2; (l) corresponds to α = 1.3, λ = 1.1; (m) corresponds to α = 1.3, λ = 1; (n) corresponds to α = 1, λ = 0.8, (o) corresponds to α = 1, λ = 0.5. Over early times each curve tends to an asymptotic line. Figure 6 : 6(Color online) The quantity ln ((∆t) α γ 0 (t)/∆) versus ln (ln (∆t)) for exp {1/e} ≤ ∆t ≤ exp {exp {2.6}}, J (ω) = ∆ (ω/∆) α exp {−λω/∆} ln 2 (ω/∆) and different values of the parameters α and λ. The curve (a) corresponds to α = 1, λ = 10000; (b) corresponds to α = 1, λ = 200, (c) corresponds to α = 1, λ = 0.01; (d) corresponds to α = 10, λ = 20; (e) corresponds to α = 10, λ = 4; (f ) corresponds to α = 10, λ = 0.01; (g) corresponds to α = 15, λ = 5; (h) corresponds to α = 15, λ = 2; (i) corresponds to α = 15, λ = 0.01, (j) corresponds to α = 20, λ = 2; (k) corresponds to α = 20, λ = 1; (l) corresponds to α = 20, λ = 0.001. Over long times each curve tends to an asymptotic line. strip is min {1, α 0 − 1} > Re s > −1, where α 0 > 0. The relationship (62) implies that for max {−3, −1 − χ 0 } < Re s < min {−1/2 − 3 , α 0 − 1}and | Im s| → +∞ the Mellin transform of the function G T (t) vanishes as follows,Ĝ 0 (s) = o |Im s| −1− 3 , H.-P Breuer, F Petruccione, The Theory of Open Quantum Systems. OxfordOxford University PressH.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford Univer- sity Press, Oxford (2002). U Weiss, Quantum Dissipative systems. SingaporeWorld Scientific3rd edU. Weiss, Quantum Dissipative systems, 3rd ed. World Scientific, Singapore (2008). . H.-P Breuer, E M Laine, J Piilo, B Vacchini, Rev. Mod. Phys. 8821002H.-P. Breuer, E.M. Laine, J. Piilo and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016). . A Rivas, S F Huelga, M B Plenio, Rep. Prog. Phys. 7794001A. Rivas, S.F. Huelga and M.B. Plenio, Rep. Prog. Phys. 77, 094001 (2014). . X.-M Lu, X Wang, C P Sun, Phys. Rev. A. 8242103X.-M. Lu, X. Wang and C.P. Sun, Phys. Rev. A 82, 042103 (2010). . R Vasile, S Maniscalco, M G A Paris, H.-P Breuer, J Piilo, Phys. Rev. A. 8452118R. Vasile, S. Maniscalco, M.G.A. Paris, H.-P. Breuer and J. Piilo, Phys. Rev. A 84, 052118 (2011). . S Luo, S Fu, H Song, Phys. Rev. A. 8644101S. Luo, S. Fu and H. Song, Phys. Rev. A 86, 044101 (2012). . H.-P Breuer, E.-M Laine, J Piilo, Phys. Rev. Lett. 103210401H.-P. Breuer, E.-M. Laine and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009); . E.-M Laine, J Piilo, H.-P Breuer, Phys. Rev. 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Johnson, Phys. Rev. A 65 , 032306 (2002). . C Addis, B Bylicka, D Chruscinski, S Maniscalco, Phys. Rev. A. 9052103C. Addis, B. Bylicka, D. Chruscinski and S. Maniscalco, Phys. Rev. A 90, 052103 (2014). . Z He, J Zou, L Li, B Shao, Phys. Rev. A. 8312108Z. He, J. Zou, L. Li and B. Shao, Phys. Rev. A 83, 012108 (2011). . J T Barreiro, M Uller, P Schindler, D Nigg, T Monz, M Chwalla, M Hennrich, C F Roos, P Zoller, R Blatt, Nature. 470486J.T. Barreiro, M. M uller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C.F. Roos, P. Zoller and R. Blatt, Nature 470, 486 (2011). . B.-H- Liu, Y.-F Huang, C.-F Li, G.-C Guo, E.-M Laine, H.-P Breuer, J Piilo, Nat. Phys. 7931B.-H-Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer and J. Piilo, Nat. Phys. 7, 931 (2011). Bath correlation functions for logarithmic spectral densities. F Giraldi, submittedF. Giraldi, Bath correlation functions for logarithmic spectral densities, submitted. . F Giraldi, Eur. Phys. J. D. 695F. Giraldi, Eur. Phys. J. D 69, 5 (2015); . Eur. Phys. J. D. 70229Eur. Phys. J. D 70, 229 (2016). . M A Cirone, G De Chiara, G M Palma, P Haikka, S Mcendoo, S Maniscalco, Phys. Rev. A. 8431602M.A. Cirone, G. De Chiara, G. M. Palma, P. Haikka, S. McEndoo and S. Maniscalco, Phys. Rev. A 84, 031602 (2011). . P Haikka, S Mcendoo, G De Chiara, G M Palma, S Maniscalco, Phys. Rev. A. 8431602P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, Phys. Rev. A 84, 031602 (2011). Asymptotic expansion of integrals. N Bleistein, R A Handelsman, Dover Publications, Inc. New YorkN. Bleistein and R.A. Handelsman, Asymptotic expansion of integrals, Dover Publica- tions, Inc. New York (1975). R Wong, Asymptotic approximations of integrals. BostonAcademic PressR. Wong, Asymptotic approximations of integrals (Academic Press, Boston, 1989). . R Wong, J F Lin, J. Math. Anal. Appl. 64173R. Wong and J.F. Lin, J. Math. Anal. Appl. 64, 173 (1978). . 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{'abstract': 'Consider local dephasing processes of a qubit that interacts with a structured reservoir of frequency modes or a thermal bath, with Ohmic-like spectral density (SD). It is known that non-Markovian evolution appears uniquely above a temperature-dependent critical value of the Ohmicity parameter, and non-Markovianity can be induced by properly engineering the external environment. In the same scenario, we find that the flow of quantum information shows regular patterns: alternate directions appear in correspondence of periodical intervals of the Ohmicity parameter α 0 . The information flows back into the system over long times for 2 + 4n < α 0 < 4 + 4n, at zero temperature, and for 3 + 4n < α 0 < 5 + 4n, at non-vanishing temperatures, where n = 0, 1, 2, . . .. Otherwise, the long time information flows into the environment. In the transition from vanishing to arbitrary non-vanishing temperature, the long time back-flow of information is stable for 3 + 4n < α 0 < 4 + 4n, while it is reverted for 2 + 4n < α 0 < 3 + 4n and 4 + 4n < α 0 < 5 + 4n. The patterns of the information flow are not altered if the low frequency Ohmic-like profiles of the SDs are perturbed with additional factors that consist in arbitrary powers of logarithmic forms. Consequently, the flow of information can be controlled, directed and reverted over long times by engineering a wide variety of reservoirs that includes and continuously departs from the Ohmic-like structure at low frequencies. Non-Markovianity and recoherence appear according to the same rules along with the back-flow of information.', 'arxivid': '1612.00813', 'author': ['Filippo Giraldi \nSchool of Chemistry and Physics\nSouth Africa Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM\nUniversity of KwaZulu-Natal and National Institute for Theoretical Physics (NITheP) Westville Campus\n4000Durban\n\nIstituto Nazionale di Alta Matematica Francesco Severi Cittá Universitaria\nPiazza Aldo Moro 500185RomaItaly\n'], 'authoraffiliation': ['School of Chemistry and Physics\nSouth Africa Gruppo Nazionale per la Fisica Matematica (GNFM-INdAM\nUniversity of KwaZulu-Natal and National Institute for Theoretical Physics (NITheP) Westville Campus\n4000Durban', 'Istituto Nazionale di Alta Matematica Francesco Severi Cittá Universitaria\nPiazza Aldo Moro 500185RomaItaly'], 'corpusid': 119087826, 'doi': '10.1103/physreva.95.022109', 'github_urls': [], 'n_tokens_mistral': 20535, 'n_tokens_neox': 17480, 'n_words': 10713, 'pdfsha': 'f315b71ab83f2fefbbae76716f95a9ba9c7d7995', 'pdfurls': ['https://arxiv.org/pdf/1612.00813v1.pdf'], 'title': ['Regular patterns in the information flow of local dephasing channels', 'Regular patterns in the information flow of local dephasing channels'], 'venue': []}

arxiv

ON THE ALGEBRAIC BOUNDARIES AMONG TYPICAL RANKS FOR REAL BINARY FORMS 23 Apr 2018 Maria Chiara Brambilla And Giovanni Staglianò ON THE ALGEBRAIC BOUNDARIES AMONG TYPICAL RANKS FOR REAL BINARY FORMS 23 Apr 2018arXiv:1804.08309v1 [math.AG] We describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight, showing that they are dual varieties of suitable coincident root loci. Introduction The study of symmetric tensors, of their rank, decomposition and identifiability is a classical problem, which received great attention recently in both pure and applied mathematics; see e.g. [Lan12] and references therein, see also [BBM14,BBO15,MMSV17,CKOV17,MMS18,ABC18]. Symmetric tensors can be interpreted as homogeneous polynomials, also called forms. The rank of a degree d form f is the minimum integer r such that there exists a decomposition f = r i=1 c i (l i ) d , where l i are linear forms and c i are scalars. In this paper we focus on the case of binary forms over the field of real numbers R. In this case it is known that the (real) rank of a general form satisfies the inequalities d+1 2 ≤ r ≤ d. Moreover all the ranks in this range are typical, that is, they occur in open subsets (with respect to the Euclidean topology) of the real vector space of degree d forms; see [Ble15]. A natural problem is to understand the geometry of the sets R d,r of forms of degree d and rank r. In particular we would like to describe the boundaries among the various sets of forms of given typical rank; more precisely, we are interested in understanding the algebraic boundaries, i.e., the Zariski closures of the topological boundaries (see Section 3 for the precise definitions). The easiest case is the maximal one, that is when the rank is equal to the degree. Indeed it is proved in [CR11,CO12] that a binary form of degree d with distinct roots has rank d if and only if all its roots are real. Hence its algebraic boundary is the discriminant hypersurface of forms with two coincident roots. The geometric description of the sets R d,r becomes much more intricate for r < d. Indeed, although the rank of a form is always greater than or equal to the number of its real distinct roots, in general the number of real distinct roots is not invariant in the region R d,r . In [LS16] the authors study the boundary of the set of forms of rank ⌈ d+1 2 ⌉, which is the minimal typical rank. They prove that the components of the boundary are dual varieties of suitable coincident root loci. We tackle the problem of describing all the intermediate boundaries in general, as proposed by Lee and Sturmfels in [LS16,Remark 4.5]. Our approach provides a unified description of all the boundaries in terms of dual varieties of coincident root loci. We recall that the cases of degree d ≤ 5 have been described in [CO12], while the case d = 6 follows by [LS16,CR11] (see Proposition 3.1 for more details). In this paper we focus on the cases d = 7 and d = 8, and we postpone a general description to future work. The paper is organized as follows. Sections 2 and 3 are devoted to preliminary results; in particular, in Proposition 3.1 we recall the known results concerning algebraic boundaries for real binary forms of degree less than or equal to 6. Section 4 and 5 contain our main results, which are Theorem 4.1 and Theorem 5.1, describing the algebraic boundaries for real binary forms of degrees respectively 7 and 8. They turn out to be dual varieties of suitable coincident root loci. Finally in Section 6 we explain some of the computational methods of which we take advantage in our study. Acknowledgements. We thank the participants to the seminar "Algebraic Geometry and Tensors" where this work has begun. In particular we are grateful to Giorgio Ottaviani for many useful discussions. Coincident root loci We recall here some known results on coincident root loci, referring to [Wey89,Kat03,Chi03,Chi04,Kur12] for details. We regard a degree d binary form f = d i=0 d i a i x d−i y i over the complex field C as a point of the projective space P(C[x, y] d ), where C[x, y] d = Sym d (C 2 ) . This space is identified with P n using homogeneous coordinates (a 0 , . . . , a d ). A partition λ = (λ 1 , . . . , λ n ) of d is a list of integers λ 1 ≥ · · · ≥λ n ≥ 1 such that n i=1 λ i = d. Given a partition λ, the coincident root locus ∆ λ ⊂ P d is the set of binary forms f of degree d which admit a factorization f = n i=1 ℓ λ i i for some linear forms ℓ 1 , . . . , ℓ n ∈ C[x, y] 1 . A partition λ can be also represented by the list of integers m 1 , . . . , m k defined as m j = |{i : λ i = j}|, and clearly k j=1 jm j = d. Then the coincident root locus ∆ λ is given by the binary forms of degree d which have m j roots of multiplicity at least j. It is classically known (see [Hil87]) that ∆ λ ⊂ P d is a variety of dimension n and degree (2.1) deg(∆ λ ) = n! m 1 !m 2 ! · · · m k ! λ 1 λ 2 · · · λ n . If λ = (2, 1 d−2 ), the corresponding coincident root locus ∆ λ = ∆ is the classical discriminant hypersurface. In the opposite case, if λ = (d) then ∆ λ is the rational normal curve C d ⊂ P d . When λ = (a, 1 d−a ), the partition is called hook, and the associated coincident root locus ∆ λ represents the tangential developable of ∆ (a+1,1 d−a−1 ) . 2.1. Singularities of ∆ λ . The singular loci of coincident root loci have been studied by Chipalkatti [Chi03] and Kurmann [Kur12]. Given a partition λ = (λ 1 , . . . , λ n ), the singular locus Sing(∆ λ ) is given by the union of ∆ µ for some suitable coarsenings µ of λ. See either [Chi03, Definition 5.2], or [Kur12, Proposition 2.1] for the precise description. In particular ∆ λ is smooth if and only if λ 1 = · · · = λ n . Otherwise the singular locus is of (not necessarily pure) codimension 1. Example 2.1. For future use, we now compute the iterate singular locus of ∆ λ , for λ = (2, 1, 1, 1) and λ = (2, 1, 1, 1, 1). Sing(∆ (2,1,1,1) ) = ∆ (3,1,1) ∪ ∆ (2,2,1) ; Sing(∆ (3,1,1) ) = ∆ (4,1) , Sing(∆ (2,2,1) ) = Sing(∆ (3,2) ); Sing(∆ (4,1) ) = Sing(∆ (3,2) ) = ∆ (5) . Sing(∆ (2,1,1,1,1) ) = ∆ (3,1,1,1) ∪ ∆ (2,2,1,1) ; Sing(∆ (3,1,1,1) ) = ∆ (4,1,1) ∪ ∆ (3,3) , Sing(∆ (2,2,1,1) ) = ∆ (3,2,1) ∪ ∆ (2,2,2) ; Sing(∆ (4,1,1) ) = ∆ (5,1) , Sing(∆ (3,2,1) ) = ∆ (3,3) ∪ ∆ (4,2) ∪ ∆ (5,1) ; Sing(∆ (4,2) ) = Sing(∆ (5,1) ) = ∆ (6) . 2.2. Duality. Consider the dual ring of differential operators C[∂ x , ∂ y ] = C[u, v], which acts on C[x, y] with the usual rules of differentiations and gives the pairing with respect to the degrees, C[x, y] d ⊗ C[u, v] k → C[x, y] d−k . The conormal variety of a coincident root locus ∆ λ is the Zariski closure of the set {(f, g) : f is a smooth point of ∆ λ , g ⊥ T f ∆ λ } ⊂ P(C[x, y] d ) × P(C[u, v] d ), where T f ∆ λ denotes the tangent space to ∆ λ at a point f . The dual variety (∆ λ ) ∨ of ∆ λ is the projection onto P(C[u, v] d ) of the conormal variety of ∆ λ . The biduality theorem (see [GKZ08]) implies that (∆ ∨ λ ) ∨ = ∆ λ . Lee and Sturmfels study duality for binary forms in [LS16]. We recall here some results which we will use in the sequel. Π n i=1 ℓ λ i −1 i (∂ x , ∂ y ) where ℓ i ∈ C[u, v] 1 . Proposition 2.3 ([LS16] ). Given λ = (λ 1 , . . . , λ n ) and ∆ λ ⊂ P(C[u, v] d ), the dual variety ∆ ∨ λ ⊂ P(C[x, y] d ) has codimension m 1 , and it is given by the join of the (n − m 1 ) coincident root loci ∆ (d−λ i +2,1 λ i −2 ) for 1 ≤ i ≤ n with λ i ≥ 2. If λ i ≥ 2 for all i, then ∆ ∨ λ is a hypersurface of degree (see [Oed12]) (2.2) (n + 1)! m 2 ! · · · m k ! (λ 1 − 1)(λ 2 − 1) · · · (λ n − 1). 2.3. Chow forms and higher associated varieties. Let G(h, m) denote the Grassmannian of projective subspaces of dimension h in P m . Let X ⊂ P m be a projective variety of dimension k. The i-th higher associated variety CH i (X) of X is defined as the closure of the set of all (m−k −1+i)-dimensional subspaces L ⊂ P m such that L∩X = ∅ and dim(L∩T x X) ≥ i for some smooth point x ∈ L ∩ X (where T x X denotes the embedded tangent space to X at x), see [GKZ08] for details. For i = 0, the associated variety CH 0 (X) ⊂ G(m−k −1, m) is the Chow hypersurface, while for i = k, we have that CH k (X) ⊂ G(m − 1, m) corresponds to the dual variety X ∨ via the Grassmannian duality G(m − 1, P m ) ≃ G(0, (P m ) ∨ ). If i = 1 and deg(X) ≥ 2, the associated variety CH 1 (X) is the Hurwitz hypersurface, see [Stu17]. The variety CH i (X) is a hypersurface if and only if i ≤ dim(X) − (m − 1 − dim(X ∨ )), see [Koh16]. In particular, if X = ∆ (λ 1 ,...,λn) is a coincident root locus, the higher associated variety CH i (X) is a hypersurface if and only if i ≤ |{j : λ j ≥ 2}|. 3. Real rank of binary forms 3.1. Typical ranks for binary forms. Given a binary form f of degree d with complex (or real) coefficients, its complex rank is the minimum integer r such that f admits a decomposition f = r i=1 (ℓ i ) d where ℓ i are linear forms with complex coefficients. The generic complex rank for binary forms of degree d (that is the rank of a general binary form of degree d) is ⌈ d+1 2 ⌉. Sylvester Theorem says that a general binary form admits a unique minimal decomposition if the degree is odd, infinitely many (parametrized by a line) if the degree is even. Consider now the polynomial ring R = R[x, y] of real binary forms. Given f ∈ R d , the real rank of f (denoted by rk(f )) is the minimum integer r such that f admits a decomposition f = r i=1 c i (l i ) d where l i ∈ R 1 and c i ∈ R; we can impose c i ∈ {1, −1} if d is even, and c i = 1 if d is odd. In the real field the notion of generic rank is replaced by the notion of typical ranks. A rank is called typical for binary forms of degree d if it occurs in an open subset of R d , with respect to the Euclidean topology. Define R d,r = {f ∈ R d : rk(f ) = r}, and denote by R d,r the interior of R d,r . Then R d,r is a semi-algebraic set in the real vector space R d , and a rank is typical exactly when R d,r is not empty. From the main result of [Ble15], a rank r is typical if and only if d+1 2 ≤ r ≤ d. Thus, from now on we assume that d+1 2 ≤ r ≤ d. We define the topological boundary ∂(R d,r ) as the set-theoretic difference of the closure of R d,r minus the interior of the closure of R d,r . It is a semi-algebraic subset of R d of pure codimension one. We define the real rank boundary ∂ alg (R d,r ) as the Zariski closure of the topological boundary ∂(R d,r ) (see also [LS16,Section 4]). The real rank boundaries ∂ alg (R d,r ) are hypersurfaces of the real space R d , that we consider as hypersurfaces of the complex projective space P(C[x, y] d ) = P d C . Real rank boundaries have been studied only in the two extreme cases, that is for maximum rank d and minimum rank r [CO12,Proposition 3.1] and [CR11, Corollary 1]); in the second case the real rank boundary ∂ alg (R d,r ) is described in [LS16, Theorem 4.1]. Hence, for d ≤ 6 we have a complete description of all the real rank boundaries, that we recall in the following: CR11,LS16]). The real rank boundaries for binary forms of degree ≤ 6 are the following hypersurfaces: = ⌈ d+1 2 ⌉. In the first case ∂ alg (R d,d ) is the dis- criminant hypersurface ∆ (2,1 d−2 ) (seeProposition 3.1 ([CO12,∂ alg (R 3,2 ) = ∂ alg (R 3,3 ) = (∆ (3) ) ∨ ; ∂ alg (R 4,3 ) = ∂ alg (R 4,4 ) = (∆ (4) ) ∨ ; ∂ alg (R 5,3 ) = (∆ (3,2) ) ∨ , ∂ alg (R 5,4 ) = (∆ (3,2) ) ∨ ∪ (∆ (5) ) ∨ , ∂ alg (R 5,5 ) = (∆ (5) ) ∨ ; ∂ alg (R 6,4 ) = (∆ (3,3) ) ∨ ∪ (∆ (4,2) ) ∨ , ∂ alg (R 6,5 ) = (∆ (3,3) ) ∨ ∪ (∆ (4,2) ) ∨ ∪ (∆ (6) ) ∨ , ∂ alg (R 6,6 ) = (∆ (6) ) ∨ . Remark 3.2. The hypersurfaces (∆ (3) ) ∨ , (∆ (4) ) ∨ , (∆ (5) ) ∨ , (∆ (6) ) ∨ coincide with the discriminant hypersurfaces for binary forms of degrees 3, 4, 5, 6 and have degrees 4, 6, 8, 10, respectively. For the other components, we have • (∆ (3,2) ) ∨ = Join(∆ (4,1) , ∆ (5) ) is a hypersurface of degree 12 (this is the apple invariant I 12 considered in [CO12]); • (∆ (3,3) ) ∨ = Join(∆ (5,1) , ∆ (5,1) ) is a hypersurface of degree 12; • (∆ (4,2) ) ∨ = Join(∆ (4,1,1) , ∆ (6) ) is a hypersurface of degree 18. 3.2. Apolarity. We recall here classical techniques, going back to Sylvester. Even if the results of this section are more general, we present them in the case of real numbers. Let R = R[x, y] be the polynomial ring of real binary forms and let D = R[∂ x , ∂ y ] = R[u, v] be the corresponding dual ring. Given l = ax + by ∈ R 1 , the apolar operator is l ⊥ = −b∂ x + a∂ y ∈ D 1 . Given a form f in R d , the apolar ideal f ⊥ ⊂ D is given by all the operators which annihilates f , that is: f ⊥ = {g(∂ x , ∂ y ) ∈ D : g ⊥ f }. A basic tool is the following: Lemma 3.3 (Apolarity lemma). Assume f ∈ R d and let l i ∈ R 1 be distinct linear forms for 1 ≤ i ≤ r. There are coefficients c i ∈ R such that f = r i=1 c i (l i ) d if and only if the operator l ⊥ 1 • · · · • l ⊥ r is in the apolar ideal f ⊥ . We will say that a form of degree d is real-rooted if it admits d distinct real roots. From Lemma 3.3, it follows that a form f has rank less than or equal to r if and only if (f ⊥ ) r = f ⊥ ∩ D r contains a real-rooted form. So the rank of f is the smallest degree r such that (f ⊥ ) r contains a real-rooted form. The following result is an elementary consequence of Lemma 3.3. Corollary 3.4. Let f be a real binary form, and let r be an integer. Then rk(F ) < r if and only if (f ⊥ ) r ⊂ D r contains a special line whose generic member is a real-rooted form. Here, we say that a line g, g ′ ⊂ D r is special if gcd(g, g ′ ) is a form of degree r − 1. The space of operators of degree r contained in f ⊥ is the kernel of the linear map A f : D r → R d−r . The catalecticant (or Hankel) matrix of f is the matrix A d,r f of size (d − r + 1) × (r + 1) that represents A f with respect to the standard basis. We denote by A d,r the generic catalecticant matrix of size (d − r + 1) × (r + 1). The following result is well-known (see e.g. [IK99]): Proposition 3.5. Assume that f ∈ R d has rank greater than or equal to 2. Then its apolar ideal f ⊥ is generated by two real forms g, g ′ such that deg g + deg g ′ = d + 2 and gcd(g, g ′ ) = 1. Conversely, any two such forms generate an ideal f ⊥ for some f ∈ R with degree deg g + deg g ′ − 2. We say that f ∈ R d is generated in generic degrees if (deg g, deg g ′ ) = (⌈ d+1 2 ⌉, ⌊ d+3 2 ⌋) . The forms that are not generated in generic degrees form a subvariety of R d . More precisely, when the degree d = 2k is even, it is the hypersurface defined by the determinant of the intermediate (k + 1) × (k + 1) catalecticant matrix A d,k ; when the degree d = 2k + 1 is odd, it is the subvariety of codimension 2 defined by the maximal minors of the intermediate (k + 1) × (k + 2) catalecticant matrix A d,k+1 . If f is a binary form of degree d, with d 2 ≤ r ≤ d and having catalecticant matrix A d,r f of maximal rank, then dim(f ⊥ ) r = 2r − d. Thus, we can consider the apolar map Ψ d,r : P d G(d − r, r) ≃ G(2r − d − 1, r) which associates to a general binary form f of degree d the projective (2r − d − 1)dimensional subspace Π f = P((f ⊥ ) r ) ⊂ P(D r ) obtained from the degree r component of the apolar ideal. In coordinates the map Ψ d,r is defined by the maximal minors of the matrix A d,r . We denote by Z d,r = Ψ d,r (P d ) ⊂ G(2r − d − 1, r) the closure of the image of Ψ d,r . Real rank boundaries of degree 7 binary forms In this section, we prove the following: Theorem 4.1. The real rank boundaries for degree 7 binary real forms are the following hypersurfaces: ∂ alg (R 7,4 ) = (∆ (3,2,2) ) ∨ ; ∂ alg (R 7,5 ) = (∆ (3,2,2) ) ∨ ∪ (∆ (4,3) ) ∨ ∪ (∆ (5,2) ) ∨ ; ∂ alg (R 7,6 ) = (∆ (4,3) ) ∨ ∪ (∆ (5,2) ) ∨ ∪ (∆ (7) ) ∨ ; ∂ alg (R 7,7 ) = (∆ (7) ) ∨ . Remark 4.2. From Proposition 2.3 and formula (2.2) we obtain: • (∆ (3,2,2) ) ∨ = Join(∆ (6,1) , ∆ (7) , ∆ (7) ) is a hypersurface of degree 24; • (∆ (4,3) ) ∨ = Join(∆ (5,1,1) , ∆ (6,1) ) is a hypersurface of degree 36; • (∆ (5,2) ) ∨ = Join(∆ (4,1,1,1) , ∆ (7) ) is a hypersurface of degree 24; • (∆ (7) ) ∨ = ∆ (2,1,1,1,1,1) is a hypersurface of degree 12. Proof. We divide the proof in several steps. The boundary ∂ alg (R 7,7 ) between ranks 7 and ≤ 6. From [CO12] and [CR11], it is know that the real rank boundary ∂ alg (R 7,7 ) is the discriminant hypersurface ∆ (2,1 5 ) in P 7 . Note that ∆ (2,1 5 ) = (∆ (7) ) ∨ = Ψ −1 7,6 (CH 0 (∆ (6) )) where Ψ 7,6 : P 7 Z 7,6 ⊂ G(4, 6) ⊂ P 20 . The boundary ∂ alg (R 7,4 ) between ranks 4 and ≥ 5. For the reader's convenience, we sketch briefly the proof given in [LS16] of the fact that ∂ alg (R 7,4 ) = (∆ (3,2,2) ) ∨ . Consider a binary form f of degree 7 with apolar ideal f ⊥ = (g 4 , g 5 ), where deg(g i ) = i. By Lemma 3.3, we have that f ∈ R 7,4 if and only if g 4 is real-rooted. When f moves toward R 7,5 ∪ R 7,6 ∪ R 7,7 and passes through the boundary ∂ alg (R 7,4 ), then (at least) two roots of g 4 must collapse and become a double root. Hence at the transition point the generator g 4 belongs to the discriminant locus ∆ (2,1,1) ⊂ P(D 4 ), and by Proposition 2.2, we get ∂ alg (R 7,4 ) ⊆ (∆ (3,2,2) ) ∨ . Now since ∂ alg (R 7,4 ) = ∅ because 4 and 5 are typical ranks, and ∆ (3,2,2) ) ∨ is irreducible, it follows that ∂ alg (R 7,4 ) = (∆ (3,2,2) ) ∨ . The boundary ∂ alg (R 7,5 ) between ranks 5 and = 5. We describe now the boundary between R 7,5 and R 7,6 ∪ R 7,7 . Let f ε be a continuous family of forms crossing the boundary ∂ alg (R 7,5 ) at the point f 0 = f , going from R 7,5 to R 7,6 ∪R 7,7 . Namely, we assume that f −ε ∈ R 7,5 and f ε ∈ R 7,6 ∪R 7,7 for any small ε with ε > 0. We can assume that for any ε the form f ε is generated in generic degree, since the locus of non generated in generic degree forms has codimension 2. In particular, we can assume f ⊥ = (g 4 , g 5 ), where deg(g 4 ) = 4 and deg(g 5 ) = 5. Let G(2, 5) be the Grassmannian of planes in P(D 5 ), and consider the apolar map (4.1) Ψ 7,5 : P 7 Z 7,5 ⊂ G(2, 5) ⊂ P 19 , which is a cubic birational map onto a subvariety Z 7,5 ⊂ P 19 of degree 84 and cut out by 42 quadric hypersurfaces. The map Ψ 7,5 sends the family f ε into a continuous family of apolar planes Π ε ; in particular, Ψ 7,5 (f ) = Π 0 = ug 4 , vg 4 , g 5 is the apolar plane of f . From Lemma 3.3, we obtain that the plane Π ε , with ε < 0, contains a real-rooted form h ε = l 1 (ε)l 2 (ε)l 3 (ε)l 4 (ε)l 5 (ε) (where l i (ε) ∈ D 1 ), while Π ε , with ε > 0, does not contain any real-rooted form. The set of real-rooted forms is a full-dimensional connected semi-algebraic subset of P 5 , and the Zariski closure of its topological boundary is the discriminant hypersurface ∆ = ∆ (2,1,1,1) . Thus the limit h 0 = lim ε→0 − h ε = l 2 1 l 2 l 3 l 4 must belong to ∆. We now analyze, taking into account also Example 2.1, the possible positions of Π 0 with respect to ∆: (1) The point h 0 is smooth and the tangent space T h 0 (∆) = l 1 u i v j : i + j = 4 contains Π 0 . This implies that Π 0 ∈ CH 2 (∆). (2) The point h 0 is smooth in a component of ∆ (3,1,1) ∪∆ (2,2,1) . We have the following subcases: (a) h 0 = l 3 1 l 2 l 3 ∈ ∆ (3,1,1) and T h 0 (∆ (3,1,1) ) = l 2 1 u i v j : i + j = 3 intersects Π 0 in a line L through h 0 . This implies that Π 0 ∈ CH 1 (∆ (3,1,1) ). (b) h 0 = l 2 1 l 2 2 l 3 ∈ ∆ (2,2,1) and T h 0 (∆ (2,2,1) ) = l 1 l 2 u i v j : i + j = 3 intersects Π 0 in a line L through h 0 . This implies that Π 0 ∈ CH 1 (∆ (2,2,1) ). (3) The point h 0 belongs to a component of ∆ (4,1) ∪∆ (3,2) , hence Π 0 ∈ CH 0 (∆ (4,1) )∪ CH 0 (∆ (3,2) ). Case (1). Clearly this case cannot occur. Indeed g 4 and g 5 would have l 1 as common divisor, and this is against our assumptions. Case (2). We show that case (2a) cannot occur. With the same argument, one sees that neither case (2b) occurs. If h 0 ∈ ug 4 , vg 4 , then we can take as degree 5 generator of the apolar ideal g 5 = h 0 . Now, every point of L is a form divisible by l 2 1 and we have that L ∩ ug 4 , vg 4 = ∅. This implies that l 1 is a common divisor of g 4 and of g 5 , which is impossible. It follows that h 0 ∈ ug 4 , vg 4 and in particular l 2 1 divides g 4 . More precisely, this implies that g 4 is of the form l 2 1 l 2 l 3 , or l 3 1 l 2 , or l 3 1 l 3 . In any cases it is obvious that f is limit of generic forms of degree 4. This implies that f is a singular point of the hypersurface ∂ alg (R 7,5 ). Hence f does not vary in a codimension 1 locus of P 7 , and Ψ −1 7,5 (CH 1 (∆ 3,1,1 )) cannot be a component of the boundary ∂ alg (R 7,5 ). Case (3). We show now that both components corresponding to this case are in the boundary. Indeed it is enough to find an example of a binary form which lies exclusively on each component and is limit of a sequence of general forms of rank 5 and a sequence of general forms of rank 6. This is done in Example 4.3 below. Recall that by Proposition 2.2, we have Ψ −1 7,5 (CH 0 (∆ (4,1) )) = (∆ (5,2) ) ∨ and Ψ −1 7,5 (CH 0 (∆ (3,2) )) = (∆ (4,3) ) ∨ . Hence we have proved that ∂ alg (R 7,5 ) \ ∂ alg (R 7,4 ) = (∆ (4,3) ) ∨ ∪ (∆ (5,2) ) ∨ . The boundary ∂ alg (R 7,6 ) between ranks 6 and = 6. At this point we know that ∂ alg (R 7,6 ) \ ∂ alg (R 7,4 ) = ∂ alg (R 7,5 ) \ ∂ alg (R 7,4 ) ∪ ∂ alg (R 7,7 ) = (∆ (4,3) ) ∨ ∪ (∆ (5,2) ) ∨ ∪ (∆ (7) ) ∨ . So, we only need to show that the boundary between R 7,4 and R 7,6 is not of codimension 1 in P 7 . Let f ε be a continuous family of forms such that f −ε ∈ R 7,4 and f ε ∈ R 7,6 for any small ε with ε > 0. The corresponding apolar plane Π ε = Ψ 7,5 (f ε ) does not contain any real-rooted form for any ε > 0. On the other hand, from Corollary 3.4, we deduce that Π ε must contain a special line L ε which is generically contained in the locus of real-rooted forms for any ε < 0. Now the limit L 0 = lim ε→0 − L ε is a special line contained in the intersection of the plane Π 0 and of the discriminant ∆ = ∆ (2,1,1,1) . By the previous analysis we deduce that the line L 0 = ug 4 , vg 4 must be contained in ∆ (4,1) ∪ ∆ (3,2) . This implies that g 4 ∈ ∆ (4) , and this forces f 0 to move in some locus of codimension ≥ 2 in P 7 , which cannot be a component of the boundary. g 4 = (u 2 + v 2 )(u 2 − v 2 ), g 5 (ε) = (u 2 + εv 2 )uv(εu + v), the degree 7 form f ε associated to the apolar ideal (g 4 , g 5 (ε)) is: ε 2 x 7 + 7(ε 2 + ε + 1)x 6 y − 21ε(ε 2 + ε + 1)x 5 y 2 − 35εx 4 y 3 + 35ε 2 x 3 y 4 + 21(ε 2 + ε + 1)x 2 y 5 − 7ε(ε 2 + ε + 1)xy 6 − εy 7 . We have rk(f ε ) = 6 for any small ε ≥ 0 and rk(f −ε ) = 5 for any small ε > 0. Moreover f 0 = x 6 y + 3x 2 y 5 belongs to (∆ (4,3) ) ∨ and it does not belong to (∆ (5,2) ) ∨ ∪ (∆ (3,2,2) ) ∨ . On the other hand, taking g 4 = (u 2 + v 2 )(2u 2 − v 2 ), g 5 (ε) = (εu 2 + v 2 )uv(εu + v), we consider the associated form f ε : ε(ε 3 + ε 2 − ε − 3)x 7 + 14(ε 2 − ε − 1)x 6 y − 42ε(ε 2 − ε − 1)x 5 y 2 − 70(ε 3 − 2)x 4 y 3 + 70ε(ε 3 − 2)x 3 y 4 − 42ε(ε 2 − 2ε + 2)x 2 y 5 + 14ε 2 (ε 2 − 2ε + 2)xy 6 − 2(3ε 3 − 2ε 2 + 2ε − 4)y 7 . Again we have rk(f ε ) = 6 for any small ε ≥ 0 and rk(f −ε ) = 5 for any small ε > 0. Moreover f 0 = 7x 6 y − 70x 4 y 3 − 4y 7 belongs to (∆ (5,2) ) ∨ and it does not belong to (∆ (4,3) ) ∨ ∪ (∆ (3,2,2) ) ∨ . For computational details, see Section 6. Real rank boundaries of degree 8 binary forms In this section, we prove the following: Theorem 5.1. The real rank boundaries for degree 8 binary real forms are the following hypersurfaces: ∂ alg (R 8,5 ) = (∆ (3,3,2) ) ∨ ∪ (∆ (4,2,2) ) ∨ ; ∂ alg (R 8,6 ) = (∆ (3,3,2) ) ∨ ∪ (∆ (4,2,2) ) ∨ ∪ (∆ (4,4) ) ∨ ∪ (∆ (5,3) ) ∨ ∪ (∆ (6,2) ) ∨ ; ∂ alg (R 8,7 ) = (∆ (4,4) ) ∨ ∪ (∆ (5,3) ) ∨ ∪ (∆ (6,2) ) ∨ ∪ (∆ (8) ) ∨ ; ∂ alg (R 8,8 ) = (∆ (8) ) ∨ . Remark 5.2. From Proposition 2.3 and formula (2.2) we obtain: • (∆ (3,3,2) ) ∨ = Join(∆ (7,1) , ∆ (7,1) , ∆ (8) ) is a hypersurface of degree 48; • (∆ (4,2,2) ) ∨ = Join(∆ (6,1,1) , ∆ (8) , ∆ (8) ) is a hypersurface of degree 36; • (∆ (4,4) ) ∨ = Join(∆ (6,1,1) , ∆ (6,1,1) ) is a hypersurface of degree 27; • (∆ (5,3) ) ∨ = Join(∆ (5,1,1,1) , ∆ (7,1) ) is a hypersurface of degree 48; • (∆ (6,2) ) ∨ = Join(∆ (4,1,1,1,1) , ∆ (8) ) is a hypersurface of degree 30; • (∆ (8) ) ∨ = ∆ (2,1,1,1,1,1,1) is a hypersurface of degree 14. Proof. From [CO12] and [CR11], we have ∂ alg (R 8,8 ) = (∆ (8) ) ∨ . On the other hand, from [LS16], we have ∂ alg (R 8,5 ) = (∆ (3,3,2) ) ∨ ∪ (∆ (4,2,2) ) ∨ . We study now the boundary between ranks 6 and ≥ 6. Let f ε be a continuous family of forms crossing the boundary ∂ alg (R 8,6 ) at the point f 0 = f , going from R 8,6 to R 8,7 ∪R 8,8 . Namely, we assume that f −ε ∈ R 8,6 and f ε ∈ R 8,7 ∪R 8,8 for any small ε with ε > 0. We can also assume that for any ε = 0 the form f ε is generated in generic degree, i.e. the apolar ideal f ⊥ ε is generated by two quintic forms g ε and g ′ ε . Moreover, since f moves in a codimension 1 locus, we may assume that f ⊥ is generated by two forms g 0 and g ′ 0 either with deg(g 0 ) = deg(g ′ 0 ) = 5, or with deg(g 0 ) = 4, deg(g ′ 0 ) = 6, and moreover g 0 ∈ ∆ (2,1,1) . Note that in the former case we have P((f ⊥ ) 6 ) = ug 0 , vg 0 , ug ′ 0 , vg ′ 0 , while in the latter case we have P((f ⊥ ) 6 ) = u 2 g 0 , uvg 0 , v 2 g 0 , g ′ 0 . Consider the apolar map (5.1) Ψ 8,6 : P 8 Z 8,6 ⊂ G(3, 6) ⊂ P 34 , which is a cubic birational map onto a subvariety Z 8,6 ⊂ P 34 of degree 686 and cut out by 186 quadric hypersurfaces. The map Ψ 8,6 sends the family f ε into the continuous family of the 3-dimensional linear spaces Π ε = P((f ⊥ ε ) 6 ). From Lemma 3.3, we obtain that Π ε , with ε < 0, contains a real-rooted form h ε = 6 i=1 l i (ε) (where l i ∈ D 1 ), while Π ε , with ε > 0, does not contain any real-rooted form. Thus the limit h 0 = lim ε→0 − h ε must belong to the discriminant hypersurface ∆ = ∆ (2,1,1,1,1) . We now analyze, recalling Example 2.1, the possible positions of Π 0 with respect to ∆: (1) The point h 0 is smooth and the tangent space T h 0 (∆) = l 1 u i v j : i + j = 5 contains Π 0 . This implies that Π 0 ∈ CH 3 (∆). (2) The point h 0 is smooth in a component of ∆ (3,1,1,1) ∪ ∆ (2,2,1,1) . We have the following subcases: (a) h 0 = l 3 1 l 2 l 3 l 4 ∈ ∆ (3,1,1,1) and T h 0 (∆ (3,1,1,1) ) = l 2 1 u i v j : i + j = 4 intersects Π 0 in a plane P through h 0 . This implies that Π 0 ∈ CH 2 (∆ (3,1,1,1) ). (b) h 0 = l 2 1 l 2 2 l 3 l 4 ∈ ∆ (2,2,1,1) and T h 0 (∆ (2,2,1,1) ) = l 1 l 2 u i v j : i + j = 4 intersects Π 0 in a plane P through h 0 . This implies that Π 0 ∈ CH 2 (∆ (2,2,1,1) ). (3) The point h 0 is smooth in a component of ∆ (3,2,1) ∪ ∆ (4,1,1) ∪ ∆ (2,2,2) . We have the following subcases: (a) h 0 = l 3 1 l 2 2 l 3 ∈ ∆ (3,2,1) and T h 0 (∆ (3,2,1) ) = l 2 1 l 2 u i v j : i + j = 3 intersects Π 0 in a line L through h 0 . This implies that Π 0 ∈ CH 1 (∆ (3,2,1) ). (b) h 0 = l 4 1 l 2 l 3 ∈ ∆ (4,1,1) and T h 0 (∆ (4,1,1) ) = l 3 1 u i v j : i + j = 3 intersects Π 0 in a line L through h 0 . This implies that Π 0 ∈ CH 1 (∆ (4,1,1) ). (c) h 0 = l 2 1 l 2 2 l 2 3 ∈ ∆ (2,2,2) and T h 0 (∆ (2,2,2) ) = l 1 l 2 l 3 u i v j : i + j = 3 intersects Π 0 in a line L through h 0 . This implies that Π 0 ∈ CH 1 (∆ (2,2,2) ). (4) The point h 0 belongs to a component of ∆ (3,3) ∪ ∆ (4,2) ∪ ∆ (5,1) , hence Π 0 ∈ CH 0 (∆ (3,3) ) ∪ CH 0 (∆ (4,2) ) ∪ CH 0 (∆ (5,1) ). In the following, we show that only the last case occurs. Case (1). This case cannot occur. Indeed from the fact that Π 0 ⊂ T h 0 (∆), we would conclude that l 1 is a common divisor of g 0 and g ′ 0 . Case (2). Consider first the case when f is not generated in generic degree, so that we have Π 0 = u 2 g 0 , uvg 0 , v 2 g 0 , g ′ 0 . If h 0 ∈ u 2 g 0 , uvg 0 , v 2 g 0 , then we can take g ′ 0 = h 0 and, since there are at least two points in P ∩ u 2 g 0 , uvg 0 , v 2 g 0 , we deduce that g 0 and g ′ 0 have a common divisor, which is a contradiction. Assume therefore that h 0 ∈ u 2 g 0 , uvg 0 , v 2 g 0 . Since g 0 ∈ ∆ (2,1,1) , the only possibility is that g 0 = l 1 l 2 l 3 l 4 . This implies that rk(f ) = 4, and it is easy to see that f is limit of a general sequence of form of rank 5. This would implies that f is not a general point of the boundary between forms of rank 6 and rank ≥ 7. Hence we can assume deg(g 0 ) = deg(g ′ 0 ) = 5, and consider the following subcases. Case (2a). The plane P meets the special lines ug 0 , vg 0 and ug ′ 0 , vg ′ 0 at points p 0 and p ′ 0 respectively. Therefore, p 0 and p ′ 0 are forms divisible by l 2 1 , and then l 1 divides both g 0 and g ′ 0 , which is a contradiction. Case (2b). Let us consider the surface Q = m∈D 1 , g∈ g 0 ,g ′ 0 mg ⊂ Π 0 ≃ P 3 swept out by all the special apolar lines of f . Using that f is generated in generic degrees, one sees that Q is a smooth quadric surface, which we will call the apolar quadric of f . The intersection P ∩ Q is a (possible reducible) plane conic, which in particular contains three noncollinear points: p 0 = m g, p ′ 0 = m ′ g ′ and p ′′ 0 = m ′′ g ′′ . We can assume g = g 0 , g ′ = g ′ 0 , and since every point of P is a form divisible by l 1 l 2 , we conclude that g 0 and g ′ 0 have a common factor, which is a contradiction. Case (3). As above we consider first the case when deg(g 0 ) = 4, deg(g ′ 0 ) = 6 and Π 0 = u 2 g 0 , uvg 0 , v 2 g 0 , g ′ 0 . If h 0 ∈ u 2 g 0 , uvg 0 , v 2 g 0 , then we can take g ′ 0 = h 0 . Moreover, since L ∩ u 2 g 0 , uvg 0 , v 2 g 0 = ∅, we deduce that g 0 and g ′ 0 have a common divisor, which is a contradiction. Thus we have that h 0 ∈ u 2 g 0 , uvg 0 , v 2 g 0 . This implies that g 0 ∈ ∆ (2,1,1) , which contradicts our assumption. Hence we can assume deg(g 0 ) = deg(g ′ 0 ) = 5, and consider the following subcases. Case (3a). Let Q be again the apolar quadric of f . We have two cases: either the line L meets Q in two distinct points mg and m ′ g ′ , or there exists a point mg ∈ L ∩ Q such that L is contained in the tangent plane T mg Q. In the former case, since l 2 1 l 2 divides mg and m ′ g ′ , we deduce that l 1 divides g and g ′ . This is a contradiction, unless we have g = g ′ and hence L is the special line gu, gv . Now, since h 0 ∈ L, we obtain that g ∈ ∆ (2,2,1) ∪ ∆ (3,1,1) , and thus f ∈ (∆ (3,3,2) ) ∨ ∪ (∆ (4,2,2) ) ∨ = ∂ alg (R 8,5 ) is also limit of generic forms of rank 5. This implies that f belongs to the singular locus of the hypersurface ∂ alg (R 7,6 ) and then Ψ −1 8,6 (CH 1 (∆ (3,2,1) )) cannot be a component of the boundary ∂ alg (R 8,6 ). In the latter case, we may assume g = g 0 and T mg Q = mg 0 , mg ′ 0 , m ′ g 0 , for some m ′ ∈ D 1 . We have h 0 = l 3 1 l 2 2 l 3 = αmg 0 + βmg ′ 0 + γm ′ g 0 , for some scalars α, β, γ, and we know that l 2 1 l 2 divides mg 0 . Since gcd(g 0 , g ′ 0 ) = 1, this implies β = 0, and then l 3 1 l 2 2 l 3 = (αm + γm ′ )g 0 . As above, from this it follows that g 0 ∈ ∆ (2,2,1) ∪ ∆ (3,1,1) and thus f does not vary in a codimension 1 locus of P 8 , Cases (3b) and (3c). Arguing as above, we deduce that f must belong to (∆ (4,2,2) ) ∨ and (∆ (3,3,2) ) ∨ , respectively, and furthermore we must have that f is a singular point of the hypersurface ∂ alg (R 8,6 ). This implies that Ψ −1 8,6 (CH 1 (∆ (4,1,1) )) and Ψ −1 8,6 (CH 1 (∆ (2,2,2) )) are not components of the boundary ∂ alg (R 8,6 ). Case (4). We show in Example 5.3 below that each of the three components corresponding to this case are in the boundary. This proves that ∂ alg (R 8,6 ) \ ∂ alg (R 8,5 ) = (∆ (4,4) ) ∨ ∪ (∆ (5,3) ) ∨ ∪ (∆ (6,2) ) ∨ . Finally we need to prove that there are no components of the boundary between R 5 and R 7 . This can be done with the same argument used at the end of the proof of Theorem 4.1. Example 5.3. Given g 0 = u 4 v − u 2 v 3 − 2v 5 , g ′ 0 = −u 5 + 2u 3 v 2 + 2uv 4 , we have ug 0 + vg ′ 0 = u 3 v 3 and the degree 8 form f 0 associated to the apolar ideal (g 0 , g ′ 0 ) is (5.2) f 0 = 8x 8 + 112x 6 y 2 + 56x 2 y 6 − y 8 . With the help of a computer, one can easily check (see Section 6) that rk(f 0 ) = 7 and f 0 ∈ (∆ (4,4) ) ∨ \ (∆ (5,3) ) ∨ ∪ (∆ (6,2) ) ∨ . Moreover, we can construct near f 0 generic degree 8 forms f ±ε having real ranks 6 and 7. Analogously, given g 0 = u 4 v − u 2 v 3 − 2v 5 , g ′ 0 = −u 5 + u 4 v + u 3 v 2 + 2uv 4 , we have ug 0 + vg ′ 0 = u 4 v 2 and the associated degree 8 form is (5.3) f 0 = x 8 + 8x 7 y + 28x 3 y 5 − 2xy 7 . One verifies that rk(f 0 ) = 7 and f 0 ∈ (∆ (5,3) ) ∨ \ (∆ (4,4) ) ∨ ∪ (∆ (6,2) ) ∨ . Finally, given g 0 = u 5 + u 4 v + 3u 3 v 2 + 3u 2 v 3 + 2uv 4 + 2v 5 , g ′ 0 = −3u 3 v 2 − 2uv 4 , we have ug 0 + (u + v)g ′ 0 = u 5 (u + v) and the associated degree 8 form is (5.4) f 0 = 8x 8 − 64x 7 y + 224x 6 y 2 − 448x 5 y 3 − 840x 4 y 4 + 672x 3 y 5 + 504x 2 y 6 − 144xy 7 − 17y 8 . One verifies that rk(f 0 ) = 7 and f 0 ∈ (∆ (6,2) ) ∨ \ (∆ (4,4) ) ∨ ∪ (∆ (5,3) ) ∨ . Computations We provide a package for Macaulay2 [GS18], named CoincidentRootLoci and available as an ancillary file to our arXiv submission, which implements methods useful to check the correctness of Examples 4.3 and 5.3. This package depends on the packages Cremona and Resultants (see [Sta17a] and [Sta17b]). In the following, we illustrate briefly some of the methods available. For technical details and examples, we refer to the documentation of the package, which can be shown using the commands installPackage and viewHelp. The method realrank computes the real rank of a binary form with rational coefficients. Indeed, Lemma 3.3 reduces the problem of computing the real rank of a binary form to that of establishing whether certain semi-algebraic sets are nonempty. The Tarski formulas defining these semi-algebraic sets can be obtained via the computation of kernels of appropriate catalecticant matrices. The problem of deciding the truth of a Tarski formula can be handled by Qepcad B via a quantifier elimination by partial cylindrical algebraic decomposition (see [Bro03]). The method calls automatically Qepcad B without requiring user intervention (provided it is installed on the system). Below, we compute the real rank of the binary form (5.2) (the run time is about 30 seconds). Macaulay2, version 1.11 with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone i1 : needsPackage "CoincidentRootLoci"; i2 : R := QQ[x,y]; i3 : F = 8*x^8+112*x^6*y^2+56*x^2*y^6-y^8; i4 : realrank F o4 = 7 The method member tests membership of a binary form in the dual variety of a coincident root locus (or in a coincident root locus). It does not pass through the hard computation of the equations but uses Proposition 2.2. Below, we verify that the binary form (5.2) lies in (∆ (4,4) ) ∨ but not in (∆ (5,3) ) ∨ ∪ (∆ (6,2) ) ∨ (the run time is less than one second). i5 : X = dual coincidentRootLocus(4,4) o5 = CRL(6,1,1) * CRL(6,1,1) (dual of CRL(4,4)) o5 : JoinOfCoincidentRootLoci i6 : member(F,X) o6 = true i7 : member(F,dual coincidentRootLocus(5,3)) or member(F,dual coincidentRootLocus(6,2)) o7 = false The method apolar computes the apolar ideal of a binary form, while recover, as the name suggests, recovers the binary form from its apolar ideal. Basically, these two methods translate to problems of computing the image or the inverse image of a point via a (bi)rational map, and then the computation is performed using tools of the package Cremona. For example, the following calculation involves the birational map (5.1) (the run time is less than one second). i8 : F == recover apolar F o8 = true For the convenience of the user, the method realRankBoundary implements Theorems 4.1 and 5.1. For example, below we get immediately the degree of the first component of ∂ alg (R 8,5 ). i9 : Y = first realRankBoundary(8,5) o9 = CRL(7,1) * CRL(7,1) * CRL(8) (dual of CRL(3,3,2)) o9 : JoinOfCoincidentRootLoci i10 : degree Y o10 = 48 Proposition 2.2 ([LS16]). Given λ = (λ 1 , . . . , λ n ) and ∆ λ ⊂ P(C[u, v] d ), the points of the dual variety ∆ ∨ λ ⊂ P(C[x, y] d ) are given by the binary forms f (x, y) that are annihilated by some order d−n operator of the form . E Angelini, C Bocci, L Chiantini, Real identifiability vs. complex identifiability. 666Linear Multilinear AlgebraE. Angelini, C. Bocci, and L. Chiantini, Real identifiability vs. complex identifiability, Linear Multilinear Algebra 66 (2018), no. 6, 1257-1267. A comparison of different notions of ranks of symmetric tensors. A Bernardi, J Brachat, B Mourrain, Linear Algebra Appl. 460A. Bernardi, J. Brachat, and B. 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Katz, How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties, Expo. Math. 21 (2003), no. 3, 219-261. Coisotropic hypersurfaces in the Grassmannian. K Kohn, K. Kohn, Coisotropic hypersurfaces in the Grassmannian, available at https://arxiv.org/abs/1607.05932, 2016. Some remarks on equations defining coincident root loci. S Kurmann, J. Algebra. 3521S. Kurmann, Some remarks on equations defining coincident root loci, J. Algebra 352 (2012), no. 1, 223-231. J M Landsberg, Tensors: geometry and applications. American Mathematical Soc128J. M. Landsberg, Tensors: geometry and applications, vol. 128, American Mathematical Soc., 2012. Duality of multiple root loci. H Lee, B Sturmfels, J. Algebra. 446H. Lee and B. Sturmfels, Duality of multiple root loci, J. Algebra 446 (2016), 499-526. Effective identifiability criteria for tensors and polynomials. A Massarenti, M Mella, G Staglianò, J. Symbolic Comput. 87A. Massarenti, M. Mella, and G. Staglianò, Effective identifiability criteria for tensors and polynomials, J. Symbolic Comput. 87 (2018), 227-237. Real rank geometry of ternary forms. M Micha Lek, H Moon, B Sturmfels, E Ventura, Ann. Mat. Pura Appl. 1963M. Micha lek, H. Moon, B. Sturmfels, and E. Ventura, Real rank geometry of ternary forms, Ann. Mat. Pura Appl. 196 (2017), no. 3, 1025-1054. Hyperdeterminants of polynomials. L Oeding, Adv. Math. 2313L. Oeding, Hyperdeterminants of polynomials, Adv. Math. 231 (2012), no. 3, 1308-1326. A Macaulay2 package for computations with rational maps. G Staglianò, G. Staglianò, A Macaulay2 package for computations with rational maps, available at https://arxiv.org/abs/1701.05329, 2017. A package for computations with classical resultants. , A package for computations with classical resultants, available at https://arxiv.org/abs/1705.01430, 2017. The Hurwitz form of a projective variety. B Sturmfels, J. Symbolic Comput. 79B. 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{'abstract': 'We describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight, showing that they are dual varieties of suitable coincident root loci.', 'arxivid': '1804.08309', 'author': ['Maria Chiara ', 'Brambilla And ', 'Giovanni Staglianò '], 'authoraffiliation': [], 'corpusid': 119317391, 'doi': '10.1016/j.laa.2018.07.036', 'github_urls': [], 'n_tokens_mistral': 17121, 'n_tokens_neox': 14919, 'n_words': 8462, 'pdfsha': '4cfbebffe7ae6d9dac9906333e26353e76fe9d1a', 'pdfurls': ['https://arxiv.org/pdf/1804.08309v2.pdf'], 'title': ['ON THE ALGEBRAIC BOUNDARIES AMONG TYPICAL RANKS FOR REAL BINARY FORMS', 'ON THE ALGEBRAIC BOUNDARIES AMONG TYPICAL RANKS FOR REAL BINARY FORMS'], 'venue': []}

arxiv

Fuzzy k-Nearest Neighbors with monotonicity constraints: Moving towards the robustness of monotonic noise 5 Mar 2020 Sergio González Department of Computer Science and Artificial Intelligence University of Granada 18071GranadaSpain Salvador García Department of Computer Science and Artificial Intelligence University of Granada 18071GranadaSpain Sheng-Tun Li stli@mail.ncku.edu.twsheng-tunli Department of Industrial and Information Management Institute of Information Management National Cheng Kung University 701TainanTaiwan Center for Innovative FinTech Business Models National Cheng Kung University 701TainanTaiwan Robert John robert.john@nottingham.ac.ukrobertjohn School of Computer Science ASAP Research Group University of Nottingham NG8 1BBNottinghamUK Francisco Herrera herrera@decsai.ugr.es Department of Computer Science and Artificial Intelligence University of Granada 18071GranadaSpain Faculty of Computing and Information Technology King Abdulaziz University JeddahSaudi Arabia Sergio González Francisco Herrera Fuzzy k-Nearest Neighbors with monotonicity constraints: Moving towards the robustness of monotonic noise 5 Mar 2020* Corresponding author:Fuzzy k-NNmonotonic constraintsordinal classificationordinal regressionclass noise This paper proposes a new model based on Fuzzy k-Nearest Neighbors for classification with monotonic constraints, Monotonic Fuzzy k-NN (MonFkNN). Real-life data-sets often do not comply with monotonic constraints due to class noise. MonFkNN incorporates a new calculation of fuzzy memberships, which increases robustness against monotonic noise without the need for relabeling. Our proposal has been designed to be adaptable to the different needs of the problem being tackled. In several experimental studies, we show significant improvements in accuracy while matching the best degree of monotonicity obtained by comparable methods. We also show that MonFkNN empirically achieves improved performance compared with Monotonic k-NN in the presence of large amounts of class noise. Introduction Monotonic constraints are prior-knowledge of some ordinal classification or regression problems about the order relationships between attributes and class labels [9]. Consider the example of house pricing. The following constraints are applied: A bigger house in the same neighborhood is constrained by higher prices as compared to smaller houses with the same features. That is, the classifier decisions should not decrease in the presence of better features while the rest remains the same. These prior constraints are required by many real-life evaluation problems, such as credit risk modeling [12] and lecturer evaluation [7]. These problems are known as Classification with Monotonic Constraints or Monotonic Classification [3]. These learning tasks have additional objectives besides accurate models, such as the monotonic consistency of predictions and minimization of the misclassification costs. The latter is also relevant since the errors between ordered classes do not hold the same importance. More metrics must be used during the learning and validation of the models. However, these other objectives may impair accuracy [4]. Hence, a fair balance must be sought between the different needs of each problem. Standard classifiers are discouraged for monotonic classification since they do not contemplate these constraints and their predictions violate the monotonicity required by certain applications. A classic example of these nonmonotonic models is the standard decision tree [3]. Standard k-Nearest Neighbors algorithm also does not take these restrictions into account, which may lead to further harm as a result of their presence in preprocessing techniques [22]. In recent years, new algorithms have been designed to minimize the number of monotonic violations in their predictions [3,23,9], i.e. the number of pairs of instances that break monotonicity [3]. To do so, some approaches focus their entire learning mechanism just on monotonicity. This strategy usually achieves completely monotonic models, but it could lead to wrong generalizations being made that are different to the knowledge of the problem. Others infer monotonic relations from the training set while maximizing their accuracy. These models have been adapted from different families of classifiers [9], such as decision trees [3,32,34], support vector machines [12], fuzzy model based classifiers [1,29], neural networks [17,40] and ensemble learning [13,36,23]. Instance-based learning has proven to be a good approach for monotonic classification [2,15,31,18]. However, some of these methods, such as Monotonic k-Nearest Neighbors [15] (MkNN), need to learn from a fully monotonic set to ensure monotonic predictions. This is rarely the case in real-life scenarios, where class noise and discrepancies are common. Therefore, data preprocessing [21,33,8,22] and relabeling strategies [35,16] must be used to remove non-monotonic samples or to change their class labels in order to force a monotonic set. In standard classification, Fuzzy k-Nearest Neighbors [25] is a very solid method with high performance, thanks to its high robustness to class noise [14]. This class noise robustness mainly lies in the extraction of the class memberships for the crisp training samples by nearest neighbor rule. In this process, the class memberships of noisy instances are shared with surrounding classes and the incorrectly assigned class looses its influence. However, these mechanisms do not consider monotonic constraints and Fuzzy k-NN cannot deal with monotonic violations or monotonic noise in the training set. In this paper, a new model designed on the basis of Fuzzy k-NN with notions of MkNN is proposed to take monotonic constraints into account, and is called Monotonic Fuzzy k-Nearest Neighbors (MonFkNN). MonFkNN has been designed with three desired features: (i) Robustness against monotonic violations. (ii) Monotonic predictions without a pure monotonic training set. (iii) Flexibility in its configurations covering different needs of performance. With these objectives in mind, MonFkNN has been designed with new mechanisms to manage monotonicity constraints and the monotonic violations in the training set. The main contributions of the MonFkNN design are: (i) The initial robustness of Fuzzy k-NN has been redesigned to mitigate the influence of monotonic violations. Firstly, the violations due to sample replicas with different classes are joined to form one class membership. Then, our approach incorporates a strictly monotonic nearest neighbor rule to the calculation of the memberships of the training examples. (ii) These monotonically constrained memberships and their medians are used in the prediction phase. The class memberships aggregation of MonFkNN is also monotonically constrained by the nearest neighbor extraction or a penalty to the contribution of non-monotonic instances. (iii) MonFkNN was built as a flexible classifier that covers different necessities of monotonicity and accuracy by tuning its parameters. It can be configured with a rigidly monotonic or standard k-NN rule if monotonicity or precision is preferred in the predictions, respectively. All these mechanisms reinforce the robustness of our proposal against monotonic noise without the need for relabeling. We understand monotonic noise as being the actual noise that can alter the class labels and, as a result, change the monotonic constraints among the samples in the data. Their parameters make our proposal adaptable to the different objectives of monotonic classification. We distinguish two different parameter configurations: a pure monotonic version in which monotonicity is prioritized, and an approximate configuration that focuses more on the prediction accuracy. We have performed several empirical studies to verify the desired features of MonFkNN. First, different behaviors of its two configurations are empirically analyzed and compared to the original FkNN. Then, our proposal is compared with 7 methods from the state-of-the-art, exhibiting substantial improvements in accuracy and maintaining the best degree of monotonicity. Finally, the robustness of our method against monotonic noise, i.e. monotonic violations, is shown in contrast to MkNN. In this last experiment, MonFkNN performs considerably better than Monotonic k-NN in scenarios with large amounts of class noise. The experimental framework used consists of 12 data-sets commonly used in monotonic classification, 7 monotonic classifiers and 3 metrics covering different aspects of performance: Accuracy, Mean Absolute Error and Non-Monotonic Index. All results are additionally validated with the non-parametric statistical Wilcoxon and Friedman rank [20,19] and Bayesian Sign tests [5]. The paper is organized as follows. In Section 2, we present the problem of classification with monotonic constraints and the methods related to our proposal: MkNN and Fuzzy k-NN. Section 3 is dedicated to explaining our model MonFkNN in detail and its algorithmic differences as compared to FkNN. The experimental framework used in the different empirical studies is presented in Section 4. In Section 5, the previously mentioned empirical studies are carried out and analyzed. Finally, the main conclusions of this study are stated in Section 6. Preliminaries In this section, we introduce the preliminaries needed: Classification with monotonicity constraints, Monotonic k-Nearest Neighbors and the original Fuzzy k-Nearest Neighbors. Monotonic Classification Monotonic classification [9] is an ordinal regression problem with monotonic constraints relating to the order of the variables and the class labels. Ordinal regression and/or classification can be seen as a nonstandard classification problem [11], which attempts to minimize the difference between the predicted labels and the real labels. Classification with monotonic constraints is also considered to be a nonstandard supervised learning problem [11]. Formally, monotonic classification aims to predict the class label y from input vector x with Q number of features, where y ∈ Y = {l 1 , l 2 , . . . , l c } and x represents an individual of our classification problem. The categories Y are arranged in an order relation ≺ as l 1 ≺ l 2 ≺ . . . ≺ l c . And, as the main property of monotonic classification, the attributes and class predictions are monotonically constrained by the problem prior-knowledge, i.e. x x → f (x) ≥ f (x ) [26], where x x implies ∀ j=1,...,Q , x j ≥ x j , that is, x dominates x . Therefore, the main objective is to build classifiers that do not violate these constraints, otherwise known as monotonic classifiers. Two different types of monotonic classifiers can be distinguished: approximate monotonic models, which minimizes the number of monotonic violations in their decisions and pure monotonic classifiers, whose predictions are always monotonic concerning the training and future examples. The latter is hard to achieve, particularly in real-life applications where the training datasets are rarely purely monotonic. To be considered monotonic, all of the pairs of instances in a data-set must be monotonic [2]: x i x j → y i ≥ y j , ∀ i,j . Monotonic k-Nearest Neighbors MkNN [15] modifies the standard nearest neighbor rule of the well-known lazy learning method to avoid monotonic violations in its predictions. To do so, MkNN computes for each new example x i the range r i = [y min , y max ] of valid class labels, which satisfies the monotonic constraints. The lowerbound y min of r i is computed as the highest class label of all instances in the training set D below the example x i . Analogously the upper-bound y max is the minimum class label of the instances in D that are higher than x i (see Eq. 1). r i = y min = max{y | (x, y) ∈ D ∧ x i x} y max = min{y | (x, y) ∈ D ∧ x x i }(1) Two different MkNN variants can be distinguished depending on how the neighbors are extracted for a new instance x i . The InRange variant considers the k nearest examples x j with their class labels y j in the range [y min , y max ]. The OutRange version extracts first the k nearest neighbors x j and then, those neighbors outside of the range r i are filtered out from the decision. If all of them are removed, a random label in r i is chosen. As in the standard k-NN method, the majority class among the k neighbors is used as the predicted label. MkNN is one of the methods that require monotonic data-sets to work properly [15]. Since, with monotonicity violations, the range r i could not be correctly computed, a relabeling technique should be used to transform the non-monotonic training data into monotonic data. These techniques intend to identify and remove the monotonicity violations by making the fewest possible changes with minimum class difference. [15,35,16]. Fuzzy k-Nearest Neighbors Fuzzy Sets [39] express the uncertainty of the example memberships to each class label. The memberships of the example x i are represented as a degree of each class belonging u i = (u i1 , u i2 , . . . , u ic ), where u il ∈ [0, 1] and c l=1 u il = 1. Nowadays, development in fuzzy sets and classifiers is still an ongoing process [37]. Fuzzy k -Nearest Neighbors algorithms [14] incorporate fuzzy concepts into the classical k-NN decision rule to learn from fuzzy sets and produce fuzzy classification rules. Recently, different approaches have been proposed based on distinct fuzzy set extensions. However, the original Fuzzy k-NN [25] (FkNN) is still one of the best approaches [14]. Recent approaches provide for the optimization of parameters in FkNN [6]. For a given new instance x i , Fuzzy k-NN [25] extracts its K nearest neighbors in the same manner as the standard k-NN. Then, its memberships for each class l are computed with the following expression: u(x, l) = K j=1 u(x j , l) * 1 ||x − x j || (m−1) K j=1 1 ||x − x j || (m−1)(2) As shown in Eq. 2, the membership u(x i , l) = u il of sample x i to class l is assigned with the product of the class membership u(x j , l) of the neighbors x j and the inverse of their distances to x i . The latter serves as a weight that biases towards the memberships of nearer samples. The parameter m determines the degree of influence of the neighbor distances. The recommended value m = 2 [25] makes the contributions of the neighboring samples reciprocal to their distances. A crisp class label for the example x i can be decided as being the label l with the greatest membership degree u il . Facing a labeled training set, Fuzzy k-NN [25] brings it into a fuzzy set with sample memberships using the nearest neighbor rule. For each training sample x i , k nearest neighbors are extracted using the leave-one-out scheme. Then memberships u(x i , l) for each class l are computed according to Eq. 3 with the number of neighbors nn l found for each class l. This transformation has proven useful against noisy samples as the memberships lose influence as they are spread to the surrounding classes (not the assigned class). u(x i , l) = 0.51 + 0.49 * (nn l /k) , if y i = l 0.49 * (nn l /k) , otherwise Monotonic Fuzzy k -Nearest Neighbors In this section, we explain our approach in detail -MonFkNN and all its mechanisms that consider monotonicity constraints. In Subsection 3.1, we explain how MonFkNN gives a final class from class memberships in a more proper manner according to monotonicity. Subsection 3.2 is dedicated to the extraction of the class memberships from the training set and redesigned to reduce the impact of monotonic noise without the need for monotonic relabeling. In Subsection 3.3, the class membership aggregation built-in MonFkNN is explained and related to the robustness and flexibility of the classifier using its parameters. Finally, we discuss the algorithmic differences between our proposal and the original FkNN in Subsection 3.4. From class memberships to the final class label Since FkNN works with class memberships, a mechanism that respects monotonicity is needed to get a final class from a vector whose elements sum up to the value of one. The class with the greatest membership is the most common decision in multiple classifiers. The original Fuzzy k-NN gives their crisp predictions as the class label with the highest membership. However, this might not be appropriate for scenarios with monotonic constraints. For example, let x i ≤ x j and their class memberships u i = (0.2, 0.2, 0.4, 0.2, 0.0) and u j = (0.0, 0.4, 0.3, 0.2, 0.1), then their final classes chosen with the highest membership break the monotonicity: argmax(u i ) = l 3 > l 2 = argmax(u j ). Even though, the instance x j has more weight values assigned to the higher labels than instance x i . In fact, u j weakly dominates u i according to the first degree stochastic dominance relation (FSD) [28] since the x i cumulative distribution function U i = (0.2, 0.4, 0.8, 1.0, 1.0) is greater, element by element, than U j = (0.0, 0.4, 0.7, 0.9, 1.0), that is, u i F SD u j ⇐⇒ (∀l ∈ Y)(U i (l) ≥ U j (l)). To make FSD applicable, class membership vectors are normalized to sum up to the value of one and treated as probability mass functions. Therefore, a cumulative distribution function U can be computed for given normalized class memberships, where FSD is defined. This transformation can be done thanks to the order relation between classes in monotonic classification. FSD is useful for defining monotonicity constraints in probabilistic classifications [31,30], with the expression x i ≤ x j =⇒ u i F SD u j . Therefore, the function that transfers a membership vector to a class label must satisfy u i F SD u j =⇒ y i ≤ y j . Centrality measures, such as mean and median, have proven to be good solutions [28,31]. Particularly, the median is applicable to ordinal problems. Following the traditional definition of median as the 50th percentile, the median is computed as the range [l m , l M ]: l m = min{l ∈ Y | U {X ≤ l} ≥ 1/2} l M = max{l ∈ Y | U {X ≥ l} ≥ 1/2}(4) where l is a class label of possible labels Y, U {X ≤ l} is the cumulative membership/probability of belonging to a class smaller or equal to l and U {X ≥ l} is the analogous definition for a class greater or equal to l. Going back to the previous example, the classes for x i and x j chosen by the median does not break monotonicity: med(u i ) = med(u j ) = 3. For l m = l M , any class label l which l m < l < l M must have a membership u(l) = 0 and U (l m ) = U (l M ) = 1/2. For example, instance x t with class memberships u t = (0.2, 0.3, 0, 0.3, 0.2) could be assigned to the classes med(u t ) = [2, 4] = 3. Class memberships robust to monotonic noise In this subsection, the class membership calculation redesigned to monotonic classification is explained. The objective of this first stage is to fix or reduce the influence of non-monotonic examples in the classification. Our method uses the robustness of the traditional Fuzzy k-NN within the knowledge of the monotonic relations between the neighbors. Algorithm 1 summarizes the procedure of obtaining robust noise class memberships for the training set. First, we have to deal with the simplest monotonic violations, that is, instances with the same input values and different classes (Lines 2-13 of Algorithm 1). These mislabels frequently appear in traditional data-sets [2] of classification with monotonic constraints as these sets are rankings or evaluations made by different experts. Therefore, MonFkNN first substitutes the replicas of any example x with one feature vector x and its memberships u(x). The membership u(x, l) of the instance x to the class l is computed with the frequency of duplicated examples x j in the training set D belonging to class l (y j = l), as shown in the following expression: u(x, l) = |{x j ∈ D|x j = x ∧ y j = l}| |{x j ∈ D|x j = x}|(5) The class label of an instance x after the elimination of its replicas is obtained by the median of the resulting memberships, as shown in Line 13 of Algorithm 1 Training class memberships extraction 1: function TrainClassMemberships({D, y} -Training data-set, k -Nearest neighbors considered, RCr -Real Class relevance) 2: for x i ∈ D do 3: for l ∈ Y do 4: if x i duplicated-in D then 5: Compute u(x i , l) with expression 5 6: for x i ∈ D do 13: y i = med(u i ) See expression 4 14: end for 15: for x i ∈ D do 16: if x i not-duplicated-in D then 17: Compute range r i with (D , y ) and expression 1 18: See Algorithm 2 19: nn = neighborsAsMkNN(x i , r i , k, inRange, D , y ) 20: for l ∈ Y do 21: Compute u(x i , l) with expression 6 22: end for 23: end if 24: end for 25: output: (D , u) 26: end function Algorithm 2 Monotonic nearest neighbor rule 1: function NeighborsAsMkNN(x -tested sample, r -range of valid classes, k -considered neighbors, typeRange -inRange or outRange, {D, y} -Training data-set) 2: initialize: nn = {} 3: for x i ∈ D do 4: if typeRange == outRange or y i ∈ r then 5: if Size(nn) < k then 6: Insert x i in nn 7: else 8: x max = arg max x j ∈nn ||x − x j || 9: if ||x − x i || < ||x − x max || then 10: Replace x max by x i in nn 11: end if 12: end if 13: end if 14: end for 15: output: nn 16: end function Algorithm 1. However, this vector will be used in the classification function with the membership aggregation as stated in the next subsection. Then, MonFkNN estimates the memberships of the remaining instances, which corresponds to Lines 13 -24 of Algorithm 1. This estimation is made using the information of the nearest neighbors of each instance. However, these nearest neighbors are extracted with a monotonic nearest neighbor rule (MkNN) instead of a traditional rule as we aim for memberships that respect monotonic constraints as much as possible. Algorithm 2 exemplifies the extraction of these monotonically constrained neighbors for a given instance x as in MkNN. In this case, Algorithm 2 is configured as an inRange variant as pointed out in Line 16 of Algorithm 1. That is, the nearest neighbors of an example x i are constrained to a range r i = [y min , y max ] of possible classes (Line 17), which preserves the monotonicity of the data-set. Once the nearest neighbors for each example x i are obtained, the information of the neighbor classes is fused into x i class memberships (Line 20). For an instance x i , the membership u(x i , l) to class l is computed with the following expression: u(x i , l) = RCr +(nn l /k) * (1 − RCr) if y i = l (nn l /k) * (1 − RCr)(6) where nn l is the number of nearest neighbors of the class l, k the total number of neighbors extracted for instance x i and y i is the original class label of the example x i . RCr is a new parameter called "Real Class relevance". Apart from the use of the monotonic nearest neighbor rule, the inclusion of RCr is another main difference between our approach MonFkNN and the original Fuzzy k-NN. RCr can be seen as the minimum membership assigned to original class y i of the instance x i , in case there are no neighbors labeled with y i . In FkNN, RCr corresponds to the value of 0.51, that is, every instance maintains its real class, even those noisy examples surrounded by other classes. By being a parameter, our method lets the user control the treatment of monotonic noise. There are some values for RCr in the range [0, 1] that have very interesting and distinct behaviors. In the case of a really noisy data-set where no labels can be trusted, RCr could be set to 0. This leaves all the responsibility to the calculation of the range of valid classes r i and the nearest neighbors. In the presence of instances with the same input values and different classes, the user could choose only to treat them with RCr = 1. Finally, if practitioners want to consider the originally labeled instances, we recommend assigning RCr to 0.5. This value ensures that the actual class is within the set of medians. In contrast to Fuzzy k-NN and its 0.51, if all neighbors belong to a same single class that is different to the current class, our method forces to choose in between these two classes. Usually, this last value (RCr = 0.5) is a good trade-off, mainly stable and with better performance. During this process, the impact of monotonic inconsistencies will be either reduced or fixed. The inconsistencies of instances with the same input vectors and different classes are completely fixed by being substituted by only a sample and class memberships with the information of their different classes. The mislabeled samples, i.e. noisy or non-monotonic examples, will have less influence towards their noisy class as they will be surrounded by more appropriated classes and their class memberships will be shared into classes in which they fit monotonically. This is the first mechanism of our method to alleviate the presence of monotonic violations, without the need for relabeling. Flexible membership aggregation After estimating the class memberships of every training instance, our algorithm is ready to predict new examples. This last phase has been designed to cover different needs of monotonic scenarios. In addition to the control of noise treatment, greater flexibility has been sought, allowing users to choose between more accurate or pure monotonic predictions. Algorithm 3 represents in pseudo-code the whole prediction procedure of our proposal MonFkNN. Particularly, the prediction of a new instance x i is detailed after having previously computed the monotonically-constrained class memberships of the training set as the previous Algorithm 1 is referred in Line 2. for x j ∈ nn do 8: if typeRange == inRange or y j ∈ r x then 9: pOR j = 1 10: else Neighbors nn out of range r x are penalized with pOR 11: pOR j = pOR 12: end if 13: end for 14: Compute class memberships u i of x i with expression 7 15: output: med(u i ) 16: end function As shown in Line 6, MonFkNN embeds another MkNN (Algorithm 2) to obtain the neighbors used in the membership aggregation and final prediction. This MkNN also has two versions, inRange and outRange versions. They are, however, substantially different when compared to original variants. The inRange alternative is based on the same idea of the original MkNN, where the neighbors of an example must belong to a set of monotonically valid classes. However, this range of classes is obtained using the medians acquired from the class memberships of the training instances constrained by monotonicity, as seen in Line 3 and Line 4. This breakthrough improves our method by increasing monotonic noise robustness. Firstly, an inRange nearest neighbor rule removes monotonic inconsistencies in the known dataset as previously shown in Algorithm 1. Then, the second MkNN uses this fixed training set (D , y ) to give monotonic predictions as seen in Algorithm 3. The outRange version of our method is completely different from the previous outRange rule. It has been designed with the intention of prioritizing to some extent the predictive ability of the classifier over monotonicity. With this purpose in mind, our method considers any example as a valid neighbor regardless of its class label. In contrast to the original model, no filtering or removal of neighbors outside the valid range is performed. However, their relevance in the membership aggregation can be reduced if needed, thanks to a penalty factor introduced in the aggregation expression. Then, for a new example x, its nearest neighbors are obtained according to the chosen variant. Their memberships are aggregated with the original FkNN formula with the addition of the penalty factor for the outRange version. The following expression shows how this parameter is integrated: u(x, l) = K j=1 u(x j , l) * pOR j ||x − x j || (m−1) K j=1 pOR j ||x − x j || (m−1)(7) As previously, the membership u(x, l) of the new sample x to the class label l is the result of the sum of the class memberships u(x j , l) of the neighbors x j inversely weighted with their distance to x. In the outRange version of our method, there is another weighting factor in the contribution to the final memberships, the parameter referred to as "penalty of outRange" (pOR). The factor pOR j is applicable only if the class y j of the neighbor x j is not in the valid class range r x of x as exemplified in Lines 7 to 13 . It can be configured with continuous values from 0 to 1. When it is assigned to 1, no penalty is applied. The value 0 means a full penalty, that is, neighbors with invalid classes will not participate in the membership aggregation. For all practical purposes, this last behavior is equivalent to the outRange MkNN. We recommend using 0.5 since it is a good balance between reducing their relevance and considering them in the decision. Finally, the class prediction of the new example x is the median of the resulting normalized class memberships. As presented, MonFkNN has been developed to be robust to monotonic noise and versatile in many scenarios. The two versions inRange and out-Range with the parameter pOR and the previously mentioned RCr help to tune the algorithm according to the necessities of different kinds of problems. Among the possibilities that offer these parameters, we have named two configurations with very distinctive behaviors: Pure Monotonic (MonFkNN-PM or PM) and Approximate Monotonic (MonFkNN-AM or AM) Fuzzy k-NN. The Pure Monotonic configuration corresponds to a value of 0.5 for the RCr parameter and the use of inRange rule to obtain the memberships of new instances. This approach aims to give predictions with the minimum violations of monotonicity. In every part of the algorithm, it prioritizes monotonicity over very accurate predictions. MonFkNN-AM prioritizes the predictive ability and relaxes the monotonic constraints. The memberships of the training set are obtained by the treatment of samples with the same feature values and different classes. Those unique examples will have a membership of 1 to the actual class and 0 for the rest. This behavior is achieved with RCr = 1. Then, as we are looking for more accurate predictions, all instances can be considered to be valid neighbors and to contribute to the final aggregation. Those instances with invalid class labels, however, will contribute with only half of their class memberships (pOR = 0.5). Our proposal MonFkNN is available at the GitHub Repository 1 . Each of the differences mentioned in Table 1 is described and explained below: Differences between standard FkNN and • The data-set used to compute the training class memberships is modified before applying the neighborhood rule. The inconsistencies of duplicates are eliminated and reduced to a single instance. The classes of the resultant instances are assigned to the median calculated with the frequency of the appearance of duplicates for each class. This procedure could not even be considered in standard classification, where there is no ordering relationship between classes. • The neighborhood considered for each training instance is constrained to the monotonicity of the data-set. Then, their resultant class memberships are also monotonically constrained. These adaptations completely modify the neighbors contributing in Eq. 3 and the whole procedure. In addition, the value of 0.51 for RCr is discouraged in MonFkNN in favor of 0.5 due to its contribution to the medians of the samples, above-mentioned in Section 3.2. • The original FkNN and MonFkNN also share the same membership aggregation, i.e. their expressions (Eq. 3 and Eq. 6) are the same for InRange and outRange (with pOR = 1) versions of MonFkNN. However, their behavior and their predictions are completely different, due to the differences in the nearest neighbor rule, in the training set and class memberships used in the aggregation procedure. As previously explained, the training class memberships extraction of MonFkNN modifies the training set fixing some monotonic inconsistencies. Duplicates are removed and some training samples might change their classes to preserve the monotonicity of the data-set. • In MonFkNN, the classes of the training samples determine the monotonically valid classes of the unlabeled instances. Thus, training samples with classes not valid for an instance x will be discarded from the neighborhood (inRange version) or penalized with the parameter pOR (outRange version). The configuration outRange version with pOR = 1 is also discouraged since the final purpose of MonFkNN is to take monotonic constraints into consideration, at least to some extent. • These mechanics acquire different neighbors to those drawn by FkNN for the same test sample, that is, different class memberships and prediction. Finally, the median as the final class of the class membership vector already implies a significant change in the behavior of the method. These differences between our proposal and the traditional FkNN are clearly supported by the experiments carried out in Section 5.1. Experimental framework This section is devoted to introducing the experimental framework used in the different empirical studies of the paper. In our experiments, we have included 12 data-sets of a good variety of problems presenting real monotonic constraints. The data-sets can be seen in Table 2, where the number of instances, attributes and classes are detailed for each data-set in the column Ins., At. and Cl., respectively. The column At. Directions indicates the monotonic direction of the relationship between each attribute and the class: direct (+) or inverse monotony (-). This information is extracted from the description of the problems involving the data-sets. The column Comparable Pairs shows the percentage of pairs of comparable samples over the total number of pairs. Two instances x i and x j are comparable if their inputs have an order relation, i.e. x i x j or x i x j . On average, one-third of the total number of pairs of these data-sets are comparable and potential violations of monotonicity in the classification process. This quite large amount cannot be neglected. These data-sets are chosen as the most frequently used in the monotonic classification literature. The classical monotonic set ERA, ESL, LEV and SWD [2] are also considered in the study. Additionally, the data-set artiset is employed for a comparative study on monotonic noise robustness of MonFkNN (see Subsection 5.4). Artiset is an artificial data-set with two attributes (x 1 , x 2 ) and nCl number of classes. For attributes x 1 , x 2 ∈ [0, 1], the class is computed as the truncation of the outcome of the following formula: f (x 1 , x 2 ) = (x 1 + x 2 2 − x 2 1 2 ) * nCl A 10-fold cross-validation scheme (10-fcv) is carried out to run the different classifiers over these sets. Their partitions have been extracted from the KEEL repository [38]. The classifiers involved in the empirical comparisons are: • Monotonic k-NN (MkNN) [15] • Ordinal Stochastic Dominance Learning (OSDL) [31] • Ordinal Learning Module (OLM) [2] • Monotonic Multi-Layer Perceptron network (MonMLP) [27] • C4.5 decision tree for monotonic induction (MID) [3] • Rank Discrimination Measure Tree (RDMT) [32] • Partially Monotonic Decision Tree (PMDT) [34] Table 3 details the parameters chosen according to the recommendations found in the original papers. As a requirement of MkNN, a relabeling technique [16] is applied to training data-sets before fitting MkNN. On the contrary, the rest of the algorithms, including MonFkNN, do not need this relabeling procedure. Therefore, all the results shown for MkNN are obtained with relabeled training sets, while other methods are trained with the original training data-sets. Algorithm Parameters MkNN [15] k = 5, distance = euclidean, neighborsType = inRange OSDL [31] balanced = No, classificationType = median, lowerBound = 0, upperBound = 1 tuneInterpolationParameter = No, weighted = No, interpolationStepSize = 10, interpolationParameter = 0.5 OLM [2] modeResolution = conservative modeClassification = conservative MonMLP [27] default parameters, hidden1 = 8 iter.max = 1000, monotonic = all att MID [3] R = 1, confidence = 0.25, items per leaf = 2 RDMT [32] H = Pessimistic rank discrimination measure, measureThreshold = 0, items per leaf = 2 PMDT [34] threshold θ = 0, items per leaf = 2 FkNN [25] k = 5, K = 9, distance = euclidean MonFkNN k = 5, K = 9, distance = euclidean Pure Monotonic RCr = 0.5, neighborsType = inRange Approximate Monotonic RCr = 1, neighborsType = outRange, pOR = 0.5 In order to evaluate the classifiers' proficiency, we have employed three measures of different aspects of their performance: predictive capability, error cost and monotonicity. Standard accuracy is used to evaluate the predictive capability of the models. Mean Absolute Error (MAE) is computed as the average differences of the true instance ranks and the predicted ranks. To evaluate monotonicity, Non-Monotonic Index (NMI) [9] measures the ratio of pairs of samples (NMP) that break monotonicity among the total of pairs, with N being the number of samples in the data-set: NMI = NMP N 2 − N These measures are computed over a set merged from the test predictions of 10-fcv sets for each data-set and classifier. Finally, the Wilcoxon statistical test, Friedman rank test [20,19] with Holm post-hoc procedure [24] and Bayesian Sign test [5] are used to validate the results of the empirical comparisons. In the Bayesian Sign test, a distribution of the differences of the results achieved by methods A and B is computed thanks to the Dirichlet Process. This distribution is shown in a graphical space divided into 3 regions: left, rope and right. The location of the majority of distribution in these sectors indicates the final decision of the pairwise Bayesian non-parametric sign test: superiority of algorithm B (left sector), statistical equivalence (rope sector) and superiority of algorithm A (right sector). For the accuracy and MAE results, we have set the inferior and superior limit of the rope region to −0.01 and 0.01, respectively. However, we have adjusted the limits to −0.0001 and 0.0001 for NMI since NMI values tend to be significantly smaller due to the big difference between the numbers of comparable instance pairs and all possible pairs. The R package rNPBST [10] has been used to extract the graphical representations of the Bayesian Sign tests analyzed in the following empirical studies. Results and analysis This section presents the results of the empirical studies and their analyses. First, the two configurations of MonFkNN are compared in Subsection 5.1, showing their different strengths. Then, our proposal is compared to methods from the state-of-the-art in terms of prediction capability and monotonicity in Subsection 5.3 and Subsection 5.3, respectively. In Subsection 5.4, the last experiment tests the noise robustness of MonFkNN in contrast to MkNN. Evaluation of Monotonic Fuzzy k-NN approaches. Pure Monotonic vs Approximate Monotonic A comparison between the Pure and Approximate Monotonic version of MonFkNN stresses the different behaviors and aspects of their performance. Additionally, the performance differences between the original FkNN and MonFkNN are analyzed. Table 4 shows the results of FkNN and the two configurations of our proposal MonFkNN in terms of Accuracy, MAE and NMI. Bold-face font indicates the best results obtained for each data-set and metric. In Table 4, the differences between both approaches (PM and AM) can be seen clearly. Just as they were designed, MonFkNN-AM has better accuracy on average, while MonFkNN-PM achieves monotonically reliable predictions. Both have good, stable results in terms of MAE, with AM coming out slightly on top. AM configuration obtains the most accurate predictions for more than 50% of the benchmark used. On the other hand, the PM model achieves better results according to monotonicity in 10 of the 12 data-sets used, with large differences in Windsorhousing and MachineCPU problems. When compared with FkNN, MonFkNN greatly improves the performance of the original algorithm. Both versions of MonFkNN (PM and AM) are better on average for each of the three different measures. Particularly, there is an overwhelmingly large difference between FkNN and MonFkNN-PM in terms of monotonicity. FkNN is better only for 3 data-sets when taking just accuracy and MAE into consideration. However, it does not outperform the monotonic predictions of MonFkNN. This improvement is also reflected in the Wilcoxon statistical test applied to the results achieved using these methods. Table 5 presents the hypothesis of equivalence of the Wilcoxon test for α = 0.1 on the pairwise comparison of FkNN (1) and our two proposals (MonFkNN-PM (2) and MonFkNN-AM (3)). As shown in Table 5 The reasons for these differences in results are clear and mainly due to their algorithmic differences. MonFkNN has learning procedures with notions in the order relation of classes and the monotonic constraints between input and output, which explain an overall better performance in terms of MAE and NMI. Additionally, MonFkNN has a greater awareness and treatment of noisy data, which helps obtain better accuracy. Since monotonicity is usually prioritized in classification with monotonic constraints, we will use MonFkNN-PM in the following empirical studies. Comparison with the State-of-the-Art: Prediction capabilities Here we evaluate the performance of our approach in comparison to methods from the state-of-the-art of monotonic classification. In this comparison, we look for a balance between accurate and monotonic predictions. Therefore, we compare the results obtained in terms of the selected metrics independently. Then, we draw our conclusions and check if our approach behaves well in the different aspects of classification with monotonic constraints. First, we evaluate the prediction capability of our method. Table 6 gathers the accuracy results for the different data-sets obtained by the tested algorithms. With these outcomes, MonFkNN-PM performs overwhelmingly better than the rest in terms of accuracy. Our approach achieves the most accurate predictions on average with a wide margin. Additionally, it obtains the best results for 5 data-sets, with particularly remarkable cases, such as balance. PMDT is the second best method in terms of accuracy and it is the only method that come close to the performance of MonFkNN-PM. However, it obtains the overall best results for one data-set only (bostonhousing). As mentioned before, we have used the Friedman rank test and the Bayesian Sign test to corroborate the significance of the differences of our approach and the selected methods. Table 7 includes the outcome of the Friedman rank and Holm tests in relation to the obtained Accuracy results. MonFkNN-PM is ranked first with a high ranking value compared to others. All the hypotheses of equivalence are rejected with small p-values with the exception of PMDT, which would be rejected for α = 0.1. The distance between the ranks of MonFkNN-PM and PMDT is still quite large. Figure 1 graphically represents the difference between MonFkNN-PM and other methods and its statistical significance in terms of accuracy. In order to save space and avoid plotting 7 heat-maps for each metric, we have only included PMDT, as it is the best and most recent algorithm among the monotonic decision trees [34]. As mentioned before, the position of the majority of the distribution in these maps determines the decision of the test: the right sector means the statistical superiority of MonFkNN-PM over the compared method, the rope sector is the statistical equivalency and the left side indicates the superiority of the other algorithm. These heat-maps clearly indicate the significant superiority of MonFkNN-PM over all methods except PMDT as the computed distributions are always located in the right region. The most significant outcome is the comparison with OLM (Figure 1c), even though it does not obtain the worst results. For MkNN ( Figure 1a) and OSDL (Figure 1b), there are a few cases where their performances are statically equivalent to MonFkNN-PM. On the contrary, MonMLP is significantly more accurate in a few data-sets, although the MonFkNN-PM is clearly superior (Figure 1d). Considering the comparison with PMDT (Figure 1d), the majority of the distribution is located in the statistical equivalence. However, it is still shifted to the right with a large number of points, indicating a better performance for MonFkNN-PM. Almost none support the performance of PMDT. Error costs could be essential for monotonic ranking problems. Table 8 shows the error in the form of MAE made by the evaluated classifiers. As was the case in accuracy performance, MonFkNN-PM clearly performs better than the rest, with the smallest error on average and for 4 of the data-sets. It also achieves similar results in problems where other algorithms come out on top, such as LEV or wisconsin. Table 9 shows the ranking of the methods and p-values obtained with the post hoc test for the MAE comparison. As in the accuracy tests, our proposal is once again ranked as the best method with a solid statistical significance as compared to almost all algorithms. PMDT still achieves similar results to MonFkNN-PM with a p-value that does not reject the hypothesis for α = 0.05, but does for α = 0.1. In this case, the p-value of PMDT is smaller and its rank difference with our proposal is larger than that obtained in terms of accuracy. Figure 2 shows the Bayesian Sign test on pairwise comparison with our method according to MAE. As shown by the distributions in the right part of the majority of the figures, MonFkNN-PM is definitely better when considering error costs. This is more statistically significant as compared to OLM (Figure 2c), where nearly the entire distribution is in the right region. MonFkNN, MkNN and OSDL share some good results, but these last two are not statistically better than the former in any circumstance as seen in Figure 2a and Figure 2b. As we have also seen in the accuracy comparison, Figure 2d points out the statistical superiority of MonFkNN-PM over MonMLP, but the latter has a better MAE in some cases. Given Figure 2e, MonFkNN-PM and PMDT can be considered to be statistically the same in terms of error costs. However, MonFkNN-PM performs better statistically than PMDT in an important part of the benchmark, as a fragment of the distribution is located on the right side and almost none are found on the left. Comparison with the State-of-the-Art: Monotonicity Now we will analyze the performance according to the monotonicity of our proposal compared to methods chosen from the state-of-the-art. Table 10 shows the NMI results achieved by the selected models. In this case, the competition is close. Monotonic decision trees (MID, RDMT, and PMDT) clearly obtain less monotonic predictions. MID has the worst behavior considering only monotonicity and PMDT is the most monotonic decision tree classifier. OLM and MonMLP are slightly better than PMDT, but they still do not come close to the best methods. MonFkNN-PM, MkNN, and OSDL perform similarly. MonFkNN-PM and OSDL are slightly better on average. It is worth mentioning the existence of simpler data-sets, such as artiset and wisconsin, in relation to monotonicity as almost every algorithm accomplishes the same good results. The best results for the more complex sets are shared by the different methods. q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 11 summarizes the comparison according to monotonicity with the Friedman statistical test results. In this case, MonFkNN-PM is barely selected as the control method. For half of the benchmark (OSDL, MkNN, MonMLP and OLM), the hypotheses of equivalence are not rejected for α = 0.05. On the contrary, all monotonic decision trees are statistically worse than MonFkNN-PM by a wide margin. The best monotonic decision tree (PMDT) does not reach good performance in terms of monotonicity of the best algorithms. This is probably due to the greedy construction of monotonic constraints into the tree. In Figure 3, the statistical comparisons of the NMI results are represented with Bayesian Sign Test heat-maps. These plots show similar conclusions extracted from the previous table with NMI results. MonFkNN is significantly superior to PMDT (Figure 3e). In Figure 3c, the right-shifted distribution points out that MonFkNN-PM is better than OLM. Although they share a part of the distribution in the rope section, OLM has too few individuals in its left section (Figure 3c). When compared with MkNN (Figure 3a shifted to the right (Figure 3a and Figure 3b), the core of the distributions are found in the rope. Then, we can roughly assume statistical equivalence. In summary, MonFkNN-PM obtains significantly better results in terms of accuracy and error cost than almost all of the considered methods. Our approach also achieves the most monotonic predictions alongside OSDL. MonFkNN-PM is slightly and non-statistically better than PMDT in terms of accuracy and error costs, but the former overwhelmingly outperforms PMDT considering monotonicity. Therefore, MonFkNN-PM is an overall better method. The main reason behind the remarkable performance of MonFkNN is its capability of not sacrificing any objective of monotonic classification. Usually, some classifiers, such as OSDL, sacrifice accurate predictions in order to accomplish monotonic models. The results of OSDL for artiset and bostonhousing statement. On the other hand, other methods, such as monotonic decision trees and particularly PMDT, achieve accurate predictions but break the monotonic constraints in their predictions more frequently. However, the MonFkNN procedure of training class membership extraction is designed to mitigate the influence of non-monotonic noisy data, without the need to aggressively modify the training data as done by relabeling in MkNN. The MonFkNN prediction stage offers the flexibility of choice for most accurate or monotonic predictions. Additionally, MonFkNN includes technologies that are more appropriate for ordinal and monotonic classification, such as median as a final class. On the robustness of Monotonic Fuzzy k-NN to monotonic noise With this last empirical study, we aim to test the robustness of MonFkNN-PM to the presence of monotonic violations or noise in the training sets as compared to MkNN. Thus, we have introduced different amounts of noisy instances in the training partitions of the artificial data-set Artiset. Then, the performance of MonFkNN-PM and MkNN is measured and compared in terms of accuracy, MAE and NMI while the noise ratio increases. In order to increase the impact of class noise, we have randomly undersampled every training set to 25% of their instances. Then, a subset of randomly selected instances is converted to noise by changing their class labels. This label modification is done according to the adjacent classes of the implicated instance. Specifically, a large number of neighbors are computed for the future noisy example x i . 15 nearest neighbors were the value used in this experiment. Next, the neighbors with the same class as x i are removed and a new class is randomly obtained in relation to the presence ratio of other classes in its filtered neighbors. This ensures a certain degree of proximity between the changed sample and its new class. This process is executed following the same cross-validation scheme mentioned earlier. Since the noise generation has a random component, the experiment was repeated three times with different seeds, averaging the obtained results. After the noise generation and before the execution of MkNN, a relabeling technique [16] was applied to the resultant data-sets. Figure 4 shows the impact of increasing noise on the number of monotonic violations in Artiset training sets. This effect is measured by the Non-Monotonic Index (NMI) over the resulting training samples. As previously mentioned, class noise significantly aggravates the monotonicity of the datasets. The increase in NMI is directly proportional to the increase in noise as clearly shown in Figure 4. Figure 5 shows the performance of MonFkNN-PM and MkNN (darker and lighter lines, respectively) on the basis of precision (5a), MAE (5b) and NMI (5c), with the progression of noise. As expected, while the amount of noise grows, the performance of both methods get worse, that is, their accuracy decreases and errors and non-monotonic predictions increases. However, there are some big differences between classifiers. Firstly, the behavior of MonFkNN-PM facing noise is clearly better than that of MkNN in every tested aspect. The black lines are always located above the lighter ones in Figure 5a, which indicates greater accuracy, and under them in Figures 5b and 5c, meaning better MAE and NMI for MonFkNN-PM. Usually, the distance between both methods is large, with the exception of the NMI results obtained for the smallest values of noise. In addition, while the noise ratio increases, their differences also increase. The slope of deterioration of MonFkNN-PM performance remains stable, even being reduced in some cases, while the MkNN slope becomes steeper as the amount of noise increases. This last event can be clearly seen when the noise ratio reaches the 25% of the instances, where the decline of MkNN is magnified, especially in terms of monotonicity (Figure 5c). On the other hand, the NMI results of MonFkNN-PM seem to increase at a slower rate by that point. This exhibits the great robustness of MonFkNN-PM to monotonic violations. Next, the behavior of both methods in relation to noise are analyzed using a graphical example. Figure 6 is a graphical representation of the predictions and classification boundaries inferred by MkNN and MonFkNN-PM for Artiset with 35% noise. Figure 6a represents the perfect class surfaces defined by Artiset generation expression (see Section 4) and the training samples. In Figure 6a, black points represent the noise artificially introduced into the data-set. In Figures 6b and 6c, the black examples are wrongly classified instances, while the right predictions are colored in white. The first clear difference between the MkNN and MonFkNN-PM performances shown in Figures 6b and 6c is the amount of black dots. MonFkNN has far fewer classification mistakes than MkNN. Additionally, MonFkNN-PM is better at conserving the right regions for the classes, while MkNN can lose nearly all the entire sections of some of them. The regions in lighter and brighter yellow are shrunk by MkNN in favor of their adjacent classes. With these experiments, MonFkNN has shown strong robustness to monotonic noise preserving the decision boundaries as precisely as possible, and hence, has performed well in terms of precision, error costs and monotonicity. This robustness is the result of all the procedures included in MonFkNN, but it may also be mainly due to the reduction of the impact of non-monotonic noise during the extraction of the class memberships of the training instances. Conclusion In this paper, we proposed a Fuzzy k-Nearest Neighbors model for classification with monotonic constraints. The final class label obtained from membership functions has been revised to respect these constraints. MonFkNN has been designed with different mechanisms to reduce the influence of monotonic violations. As a demonstration of its flexibility, two different model configurations with different behaviors have been presented. Over the course of the experimental analyses, the great potential of both proposed versions, namely Pure and Approximate Monotonic Fuzzy k-NN, has been shown in relation to monotonicity and accuracy, respectively. Compared to other methods, MonFkNNN is significantly better in terms of accuracy and error cost, matching the best NMI results. In addition, it has shown its robustness to large amounts of noise while preserving its good performance. Future proposals should be robust to monotonic noise in order to obtain accurate and monotonic predictions. MonFkNN, as an example, opens possibilities to other fuzzy approaches since they are also potentially reliable against noise. Additionally, fuzzy techniques may be useful when defining different levels of constraints between input and output attributes. That is, some attributes may be more important than others regarding monotonicity. This problem representation may be very useful for monotonic classifiers. Algorithm 3 3MonFkNN: Prediction stage 1: function MonFkNN(x i -sample to predict, {D, y} -training data-set, k -neighbors considered for training class memberships, RCr -Real Class relevance, K -neighbors considered for prediction, typeRange -inRange or outRange, pOR -out-of-range penalty) 2: (D , u ) = TrainClassMemberships(D, y, k, RCr) 3: Obtain medians y of each sample in D with u and expression 4 4: Compute range r i with expression 1 and (D , y ) neighborsAsMkNN(x i , r i , K, typeRange, D , y ) 7: Figure 1 : 1Bayesian Sign Test heat-map for MonFkNN-PM vs. the rest in terms of accuracy. Figure 2 : 2Bayesian Sign Test heat-map for MonFkNN-PM vs. the rest in terms of MAE. ), OSDL (Figure 3b) and MonMLP (Figure 3d), big parts of the distributions are located in all the decision sectors. Even though their distributions are slightly Figure 3 : 3and the outcome of MkNN for balance are good examples of this Bayesian Sign Test heat-map for MonFkNN-PM vs. the rest in terms of NMI. Figure 4 : 4Impact of the addition of class noise in Artiset on monotonic violations measured by NMI . Noise effect in terms of NMI. Figure 5 : 5Comparison of MonFkNN-PM and MkNN performance on Artiset data-set with the different amounts of noisy samples. Figure 6 : 6Classification boundaries inferred by MkNN and MonFkNN-PM from the plotted Artiset with 35% noisy instances. Black points represent the instances wrongly classified by the decision surfaces shown. MonFkNN: Theoretical discussion Standard FkNN and MonFkNN have a similar mathematical formulation. In other words, the expressions used by MonFkNN in the training class membership extraction (Eq. 6) and in the membership aggregation (Eq. 7) are the same as those used by FkNN (Eq. 3 and Eq. 2), for RCr = 0.51and pOR = 1. The global behavior of our method is however still completely different to the standard FkNN, due to significant algorithmic differences.Table 1summarizes the main differences between standard FkNN and our proposal MonFkNN. Standard class membership aggregation. Monotonically constrained membership aggregation.No penalty to any neighbors in Eq. 2 pOR Penalty to out-of-range neighbors in Eq. 7.Final class as highest membershipFinal class as median of class membershipsFkNN MonFkNN No special treatment of duplicates. Duplicates are reduced to a single instance. Standard nearest neighbor rules. Monotonic nearest neighbor rules. Standard training membership extraction. Monotonically constrained class memberships. Conservation of original classes in the Loss of influence of original class towards training class membership extraction. monotonicity with RCr <= 0.5. Value 0.51 in Eq. 3 Parameter RCr in Eq. 6 Table 1 : 1Summary of algorithmic differences between standard FkNN and MonFkNN. Table 2 : 2Description of the 12 data-sets used.Data-set Ins. At. Cl. At. Directions Comparable Pairs artiset 1000 2 10 All direct directions 49.79% balance 625 4 3 {-, -, +, +} 25.64% bostonhousing4cl 506 13 4 {-, +, -, +, -, +, -, +, -, -, -, +, -} 14.85% car 1728 6 4 All direct directions 14.36% ERA 1000 4 9 All direct directions 16.77% ESL 488 4 9 All direct directions 70.65% LEV 1000 4 5 All direct directions 24.08% machineCPU 209 6 4 {-, +, +, +, +, +} 49.53% qualitative bankruptcy 250 6 2 All inverse directions 43.77% SWD 1000 10 4 All direct directions 12.62% windsorhousing 546 11 2 All direct directions 27.07% wisconsin 683 9 2 All direct directions 58.04% Table 3 : 3Parameters considered for the algorithms compared. Table 4 : 4Results for the Pure and Approximate Monotonic Fuzzy k-NNAccuracy MAE NMI FkNN MonFkNN-PM MonFkNN-AM FkNN MonFkNN-PM MonFkNN-AM FkNN MonFkNN-PM MonFkNN-AM artiset 0.9339 0.9309 0.9349 0.0661 0.0691 0.0651 0.0000 0.0000 0.0000 balance 0.8896 0.9307 0.9008 0.1424 0.0853 0.1168 0.0000 0.0000 0.0001 bostonhousing4cl 0.7174 0.6561 0.7134 0.3241 0.3972 0.3261 0.0004 0.0000 0.0001 car 0.9311 0.9740 0.9834 0.0793 0.0295 0.0195 0.0002 0.0000 0.0000 ERA 0.1730 0.2420 0.2430 1.6660 1.2813 1.2993 0.0141 0.0052 0.0052 ESL 0.6783 0.7036 0.7131 0.3484 0.3149 0.3053 0.0014 0.0004 0.0003 LEV 0.6020 0.6377 0.6110 0.4330 0.3927 0.4223 0.0021 0.0004 0.0009 machineCPU 0.6699 0.7033 0.6699 0.3589 0.3158 0.3493 0.0058 0.0002 0.0017 qualitative bankruptcy 0.9960 0.9960 0.9960 0.0040 0.0040 0.0040 0.0000 0.0000 0.0000 SWD 0.5350 0.5807 0.5833 0.5180 0.4370 0.4380 0.0027 0.0007 0.0003 windsorhousing 0.7857 0.7576 0.7839 0.2143 0.2424 0.2161 0.0062 0.0005 0.0051 wisconsin 0.9678 0.9653 0.9663 0.0322 0.0347 0.0337 0.0000 0.0000 0.0000 Avg: 0.7400 0.7565 0.7583 0.3489 0.3003 0.2996 0.0027 0.0006 0.0012 , MonFkNN-AM is statistically better than FkNN in terms of accuracy and MAE with p-Values under 0.1. Considering monotonicity, MonFkNN-PM and -AM statistically outperform FkNN with very low p-Values. Overall, MonFkNN is clearly superior to FkNN in scenarios with monotonic constraints. Table 5 : 5Wilcoxon test applied to the results obtained by Fuzzy k-NN algorithms: FkNN (1), MonFkNN-PM (2) and MonFkNN-AM (3)Comparison R + R − Hypothesis (α = 0.1) p-ValueAccuracy: (2) vs. (1) 49.0 17.0 Not Rejected 0.1748 (3) vs. (1) 61.5 16.5 Rejected 0.0847 MAE: (2) vs. (1) 51.0 15.0 Not Rejected 0.1230 (3) vs. (1) 57.0 9.0 Rejected 0.0322 NMI: (2) vs. (1) 76.5 1.50 Rejected 0.0012 (3) vs. (1) 72.5 5.50 Rejected 0.0059 Table 6 : 6Results in terms of Accuracy achieved by the tested algorithmsMonFkNN-PM MkNN OSDL OLM MonMLP MID RDMT PMDT artiset 0.9309 0.9199 0.1952 0.7948 0.9463 0.7237 0.8749 0.8539 balance 0.9307 0.8624 0.6352 0.8320 0.9131 0.7808 0.7216 0.7792 bostonhousing4cl 0.6561 0.6126 0.2787 0.5277 0.3979 0.6739 0.6304 0.6739 car 0.9740 0.9711 0.9549 0.9543 0.8474 0.8027 0.7297 0.9682 ERA 0.2420 0.1990 0.2320 0.1690 0.2380 0.2760 0.2390 0.2430 ESL 0.7036 0.6332 0.6721 0.5738 0.7234 0.6414 0.5635 0.6598 LEV 0.6377 0.4630 0.6400 0.4250 0.6167 0.6070 0.5210 0.6370 machineCPU 0.7033 0.6890 0.2919 0.6746 0.6730 0.6220 0.6555 0.6507 qualitative bankruptcy 0.9960 0.9960 0.9160 0.9800 0.6427 0.9840 0.9840 0.9920 SWD 0.5807 0.5200 0.5840 0.4160 0.5063 0.5540 0.5180 0.5830 windsorhousing 0.7576 0.5861 0.4927 0.7564 0.7790 0.8205 0.8022 0.7564 wisconsin 0.9653 0.9649 0.9590 0.8873 0.8604 0.9517 0.9502 0.9561 Avg: 0.7565 0.7014 0.5710 0.6659 0.6787 0.7031 0.6825 0.7294 Table 7 : 7Holm test applied to the Accuracy results among the tested algorithmsControl Method: MonFkNN-PM (2.04) i Algorithm (Rank) Z p-Value Hypothesis (α = 0.05) 7 OLM (6.13) 4.083 0.00004 Rejected 6 OSDL (5.42) 3.375 0.00073 Rejected 5 RDMT (5.38) 3.333 0.00085 Rejected 4 MonMLP (4.67) 2.625 0.00866 Rejected 3 MID (4.42) 2.375 0.01754 Rejected 2 MkNN (4.21) 2.167 0.03026 Rejected 1 PMDT (3.75) 1.708 0.08757 Not Rejected Table 8 : 8Results in terms of MAE achieved by the tested algorithmsMonFkNN-PM MkNN OSDL OLM MonMLP MID RDMT PMDT artiset 0.0691 0.0771 1.6897 0.2082 0.0537 0.3123 0.1251 0.1471 balance 0.0853 0.1504 0.4912 0.1920 0.0992 0.3360 0.3840 0.2560 bostonhousing4cl 0.3972 0.4901 0.9368 0.5988 0.7655 0.3893 0.4249 0.3676 car 0.0295 0.0359 0.0475 0.0538 0.1599 0.2506 0.3079 0.0365 ERA 1.2813 1.4270 1.2850 2.1500 1.2317 1.2970 1.3060 1.2870 ESL 0.3149 0.3791 0.3607 0.4734 0.2910 0.3934 0.4918 0.3750 LEV 0.3927 0.5740 0.3920 0.6680 0.4170 0.4290 0.5430 0.3940 machineCPU 0.3158 0.3301 0.9043 0.3589 0.3413 0.4211 0.3589 0.3732 qualitative bankruptcy 0.0040 0.0040 0.0840 0.0200 0.3573 0.0160 0.0160 0.0080 SWD 0.4370 0.4840 0.4370 0.7630 0.5167 0.4750 0.4990 0.4340 windsorhousing 0.2424 0.4304 0.5073 0.2436 0.2210 0.1795 0.1978 0.2436 wisconsin 0.0347 0.0337 0.0410 0.1127 0.1396 0.0483 0.0498 0.0439 Avg: 0.3003 0.3680 0.5980 0.4869 0.3828 0.3790 0.3920 0.3305 Table 9 : 9Holm test applied to the MAE results among the tested algorithmsControl Method: MonFkNN-PM (2.00) i Algorithm (Rank) Z p-Value Hypothesis (α = 0.05) 7 OLM (6.17) 4.167 0.00003 Rejected 6 RDMT (5.54) 3.542 0.00040 Rejected 5 OSDL (5.29) 3.292 0.00099 Rejected 4 MID (4.96) 2.958 0.00309 Rejected 3 MonMLP (4.25) 2.250 0.02445 Rejected 2 MkNN (4.04) 2.042 0.04119 Rejected 1 PMDT (3.75) 1.750 0.08011 Not Rejected Table 10 : 10Results in terms of NMI achieved by the tested algorithms Table 11 : 11Holm test applied to the NMI results among the tested algorithmsControl Method: MonFkNN-PM (2.9583) i Algorithm (Rank) Z p-Value Hypothesis (α = 0.05) 7 MID (7.00) 4.042 0.00005 Rejected 6 RDMT (6.33) 3.375 0.00074 Rejected 5 PMDT (5.75) 2.792 0.00524 Rejected 4 OLM (4.13) 1.167 0.24335 Not Rejected 3 MonMLP (3.63) 0.667 0.50499 Not Rejected 2 MkNN (3.13) 0.167 0.86763 Not Rejected 1 OSDL (3.08) 0.125 0.90052 Not Rejected https://github.com/sergiogvz/MonFkNN AcknowledgementsThis work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2017-89517-P and a research scholarship (FPU) given to Sergio González by the Spanish Ministry of Education, Culture and Sports. 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{'abstract': 'This paper proposes a new model based on Fuzzy k-Nearest Neighbors for classification with monotonic constraints, Monotonic Fuzzy k-NN (MonFkNN). Real-life data-sets often do not comply with monotonic constraints due to class noise. MonFkNN incorporates a new calculation of fuzzy memberships, which increases robustness against monotonic noise without the need for relabeling. Our proposal has been designed to be adaptable to the different needs of the problem being tackled. In several experimental studies, we show significant improvements in accuracy while matching the best degree of monotonicity obtained by comparable methods. We also show that MonFkNN empirically achieves improved performance compared with Monotonic k-NN in the presence of large amounts of class noise.', 'arxivid': '2003.02601', 'author': ['Sergio González \nDepartment of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain\n', 'Salvador García \nDepartment of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain\n', 'Sheng-Tun Li stli@mail.ncku.edu.twsheng-tunli \nDepartment of Industrial and Information Management\nInstitute of Information Management\nNational Cheng Kung University\n701TainanTaiwan\n\nCenter for Innovative FinTech Business Models\nNational Cheng Kung University\n701TainanTaiwan\n', 'Robert John robert.john@nottingham.ac.ukrobertjohn \nSchool of Computer Science\nASAP Research Group\nUniversity of Nottingham\nNG8 1BBNottinghamUK\n', 'Francisco Herrera herrera@decsai.ugr.es \nDepartment of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain\n\nFaculty of Computing and Information Technology\nKing Abdulaziz University\nJeddahSaudi Arabia\n', 'Sergio González ', 'Francisco Herrera '], 'authoraffiliation': ['Department of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain', 'Department of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain', 'Department of Industrial and Information Management\nInstitute of Information Management\nNational Cheng Kung University\n701TainanTaiwan', 'Center for Innovative FinTech Business Models\nNational Cheng Kung University\n701TainanTaiwan', 'School of Computer Science\nASAP Research Group\nUniversity of Nottingham\nNG8 1BBNottinghamUK', 'Department of Computer Science and Artificial Intelligence\nUniversity of Granada\n18071GranadaSpain', 'Faculty of Computing and Information Technology\nKing Abdulaziz University\nJeddahSaudi Arabia'], 'corpusid': 212415141, 'doi': '10.1016/j.neucom.2019.12.152', 'github_urls': ['https://github.com/sergiogvz/MonFkNN'], 'n_tokens_mistral': 26332, 'n_tokens_neox': 22913, 'n_words': 14756, 'pdfsha': '9936f70be81d7458473d819d6231f84ff585870d', 'pdfurls': ['https://arxiv.org/pdf/2003.02601v1.pdf'], 'title': ['Fuzzy k-Nearest Neighbors with monotonicity constraints: Moving towards the robustness of monotonic noise', 'Fuzzy k-Nearest Neighbors with monotonicity constraints: Moving towards the robustness of monotonic noise'], 'venue': []}

arxiv

The envelope of a complex Gaussian random variable May 2023 1 Sattwik Ghosal Ranjan Maitra The envelope of a complex Gaussian random variable May 2023 1Index Terms associated Laguerre polynomialBeckmann distributionconfluent hypergeometric functiongeneralized Beckmann distribu- tionHoyt distributionhypergeometric functionKummel functionLaguerre polynomialgeneralized Marcum functionNakagami distributionRayleigh distributionRice distributionWhittaker function This article explicitly characterizes the distribution of the envelope of an elliplical Gaussian complex vector, or equivalently, the norm of a bivariate normal random vector with general covariance structure. The probability density and cumulative distribution functions are explicitly derived. Some properties of the distribution, specifically, its moments and moment generating functions, are also derived and shown to exist. These functions and expressions are exploited to also characterize the special case distributions where the bivariate Gaussian mean vector and covariance matrix have some simpler structure. I. INTRODUCTION The envelope, amplitude or norm, of a complex Gaussian random variable has applications in many engineering and scientific disciplines, for example, in radar and communications [1], [2], [3], [4], [5], position localization [6] and navigation [7], or in magnitude resonance imaging [8], [9]. It is defined to be R= X 2 1 +X 2 2 , for the complex gain X 1 +iX 2 , where i= √ −1 and (X 1 , X 2 ) have a bivariate normal distribution N 2 (µ, Σ) with bivariate mean vector µ and 2×2 dispersion matrix Σ. Various cases of the envelope distribution have received special names, definitions and treatments. For instance, R is said to have the Beckmann distribution [1], [2] when Σ is a diagonal matrix. This distribution reduces further to the Hoyt/Nakagami-q distribution [10], [11], [12] when additionally µ=0. On the other hand, we get the Rice distribution [13], [14] when µ =0 but Σ∝I 2 , the identity matrix. For µ=0 and Σ∝I 2 , R has the Rayleigh distribution [15]. In the most general scenario, R itself is a special case of the generalized Beckmann distribution [16] that is the distribution of the Euclidean norm of a p-variate Gaussian random vector 1 . To fix context, we call the p=2 case the second order generalized Beckmann distribution. Complementing the envelope of a complex Gaussian random variable is its phase that also has applications in signal processing [17], [18], [19] and in other areas [20], [21]. The phase distribution has received attention in the communications literature [22], and more extensively, in statistics [23], [24], [25], but the same is not true for the envelope. Only the Rayleigh, Rice, Hoyt/Nakagami-q and Beckmann distributions have been well-characterized or studied [12], [14], [26], [27], [28], [29], but similar characterizations do not exist for the second order generalized Beckmann distribution with general Σ. Indeed, the exact probability density function (PDF) is the only property of the second order generalized Beckmann distribution that has been derived [22], [30]. Nevertheless, there are situations [22], [30] where nonhomogeneous receiver quadrature error or I-Q gain mismatch, and correlated Gaussian noise [31] can yield envelopes from this distribution, there it is of importance to characterize its properties. Therefore, in this paper, we explicitly derive, in Section II, the cumulative distribution function (CDF) of the second order generalized Beckmann distribution. For general p, [16] provided upper and lower bounds for the CDF of the generalized Beckmann distribution, but we provide explicit representations of the CDF here. We also use this opportunity to lay out a detailed derivation of the PDF to supplement the sketch provided of a similar specification [22], and that itself is of an alternative form to the one in [30]. We next provide the moment generating function (MGF) M R (t) of the second order generalized Beckmann distribution, after showing that it exists for any finite t ∈ R. Formulae for raw moments are also provided here. The derived formulae for the PDF, CDF, MGF, and the moments are then applied in Section III in the case of the specialized envelope distributions, namely, the Rayleigh, Rice, Beckmann and Hoyt/Nakagami-q distributions. In some cases, our formulae match existing formulae obtained through other means, while in other cases, our methods provide properties of these distributions not hitherto derived. We conclude with some discussion. Our article also has appendices containing some necessary technical details. II. MAIN RESULTS A. The probability density function The PDF of the second order generalized Beckmann distribution has been specified in two alternative ways [22], [30]. Our version is similar to that in [22] who only provided a very terse sketch, so we use this opportunity to rewrite the PDF and provide a formal proof for a fuller reference. Result 1. Let X=(X 1 , X 2 )∼N 2 (µ, Σ), where µ=(µ 1 , µ 2 ) ⊤ , and Σ has diagonal elements σ 2 1 and σ 2 2 and off-diagonal element ρσ 1 σ 2 . The PDF of R= X 2 1 +X 2 2 is f R (r; µ, Σ)=αr exp (−βr 2 ) ∞ j=0 ǫ j I 2j (−ψr)I j (ηr 2 ) cos 2jδ 1[r>0],(1) where 1(·) is the indicator function, ǫ j =2 1[j>0] , I m (.) is the modified Bessel function of the first kind of the mth order, δ=(φ−φ/2) mod π, andφ = arctan µ 1 σ 2 2 −ρµ 2 σ 1 σ 2 µ 2 σ 2 1 −ρµ 1 σ 2 σ 1 , φ= arctan 2ρσ 2 σ 2 σ 2 2 −σ 2 1 , α= 1 σ 1 σ 2 1−ρ 2 exp − µ 2 1 σ 2 2 +σ 2 1 µ 2 2 + 2ρσ 1 σ 2 µ 1 µ 2 2σ 2 1 σ 2 2 (1−ρ 2 ) > 0, β= σ 2 1 +σ 2 2 4σ 2 1 σ 2 2 (1−ρ 2 ) > 0, η= (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 4σ 2 1 σ 2 2 (1−ρ 2 ) ≥ 0. ψ= σ 2 1 (ρµ 1 σ 2 −µ 2 σ 1 ) 2 +σ 2 2 (ρµ 2 σ 1 −µ 1 σ 2 ) 2 σ 2 1 σ 2 2 (1−ρ 2 ) ≥ 0, Proof. See Appendix A. B. The cumulative distribution function We now provide explicit forms of the CDF of a generalized Beckmann random variable. Theorem 2. Let ℓ (j) k1,k2 . =k 1 +2k 2 +2j, while (n) (l) . =n!/(n−l)! denotes a falling factorial [32,Page 48], and C(k 1 , k 2 , j; ψ, η)= ψ 2k1 η 2k2 4 k1+k2 k 1 !(k 1 + 2j)!k 2 !(k 2 + j)! , T 1 (k 1 , k 2 , j; ψ, η)=C(k 1 , k 2 , j; ψ, η) ℓ (j) k1,k2 ! β ℓ (j) k 1 ,k 2 , and T 2 (u, k 1 , k 2 , j, k; ψ, η)=C(k 1 , k 2 , j; ψ, η) (ℓ (j) k1,k2 ) (k−1) β k−1 u 2ℓ (j) k 1 ,k 2 +2−2k . Under the definitions and setting of Result 1, R has CDF F R (u; µ, Σ)= α 2β ∞ j=0 ǫ j ηψ 2 8 j cos 2jδ ∞ k1=0 ∞ k2=0 T 1 (k 1 , k 2 , j; ψ, η) − exp (−βu 2 ) ∞ k1=0 ∞ k2=0 ℓ (j) k 1 ,k 2 +1 k=1 T 2 (u, k 1 , k 2 , j, k; ψ, η) ,(2) for u > 0, and is zero for u≤0. Proof. By definition, F R (u; µ, Σ)= u 0 f R (r; µ, Σ)dr, where f R (r; µ, Σ) is as in (1). Define ∆ u n = u 0 x n exp (−βx 2 )dx. Integrating by parts yields the recursive relation ∆ u n = u 0 x n exp (−βx 2 )dx=− u n−1 exp (−βu 2 ) 2β + n−1 2β ∆ u n−2 . Then, with i [k] . =i(i−2)(i−4) . . . (i−2k+2), for even n, ∆ u n =− 1 2β exp (−βu 2 ) n 2 k=1 u n+1−2k (2β) k−1 (n−1) [k−1] + (n−1) [ n 2 −1] (2β) n 2 π β [Φ(u 2β)−1/2], = − 1 2β exp (−βu 2 ) n 2 k=1 u n+1−2k (2β) k−1 (n − 1) [k−1] + Γ n+1 2 2β n+1 2 [Φ(u 2β) − 1/2](3) while, for odd n, ∆ u n = n−1 2 ! 2β n+1 2 − 1 2β exp (−βu 2 ) n+1 2 k=1 u n+1−2k β k−1 n−1 2 (k−1) = Γ n+1 2 2β n+1 2 − 1 2β exp (−βu 2 ) n+1 2 k=1 u n+1−2k β k−1 n−1 2 (k−1)(4) For any integer j≥0, I j (z)= z 2 j ∞ k=0 z 2k 4 k k! Γ(k + j + 1) . Therefore, I 2j (−ψr)I j (ηr 2 ) = 1 8 ηψ 2 r 4 j ∞ k1=0 ∞ k2=0 ( 1 4 ψ 2 r 2 ) k1 ( 1 4 η 2 r 4 ) k2 k 1 ! k 2 ! Γ(k 1 +2j+1)Γ(k 2 +j+1) = 1 8 ηψ 2 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j; ψ, η)r 2k1+2k2+4j(5) which, upon combining with (4), yields u 0 αr exp (−βr 2 )I 2j (−ψr)I j (ηr 2 )dr = ∞ k1=0 ∞ k2=0 α 1 8 ηψ 2 j C(k 1 , k 2 , j; ψ, η)∆ u 2ℓ (j) k 1 ,k 2 +1 = α 2β 1 8 ηψ 2 j ∞ k1=0 ∞ k2=0 T 1 (k 1 , k 2 , j; ψ, η) − exp (−βu 2 ) ∞ k1=0 ∞ k2=0 ℓ (j) k 1 ,k 2 +1 k=1 T 2 (u, k 1 , k 2 , j, k; ψ, η) ,(6) from where we get (2), after multiplying each term with ǫ j cos 2jδ and summing over j ∈ {0, 1, 2 . . .}. Proposition 3. Under the framework of Theorem 2, the CDF of R is equivalently, F R (u; µ, Σ) = α 2 ∞ j=0 ǫ j cos2jδ ηψ 2 8 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j, ψ, η) u 2l (j) k 1 ,k 2 +2 ℓ (j) k1,k2 +1 1 F 1 ℓ (j) k1,k2 +1, l (j) k1,k2 +2, −βu 2(7) for u > 0 and zero everywhere else. Here, 1 F 1 (·, ·, ·) is the confluent hypergeometric function, or Kummer's function, of the first kind [33], [34]. Proof. Note that ∆ u n = γ(n,βu 2 ) 2β n−1 2 , where γ(a, x)= x 0 t a−1 exp (−t)dt, the lower incomplete gamma function. Also, from (13.6.10) of [34], γ(a, x)=a −1 x a 1 F 1 (a, a + 1, −x). Then, ∆ u 2ℓ (j) k 1 ,k 2 +1 = u 2l (j) k 1 ,k 2 +2 2(ℓ (j) k1,k2 +1) 1 F 1 ℓ (j) k1,k2 +1, l (j) k1,k2 +2, −βu 2 , and the first line in the right hand side of (6) is also expressed as u 0 αr exp (−βr 2 )I 2j (−ψr)I j (ηr 2 )dr = α 2β ηψ 2 8 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j, ψ, η) u 2l (j) k 1 ,k 2 +2 ℓ (j) k1,k2 +1 1 F 1 ℓ (j) k1,k2 +1, ℓ (j) k1,k2 +2, −βu 2 ,(8) from where we get (7), in the same manner as (2) is obtained from (7). Remark 4. Theorem 2 and Proposition 3 provide two alternative versions of the CDF. In general, (7) involves fewer terms to calculate in the third series, because of reductions obtained by analytical integration of ∆ u 2ℓ (j) k 1 ,k 2 +1 . However, as seen, for example, in Section III-B or in Section III-F, there are some special cases where Proposition 3 may provide faster calculations because of the direct calculation of (7) through high-precision numerical algorithms in standard software libraries. C. The Moment Generating Function Theorem 5. Under the framework and definitions of Result 1 and Theorem 2. the MGF M R (t), of R exists ∀t ∈ R, and is M R (t)=α ∞ j=0 ǫ j cos 2jδ 1 8 ηψ 2 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j; ψ, η)I (β) 2ℓ (j) k 1 ,k 2 +1 (t),(9) where, for any odd integer m, we define I (β) m (t)= Γ m+1 2 2β m+1 2 1 F 1 m+1 2 , 1 2 , t 2 4β +t Γ m 2 +1 2β m/2+1 1 F 1 m 2 +1, 3 2 , t 2 4β ,(10) which admits an alternate representation given by I (β) m (t)= √ π exp t 2 4β 2β m+1 2 Γ m 2 + 1 L (−1/2) m 2 − t 2 4β + t 2 √ β Γ m + 1 2 L (1/2) m−1 2 − t 2 4β .(11) Here, L ν (x) is more commonly known as an associated Laguerre polynomial. Proof. We have M R (t)=E{exp (tR)}= ∞ 0 exp (tr)f R (r)dr, with f R (r) as in (1). Therefore, M R (t)=α ∞ j=1 ǫ j cos 2jδ ∞ 0 r exp (tr−βr 2 ){I 2j (−ψr)I j (ηr 2 )}dr.(12) We begin by proving existence. Note that I j (κ)≤I 0 (κ), ∀j ≥ 1. Also | cos 2jδ| < 1, and the integrands are all nonnegative. Hence, |M R (t)| ≤ α ∞ 0 r exp (tr−βr 2 )I 0 (−ψr) ∞ j=1 ǫ j I j (ηr 2 )dr. As I 0 (−ψr) is an even function, I 0 (−ψr)=I 0 (ψr)< exp (ψr). Also ∞ j=1 ǫ j I j (ηr 2 )= exp (ηr 2 ), and β≥η since |ρ|≤1. Therefore, |M R (t)|≤ ∞ 0 αr exp (tr−βr 2 +ψr+ηr 2 )dr = α exp (t+ψ) 2 4(β−η) ∞ 0 r exp [−{r− t+ψ 2 √ β−η } 2 ]dr<∞(13) for t ∈ R, since the integral is proportional to E(W 1[W ≥ 0]), with W ∼N ( t+ψ 2 √ β−η , 1 2 ) . It remains to show (9). From (5), ∞ 0 r exp (tr−βr 2 )I 2j (−ψr)I j (ηr 2 )dr= 1 8 ηψ 2 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j; ψ, η)I 2ℓ (j) k 1 ,k 2 +1 (t; β), where I (β) m (t)= ∞ 0 r m exp (tr−βr 2 )dr= ∞ k=0 t k k! ∆ ∞ m+k ,(14) and m is odd. Splitting the series in terms of series of odd and even terms, and using ∆ ∞ n = Γ( n+1 2 ) 2β n+1 2 for even n, from (3), and for odd n from (4), we get I (β) m (t)= ∞ k=0 t 2k (2k)! ( m−1 2 +k)! 2β m+1 2 +k + √ π ∞ k=0 t 2k+1 (2k+1)! (m+2k+1)! ( m−1 2 +k+1)! 1 2 m+2k+2 β m 2 +k+1 = m−1 2 ! 2β m+1 2 1 F 1 m+1 2 , 1 2 , t 2 4β +t √ π (4β) m/2+1 (m + 1)! m+1 2 ! 1 F 1 m 2 +1, 3 2 , t 2 4β = Γ m+1 2 2β m+1 2 1 F 1 m+1 2 , 1 2 , t 2 4β +t Γ m 2 +1 2β m/2+1 1 F 1 m 2 +1, 3 2 , t 2 4β ,(15) from where (9) follows. To obtain the alternative representation in (11), we note that from (13.1.27) of [34], we get the relations: 1 F 1 m + 1 2 , 1 2 , t 2 4β = exp t 2 4β 1 F 1 − m 2 , 1 2 , − t 2 4β , and 1 F 1 m 2 +1, 3 2 , t 2 4β = exp t 2 4β 1 F 1 − m − 1 2 , 3 2 , − t 2 4β .(16) From §16.1 of [36], and then (2.8) of [35] 1 F 1 (a, b, x) = exp x 2 x − b 2 M b 2 −a, b−1 2 (x) = Γ(b)Γ(1 − a) Γ(b − a) L (b−1) −a (x), where M κ,µ (x) is one of the two Whittaker functions [37]. Therefore, 1 F 1 − m 2 , 1 2 , − t 2 4β = Γ 1 2 Γ m 2 + 1 Γ m+1 2 L (−1/2) m 2 − t 2 4β , and 1 F 1 − m − 1 2 , 3 2 , − t 2 4β = Γ 3 2 Γ m+1 2 Γ m 2 + 1 L (1/2) m−1 2 t 2 4β .(17) Combining (16) and (17) and inserting into (15) yields (11). Remark 6. We make two comments on our results. 1) Our Laguerre functions are either associated Laguerre polynomials of nonnegative integer order (with α = 1 2 ), or Laguerre functions of half-integer order (with α = − 1 2 ), for which analytical expressions exist and aid computations. 2) Appendix B derives an alternative form of the MGF of the second order GBD. However, while that form does not require the evaluation of first order confluent hypergeometric functions, it requires the calculation of more terms, and so we consider Theorem 5 to be the preferred approach, with the Laguerre functions or 1 F 1 (·, ·, ·) readily evaluated in high precision using standard software libraries. D. Moments The moments are obtained as a corollary to Theorem 5. µ s = α 2β ∞ j=0 ǫ j cos 2jδ ηψ 2 8 j ∞ k=0 C(0, k, j; ψ, η) Γ s 2 +ℓ (j) 0,k2 +1 β s 2 +ℓ (j) 0,k 1 F 1 ℓ (j) 0,k + s 2 +1, 2j+1, ψ 2 4β .(18) Proof. We obtain the sth raw moment from the MGF from µ s = d s dt s M R (t) t=0 . The only part of M R (t) involving t in (9) is I (β) 2ℓ (j) k 1 ,k 2 +1 (t). Let ⌈w⌉ be the smallest integer not below w. From the second equality in (15), d s dt s I (β) m (t)= ∞ k=⌈ s 2 ⌉ t 2k−s (2k−s)! ( m−1 2 +k)! 2β m+1 2 +k + ∞ k=⌈ s−1 2 ⌉ t 2k+1−s (2k+1−s)! (m+2k+1)! ( m−1 2 +k+1)! β −( m 2 +k+1) 2 m+2k+1 .(19) Evaluated at t = 0, the first term in the first series in (19) is the only one that survives for even s, and yields ( m+s−1 2 )!/(2β m+s+1 2 ), while for odd s, only the first term in the second series makes it and is √ π(m + s)!/{( m+s 2 )!β ( m+s+1 2 ) 2 (m+s+1) }. In both cases, the surviving term can be re-expressed as Γ( m+s+1 2 ) 2β m+s+1 2 , and so, since m = 2ℓ (j) k1,k2 +1, we get µ s = α 2β ∞ j=0 ǫ j cos 2jδ ηψ 2 8 j ∞ k1=0 ∞ k2=0 C(k 1 , k 2 , j; ψ, η) Γ s 2 +ℓ (j) k1,k2 +1 β s 2 +ℓ (j) k 1 ,k 2 = α 2β ∞ j=0 ǫ j cos 2jδ ηψ 2 8 j ∞ k=0 C(0, k, j; ψ, η) Γ s 2 +ℓ (j) 0,k2 +1 β s 2 +ℓ (j) 0,k 1 F 1 ℓ (j) 0,k + s 2 +1, 2j+1, ψ 2 4β . III. APPLICATION TO SPECIAL DISTRIBUTIONS We now illustrate our derivations on Section II on some envelope distributions that are special cases of the generalized Beckmann distribution. In some of these special cases, the properties are known and our derivations provide the same answer or an alternative formula, while in other cases, our derivations provide additional characterization of these distributions. A. The Rayleigh distribution In this case, we have µ 1 =µ 2 =0, ρ=0 and σ 1 =σ 2 =0. Then,φ=0, φ=π/2, α=1/σ 2 , β=1/(2σ 2 ), η=ψ=0, I 0 (0)=1 and I s (0)=0 ∀ s≥1. Using Result 1, we get f R (r; µ, σ)= r σ 2 exp − r 2 2σ 2 1(r>0), which is the known directly calculated PDF of the Rice distribution. Further, C(k 1 , k 2 , j; 0, 0)=0, unless k 1 =k 2 =j=0, in which case, C(0, 0, 0; 0, 0)=1. Then, F R (u; µ, Σ)=1 − exp {−u 2 /(2σ 2 )}, for u≥0, which we know is the directly calculated CDF of the Rayleigh distribution with scale parameter σ. The only non-zero terms in the MGF of (9) are, from when k 1 =k 2 =j=0. Then, from (18), I (β) 1 (t) = 1 F 1 (1, 1 2 , σ 2 t 2 2 )+σt π 2 1 F 1 ( 3 2 , 3 2 , σ 2 t 2 2 ), and M R (t)=αI (β) 1 (t) . We state and prove Lemma 9 that shows 1 F 1 (1, 1 2 , σ 2 t 2 2 ) = 1+σ 2 t 2 1 F 1 (1, 3 2 , σ 2 t 2 2 ). Also, from (13.1.27) and (13.6.19) of [34], 1 F 1 (1, 3 2 , σ 2 t 2 2 ) = exp σ 2 t 2 2 1 F 1 ( 1 2 , 3 2 , − σ 2 t 2 2 ), and 1 F 1 ( 1 2 , 3 2 , − σ 2 t 2 2 ) = π 2 1 σt erf σt √ 2 , while 1 F 1 ( 3 2 , 3 2 , σ 2 t 2 2 ) = exp σ 2 t 2 2 from (13.6.12) of [34]. Combining, M R (t) = 1 + σt π 2 exp σ 2 t 2 2 erf σt √ 2 + 1 , which is the known directly calculated MGF of the Rayleigh distribution. Concluding, the sth Rayleigh raw moment is µ s = (α/2β)C(0, 0, 0, 0, 0)Γ(s/2 + 1)/β s 2 = 2 s/2 σ s Γ(s/2 + 1), since 1 F 1 (a, b, 0) = 1. B. The Rice distribution In this case, ρ=0, σ 1 =σ 2 =σ, and µ 1 = ν cos ξ, µ 2 = ν sin ξ in polar form. Then α = 1 σ 2 exp − ν 2 2σ 2 , β = 1 2σ 2 , ψ = ν/σ, η=0, and δ = {arctan (cot ξ)} mod π, η=0 and so f R (r; µ, Σ)= r σ 2 exp − ν 2 +r 2 2σ 2 I 0 − νr σ 1[r>0], the known directly calculated Rice(σ, ν) PDF. Also, as in Section III-A, C(k 1 , k 2 , j; ψ, η) makes a positive contribution to the series in (2), (12) or (18) only when k 2 =j=0. Now, C(k 1 , 0, 0; ψ, 0)= ψ 2k 1 4 k 1 (k1!) 2 , so that T 1 (k 1 , 0, 0; ψ, 0)= ψ 2k 1 4 k 1 k1!β k 1 and T 2 (u, k 1 , 0, 0, k; ψ, 0)= ψ 2k 1 4 k 1 k1!(k1−k+1)!β k−1 u 2(k1+1−k) . From (2), F R (u; σ, ν)= exp − ν 2 2σ 2 ∞ k1=0 ψ 2k1 4 k1 k 1 !β k1 − exp − u 2 2σ 2 ∞ k1=0 k1+1 k=1 ψ 2k1 u 2(k1+1−k) 4 k1 k 1 !(k 1 −k+1)!β k−1 .(20) Also, from the alternative representation of the CDF in (7), we get F R (u; σ, ν)= α 2β ∞ k1=0 C(k 1 , 0, 0, ψ, 0) β k1 γ(k 1 +1, βu 2 ) = exp − ν 2 2σ 2 ∞ k1=0 1 k! γ(k + 1, u 2 2σ 2 ) Γ(k + 1) ν 2k 2 k σ 2k = 1−Q 1 ν σ , u σ ,(21) where Q 1 (·, ·) is the generalized Marcum Q-function of the first order [38], and the reduction to it follows from (2.12) of [39]. The CDF (21) is preferred over the one in (20) because high-precision algorithms for Q 1 (·, ·) exist in many standard software. For the MGF, we have from (9), for all t∈R, M R (t) = exp − ν 2 2σ 2 ∞ k=0 ( ν σ ) 2k 2 k k! 1 F 1 k+1, 1 2 , σ 2 t 2 2 + π 2 σt 2 ∞ k=0 ( ν σ ) 2k 8 k (k!) 2 (2k+2)! (k+1)! 1 F 1 2k+3 2 , 3 2 , σ 2 t 2 2 , and, from (18), µ s = exp − ν 2 2σ 2 ∞ k1=0 ν 2k1 4 k1 σ 4k1 k 1 ! 2 Γ(k 1 + 1 + s 2 ) β s 2 +k1 = β − s 2 exp − ν 2 2σ 2 Γ s 2 +1 1 F 1 s 2 +1, 1, ν 2 2σ 2 . From (13.1.27) and (13.6.9) of [34], exp (− ν 2 2σ 2 ) 1 F 1 s 2 + 1, 1, ν 2 2σ 2 = 1 F 1 − s 2 , 1, − ν 2 2σ 2 = L s 2 − ν 2 2σ 2 , where Lq(·) is the qth (simple) Laguerre polynomial. Therefore, µ s = 2 s 2 σ s Γ s 2 + 1 L s 2 − ν 2 2σ 2 . 7 C. The Hoyt distribution The Hoyt distribution is when µ 1 =µ 2 =0, ρ=0, but σ 2 ≥ σ 1 . Thus, it is a slight generalization of the Rayleigh distribution. For this distribution, therefore, we have α = 1 σ1σ2 , β = 1 4 1 σ 2 1 + 1 σ 2 2 , ψ = 0 η = σ 2 2 −σ 2 1 4σ 2 1 σ 2 2 and δ = π 2 . This means that the only terms that show up in the series in the PDF is when j=0. Therefore, using our formula, the PDF of the Hoyt distrbution is f R (r) = r σ 1 σ 2 exp − 1 4 1 σ 2 1 + 1 σ 2 2 r 2 I 0 σ 2 2 −σ 2 1 4σ 2 1 σ 2 2 r 2 1[r>0], which is similar to the one specified in [30]. Further, with regard to the CDF, the MGF or the moments is when k 1 =j=0 and k 2 ≥ 0. In this case, C(0, k 2 , 0; ψ, η) = 1 4 k 2 k2! 2 (σ 2 2 −σ 2 1 ) 2 16σ 4 1 σ 4 2 k2 . Further, ℓ (j) k1,k2 = 2k 2 , so from Proposition 3, we get the CDF to be F R (r; σ 1 , σ 2 ) = 1 2σ 1 σ 2 ∞ k2=0 1 4 k2 k 2 ! 2 (σ 2 2 − σ 2 1 ) 2 16σ 4 1 σ 4 2 k2 u 4k2+2 2k 2 + 1 1 F 1 2k + 1, 2k + 2, − 1 4 1 σ 2 1 + 1 σ 2 2 u 2 , while the MGF, from Theorem 5, is M R (t) = 2σ 1 σ 2 σ 2 1 + σ 2 2 ∞ k=0 (σ 2 2 − σ 2 1 ) 2 4(σ 2 1 + σ 2 2 ) 2 k Γ(2k + 1) k! 2 1 F 1   2k + 1, 1 2 , t 2 1 σ 2 1 + 1 σ 2 2   + 4tσ 2 1 σ 2 2 (σ 1 + σ 2 ) 3/2 ∞ k=0 (σ 2 2 − σ 2 1 ) 2 4(σ 2 1 + σ 2 2 ) 2 k Γ 2k+3 2 k! 2 1 F 1   2k + 3 2 , 3 2 , t 2 1 σ 2 1 + 1 σ 2 2   . Further, from Corollary 7, since ψ=) and 1 F 1 (a, b, 0)=1 a, b, c, z) is the hypergeometric function [36], [40], and we thus arrive at a formula that matches the independentlyderived formula in (7) of [27]. µ s = 1 σ 1 σ 2 ∞ k=0 (σ 2 2 −σ 2 1 ) 2 64σ 4 1 σ 4 2 k k! 2 Γ 2k+ s 2 +1 2 4k+s+1 1 σ 2 1 + 1 σ 2 2 −(2k+1+ s 2 ) = 1 σ 1 σ 2 2 s+1 Γ s 2 +1 1 σ 2 1 + 1 σ 2 2 1+ s 2 ∞ k=0 ( s 2 +1)( s 2 +2) . . . ( s 2 +2k) k! (σ 2 2 −σ 2 1 ) 2 4(σ 2 1 +σ 2 2 ) 2 k k! = (2σ 1 σ 2 ) s+1 (σ 2 1 +σ 2 2 ) 1+ s 2 Γ s 2 +1 ∞ k=0 2 2k ( s 4 + 1 2 )( s 4 +1) . . . ( s 4 +k) k! (σ 2 2 −σ 2 1 ) 2 4(σ 2 1 +σ 2 2 ) 2 k k! = (2σ 1 σ 2 ) s+1 (σ 2 1 +σ 2 2 ) 1+ s 2 Γ s 2 +1 ∞ k=0 ( s 4 + 1 2 )( s 4 + 3 2 ) . . . ( s 4 +k− 1 2 ) ( s 4 +1)( s 4 +2) . . . ( s 4 +k) k! (σ 2 2 −σ 2 1 ) 2 (σ 2 1 +σ 2 2 ) 2 k k! = (2σ 1 σ 2 ) s+1 (σ 2 1 +σ 2 2 ) 1+ s 2 Γ s 2 +1 2 F 1 s 4 + 1 2 , s 2 +1, 1, (σ 2 2 −σ 2 1 ) 2 (σ 2 1 +σ 2 2 ) 2 , where 2 F 1 ( D. The Beckmann distribution The Beckmann distribution is a special case of the generalized form, and happens when ρ=0. Without loss of generality, let us assume that σ 2 ≥σ 1 . Then we have δ= arctan . In this case, all terms involving all k 1 , k 2 , j contribute to the series. Further, C(k 1 , k 2 , j; ψ, η)= 1 4 k1+k2 k 1 !(k 1 +2j)!k 2 !(k 2 +j)! µ 2 1 σ 4 2 +µ 2 2 σ 4 1 σ 4 1 σ 4 2 k1 (σ 2 2 −σ 2 1 ) 2 16σ 4 1 σ 4 2 k2 .(22) There are no further simplifications of the formulae for the PDF, CDF, MGF or moments possible here, so we point back Section II for the specific formulate for each of these quantities, with the above values inserted in those formulae. E. Identical quadrature components model This case occurs when the underlying complex Gaussian random variable has identical but correlated quadratures, that is, σ 1 =σ 2 ≡σ. In this case, α= 1 σ 2 √ 1−ρ 2 exp − µ 2 1 +µ 2 2 −2ρµ1µ2 2σ 2 (1−ρ 2 ) , β= 1 2σ 2 (1−ρ 2 ) , ψ= √ (ρµ1−µ2) 2 +(ρµ2−µ1) 2 σ 2 (1−ρ 2 ) , η= ρ 2σ 2 (1−ρ 2 ) , and δ= arctan µ1−ρµ2 µ2−ρµ1 − π 4 mod(π). With these quantities, we write C(k 1 , k 2 , j; ψ, η)= 1 4 k1+k2 k 1 !(k 1 +2j)!k 2 !(k 2 +j)! (ρµ 1 −µ 2 ) 2 +(ρµ 2 −µ 1 ) 2 σ 4 (1−ρ 2 ) 2 k1 ρ 2 4σ 4 (1−ρ 2 ) 2 k2 . In general, as in the Beckman case, no further simplifications in the formulae from Section II are possible and these values are inserted to get expressions for the different quantities. However, when additionally, µ 1 =µ 2 ≡µ, then α= 1 σ 2 √ 1−ρ 2 exp − µ 2 σ 2 (1+ρ) , β= 1 2σ 2 (1−ρ 2 ) , ψ= µ σ 2 (1+ρ) , η= ρ 2σ 2 (1−ρ 2 ) , δ=0, and C(k 1 , k 2 , j; ψ, η)= µ 2k 1 ρ 2k 2 4 k 1 +2k 2 k1!k2!(k1+j)!(k2+2j)!σ 4k 1 +k 2 (1−ρ 2 ) k 1 +k 2 . In this case, the PDF is f R (r; µ, σ, ρ)= r σ 2 1−ρ 2 exp − µ 2 (1−ρ)+r 2 2σ 2 (1−ρ) 2 ∞ j=0 I 2j −µr σ 2 (1+ρ) I j ρr 2 2σ 2 (1−ρ 2 ) 1[r>0]. When additionally µ=0, we get another slight generalization of the Rayleigh distribution, and because then the only terms involving j in the series contribute when j=0. In that case, the PDF is, f R (r; σ, ρ)= r σ 2 1−ρ 2 exp − r 2 2σ 2 (1−ρ) 2 I 0 ρr 2 2σ 2 (1−ρ 2 ) 1[r>0], or an alternative form of the Rice PDF derived by [41]. We subsume further discussion of this case in the next section. F. Background signal-free model The background case is a slight generalization of the Hoyt density, and occurs when µ 1 =µ 2 =0 in our generalized Beckmann setup. Then, α= 1 σ1σ2 √ 1−ρ 2 , β= σ 2 1 +σ 2 2 4σ 2 1 σ 2 2 (1−ρ 2 ) , η= √ (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 4σ 2 1 σ 2 2 (1−ρ 2 ) , and ψ=0. Alsoφ= π 2 and φ=arctan 2ρσ1σ2 σ 2 2 −σ 2 1 . Then k 1 =0, j=0 is the only time that the terms in the series contribute. Therefore, f R (r; σ 1 , σ 2 , ρ) = r σ 1 σ 2 1−ρ 2 , exp − (σ 2 1 +σ 2 2 )r 2 4σ 2 1 σ 2 2 (1−ρ 2 ) I 0 r 2 (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 4σ 2 1 σ 2 2 (1−ρ 2 ) 1[r>0]. Also C(0, k, 0; 0, η)= 1 64 k (k!) 2 {(σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 4 2 ρ 2 } k σ 4k 1 σ 4k 2 (1−ρ 2 ) 2k , and from Proposition 3, we get the CDF F R (u; σ 1 , σ 2 , ρ) = u 2 2σ 1 σ 2 1−ρ 2 ∞ k=0 u 4k 64 k (k!) 2 (2k + 1) (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 4 2 ρ 2 k σ 4k 1 σ 4k 2 (1−ρ 2 ) 2k 1 F 1 2k + 1, 2k + 2, − u 2 (σ 2 1 +σ 2 2 ) 4σ 2 1 σ 2 2 (1 − ρ 2 ) ,(23) and the MGF, from Theorem 5, is M R (t) = 2σ 1 σ 2 (1 − ρ 2 ) σ 2 1 + σ 2 2 ∞ k=0 (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 k 4 k (k!) 2 (σ 2 1 +σ 2 2 ) 2k Γ(2k+1) 1 F 1 2k+1, 1 2 , t 2 σ 2 1 σ 2 2 (1−ρ 2 ) σ 2 1 +σ 2 2 + 2tσ 1 σ 2 (1 − ρ 2 ) 1/2 (σ 2 1 + σ 2 2 ) Γ 2k+ 3 2 1 F 1 2k+ 3 2 , 3 2 , t 2 σ 2 1 σ 2 2 (1−ρ 2 ) σ 2 1 +σ 2 2 . Finally, Corollary 7 gives µ s = 2 s+1 (σ 1 σ 2 ) 1+s (1−ρ 2 ) 1− s 2 (σ 2 1 +σ 2 2 ) s 2 +1 ∞ k=0 1 4 k (k!) 2 (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 k (σ 2 1 + σ 2 2 ) 2k Γ 2k+ s 2 +1 . = 2 s+1 (σ 1 σ 2 ) 1+s (1−ρ 2 ) 1− s 2 (σ 2 1 +σ 2 2 ) s 2 +1 Γ s 2 +1 2 F 1 s 4 + 1 2 , s 4 +1, 1, (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 (σ 2 1 + σ 2 2 ) 2 Note also that for ρ=0, the above reduces to the Hoyt moments in [27], or in Section III-C. The same holds for the PDF, the CDF and MGFs which reduce to the forms in Section III-C. Further, for the special case of σ 1 =σ 2 ≡σ, and non-zero ρ, the CDF in (23) reduces to F R (u; σ, ρ) = u 2 2σ 2 1−ρ 2 ∞ k=0 1 4 k (k!) 2 u 4k 2k + 1 ρ 2k 1 F 1 2k + 1, 2k + 2, − u 2 2σ 2 (1 − ρ 2 ) ,(24) while the MGF is M R (t)= (1 − ρ 2 ) ∞ k=0 ρ 2k 4 k (k!) 2 Γ(2k+1) 1 F 1 2k+1, 1 2 , t 2 σ 2 (1−ρ 2 ) 2 +tσ 2(1 − ρ 2 )Γ 2k+ 3 2 1 F 1 2k+ 3 2 , 3 2 , t 2 σ 2 (1−ρ 2 ) 2 ,(25) and the raw moments are µ s = 2 s 2 (1−ρ 2 ) 1− s 2 σ s ∞ k=0 ρ 2k 4 k (k!) 2 Γ 2k+ s 2 +1 = 2 s 2 (1−ρ 2 ) 1− s 2 σ s Γ s 2 +1 2 F 1 s 4 + 1 2 , s 4 +1, 1, ρ 2 .(26) We thus obtain, through (24), (25) and (26), further characterization of the alternative form of the Rice distribution of [41]. IV. DISCUSSION This article provides a full characterization of the envelope distribution of a complex Gaussian random variable of general form. We explicitly derive the CDF, the MGF and the moments, all of which are shown to exist. Our derivations reduce to the forms for the special cases of Rice, Rayleigh and Nakagami-q/Hoyt distributions. We also investigate reductions for the case of the Beckmann, identical quadrature components models and signal-free models. Our reductions in some cases provide further characterization of these special case. Given the importance of the envelope of the complex Gaussian random variable in communications and signal processing, we expect our derivations to further the analysis of the performance these systems by more accurate modeling using our envelope distributions. Finally, we note that our derivations are in the context of the second order generalized Beckmann distribution, but of interest would be the general case of the generalized Beckmann distribution. APPENDIX A PROOF OF RESULT 1 We provide a detailed proof of Result 1. Proof. Without loss of generality, let σ 2 >σ 1 . Transforming X to polar form, i.e., X 1 =R cos Θ and X 2 =R sin Θ, with R>0 and 0<Θ<2π, the PDF of (R, Θ) is f R,Θ (r, θ)= r 2π|Σ| 1/2 exp g(r;µ,Σ) 1−ρ 2 where g(r; µ, Σ)= − µ 2 1 2σ 2 1 − µ 2 2 2σ 2 2 − ρµ 1 µ 2 σ 1 σ 2 − r 2 4σ 2 1 (1− cos 2θ)− r 2 4σ 2 2 (1+ cos 2θ)+ r 2 ρ 2σ 1 σ 2 sin 2θ + r( µ 2 σ 2 2 − µ 1 ρ σ 1 σ 2 ) cos θ+r( µ 1 σ 1 2 − µ 2 ρ σ 1 σ 2 ) sin θ, which when simplifying notation to reduce clutter, gives f R,Θ (r, θ)= αr exp(−βr 2 ) 2π exp A φ cos(2θ−φ)+Aφcos(θ−φ) , with Aφ= r √ σ 2 1 (ρµ1σ2−µ2σ1) 2 +σ 2 2 (ρµ2σ1−µ1σ2) 2 4σ 2 1 σ 2 2 (1−ρ 2 ) , and A φ = r 2 √ (σ 2 2 −σ 2 1 ) 2 +4σ 2 1 σ 2 2 ρ 2 4σ 2 1 σ 2 2 (1−ρ 2 ) . It remains to integrate f R,Θ (r, θ) over θ ∈ [0, 2π). We have 1 2π ǫ k I k (κ 2 ) cos 2k(θ+δ) dθ =I 0 (κ 1 )I 0 (κ 2 )+ 4 2π ∞ j=1 ∞ k=1 I j (κ 1 )I k (κ 2 ) 2π 0 cosjθcos 2k(θ+δ)dθ since 2π 0 cos jθdθ=0 and 2π 0 cos 2k(θ + δ)dθ=0 for any integer j, k. Expanding cos 2k(θ + δ) and then integrating using the change of variables θ → θ−π, gives 2π 0 cos jθ cos 2k(θ + δ)dθ= cos 2kδ(−1) j π −π cos jθ cos 2kθdθ− sin 2kδ(−1) j π −π cos jθ sin 2kθdθ =(−1) j cos 2kδ π −π cos jθ cos 2kθdθ, since π −π cos jθ sin 2kθdθ=0, as cos jθ sin 2kθ is an odd function in θ∈(−π, π). For j =2k, we have ǫ j I 2j (κ 1 )I j (κ 2 ) cos 2jδ, and the result follows. APPENDIX B AN ALTERNATIVE FORM OF THE MGF We provide an alternative to (9) in Theorem 5. We express I ν (x) is a Laguerre function, as introduced by[35], for unrestricted ν. When ν is a nonnegative integer, L (a) Corollary 7 . 7Under the framework and definitions of Result 1 and Theorems 2 and 5, the sth raw moment of R is µ s =E(R s ), A φ cos 2(θ+δ)−Aφ cos θ]dθ=G 0 (δ, −Aφ, A φ ), with G 0 (δ, κ 1 , κ 2 ) {κ 1 cos θ + κ 2 cos(θ + δ)}dθ. By Fourier series expansion, we haveexp(κ 1 cos θ)=a 0 + ∞ j=1 a j cos jθ+ ∞ j=1 b j sin jθ,with coefficients b j = 1 2π 2π 0 exp (κ 1 cos θ) sin jθdθ=0, and a j = 1 2π 2π 0 exp (κ 1 cos θ) cosjθdθ=ǫ j I j (κ 1 ). Consequently, exp(κ 1 cos θ)= ∞ j=0 ǫ j I j (κ 1 ) cos jθ, +1 exp (tr − βr 2 )dr. m (t) differently than in(14). Specifically, we writeI (β) m (t) = ∞ 0 r m exp tr − βr 2 dr= exp t 2 4β ∞ 0 r m exp − βr − t 2 √ β 2 dr, Some authors call the generalized Beckmann distribution the generalized Rice distribution, but we feel that the former name offers greater clarity, referring to the distribution of the Euclidean norm of a Gaussian random vector in its most general formulation. where erf(·) is the error function. Hence, the result follows.APPENDIX C AN IDENTITY FOR 1 F 1 (1, s, z) Lemma 9. Let 1 F 1 (a, b, x) be the confluent hypergeometric function of the first kind[33]. Then, the following identity holds:Proof. 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{'abstract': 'This article explicitly characterizes the distribution of the envelope of an elliplical Gaussian complex vector, or equivalently, the norm of a bivariate normal random vector with general covariance structure. The probability density and cumulative distribution functions are explicitly derived. Some properties of the distribution, specifically, its moments and moment generating functions, are also derived and shown to exist. These functions and expressions are exploited to also characterize the special case distributions where the bivariate Gaussian mean vector and covariance matrix have some simpler structure.', 'arxivid': '2305.03038', 'author': ['Sattwik Ghosal ', 'Ranjan Maitra '], 'authoraffiliation': [], 'corpusid': 258480237, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18624, 'n_tokens_neox': 15254, 'n_words': 8624, 'pdfsha': '2d362130b4fcc8c34a416768067cd0c71abdbe40', 'pdfurls': ['https://export.arxiv.org/pdf/2305.03038v2.pdf'], 'title': ['The envelope of a complex Gaussian random variable', 'The envelope of a complex Gaussian random variable'], 'venue': []}

arxiv

VBS W ± W ± H production at the HL-LHC and a 100 TeV pp-collider 7 Feb 2017 Christoph Englert christoph.englert@glasgow.ac.uk SUPA School of Physics and Astronomy University of Glasgow G12 8QQGlasgowUK Qiang Li Department of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina CAS Center for Excellence in Particle Physics 100049BeijingChina Michael Spannowsky michael.spannowsky@durham.ac.uk Institute for Particle Physics Phenomenlogy Department of Physics Durham University DH1 3LEUK Mengmeng Wang mengmeng.wang@cern.ch Department of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina Lei Wang Department of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina VBS W ± W ± H production at the HL-LHC and a 100 TeV pp-collider 7 Feb 2017Prepared for submission to JHEPLHCVector Boson ScatteringWWHH coupling W ± W ± H production at hadron colliders through vector boson scattering is a so far unconsidered process, which leads to a clean signature of two same-sign charged leptons and two widely separated jets. This process is sensitive to the HHH and WWHH couplings and any deviation of these couplings from their SM predictions serves as direct evidence of new physics beyond the SM. In this paper we perform a Monte Carlo study of this process for the √ s = 14 TeV LHC and a 100 TeV pp-collider, and provide projections of the constraints on the triple-Higgs and WWHH quartic couplings for these environments.In particular, we consider the impact of pileup on the expected sensitivity in this channel. Our analysis demonstrates that although the sensitivity to the HHH coupling is rather low, the WWHH coupling can be constrained in this channel within ∼ 100% and ∼ 20% at 95% confidence level around the SM prediction at the HL-LHC and a 100 TeV pp-collider, respectively. Introduction After the discovery of the 125 GeV Higgs-like boson [1][2][3][4], one of the primary goals of present collider phenomenology is to formulate ways to pave the way to a better understanding of the mechanism of electroweak symmetry breaking (EWSB). In particular, the trilinear Higgs HHH and the quartic VVHH vertices (with V representing the W and Z vector bosons) are key parameters, which are also directly linked to radiative instability of the TeV scale [5], as well as to a potential radiative nature of EWSB [6][7][8]. In the Standard Model (SM), the WWHH coupling is determined by electroweak gauge invariance, which enforces g WWHH = e 2 /(2s 2 w ), with s w denoting the sine of the Weinberg angle and e the electric charge, respectively. Any deviation from this value indicates the necessary existence of new physics beyond the SM, as a departure from the gauge-relations directly induces (perturbative) unitarity violation [9], unless new resonant states mend the dangerous growth of the WW → HH amplitude. Models with extra dimensions and their holographic interpretation in terms of composite theories are well-known examples of how such coupling modifications can appear in the low energy formulation of strongly interacting scenarios (for a recent review see [10]). Modifications of unitarity sum rules can be used to predict some properties of new composite states [11][12][13]. In such scenarios, only measuring the trilinear gauge couplings is not necessarily indicative of the quartic gauge couplings in the low energy effective field theory (EFT), as new states are crucial to enforce d > 4 gauge invariance in the dual holographic picture. Bearing scenarios like this in mind, there is motivation to isolate the sensitivity to the quartic couplings in collider processes. 1 Aiming to probe the VVHH couplings at hadron colliders, one usually thinks of exploiting processes with two final state Higgs bosons. This final state has been investigated in Refs. [19,20] (see also [21]), which have shown that focussing on the vector boson scattering (VBS) component of HH+2 jets production can in principle constrain the quartic gauge-Higgs coupling within ∼ 50% around the SM prediction. One of the shortcomings of such an analysis is that all VVHH couplings contribute coherently. Systematically distinguishing between the contributing couplings as would be required to phenomenologically reverse-engineer, e.g., the Veltman condition is not possible, in particular given the low statistical yield. In this paper, we focus on a so far unconsidered process, pp → W ± W ± H+2 jets, with W ± decaying to leptons, which is predominantly sensitive to the WWHH coupling exclusively as it does not involve the ZZHH coupling at leading order. This way, a successful analysis of this final state at present or future hadron colliders will not only provide additional information to a κ-framework analysis [37] (which we will limit ourselves to in this first study), but is also likely to provide complementary information for a more comprehensive SM-EFT analysis (in particular by accessing different kinematical regimes than final states with on-shell Higgs bosons [19,20]). Furthermore, it provides a relatively clean signal of two same-sign leptons and two VBS jets, analogous to the standard VBS paradigm [22]. However, due to the small production rates of this process at the current energy frontier of the LHC, as well as relatively large expected backgrounds, one must go beyond the current LHC scope to Higher Luminosity (HL) and increased collision energy. The so called HL-LHC is designed to reach the LHC design energy of 14 TeV and will include upgrades to the LHC accelerator and detector environments, allowing the machine to eventually take around 3000 fb −1 of data [24]. Another option, which has received considerable interest, is a 100 TeV pp-collider [25][26][27]. The large statistics that both options can accumulate will allow us to also access rare processes (including the one we are interested in) and set constraints on their potential deviations from the SM. In this work we provide a first detailed MC feasibility study of measuring VBS W ± W ± H production, and probing the quartic coupling of WWHH, at the HL-LHC and 100 TeV pp-collider. Our work takes into account the effects from parton showering, detector simulation, as well as pileup; we also comment on the sensitivity of this process to the trilinear Higgs coupling. The work is organised as follows: We describe the framework of our simulation studies in Secs. 2.1 and 3.1, and present the numerical results in Secs. 2.2 and 3.2, for the 14 TeV HL-LHC and 100 TeV pp-collider, respectively. We present our conclusions in Sec. 4. 2 VBS W ± W ± H production at the 14 TeV LHC Event Simulation and Selection The characteristic signal that we are interested in contains two well-identified leptons (electrons e, or muons µ) with same charge, in association with 2 VBS jets and 2 btagged jets. In Fig. 1, we show representative Feynman diagrams contributing to the VBS W ± W ± H + jj production at the LHC. We plot the VBS jets' invariant mass M jj and ❲ ✝ ❲ ✝ ❍ ❲ ✝ ❲ ✝ ❍ q q ✵ q q ✵ q q q ✵ q ✵ Figure 1. Representative Feynman diagrams for VBS same-sign W ± W ± H productions at the LHC, which involve the HHH and WWHH vertices. pseudo-rapidity separation |∆η jj | at parton level, in the SM and also the cases of varied g WWHH in Fig. 2. As expected [22], the VBS-type topology leads to a sizable rapidity gap between the forward tagging jets with all weak boson-associated decay products focussed in the central region of the detector. This can be used to suppress the expected backgrounds. As can be seen from Fig. 2, not only the total normalization of signal depends on the value of the quartic coupling, but also VBS W ± W ± H production tends to have harder M jj and, consequently, more separated |∆η jj | distributions. We follow the Snowmass Energy Frontier studies [28][29][30] for our signal and background simulations. We take existing samples directly from Snowmass [28][29][30], including tt, tt + B (B = γ, W, Z or H), B+jets and single top. These MC samples are generated with MadGraph/MadEvent [31], interfaced with Pythia 6 [32] for parton showering and hadronization, and Delphes version 3 [33] for detector simulation with the so-called 'Combined Snowmass Detector' configuration [28]. In Delphes, we consider no pileup (No-PU) and mean 50 pileup (PU50) scenarios at the 14 TeV LHC, and no pileup (No-PU) and 140 pileup (PU140) for the future 100 TeV pp-collider option, owing to the larger expected pileup contribution when moving from 14 to 100 TeV collisions. It is worth mentioning that the b-tagging efficiency is rather low in the Snowmass configuration, around 20%−30%, when the b jets' transverse momentum is around 30 GeV [28]. However, the b-tagging efficiency could reach 70% when the b jets' P T extends to 100 GeV. Consequently, to enlarge the signal selection efficiency, both 2 b-tagged and 1 b-tagged jet categories should be considered, see below. For the samples not included in the Snowmass studies, i.e. our signal VBS W ± W ± H+2 jets and background W ± W ± + QCD jets and VBS W ± W ± (or WZ) + jets, we produce them exactly following the description above. Finally, the analysis is based on the ExRoot-Analysis [34] and ROOT [35] packages. The Snowmass samples have associated NLO QCD weight factors at event level and thus are normalized beyond LO [28]. For tt process, we further apply a reweighting factor related to the most accurate prediction of Ref. [36]. The theoretical uncertainties at the 14 TeV LHC, are at around 5%, 15% and 5% level, for tt [36], ttH [37] and single top processes [38], respectively. We therefore assume a 20% overall uncertainty on the background yields to compare with the nominal results without such systematic included, as will be shown below. In our selection we require exactly 2 isolated leptons with identical charge, in addition to 2 VBS jets as well as 2 jets with a "b-tag" as defined below. We apply the following cuts: 1.) require exactly 2 leptons with same-sign charge with P T l ≥ 20 GeV, |η l | < 2.5 and R ll = ∆η 2 ll + ∆φ 2 ll > 0.4,2. ) require at least 4 jets. Among those we require that there is at least 1 b-tagged jet with P T b ≥ 25 GeV, |η b | < 2.5, and at least 2 non b-tagged jets, with P T j ≥ 25 GeV, |η j | < 4.7. a.) If there are 2 b-tagged jets, we choose 2 VBS jets as the leading 2 non-b jets, b.) if there is only 1 b-tagged jet, we loop over the leading 3 non b-tagged jets, select the 2 VBS jets on the basis of the largest invariant mass M jj , and then choose the remaining jet (with additional selection |η| < 2.5) to be combined with the b-tagged jet to reconstruct Higgs (we will label this with "b" in the following although there might not be a positive tag). 3 shows the M bb distributions of signal and background after cuts 1.)-6.) have been applied for the HL-LHC 14 TeV and 100 TeV pp collider including pileup. One can see that although the Higgs peak can be reconstructed around 120-125 GeV for the signal, it is considerably washed out due to pileup and mistag effects. Thus we decide to choose a wide mass window in 7.) as listed above. A cut flow for our analysis can be found in Tab. 1, which gives results for √ s = 14 TeV PU50. Each number represents the efficiency passing that single step's selection. One can clearly see the power of VBF selections which can suppress backgrounds by more than two orders of magnitudes than signal. Numerical Results at HL-LHC In Tab One can see that our signal is not sensitive to the rescalings of trilinear Higgs coupling λ HHH , while there is sensitivity to g WWHH . With an integrated luminosity of 3000 fb −1 at the 14 TeV LHC, we expect that the gauge-Higgs quartic coupling g WWHH can be constrained to be smaller than ∼ 2 -2.5 times of SM value at 95% confidence level (CL) κ WWHH = g WWHH g SM WWHH = 1 +1.2(1.4) −1 ,(2.1) without (with) pileup effects included. The significance distribution that underpins this result is shown in Fig. 4 (see also [39]), and calculated using σ = 2 ln(Q) , Q = (1 + N s /N b ) N obs exp(−N s ) ,(2.2) which corresponds to 2 ln[L(S + B)/L(B)]. L symbolizes the Poisson likelihood: N b is the total background yield including also the SM VBS W ± W ± H prediction, while N s is the signal yield defined as the excess of the signal with non-SM g WWHH over the SM one. σ is related to log likelihood ratio, and a value of 1.96 corresponds to the 95% confidence level exclusion limit in the case of only one degree of freedom. We have also compared scenarios with cuts 9.) (A) and 9.) (B), with and without pileup. For the No-PU case, the more stringent cut of 9.) (B) yields a better performance. However, pileup significantly impacts both options. As mentioned above, we include an additional sensitivity projection to Fig. 4 to be compared to 9.) (B), which includes the effect of 20% systematics on background yields (we follow the procedure as suggested in [40]). The sensitivities do change only slightly, as the results are dominated by statistical errors. 3 VBS W ± W ± H production at a at 100 TeV pp-collider Event Simulation and Selection For the 100 TeV analysis we largely follow the cut scenario described in the above Sec. 2.1. However, we include some modifications which optimize the cut flow for the more energetic final states compared to the HL-LHC: (1) the lepton requirement is tightened to P T l ≥ 50 GeV, and (2) selection cuts 8.) are changed to (A * ) |∆η jj | > 6, and M jj > 2 TeV , (B * ) |∆η jj | > 7, and M jj > 2 TeV . As for 14 TeV, we again include the impact of pileup to our discussion of results. As pileup will increase at 100 TeV compared to the 14 TeV collisions, we concentrate on the No-PU and PU140 scenarios. Numerical Results In Fig. 5 The projected sensitivity to g WWHH is shown in Fig. 6, from which one can see that g WWHH can now be further constrained to be κ WWHH = g WWHH g SM WWHH = 1 +0.2(0.4) −0.1(0.3) ,(3.1) without (with) pileup effects included, i.e. within ∼ 20−30% around SM prediction at 95% CL. This is a significant improvement over the LHC projection. We have also compared scenarios with cuts 8.) (A * ) and 8.) (B * ), and with or without pileup. For both these cases, the more stringent option 8.) (B * ) gives better performance than 8.) (A * ), owing to the high energetic final states that can be accessed at the 100 TeV machine. Summary and Conclusions VBS W ± W ± H+2 jet production is a so far unconsidered process with the potential to add sensitivity to the current Higgs characterization program. The same-sign leptonic Signif. g WWHH /g SM WWHH Selection A * without PU Selection A * with PU140 Selection B * without PU Selection B * with PU140 95% C.L. exclusion line Figure 6. Significance dependence on g WWHH , at the 100 TeV pp-collider with an integrated luminosity of 3000 fb −1 . Again the significance follows Eq. (2.2). final state is particularly clean on top of good additional background suppression handles motivated from VBS Higgs+2 jet production. Our results show that at the high luminosity LHC with a target of 3000 fb −1 we can expect a similar sensitivity to the quartic WWHH coupling as provided by VBS HH production, for which we expect κ V V HH ≃ 1.6 [20]. 2 Therefore, VBS W ± W ± H+2 jet can assist in disentangling the individual contributions of the quartic gauge-Higgs vertices. In the search region selected by a maximum background rejection, modifications of the trilinear Higgs coupling have no significant impact on the signal yield. Adapting our study to the 100 TeV pp-collider we find that the g WWHH coupling can be constrained significantly better within ∼ 20% around SM prediction at 95% CL for a comparable luminosity as the HL-LHC. Therefore, this process and its impact can be considered as another motivation to push the high energy frontier. [ 3.) We furthermore impose R bb,bj,bl > 0.4, and R jj,jl > 0.4, 4.) and require a significant amount of missing energy / E T > 30 GeV,5.) require |M ll − M Z | > 15GeV for same flavor lepton category, to suppress Drell-Yan backgrounds, 6.) require M ll > 50 GeV to suppress soft lepton contributions from heavy flavor decays in W+jets and top-quark backgrounds, 7.) and impose compatibility with the Higgs mass |M bb − M H | < 20 GeV, Figure 3 . 3M bb distributions of signal and background for the HL-LHC at 14 TeV with 50 pileup scenario and 100 TeV with 140 pileup scenario. Figure 4 . 4Dependence of the significance of Eq. (2.2) on g WWHH , at the 14 TeV HL-LHC with an integrated luminosity of 3000 fb −1 . Figure 5 . 5M jj and ∆η jj distributions for W ± W ± H productions at the 100 TeV pp-collider, in the SM and for varied g WWHH cases. Figure 2. M jj and |∆η jj | distributions for W ± W ± H productions at the 14 TeV LHC, in the SM or varied g WWHH cases, at parton level, with default parton level setting.GeV] jj M 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 [pb/bin] jj /dM σ d -6 10 -5 10 -4 10 SM WWHH 3*g SM WWHH 2*g SM WWHH g H + 2jets ± W ± 14TeV LHC, VBF W | jj η ∆ | 0 2 4 6 8 10 12 14 [pb/bin] jj η ∆ /d σ d -6 10 -5 10 -4 10 SM WWHH 3*g SM WWHH 2*g SM WWHH g H + 2jets ± W ± 14TeV LHC, VBF W We then focus on two different final cut scenarios for comparison, to further enhance the VBS signal: (A) |∆η jj | > 5 and M jj > 1.5 TeV, or, (B) |∆η jj | > 6 and M jj > 2 TeV.Cut Flow Table 1.) 2.) 3.) 4.) 5.) 6.) 7.) 8. A) 8. B) tt 0.02% 26.1% 99.9% 81.1% 92.8% 65.5% 19.5% 0.01% 0.01% tt + B 0.49% 48% 99.9% 91.8% 90.3% 87.5% 22.1% 0.3% 0.02% Single Top 0.01% 12.4% 99.9% 88.8% 87.3% 81.4% 23.5% 0.8% 0.48% B/BB+ jets 0.03% 0.9% 100% 86.4% 91.3% 88.8% 16.9% 0.03% 0.01% Signal 2.83% 25.2% 100% 87.4% 92% 89.6% 39.8% 34.1% 17.1% Signal (2 × g WWHH ) 4.11% 20% 100% 92.7% 96.7% 97.3% 46.5% 40.3% 25.4% Signal (3 × g WWHH ) 4.38% 23.4% 100% 98% 99% 98% 40.5% 32.9% 20.3% Signal (5 × λ HHH ) 2.91% 24.6% 100% 93.7% 91% 90.1% 34.5% 34.2% 23.7% Table 1. Cut chain table for backgrounds and signals at the LHC with √ s = 14 TeV in the 50 pileup scenario. 8.) Fig. . 2, we show the signal and background yields at the HL-LHC with √ s = 14 TeV and integrated luminosity of 3000 fb −1 , after the selection cuts as listed in Sec. 2.1. Numbers are provided for both no pileup and mean 50 pileup scenarios. The largest background contribution results from tt + B. The remaining contributions are all found to be small. This applies to the Snowmass B+jets, our produced W ± W ± + QCD jets, and the VBS W ± W ± (or WZ) + jets contributions.Table 2. Yields for backgrounds and signals at the LHC with √ s = 14 TeV and integrated luminosity of 3000 fb −1 .Processes ∆η jj > 5 and M jj > 1.5 TeV ∆η jj > 6 and M jj > 2 TeV (A) No-PU (B) PU50 (A) No-PU (B) PU50 tt 0.0 0.86 0.0 0.86 tt + B 10.38 13.5 2.79 1.13 Single Top 0.156 7.2 0.06 4.4 B/BB+ jets 0.89 0.07 0.0 0.03 total bkg 11.43 21.6 2.85 6.42 Signal 0.52 0.73 0.1 0.37 Signal (2 × g WWHH ) 5.61 6.89 3.8 4.3 Signal (3 × g WWHH ) 22.04 22.03 11.87 13.56 Signal (5 × λ HHH ) 1.1 0.8 0.5 0.8 Processes ∆η jj > 6 and M jj > 2 TeV ∆η jj > 7 and M jj > 2 T eVTable 3. Signal and background yields at the 100 TeV pp-collider with an integrated luminosity of 3000 fb −1 ., we show the VBS jets' invariant mass M jj and their pseudo-rapidity gap |∆η jj | at parton level for three different g WWHH values. One can see that both distributions are shifted to higher values compared with the HL-LHC case. Tab. 3 provides signal and background yields after the full selection at a 100 TeV pp-collider for an integrated luminosity of 3000 fb −1 . Note that for the 100 TeV center-of-mass energy, tt becomes the most dominating background, as non-prompt leptons from hadron decays can be energetic now and pass respective selections. (A * ) No-PU (B * ) PU140 (A * ) No-PU (B * ) PU140 tt 121 17240 0 975 tt + B 618 1432 207 314 Single Top 113 6157 0 270 B/BB+jets 0.96 239 0 0 total bkg 853 25068 207 1559 Signal 15.2 13.4 8.94 8.05 Signal (2 × g WWHH ) 927 948 625 689 Signal (3 × g WWHH ) 3457 3553 2785 2881 Signal (5 × λ HHH ) 0.75 0.69 0.27 0.48 [GeV] jj M 0 5000 10000 15000 20000 25000 30000 [pb/bin] jj /dM σ d -4 10 -3 10 -2 10 -1 10 SM WWHH 3*g SM WWHH 2*g SM WWHH g H + 2jets ± W ± 100TeV LHC, VBF W | jj η ∆ | 0 2 4 6 8 10 12 14 [pb/bin] jj η ∆ /d σ d -4 10 -3 10 -2 10 -1 10 SM WWHH 3*g SM WWHH 2*g SM WWHH g H + 2jets ± W ± 100TeV LHC, VBF W Integrating out extra states leads to a plethora of modified couplings that can be investigated in a global non-linear SM EFT fits[14][15][16][17][18]; the focus of our work is to discuss the sensitivity of a particular process that could be exploited in this direction as well. 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J Anderson, A Avetisyan, R Brock, S Chekanov, T Cohen, N Dhingra, J Dolen, J Hirschauer, arXiv:1309.1057hep-exJ. Anderson, A. Avetisyan, R. Brock, S. Chekanov, T. Cohen, N. Dhingra, J. Dolen and J. Hirschauer et al., arXiv:1309.1057 [hep-ex]. . A Avetisyan, arXiv:1308.1636hep-exA. Avetisyan et al., arXiv:1308.1636 [hep-ex]. . J Alwall, M Herquet, F Maltoni, O Mattelaer, T Stelzer, arXiv:1106.0522JHEP. 1106128hep-phJ. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP 1106 (2011) 128 [arXiv:1106.0522 [hep-ph]]. . T Sjostrand, L Lonnblad, S Mrenna, P Z Skands, hep-ph/0308153T. Sjostrand, L. Lonnblad, S. Mrenna and P. Z. Skands, hep-ph/0308153. . J De Favereau, DELPHES 3 CollaborationarXiv:1307.6346JHEP. 140257hep-exJ. de Favereau et al. [DELPHES 3 Collaboration], JHEP 1402, 057 (2014) [arXiv:1307.6346 [hep-ex]]. . R Brun, F Rademakers, Nucl. Instrum. Meth. A. 38981R. Brun and F. Rademakers, Nucl. Instrum. Meth. A 389 (1997) 81. Top++: a program for the calculation of the top-pair cross-section at hadron colliders. M Czakon, A Mitov, Comput.Phys.Commun. 1852930M. Czakon and A. Mitov, Top++: a program for the calculation of the top-pair cross-section at hadron colliders, Comput.Phys.Commun. 185 (2014) 2930. . S Heinemeyer, 10.5170/CERN-2013-004arXiv:1307.1347LHC Higgs Cross Section Working Group. hep-phS. Heinemeyer et al. [LHC Higgs Cross Section Working Group], doi:10.5170/CERN-2013-004 arXiv:1307.1347 [hep-ph]. . N Kidonakis, arXiv:1609.07404hep-phN. Kidonakis, arXiv:1609.07404 [hep-ph]. . G Aad, The ATLAS CollaborationarXiv:0901.0512hep-exG. Aad et al. [The ATLAS Collaboration], arXiv:0901.0512 [hep-ex]. . Robert D Cousins, James T Linnemann, Jordan Tucker, arXiv:physics/0702156Nuclear Instruments and Methods in Physics Research A. 595480Robert D. Cousins, James T. Linnemann and Jordan Tucker, Nuclear Instruments and Methods in Physics Research A 595 (2008) 480, arXiv:physics/0702156.

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{'abstract': 'W ± W ± H production at hadron colliders through vector boson scattering is a so far unconsidered process, which leads to a clean signature of two same-sign charged leptons and two widely separated jets. This process is sensitive to the HHH and WWHH couplings and any deviation of these couplings from their SM predictions serves as direct evidence of new physics beyond the SM. In this paper we perform a Monte Carlo study of this process for the √ s = 14 TeV LHC and a 100 TeV pp-collider, and provide projections of the constraints on the triple-Higgs and WWHH quartic couplings for these environments.In particular, we consider the impact of pileup on the expected sensitivity in this channel. Our analysis demonstrates that although the sensitivity to the HHH coupling is rather low, the WWHH coupling can be constrained in this channel within ∼ 100% and ∼ 20% at 95% confidence level around the SM prediction at the HL-LHC and a 100 TeV pp-collider, respectively.', 'arxivid': '1702.01930', 'author': ['Christoph Englert christoph.englert@glasgow.ac.uk \nSUPA\nSchool of Physics and Astronomy\nUniversity of Glasgow\nG12 8QQGlasgowUK\n', 'Qiang Li \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n\nCAS Center for Excellence in Particle Physics\n100049BeijingChina\n', 'Michael Spannowsky michael.spannowsky@durham.ac.uk \nInstitute for Particle Physics Phenomenlogy\nDepartment of Physics\nDurham University\nDH1 3LEUK\n', 'Mengmeng Wang mengmeng.wang@cern.ch \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n', 'Lei Wang \nDepartment of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n'], 'authoraffiliation': ['SUPA\nSchool of Physics and Astronomy\nUniversity of Glasgow\nG12 8QQGlasgowUK', 'Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina', 'CAS Center for Excellence in Particle Physics\n100049BeijingChina', 'Institute for Particle Physics Phenomenlogy\nDepartment of Physics\nDurham University\nDH1 3LEUK', 'Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina', 'Department of Physics and State Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina'], 'corpusid': 119537021, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11993, 'n_tokens_neox': 9407, 'n_words': 4650, 'pdfsha': '304866e7324c8ef4ffd217089e4ed0da1a61e020', 'pdfurls': ['https://arxiv.org/pdf/1702.01930v1.pdf'], 'title': ['VBS W ± W ± H production at the HL-LHC and a 100 TeV pp-collider', 'VBS W ± W ± H production at the HL-LHC and a 100 TeV pp-collider'], 'venue': []}

arxiv

Lattice dynamics and thermodynamics of bcc iron at pressure: first- principles linear response study Xianwei Sha Carnegie Institution of Washington 5251 Broad Branch Road20015WashingtonNW, DC R E Cohen Carnegie Institution of Washington 5251 Broad Branch Road20015WashingtonNW, DC Lattice dynamics and thermodynamics of bcc iron at pressure: first- principles linear response study We compute the lattice-dynamical and thermal equation of state properties of ferromagnetic bcc iron using the first principles linear response linear-muffin-tin-orbital method in the generalizedgradient approximation. The calculated phonon dispersion and phonon density of states, both at ambient and high pressures, show good agreement with inelastic neutron scattering data. We find the free energy as a function of volume and temperature, including both electronic excitations and phonon contributions, and we have derived various thermodynamic properties at high pressure and temperature. The thermal equation of state at ambient temperature agrees well with diamond-anvil-cell measurements. We have performed detailed investigations on the behavior of various thermal equation of state parameters, such as the bulk modulus K, the thermal expansivity α, the Anderson-Grüneisen parameter δ T , the Grüneisen ratio γ, and the heat capacity C V as function of temperature and pressure. A detailed comparison has been made with available experimental measurements, as well as results from similar theoretical studies on nonmagnetic bcc Tantalum.PACS number(s): 05.70. Ce, 64.30.+t, 71.20.Be 1 Introduction During the past decade, tremendous experimental 1-7 and theoretical [8][9][10][11][12][13][14] efforts have been devoted to investigate various properties of iron, especially for those at high pressure and temperature conditions. Body-center-cubic (bcc) is the ground state structure for iron at ambient conditions. It transforms to face-center-cubic (fcc) structure at elevated temperature (~1150K at ambient pressure), and to hexagonal-close-packed (hcp) structure at increased pressure (~11GPa at room temperature) 15,16 . Iron also shows very interesting and complex magnetic behavior under different temperature and pressure conditions. It has become a fundamental problem in material science to understand the mechanism of these solid-state phase transitions, the magnetism, and the phase diagram of Fe. The study of iron is also of great geophysical interest, because the Earth's core consists mainly of this element. Various lattice dynamical and thermodynamic properties as a function of temperature and pressure may provide important information to understand the phase transitions, phase diagram and dynamic response of materials. Several first-principle calculations 13, 14,17 have been performed to study the lattice dynamical properties of bcc iron, generally giving good agreement with experiment 18 . Here we concentrate on the thermal equation of state of ferromagnetic bcc iron, using quasiharmonic lattice dynamics with an all-electron method that does not depend on pseudopotentials. In section II we detailed our methods to perform first principles calculations, as well as the theoretical techniques to obtain thermal properties. We present our results and related discussions about lattice dynamics in section III, and about the thermal equation of state properties in section IV. We conclude with a summary in Section V. Theoretical methods For many metals and alloys, the Helmholtz free energy F of a solid has three major contributions 19 : F(V,T)=E static (V)+F el (V,T)+F vib (V,T)(1) With V as the volume, and T as the temperature. E static (V) is the energy of a static lattice at absolute zero temperature, F el (v,T) is the electronic thermal free energy arising from electronic excitations, and F vib (V,T) is the vibrational energy contribution. We assume that the existence of lattice vibrations does not significantly affect the electronic contribution for the thermal properties, and all the terms here are calculated for the ideal lattice. E static (V) and F el (v,T) can be obtained from first-principles calculations directly. There are several ways to examine the lattice vibrational contribution, including the linear response (LR) lattice dynamics, particle-in-cell (PIC) model and molecular dynamics. First-principles LR calculations can give important lattice dynamics information, but it is computationally expensive. Additionally, it usually requires use of the quasi-harmonic approximation, and anharmonic effects are usually neglected. The PIC model is a mean field approximation to the thermal contribution 20,21 . It includes on-site anharmonicity and requires less computer power when combined with first-principles totalenergy calculations. However, it neglects the interatomic correlations and diffusion. The accuracy and reliability of molecular dynamics methods strongly depends on the quality and versatility of the interatomic potential. It is computational expensive to study the thermal equation of state properties rigorously from first-principles molecular dynamics calculations directly, and one obtains only the classical contributions, so that properties at room temperature and below are not reliable. In the present paper, we use first principles full potential Linear- In the interstitial region the basis functions are expanded in plane waves with the energy up to the cutoff corresponding to 78, 140, and 224 plane waves per s, p and d orbital, respectively. The induced charge densities, the screened potentials and the envelope functions are represented by spherical harmonics up to l max =6 within the MT spheres and by plane waves in the interstitial region with cutoff corresponding to the 16×16×16 fast-Fourier-transform grid in the unit cell of direct space for the bcc structure. The k-space integration needed for constructing the induced charge density is performed over the 16×16×16 grid (145 k points in the irreducible wedge of the Brillouin zone (BZ)). The improved tetrahedron method was used for the k-point sampling 24 . An earlier first-principle linear response study showed that phonon dispersion in Fe at ambient conditions can be well reproduced by the combined use of the generalized gradient approximations (GGA), spin polarization and ultrasoft pseudopotentials, and the local spin density approximation (LSDA) phonon frequencies are systematically higher than experimental data 17 . In the present work, we use the Perdew-Burke-Ernzerhof (PBE) GGA 25 for the exchange and correlation energy. B. Electronic contribution The electronic free energy can be written as (2) ) , ( ) , ( ) , ( T V TS T V E T V F el el el − = The electronic entropy is given: (3) ∑ − − + − = i i i i i B el f f f f k T V S ) 1 ln( ) 1 ( ln 2 ) , ( Where (4) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = T k T f B i i )] ( [ exp 1 1 µ ε is the Fermi-Dirac occupation number. The parameter µ is the chemical potential, k B is the Boltzmann constant, and ε i are the eigenvalues. The sum is over all energy levels, with both the occupied and unoccupied states (up to 1Ry above the static Fermi energy) included. The chemical potential is determined from the particle conservation equation (5) where N el is the total number of the electrons in the systems. el i i N f = ∑ The current method to calculate the electronic free energy F el and electronic entropy S el assumes that the eigenvalues are temperature-independent for given lattice and nuclear positions; only the occupation numbers change with the temperature through the Fermi-Dirac distribution. Wasserman et. al. found that the electronic entropies of hcp Fe calculated from tight-binding total energy calculations based on static eigenvalues agree within 1% with the values from self-consistent high temperature Linear Augmented Plane Wave (LAPW) method over the whole temperature (6000-9000K) and volume (40-90 bohr 3 /atom) ranges 8 , which clearly indicates that the eigenvalue approximation is well justified for transition metals. C. Vibrational contribution The linear response method based on density functional theory (DFT) and the density functional perturbation theory (DFPT) has already been successfully applied to calculate the lattice vibrational contribution to the free energy and other thermodynamic properties of many materials [26][27][28][29] . In the framework of the linear response LMTO method, the dynamical matrix and the phonon linewidths are determined as a function of wave vector for a set of irreducible q points at the 8×8×8 reciprocal lattice grid (29 irreducible q points) for the bcc structure. The perturbative approach is employed for calculating the self-consistent change in the potential 28,29 . Careful tests have been done against k and q point grids and many other parameters to make sure all the results are well converged. The self-consistent calculation is terminated when the total energy change is less than 10 -9 Ry in the full potential LMTO, and when the change in the charge density is less than 10 -7 in the linear response. The vibrational contribution to the free energy is calculated by combination of the linear response and the quasi-harmonic approximations. Once the phonon dispersion relation and/or phonon density of state is obtained from the linear response lattice dynamics calculations, the phonon internal energy (u ph ) and phonon free energy (f ph ) are computed from 32 : ] 1 1 )[ , ,( ) , ( , 2 1 −+ = ∑ T k i q i ph B i e T V q T V u ω ω h h (6) ∑ = i q B i B ph T k T V q T k T V f , ] 2 ) , , ( sinh( 2 ln[ ) , ( ω h(7) Within the quasiharmonic approximation for insulators, at a given q vector, the frequency ω is solely a function of volume and temperature independent. Although ω does depend on temperature for transition metals because of the electron-phonon coupling, the normal quasiharmonic treatment will not cause any serious problems because the thermal electronic excitation is usually small 26,27 . In the present paper, we assume that the phonon frequencies ω at a fixed volume are independent of the temperature. Lattice dynamics We full potential LMTO calculations are in excellent agreement at ambient conditions. Both also agree well with the experiment. On the other hand, the symmetrical discrepancies between the LDA pseudopotential calculations and the experiment suggests that the LDA approximation not only fails to give the correct ground state structure for Fe 33 , but also fails to accurately describe the phonons. In figure 2 40 . Such dynamical precursor effects of the lattice instability have also been found in group-IV transition metals such as Ti, Zr and Hf [41][42][43] . However, the bcc-hcp transition in Fe seems to be quite different. Klotz and Barden measured the phonon dispersion at 0 and 9.8 GPa by inelastic neutron scattering, and demonstrated that such effects are definitely absent 18 . As shown in Fig.1 and 2 Where F 0 (T) and V 0 (T) are the zero pressure equilibrium energy and volume, x = (V/V 0 ) 1/3 , K 0 (T) is the bulk modulus, ) 1 ( 2 3 ' 0 − = K ξ and 0 ' 0 ] ) ( [ P T K K ∂ ∂ = . The subscript 0 throughout represents the standard state P=0 GPa. Pressure can be obtained analytically as: In Fig. 4 )}}] 1 ( exp{ } 1 ) 1 ( { 1 { ) ( ) ( 9 ) ( ) , ( 2 0 0 0 x x T V T K T F T V F − − − + + = ξ ξ ξ )} 1 ( exp{ } ) 1 )( ( 3 { ) , ( 2 0 x x x T K T V P − − = ξ phase at T≈1150K, and to the hcp phase at G≈11GPa. There are several reasons to include results beyond this thermodynamically stable field. Bcc phase is metastable in these regions, which might be approached in some shock experiment. The bcc phase becomes stable just before melting at zero pressure, and could become entropically stabilized again at very high temperatures and pressures, although this seems not to happen in pure iron 13 The thermal pressure can be obtained from the pressure difference between isotherms. The thermal pressures as function of volume and temperature are shown in Fig. 6. At low temperature, the thermal pressures are small, and show little volume-dependence. At elevated temperature, the magnitude of thermal pressure increase significantly, and their values first show a decrease with increase of pressure, and then show a strong increase for volumes smaller than 70 bohr 3 /atom. This is different from bcc Ta, where the volume dependence of the thermal pressure is weak up to 80% compression for a large temperature range (947-9947K) 44 . The different thermal pressure behavior of bcc Fe might be partly due to the pressure dependence of the magnetic moment. At a given volume, the thermal pressure shows a linear increase with temperature. The pressure change at a given volume is: dT K T V P T V P T T T ∫ = − 0 ) , ( ) , ( 0 α(10) where α is the thermal expansion coefficient and K T is the bulk modulus. For many materials, αK T is constant in the classical regime. For bcc Ta, for almost all the volumes, the thermal pressure has a slope of ~0.00442GPa/K 44 . However, the slopes for bcc Fe show a strong volume dependence, which might be attributed to its different magnetic moments with volumes 33 . At low temperature, the Helmholtz free energy in the Debye approximation is 19 , )] ( ) 1 ln( 3 ) ( 8 9 [ / T D e T RT E F D T D static D θ θ θ − − + + = − (11) Debye function D(θ D /T) is ∫ − = T z D D D e dz z T T D / 0 3 3 1 ) ( 3 ) ( θ θ θ(12) In the classical regime, for temperatures above Debye temperature θ D , an accurate ∑ = = = = − = 3 , 3 0 , 1 ln 3 j i j i B j i ij th T T k V T A F (13) The term TlnT is necessary to give the proper classical behavior at low temperature. A global fit to the calculated thermal Helmholtz free energies can be performed to determine the parameters A ij . The above thermal free energy functional gives a good description of various thermal equation of state parameters for bcc Ta at high temperatures 44 . Debye temperatures θ D (T) at 0K can be calculated by numerical integration of phonon density of state, and can be obtained according to equation (11) for other temperatures. In Fig. 7 we show the calculated θ D (T) as a function of temperature at several different volumes. With increase of pressure (decrease of volume), θ D (T) shows a strong increase. At low temperature, θ D (T) drops with increase of T. However, with further increase (T > 250K), θ D (T) shows almost no temperature dependence. All these are consistent with recent neutron scattering experiment 18 . The calculated θ D (T), at both ambient pressure (V=79.6 bohr 3 /atom) and 9.8 GPa (V=75 bohr 3 /atom), usually agrees within 10-15K (~2-3%) with the measured data. The calculated and fitted thermal free energies as function of temperature and volume are compared in Fig. 8. At both low (a) and high (b) temperature regimes, the fit gives good agreement with the calculated data, with rms deviations of ~0.2 mRy. For the high temperature fitting, the residuals (Fig. 8(c)) are small over the whole temperature range for all the volumes studied, with values less than 0.6 mRy. This is different from bcc Ta, where the residuals are much larger due to the electronic topological transition 44 . The various thermal equation of state properties can be derived analytically from the Helmholtz free energy. The thermal expansion coefficient α is: T P V F V T F V T V V ) /( ) ( 1 ) ( 1 2 2 2 ∂ ∂ ∂ ∂ ∂ − = ∂ ∂ = α (14) At low temperatures, the Debye expression for α is 19 : ] 1 ) / ( 3 ) ( 4 [ 3 / − − = T T D e T T D V K R θ θ θ γ α (15) The thermal expansion coefficient of bcc Fe agrees well with experiment at low temperatures [ Fig. 9(a)]. At high temperature, α shows a linear increase under several different pressures [ Fig. 9(b)], as predicted by the quasiharmonic approximation at the high T limit 19 . Thermal expansivity is a very sensitive parameter, and the discrepancy between the calculated α at ambient pressure and high temperature with the experiment 49,50 might be attributed to several factors: the errors in the first-principles calculations, anharmonic effects, and most likely magnetic fluctuations. It should be noted that the calculated α shows good agreement with experiment even at high temperature when applying similar theoretical approaches to nonmagnetic bcc vanadium. The calculated thermal expansion coefficient shows a rapid drop with increasing pressure [ Fig. 9(c)]. The relationship between α and pressure are characterized by the Anderson-Grüneisen parameter δ T 19 : P T T T T T K K V ) ( 1 ) ln ln ( ∂ ∂ − = ∂ ∂ = α α δ(16) The calculated δ T of bcc Fe shows quite complex behavior as a function of pressure and temperature [ Fig. 10]. At a given pressure, δ T first decreases with temperature, and then shows a slight increase, similar to the behavior in bcc Ta 44 . At all temperatures, δ T shows a strong decrease with pressure. For many materials, the parameter δ T can be fitted to a form as a function of volume 51 : κ η η δ δ × = = ) 1 ( T T(17) where η=V/V 0 (T 0 ). For bcc Ta, the average δ T shows δ T (η)=4.56×η 1.29 for temperature 0-6000K 44 , and δ T (η)=4.56×η 1.29 has been reported for MgO at 1000K 51 . However, bcc Fe shows different behavior. Although the parameter δ T shows a strong decrease with compression at all temperatures, it does not drop that rapidly as power order when the pressure is high [ Fig. 10(c)]. Similar behavior has also been reported for fcc and hcp Fe 8 . The Grüneisen ratio γ is an important thermodynamic parameter used to quantify the relationship between the thermal and elastic properties of a solid, particularly for understanding shock dynamics 19 . V V T V T U T V F V C V K U P V ) ( ) ( 2 ∂ ∂ ∂ ∂ ∂ = = ∂ ∂ = α γ(18) where U is the internal energy. As shown in Fig. 11, at a given pressure, γ of bcc Fe first shows an increase with increasing temperature, and then a rapid decrease when T > ~ 1000K. On the other hand, the variation of γ with pressure is moderate. This is significantly different from bcc Ta, where the temperature dependence is moderate, but the pressure dependence is not. The calculated Grüneisen ratio at 500K shows good agreement with experiment using the adiabatic decompression method 52 . The volume dependence of the Grüneisen ratio is given by the parameter q: V q ln ln ∂ ∂ = γ (19) The parameter q is often assumed to be a constant, for example, 0.6 for bcc Fe 52 , and 0.7-1.62 for hcp Fe depending on the pressure range and measuring methods 53 . However, our calculations show that q is both temperature and pressure dependent [ Fig. 12]. The parameter q decreases significantly with pressure, but its temperature behavior is quite complex. At ambient pressure, q shows a slight increase with temperature. However, at high pressure, q first shows a strong decrease with increasing temperature, and then a slight increase when T > ~ 1500K. Similar complex behavior the parameter q has also been reported for bcc Ta 44 . At low temperatures, the Debye heat capacity at constant volume C V is: (20) and is in good agreement with experiment [ Fig. 13(a)]. At high temperatures [ Fig. 13 (b) and (c)], C V is pressure independent at a given temperature. When the temperature is less than ~1200K, C V only shows slight increase with temperature. At higher temperature, the increase of C V becomes more noticeable. This increase comes mainly from the electronic excitation contribution. ] 1 ) / ( 3 ) ( 4 [ 3 / − − = T V e T T D R C θ θ θ Conclusions In summary, we have performed detailed first principles linear response calculations to study the calculate lattice dynamical properties of ferromagnetic bcc Fe at 6 different volumes, 65, 75, 75, 79.6, 85 and 90 bohr 3 /atom. For bcc Fe at volumes significantly beyond the above regimes, some phonon branches will soften and become unstable, consistent with earlier calculatations 13 . In Fig. 1 we show our calculated phonon dispersion curves (solid lines) of ferromagnetic bcc iron at the experimental equilibrium volume (79.6 bohr 3 /atom) at ambient conditions. For both the transverse (TA) and longitudinal (LA) acoustical branches along several high-symmetry directions in the Brillouin zone, the theoretically determined phonon frequencies agree well with experimental data from recent inelastic neutron scattering measurements at 300K (dots) 18 . There are no phonon anomalies. Dal Corso and Gironcoli performed first principles linear response calculations on magnetic bcc Fe using an ultrasoft pseudopotential; their computed phonon dispersion curves 17 using PBE GGA and LDA approximations are shown as the dashed and dotted lines in Fig. 1. Using linear response and the same PBE GGA approximations for exchange and correlation functional, their ultrasoft pseudopotential and our . The computed lattice dynamical and thermal equation of state of bcc phases in these regions may provide crucial information in understanding the phase diagram at extreme conditions. We show the Vinet equation of state fitting parameters, F 0 (T), V 0 (T), K 0 (T), and K 0 '(T) as functions of temperature in Fig.5 and Table 1. Although both LAPW 33 and LMTO calculations underestimate the experimental equilibrium volume 48 , GGA shows a dramatic improvement over the LSDA results. lattice dynamics and thermal equation of state properties of ferromagnetic bcc Fe. The calculated phonon dispersion and phonon density of state, both at ambient and high pressures, agree well with inelastic neutron scattering experiment. No dynamic precursor effects of lattice instability exist for the bcc-hcp phase transition. The calculated free energies have been treated by three different forms: Vinet equation of state, simple linear thermal pressure and a global fit to the thermal Helmholtz free energy. The calculated thermal equation of state agrees well with experiment. The thermal expansion coefficient agrees well with the experiment at low temperature. The difference of high temperature might be attributed to the influence of magnetic fluctuations. The calculated Grüneisen ratio and heat capacity C V show little pressure dependence. Thermal electronic excitations contribute significantly to the temperature dependence of C V at high temperature. P. Vinet, J. H. Rose, J. Ferrante, and J. R. Smith, J. Phys.-Condes. Matter 1, 1941 (1989). 48 A. P. Jephcoat, H. K. Mao, and P. M. Bell, J. Geophys. Res.-Solid Earth 91, 4677 (1986). 49 D. E. Gray, American Institute of Physics handbook (McGraw-Hill, New York, 1982). Figure 1 .Figure 2 . 12Calculated phonon dispersion curves (solid lines) for ferromagnetic bcc Fe at 79.6 bohr 3 /atom, in comparison to neutron inelastic scattering measurements (dots, Ref.18 ) and first principles linear response calculations using ultrasoft pseudopotential with GGA (dashed lines) and LDA (dot lines) approximations computed at experimental lattice constant (Ref. 17). Calculated phonon dispersion curves for ferromagnetic bcc Fe at two compressed volumes, V=75 bohr 3 /atom (solid lines) and V=70 bohr 3 /atom (dotted lines), in comparison to the neutron inelastic scattering data at 9.8 GPa (dots, Ref. 18). Figure 3 .Figure 4 .Figure 5 . 345Calculated phonon density of states (dos, lines) of ferromagnetic bcc Fe at five different volumes, 70, 75, 79.6, 85 and 90 bohr 3 /atom, in comparison to the neutron inelastic scattering data at 0 and 9.8 GPa (dots, Ref. 18). Calculated thermal equation of state (lines) at temperatures between 250K and 2250K. Experimental equation of state, measured in a diamond anvil cell at the room temperature, is shown as dots (Ref. 48). Fitted Vinet equation of state parameters, F 0 (T) Figure 6 .Figure 7 . 67Thermal pressure of bcc Fe as function of volume (a) and temperature (Calculated Debye temperature Θ D (T) as a function of temperature at several volumes. Θ D (T) shows a rapid increase with pressure (decrease of volume). With increase of temperature, Θ D (T) drops rapidly at low temperature, but shows little temperature dependence when T>250K. Neutron scattering experimental Θ D (T) at 0 and 9.8 GPa are shown as filled circles and open diamonds, respectively (from Ref. 18). Figure 8 Figure 9 89Global fit of thermal free energies to the calculated first principles data at several different volumes and temperatures. At both low (a) and high (b) temperature regimes, the fitted (lines) thermal energies agrees well with the computed data (symbols); and the residuals at high temperatures(c) are very small. The thermal expansion coefficient of bcc Fe as function of temperature (a and b) and pressure(c). Experimental data (dots) are from Ref. 49 and 50. Figure 10 TheFigure 11 The 1011Anderson-Grüneisen parameter δ T as a function of (a) T, (b) P (d) η= V/Grüneisen ratio γ as a function of (a) temperature and (b) pressure.Experimental data (dots) are from Ref.53. Figure 12 12Varation of the parameter q as a function of (a) temperature and (b) pressure. Figure 13 13Heat capacity at constant volume . Calculated C V agrees with experiment (dots, Ref 49) at low temperature (a). At high temperature, C v increases significantly due electronic thermal excitation(b), but shows little pressure dependence (c). Muffin-Tin-Orbital (LMTO) calculations to evaluate E static (V) and F el (v,T), and linear response LMTO calculations for the lattice vibrational properties.Since bcc Fe is ferromagnetic, magnetic fluctuations at high temperatures contribute to the Helmholtz free energies. We are currently performing Monte Carlo simulations based on an effective Hamiltonian to examine the contribution from magnetic fluctuations in bcc Fe, and the results will be reported in the future. and the remaining interstitial region. Non-overlapping muffin-tin spheres with radii of 2.18 bohr have been used in all the calculations. The self-consistent calculations are performed using 3κspd-LMTO basis set with one-center expansions performed inside the MT spheres up to l max =6.A. Static lattice energy We use multi-κ basis sets and two energy panels in the LMTO method 22, 23 . Space is divided into the non-overlapping Muffin-Tin (MT) spheres surrounding each individual atom we show the calculated phonon dispersion curves at two compressed volumes: V= 75 bohr 3 /atom (solid lines) and V=70 bohr 3 /atom (dotted lines). The calculated phonon frequencies show a significant increase of 6%-15% when the volume decreases from 79.6 to 75.0 bohr 3 /atom. This is consistent with the inelastic neutron scattering experiment, where all measured phonon frequencies increase 5%-10% for a volume reduction corresponding to 5%18 . With further volume compression from 75 to 70 bohr 3 /atom, most phonon frequencies show another 6%-14% increase, except for q vectors close to the H point in the BZ boundary, where the increase is much less significant, with only a 1% increase at H point. With increase of pressure, the phonon dispersion shows more complex and anomalous behavior near the H point. At V=75 bohr 3 /atom, a noticeable dip of the longitudinal mode near [3/4,0,0] appears. The dip becomes more significant at V=70 bohr 3 /atom, with an additional pronounced dip of both the Recent developments in high pressure and inelastic neutron scattering techniques by Klotz et. al. have made it possible to measure the lattice vibrational properties under high pressure accurately 34, 35 , and their measured phonon dispersion data for bcc Fe at 9.8 GPa 18 are shown as dots in Fig. 2. According to the thermal equation of state (see the following section), the equilibrium volume of bcc Fe at 9.8GPa is close to 75 bohr 3 /atom, thus we can compare our calculated phonon dispersions at 75 bohr 3 /atom with the experiment directly. As seen in Fig. 2, the experiment and the calculations agree well at high pressure, often within a few percent over the whole BZ. Bcc iron transforms under pressure to the hcp phase at around 10~16GPa 36, 37 . Lattice dynamics of bcc Fe under pressure can provide important information in understanding the transition mechanism. According to the well-known Burger mechanism, the bcc to hcp structural transformation can be achieved by two simultaneous distortions: (1) opposite displacements of the bcc (110) planes in the [110] directions, which corresponds to the TA1(N) phonon mode at the bcc BZ boundary 38, 39 ; (2) shear deformation in the [001] direction while keeping the volume and the bcc (110) planes unchanged. If the phase transition is dominated by the Burger mechanism, the frequencies of T1(N) phonon mode should show anomalous behavior close to the transition. For the bcc-hcp transition of Ba, first-principles calculations clearly indicate alongitudinal and transverse branches for q vector close to [3/4,3/4,3/4]. substantial softening of this T1 N-point phonon mode when approaching the transition pressure (~4GPa) we show the calculated thermal equation of state of bcc Fe at temperatures between 250K to 2250K. The calculated thermal equation of state at 250K shows good agreement with diamond-anvil-cell measurements (dots) at the room temperature 48 . While the bcc phase is dynamically stable over the whole pressure and temperature ranges shown here, it is only thermodynamically stable over a small region. Ferromagnetic bcc iron transforms to the fcc high-temperature global equation of state can be formed from the T=0K Vinet isotherm plus avolume-dependent thermal free energy F th 44 : Table 1 1Equation-of-state parameters determined by fitting a Vinet equation to the computed binding energy curves. LAPW results are from Ref. 33, and experimental data are from Ref. 48.V 0 (bhor 3 ) K 0 (GPa) K 0 ' LMTO-GGA 75.36 178 4.7 LAPW-GGA 76.84 189 4.9 LAPW-LSDA 70.73 245 4.6 Expt. 79.51 172 5.0 ACKNOWLEDGEMENTSMuch thanks to S. Y. Savrasov for kind agreement to use his LMTO codes and many helpful discussions. This work was supported by DOE ASCI/ASAP subcontract B341492 to Caltech DOE w-7405-ENG-48. 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{'abstract': 'We compute the lattice-dynamical and thermal equation of state properties of ferromagnetic bcc iron using the first principles linear response linear-muffin-tin-orbital method in the generalizedgradient approximation. The calculated phonon dispersion and phonon density of states, both at ambient and high pressures, show good agreement with inelastic neutron scattering data. We find the free energy as a function of volume and temperature, including both electronic excitations and phonon contributions, and we have derived various thermodynamic properties at high pressure and temperature. The thermal equation of state at ambient temperature agrees well with diamond-anvil-cell measurements. We have performed detailed investigations on the behavior of various thermal equation of state parameters, such as the bulk modulus K, the thermal expansivity α, the Anderson-Grüneisen parameter δ T , the Grüneisen ratio γ, and the heat capacity C V as function of temperature and pressure. A detailed comparison has been made with available experimental measurements, as well as results from similar theoretical studies on nonmagnetic bcc Tantalum.PACS number(s): 05.70. Ce, 64.30.+t, 71.20.Be 1', 'arxivid': 'cond-mat/0512469', 'author': ['Xianwei Sha \nCarnegie Institution of Washington\n5251 Broad Branch Road20015WashingtonNW, DC\n', 'R E Cohen \nCarnegie Institution of Washington\n5251 Broad Branch Road20015WashingtonNW, DC\n'], 'authoraffiliation': ['Carnegie Institution of Washington\n5251 Broad Branch Road20015WashingtonNW, DC', 'Carnegie Institution of Washington\n5251 Broad Branch Road20015WashingtonNW, DC'], 'corpusid': 18747506, 'doi': '10.1103/physrevb.73.104303', 'github_urls': [], 'n_tokens_mistral': 12089, 'n_tokens_neox': 10223, 'n_words': 6223, 'pdfsha': '67bb4f396b2523d5e08dbbc0ba256ea96f369fc3', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0512469v1.pdf'], 'title': ['Lattice dynamics and thermodynamics of bcc iron at pressure: first- principles linear response study', 'Lattice dynamics and thermodynamics of bcc iron at pressure: first- principles linear response study'], 'venue': []}

arxiv

Observation of non-Hermitian edge burst in quantum dynamics 22 Mar 2023 Lei Xiao Beijing Computational Science Research Center 100084BeijingChina Wen-Tan Xue Institute for Advanced Study Tsinghua University 100084BeijingChina Fei Song Institute for Advanced Study Tsinghua University 100084BeijingChina Yu-Min Hu Institute for Advanced Study Tsinghua University 100084BeijingChina Wei Yi CAS Key Laboratory of Quantum Information University of Science and Technology of China 230026HefeiChina CAS Center For Excellence in Quantum Information and Quantum Physics 230026HefeiChina Zhong Wang Institute for Advanced Study Tsinghua University 100084BeijingChina Peng Xue Beijing Computational Science Research Center 100084BeijingChina Observation of non-Hermitian edge burst in quantum dynamics 22 Mar 2023 The non-Hermitian skin effect, by which the eigenstates of Hamiltonian are predominantly localized at the boundary, has revealed a strong sensitivity of non-Hermitian systems to the boundary condition. Here we experimentally observe a striking boundary-induced dynamical phenomenon known as the non-Hermitian edge burst, which is characterized by a sharp boundary accumulation of loss in non-Hermitian time evolutions. In contrast to the eigenstate localization, the edge burst represents a generic non-Hermitian dynamical phenomenon that occurs in real time. Our experiment, based on photonic quantum walks, not only confirms the prediction of the phenomenon, but also unveils its complete space-time dynamics. Our observation of edge burst paves the way for studying the rich real-time dynamics in non-Hermitian topological systems. Non-Hermitian physics has attracted increasing attention in a vast variety of contexts ranging from classical waves to open quantum systems [1,2]. Intriguingly, the spatial boundary plays a much more dramatic role in non-Hermitian systems than in Hermitian ones. In particular, for certain non-Hermitian systems, the eigenstates concentrate predominantly at the boundary, which is known as the non-Hermitian skin effect (NHSE) [3][4][5][6][7][8][9][10][11][12][13][14]. Among many other consequences, it implies a fundamental revision of the principle of bulk-boundary correspondence [11,12]. Whereas the NHSE has revealed intriguing static properties such as novel behaviors of eigenstates and energy spectra, in this work we unveil a striking dynamic boundary effect in non-Hermitian systems. We experimentally observe that in a class of lossy quantum walk of single photons, the loss rate is drastically enhanced at the boundary. Specifically, for a lossy particle initially located at a position far from the boundary of a lattice system, the space-resolved loss has a surprisingly high boundary peak, in sharp contrast to the common expectation that the particle loss should decay away from the initial position. Remarkably, the relative height of the edge peak even grows as the distance between the initial position and boundary increases. This striking phenomenon, dubbed non-Hermitian edge burst, has been predicted in recent theories [15,16]. Since both the NHSE and edge burst involve boundary localization, it is tempting to attribute the latter to the former. However, it turns out that NHSE does not guarantee the emergence of edge burst. Closing the gap of the imaginary part of energy spectrum (i.e., the imaginary gap or dissipative gap) is the other necessary condition, which highlights the rich implication of spectral profile and topology in non-Hermitian systems [9,10]. At a deeper level, a novel dynamic bulk-edge scaling relation has been suggested as the origin of edge burst [15]. Thus, the edge burst signifies an unprecedented interplay between non-Hermitian topological physics and non-Hermitian dynamical phenomena. Lossy quantum walk.-To study the non-Hermitian edge burst, we design a one-dimensional quantum walk [17][18][19][20] with the Floquet operator U = R θ 2 2 SR θ 1 2 L(γ).(1) The shift operator S = x |x − 1 x| ⊗ |0 0| + |x + 1 x|⊗|1 1|, so that the walker's position is shifted from the site x to x − 1 or x + 1 according to the coin state |0 or |1 . The coin state is rotated along the y axis by R(θ) = 1 w ⊗e −iθσy , where 1 w = x |x x| is the identity operator. The operator L(γ) = 1 w ⊗ 1 0 0 e −2γ generates a state-selective loss. For our photonic platform, it is more convenient to create a domain wall instead of an open boundary [see Fig. 1(a)]. The left (L) and right (R) regions are characterized by coin parameters θ L 1,2 and θ R 1,2 , respectively. The dynamics of the non-Hermitian quantum walk follows |ψ(t) = U t |ψ(0) ,(2) where |ψ(0) is the initial state and t is the integer discrete time. One can also define an effective non-Hermitian Hamiltonian H eff by U = exp(−iH eff ), which shares the same eigenstates as U . The Floquet operator U defined in Eq. (1) and the associated H eff exhibit the NHSE, which originates from the state-dependent directional hoppings built in the model (akin to Refs. [11,21]). In the presence of a domain wall [ Fig. 1(a)], all the eigenstates of U exhibit localization at the domain wall when the non-Hermiticity is nonzero, i.e., γ = 0. Accordingly, the generalized Brillouin zone (GBZ) deviates from the unit circle [see Figs. 2(a) and (b)] [3,22,23]. Here, we focus on two sets of parameters, θ R 2 = 0.12π and θ R 2 = 0.48π, with other parameters fixed as θ L 1,2 = 0.85π, θ R 1 = 0.12π, and γ = 0. 8 gap (the gap between 0 and the maximum imaginary part of the spectrum) is zero for θ R 2 = 0.12π but nonzero for θ R 2 = 0.48π. In fact, the imaginary gap vanishes along the lines θ 1 = 2πn ± θ 2 (n ∈ Z) (see Supplementary Information). Observation of edge burst.-In our experiment, a walker is initialized at a site x 0 , which evolves under Eq. (2) in discrete time steps. The key quantity for edge burst is the probability P (x) that the walker escapes from the position x. In practice, one can measure the spacetime-resolved loss p(x, t) from t = 1 to t = T , with T being a large integer so that the loss is almost complete. The sum over t then gives P (x) = T t=1 p(x, t).(3) According to the specific form of loss adopted here, we have p(x, t) = (1 − e −4γ )| 1| ⊗ x| ψ(t − 1) | 2 .(4) It may also be written as p( x, t) = | 1| ⊗ x| M |ψ(t − 1) | 2 with M = 1 w ⊗ 0 0 0 √ 1 − e −4γ , which can be implemented by a partial measurement via the PPBS [see Fig. 1(a)] at the time step t. We also define a timedependent total loss probability P (t) = t t =1 x p(x, t ),(5) so that the survival probability after a t-step evolution is 1 − P (t). In our quantum-walk platform, p(x, t) can be readily extracted from photon-number measurements (see Methods), and P (x), P (t) can be obtained from Eqs. (3)(5). We implement a 14-step (T = 14) quantum walk with initial walker location x 0 = 10. The space-resolved loss probability P (x) is shown in Figs. 2(e) and (f) for the aforementioned two sets of parameters. In both (e) (θ R 2 = 0.12π) and (f)(θ R 2 = 0.48π), we observe that the loss probability initially decays away from x 0 . Moreover, the P (x) profile is asymmetric around x 0 , which can be naturally attributed to the NHSE. The surprising feature is an exceptionally high peak emerging at the domain wall in Fig. 2(e). Intuitively, one may resort to the NHSE to explain this edge burst. However, the NHSE is also strong for the parameters of Fig. 2(f), yet the edge burst is not seen there. Therefore, the origin of edge burst cannot be explained by the NHSE alone. In fact, the imaginary gap plays an essential role here [15]. The corresponding imaginary gap, shown in Figs. 2(c) and (d), is zero and nonzero for Figs. 2(e) and (f), respectively. To unveil the space-time profile of walker's loss, we plot p(x, t) for the above two sets of parameters. Figs. 2(g) and (h) show that the walker propagates almost ballistically with concurrent loss along the trajectory. In the case of edge burst, a large loss peak in p(x, t) emerges when the walker hits the domain wall. It also indicates that the burst occurs around a particular time, before which it is indiscernible. Im(E) Furthermore, we vary the initial position x 0 = 5, 6, 7, 8, 9, 10 and measure the time-dependent loss probability P (t). As shown in Fig. 3(a), for θ R 2 = 0.12π (with edge burst), P (t) suddenly increases near the domain wall. In contrast, in Fig. 3(b), for θ R 2 = 0.48π (without edge burst), P (t) increases steadily with t without sudden change. Similarly, the space-resolved survival probability |ψ(x = −1, t)| 2 at the domain wall at each step t behaves differently with and without the edge burst [see Fig. 3(c)]. The value of |ψ(x = −1, t)| 2 is significantly larger in the presence of edge burst. In Fig. 3(d), we show that the edge burst remains robust when the starting position varies. In contrast, when the edge burst is absent, P (x) decays rapidly as x 0 moves away from the domain wall [see Fig. 3 -2 0 2 Re(E) 2 0 -2 Im(β) -2 0 2 Re(β) Im(E) -2 0 2 Re(E) 2 0 -2 Im(β) -2 0 2 Re(β) (f) (h) (d) (b) 0.0 -0.5 -1.0 -1.5 π θ R 2 =0.48 π θ R 2 =0.12 π θ R 2 =0.48 π θ R 2 =0.12 π θ R 2 =0.48 π θ R 2 = 0 .1 2 π θ R 2 = 0(e)]. To further characterize the edge burst, we measure the relative height P domain /P min , where P domain ≡ P (x = −1) is the probability that the photon escapes from the domain wall x = −1, and P min ≡ min x=−1,··· ,x0 {P (x)} is the minimum of P (x) in the interval between the initial location x 0 and the domain wall location x = −1. The edge burst is characterized by P domain /P min 1, while its absence means that P domain /P min is on the order of unity. As shown in Fig. 3(f), for θ R 2 = 0.48π, the measured relative height remains close to 1 as x 0 increases. In stark contrast, for θ R 2 = 0.12π, the relative height increases with x 0 and fits well with a linear relation P domain /P min ∼ x 0 . Thus, the relative height grows as the initial walker position moves away from the domain wall. While counterintuitive, this behavior is a consequence of a novel bulk-edge scaling relation [15]. Discussions.-We present the first experimental observation of the non-Hermitian edge burst by using discretetime non-Hermitian quantum walk of photons. Our experiment not only demonstrates that edge burst originates from the intriguing interplay between two unique non-Hermitian concepts, the NHSE and imaginary gap, but also unveils the real-time dynamics of this phenomenon. The observation of non-Hermitian edge burst paves the way for investigating the real-time dynamics in non-Hermitian topological systems, which remains largely unexplored. From a practical perspective, the edge burst may offer a promising non-Hermitian approach for the on-demand harvesting of light or particles at a prescribed position. Methods Implementation.-For the experimental implementation, we adopt the scheme of single-photon discrete-time quantum walks illustrated in Fig. 1(b). Photon pairs are created by spontaneous parametric down conversion, where a 20mm type-II periodically poled potassium titanyl phosphate (PPKTP) crystal is pumped by a 405nm continuous wave diode laser with the power of 1mW. One photon serves as a trigger, and the other as a heralded single photon undertaking the quantum walk. The photon polarizations are adopted as the coin state. The walker photon is initialized in the spatial mode |x 0 with the internal state |0 , i.e. |ψ(0) = |x 0 ⊗ |0 . The localized initial state is prepared by passing the walker photons through a half-wave plate (HWP) and a polarizing beam splitter (PBS). x 0 =5 x 0 =6 x 0 =7 x 0 =8 x 0 =9 x 0 =10 x 0 =6 (d) (e)(f)π θ R 2 =0.12 π θ R 2 =0.12 π θ R 2 =0.48 π θ R 2 =0.48 π θ R 2 =0.48 π θ R 2 =0 .48 π θ R 2 =0 .12 For the quantum-walk dynamics, the shift operator S is implemented by a beam displacer (BD) whose optical axis is cut in the way so that the vertically polarized photons are directly transmitted and the horizontally polarized photons are laterally displaced into a neighboring mode. The coin rotation R( θ 1(2) 2 ) is realized by two HWPs at 0 and θ 1(2) 4 , respectively. The loss operator L(γ) is realized by a partially polarizing beam splitter (PPBS), which completely transmits the coin state |0 but reflects the coin state |1 with a probability e −4γ . At last, avalanche photodiodes (APDs) are used to detect the walker photons coinciding with the trigger photons. The total number of coincidences is approximately 23000. The measurements are based on photon-number counting. The space-time-resolved probability p(x, t) can be calculated from the photon number through p(x, t) = N (x, t) x N (x , t) + t t =1 x N (x , t ) ,(6) where N (x, t) is the number of photons escaping from the position x at the time step t, and N (x, t) is the number of remaining photons at x after a t-step evolution. Finally, the space-resolved survival probability at x can be calculated as |ψ(x, t)| 2 = N (x, t) x N (x , t) + t t =1 x N (x , t ) .(7) Note. After completing this work, we learned of a related experiment by a team at Southern University of Science and Technology. Supplemental Material for "Observation of non-Hermitian edge burst in quantum dynamics" Effective Hamiltonian and generalized Brillouin zone In this section, we derive an expression for the effective Hamiltonian H eff in momentum space. First, we transform the real-space nonunitary Floquet operator U and its conjugate transpose U † into the momentum-space U k and U † k : U k = d 0 σ 0 − id 1 σ 1 − id 2 σ 2 − id 3 σ 3 , U † k = d * 0 σ 0 + id * 1 σ 1 + id * 2 σ 2 + id * 3 σ 3 (S1) where σ 1,2,3 are the Pauli matrices and σ 0 is the identity matrix, and d 0 = e −γ (cosh γ cos k cos θ 1 + θ 2 2 + i sinh γ sin k cos θ 1 − θ 2 2 ), d 1 = e −γ (cosh γ sin k sin θ 1 − θ 2 2 + i sinh γ cos k sin θ 1 + θ 2 2 ), d 2 = e −γ (cosh γ cos k sin θ 1 + θ 2 2 − i sinh γ sin k sin θ 1 − θ 2 2 ), d 3 = e −γ (− cosh γ sin k cos θ 1 − θ 2 2 + i sinh γ cos k cos θ 1 + θ 2 2 ). (S2) Note that the relation d 2 0 + d 2 1 + d 2 2 + d 2 3 = e −γ is satisfied. The eigenvalue and eigenvector can be derived from U k |ψ ± = λ ± |ψ ± , U † k |χ ± = λ * ± |χ ± .(S3) Straightforward calculations lead to λ ± = d 0 ± it 0 , λ * ± = d * 0 ∓ it * 0 ,(S4)|ψ ± = 1 d 1 + id 2 d 3 ∓ t 0 d 1 + id 2 , χ ± | = 1 d 1 − id 2 (d 3 ∓ t 0 , d 1 − id 2 ),(S5) where t 0 = e −2γ − d 2 0 . Since the effective Hamiltonian H eff (k) is related to U k through U k = e −iH eff , the quasienergy spectrum of H eff (k) is E ± (k) = i ln λ ± (k) = ± arccos(cosh γ cos k cos θ 1 + θ 2 2 + i sinh γ sin k cos θ 1 − θ 2 2 ) − iγ.(S6) Specifically, for θ 1 = θ 2 , we have E − (k = π/2) = −arccos(i sinh γ) − iγ = −arccos(sin iγ) − iγ = −π/2,(S7) so that Im[E − (k = π/2)] = 0, i.e., the imaginary gap closes at k = π/2. For θ 1 = −θ 2 , the imaginary gap closes at k = 0 because E + (k = 0) = arccos(cosh γ) − iγ = arccos(cos(iγ)) − iγ = 0. (S8) Thus, the imaginary gap closes when θ 1 = 2πn ± θ 2 (n ∈ Z). While the eigenvectors in Eq. (S5) are not orthogonal, one can derive a set of bi-orthonormal eigenvectors {|ψ ± , |χ ± }: |ψ ± = |ψ ± χ ± |ψ ± = 1 2t 0 (t 0 ∓ d 3 ) d 3 ∓ t 0 d 1 + id 2 , χ ± | = χ ± | χ ± |ψ ± = 1 2t 0 (t 0 ∓ d 3 ) (d 3 ∓ t 0 , d 1 − id 2 ),(S9) which satisfy χ m |ψ n = δ mn , m=+,− |ψ m χ m | = 1. (S10) It follows that U k = λ + |ψ + χ + | + λ − |ψ − χ − |,(S11) and the effective Hamiltonian H eff (k) can be written as H eff = i ln λ + |ψ + χ + | + i ln λ − |ψ − χ − |.(S12) To derive the generalized Brillouin zone (GBZ) [3,22], we rewrite the Floquet operator U as U = x |x − 1 x| ⊗ A 0 + |x + 1 x| ⊗ A 1 ,(S13) where A 0 =R c ( θ 2 2 )P 0 R c ( θ 1 2 )L c (γ), A 1 =R c ( θ 2 2 )P 1 R c ( θ 1 2 )L c (γ),(S14) with L c (γ) = 1 0 0 e −2γ , R c (θ) = e −iθσy , P 0 = |0 0| and P 1 = |1 1|. In view of the translational symmetry inside the bulk, the eigenstate |ϕ of U can be expressed as |ϕ = x,j β x j |x ⊗ |φ j c ,(S15) where |φ j c is the coin state and β j is the spatial-mode function. Inserting Eq. (S15) into eigen-equation U |ϕ = λ|ϕ , we obtain (A 0 β + A 1 β − λ)|φ c = 0,(S16) which has nontrivial solutions only when det[A 0 β + A 1 β − λ] = 0.(S17) In an explicit form, Eq. (S17) is a quadratic equation of β: [sin( θ 1 2 ) sin( θ 2 2 ) − e 2γ cos( θ 1 2 ) cos( θ 2 2 )]β 2 + ( 1 λ + e 2γ λ)β + e 2γ sin( θ 1 2 ) sin( θ 2 2 ) − cos( θ 1 2 ) cos( θ 2 2 ) = 0.(S18) In the thermodynamic limit, the GBZ equation is determined by |β 1 (λ)| = |β 2 (λ)| [3,22]. Thus, we obtain |β 1 | = |β 2 | = | e 2γ sin( θ1 2 ) sin( θ2 2 ) − cos( θ1 2 ) cos( θ2 2 ) sin( θ1 2 ) sin( θ2 2 ) − e 2γ cos( θ1 2 ) cos( θ2 2 ) | = | cosh γ cos θ1+θ2 2 − sinh γ cos θ2−θ1 2 cosh γ cos θ1+θ2 2 + sinh γ cos θ2−θ1 2 |.(S19) Therefore, the GBZ is a circle in the complex plane, as shown in Fig. 2 in the main text. When |β| < 1(|β| > 1), the skin modes are localized at the left (right) edge. According to Eq. (S19), when cos θ1+θ2 FIG. S1. Numerical simulations for the loss probabilities P (−1) (for the domain wall) and P (x) (for the bulk), and the relative height P domain /Pmin . The coin parameters are fixed as θ L 1,2 = 0.85π and θ R 1 = 0.12π. For the upper row (red), θ R 2 = 0.12π, and the edge burst is present; for the lower row (blue), θ R 2 = 0.48π, and the edge burst is absent. (a)(b) The loss probability P (x = −1) versus x0. (c)(d) P (x) versus x0 − x. (e)(f) The relative height P domain /Pmin. The dots are from numerical simulations, and the black solid lines are the fitting results. Numerical fitting for larger time steps In the experiment, we have found that the relative height can be well fitted by P domain /P min ∼ x 0 . Thus, the relative height grows as x 0 increases. In this section, we add numerical simulations with more steps to further demonstrate this behavior. As illustrated in Fig. S1, we fit the loss probability P (x = −1) at the domain wall, P (x) in bulk, and the relative height P domain /P min . The results show that when the edge burst exists [ Fig. S1(a,c,e)], both P (x = −1) and P (x) follow power laws: P (x = −1) ∼ x −α d 0 and P (x) ∼ (x 0 − x) −α b , with certain α d and α b . The fitting for the relative height is P domain /P min ∼ x 1.0802 0 , which is close to the P domain /P min ∼ x 0 behavior predicted by theory and supported by our experiment. Notably, the fitting for α d,b are α d = 0.4717 and α b = 1.4751, so that α b − α d = 1.0034, which agrees well with the predicted bulk-edge scaling relation in Ref. [15]. When the edge burst is absent [ Fig. S1(b,d,f)], the fitting turns out to be exponential: P (x = −1) ∼ β x0−x d and P (x) ∼ β x0−x b , with certain β b and β d that are approximately equal. The relative height P domain /P min is almost constant as x 0 varies. FIG. 1 . 1Experimental implementation. (a) The domain-wall geometry of the non-Hermitian quantum walk. The operations of S, R, L contained in U are pictorially shown. (b) Experimental setup. Photon pairs are created by the spontaneous parametric down conversion process in a type-II cut PPKTP crystal. One of the photon is injected into the quantum-walk interferometric network, and the other is used as the trigger. The walker photon passes the polarizing beam splitter (PBS) and the half-wave plate (HWP), so that its polarization is prepared in the coin state |0 . It then undertakes the quantum walk through the network containing partially polarizing beam splitters (PPBSs), HWPs, beam displacers (BDs). Finally, avalanche photodiodes (APDs) are used to detect the walker photons that coincide with the trigger photons. .4 8 FIG. 2 . 82Edge burst in non-Hermitian quantum walks. The fixed parameters are θ L 1,2 = 0.85π, θ R 1 = 0.12π and γ = 0.8. (a)(b) Brillouin zone (BZ) and generalized Brillouin zone (GBZ) for θ R 2 = 0.12π and θ R 2 = 0.48π. (c)(d) Energy spectra (for the right region in which the walker is initialized) under the periodic boundary condition (PBC) for two indicated values of θ R 2 . (e)(f) Experimentally measured P (x) of a 14-step non-Hermitian quantum walk with the initial state |x0 = 10 ⊗ |0 . (g)(h) The space-time-resolved loss probability p(x, t) for the two values of θ R 2 . Error bars represent the statistical uncertainty under the assumption of Poissonian statistics. FIG. 3 . 3(a)(b) Experimentally measured time-dependent total loss probability P (t) for different starting positions x0 = 5, 6, 7, 8, 9, 10, respectively. For (a), θ R 2 = 0.12π; for (b), θ R 2 = 0.48π. Other parameters are the same as those in Fig. 2. (c) The measured survival probability at the domain wall, |ψ(x = −1, t)| 2 , for different starting positions. θ R 2 = 0.12π (upper panel) and 0.48π (lower panel). 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{'abstract': 'The non-Hermitian skin effect, by which the eigenstates of Hamiltonian are predominantly localized at the boundary, has revealed a strong sensitivity of non-Hermitian systems to the boundary condition. Here we experimentally observe a striking boundary-induced dynamical phenomenon known as the non-Hermitian edge burst, which is characterized by a sharp boundary accumulation of loss in non-Hermitian time evolutions. In contrast to the eigenstate localization, the edge burst represents a generic non-Hermitian dynamical phenomenon that occurs in real time. Our experiment, based on photonic quantum walks, not only confirms the prediction of the phenomenon, but also unveils its complete space-time dynamics. Our observation of edge burst paves the way for studying the rich real-time dynamics in non-Hermitian topological systems.', 'arxivid': '2303.12831', 'author': ['Lei Xiao \nBeijing Computational Science Research Center\n100084BeijingChina\n', 'Wen-Tan Xue \nInstitute for Advanced Study\nTsinghua University\n100084BeijingChina\n', 'Fei Song \nInstitute for Advanced Study\nTsinghua University\n100084BeijingChina\n', 'Yu-Min Hu \nInstitute for Advanced Study\nTsinghua University\n100084BeijingChina\n', 'Wei Yi \nCAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiChina\n\nCAS Center For Excellence in Quantum Information and Quantum Physics\n230026HefeiChina\n', 'Zhong Wang \nInstitute for Advanced Study\nTsinghua University\n100084BeijingChina\n', 'Peng Xue \nBeijing Computational Science Research Center\n100084BeijingChina\n'], 'authoraffiliation': ['Beijing Computational Science Research Center\n100084BeijingChina', 'Institute for Advanced Study\nTsinghua University\n100084BeijingChina', 'Institute for Advanced Study\nTsinghua University\n100084BeijingChina', 'Institute for Advanced Study\nTsinghua University\n100084BeijingChina', 'CAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiChina', 'CAS Center For Excellence in Quantum Information and Quantum Physics\n230026HefeiChina', 'Institute for Advanced Study\nTsinghua University\n100084BeijingChina', 'Beijing Computational Science Research Center\n100084BeijingChina'], 'corpusid': 257687483, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10654, 'n_tokens_neox': 8856, 'n_words': 4883, 'pdfsha': '1b95a59aa7d23ad8234b073de924dd7d442d9396', 'pdfurls': ['https://export.arxiv.org/pdf/2303.12831v1.pdf'], 'title': ['Observation of non-Hermitian edge burst in quantum dynamics', 'Observation of non-Hermitian edge burst in quantum dynamics'], 'venue': []}

arxiv

On the Lie superalgebra gl(m|n) weight system Jul 2022 Zhuoke Yang zetadkyzk@gmail.com On the Lie superalgebra gl(m|n) weight system Jul 2022 To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation of these weight systems is complicated. In the recent paper by the present author, an extension of the gl(N )-weight system to arbitrary permutations is defined, which allows one to develop a recurrence relation for an efficient computation of its values. In addition, the result proves to be universal, valid for all values of N and allowing thus to define a unifying gl-weight system taking values in the ring of polynomials in infinitely many variables C0 = N, C1, C2, . . . . In the present paper, we extend this construction to the weight system associated to the Lie superalgebra gl(m|n). Then we prove that the gl(m|n)-weight system is equivalent to the gl-one, under the substitution C0 = m − n. Introduction In V. A. Vassiliev's theory of finite type knot invariants [15], a weight system can be associated to each such invariant. A weight system is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, according to a Kontsevich theorem [10], to each weight system taking values in a field of characteristic 0, a finite type knot invariant can be associated in a canonical way. This makes studying weight systems an important part of knot theory. There is a number of approaches to constructing weight systems. In particular, a huge class of weight systems can be constructed from metrized finite dimensional Lie algebras. In spite of the fact that the construction is straightforward, explicit computations are elaborative, and until recently no efficient way to implement them was known. In a recent paper [17], the present author, following a suggestion of M. Kazarian, extended the weight system corresponding to the Lie algebra gl(N ) to arbitrary permutations, which allowed for proving a recurrence relation for it, whence computing its values explicitly. By means of the recurrence relation, we have defined a universal gl-weight system, which contains in itself all the gl(N )-weight systems, for arbitrary N . In the present paper, we do a similar thing for the weight system corresponding to the Lie superalgebra gl(m|n). We prove that it is a specialization of the gl-weight system, for C 0 = m − n. The original references to the Lie superalgebras can be found in [8]. Weight systems arising from Lie superalgebras are defined in [14]. The straightforward approach to computing the values of a Lie superalgebra weight system on a general chord diagram amounts to elaborating calculations in the noncommutative universal enveloping algebra, in spite of the fact that the result belongs to the center of the latter. This approach is rather inefficient even for the simplest noncommutative Lie superalgebra gl(1|1). For this Lie Superalgebra, however, there is a recurrence relation due to Figueroa-O'Farrill, T. Kimura and A. Vaintrob [7]. Much less is known about other Lie superalgebras; the goal of the present paper is to establish an efficient way to compute the gl(m|n)-weight system, for arbitrary m and n, and to prove that it is equivalent to the gl-weight system. The approach is based on defining an invariant of permutations taking values in the center of the universal enveloping algebra of gl(m|n). The restriction of this invariant to involutions without fixed points (such an involution determines a chord diagram) coincides with the value of the gl(m|n)-weight system on this chord diagram. We prove the recursion for gl(m|n)-weight system, which is the same as the recursion for the gl-one with C 0 = m − n. The paper is organized as follows. In Sec. 2, we recall necessary information about Lie superalgebras, especially, about the Lie superalgebra gl(m|n). In Sec. 3, we review the general definitions of Lie algebra/Lie superalgebra weight systems. In Sec. 4, we review the gl-weight system and the recurrence rule we have introduced in our previous paper. Sec. 5 is devoted to the definition of the extension of the gl(m|n)-weight system to permutations and calculations of some small examples. In Sec. 6, we prove another main theorem, which asserts that the gl(m|n)-weight system is equivalent to the gl-one. The author is grateful to M. Kazarian G. I. Olshanskii and A. N. Sergeev for valuable suggestions, and to S. Lando for permanent attention. Lie superalgebras First we recall the notion of Lie superalgebra, more details can be found in [8]. Everywhere in the paper, the ground field is C, the field of complex numbers. A super vector space, or a Z 2 -graded vector space, is a vector space decomposed as a direct sum V = V 0 ⊕ V 1 . The indices (or degrees) 0 and 1 are thought of as elements of Z 2 ; V 0 is called the even part of V , and V 1 is the odd part of V . An element x ∈ V is homogeneous if it belongs to either V 0 or V 1 . For x homogeneous, we write |x| for the degree of x. The (super) dimension of V is the pair (dim V 0 | dim V 1 ) also sometimes written as dim V 0 + dim V 1 . The vector space gl(V ) of all endomorphisms of a super vector space V is a super vector space itself: the subspace gl(V ) i , i = 0, 1, consists of maps f : V → V such that f (V j ) ⊆ V j+i ; each f ∈ gl(V ) can be written as a sum f 0 + f 1 with f i ∈ gl(V ) i . If V is finite-dimensional, then the supertrace of f is defined as sTrf = Tr f 0 − Tr f 1 . A superalgebra is a super vector space A together with a bilinear product which respects the degree: |xy| = |x| + |y| for all homogeneous x and y in A. The supercommutator in a superalgebra A is a bilinear operation defined on homogeneous x, y ∈ A by [x, y] = xy − (−1) |x| |y| yx. The elements of A whose supercommutator with the whole of A is zero form the super center of A. The supercommutator satisfies the following identities: where x, y, z are homogeneous. A super vector space with a bilinear bracket [·, ·] satisfying these identities is called a Lie superalgebra. Each Lie superalgebra g can be thought of as a subspace of its universal enveloping superalgebra U (g) defined as the quotient of the tensor algebra T (g) by the ideal J generated by the elements of the form |[x, y]| = |x| + |y|,x ⊗ y − (−1) |x| |y| y ⊗ x − [x, y], where x and y are arbitrary homogeneous elements of g. Note that the tensor algebra T (g) inherits a Z 2 -grading from g. Since the ideal J is generated by homogeneous elements, it follows that U (g) is also Z 2graded and the supercommutator in U (g) induces the bracket in g. We remark that ZU (g), the center of U (g), can be defined as follows ZU (g) = {x ∈ U (g)|xy = yx for all y ∈ g}. The theory of Lie superalgebras was developed by V. Kac [8]; it closely parallels the usual Lie theory. Let g = gl(m|n) be the vector space of all block square (m + n) × (m + n)matrices of the form X = A B C D , where A is a square m × m-matrix and D is a square n × n-matrix. Let g 0 denote the subspace of all such matrices with B = C = 0 and g 1 the subspace of all such matrices with A = D = 0. Then g = g 0 ⊕ g 1 is a Z 2 -graded associative algebra with respect to ordinary matrix multiplication, and g i is the set of all homogeneous elements of degree i, i = 0, 1. Elements of g 0 are called even, those of g 1 odd. Throughout what follows, if |a| occurs in an expression, then it is assumed that a is homogeneous, so that |a| = 0 provided a is even, and |a| = 1 if a is odd. The vector space gl(m|n) becomes a Lie superalgebra where the bracket is defined in terms of the usual matrix product by [a, b] = ab − (−1) |a||b| ba. Similarly, if V = V 0 ⊕ V 1 is a Z 2 -graded vector space, then End(V ) becomes a Lie superalgebra which we denote gl(V ). If dim V 0 = m and dim V 1 = n, then by choosing a homogeneous basis, we identify gl(V ) ∼ = gl(m|n). If X is as in (2), we define the supertrace of X, denoted sTr(X), by sTr(X) = Tr(A) − Tr(D) and the bilinear form ·, · on gl(m|n) is defined as a, b = sTr(ab). Clearly, this bilinear form is nondegenerate. We denote by E ij , i, j = 1, . . . , m + n, the standard basis of the Lie superalgebra gl(m|n) consisting of matrix units. The Z 2 -grading on gl(m|n) is defined by |E ij | =ī +j, whereī is an element of Z 2 which equals0 or1 depending on whether i ≤ m or i > m. The commutation relations in this basis are given by [E ij , E kl ] = δ jk E il − (−1) (ī+j)(k+l) δ il E kj . Below, we recall the result about the images of the Casimir elements C k ∈ U (gl(m|n)) under the Harish-Chandra isomorphism related with supersymmetric functions. p k = m i=1 x k i − (−1) k m+n j=m+1 x k j . We have f ∈ S(x 1 , . . . , x m |x m+1 , . . . , x m+n ) iff f is symmetric separately in x 1 , . . . , x m and x m+1 , . . . , x m+n , and if substituting x m = t, x m+n = −t in f provides a function independent of t. Theorem 2.2 (Casimir elements C k in U (gl(m|n)) [13]) The center ZU (gl(m|n)) of the universal enveloping algebra of the Lie superalgebra gl(m|n) is a polynomial algebra generated by the Casimir elements C k , k = 1, 2, . . . , defined as C k = m+n i1,...,i k (−1)ī 2 +ī3···+ī k E i1i2 E i2i3 . . . E i k−1 i k E i k i1 , where we omit the tensor product sign ⊗ between the matrix units. Below, we omit the tensor product sign provided this causes no confusion with the matrix product. The triangular decomposition of the Lie superalgebra gl(m|n) gives a vector space decomposition g = n − ⊕ h ⊕ n + where n − and n + are the nilpotent subalgebras of, respectively, upper and lower triangular matrices in gl(m|n), and h is the subalgebra of diagonal matrices. The universal enveloping algebra U (gl(m|n)) of the Lie superalgebra gl(m|n) admits the direct sum decomposition U (gl(m|n)) = (n − U (gl(m|n)) + U (gl(m|n))n + ) ⊕ U (h), The Harish-Chandra projection for U (gl(m|n)) is the projection to the second summand φ : U (gl(m|n)) → U (h) = C[E 1,1 , E 2,2 , · · · , E m+n,m+n ], where E 1,1 , E 2,2 , · · · , E m+n,m+n are the diagonal matrix units in gl(m|n); they commute with one another. [13]) The Harish-Chandra projection, when restricted to the center ZU (gl(m|n)), is an algebra isomorphism to the polynomial algebra of supersymmetric functions S(x 1 , . . . , x m |x m+1 , . . . , x m+n ) in the shifted generators x i = E ii + r i , where r i = j>i (−1)ī +j + 1 2 1 − (−1)ī . Explicitly, we have 1 − ∞ k=0 φ(C k )z k+1 = m+n i=1 1 − z 1 − z(−1)īx i (−1)ī Remark The center ZU (gl(m|n)) of the universal enveloping algebra of the Lie superalgebra gl(m|n) is not a finitely generated polynomial algebra. However, for fixed m, n, one can express the variables x i in terms of the first m+n Casimirs C 1 , . . . , C m+n . After substituting these expressions in the formula above, one can write the higher Casimirs C k , for k > m + n, as rational functions in the variables C 1 , . . . , C m+n . Chord diagrams and weight systems Below, we use standard notions from the theory of finite order knot invariants; see, e.g. [5,12]. A chord diagram of order n is an oriented circle (called the Wilson loop) endowed with 2n pairwise distinct points split into n disjoint pairs, considered up to orientation-preserving diffeomorphisms of the circle. A weight system is a function w on chord diagrams satisfying the 4-term relations; see Fig. 1. w( ) − w( ) = w( ) − w( ) Figure 1: 4-term relations In figures, the outer circle of the chord diagram is always assumed to be oriented counterclockwise. Dashed arcs may contain ends of arbitrary sets of chords, same for all the four terms in the picture. Let us recall the construction of the gl(N )-weight system for permutations, as introduced in [17]. Given a Lie algebra g equipped with a non-degenerate invariant bilinear form, one can construct a weight system with values in the center of its universal enveloping algebra U (g). This is the form M. Kontsevich [10] gave to a construction due to D. Bar-Natan [1]. Kontsevich's construction proceeds as follows. Given a chord diagram D with n chords, we first choose a base point on the circle, away from the ends of the chords of D. This gives a linear order on the endpoints of the chords, increasing in the positive direction of the Wilson loop. We order the chords of D according to the order of their left endpoints. Let us number the chords from 1 to n, and their endpoints from 1 to 2n, in the increasing order. Then D gives a permutation σ D of the set {1, 2, . . . , 2n} as follows. For 1 ≤ i ≤ n the permutation σ D sends 2i − 1 to the (number of the) left endpoint of the ith chord, and 2i to the (number of the) right endpoint of the same chord. The permutation σ D is exactly the re-arrangement, which sends the endpoints of the diagram with n consecutive isolated chords into D. Definition 3.1 (Universal Lie algebra weight system) Let g be a metrized Lie algebra over C, that is, a Lie algebra with an ad-invariant non-degenerate bilinear form ·, · . The bilinear form ·, · on g is a tensor in g * ⊗ g * . The algebra g being metrized, we can identify g * with g and think of ·, · as of an element of g⊗g. The permutation σ D acts on g ⊗2n by interchanging the factors. The value of the universal Lie algebra weight system w g (D) is the image of the nth tensor power ·, · ⊗n under the map g ⊗2n σD − − → g ⊗2n − − → U (g), where the second map is the restriction of the natural projection of the tensor algebra on g to its universal enveloping algebra. This construction of Lie algebra weight systems works also for Lie superalgebras, which are more general than Lie algebras. Let us recall the definition of the Lie superalgebra weight system on chord diagrams. If g is a metrized Lie superalgebra, the very same construction works with only one modification: re-arranging the factors in the final step should be done with certain care. Instead of simply permuting the factors in the tensor product one should use a representation of the symmetric group S k on k letters, which acts on the k th tensor power of any super vector space. This representation is defined as follows. Define S : g ⊗ g → g ⊗ g; S : x ⊗ y → (−1) |x||y| y ⊗ x. The map S is an involution; in other words, it defines a representation of the symmetric group S 2 on the vector space g ⊗2 . More generally, the representation of S k on g ⊗k is defined by sending the elementary transposition (i, i + 1) to id ⊗i−1 ⊗ S ⊗ id ⊗k−i−1 , i = 1, . . . , k − 1. Definition 3.2 (Universal Lie superalgebra weight system) Let g be a metrized Lie superalgebra over C, that is, a Lie superalgebra with an adinvariant non-degenerate bilinear form ·, · . The bilinear form ·, · on g is a tensor in g * ⊗ g * . The algebra g being metrized, we can identify g * with g and think of ·, · as of an element of g ⊗ g. The permutation σ D acts on g ⊗2n by interchanging the factors. The value of the universal Lie superalgebra weight system w g (D) is the image of the nth tensor power ·, · ⊗n under the map g ⊗2n S2n•σD − −−−− → g ⊗2n − −−−− → U (g), where the second map is the restriction of the natural projection of the tensor algebra on g to its universal enveloping algebra. its image belongs to the ad-invariant subspace U (g) g = {x ∈ U (g)|xy = yx for all y ∈ g} = ZU (g); this map from chord diagrams to ZU (g) satisfies the 4-term relations. Therefore, w g is a weight system taking values in ZU (g). Review of the gl-weight system and the Recurrence Rule Let us recall the construction of the gl(N )-weight system for permutations, as introduced in [17]. For a permutation σ ∈ S k , set w gl(N ) (σ) = N i1,··· ,i k =1 E i1i σ(1) E i2i σ(2) · · · E i k i σ(k) ∈ U (gl(N )). For example, the standard Casimir generator C k = N i1,··· ,i k =1 E i1i2 E i2i3 · · · E i k−1 i k m E i k i1 corresponds to the cyclic permutation 1 → 2 → · · · → k → 1 ∈ S k . It is shown in [17] that • the value of w gl(N ) on any permutation lies in the center of U (gl(N )); • this element is invariant under conjugation by the standard cyclic permutation, that is w gl(N ) (σ) = N i1,··· ,i k =1 E i2i σ(2) · · · E i k i σ(k) E i1i σ(1) . Definition 4.1 (digraph of a permutation) Let us represent a permutation as an oriented graph. The k vertices of the graph correspond to the permuted elements. They are ordered and are placed on a horizontal arrow looking right. The arc arrows show the action of the permutation (so that each vertex is incident with exactly one incoming and one outgoing arc edge). The digraph of a permutation σ ∈ S k consists of these k vertices and k oriented edges, for example: ((1 n + 1)(2 n + 2) · · · (n 2n)) = 1 2 n n+1n+2 2n · · · · · · Theorem 4.2 ( [17]) The value of the w gl(N ) invariant of permutations possesses the following properties: • for the empty graph (with no vertices) the value of w gl(N ) is equal to 1, w gl(N ) (○) = 1; • w gl(N ) is multiplicative with respect to concatenation of permutations; • for a cyclic permutation (with the cyclic order on the set of permuted elements compatible with the permutation), the value of w gl(N ) is the standard generator, w gl(N ) (1 → 2 → · · · → k → 1) = C k . For the special case σ(l + 1) = l, the recurrence looks like follows: l l+1 a b − l+1 l a b = C 1 × a b − N × l' a b These relations are indeed a recursion, that is, they allow one to replace the computation of w gl(N ) on a permutation with its computation on simpler permutations. The recursion rule of the theorem defines a weight system on permutations taking values in the ring of polynomials in infinitely many variables C[C 0 = N, C 1 , C 2 , . . . ]. We denote this universal gl-weight system by w gl , where C 0 coincides with the number N in the second recursion rule and C k corresponds to the standard length k cyclic permutation. The recursion in the theorem allows one to compute this weight system effectively. Extension of the gl(m|n)-weight system to permutations We define w gl(m|n) on permutations in the following way, which is similar to the definition for w gl(N ) . For a permutation σ ∈ S k , set w gl(m|n) (σ) = m+n i1,··· ,i k =1 (−1) fσ E i1i σ(1) E i2i σ(2) · · · E i k i σ(k) , where f σ is the sign function which is a polynomial inī 1 ,ī 2 , . . . ,ī k in the field Z 2 defined below. The sign function f σ is a polynomial that has linear and quadratic terms only. For example, for the standard cyclic permutation (12 . . . k) : 1 → 2 → · · · → k → 1, we have f (12...k) =ī 2 + · · · +ī k . We say that an index a, 1 ≤ a ≤ k, is distinguished with respect to σ if σ −1 (a) < a. The set of distinguished indices is denoted by P 1 (σ) ⊂ {1, . . . , k}. We say that a pair of indices (a, b), 1 ≤ a < b ≤ k, is distinguished if the two pairs of distinct real numbers (σ −1 (a) + ǫ, a − ǫ) and (σ −1 (b) + ǫ, b − ǫ) alternate; here ǫ > 0 is a small real number, say, ǫ = 1 3 . The set of distinguished pairs of indices is denoted by P 2 (σ) ⊂ {1, . . . , k} × {1, . . . , k}. Definition 5.1 The sign function f σ of a permutation σ ∈ S k is defined by f σ (ī 1 ,ī 2 , . . . ) = a∈P1(σ)ī a + (a,b)∈P2(σ)ī aīb . A more convenient treatment of the invariant w gl(m|n) (σ) and the sign function uses the language of digraphs from the previous section. The set of indices participating in the summation will be labelled by the edges (rather than by vertices). For each vertex i, we denote by in(i) and out(i) the incoming edge and outcoming edge incident to the vertex i, respectively. With this notation, we have w gl(m|n) (σ) = m+n i1,··· ,i k =1 (−1) fσ E i in(1) i out(1) · · · E i in(k) i out(k) . The original formula corresponds to the numbering of the edges such that the edge i → j is numbered j. The result is obviously independent of the numbering. With this notation, an edge is distinguished if it is directed from left to right. A pair of edges with pairwise distinct ends is distinguished if the corresponding pairs of vertices alternate. If the edges have common vertices, we first bring them to a general position by shifting slightly the beginning of each edge to the right and the endpoint of each edge to the left, and then check whether the pairs of ends of the shifted edges do alternate. Claim 5.2 For the cyclic permutation σ = (12 · · · k), the diagram is 1 2 3 k-2 k-1 k · · · ; and we have f (12···k) =ī 2 + · · · +ī k . Assume that two permutations σ and σ ′ are conjugate by a transposition of two neighboring elements. Then these two elements are the endpoints of the four edges a, b, c, d as shown in the picture below (among the edges a, b, c, d there could be pairs of coincident ones). f σ ′ = f σ + (ī a +ī d )(ī b +ī c ). In other words, each of the four pairs of edges (a, c), (a, d), (b, c), (b, d) changes the property of being distinguished when one passes from the permutation σ to σ ′ . Since the sign function f σ matches the sign in the Casimir elements and this lemma says the sign function f σ matches the involution operation S, we have Claim 5.4 The gl(m|n)-weight system for chord diagrams in [14,7] is a special case of the gl(m|n)-weight system for permutations, where we treat a chord diagram with k chords as an involution without fixed points on the set of 2k elements. Example 5.5 Let σ = (132) ∈ S 3 . According to the definition, w gl(m|n) ((132)) = m+n i1,i2,i3=1 (−1) f (132) E i1i3 E i2i1 E i3i2 , where f (132) =ī 3 +ī 1ī3 +ī 3ī2 +ī 1ī2 , and we have the Lie superbracket [E ij , E kl ] = δ jk E il − (−1) (ī+j)(k+l) δ il E kj . Now, C 3 − w gl(m|n) ((132)) = m+n i1,i2,i3=1 (−1)ī 3+ī2 E i1i3 E i3i2 E i2i1 − (−1) (ī3+ī2)(ī1+ī2) E i3i2 E i2i1 = m+n i1,i2,i3=1 (−1)ī 3+ī2 E i1i3 [E i3i2 , E i2i1 ] = m+n i1,i2,i3=1 (−1)ī 3+ī2 E i1i3 (δ i2i2 E i3i1 − δ i3i1 (−1) (i3+ī2)(i1+ī2) E i2i2 ) = (m − n) m+n i1,i3=1 (−1)ī 3 E i1i3 E i3i1 − m+n i1,i2=1 E i1i1 E i2i2 = (m − n)C 2 − C 2 1 Finally, we get w gl(m|n) ((132)) = C 3 − (m − n)C 2 + C 2 1 . Result k σ f σ polynomial in Casimir elements 2 Id 0 C 2 1 (1 2)ī 2 C 2 3 Id 0 C 3 1 (1 2)ī 2 C 1 C 2 (2 3)ī 3 C 1 C 2 (1 3)ī 3 C 1 C 2 (1 2 3)ī 2 +ī 3 C 3 (1 3 2)ī 3 +ī 2 + (ī 3 +ī 2 )(ī 1 +ī 2 ) C 3 − (m − n)C 2 + C 2 1 4 Id 0 C 4 1 (1 2)ī 2 C 2 1 C 2 (2 3)ī 3 C 2 1 C 2 (1 3)ī 3 C 2 1 C 2 (1 4)ī 4 C 2 1 C 2 (2 4)ī 4 C 2 1 C 2 (3 4)ī 4 C 2 1 C 2 (1 2)(3 4)ī 2 +ī 4 C 2 2 (1 4)(2 3)ī 3 +ī 4 C 2 2 (1 3)(2 4)ī 3 +ī 4 + (ī 2 +ī 4 )(ī 1 +ī 3 ) C 2 2 − (m − n)C 2 + C 2 1 (1 2 3)ī 2 +ī 3 C 1 C 3 (1 2 4)ī 2 +ī 4 C 1 C 3 (1 3 4)ī 3 +ī 4 C 1 C 3 (2 3 4)ī 3 +ī 4 C 1 C 3 (1 3 2) (ī 2 +ī 3 )(ī 1 +ī 3 ) C 1 (C 3 − (m − n)C 2 + C 2 1 ) (1 4 2) (ī 2 +ī 4 )(ī 1 +ī 4 ) C 1 (C 3 − (m − n)C 2 + C 2 1 ) (1 4 3) (ī 3 +ī 4 )(ī 1 +ī 4 ) C 1 (C 3 − (m − n)C 2 + C 2 1 ) (2 4 3) (ī 2 +ī 4 )(ī 3 +ī 4 ) C 1 (C 3 − (m − n)C 2 + C 2 1 ) (1 2 3 4)ī 2 +ī 3 +ī 4 C 4 (1 2 4 3)ī 2 +ī 4 +ī 1ī4 +ī 1ī3 +ī 4ī3 C 4 − (m − n)C 3 + C 1 C 2 (1 3 2 4) C 4 − (m − n)C 3 + C 1 C 2 (1 3 4 2) C 4 − (m − n)C 3 + C 1 C 2 (1 4 2 3) C 4 − (m − n)C 3 + C 1 C 2 (1 4 3 2)ī 4 + (ī 1 +ī 3 )(ī 4 +ī 2 ) C 4 − 2(m − n)C 3 + 2C 1 C 2 + +(m − n) 2 C 2 − (m − n)C 1 In all the above examples, the value of the gl(m|n)-weight system is a polynomial in the difference m−n. The following stronger theorem, which is another main result of the present paper, asserts that this is always true. Theorem 5.6 The weight system w gl(m|n) for permutations is the result of substituting m − n for C 0 , and the k th Casimir element in gl(m|n) for C k , k > 0, in the weight system w gl . The proof of this theorem is given in the next section. Example 5.7 In [7], a recurrence relation for computing the values of the gl(1|1)-weight system is given. Our approach suggests another recurrence for this weight system extended to permutations. Setting C 0 = 1 − 1 = 0 and using Theorem 2.3 we can express higher Casimirs in ZU (gl(1|1)) in terms of C 1 , C 2 . Namely, we have 1 − ∞ k=0 ϕ(C k )z k+1 = 1 − z 1−zx1 1 − z 1+zx2 , which gives ϕ(C 1 ) = x 1 + x 2 , ϕ(C 2 ) = (x 1 + x 2 )(x 1 − x 2 + 1), ϕ(C 3 ) = (x 1 + x 2 )(x 2 1 − x 1 x 2 + x 2 2 + x 1 − 2x 2 + 1) = ϕ(C 1 )( 3ϕ(C 2 ) 2 4ϕ(C 1 ) 2 + ϕ(C 1 ) 2 4 − ϕ(C 1 ) 2 + 1 4 ), . . . so that we have C 3 = 3C 2 2 4C 1 + C 3 1 4 − C 2 1 2 + C 1 4 , . . . and, more generally x = C 2 1 − C 1 + C 2 2C 1 , y = C 2 1 + C 1 − C 2 2C 1 , ∞ k=0 C k z k = 1 − (1−(x+1)z)(yz+1) (1−xz)(1−(1−y)z) z = C 1 z 1 − (−C 2 1 +C1+C2)z 2C1 1 − (C 2 1 −C1+C2)z 2C1 For example, if we make this substitution in the explicit formulas for the values of the w gl -weight system on chord diagrams whose intersection graph is a complete graph given in [17], we obtain the following values of the gl(1|1)weight system on these diagrams: These results are worth to be compared with the values of the skew characteristic polynomial of complete graphs from [6]. 6 Proof of theorem 5.6 We prove the theorem by directly proving that w gl(m|n) satisfies the same Recurrence Rule as w gl with C 0 = m − n. Assuming the permutation σ is as shown before, and supposeσ merges the two nodes and connects the edges a and c, Using Lemma 5.3 and Lemma 6.1, we obtain (−1) fσ · · · E ij E kl · · · − (−1) f σ ′ · · · E kl E ij · · · = (−1) fσ · · · E il · · · − (−1) fσ′ · · · E kj · · · ′ which is w gl(m|n) (σ) − w gl(m|n) (σ ′ ) = w gl(m|n) (σ) − w gl(m|n) (σ ′ ). It is the same Recurrence Rule as for w gl(m−n) . For the special case σ(k + 1) = k, the recurrence looks like k+1 k a b σ − k k+1 a b σ ′ = (m − n) × k' a b σ − C 1 × a b σ ′ We have the relationship E ij E jk − (−1) (ī+j)(j+k) E jk E ij = δ jj E ik − (−1) (ī+j)(j+k) δ ik E jj We produce everything we need to let the first term be w gl(m|n) (σ): (−1) fσ · · · E ij E jk · · · − (−1) fσ +(ī+j)(j+k) · · · E jk E ij · · · = δ jj (−1) fσ · · · E ik · · · − δ ik (−1) fσ+(ī+j)(j+k) · · · E jj · · · We have (−1) fσ · · · E ij E jk · · · − (−1) f σ ′ · · · E jk E ij · · · = (−1)j +fσ · · · E ik · · · − (−1) fσ′ · · · E jj · · · =(m − n) (−1) fσ · · · E ik · · · − C 1 (−1) fσ′ · · · 1 · · · , hence w gl(m|n) satisfies the special case of the Recurrence Rule with number m − n. Finally, w gl(m|n) obeys the same Recurrence Rule as w gl with C 0 = m − n. [x, y] = −(−1) |x| |y| [y, x] and (−1) |z| |x| [x, [y, z]] + (−1) |y| |z| [z, [x, y]] + (−1) |x| |y| [y, [z, x]] = 0, Definition 2.1 (supersymmetric functions)The ring of supersymmetric functions S(x 1 , . . . , x m |x m+1 , . . . , x m+n ) is defined as the subring of C[x 1 , . . . , x m+n ] generated by the homogeneous generators p k given by Claim 3.3 [5, 10 ] 10The function w g : D → w g (D) on chord diagrams has the following properties: 1. the element w g (D) does not depend on the choice of the base point on the diagram; • (Recurrence Rule) For the graph of an arbitrary permutation σ in S k , and for any two neighboring elements l, l+1, of the permuted set {1, 2, . . . , k}, we have for the values of the w gl(N ) weight system diagrams on the left, two horizontally neighboring vertices and the edges incident to them are depicted, while on the right these two vertices are replaced with a single one; the other vertices are placed somewhere on the circle and their positions are the same on all diagrams participating in the relations, but the numbers of the vertices to the right of the latter are to be decreased by 1. The sign functions f σ and f σ ′ are related by 3. The other edges making a distinguished pair with only a or c will also make a distinguished pair with a/c. And the edges making distinguished pairs both with a and c will not make a distinguished pair with a/c. However, sinceī a =ī c and the field is Z 2 , we haveī xīa +ī xīc = 0. Therefore, these cases do not differ.4. We only need to consider the relationship between a, b, c, d (a) the linear term: since the edges a and c turn into a longer edge, the difference in the linear term isī a .(b) the difference in the quadratic term isī aīc =ī a as well.Summing everything together, we obtain no difference.For the other arrangements of the end points of the four arrows the calculation similar, and we skip the rest of the proof.We produce everything we need to let the first term be w gl(m|n) (σ):(−1) fσ · · · E ij E kl · · · − (−1) fσ +(ī+j)(k+l) · · · E kl E ij · · · = δ jk (−1) fσ · · · E il · · · − δ il (−1) fσ +(ī+j)(k+l) · · · E kj · · · Dror Bar-Natan, On the Vassiliev knot invariants. 34Dror Bar-Natan, On the Vassiliev knot invariants, Topol- ogy, 34(2):423-472, 1995. (an updated version available at http://www.math.toronto.edu/~drorbn/papers). Dror Bar-Natan, Weights of Feynman diagrams and the Vassiliev knot invariants. preprintDror Bar-Natan, Weights of Feynman diagrams and the Vassiliev knot invariants, preprint, February 1991. (an updated version available at http://www.math.toronto.edu/~drorbn/papers). Mutant knots and intersection graphs. S Chmutov, S Lando, Algebr. Geom. Topol. 7S. Chmutov, S. Lando, Mutant knots and intersection graphs, Algebr. Geom. Topol. 2007, 7, 3, 1579-1598 Remarks on the Vassiliev knot invariants coming from sl 2 , Topology. S Chmutov, A Varchenko, 36S. Chmutov, A. Varchenko, Remarks on the Vassiliev knot invariants com- ing from sl 2 , Topology 1997, 36, 1, 153-178 S Chmutov, S Duzhin, J Mostovoy, Introduction to Vassiliev Knot Invariants. Cambridge University PressS. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press, May 2012. R Dogra, S Lando, arXiv:2201.07084Skew characteristic polynomial of graphs and embedded graphs. math.COR. Dogra, S. Lando, Skew characteristic polynomial of graphs and embedded graphs, arXiv:2201.07084 [math.CO] The Universal Vassiliev Invariant for the Lie Superalgebra gl(1|1). T Figueroa-O&apos;farrill, A Kimura, Vaintrob, Comm. Math. Phys. 185Figueroa-O'Farrill, T. Kimura, A. Vaintrob, The Universal Vassiliev In- variant for the Lie Superalgebra gl(1|1), Comm. Math. Phys., 1997, 185, 93-127 Lie Superalgebras. V G Kac, 10.1016/0001-8708(77)90017-2Adv.Math. 26V. G. Kac, Lie Superalgebras, Adv.Math. 26 (1977) 8-96 DOI: 10.1016/0001-8708(77)90017-2 Laplace operators of infinite-dimensional Lie algebras and theta functions. V G Kac, Proceedings of the National Academy of Sciences. 812V. G. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proceedings of the National Academy of Sciences 81, no. 2 (1984): 645-647. M Kontsevich, Vassiliev knot invariants. 16Advances in Soviet Math.M. Kontsevich, Vassiliev knot invariants, in: Advances in Soviet Math., 16(2):137-150, 1993. On a Hopf algebra in graph theory. Sergei K Lando, Journal of Combinatorial Theory, Series B. 801Sergei K. Lando, On a Hopf algebra in graph theory, Journal of Combina- torial Theory, Series B, 80(1):104-121, 2000. Sergei K Lando, Alexander K Zvonkin, Graphs on surfaces and their applications. Springer Science & Business Media141Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, volume 141. Springer Science & Business Media, 2013. New expressions for the eigenvalues of the invariant operators of the general linear and the orthosymplectic Lie superalgebras. C O Nwachuku, M A Rashid, Journal of mathematical physics. 268C. O. Nwachuku and M. A. Rashid, New expressions for the eigenvalues of the invariant operators of the general linear and the orthosymplectic Lie superalgebras, Journal of mathematical physics 26, no. 8 (1985): 1914-1920. A Vaintrob, Vassiliev knot invariants and Lie S-algebras Mathematical Research Letters. 1A. Vaintrob, Vassiliev knot invariants and Lie S-algebras Mathematical Research Letters, 1, 579-595.(1994) Cohomology of knot spaces. V A Vassiliev, Advances in Soviet Math. 1V. A. Vassiliev, Cohomology of knot spaces. in: Advances in Soviet Math., bf 1, 1990, 23-69. Z Yang, arXiv:2102.00888On values of sl 3 weight system on chord diagrams whose intersection graph is complete bipartite. math.COZ. Yang, On values of sl 3 weight system on chord diagrams whose inter- section graph is complete bipartite, arXiv:2102.00888 [math.CO] Z Yang, arXiv:2202.12225New approaches to gl N weight system. math.COZ. Yang, New approaches to gl N weight system, arXiv:2202.12225 [math.CO]

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{'abstract': 'To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation of these weight systems is complicated. In the recent paper by the present author, an extension of the gl(N )-weight system to arbitrary permutations is defined, which allows one to develop a recurrence relation for an efficient computation of its values. In addition, the result proves to be universal, valid for all values of N and allowing thus to define a unifying gl-weight system taking values in the ring of polynomials in infinitely many variables C0 = N, C1, C2, . . . . In the present paper, we extend this construction to the weight system associated to the Lie superalgebra gl(m|n). Then we prove that the gl(m|n)-weight system is equivalent to the gl-one, under the substitution C0 = m − n.', 'arxivid': '2207.00327', 'author': ['Zhuoke Yang zetadkyzk@gmail.com ', 'Zhuoke Yang zetadkyzk@gmail.com '], 'authoraffiliation': [], 'corpusid': 250243920, 'doi': '10.1016/j.geomphys.2023.104808', 'github_urls': [], 'n_tokens_mistral': 11613, 'n_tokens_neox': 10212, 'n_words': 6294, 'pdfsha': '50ffcd78f7d33fbe70b932290855d83fb2bdc8ea', 'pdfurls': ['https://export.arxiv.org/pdf/2207.00327v1.pdf'], 'title': ['On the Lie superalgebra gl(m|n) weight system', 'On the Lie superalgebra gl(m|n) weight system', 'On the Lie superalgebra gl(m|n) weight system', 'On the Lie superalgebra gl(m|n) weight system'], 'venue': []}

arxiv

The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion 21 Oct 2022 Barak Budnick Racah Institute of Physics The Hebrew University 9190401JerusalemIsrael Ofer Biham Racah Institute of Physics The Hebrew University 9190401JerusalemIsrael Eytan Katzav Racah Institute of Physics The Hebrew University 9190401JerusalemIsrael The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion 21 Oct 2022 We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion).To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability P add and a random node deletion step takes place with probability P del = 1 − P add . The balance between the growth and contraction processes is captured by the parameter η = P add − P del . The case of pure network growth is described by η = 1.In case that 0 < η < 1 the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where −1 < η < 0, the overall process is of network contraction, while in the special case of η = 0 the expected size of the network remains fixed, apart from fluctuations. Using the master equation and the generating function formalism we obtain a closed form expression for the time dependent degree distribution P t (k). The degree distribution P t (k) includes a term that depends on the initial degree distribution P 0 (k), which decays as time evolves, and an asymptotic distribution P st (k) which is independent of the initial condition. In the case of pure network growth (η = 1) the asymptotic distribution P st (k) follows an exponential distribution, while for −1 < η < 1 it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth (0 < η < 1) the degree distribution P t (k) eventually converges to P st (k). In the case of overall network contraction (−1 < η < 0) we identify two different regimes. For −1/3 < η < 0 the degree distribution P t (k) quickly converges towards P st (k). In contrast, for −1 < η < −1/3 the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. Thus, the model exhibits three phase transitions: a structural transition between two functional forms of P st (k) at η = 1, a transition between an overall growth and overall contraction at η = 0 and a dynamical transition between fast and slow convergence towards P st (k) at η = −1/3. The analytical results are found to be in very good agreement with the results obtained from computer simulations. I. INTRODUCTION In the past 25 years or so, the field of network research has emerged as a major field of study, which significantly contributed to the understanding of the structure and dynamics of biological, social and technological networks [1][2][3][4][5]. It was found that empirical networks are typically small-world networks that exhibit fat-tailed degree distributions with scale free structures [6][7][8]. Much theoretical effort has focused on generic processes of network expansion or growth. It was found that newly formed nodes tend to connect preferentially to nodes of high degree, and that this property leads to the emergence of scale-free networks with power-law degree distributions of the form P (k) ∼ k −γ , where 2 < γ ≤ 3 and the second moment of the degree distribution diverges [7][8][9][10]. In particular, the Barabási-Albert (BA) model exhibits a scale-free structure that emerges from the preferential-attachment process [7]. In this model, at each time step a new node is added to the network and forms links to m of the existing nodes, such that the probability of an existing node of degree k to gain a link to the new node is proportional to k. The degree distribution of the BA network exhibits a power-law tail with γ = 3. Variants of the BA model were shown to yield powerlaw distributions with exponents in the range 2 < γ ≤ 3 [9][10][11]. Another important class of network growth models is based on the duplication of existing nodes, where a new (daughter) node is connected to each neighbor of the duplicated (mother) node with probability p, and in some cases it is also connected to the mother node itself [12][13][14][15][16][17][18][19][20]. The degree distributions of node duplication networks follow a power-law distribution, where γ is a monotonically decreasing function of p [13,15,18,19]. The opposite scenario of network contraction has attracted increasing attention in recent years. For example, the contraction processes of social networks was recently studied [21,22]. Such networks may lose users due to loss of interest, concerns about privacy or due to their migration to other social networks. Another example is the evolution of gene networks, in which it was recently found that the process of gene loss plays a significant role [23]. A different context of great practical importance is the cascading failure of powergrids [24,25], in which the functional part of the network quickly contracts. Infectious processes such as epidemics that spread in a network [26,27] lead to the contraction of the subnetwork of the susceptible (or uninfected) nodes, and may thus be considered as network contraction processes. Similarly, network immunization schemes [28] also belong to the class of network contraction processes because they induce the contraction of the subnetwork of susceptible nodes. The framework of network contraction is especially relevant in the context of neurodegeneration, which is the progressive loss of structure and function of neurons in the brain. Such processes occur in normal aging [29] as well as in a large number of incurable neurodegenerative diseases such as Alzheimer, Parkinson, Huntington and Amylotrophic Lateral Sclerosis, which result in a gradual loss of cognitive and motoric functions [30]. These diseases differ in the specific brain regions or circuits in which the degeneration occurs. The analysis of the evolving structure may provide useful insight into the structural aspects of the loss of neurons and synapses in neurodegenerative processes [31]. Network contraction processes, which may result from inadvertent failures or from deliberate attacks, were studied using the framework of percolation theory [32][33][34][35][36][37][38][39][40][41][42][43]. It was shown that scale-free networks are resilient to attacks targeting random nodes [32], but are vulnerable to attacks that target high degree nodes or hubs [33]. In both cases, when the number of deleted nodes exceeds some threshold, the network breaks down into disconnected components [32][33][34][44][45][46][47]. This analysis provided important insights on the final stages of network collapse. However, until recently the evolution of complex networks in the early and intermediate stages of their contraction process, before fragmentation, has not been studied in sufficient detail. Understanding the patterns that emerge in the early and intermediate stages of network failures or attacks is crucial for their detection and for devising ways to fix the network and block such attacks. Recently we considered the evolution of complex networks during generic contraction and collapse scenarios [48,49]. These scenarios include random node deletion, preferential node deletion and propagating node deletion. The random node deletion process describes random failures or random attacks that do not target any specific type of nodes. The process of preferential node deletion describes attacks that preferentially target high degree nodes, while propagating node deletion describes processes that propagate from an infected node to its neighbors. To analyze these processes we derived a master equation for the time dependence of the degree distribution P t (k) in each one of the three network contraction scenarios. In the scenario of random node deletion, the master equation is exact for any ensemble of initial networks, while in the scenarios of preferential and propagating node deletion it is exact for the case of configuration model networks, in which there are no degree-degree correlations [50][51][52][53][54]. However, it was shown to provide reasonably accurate results for the time-dependent degree distributions even in networks that exhibit degreedegree correlations. Using the master equation we established that when networks contract via any of the node deletion scenarios described above, their degree distributions evolve towards a Poisson distribution, namely they become Erdős-Rényi (ER) networks [55][56][57]. These networks belong to an ensemble of maximum entropy random graphs [51]. The emerging structure of networks that evolve under a combination of growth and contraction processes was studied in Refs. [58][59][60]. These papers focus on the regime in which the overall process is of network growth. A particularly interesting case is of networks that grow via a combination of preferential attachment and random attachment, which exhibit a degree distribution with a power-law tail. It was found that under low rate of random node deletion the degree distribution maintains its power-law tail. However, above some threshold (that depends on the mixture of random attachment and preferential attachment) the power-law tail is lost and is replaced by a discrete exponential degree distribution (which is also known as a geometric distribution). The phase boundary between the two phases was calculated (using different parameterizations), giving rise to highly insightful phase diagrams [59,60]. The combination of growth via node addition and random attachment and contraction via random node deletion was also studied [58]. In the limit of pure growth this model gives rise to networks that exhibit an exponential (geometric) degree distribution [20,58]. As mentioned above, Refs. [58][59][60] focus on the steady state solution of the degree distribution in case that the overall process is of network growth. The complementary regime in which the rate of node deletion exceeds the rate of node addition has not been studied. In this paper we analyze the emerging structure of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion). We derive a master equation for the time dependence of the degree distribution under this combination of growth and contraction processes. Using the generating function formalism we obtain a closed form expression for the degree distribution P t (k). It includes a term that depends on the initial condition, which decays as time evolves, and an asymptotic term which is an attractive fixed point. We identify a phase transition between the phase of pure network growth and the phase that combines growth and contraction. This transition implies that even the slightest rate of node deletion leads to a qualitative change in the nature of the degree distribution. In the regime of overall network growth, P t (k) eventually converges towards the asymptotic steady state form P st (k). In contrast, in the regime of overall network contraction the asymptotic degree distribution is not always reached due to the finite life-time of the network. This gives rise to a second phase transition, between the phase of overall network growth and the phase of overall network contraction. In the phase of overall network contraction we identify a third transition, between the case of low deletion rate, in which the degree distribution P t (k) quickly approaches P st (k), and the case of high deletion rate, in which the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. The analytical results are found to be in very good agreement with the results obtained from computer simulations. The paper is organized as follows. In Sec. II we describe the dynamical model that combines growth (via node addition and random attachment) and contraction (via random node deletion). In Sec. III we derive a master equation for the time dependent degree distribution P t (k). In Sec. IV we use the master equation to derive a differential equation for the generating function G t (u) of the degree distribution and present its time-dependent solution. In Sec. V we present a closed-form expression for the degree distribution P t (k), obtained from G t (u). In Sec. VI we calculate the mean and variance of the degree distribution. The results are summarized and discussed in Sec. VII. In Appendix A we solve the differential equation for G t (u) and extract the degree distribution P t (k). In Appendix B we calculate the degree distribution P t (k) in the special case of pure network growth. II. THE MODEL Consider a network that evolves as follows. At each time step, one of two possible processes takes place: (a) growth step: with probability P add an isolated node (of degree k = 0) is added to the network. The node addition is followed by the addition of m edges between pairs of random nodes (which have not been connected before). This is done by repeating the following step m times: each time two random nodes (which have not been connected before) are selected and are connected to each other by an edge; (b) contraction step: with probability P del = 1 − P add a random node is deleted, together with its edges. When a growth step is selected at time t, the network size increases according to N t+1 = N t + 1, while the degrees of the m pairs of newly connected nodes increase from k i to k i + 1. When a contraction step is selected at time t, the network size decreases according to N t+1 = N t −1. Consider a node of degree k, whose neighbors are of degrees k ′ r , r = 1, 2, . . . , k. Upon deletion of such node the degrees of its neighbors are reduced to k ′ r − 1, r = 1, 2, . . . , k. We denote the initial number of nodes in the network at time t = 0 by N 0 . The expectation value of the number of nodes in the network at time t is N t = N 0 + ηt,(1) where η = P add − P del .(2) The parameter η provides a convenient classification of the possible scenarios. The case of pure growth is described by η = 1. For 0 < η < 1 the overall process is of network growth, while for −1 ≤ η < 0 the overall process is of network contraction. In the special case of η = 0 the network size remains the same, apart from possible fluctuations. It is convenient to express the probabilities P add and P del in terms of the parameter η, namely P add = 1 + η 2(3) and P del = 1 − η 2 .(4) In the case of −1 < η < 0 it is convenient to define the normalized time variable τ = |η|t N 0 ,(5) that measures the fraction of nodes that are deleted from the network up to time t. The expected size of the contracting network at time t can be expressed by N t = N 0 (1 − τ ). Note that the network vanishes at τ = 1. In the model considered here the m edges added at time t connect pairs of existing random nodes. This model is different from the random attachment model studied in Ref. [58], in which the new edges connect the new node to m random nodes in the network. Thus, in the model of Ref. [58] the degree of the new node upon its addition to the network is k = m. As a result, the degree distribution exhibits a cusp at k = m, separating between the regime of low degrees, k < m, and the regime of high degrees, k > m. In the model studied here the new node is added with degree k = 0 and gains links one at a time in subsequent time steps. As a result, the degree distribution exhibits the same functional form over the whole range of possible values of k. In that sense, the model studied here is somewhat simpler, while fundamentally belonging to the same class of random attachment models. III. THE MASTER EQUATION Consider an ensemble of networks of size N 0 at time t = 0, whose initial degree distribution is given by P 0 (k). The networks evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion). Below we derive a master equation [61,62] that describes the time evolution of the degree distribution P t (k) = N t (k) N t ,(6) where N t (k), k = 0, 1, . . . , is the number of nodes of degree k at time t and N t = k N t (k) is the network size at time t. The master equation formulation was used before in network growth processes [9,10] and in processes that combine growth and contraction [58][59][60]. In general, the master equation accounts for the time evolution of the degree distribution P t (k) over an ensemble of networks of the same initial size N 0 and initial degree distribution P 0 (k), which are exposed to the same dynamical processes. In order to derive the master equation, we first consider the time evolution of N t (k), which can be expressed in terms of the forward difference ∆ t N t (k) = N t+1 (k) − N t (k).(7) In the case of a growth step, the addition of an isolated node increases by 1 the number of nodes of degree k = 0, namely N t (0) → N t (0) + 1. The contribution of this process to the evolution of N t (k) is given by A t (k) = P add δ k,0 ,(8) where δ i,j is the Kronecker delta symbol. The probability that a random node of degree k will gain an additional edge at time t is given by U t (k → k + 1) = 2mP add N t (k) N t .(9) Similarly, the probability that a random node of degree k − 1 will gain an additional edge is U t (k − 1 → k) = 2mP add N t (k − 1) N t .(10) Here we use the convention that N t (−1) = 0. In the case of a contraction step, the probability that the node selected for deletion at time t is of degree k is given by N t (k)/N t . Thus, the rate of change of N t (k) due to a deletion of a node of degree k is given by D t (k) = −P del N t (k) N t .(11) Consider the case in which the process that takes place at time t is the deletion of a random node. In case that the deleted node is of degree k ′ , it affects k ′ adjacent nodes, which lose one link each. The probability of each one of these k ′ nodes to be of degree k is given by kN t (k)/[N t K t ], where K t is the mean degree. We denote by W t (k → k − 1) the expectation value of the number of nodes of degree k that lose a link at time t and are reduced to degree k − 1. Summing up over all possible values of k ′ , we find that the effect of node deletion on neighboring nodes of degree k is given by W t (k → k − 1) = P del kN t (k) N t .(12) Similarly, the effect on neighboring nodes of degree k + 1 accounts to W t (k + 1 → k) = P del (k + 1)N t (k + 1) N t .(13) Combining the effects on the time dependence of N t (k) we obtain ∆ t N t (k) = A t (k) + [U t (k − 1 → k) − U t (k → k + 1)] + D t (k) + [W t (k + 1 → k) − W t (k → k − 1)] .(14) Inserting the expressions for A t (k), D t (k), U t (k − 1 → k), U t (k → k + 1), W t (k → k − 1) and W t (k + 1 → k), from Eqs. (8), (11), (9), (10), (12) and (13), respectively, we obtain ∆ t N t (k) = P add δ k,0 + 2m N t (k − 1) − N t (k) N t + P del (k + 1)[N t (k + 1) − N t (k)] N t .(15) Since nodes are discrete entities the processes of node addition and deletion are intrinsically discrete. Therefore, the replacement of the forward difference ∆ t N t (k) by a time derivative of the form dN t (k)/dt involves an approximation. The error associated with this approximation was shown to be of order 1/N 2 t , which quickly vanishes for sufficiently large networks [48]. Therefore, the difference equation (15) can be replaced by the differential equation d dt N t (k) = P add δ k,0 + 2m N t (k − 1) − N t (k) N t + P del (k + 1)[N t (k + 1) − N t (k)] N t .(16) The derivation of the master equation is completed by taking the time derivative of Eq. (6), which is given by d dt P t (k) = 1 N t d dt N t (k) − N t (k) N 2 t d dt N t .(17) Inserting the time derivative of N t (k) from Eq. (16) and using the fact that dN t /dt = η [from Eq. (1)], we obtain the following master equation d dt P t (k) = 1 + η 2N t [δ k,0 − P t (k)] + m(1 + η) N t [P t (k − 1) − P t (k)] + 1 − η 2N t [(k + 1)P t (k + 1) − kP t (k)] ,(18) where we have also expressed P add and P del in terms of η, using Eqs. (3) and (4). In essence, the master equation consists of a set of coupled ordinary differential equations for P t (k), k = 0, 1, 2, . . . . In Eq. (18) we use the convention that P t (−1) = 0. For a given initial size N 0 and initial degree distribution P 0 (k), the master equation can be solved by direct numerical integration. In the case of pure growth (η = 1) the master equation is reduced to the form d dt P t (k) = 1 N t [δ k,0 − P t (k)] + 2m N t [P t (k − 1) − P t (k)].(19) IV. THE GENERATING FUNCTION Below we solve the master equation using the generating function formalism. We denote the generating function by G t (u) = ∞ k=0 u k P t (k),(20) which is the Z-transform of the degree distribution P t (k) [63]. Multiplying Eq. (18) by u k and summing up over k, we obtain a partial differential equation for the generating function, which is given by N 0 1 + ηt N 0 ∂G t (u) ∂t − 1 − η 2 (1 − u) ∂G t (u) ∂u + 1 + η 2 [2m(1 − u) + 1] G t (u) = 1 + η 2 . (21) This is a first order inhomogeneous linear partial differential equation of two variables. Note that η = 1 is a singular point of this differential equation. At η = 1 the coefficient of the term that includes the derivative of G t (u) with respect to u vanishes, thus reducing the order of the equation. This is reflected in the fact that for η = 1 the steady-state solution of Eq. (21) is of a different nature than the solution for −1 < η < 1, implying a structural phase transition at η = 1. For the analysis of Eq. (21) it is useful to define the parameter r = 1 + η 1 − η .(22) In the regime of overall network growth, in which 0 < η < 1, the parameter r is a monotonically increasing function of η, which rises from r = 1 for η = 0 to r → ∞ at η → 1. In the regime of overall network contraction, where −1 < η < 0, r is also a monotonically increasing function of η, which rises from r = 0 at η = −1 to r = 1 at η = 0. In Appendix A we use the method of characteristics to solve Eq. (21) and obtain the generating function G t (u) for −1 ≤ η < 1. It is given by G t (u) = α r t e −2rm(1−αt)(1−u) G 0 [1 − α t (1 − u)] + r 1 αt y r−1 e −2rm(1−u)(1−y) dy,(23) where G 0 (x) is the generating function of the initial degree distribution P 0 (k) and α t =            1 + ηt N 0 − 1−η 2η 0 < η < 1 exp − t 2N 0 η = 0 1 − |η|t N 0 1+|η| 2|η| −1 ≤ η < 0.(24) The generating function G t (u), given by Eq. (23), consists of two terms. The first term depends on the degree distribution of the initial network while the second term does not depend on the properties of the initial network. Note that G t (1) = 1, reflecting the normalization of the distribution P t (k). Plugging u = 1 in the first term of Eq. (23) shows that the weight of the first term is equal to w t = α r t ,(25) where α t decreases monotonically as time evolves (from its initial value of α 0 = 1). Therefore, the decay of w t as time evolves controls the rate at which the information about the initial network structure is lost. Note that in Eq. (24) the expression α t = (1+ηt/N 0 ) − 1−η 2η is valid for any η = 0. However, there is a qualitative difference in the behavior of α t between the regime of overall network growth (η > 0) and the regime of overall network contraction (η < 0). This difference is emphasized by the presentation of Eq. (24), where we express it somewhat differently in the two regimes. More specifically, in the regime of overall network growth the parameter α t gradually decreases towards zero as time evolves and the network continues to grow for an unlimited period of time. In contrast, in the regime of overall network contraction, α t reaches zero after a finite time, namely at t vanish = N 0 |η| ,(26) which is the time it takes for the network to vanish completely. r is given by Eq. (22). In case that η ≥ 0 the coefficient w t decreases monotonically as a function of t but converges towards 0 only asymptotically. In case that η < 0 the coefficient w t vanishes at a finite time t vanish = N 0 /|η|. The curve of w t vs. t/N 0 is convex for −1/3 < η < 0 and concave for −1 < η < −1/3. In Fig. 1 we present the coefficient w t as a function of t/N 0 for networks that evolve under a combination of growth (via random node addition and random attachment) and contraction (via random node deletion) for (a) 0 ≤ η < 1; and (b) −1 < η < 0, obtained from Eq. (24), where r is given by Eq. (22). In case that η ≥ 0 the coefficient w t decreases monotonically as a function of t but converges towards 0 only asymptotically. In case that η < 0, the coefficient w t vanishes after a finite time t vanish , given by Eq. (26). For −1 < η < 0 the weight w t can be expressed in the form w t = 1 − t t vanish 1−|η| 2|η| .(27) In this range the time derivative of w t is given by dw t dt = − 1 − |η| 2|η|t vanish 1 − t t vanish 1−3|η| 2|η| .(28) This derivative represents the rate at which the memory of the initial network is lost. For −1/3 < η < 0 the exponent in Eq. (28) is positive, while for −1 < η < −1/3 it is negative. Therefore, as η crosses −1/3 the derivative dw t /dt| t=t vanish changes discontinuously from 0 to −∞. Such discontinuous changes represent a typical behavior at a phase transition. In Fig. 2 we present the coefficient w t as a function of t/t vanish for networks that evolve under a combination of growth (via random node addition and random attachment) and contraction (via random node deletion) for −1 < η < 0. As t → t vanish the slope dw t /dt vanishes for −1/3 < η < 0 and diverges for −1 < η < −1/3. As time evolves, the first term in Eq. (23) decreases while the second term increases and flows towards an asymptotic state, given by G st (u) = r 1 0 y r−1 e −2rm(1−u)(1−y) dy.(29) Expressing the integral in terms of the lower incomplete gamma function γ(s, x), given by Eq. (A8) in Appendix A, we obtain G st (u) = re −2rm(1−u) [−2rm(1 − u)] −r γ[r, −2rm(1 − u)].(30) Using this notation, one can express Eq. (23) in the form G t (u) = α r t e −2rm(1−αt)(1−u) G 0 [1 − α t (1 − u)] + 1 − γ[r, −2rmα t (1 − u)] γ[r, −2rm(1 − u)] G st (u),(31) where the first term captures the memory of the degree distribution of the initial network while the second term includes the components that do not depend on the initial degree distribution. As time evolves, the first term decays while the second term converges towards the asymptotic form, given by Eq. (30). V. THE DEGREE DISTRIBUTION In Appendix A we extract the time dependent degree distribution P t (k) from the generating function G t (u). It is given by P t (k) = α r t e −2rm(1−αt) k! k i=0 k i α i t d i G 0 (u) du i u=1−αt [2rm(1 − α t )] k−i + re −2rm (2rm) k k! 1 αt y r−1 e 2rmy (1 − y) k dy.(32) The dependence of P t (k) on the initial degree distribution P 0 (k) is captured by first term of Eq. (32), while the second term is an asymptotic solution that does not depend on the initial condition. This asymptotic solution is essentially an attractive fixed point. The rate of convergence depends on the parameter η. More precisely, it is regulated by the coefficient w t = α r t which appears in front of the term that captures the initial condition. As mentioned in the previous section, the dependence of w t on time is different in the regime of overall network growth (η > 0) and the regime of overall network contraction (η < 0). For η > 0 the coefficient w t decays asymptotically like w t ∼ t − r r−1 .(33) Thus, for sufficiently long times the memory of the initial degree distribution is completely lost and P t (k) approaches its asymptotic form. In the case of η < 0 the coefficient w t decays as time evolves until it vanishes at a finite time t vanish . At the point η = −1/3 there is transition from a convex shape of w t as a function of the time t (for −1/3 < η < 0) to a concave shape (for −1 ≤ η < −1/3), as can be seen in Fig. 2. For η > −1/3, as t → t vanish the derivative dw t /dt → 0. In contrast, for η < −1/3, as t → t vanish the derivative dw t /dt → −∞. This sharp discontinuity in dw t /dt| t vanish at η = −1/3 pinpoints the location of the dynamical transition. Note that the value of η = −1/3 corresponds to the situation in which P add = 1/3 and P del = 2/3, namely on average there are two node deletion steps for each node addition step. From Eq. (32) one observes that on top of the overall dependence on w t , the rate of convergence of P t (k) towards its asymptotic value depends on the degree k. The asymptotic form of P t (k) in the long time limit can be obtained by inserting α t = 0 in Eq. (32). It yields P st (k) = re −2rm (2rm) k k! 1 0 y r−1 e 2rmy (1 − y) k dy.(34) The right hand side of Eq. (34) can be expressed in the form P st (k) = e −2rm (2rm) k k! rB(k + 1, r) 1 F 1   r k + r + 1 2rm   ,(35) where B(m, n) is the beta function and 1 F 1 (·) is the confluent hypergeometric function [64]. The tail of the steady state degree distribution P st (k), where k ≫ r can be reduced to P st (k) ≃ Γ(r + 1)k −r e −2rm (2rm) k k! .(36) This tail resembles the Poisson distribution in the sense that it satisfies the condition that P st (k)/P st (k − 1) ∝ 1/k. In the special case of η = 0 (where r = 1), which represents a perfect balance between the growth and contraction processes, the distribution P st (k) takes a particularly simple form P st (k; η = 0) = 1 2m 1 − Γ(k + 1, 2m) Γ(k + 1) ,(37) where Γ(s, x) is the upper incomplete gamma function, which can be expressed in terms of the lower incomplete gamma function, in the form Γ(s, x) = Γ(s) − γ(s, x). The steady state degree distribution for the special case of balanced growth and contraction was calculated in Ref. [58] for a slightly different model. The degree distribution P st (k; η = 0), given by Eq. (37), resembles the degree distribution presented in Eq. (20) of Ref. [58]. The difference in the pre-factors reflects the variation in the details of the growth mechanism between the two models. The discontinuity in the derivative dw t /dt| t vanish across η = −1/3 has interesting implications on the evolution of the degree distribution P t (k) in the late stages of the contraction process. For η > −1/3 there is a significant time window in which w t is small and thus the time dependent degree distribution P t (k) is in the vicinity of P st (k). In contrast, for η < −1/3 the weight w t decreases slowly until the very late stages of the contraction process and then falls down sharply as the time t vanish is approached. Therefore, there is only an extremely short time window in which P t (k) is in the vicinity of P st (k). As discussed in Sec. IV, the case of η = 1 corresponds to a singular point of the equation for the generating function G t (u) [Eq. (21)]. Therefore, this case requires a special treatment. In Appendix B we solve the master equation for the special case of pure growth (η = 1) and obtain the time dependent degree distribution P t (k) in this case too. It is given by P t (k; η = 1) = β 2m+1 t P 0 (k) + k i=1 β 2m+1 t i! (−2m ln β t ) i [P 0 (k − i) − P st (k − i; η = 1)] + 1 − β 2m+1 t P st (k; η = 1),(38) where β t is given by Eq. (B7) and P st (k; η = 1) = 1 2m + 1 2m 2m + 1 k(39) is the steady state degree distribution obtained at long times. Comparing Eq. (36) to Eq. (39) describing the degree distribution in the case of pure growth, we conclude that there is a phase transition at η = 1. In the case of pure growth (η = 1) the degree distribution follows an exponential distribution, whose tail decays more slowly than Eq. (36) that applies in the range of −1 < η < 1. Consider the special case in which the initial network is generated using the random attachment model. This model is obtained by choosing η = 1, where the number of edges added in each growth step is denoted by m 0 until the network size reaches N 0 nodes. Using the results of Appendix B, it is found that for a sufficiently large network size N 0 the generating function of the resulting network converges towards its steady state form, which is given by G 0 (u) = 1 2m 0 (1 − u) + 1 .(40) The initial network is then exposed to a combination of node addition with random attachment and random node deletion, characterized by −1 < η < 1, where the number of edges added in each growth step is m. Inserting G 0 (u) from Eq. (40) into Eq. (32) and carrying out the differentiation, we obtain P t (k) = α r t e −2rm(1−αt) 2m 0 α t + 1 k i=0 2m 0 α t 2m 0 α t + 1 i [2rm(1 − α t )] k−i (k − i)! + re −2rm (2rm) k k! 1 αt y r−1 e 2rmy (1 − y) k dy.(41) Interestingly, the sum in Eq. (41) takes the form of a convolution between an exponential distribution and a Poisson distribution. The mean of the exponential distribution is equal to 2m 0 α t , while the mean of the Poisson distribution is 2rm(1 − α t ). The exponential distribution descends from the intial degree distribution, which is given by Eq. Consider the case in which the initial network is an Erdős-Rényi network with mean degree c, whose degree distribution is known to be a Poisson distribution. In this case the time-dependent degree distribution takes a particularly simple form, namely P t (k) = α r t e −[αtc+2rm(1−αt)] [α t c + 2rm(1 − α t )] k k! + re −2rm (2rm) k k! 1 αt y r−1 e 2rmy (1 − y) k dy.(42) The first term in Eq. (42) represents a Poisson distribution whose mean degree evolves in time, extrapolating between the initial value of the mean degree, c, and a final value of 2rm. The second term does not depend on the initial network and is identical to the corresponding term that is obtained for other initial conditions. In this case the initial network is a maximal entropy network. For overall network contraction, under conditions of sufficiently high deletion rate (−1 < η < −1/3) the first term of Eq. (42) maintains this property for a long time window with a decreasing mean degree. This resembles the behavior in the limit of pure network contraction, discussed in Refs. [48,49]. In node addition and random attachment) and contraction (via random node deletion), in the special case of η = 0 in which the network size is fixed, apart from possible fluctuations. We also present simulation results (circles). The initial network is an ER network of size N 0 = 10 4 with mean degree c = 3. The analytical results are in very good agreement with the simulation results (circles), which are shown for t = 6N 0 , where the degree distribution has already converged to its asymptotic form P st (k). In Fig. 6 we present analytical results (solid lines) for the time-dependent degree distri- nodes. Thus, the initial degree distribution P 0 (k) is given by Eq. (39), with m replaced by m 0 . The simulation results (circles) are in very good agreement with the corresponding analytical results. As time evolves the time dependent degree distribution P t (k) converges towards the asymptotic distribution P st (k). For η = −1/4 the degree distribution P t (k) approaches P st (k) when a significant fraction of the network is still in place. In contrast, for η = −1/2 and −3/4 the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. The transition between the two dynamical behaviors takes place at η = −1/3. VI. THE MEAN AND VARIANCE OF THE DEGREE DISTRIBUTION The mean degree at time t can be obtained from K t = d du G t (u) u=1 .(43) Inserting G t (u) from Eq. (23) into Eq. (43), we obtain K t = α r+1 t K 0 + (1 − α r+1 t ) K st ,(44) where K st = 2rm r + 1 .(45) In Fig. 7 we present analytical results (solid lines), obtained from Eq. (44), for the mean degree K t vs. time t for networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) for (a) 0 ≤ η < 1; and (b) −1 < η < 0. The mean degree of the initial network is K 0 = 16. In case that η > 0 the mean degree gradually converges towards its asymptotic value. In case that η < 0 the network vanishes at a finite time t vanish = N 0 /|η|. In Fig. 8 we present analytical results (solid lines), for the mean degree K t vs. t/t vanish for networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) for −1 < η < 0. To obtain the variance Var t (K) we use the cumulant generating function, which is given by F t (x) = ln G t (e x ).(46) The variance is obtained from Inserting F t (x) from Eq. (46) into Eq. (47) we obtain Var t (K) = d 2 dx 2 F t (x) x=0 .(47)Var t (K) = α r+2 t Var 0 (K) + α r+1 t (α t − 1) K 2 0 + (α r+1 t − 2α t + 1) K 0 − α r+1 t (α r+1 t − 1) ( K 0 − K st ) 2 + 2α r+1 t (α t − 1)(r + 1) r + 1 r + 2 K st − K 0 K st + (1 − α r+1 t )Var st (K),(48) where Var st (K) = 2rm[(2m + 1)r 2 + 3r + 2] (r + 1) 2 (r + 2)(49) is the variance of P st (k), given by Eq. (35). Note that at t = 0 the right hand side of Eq. (48) is reduced to Var 0 (K) while in the long time limit it converges towards Var st (K). The mean K t (η = 1) and variance Var t (K; η = 1) of the degree distribution P t (k; η = 1) in the case of η = 1 are calculated in Appendix B. The steady state results K st (η = 1) and Var st (K; η = 1) coincide with those obtained from K t and Var t (K), respectively, in the limit of η → 1 (r → ∞). VII. SUMMARY AND DISCUSSION We presented analytical results for the time-dependent degree distribution P t (k) of networks that evolve under the combination of growth (via node addition and random attachment) and contraction (via random node deletion). In case that the rate of node addition exceeds the rate of node deletion, the overall process is of network growth, while in the opposite case the overall process is of network contraction. Using the master equation and the generating function formalism we obtained a closed form expression for the degree distribution P t (k). It includes a term that depends on the initial condition P 0 (k), which decays as time evolves, and a long-time asymptotic term P st (k), which is an attractive fixed point. Interestingly, the expression for P t (k) is identical in the regimes of overall growth and overall contraction. The model of network growth via node addition and random attachment can be considered as the simplest network growth model. It gives rise to networks that exhibit an exponential degree distribution. Similarly, the model of network contraction via random node deletion can be considered as the simplest network contraction model. The contracting networks converge towards the ER structure, which exhibits a Poisson degree distribution whose mean degree decreases as time proceeds. The combination of growth via node addition and random attachment and contraction via random node deletion yields novel structures which depend on the balance between the rates of the two processes. In Fig. 9 we present the phase diagram of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion), in terms of the growth rate −1 ≤ η ≤ 1. The case of η = 1 represents pure network growth via node addition and random attachment. The case of 0 < η < 1 represents a combination of growth and contraction where the overall process is of network growth. The case of η = 0 represents a balance between the growth and contraction processes such that on average the network size remains fixed. The case of −1 < η < 0 represents a combination of growth and contraction where the overall process is of network contraction. The case of η = −1 corresponds to pure contraction via random node deletion. At η = 1 there is a structural phase transition between the steady-state degree distribution at η = 1, which follows an exponential distribution, given by Eq. (39), and the steady-state degree distribution in the regime of 0 < η < 1, given by Eq. (35), which where the overall process is of network growth. The case of η = 0 represents a balance between the growth and contraction processes such that on average the network size remains fixed. The case of −1 < η < 0 represents a combination of growth and contraction where the overall process is of network contraction. The case of η = −1 corresponds to pure contraction via random node deletion. At η = 1 there is a structural phase transition between the exponential degree distribution in the asymptotic state for η = 1 and the asymptotic Poisson-like degree distribution in the regime of 0 < η < 1, whose tail decays faster than the exponential distribution. At η = 0 there is a phase transition between the η > 0 phase which exhibits an ever growing network whose degree distribution converges to an asymptotic form and the η < 0 phase in which the network vanishes after a finite time t vanish . At η = −1/3 there is a dynamical transition. For −1/3 < η < 0 the degree distribution P t (k) quickly converges towards P st (k). In contrast, for −1 < η < −1/3 the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. The phase transition at η = 1 essentially emanates from this singularity. At η = 0 there is a phase transition between the η > 0 phase which exhibits an ever growing network and the η < 0 phase in which the network vanishes after a finite time. Surprisingly, the expression for the time dependent degree distribution P t (k), given by Eq. (41), is identical on both sides of the transition. However, the qualitative behavior of the coefficient α t is fundamentally different on both sides. For η > 0 the coefficient α t gradually decays as time evolves but remains positive at any finite time. In contrast, for η < 0 it decays to zero after a finite time t vanish , at which the whole network vanishes. At η = −1/3 there is a dynamical transition between a phase of slow network contraction for −1/3 < η < 0 and a fast contracting phase for −1 ≤ η < −1/3. In the phase of slow contraction the degree distribution converges towards P st (k) and remains in its vicinity for a finite time window, before the network vanishes. In the fast contracting phase the network size quickly decreases and it vanishes before the weight of P st (k) becomes significant. In this case, the evolution of the degree distribution P t (k) during the contraction process qualitatively resembles the case of pure network contraction via random node deletion (η = −1), considered in Refs. [48,49]. The behavior of the degree distribution P t (k) in the scenario of overall network contraction −1 < η < 0 can be considered in the context of dynamical processes that exhibit intermediate asymptotic states [65,66]. These are states that appear at intermediate time scales, which are sufficiently long for such structures to build up, but shorter than the time scales at which the whole system disintegrates. The intermediate time scales can be made arbitrarily long by increasing the initial size of the system, justifying the term 'asymptotic'. More specifically, in the regime of −1/3 < η < 0 the intermediate asymptotic state exhibits the degree distribution P st (k), while in the regime of −1 ≤ η < −1/3 the intermediate asymptotic degree distribution is dominated by the first term of P t (k), given by Eq. (32). This work was supported by grant no. 2020720 from the United States-Israel Binational Science Foundation (BSF). Appendix A: Calculation of the degree distribution P t (k) In this Appendix we solve the master equation [Eq. (18)] for −1 ≤ η < 1 and obtain the time dependent degree distribution P t (k). In the first step we solve the differential equation (21) using the method of characteristics and obtain the time dependent generating function G t (u). The method of characteristics applies to hyperbolic partial differential equations. In this method the partial differential equation is reduced to a set of ordinary differential equations called characteristic equations. The characteristic equations of Eq. (21) can be written as du dt = − 1 − η 2 1 − u N 0 + ηt (A1) and dG t (u) du = 1 + η 1 − η 2m + 1 1 − u G t (u) − 1 1 − u . (A2) Solving Eq. (A1), one obtains a relation between u and t, via an integration constant C 1 . In the case of η = 0, it is given by C 1 = (1 − u) 2η 1−η N 0 + ηt ,(A3) while in the case of η = 0 it is given by C 1 = (1 − u)e −t/2N 0 .(A4) In order to solve Eq. (A2), we express the generating function in the form G t (u) = G (h) t (u) + G (p) t (u),(A5) where G t is the inhomogeneous part of G t (u). Solving for the homogeneous part, we obtain G (h) t (u) = C 2 e 2rmu (1 − u) −r ,(A6) where C 2 is an integration constant, and r is defined in Eq. (22). Solving Eq. (A2) for the inhomogeneous part of G t (u), we obtain G (p) t (u) = re −2rm(1−u) γ[r, −2rm(1 − u)] [−2rm(1 − u)] r ,(A7) where γ(s, x) = x 0 t s−1 e −t dt (A8) is the lower incomplete gamma function [64]. Inserting G (h) t (u) from Eq. (A6) and G (p) t (u) from Eq. (A7) into Eq. (A5) and extracting the integration constant C 2 , we obtain C 2 = e −2rmu (1 − u) r G t (u) − re −2rm (1 − u) r γ[r, −2rm(1 − u)] (−2rm) r .(A9) Starting with the case of η = 0, we combine the solutions of the two characteristic equations and obtain the solution of Eq. (21), which is given by G t (u) = e 2rmu (1 − u) −r F (1 − u) 2η 1−η N 0 + ηt + re −2rm(1−u) γ[r, −2rm(1 − u)] [−2rm(1 − u)] r ,(A10) where F is an arbitrary function. In order to impose the initial condition G 0 (u) we set t = 0 in Eq. (A10) and obtain G 0 (u) = e 2rmu (1 − u) −r F (1 − u) 2η 1−η N 0 + re −2rm(1−u) γ[r, −2rm(1 − u)] [−2rm(1 − u)] r .(A11) Solving for the arbitrary function F , we obtain F (1 − u) 2η 1−η N 0 = e −2rmu (1 − u) r G 0 (u) − re −2rm γ[r, −2rm(1 − u)] (−2rm) r .(A12) We introduce the variable z = (1 − u) 2η 1−η N 0 .(A13) Expressing u in terms of z, we obtain Inserting F (z) from Eq. (A15) into Eq. (A10), we obtain G t (u) = α r t e −2rm(1−u)(1−αt ) G 0 [1 − α t (1 − u)] + re −2rm(1−u) γ[r, −2rm(1 − u)] − γ[r, −2rmα t (1 − u)] [−2rm(1 − u)] r ,(A16) where α t = 1 + ηt N 0 − 1−η 2η .(A17) A similar analysis applies to the special case of η = 0. In this case one needs to use the special expression for C 1 , given by Eq. (A4). It yields the same form of G t (u), given by Eq. (A16), but with a different expression for α t , which in the case of η = 0 is given by α t = exp − t 2N 0 .(A18) The time dependent degree distribution is obtained by differentiating the generating function G t (u): P t (k) = 1 k! ∂ k G t (u) ∂u k u=0 . (A23) Inserting G t (u) from Eq. (A22) into Eq. (A23), we obtain the main result of this Appendix, namely P t (k) = α r t e −2rm(1−αt) k! k i=0 k i α i t d i G 0 (u) du i u=1−αt [2rm(1 − α t )] k−i + re −2rm (2rm) k k! 1 αt y r−1 e 2rmy (1 − y) k dy.(A24) This is a closed form analytical expression for the time dependent degree distribution P t (k). It is based on the initial degree distribution P 0 (k), which is encoded in the generating function at time t = 0, G 0 (u). Appendix B: Calculation of P t (k) in the case of pure network growth The case of pure network growth via node addition and random attachment is obtained for η = 1. Inserting η = 1 in Eq. which is an exponential distribution. The mean of the distribution P st (k; η = 1) is given by K st (η = 1) = 2m,(B10) and its variance is given by Var st (K; η = 1) = 2m(2m + 1). The time dependent degree distribution is obtained by expanding the right hand side of Eq. (B6) in powers of u. It yields P t (k; η = 1) = β 2m+1 t P 0 (k) + β 2m+1 t k i=1 [2m ln β t ] i i! [P 0 (k − i) − P st (k − i; η = 1)] + 1 − β 2m+1 t P st (k; η = 1).(B12) The mean degree can be obtained from Eq. (43), where G t (u; η = 1) is taken from Eq. (B6). It is given by K t (η = 1) = β t K 0 + (1 − β t )2m.(B13) To obtain the variance Var t (K) we use the cumulant generating function, which is given by F t (x; η = 1) = ln G t (e x ; η = 1). The variance is obtained from Var t (K; η = 1) = d 2 dx 2 F t (x; η = 1) x=0 . (B15) Inserting FIG. 1 . 1(Color online) The coefficient w t as a function of t/N 0 for networks that evolve under a combination of growth via random node addition and random attachment and contraction via random node deletion for (a) 0 ≤ η < 1; and (b) −1 < η < 0, obtained from Eqs. (24)-(25), where FIG. 2 . 2(Color online) The coefficient w t as a function of t/t vanish for networks that evolve under a combination of growth via random node addition and random attachment and contraction via random node deletion for η = −1/11, −1/5, −1/3, −1/2 and −5/7 (from left to right), obtained from Eq.(27), which is valid for η < 0. The curve of w t vs. t/N 0 is convex for −1/3 < η < 0 and concave for −1 < η < −1/3, while for η = −1/3 it follows a straight line. (39), while the Poisson distribution emerges from the dynamics of the attachment and deletion processes.The Poisson distribution describes the degree distribution of an Erdős-Rényi network, which is a maximal entropy network with a given value of the mean degree. Therefore, the Poisson distribution in Eq. (41) reflects the randomization of the degrees as the network evolves in time. Fig. 3 FIG. 3 .FIG. 4 . 334we present analytical results (solid line), obtained from Eq.(39), for the steadystate degree distribution P st (k) of networks that evolve under conditions of pure growth (η = 1) via node addition and random attachment with m = 4. To examine the convergence towards the steady-state degree distribution, we also present simulation results (circles) for the time-dependent degree distribution P t (k) for a network grown from an initial ER network of size N 0 = 100 with mean degree c = 3 up to a size of N = 10 4 . The tail of the degree distribution obtained from the simulations deviates from the steady state distribution. This deviation is due to the slow convergence of P t (k) towards P st (k) in the case η = 1. This conclusion is supported by the very good agreement between the simulation results (circles) and the corresponding analytical results (dashed line) for P t (k) at t = N − N 0 , obtained from Eq.(38).InFig. 4we present analytical results (solid lines), obtained from Eq.(35), for the steadystate degree distributions P st (k) of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) in the regime of overall network growth (0 < η < 1). Results are presented for (a) η = 3/4, (b) η = 1/2 and (c) η = 1/4. We also present simulation results (circles), which are shown for (Color online) Analytical results (solid line) for the asymptotic degree distribution P st (k) of networks that evolves under conditions of pure growth (η = 1) via node addition and random attachment with m = 4, obtained from Eq.(39). To examine the convergence towards the steady state, we also present simulation results (circles) for the time dependent degree distribution P t (k) for a network grown from an initial ER network of size N 0 = 100 with mean degree c = 3 up to a size of N = 10 4 . The tail of the degree distribution obtained from the simulations deviates from the steady state distribution. This deviation is due to the slow convergence of P t (k) towards P st (k) in the case η = 1. This conclusion is supported by the very good agreement between the simulation results (circles) and the corresponding analytical results (dashed line) for P t (k) at t = N − N 0 , obtained from Eq.(38). N = 10, 000. The initial network used in the simulations is an ER network of size N 0 = 100 with mean degree c = 3. In the case of η = 1/2 and η = 1/4 the analytical results are in very good agreement with the simulation results, which means that the degree distribution in the simulation has already converged to its steady-state form P st (k). In the case of η = 3/4 one finds that at N = 10, 000 the tail of the degree distribution P t (k) deviates from the steady-state distribution P st (k). This deviation is due to the slow convergence of P t (k) as η is increased towards 1. To justify this conclusion, we also present analytical results (dashed line) for P t (k), obtained from Eq. (42) at t = (N − N 0 )/η, which are in very good agreement with the simulation results (circles).InFig. 5we present analytical results (solid lines), obtained from Eq. (37), for the steadystate degree distribution P st (k) of networks that evolve under a combination of growth (Color online) Analytical results (solid lines), obtained from Eq. (35), for the steadystate degree distributions P st (k) of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) in the regime of overall network growth (0 < η < 1). Results are presented for (a) η = 3/4, (b) η = 1/2 and (c) η = 1/4. We also present simulation results (circles), which are shown for N = 10, 000. The initial network used in the simulations is an ER network of size N 0 = 100 with mean degree c = 3. In the case of η = 1/2 and η = 1/4 the analytical results are in very good agreement with the simulation results, which means that the degree distribution in the simulation has already converged to its steady-state form P st (k). In the case of η = 3/4 one finds that at N = 10, 000 the tail of the degree distribution P t (k) deviates from the steady-state distribution P st (k). This deviation is due to the slow convergence of P t (k) as η is increased towards 1. To justify this conclusion, we also present analytical results (dashed line) for P t (k), obtained from Eq. (42) at t = (N − N 0 )/η, which are in very good agreement with the simulation results (circles). online) Analytical results (solid lines) for the asymptotic degree distributions P st (k) of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) in the special case of η = 0 in which the network size is fixed, apart from possible fluctuations. The initial network is an ER network of size N 0 = 10 4 with mean degree c = 3. The analytical results for P st (k) are obtained from Eq. (37).The analytical results are in very good agreement with the simulation results (circles), which are shown for t = 6N 0 , where the degree distribution has already converged to its asymptotic form P st (k). butions P t (k) of networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) in the regime of overallnetwork contraction for (a) η = −1/4, (b) η = −1/2 and (c) η = −3/4. In each frame the degree distribution P t (k), obtained from Eq. (41), is shown (right to left) for τ = 0, τ = 1/4, τ = 1/2 and τ = 3/4, where the normalized time τ is the fraction of nodes that have been deleted [Eq. (5)]. The long-time degree distribution P st (k), obtained from Eq. (35), is also shown (dashed lines). The initial condition at t = 0 is a network obtained from random node addition and random attachment with m 0 = 8 and it consists of N = 12, 500 FIG. 6 . 6(Color online) Analytical results (solid lines) for the degree distributions of networks that evolve under a combination of growth via random node addition and random attachment and contraction via random node deletion in the regime of overall network contraction for (a) η = −1/4, (b) η = −1/2 and (c) η = −3/4. In each frame the degree distribution P t (k) is shown (right to left) for τ = 0, τ = 1/4, τ = 1/2 and τ = 3/4, where the normalized time τ is the fraction of nodes that have been deleted [Eq. (5)]. The asymptotic distribution P st (k) is also shown (dashed lines). The initial network is obtained from random node addition and random attachment with m 0 = 8 and it consists of N 0 = 12, 500 nodes. The analytical results for P t (k), are obtained from Eq. (41). The simulation results (circles) are in very good agreement with the corresponding analytical results. As time evolves the time dependent degree distribution P t (k) converges towards the asymptotic distribution P st (k). For η = −1/4, the degree distribution P t (k) approaches P st (k)when a significant fraction of the network is still in place. In contrast, for η = −1/2 and −3/4 the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. The transition between the two dynamical behaviors takes place at η = −1/3. FIG. 7 . 7(Color online) Analytical results (solid lines), obtained from Eq. (44), for the mean degree K t vs. time t for networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) for (a) from top to bottom); and (b) η = −1/10, −1/4, −1/2 and −3/4 (from top to bottom). In all cases the initial network has a mean degree of K 0 = 16. In case that η > 0 the mean degree gradually converges towards its asymptotic value. In case that η < 0 the network vanishes at a finite time t vanish = N 0 /|η|. FIG. 8 . 8(Color online) Analytical results (solid lines), obtained from Eq. (44), for the mean degree K t vs. t/t vanish for networks that evolve under a combination of growth (via node addition and random attachment) and contraction (via random node deletion) for η = 10 (from top to bottom). The initial network has a mean degree of K 0 = 16. FIG. 9 . 9The phase diagram of networks that evolve under a combination of growth via random node addition and random attachment and contraction via random node deletion, in terms of the growth rate −1 ≤ η ≤ 1. The case of η = 1 represents pure network growth via node addition and random attachment. The case of 0 < η < 1 represents a combination of growth and contraction decays like a Poisson distribution. This degree distribution essentially consists of a linear combination of Poisson distributions. Its tail is dominated by the Poisson component with the largest mean degree, given by Eq. (36). This transition implies that even the slightest rate of node deletion leads to a qualitative change in the nature of the steady state degree distribution. From a technical point of view, η = 1 is a singular point in the differential equation (21) for the generating function G t (u), where the order of the equation changes. . (A12) in terms of the variable z, we obtainF (z) = e −2rm 1−(zN 0 ) 1−η 2η (zN 0 ) r(1−η) 2η G 0 1 − (zN 0 ) 1−η 2η− re −2rm γ r, −2rm(zN 0 ) To simplify Eq. (A16) we first denoteS(u) = γ[r, −2rm(1 − u)] − γ[r, −2rmα t (1 − u)].(A19)Replacing γ(s, x) by its integral representation (A8), one can express S(u) in the form = −2rm(1 − u)y in Eq. (A20), we obtainS(u) = [−2rm(1 − u)] r 1 αt y r−1 e 2rm(1−u)y dy. (A21) Plugging S(u) from Eq. (A21) into Eq. 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Exact solutions for models of evolving networks with addition and deletion of nodes. C Moore, G Ghoshal, M E J Newman, Phys. Rev. E. 7436121C. Moore, G. Ghoshal and M.E.J. Newman, Exact solutions for models of evolving networks with addition and deletion of nodes, Phys. Rev. E 74, 036121 (2006). Uncovering the role of elementary processes in network evolution. G Ghoshal, L Chi, A.-L Barabási, Scientific Reports. 32920G. Ghoshal, L. Chi and A.-L. Barabási, Uncovering the role of elementary processes in network evolution, Scientific Reports 3, 2920 (2013). Topological phase transition in a network model with preferential attachment and node removal. H Bauke, C Moore, J B Rouquier, D Sherrington, Eur. Phys. J. B. 83519H. Bauke, C. Moore, J.B. Rouquier and D. Sherrington, Topological phase transition in a network model with preferential attachment and node removal, Eur. Phys. J. B 83, 519 (2011). N G Van Kampen, Stochastic Processes in Physics and Chemistry. AmsterdamNorth Holland3rd EditionN.G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd Edition (North Hol- land, Amsterdam, 2007). C Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. BerlinSpringer-Verlag3rd editionC. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sci- ences, 3rd edition, (Springer-Verlag, Berlin, 2004). C L Phillips, H T Nagle, Chakrabortty, Digital Control System: Analysis and Design. HarlowPearson EducationFourth EditionC.L. Phillips, H.T. Nagle and A Chakrabortty, Digital Control System: Analysis and Design, Fourth Edition (Pearson Education, Harlow, 2015). F W J Olver, D M Lozier, R R Boisvert, C W Clark, NIST Handbook of Mathematical Functions. CambridgeCambridge University PressF.W.J. Olver, D.M. Lozier, R.R. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010). . 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{'abstract': 'We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion).To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability P add and a random node deletion step takes place with probability P del = 1 − P add . The balance between the growth and contraction processes is captured by the parameter η = P add − P del . The case of pure network growth is described by η = 1.In case that 0 < η < 1 the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where −1 < η < 0, the overall process is of network contraction, while in the special case of η = 0 the expected size of the network remains fixed, apart from fluctuations. Using the master equation and the generating function formalism we obtain a closed form expression for the time dependent degree distribution P t (k). The degree distribution P t (k) includes a term that depends on the initial degree distribution P 0 (k), which decays as time evolves, and an asymptotic distribution P st (k) which is independent of the initial condition. In the case of pure network growth (η = 1) the asymptotic distribution P st (k) follows an exponential distribution, while for −1 < η < 1 it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth (0 < η < 1) the degree distribution P t (k) eventually converges to P st (k). In the case of overall network contraction (−1 < η < 0) we identify two different regimes. For −1/3 < η < 0 the degree distribution P t (k) quickly converges towards P st (k). In contrast, for −1 < η < −1/3 the convergence of P t (k) is initially very slow and it gets closer to P st (k) only shortly before the network vanishes. Thus, the model exhibits three phase transitions: a structural transition between two functional forms of P st (k) at η = 1, a transition between an overall growth and overall contraction at η = 0 and a dynamical transition between fast and slow convergence towards P st (k) at η = −1/3. The analytical results are found to be in very good agreement with the results obtained from computer simulations.', 'arxivid': '2209.10027', 'author': ['Barak Budnick \nRacah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael\n', 'Ofer Biham \nRacah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael\n', 'Eytan Katzav \nRacah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael\n'], 'authoraffiliation': ['Racah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael', 'Racah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael', 'Racah Institute of Physics\nThe Hebrew University\n9190401JerusalemIsrael'], 'corpusid': 252407696, 'doi': '10.1103/physreve.106.044305', 'github_urls': [], 'n_tokens_mistral': 23960, 'n_tokens_neox': 20968, 'n_words': 13645, 'pdfsha': '04f115a103d1ea6e0f73624c94f617af514dc435', 'pdfurls': ['https://export.arxiv.org/pdf/2209.10027v2.pdf'], 'title': ['The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion', 'The structure of networks that evolve under a combination of growth, via node addition and random attachment, and contraction, via random node deletion'], 'venue': []}

arxiv

Photons, neutrinos, and optical activity 15 Jul 2001 Ali Abbasabadi Department of Physical Sciences Ferris State University 49307Big RapidsMichiganUSA Wayne W Repko Department of Physics and Astronomy Michigan State University 48824East LansingMichiganUSA Photons, neutrinos, and optical activity 15 Jul 2001(Dated: March 25, 2022)numbers: 1315+g1460Lm1470Bh9530Cq We compute the one-loop helicity amplitudes for low-energy νγ → νγ scattering and its crossed channels in the standard model with massless neutrinos. In the center of mass, with √ s = 2ω ≪ 2me, the cross sections for these 2 → 2 channels grow roughly as ω 6 . The scattered photons in the elastic channel are circularly polarized and the net value of the polarization is non-zero. We also present a discussion of the optical activity of a sea of neutrinos and estimate the values of its index of refraction and rotary power. INTRODUCTION Neutrino-photon elastic scattering νγ → νγ and its crossed channels, which are of interest in astrophysical applications, have been studied using a variety of models for the weak interaction [1,2,3,4,5,6]. When the center of mass energy √ s is much less than twice the electron mass 2m e , the amplitudes for these processes with massless neutrinos are of leading order s 2 /m 4 W , where m W is W -boson mass [5]. This dependence leads to low-energy cross sections which grow as s 3 , with a scale set by m W . Apart from a factor ln(m 2 W /m 2 e ), there is no dependence on m e in leading order. Here, we present expressions for the low-energy neutrino-photon helicity amplitudes which are valid to higher orders in s, and contain terms whose scale is set by m 2 e . We find that the helicity flip amplitudes for the νγ → νγ and the helicity non-flip amplitudes for the γγ → νν, which vanish in leading order, are non-zero. Furthermore, the inclusion of higher powers of s, such as s 3 /m 6 W , enables us to use the forward elastic scattering amplitudes for νγ → νγ to study the optical activity of a sea of neutrinos [7]. In the next section, we use invariant decompositions of the amplitudes for νγ → νγ and its crossed channels to obtain properties of and restrictions on the corresponding helicity amplitudes. Section III gives the numerical results for the complete one-loop helicity dependent differential and total cross sections. This is followed by a discussion and conclusions, which include a treatment of the production of circularly polarized photons in low energy νγ → νγ scattering. We also calculate the index of refraction and the rotary power for a sea of neutrinos as a function of the energy of an incident photon and the temperature of the neutrinos. HELICITY AMPLITUDES The general expressions for the Lorentz-invariant, Bose symmetric, and time reversal invariant helicity amplitudes, A νγ→νγ λ1λ2 (s, t, u), for the process νγ → νγ can be found in Refs. [8,9]. They are A νγ→νγ ++ (s, t, u) = su cos(θ/2)F (s, t, u) ,(1)A νγ→νγ −− (s, t, u) = −s 2 cos(θ/2)F (u, t, s) ,(2)A νγ→νγ +− (s, t, u) = st cos(θ/2)[G(s, t, u) − G(u, t, s)] ,(3) A νγ→νγ −+ (s, t, u) = A νγ→νγ +− (s, t, u) ,(4) where the Mandelstam variables s, t, and u are defined by s = ( p 1 + k 1 ) 2 , t = (p 1 − p 2 ) 2 = − 1 2 s(1 − z), u = (p 1 − k 2 ) 2 = − 1 2 s(1 + z) , and z = cos θ, with θ the angle between the incoming neutrino, which is moving in the +z direction, and the outgoing neutrino. The helicity of the incoming photon is λ 1 = ±1 and the helicity of the outgoing photon is λ 2 = ±1. Here, the 4-momenta of the incoming neutrino and photon are p 1 and k 1 , respectively, with p 2 and k 2 denoting the corresponding outgoing momenta. To ensure the conservation for the angular momentum in Eqs. (1)-(3), it is necessary to require that the functions F (s, t, u), F (u, t, s), and [G(s, t, u) − G(u, t, s)] be non-singular in the limit u → 0 (backward scattering). In addition, the function [G(s, t, u) − G(u, t, s)] must also be non-singular in the limit t → 0 (forward scattering). From the Eqs. (1)-(4), the interchange of s and u results in the following relation A νγ→νγ λ1λ2 (s, t, u) = A νγ→νγ −λ1−λ2 (u, t, s) ,(5) where, under this interchange, we have changed the fac- tor s cos(θ/2) = s −u/s to u −s/u = −s −u/s = −s cos(θ/2). A similar decomposition of the helicity amplitudes A γγ→νν λ1λ2 (s, t, u) for the crossed channel process γγ → νν, expressions are A γγ→νν −+ (s, t, u) = 1 2 su sin θF (t, s, u) ,(6)A γγ→νν +− (s, t, u) = − 1 2 st sin θF (u, s, t) ,(7)A γγ→νν −− (s, t, u) = 1 2 s 2 sin θ[G(t, s, u) − G(u, s, t)] , (8) A γγ→νν ++ (s, t, u) = A γγ→νν −− (s, t, u) ,(9) with s, t, and u defined by s = (k 1 +k 2 ) 2 , t = (k 1 −p 1 ) 2 = − 1 2 s(1 − z), u = (k 1 − p 2 ) 2 = − 1 2 s(1 + z) , and z = cos θ, where θ is the angle between the incoming photon 1, which is moving in the +z direction, and the outgoing neutrino. The helicities of the incoming photons are λ 1 = ±1 and λ 2 = ±1, and the incoming photons have 4momenta k 1 and k 2 , while p 1 and p 2 are the momenta of the outgoing neutrino and anti-neutrino, respectively. In this case, conservation of angular momentum in Eqs. (6)- (8) requires that the functions F (t, s, u), F (u, s, t), and [G(t, s, u) − G(u, s, t)] be non-singular in the limit u → 0. This is sufficient to make these functions non-singular in the limit t → 0. Using Eqs. (6)-(9), the interchange of t and u results in the following relation A γγ→νν λ1λ2 (s, t, u) = −A γγ→νν −λ1−λ2 (s, u, t) .(10) In addition, the comparison of Eqs. (1)-(4) with Eqs. (6)- (9) shows that A γγ→νν λ1λ2 (t, s, u) = −iA νγ→νγ −λ1λ2 (s, t, u) .(11) Here, under the interchange of s and t, we have changed the factor s sin θ = 2s tu/s 2 to 2t su/t 2 = −2 √ su = −2is (1 + z)/2 = −2is cos(θ/2). Using the invariance of the helicity amplitudes under the CPT operation, we obtain the helicity amplitudes for the processesνγ →νγ and νν → γγ from those of νγ → νγ and γγ → νν, respectively. The results are Aν γ→νγ λ1λ2 (s, t, u) = A νγ→νγ −λ1−λ2 (s, t, u) ,(12) A νν→γγ λ1λ2 (s, t, u) = A γγ→νν −λ1−λ2 (s, t, u) . We have calculated the amplitudes for the diagrams of Fig. 1, for νγ → νγ and its crossed channels, in a nonlinear R ξ gauge such that the coupling between the photon, the W -boson, and the Goldstone boson (φ) vanishes [5,11,12]. Since the Goldstone boson-electron couplings introduce a factor m 2 e /m 2 W , and we are keeping terms of this order in our amplitudes, the contribution from the Goldstone boson, in the diagrams of Fig. 1, must be included. For zero neutrino mass, the two sets of Wexchange and φ-exchange diagrams are separately gauge invariant. Also, the contributions of these two sets of diagrams to the helicity amplitudes for νγ → νγ separately have the structure of Eqs. (1)-(4). This is also true for Using the algebraic manipulation software form [13] and schoonschip [14], we have expressed the diagrams in terms of Feynman parameter integrals, and for √ s ≪ 2m e , have expanded these amplitudes in a power series in s/m 2 e , t/m 2 e , and m 2 e /m 2 W . The results of the calculation for the functions F (s, t, u) and G(s, t, u) − G(u, t, s), are F (s, t, u) = α 2 8m 4 W sin 2 θ W f (s, t, u) , (14) G(s, t, u) − G(u, t, s) = α 2 8m 4 W sin 2 θ W g(s, t, u) ,(15) Here, θ W is the weak mixing angle, α is the fine structure constant, and we have neglected higher powers of s/m 2 e , t/m 2 e , and m 2 e /m 2 W . The first two terms of f (s, t, u) in Eq. (16) were previously derived in the Ref. [5]. These results show that there are many higher order terms whose scale is set by m 2 e . Note, however, that in the forward direction (t = 0) the scale in Eq. (16) is set by m 2 W . This suggests that Eq. (16) is valid even for √ s > m e when t = 0. We can confirm the validity of the forward limit of Eq. (16) for the range √ s ≪ m W by using the dispersion relation A νγ→νγ λλ (s, 0, −s) = s 2 π ∞ (mW +me) 2 ds ′ s ′ × σ λ (s ′ ) s ′ − s + σ λ (−s ′ ) s ′ + s ,(18) to obtain the exact value of the non-flip forward helicity amplitude A νγ→νγ λλ (s, 0, −s) for s < (m W + m e ) 2 . The νγ→e − W + for the process νγ → e − W + , after summation over the helicities of the W -boson and the electron (λ is helicity of the photon). Using Eqs. (5) and (18), it can be shown that the following symmetry relation must exist σ λ (−s ′ ) = σ −λ (s ′ ) .(19) Therefore, we can write [8] A νγ→νγ λλ (s, 0, −s) = s 2 π ∞ (mW +me) 2 ds ′ s ′ × σ λ (s ′ ) s ′ − s + σ −λ (s ′ ) s ′ + s .(20) A calculation of σ λ (s) gives σ λ (s) = √ 2G F α 2 s − λ m 2 e s 2 + 6λ m 2 W s 2 − 2 m 2 e m 2 W s 3 −2 m 4 e s 3 + 4 m 4 W s 3 µ(s, m 2 e , m 2 W ) + −2λ m 2 W s + m 2 e m 2 W s 2 + 3λ m 2 e m 2 W s 2 + m 4 e s 2 −2 m 4 W s 2 + 2λ m 4 W s 2 + m 2 e m 4 W s 3 − 2 m 4 e m 2 W s 3 − m 6 e s 3 + 2 m 6 W s 3 ℓ(s, m 2 e , m 2 W ) + 1 2 m 2 e s + λ 2 m 2 e s + m 2 W s − λ m 2 W s + m 2 e m 2 W s 2 + 3λ m 2 e m 2 W s 2 − 2 m 4 W s 2 + 2λ m 4 W s 2 + m 4 e s 2 + m 2 e m 4 W s 3 − 2 m 4 e m 2 W s 3 − m 6 e s 3 + 2 m 6 W s 3 ℓ(s, m 2 W , m 2 e ) ,(21) where µ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz , (22) ℓ(x, y, z) = ln x − y + z + µ(x, y, z) x − y + z − µ(x, y, z) ,(23) and G F = πα/( √ 2 m 2 W sin 2 θ W ) is the Fermi coupling. After setting the electron mass to zero everywhere but in the logarithm, our spin averaged cross section (σ + + σ − )/2 agrees with the result previously obtained by Seckel [15]. 5) and (16), and although Eq. (16) was derived for √ s ≪ 2m e , its range of validity for t = 0, as suggested above, extends to √ s ≪ m W . Assuming √ s ≪ m W , an expansion of the dispersion integral Eq. (20) to order s 3 /m 6 W gives A νγ→νγ ++ (s, 0, −s) = −α 2 s 2 8m 4 W sin 2 θ W f (s, 0, −s) , (24) A νγ→νγ −− (s, 0, −s) = −α 2 s 2 4 2 f (−s, 0, s) , (25) DIFFERENTIAL AND TOTAL CROSS SECTIONS In Fig. 2, we show the differential cross sections, for νγ → νγ, using dσ νγ→νγ λ1λ2 dz = 1 32πs |A νγ→νγ λ1λ2 | 2 ,(27) where λ 1 and λ 2 are the helicities of the incoming and outgoing photons, respectively. Here, z = cos θ, and θ is the angle between the incoming neutrino, which is moving in the +z direction, and the outgoing neutrino. The total cross sections for helicities λ 1 and λ 2 are given by σ νγ→νγ λ1λ2 = 1 −1 dσ νγ→νγ λ1λ2 dz dz ,(28) and are plotted in Fig. 3. Also shown in dots is the helicity flip cross section, which can be seen to be much smaller than the cross sections for helicity non-flip. This feature seems not to be a consequence of any symmetry. Figure 3 illustrates the roughly s 3 behavior of the total cross section for helicity non-flip, and s 5 behavior for the helicity flip, at photon energies ω ≪ m e . A fit to the points in Fig. 3 gives σ νγ→νγ −− = 3.9 × 10 −32 ω m e 6 pb ,(29)σ νγ→νγ ++ = 2.0 × 10 −32 ω m e 6 pb ,(30)σ νγ→νγ −+ = 2.2 × 10 −38 ω 10 pb ,(31) with σ νγ→νγ +− = σ νγ→νγ −+ .(32) Here, ω = √ s/2 is the energy of a photon (or a neutrino), and m e is the mass of the electron. Therefore, the total cross section for an unpolarized initial photon can be approximated as σ νγ→νγ = 3.0 × 10 −32 ω m e 6 pb , ω ≪ m e .(33) This is the same ω 6 behavior as found in the numerical calculations of Ref. [8] for the region m e ≪ ω ≪ m W , but with a larger slope. In Fig. 4, we show the differential cross sections for γγ → νν, using dσ γγ→νν λ1λ2 dz = 1 32πs |A γγ→νν λ1λ2 | 2 ,(34) where λ 1 and λ 2 are the helicities of the incoming photons. In this case, z = cos θ, with θ being the angle between the incoming photon 1, which is moving in the +z direction, and the outgoing neutrino. The identity of the two helicity amplitudes A γγ→νν with σ γγ→νν ++ = σ γγ→νν −− ,(38)σ γγ→νν +− = σ γγ→νν −+ ,(39) which are the consequences of Eq. (10). The total cross section for the unpolarized initial photons can be approximated as σ γγ→νν = 2.0 × 10 −33 ω 6 pb , ω ≪ m e .(40) In this case, too, the ω 6 dependence is the same as that found in Ref. [10] for m e ≪ ω ≪ m W , but with a large slope. The differential cross sections for the processνν → γγ, for photons with helicities λ 1 and λ 2 , can be obtained from the following relation dσν ν→γγ λ1λ2 dz = dσ γγ→νν −λ1−λ2 dz ,(41) where z = cos θ, and on the left side of this equation, θ is the angle between the incoming anti-neutrino, which is moving in the +z direction, and the outgoing photon 1 with helicity λ 1 . The various helicity-dependent total cross sections are related as σν ν→γγ λλ = 1 2 σ γγ→νν −λ−λ ,(42)σν ν→γγ λ−λ = σ γγ→νν −λλ ,(43) and total cross section for the production of a pair of back-to-back photons can be obtained from σν ν→γγ = 1 2! λ1λ2 +1 −1 dσν ν→γγ λ1λ2 dz dz .(44) In view of Eq. (41), we have σν ν→γγ = 2 σ γγ→νν .(45) DISCUSSION AND CONCLUSIONS We have shown that the energy dependence of the cross sections σ νγ→νγ and σ γγ→νν at low energies, ω ≪ m e , is the same as that in the energy region, m e ≪ ω ≪ m W . As a result, the effective interaction introduced in Refs. [8] and [16] contains all the essential of the features of these cross sections. This includes the prediction that the final photons in the channel νγ → νγ acquire (parity violating) circular polarization. To investigate the degree of circular polarization of the final photon in the process νγ → νγ, we define the polarization P as P = σ −− + σ +− − σ −+ − σ ++ σ −− + σ +− + σ −+ + σ ++ ,(46) where σ λ1λ2 ≡ σ νγ→νγ λ1λ2 is defined in Eq. (28). It is clear from Eqs. (29)-(32) that σ +− = σ −+ ≪ σ −− , and σ −− ≃ 2 σ ++ . Therefore, for photons with the energies ω ≪ m e , Eq. (46) gives P ≃ 1 3 ,(47) which is independent of the ω. This result in comparable about 0.3 for photons with energies 1 GeV < ∼ ω < ∼ 10 GeV. The angular dependence, P(z), of the final photon's polarization in the process νγ → νγ, can be obtained from the differential form of Eq. (46), P(z) = dσ −− /dz − dσ ++ /dz dσ −− /dz + 2dσ +− /dz + dσ ++ /dz .(48) where the dσ λ1λ2 /dz ≡ dσ νγ→νγ λ1λ2 /dz are defined in the Eq. (27), and we have used the equality dσ +− /dz = dσ −+ /dz. The polarization P(z) is plotted in Fig. 6 as a function of z = cos θ, where θ is the angle between the incoming and the outgoing neutrinos. In this figure, the solid line is for photons of energy ω = m e /2, and the dashed line, which is taken from the Ref. [8], is for photons of energy ω = 10 GeV . It is clear from Fig. 6 that the polarization P(z) remains effectively unchanged for wide range of energies ω < ∼ 10 GeV. Notice that Eq. (48) can be approximated as P(z) ≃ 4 − (1 + z) 2 4 + (1 + z) 2 ,(49) which is independent of the energy of photon. Our low-energy helicity non-flip amplitudes, obtained from Eqs. (1) and (2) using Eq. (16), enable us to discuss the optical activity of a sea of neutrinos [7]. To do this, it is necessary to establish a relationship between the forward-scattering amplitude and the Lorentz-invariant helicity amplitude for the general case where photons and neutrinos are colliding non-collinearly. Following Møller [17,18], we define the Lorentz-invariant differential cross section dσ, for the general non-collinear process 1 + 2 → 3 + 4 as [19] dσ = |A(s, t, u)| 2 F dQ ,(50) where A(s, t, u) is the Lorentz-invariant amplitude, F is the Lorentz-invariant flux F = 4 (p 1 · p 2 ) 2 − m 2 1 m 2 2 ,(51) and dQ is the Lorentz-invariant phase space differential element dQ = (2π) 4 δ (4) (p 1 + p 2 − p 3 − p 4 ) × d 3 p 3 (2π) 3 2E 3 d 3 p 4 (2π) 3 2E 4 .(52) After integration over d 3 p 4 , we have dQ = p 2 3 16π 2 E 3 E 4 dΩ 3 | p 3 | E 3 − p 3 . p 4 | p 3 |E 4 .(53) Here, m i , E i , and p i are the mass, energy, and mo- The resulting dσ is essentially identical to Eq. (93) of Ref. [18]. From Eqs. (50), (51), and (53), it is clear that, in the case of forward elastic scattering (m 3 = m 1 , m 4 = m 2 ; p 3 = p 1 , p 4 = p 2 ) and for massless particles (m i = 0, i = 1, 2, 3, 4), we have dσ dΩ 3 t=0 = E 2 1 |A(s, 0, −s)| 2 16π 2 s 2 ,(54) where s = (p 1 + p 2 ) 2 = 4E 1 E 2 sin 2 (θ 12 /2), and θ 12 is the angle between the momenta p 1 and p 2 of the initial incoming particles 1 and 2. A comparison of Eq. (54) with the cusomary definition of the scattering amplitude, written for forward scattering as dσ dΩ 3 t=0 = |f (0)| 2 ,(55) gives f (0) = E 1 4πs A(s, 0, −s) .(56) To compute the optical activity of a neutrino sea, we consider a photon of helicity λ and energy ω traversing a bath of neutrinos that are in thermal equilibrium at the temperature T ν . To give an order of magnitude estimate of the index of refraction n λ of this sea, we write [20,21] n λ − 1 = 2π ω 2 dN ν f νγ→νγ λλ (0) ,(57) where the forward-scattering amplitude f νγ→νγ λλ (0), from the Eq. (56), is f νγ→νγ λλ (0) = ω 4πs A νγ→νγ λλ (s, 0, −s) .(58) The Fermi-Dirac distribution, dN ν , is dN ν = 1 (2π) 3 d 3 p ν e Eν /Tν + 1 ,(59) and we have neglected the chemical potential for the neutrinos [22]. Here, A νγ→νγ λλ (s, 0, −s) is given in the Eqs. (24)-(26), p ν and E ν are the momentum and energy of a neutrino, and s = 4ωE ν sin 2 (θ νγ /2), where θ νγ is the angle between the incoming photon and the incoming neutrino. Therefore, Eqs. (57) and (58) give n λ − 1 = dN ν 2ωs A νγ→νγ λλ (s, 0, −s) .(60) After using Eq. (59) for dN ν , Eqs. (24)-(26) for the amplitudes A νγ→νγ λλ (s, 0, −s), and performing integration in the Eq. (60), we obtain n + − 1 = T 4 ν m 4 W c 0 + ωT 5 ν m 6 W c 1 ,(61)n − − 1 = T 4 ν 4 c 0 − ωT 5 ν 6 c 1 ,(62) where (for α = 1/137) c 0 = 7α 2 ζ(4) 4π 2 sin 2 θ W ln m 2 W m 2 e + 3 4 ≃ 1.1 × 10 −3 ,(63)c 1 = 15α 2 ζ(5) 4π 2 sin 2 θ W ln m 2 W m 2 e − 8 3 ≃ 2.0 × 10 −3 ,(64) and ζ(x) is the Riemann zeta function. The range of the validity of the above relations for the index of refraction, as far as energy is concerned, is related to that of Eqs. (24)-(26), which is s = 4ωE ν sin 2 (θ νγ /2) ≪ m 2 W . In Eq. (60), if we change the upper limit of the integration on E ν from the infinity to f T ν , the contributions of this integral to c 0 and c 1 in the Eqs. (61) and (62) change by 9% and 18%, respectively, if we use f = 7 (for f = 8, the corresponding changes are 4% and 10%). Here, we set the following criterion 4ωf T ν ≪ m 2 W ,(65) which for f = 7 is ωT ν ≪ 2.7 × 10 15 GeV · K ,(66) where ω is the photon energy in GeV, and T ν is the neutrino temperature in Kelvin. From Eqs. (61)-(64), we have n + − n − = 2 ωT 5 ν m 6 W c 1 ≃ 7.0 × 10 −80 ωT 5 ν ,(67) and the following approximate relation n + − 1 ≃ n − − 1 ≃ T 4 ν m 4 W c 0 ≃ 1.5 × 10 −63 T 4 ν .(68) Equation (68) implies that the index of refraction is independent of the helicity and the energy of the incident photon, as long as Eq. (66) is satisfied. When linearly polarized light propagates through a medium that has different indices of refraction for positive and negative helicities (n + = n − ), the plane of polarization of the light rotates by an angle φ, which is [23] φ = π λ γ (n + − n − ) l = ω 2 (n + − n − ) l ,(69) where ω and λ γ = 2π/ω are the energy and wavelength of the photon and l is the distance traveled by photons in the medium. To estimate the specific rotary power, φ/l, for a sea of neutrinos, we use Eqs. (67) and (69) to obtain φ l = ω 2 T 5 ν m 6 W c 1 ≃ 1.8 × 10 −64 ω 2 T 5 ν rad/m .(70) A positive angle of rotation, φ > 0, that is n + > n − , corresponds to a clockwise rotation (dextrorotation) of the plane of polarization of the linearly polarized incident photons, as viewed by an observer that is detecting the forward-scattered light. Thus, the optical activity of a neutrino sea is that of a dextrorotary medium. In addition, it is clear from Eq. (70) that the rotary power, φ/l, varies as 1/λ 2 γ , which is the same as that of quartz and most transparent substances for visible light. To get a rough estimate of rotation angle φ for linearly polarized photons propagating through the relic neutrino sea, we use Eq. (70) with l = ct, c = 3 × 10 8 m/s, t ∼ 15 × 10 9 years, T ν ∼ 2 K, and ω ∼ 10 20 eV, and for the neutrino part of the sea we find φ ∼ 8 × 10 −15 rad , which is exceedingly small. The antineutrino part of the sea gives a rotation with opposite sign, such that if the asymmetry parameter [24], L ≡ (N ν − Nν)/N γ , is zero, the resultant angle of rotation φ will be zero. The proper treatment for the case L = 0, is to include the chemical potential in the neutrino and antineutrino distribution functions. √ s = 20 GeV, which is taken from the Ref. [8]. have neglected terms in higher powers of m 2 e /m 2 W . The function f (−s, 0, s) can be found from Eq. (26) by replacing s by −s. Notice that Eqs. (24)-(26) are consistent with Eqs. ( Eq.(4). It also shows the vanishing of the amplitudes for backward scattering, and vanishing of the flip amplitudes for forward scattering, as given in Eqs. (1)-(4). 5 5Eq.(9), is also shown in this figure, as is the vanishing of the amplitudes for the forward and backward scattering, implied by Eqs. (6)-(9). Notice also that this figure clearly exhibits the symmetry relations of Eq. (10).The total cross sections for γγ → νν are plotted inFig. dotsis the cross section for the helicity non-flip, which can be seen to be much smaller than the helicity flip cross sections. Again, we see a roughly s 3 behavior of the dominant contributions to the total cross section, and a s 5 behavior of the non-leading contributions, at photon energies ω ≪ m e . Fits to the points in this case give FIG. 1 : 1Diagrams for νeγ → νeγ or γγ → νeνe. The diagram (d) will give zero contribution. For each of (a), (b), (c) there is also a diagram with the photons interchanged.FIG. 2: The helicity dependent differential cross sections for νγ → νγ are shown for √ s = me. The solid line is dσ−−/dz, the dashed line is dσ++/dz, and the dotted line is dσ+−/dz. The dσ−+/dz is the same as dσ+−/dz. FIG. 3: The helicity dependent total cross sections for νγ → νγ are shown. The solid line is σ−−, the dashed line is σ++, and the dotted line is σ+−. The σ−+ is the same as σ+−. FIG. 4: The helicity dependent differential cross sections for γγ → νν are shown for √ s = me. The solid line is dσ+−/dz, the dashed line is dσ−+/dz, and the dotted line is dσ++/dz. The dσ−−/dz is the same as dσ++/dz. FIG. 5 : 5The helicity dependent total cross sections for γγ → νν are shown. The solid line is σ+− and the dotted line is σ++. The σ−+ is the same as σ+−, and σ−− is the same as σ++.FIG. 6: The polarization P(z) of the final photons in the process νγ → νγ, as it is defined in the Eq. (48), is shown. The solid line is polarization for the center of mass energy √ s = me, while the dashed line is for the center of mass energy AcknowledgmentsWe wish to thank Duane Dicus for numerous helpful discussions. One of us (A.A.) wishes to thank the Department of Physics and Astronomy at Michigan State University for its hospitality and computer resources. This work was supported in part by the National Science Foundation under Grant No. PHY-0070443. . H.-Y Chiu, P Morrison, Phys. Rev. Lett. 5573H.-Y. Chiu and P. Morrison, Phys. Rev. Lett. 5, 573 (1960). . M J Levine, Nuovo Cimento 48A. 67M. J. Levine, Nuovo Cimento 48A, 67 (1967). . L F Landovitz, W M Schreiber, Nuovo Cimento 2A. 359L. F. Landovitz and W. M. Schreiber, Nuovo Cimento 2A, 359 (1971). . V K Cung, M Yoshimura, Nuovo Cimento 29A. V. K. Cung and M. Yoshimura, Nuovo Cimento 29A, . D A Dicus, W W Repko, Phys. Rev. D. 485106D. A. Dicus and W. W. Repko, Phys. Rev. D 48, 5106 (1993). . J Liu, Phys. Rev. D. 442879J. Liu, Phys. Rev. D 44, 2879 (1991). Optical activity of a neutrino sea in the Standard Model. G Karl, V Novikov, hep-ph/0009012G. Karl and V. Novikov, "Optical activity of a neutrino sea in the Standard Model," hep-ph/0009012. . A Abbasabadi, A Devoto, W W Repko, Phys. Rev. A. Abbasabadi, A. Devoto, and W. W. Repko, Phys. Rev. For a related decomposition in the case ν ′ → νγγ, see. J F Nieves, Phys. Rev. D. 281664For a related decomposition in the case ν ′ → νγγ, see J. F. Nieves, Phys. Rev. D 28, 1664 (1983). . A Abbasabadi, A Devoto, D A Dicus, W W Repko, Phys. Rev. D. 5913012A. Abbasabadi, A. Devoto, D. A. Dicus, and W. W. Repko, Phys. Rev. D 59, 013012 (1999). . M B Gavela, G Girardi, C Malleville, P Sorba, Nucl. Phys. 193257M. B. Gavela, G. Girardi, C. Malleville, and P. Sorba, Nucl. Phys. B193, 257 (1981); . M Bace, N D Hari Dass, Ann. of Phys. 94349M. Bace and N. D. Hari Dass, Ann. of Phys. 94, 349 (1975). . J F Nieves, P B Pal, D G Unger, Phys. Rev. D. 28908J. F. Nieves, P. B. Pal, and D. G. Unger, Phys. Rev. D 28, 908 (1983). New features of form. J A M Vermaseren, math- ph/0010025J. A. M. Vermaseren, "New features of form," math- ph/0010025. M J G Veltman ; Veltman, D N Williams, schoonschip A Program for Symbol Handling. University of MichiganSchoonschip '91," hep-ph/9306228M. J. G. Veltman, "schoonschip A Program for Sym- bol Handling," University of Michigan, report, 1984 (un- published). See also, M. Veltman and D. N. Williams, "Schoonschip '91," hep-ph/9306228. . D Seckel, Phys. Rev. Lett. 80900D. Seckel, Phys. Rev. Lett. 80, 900 (1998). . D A Dicus, K Kovner, W W Repko, Phys. Rev. D. 6253013D. A. Dicus, K. Kovner and W. W. Repko, Phys. Rev. D 62, 053013 (2000). . C Møller, Danske Videnskab, 23C. Møller, Danske Videnskab. Selskab 23, No. 1 (1945). For a reprint of the Ref. M H Ross, Quantum Scattering Theory. Bloomington, IndIndiana University Press109For a reprint of the Ref.[17], see M. H. Ross, Quantum Scattering Theory (Indiana University Press, Blooming- ton, Ind. 1963), page 109. For a brief disscusion of differential cross section for noncollinear scattering, see. R G Newton, Scattering Theory of Waves and Particles. NYMcGraw-Hillpage 220. See also page 91 of the Ref. [21For a brief disscusion of differential cross section for non- collinear scattering, see R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, NY, 1966), page 220. See also page 91 of the Ref. [21]. 57) is based on the standard Lorentz relation that connects the index of refraction to the forward-scattering amplitude, for a medium that is at T = 0 temperature. Here, we make a nontrivial assumption that the introduction of finite temperature in the medium, will not spoil the coherent condition that was necessary to derive the Lorentz relation. For a similar approach, see. P Langacker, J Liu, Notice that Eq. 464140Phys. Rev. DNotice that Eq. (57) is based on the standard Lorentz relation that connects the index of refraction to the forward-scattering amplitude, for a medium that is at T = 0 temperature. Here, we make a nontrivial assump- tion that the introduction of finite temperature in the medium, will not spoil the coherent condition that was necessary to derive the Lorentz relation. For a similar ap- proach, see P. Langacker and J. Liu, Phys. Rev. D 46, 4140 (1992). M L Goldberger, K M Watson, Collision Theory. NYWiley771M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, NY, 1964), page 771. P J E Peebles, Principles of Physical Cosmology. Princeton, NJPrinceton University Press160P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, NJ, 1993), page 160. For a discussion of the rotation of the polarization plane of a linearly polarized light in a gas, see the discussion leading to the Eq. V G Baryshevsky, hep-ph/0007353J. P. Mathieu. 8235Pergamon PressOpticsFor a discussion of the rotation of the polarization plane of a linearly polarized light in a gas, see the discussion leading to the Eq. (8) of V. G. Baryshevsky, "Time- reversal-violating birefringence of photon in a medium exposed to electric and magnetic field", hep-ph/0007353. For a general discussion of rotary power, see J. P. Math- ieu, Optics (Pergamon Press, NY, 1975), page 235; . M V Klein, Optics. 502John Wiley & SonsM. V. Klein, Optics (John Wiley & Sons, NY, 1970), page 502. . J Lesgourgues, S Pastor, Phys. Rev. D. 60103521J. Lesgourgues and S. Pastor, Phys. Rev. D 60, 103521 (1999).

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{'abstract': 'We compute the one-loop helicity amplitudes for low-energy νγ → νγ scattering and its crossed channels in the standard model with massless neutrinos. In the center of mass, with √ s = 2ω ≪ 2me, the cross sections for these 2 → 2 channels grow roughly as ω 6 . The scattered photons in the elastic channel are circularly polarized and the net value of the polarization is non-zero. We also present a discussion of the optical activity of a sea of neutrinos and estimate the values of its index of refraction and rotary power.', 'arxivid': 'hep-ph/0107166', 'author': ['Ali Abbasabadi \nDepartment of Physical Sciences\nFerris State University\n49307Big RapidsMichiganUSA\n', 'Wayne W Repko \nDepartment of Physics and Astronomy\nMichigan State University\n48824East LansingMichiganUSA\n', 'Ali Abbasabadi \nDepartment of Physical Sciences\nFerris State University\n49307Big RapidsMichiganUSA\n', 'Wayne W Repko \nDepartment of Physics and Astronomy\nMichigan State University\n48824East LansingMichiganUSA\n'], 'authoraffiliation': ['Department of Physical Sciences\nFerris State University\n49307Big RapidsMichiganUSA', 'Department of Physics and Astronomy\nMichigan State University\n48824East LansingMichiganUSA', 'Department of Physical Sciences\nFerris State University\n49307Big RapidsMichiganUSA', 'Department of Physics and Astronomy\nMichigan State University\n48824East LansingMichiganUSA'], 'corpusid': 119067197, 'doi': '10.1103/physrevd.64.113007', 'github_urls': [], 'n_tokens_mistral': 11128, 'n_tokens_neox': 9298, 'n_words': 5616, 'pdfsha': '3f2267fefb7382b5d1c02514122e29f09fb5524c', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/0107166v1.pdf'], 'title': ['Photons, neutrinos, and optical activity', 'Photons, neutrinos, and optical activity', 'Photons, neutrinos, and optical activity', 'Photons, neutrinos, and optical activity'], 'venue': []}

arxiv

EFFET DU CHLORURE DE SODIUM (NaCl) SUR LA CROISSANCE DE SIX ESPECES d'Acacia EFFECT OF SODIUM CHLORIDE (NaCl) ON THE GROWTH OF SIX Corresponding and Author |Received | 01 March 2017| |Accepted | 29 March 2017| |Published 10 April 2017 | |khalil Chérifi Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University | P.O. Box 8106 | 8000Agadir| Morocco | |abdelmjid Anagri Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University | P.O. Box 8106 | 8000Agadir| Morocco | | El Houssine Boufous American Journal of Innovative Research and Applied Sciences. ISSN Department of Biochemistry and Microbiology | Laval University | Quebec City (Quebec) 2429-5396 I www.american-jirasCanada | | | Abelhamid El Mousadik Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University | P.O. Box 8106 | 8000Agadir| Morocco | Acacia Species American Journal of Innovative Research and Applied Sciences. ISSN 2429-5396I www.american-jiras EFFET DU CHLORURE DE SODIUM (NaCl) SUR LA CROISSANCE DE SIX ESPECES d'Acacia EFFECT OF SODIUM CHLORIDE (NaCl) ON THE GROWTH OF SIX Corresponding and Author |Received | 01 March 2017| |Accepted | 29 March 2017| |Published 10 April 2017 |105 ORIGINAL ARTICLE See: http://creativecommons.org/licenses/by-nc/4.0/Mots-clés: Tolérance à la salinitéVariabilitéAcaciaAmélioration des plantesRéhabilitation Keywords: Salt toleranceVariabilityAcaciaPlant breedingRehabilitation RESUMEIntroduction : Au cours de ces dernières décennies on assiste à une diminution progressive des superficies cultivables dans les régions arides et semi-arides à cause de l'accumulation des sels liée à la rareté des précipitations, au mauvais drainage, à la sècheresse prolongées et à l'absorption de l'eau par les plantes. Contexte : Devant l'ampleur de ce problème, il s'avère donc nécessaire de proposer des programmes d'évaluation et de conservation des espèces menacées d'extinction. Le repérage d'espèces plus adaptées et la sélection des variétés tolérantes à la salinité resteraient la voie économique la plus efficace pour l'exploitation des terrains affectés. Objectifs : L'objectif de cette étude est de déterminer la capacité de tolérance à la salinité au cours du développement végétatif chez six espèces appartenant au genre Acacia. Ceci dans le but d'élaborer une classification des seuils de tolérance au stress salin, critère important dans le choix des espèces à retenir dans un programme de mise en valeur des zones affectées par la salinité. Méthodes : L'effet du stress salin a été abordé sur un certain nombre de caractères agro-morphologiques en conditions contrôlées. Les concentrations de NaCl appliquées, en plus du témoin, sont : 100 mM, 200 mM, 300 mM et 400 mM. Résultats : Les résultats ont montré une variabilité non négligeable dans le comportement des plantes des différentes espèces en fonction du stress salin. Chez les six espèces d'Acacia, le sel a exercé un effet dépressif sur tous les paramètres de croissance étudiés. Toutefois, le taux de réduction diffère selon l'intensité de stress salin et le degré de sensibilité ou de tolérance de l'espèce. La croissance en hauteur, le nombre de feuilles et la biomasse sèche totale sont vraisemblablement les paramètres les plus affectés. Cependant, il est important de signaler que toutes les espèces d'Acacia considérées dans ce travail ont survécu, même à 400 mM de NaCl, et ont présenté différents degrés de tolérance à la salinité. Dans cette situation, les espèces A. horrida et A. raddiana s'avèrent globalement les plus performantes au stade végétatif. Conclusions : La variabilité génétique, dévoilée par ces espèces dans les différentes conditions de stress salin, permettrait un choix d'écotypes pouvant entrer dans des schémas de sélection et d'amélioration variétale pour la réhabilitation des parcours dégradés surtout en zones affectées par la salinité.ABSTRACTBackground: Salinity is one of the major abiotic stresses affecting plant production in arid and semi-arid regions. It causes reduction of cultivable area and combined with other factors, presents a serious threat to food stability in these areas. Context: In front of this problem, the selection of salt tolerant species and varieties remains the best economic approach for exploitation and rehabilitation of salt-affected regions. Objective: The purpose of this study was to assess and compare the seed germination response of six Acacia species under different NaCl concentrations in order to explore opportunities for selection and breeding salt tolerant genotypes. Methods: The salinity effect was examined by measuring some agro-morphological parameters in controlled growth environment using five treatment levels: 0, 100, 200, 300 and 400 mM of NaCl. Results: The analyzed data revealed significant variability in salt response within and between species. All growth parameters were progressively reduced by increased NaCl concentrations. Growth in height, leaf number and total plant dry weight were considered as the most sensitive parameters. However, the growth reduction varied among species in accordance with their tolerance level. It is important to note that all species survived at the highest salinity (400 mM). Whereas A. horrida and A. raddiana were proved to be often the best tolerant, they recorded the lowest reduction percentage at this stage. Conclusion: The genetic variability found in the studied species at seedling stage may be used to select genotypes particularly suitable for rehabilitation and exploitation of lands affected by salinity. INTRODUCTION La salinisation est un processus important de dégradation des sols. Elle constitue un facteur limitant à la croissance et au développement des plantes. Les conséquences de ce phénomène qui ne cesse de prendre de l'ampleur, se manifestent par la toxicité directe due à l'accumulation excessive des ions (Na + et Cl -) dans les tissus des organes, et à un déséquilibre nutritionnel imputable essentiellement à des compétitions entre les éléments minéraux, tel que le sodium avec le potassium et le calcium, le chlorure avec le nitrate, le phosphate et le sulfate [1,2]. En conséquence, les glycophytes les plus tolérantes seront celles qui, tout en utilisant le Na + comme osmoticum, conserveront une forte sélectivité vis à vis du K + [3]. La stratégie utilisée par les végétaux pour éviter les problèmes d'excès d'ions tout en réalisant leur équilibre osmotique est la compartimentation cellulaire, qui se traduit par une accumulation préférentielle du Na + dans la vacuole [4]. Cependant, chez les glycophytes tolérantes, on discerne également une compartimentation à l'échelle de la plante, surtout dans les organes jeunes où la teneur en Na+ reste faible [5,6]. La variabilité pour la tolérance à la salinité a été étudiée chez beaucoup d'espèces [5,[7][8][9][10][11]. Pour des raisons pratiques, de nombreuses explorations de cette variabilité ont été abordées au stade végétatif sur des plantes très jeunes. Cette approche est justifiée par le fait que la réponse des plantules est parfois fortement prédictive de celle des plantes adultes [12][13][14]. Dans certains cas, l'écart entre la tolérance au stade plantule et celle au stade adulte peut justifier les différences entre les mécanismes impliqués d'un stade de développement à un autre [15][16][17][18][19]. Plusieurs recherches ont montré que la croissance en hauteur [20,21], la production de biomasse des tiges et des racines [22,23] est négativement affectée par l'augmentation de la salinité. Face à la salinisation des sols qui constitue l'un des facteurs abiotiques majeurs réduisant le rendement agricole, l'introduction d'espèces végétales tolérantes à la salinité est une stratégie alternative recommandée pour valoriser les sols touchés par ce phénomène. Cette approche, permettraient d'améliorer le couvert végétal et résoudre les problèmes de régénération de certaines espèces forestières en zones arides et semi-arides, particulièrement celles appartenant au genre Acacia qui représentent certainement une richesse écologique menacée en Afrique du Nord [24]. Acacia, appartenant à la famille des Fabacées, est identifié comme étant un genre cosmopolite, varié et riche. Ces espèces à usage multiple peuvent coloniser des sols pauvres grâce à leur capacité de fixer l'azote atmosphérique par leur association symbiotique avec le Rhizobium des nodosités racinaires. Au Maroc les écosystèmes à base d'Acacia représentent un enjeu stratégique pour les régions semi-arides, arides et sahariennes du pays aussi bien sur le plan écologique que socio-économique. Ces écosystèmes rares et originaux, peuvent constituer une protection naturel contre la désertification.et fournir un intérêt multifonctionnel et multi-usager, tels que leurs utilisations dans la production de bois d'énergie et de service, de gomme arabique, de substances pharmaceutiques, de fourrage, de produits mellifères, ainsi que leur utilisation en reboisement et en foresterie urbaine [25]. La maîtrise des exigences de croissance des plantules est une étape importante dans le succès des opérations de reboisement de ces espèces. Ce stade, très important pour le développement des plantes, est surtout contrôlé par des facteurs génétiques et environnementales, en particulier la salinité [26]. Malheureusement, au Maroc, peu de travaux de recherche sur le degré de tolérance à la salinité chez les Acacia ont été effectués. Notre présente étude s'inscrit dans le cadre d'une évaluation de la variabilité des réponses au stade plantule de six espèces d'Acacia soumises à des doses croissantes de NaCl. L'effet du stress salin a été abordé sur un certain nombre de caractères agro-morphologiques en conditions contrôlées. Ceci dans le but d'identifier le matériel végétal le plus performant pour des programmes de restauration et de valorisation de la productivité végétale, particulièrement pour la réhabilitation des parcours dégradés, c'est le cas d'Acacia gummifera et Acacia raddiana, ainsi que pour des projets de reboisements en milieux affectés par la salinité, dans le cas des espèces exotiques, comprenant Acacia eburnea, Acacia cyanophylla, Acacia cyclops et Acacia horrida. MATERIELS ET METHODES Matériel végétal L'analyse de la diversité de la tolérance à la salinité a porté sur deux espèces autochtones, représentées par Acacia gummifera et Acacia raddiana et quatre espèces exotiques, représentées par Acacia eburnea, Acacia cyanophylla, Acacia cyclops et Acacia horrida. La majeure partie des graines des différentes espèces testées ont été collectées sous forme de gousses dans différentes régions du sud-ouest marocain (Tableau 1). Elles nous ont été aimablement fournies par la station régionale des semences forestières de Marrakech et ont été conservées au froid (4°C) jusqu'à l'analyse. Caractères mesurés Après huit semaines de culture, quatre paramètres ont été mesurés dans différentes conditions de stress salin (Tableau 2). Les caractères retenus se rapportent au développement végétatif des plantes ainsi qu'à l'estimation de la valeur fourragère :  Nombre totale de feuilles (Nbr.Fll). Ces mêmes critères ont été aussi utilisés par d'autres auteurs dans l'estimation de la croissance de la biomasse chez certaines espèces d'Acacia cultivées sous stress salin [29,30]. Tableau 2: Le tableau montre les paramètres mesurés dans l'évaluation de l'effet de la salinité au stade végétatif. Code des caractères Signification Analyses statistiques Les six espèces ont été traitées selon un dispositif complètement randomisé, à raison de 10 plantes par population et par traitement. Les données relatives aux pourcentages de réduction des différents paramètres de croissance ont été transformées en arcsin racine carrée avant d'être soumises à l'analyse de variance à deux critères de classification (espèce et [NaCl]). La comparaison des moyennes entre les différentes espèces, pour chaque traitement, a été réalisée par le test de Newman et Keuls. Pour chaque concentration, les espèces dont les moyennes ne sont pas significativement différentes ont été regroupées dans une même ellipse sur les graphiques. Les traitements des données ont été réalisés par le logiciel Statistica (Version 6) [31]. RESULTATS Le tableau 3 résume l'analyse de variance de l'effet espèce et de l'effet NaCl ainsi que leur interaction. Pour l'ensemble des caractères, le teste ANOVA montre un effet très hautement significatif entres les espèces et entre les différents traitements de sel. Ces différences sont plus marquées dans le cas des caractères se rapportant à la croissance en longueur des plantules. De la même manière, l'interaction (NaCl * Espèce) révèle aussi un effet très hautement significatif pour les caractères longueur de la tige et le nombre de feuilles et un effet significatif si on considère le poids total de la matière sèche tandis que pour le diamètre du collet l'interaction n'est pas significative. En conséquence, pour ce dernier critère, la salinité agit sur les différentes espèces de la même manière, quel que soit la concentration. Pour le reste des paramètres étudiés, l'effet de NaCl diffère d'une espèce à l'autre. Tableau 3: Le tableau montre l'analyse de la variance à deux critères de classification abordée sur les différents caractères végétatifs mesurés chez les espèces d'Acacia étudiées. Caractères Diamètre du collet (D.Coll) L'augmentation de la concentration en sel dans la solution d'eau d'irrigation, diminue de façon significative le taux de croissance relative du diamètre du collet chez toutes les espèces considérées (figure 4). Pour ce critère on note une réduction plus faible par rapport aux autres caractères étudiés. À des concentrations allant de 200 à 400 mM, la réduction du diamètre du collet chez A. gummifera est significativement plus importante que celles des autres espèces. En effet, à 400 mM on note une régression qui peut atteindre 52% chez cette espèce comparativement à A. horrida chez laquelle la régression n'a été que de 14%. Les espèces A. horrida, A. raddiana et A. cyanophylla forment un groupe homogène et réagissent de la même manière au stress salin avec les réductions les plus faibles. Nombre de feuilles (Nbr.Fll) Pour ce caractère, on note une classification moins nette des différentes espèces surtout au niveau des concentrations modérées en sel (figure 5). L'espèce A. cyanophylla semble encore une fois la plus affectée par le sel et montre une réduction de sa production foliaire atteignant les 86% à la concentration 400 mM de NaCl. En générale cette espèce exhibe la plus grande sensibilité à la salinité à ce stade de développement. D'un autre côté, A. raddiana a présenté une faible réduction du nombre de feuilles par plante à la concentration 300 mM de NaCl. On note, dans ce cas, une réduction de 33% chez l'espèce la plus tolérante contre 63% chez A. cyanophylla pour atteindre à la concentration de 400 mM une réduction de 48% chez A. raddiana et 86% chez l'espèce la plus sensible. Biomasse totale (PS) L'augmentation de la concentration de NaCl a un effet significatif sur la biomasse sèche de toutes les parties de la plante (feuilles, tiges et racines) des espèces testées (figure 6). La plus grande variabilité a été observée pour les deux concentrations 300 et 400 mM de NaCl. Comme pour les caractères nombre de feuilles et longueur de la tige, A. cyanophylla se distingue, encore une fois, des autres espèces en affichant pour ce critère, toutes concentrations confondues, les réductions les plus élevées. Par ailleurs, A. horrida et A. raddiana semblent être, en générale, moins affectées pour ce caractère, elles ont montrée de ce fait les plus faibles régressions de la matière sèche totale élaborée, variant en moyenne entre 13 et 34% contre 40 et 77% pour A cyanophylla, la plus sensible. Les autres espèces occupent une situation intermédiaire et montrent une grande variabilité au niveau des doses élevées en NaCl (400 mM) par rapport aux concentrations faibles. DISCUSSIONS Les résultats présentés dans cette partie, montrent que la salinité réduit en générale la croissance des plantules chez l'ensemble des espèces étudiées. Néanmoins, une grande variabilité entre les espèces a été révélée à ce stade. Les interactions très hautement significatives entre les deux effets (espèce*salinité), observées dans notre cas, montre la possibilité d'une sélection essentiellement sur la base des caractères : Longueur de la tige, nombre de feuille et la biomasse totale. Toutefois, on a constaté qu'une espèce performante pour un caractère donné n'est pas forcément la meilleure pour un autre critère. C'est le cas d'A. cyanophylla dont le nombre de feuilles, la biomasse totale et la longueur de la tige paraissaient relativement affectée par le sel, mais dans le cas du caractère relatif au diamètre du collet elle était classée parmi les espèces les plus tolérantes. Nos résultats sont en concordance avec les travaux de Nguyen et al. [32] dans lesquels ils ont révélés que les deux espèces Acacia auriculiformis et Acacia mangium ont réagi également par une réduction de la croissance de la partie aérienne en réponse au stress salin. Cet effet est fréquent chez les glycophytes [33], où la diminution de la croissance de l'appareil végétatif observée peut être expliquée par une augmentation de la pression osmotique provoquée par NaCl, ce qui bloque l'absorption de l'eau par les racines. Les plantes s'adaptent ainsi au stress salin par la réduction de leur croissance afin d'éviter les dommages causés par le sel [34,35]. Les effets de la salinité sur la croissance des plantules cultivés en conditions semi contrôlées, dépendent de plusieurs facteurs. Ils varient selon la teneur de NaCl appliquée, l'espèce, la provenance, le stade végétatif et la partie de la plante [2]. Les effets de la salinité se manifestent principalement par un ralentissement de la croissance de l'appareil végétatif. D'autre part, il est important de signaler que toutes les espèces d'Acacia considérées dans ce travail ont survécu, même à 400 mM de NaCl, alors que selon Ghulam et al. [36], ce niveau de salinité a été nuisible pour A. nilotica tandis que A. ampliceps tolère cette concentration. La comparaison de la tolérance à la salinité chez cinq espèces d'Acacia : A. ampliceps, A. salicina, A. ligulata, A. holosericea et A. mangium a révélé que A. ampliceps était la plus tolérante et a survécu même à 428 mM en NaCl, concentration à laquelle toutes les autres espèces ont été sévèrement affectées [37]. Pour le paramètre matière sèche, les réductions les plus faibles sont enregistrées chez A. horrida et A. raddiana. Le pourcentage de réduction de la matière sèche, généralement considéré comme indice de sensibilité des plantes vis-à-vis du stress salin, montre que la concentration 400 mM NaCl est insuffisante pour engendrer une réduction relative de 50 % par rapport aux témoins, seuil très utilisé pour le classement de la tolérance des plantes [21]. En outre, même à une concentration plus faible (300 mM) les autres espèces manifestent des réductions plus marquées. Les feuilles sont les parties les plus sensibles de la plante au stress salin. Cependant, chez l'ensemble des espèces on assiste à une réduction significative du nombre de feuilles par rapport au témoin surtout à partir de 300 mM de NaCl. Des résultats similaires ont été rapportés sur d'autres espèces par [38][39][40][41]. D'autre part, au cours de l'expérimentation on a constaté que la croissance foliaire est également très affectée par l'augmentation du stress salin quelle que soit l'espèce. L'expansion des feuilles est considérablement inhibée par le sel, les nouvelles feuilles se développent lentement et le vieillissement des feuilles âgées s'accélère. D'ailleurs, quand la surface foliaire est réduite par la salinité, la production de carbohydrates devient insuffisante pour supporter la croissance et le rendement [42]. La réduction de la croissance, dans les conditions d'un stress salin est attribuée à plusieurs facteurs, parmi lesquelles l'accumulation des ions, aussi bien en Na + qu'en Clà des teneurs élevées dans les tissus foliaires qui est la cause principale des contraintes ioniques au niveau des tissus de la plante [43]. Selon ces auteurs, le stress salin cause un déséquilibre nutritionnel qui en résulte l'inhibition de l'absorption des éléments nutritifs essentiels comme le Ca 2+ , K + , Mg 2+ , NO3par les phénomènes de compétition minérale de fixation apoplasmique [44]. En outre, il est établi qu'un supplément de Ca 2+ dans le milieu de culture améliore les conditions de croissance sous stress salin [45]. Le dérèglement de l'absorption du calcium inhibe également l'établissement de la nodulation et la fixation d'azote chez les légumineuses [46]. Il parait que cet ion est impliqué dans le processus de la reconnaissance Rhizobium-poil absorbant [47]. Par ailleurs, la diminution de la croissance des parties aérienne peut aussi être expliquer par des perturbations des taux des régulateurs de croissance dans les tissus, particulièrement l'acide abscissique et les cytokinines induites par le sel [48], mais aussi à une diminution de la capacité photosynthétique provoqué par la diminution de la conductance stomatique de CO 2 sous la contrainte saline. Dans tous les cas, cette réduction de la croissance des différentes parties aériennes est considérée comme une stratégie adaptative nécessaire à la survie des plantes exposées à la salinité [49]. Ceci permet à la plante d'emmagasiner de l'énergie nécessaire pour faire face au stress afin de réduire les dommages irréversibles occasionnés, quand le seuil de la concentration létale est atteint [39]. De plus, il a été constaté que la tolérance au stade germination, dans les conditions de nos expériences, ne reflète pas dans tous les cas celle au stade végétatif. En effet, l'espèce A. horrida, classée parmi les plus sensibles au sel durant la germination, manifeste une tolérance importante vis-à-vis de NaCl au stade plantule. Par contre l'espèce A. raddiana, reconnue par sa tolérance à la salinité, très marquée, au stade germination, conserve globalement sa performance aux stades avancés. Cependant, la germination sous contrainte saline n'est pas suffisante pour identifier des espèces tolérantes au sel [26,50]. Dans ce contexte, de nombreux auteurs ont montré que la réponse à la salinité variait selon le stade de développement de la plante [51][52][53]. Toutefois, la germination et les premiers stades de la croissance seraient les phases les plus sensibles [54]. CONCLUSION Le stress salin exerce chez les Six espèces d'Acacia un effet dépressif sur tous les paramètres de croissance étudiés. Toutefois, le taux de réduction diffère selon l'intensité de stress salin et le degré de sensibilité ou de tolérance de l'espèce. La croissance en hauteur, le nombre de feuilles et la biomasse sèche totale sont vraisemblablement les plus affectés. Cependant, toutes les espèces d'Acacia considérées dans ce travail ont survécu, même à 400 mM de NaCl et présentent différents degrés de tolérance à la salinité. Les espèces A. horrida et A. raddiana s'avèrent globalement les plus performantes. Cette variabilité génétique dévoilée par ces espèces dans les différentes conditions de stress salin, surtout sous des seuils élevés en NaCl allant jusqu'à 400 mM, constitue un atout intéressant qui peut être utilisé aussi bien dans le choix des espèces à retenir pour améliorer la tolérance à la salinité que dans les programmes de valorisation et de réhabilitation des sols salés. Par ailleurs, ces recherches doivent être poursuivies par des expériences à des stades végétatifs plus avancés afin de confirmer les résultats constatés aux stades germination et juvénile. Toutefois, le dosage et l'identification des osmorégulateurs tels que la proline, pourrait mieux éclaircir les mécanismes d'ajustement osmotique nécessaires à ces plantes pour s'adapter au stress salin. Cela pourrait expliquer, par la même, leur tendance à l'halophilie, observée au cours des deux stades étudiés. Reconnaissance : Nous exprimons toute notre reconnaissance et nos remerciements aux responsables de la Station Régionale des Semences Forestières de Marrakech (Maroc) pour leurs informations et orientations au cours des prospections pour la récolte des gousses.  Diamètre au collet (D.Coll) : Le diamètre au collet en (mm), mesuré à l'aide d'un pied à coulisse (figure 1).  Longueur finale de la tige (Lg.Tige) : Mesurée en utilisant une règle graduée du collet à l'insertion du méristème apicale (figure 2).  Poids de la matière sèche (PS) : Ce paramètre est nettement plus fiable et plus simple. La biomasse totale (en mg de matière sèche) est séchée dans l'étuve à 87°C puis pesée 48 heures plus tard. Figure 1 : 1Mesure du diamètre au collet à l'aide d'un pied à coulisse. Figure 2 : 2Mesure de la longueur de la tige à l'aide d'une règle graduée. 3. 1 . 1Etude des caractères pris séparément pour l'ensemble des espèces étudiées 3.1.1. Longueur de la tige (Lg.Tige) L'analyse de variance pour la longueur de la tige révèle un effet très hautement significatif des deux facteurs, salinité et espèce ainsi que leur interaction. Le classement des différentes espèces par le test de Newman et Kheuls selon leur tolérance à la salinité, montre une régression de la croissance en hauteur de la tige des plantules chez les différentes espèces étudiées (figure 3). Cependant, les réductions les plus importantes ont été notées pour A. cyanophylla, surtout dans le cas des fortes concentrations. Elle enregistre jusqu'à 92% pour une concentration de 400 mM. Par ailleurs, A. horrida semble la moins perturbée par le sel, du moins pour ce caractères, et affiche les valeurs les plus faibles. Les pourcentages de réduction varient dans ce cas entre 10 % et 47 %, respectivement pour les concentrations 100 et 400 mM. Le reste des espèces réagissent avec modération et occupent une situation relativement intermédiaire entre ces deux dernières espèces. Figure 3 : 3Représentation des moyennes calculées pour chaque espèce étudiées par traitement de NaCl pour la longueur de la tige. (Les moyennes des espèces groupées dans la même ellipse ne sont pas significativement différentes selon le test de Newman et Keuls à 5%). Figure 4 : 4Représentation des moyennes calculées pour chaque espèce étudiées par traitement de NaCl pour le diamètre du collet. (Les moyennes des espèces groupées dans la même ellipse ne sont pas significativement différentes selon le test de Newman et Keuls à 5%). Figure 5 : 5Représentation des moyennes calculées pour l'ensemble des espèces étudiées par traitement en NaCl pour le nombre de feuilles. (Les moyennes des espèces groupées dans la même ellipse ne sont pas significativement différentes selon le test de Newman et Keuls à 5%). Figure 6 : 6Représentation des moyennes calculées pour l'ensemble des espèces étudiées par traitement en NaCl pour la biomasse totale (PS). (Les moyennes des espèces groupées dans la même ellipse ne sont pas significativement différentes selon le test de Newman et Keuls à 5%). Tableau 1 : 1Le tableau montre la localisation géographique des différents échantillons d'espèces étudiées.Espèces Provenance Région de provenance A. gummifera Reserve de faune de Rmila (Marrakech) Haut Atlas occidental A. raddiana Reserve de faune de Rmila (Marrakech) Haut Atlas occidental A. cyclops Région d'Essaouira Souss Nord A. cyanophylla Canal de Rocade (Marrakech) Haut Atlas occidental A. horrida Commune rurale de Saada (Marrakech) Haut Atlas occidental A. eburnea Commune rurale de Saada (Marrakech) Haut Atlas occidental Lg.Tige D.Coll Nbr.Fll PS Longueur de la tige Diamètre au collet Nombre totale de feuilles Biomasse totale (en mg de matière sèche) CM : Carré moyen ; NS : test non significatif ; * : test significatif ; *** : test très hautement significatifCM Espèce CM Sel CM Interaction F Espèce F Sel F Interaction Lg.Tige 11439,2 20596,0 555,2 121,697*** 219,111*** 5,907*** D.Coll 6051,7 2732,7 114,2 19,932*** 9,000*** 0,376 NS Nbr.Fll 3425,2 11918,6 317,0 32,291*** 112,363*** 2,988*** PS 6379,6 10101,6 279,3 42,478*** 67,261*** 1,860* This is an Open Access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. 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{'abstract': "RESUMEIntroduction : Au cours de ces dernières décennies on assiste à une diminution progressive des superficies cultivables dans les régions arides et semi-arides à cause de l'accumulation des sels liée à la rareté des précipitations, au mauvais drainage, à la sècheresse prolongées et à l'absorption de l'eau par les plantes. Contexte : Devant l'ampleur de ce problème, il s'avère donc nécessaire de proposer des programmes d'évaluation et de conservation des espèces menacées d'extinction. Le repérage d'espèces plus adaptées et la sélection des variétés tolérantes à la salinité resteraient la voie économique la plus efficace pour l'exploitation des terrains affectés. Objectifs : L'objectif de cette étude est de déterminer la capacité de tolérance à la salinité au cours du développement végétatif chez six espèces appartenant au genre Acacia. Ceci dans le but d'élaborer une classification des seuils de tolérance au stress salin, critère important dans le choix des espèces à retenir dans un programme de mise en valeur des zones affectées par la salinité. Méthodes : L'effet du stress salin a été abordé sur un certain nombre de caractères agro-morphologiques en conditions contrôlées. Les concentrations de NaCl appliquées, en plus du témoin, sont : 100 mM, 200 mM, 300 mM et 400 mM. Résultats : Les résultats ont montré une variabilité non négligeable dans le comportement des plantes des différentes espèces en fonction du stress salin. Chez les six espèces d'Acacia, le sel a exercé un effet dépressif sur tous les paramètres de croissance étudiés. Toutefois, le taux de réduction diffère selon l'intensité de stress salin et le degré de sensibilité ou de tolérance de l'espèce. La croissance en hauteur, le nombre de feuilles et la biomasse sèche totale sont vraisemblablement les paramètres les plus affectés. Cependant, il est important de signaler que toutes les espèces d'Acacia considérées dans ce travail ont survécu, même à 400 mM de NaCl, et ont présenté différents degrés de tolérance à la salinité. Dans cette situation, les espèces A. horrida et A. raddiana s'avèrent globalement les plus performantes au stade végétatif. Conclusions : La variabilité génétique, dévoilée par ces espèces dans les différentes conditions de stress salin, permettrait un choix d'écotypes pouvant entrer dans des schémas de sélection et d'amélioration variétale pour la réhabilitation des parcours dégradés surtout en zones affectées par la salinité.ABSTRACTBackground: Salinity is one of the major abiotic stresses affecting plant production in arid and semi-arid regions. It causes reduction of cultivable area and combined with other factors, presents a serious threat to food stability in these areas. Context: In front of this problem, the selection of salt tolerant species and varieties remains the best economic approach for exploitation and rehabilitation of salt-affected regions. Objective: The purpose of this study was to assess and compare the seed germination response of six Acacia species under different NaCl concentrations in order to explore opportunities for selection and breeding salt tolerant genotypes. Methods: The salinity effect was examined by measuring some agro-morphological parameters in controlled growth environment using five treatment levels: 0, 100, 200, 300 and 400 mM of NaCl. Results: The analyzed data revealed significant variability in salt response within and between species. All growth parameters were progressively reduced by increased NaCl concentrations. Growth in height, leaf number and total plant dry weight were considered as the most sensitive parameters. However, the growth reduction varied among species in accordance with their tolerance level. It is important to note that all species survived at the highest salinity (400 mM). Whereas A. horrida and A. raddiana were proved to be often the best tolerant, they recorded the lowest reduction percentage at this stage. Conclusion: The genetic variability found in the studied species at seedling stage may be used to select genotypes particularly suitable for rehabilitation and exploitation of lands affected by salinity.", 'arxivid': '1711.08064', 'author': ['|khalil Chérifi \nLaboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |\n', '|abdelmjid Anagri \nLaboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |\n', '| El ', 'Houssine Boufous \nAmerican Journal of Innovative Research and Applied Sciences. ISSN\nDepartment of Biochemistry and Microbiology | Laval University | Quebec City (Quebec)\n2429-5396 I www.american-jirasCanada |\n', '| ', '| Abelhamid ', 'El Mousadik \nLaboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |\n', 'Acacia Species ', '\nAmerican Journal of Innovative Research and Applied Sciences. ISSN\n2429-5396I www.american-jiras\n'], 'authoraffiliation': ['Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |', 'Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |', 'American Journal of Innovative Research and Applied Sciences. ISSN\nDepartment of Biochemistry and Microbiology | Laval University | Quebec City (Quebec)\n2429-5396 I www.american-jirasCanada |', 'Laboratory of Biotechnology and Valorization of Natural Resources | Faculty of sciences | Ibn Zohr University |\nP.O. Box 8106 | 8000Agadir| Morocco |', 'American Journal of Innovative Research and Applied Sciences. ISSN\n2429-5396I www.american-jiras'], 'corpusid': 19456672, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18872, 'n_tokens_neox': 16808, 'n_words': 7550, 'pdfsha': 'd20ec1d48d2c5fa9bb5033e541c82579ae133522', 'pdfurls': ['https://arxiv.org/pdf/1711.08064v1.pdf'], 'title': ["EFFET DU CHLORURE DE SODIUM (NaCl) SUR LA CROISSANCE DE SIX ESPECES d'Acacia EFFECT OF SODIUM CHLORIDE (NaCl) ON THE GROWTH OF SIX Corresponding and Author", "EFFET DU CHLORURE DE SODIUM (NaCl) SUR LA CROISSANCE DE SIX ESPECES d'Acacia EFFECT OF SODIUM CHLORIDE (NaCl) ON THE GROWTH OF SIX Corresponding and Author"], 'venue': []}

arxiv

Vertex functions and their flow equations from the 2PI effective action 21 Nov 2022 Peter Millington Paul M Saffin Department of Physics and Astronomy School of Physics and Astronomy University of Manchester M13 9PLManchesterUnited Kingdom University of Nottingham NG7 2RDNottinghamUnited Kingdom Vertex functions and their flow equations from the 2PI effective action 21 Nov 2022(Dated: 8 November 2022) By exploiting the convexity of the two-particle-irreducible (2PI) effective action, we describe a procedure for extracting n-point vertex functions. This procedure is developed within the context of a zero-dimensional "quantum field theory" and subsequently extended to higher dimensions. These results extend the practicability and utility of a recent, alternative approach to the functional renormalization group programme [see Phys. Rev. D 104 (2021) 069906; J. Phys. A 54 (2021) 465401], and clarify the relationship between the flow equations for coupling parameters and vertices. This is an author-prepared post-print of J. Phys. A: Math. Theor. 55 (2022) 435402, published by IOP Publising under the terms of the CC BY 4.0 license. * Electronic address: peter.millington@manchester.ac.uk † Electronic address: paul.saffin@nottingham.ac.uk early applications of higher-order Legendre transforms in non-relativistic statistical mechanics, e.g., by De Dominicis and Martin [2, 3]. 2 The importance of identifying the true convex-conjugate variables, in terms of the non-connected functions, was emphasised earlier in Ref. [6]. 3 Recently, zero-dimensional quantum field theories have also been used to study the functional renormalization group in the context of O(N ) models [25-27]. I. INTRODUCTION In a series of works, we have used a zero-dimensional "quantum field theory" to provide explicit expositions of a number of features of the two-particle irreducible (2PI) effective action. 1 This has included an exposition of the behaviours of the external sources in its definition (see also Ref. [4]), as well as the true convex-conjugate pairs of variables that correspond respectively to the one-and non-connected two-point functions [5]. 2 In the same work, we were able to visualise the convexity of the 2PI effective action, illustrate how nonconvexity emerges for particular constraints on the two-point function, describe how the number and type of the saddle points is fixed by these constraints, and recover the Maxwell construction for a classical action with two local minima (and mirroring the analysis of Ref. [7]). Subsequently, by exploiting the convexity of the 2PI effective action, we were able to make use of the identities reported in footnote 11 of Coleman, Jackiw and Tomboulis [1] to derive an expression for the two-point function in terms of partial derivatives of the 2PI effective action [8]. Similar expressions can be found in, e.g., Ref. [9], (cf. the 'external propagator' of Ref. [10]), and many of the functional identities reported in this work can also be found (again derived by means of convexity) in the much earlier and little-known series of works by Vasil'ev [6], and Vasil'ev and Kazanskii [11,12]. The resulting expression for the two-point function allowed us to provide a detailed comparison of a new approach to deriving the exact flow equations of the functional renormalization group based on the 2PI effective action [13] with the well-known approach based on the average one-particle-irreducible (1PI) effective action [14][15][16] (see also Ref. [17] in the context of gravity, and, e.g., Refs. [18][19][20][21][22][23][24] for reviews). 3 However, in order for this new approach to be of utility, it is necessary to formulate a procedure for extracting the n-point vertex functions from the 2PI effective action, and this is the focus of the present work. As we will see, this involves taking partial derivatives of the 2PI effective action with respect 1 The relativistic 2PI effective action was described by Coleman, Jackiw and Tomboulis [1], building on to both the one-and two-point functions, differing therefore from the corresponding procedure for the 1PI [28] or average 1PI [29] effective actions (as is relevant to the functional renormalization group programme). By this means, we are able to resolve an observation made in Ref. [8], in the case of a zero-dimensional φ 4 theory, that the flow equation for the coefficient of the term quartic in the would-be scalar field φ does not match the expected flow equation for the four-point vertex. The remainder of this article is organised as follows. In Sec. II, we first review the definition of the 2PI effective action, its convexity, and the extraction of the two-point function, all in the context of a zero-dimensional, scalar quantum field theory. We then proceed in Sec. III to describe a procedure for extracting n-point vertices from partial derivatives of the 2PI effective action, before applying it to the derivation of the flow equation for the four-point vertex at second-order in perturbation theory in Sec. IV within the approach of Refs. [8,13]. The generalisation of these results to non-zero dimensions is presented in Sec. V. Our concluding remarks are provided in Sec. VI. II. 2PI EFFECTIVE ACTION The two-particle-irreducible (2PI) effective action [1] has the form Γ(φ, ∆) = W(J , K) + J φ + 1 2 K(φ 2 + ∆),(1) wherein W(J, K) = − ln Z(J, K) and, for our zero-dimensional example, Z(J, K) = N dΦ exp − 1 S(Φ) − JΦ − 1 2 KΦ 2(2) is the path integral in the presence of sources J and K with normalization N . For definiteness, we take the classical action to have the form S(Φ) = 1 2! Φ 2 + λ 4! Φ 4 ,(3) with λ > 0. The particular sources J ≡ J (φ, ∆) and K ≡ K(φ, ∆) are fixed by extremisation of the effective action such that they are functions of the "one-" and "two-point" variables φ and ∆ (as emphasized in Refs. [4,6,11,12]). The latter are given by φ = − ∂ ∂J W(J , K),(4a)∆ = −2 ∂ ∂K W(J , K) − φ 2 = − ∂ 2 ∂J 2 W(J , K). (4b) The equations of motion (algebraic equations, in the zero-dimensional case) for φ and ∆ are obtained by taking partial derivatives of the effective action: ∂ ∂φ Γ(φ, ∆) = J (φ, ∆) + K(φ, ∆)φ,(5a)∂ ∂∆ Γ(φ, ∆) = 2 K(φ, ∆),(5b) where partial derivatives with respect to φ are understood at fixed ∆ and vice versa. The 2PI effective action is a double Legendre transform of the Schwinger function W(J, K) with respect to the sources J and K. By inspection of Eq. (1), we should anticipate that the convex-conjugate pairs of variables are J ′ = J ,(6a)K ′ = 1 2 K,(6b) and φ ′ (φ, ∆) = φ,(7a)∆ ′ (φ, ∆) = φ 2 + ∆ (7b) (not φ and ∆). These definitions lead to ∂ ∂φ ′ = ∂ ∂φ − 2 φ ∂ ∂∆ ,(8a)∂ ∂∆ ′ = 1 ∂ ∂∆ ,(8b) and it can be proven that Γ(φ, ∆) is a convex function of (φ ′ , ∆ ′ ) [5]. It is important to note that the 2PI effective action is not, in general, convex with respect to the pair (φ, ∆) [5]. The convexity of the 2PI effective action leads to four identities, encoded in the product [11,12] Hess(−W)Hess(Γ) = 1, where the Hessian of −W is taken with respect to J ′ and K ′ , and the Hessian of Γ is taken with respect to φ ′ and ∆ ′ . These four identities, along with the definitions of the convex conjugate pairs of variables in Eq. (7), allow us to express the partial derivatives of W with respect to the original sources J and K in terms of the derivatives of the 2PI effective action with respect to the variables φ and ∆ [8] (equivalent identities appear in footnote 11 of Ref. [1]): ∂ 2 W ∂J 2 = −∆ = − ∂ 2 Γ ∂φ 2 − 2 ∂Γ ∂∆ − ∂ 2 Γ ∂φ∂∆ ∂ 2 Γ ∂∆ 2 −1 ∂ 2 Γ ∂φ∂∆ −1 , (10a) ∂ 2 W ∂J ∂K = 2 ∂ 2 Γ ∂∆ 2 −1 ∂ 2 Γ ∂φ∂∆ − 2 φ ∂ 2 Γ ∂∆ 2 ∆,(10b)∂ 2 W ∂K 2 = − 2 4 ∂ 2 Γ ∂∆ 2 −1 ∆ ∂ 2 Γ ∂φ 2 − 2 ∂Γ ∂∆ − 4 φ ∂ 2 Γ ∂φ∂∆ + 4 2 φ 2 ∂ 2 Γ ∂∆ 2 . (10c) III. EXTRACTING n-POINT VERTEX FUNCTIONS By application of the chain rule, we can show that   ∂ ∂J ′ ∂ ∂K ′   =   ∂φ ′ ∂J ′ ∂∆ ′ ∂J ′ ∂φ ′ ∂K ′ ∂∆ ′ ∂K ′     ∂ ∂φ ′ ∂ ∂∆ ′   = Hess(−W)   ∂ ∂φ ′ ∂ ∂∆ ′   .(11) Similarly,   ∂ ∂φ ′ ∂ ∂∆ ′   = Hess(Γ)   ∂ ∂J ′ ∂ ∂K ′   .(12) These expressions are mutually consistent by virtue of Eq. (9). Introducing the coordinates f a ≡ (φ ′ , ∆ ′ ) and f a ≡ (J ′ , K ′ ), and the derivatives ∂ a ≡ ∂ ∂f a = ∂ ∂φ ′ , ∂ ∂∆ ′ , (13a) ∂ a ≡ ∂ ∂f a = ∂ ∂J ′ , ∂ ∂K ′ ,(13b) we can write ∂ a = M ab ∂ b ,(14a)∂ a = M ab ∂ b ,(14b) where M ab = Hess(−W) ab ,(15a)M ab = Hess(Γ) ab ,(15b) with [from Eq. (9)] M ab M bc = δ c a .(16) Note that M behaves like a metric on the configuration space and the dual space of convexconjugate variables. This suggests an intriguing methodology for dealing with the functional identities arising from the nPI effective action that may be related to the geometry of Hessian manifolds (see, e.g., Ref. [30]) and information manifolds (see, e.g., Ref. [31]), and we leave further discussion of this to future work. By the above means, and using Eqs. (8a) and (8b), we can express the derivative with respect to J ′ in terms of derivatives with respect to φ and ∆: ∂ ∂J ≡ ∂ ∂J ′ = ∂φ ′ ∂J ′ ∂ ∂φ ′ + ∂∆ ′ ∂J ′ ∂ ∂∆ ′ = − ∂ 2 W ∂J 2 ∂ ∂φ ′ − 2 ∂ 2 W ∂J ∂K ∂ ∂∆ ′ = ∆ ∂ ∂φ − 2 φ ∂ ∂∆ − 2 2 ∂ 2 Γ ∂∆ 2 −1 ∂ 2 Γ ∂φ∂∆ − 2 φ ∂ 2 Γ ∂∆ 2 ∆ 1 ∂ ∂∆ = ∆ ∂ ∂φ − ∂ 2 Γ ∂φ∂∆ ∂ 2 Γ ∂∆ 2 −1 ∂ ∂∆ .(17) An equivalent operator first appeared in Ref. [11]. The two-point function is given in Eq. (10a), and the various n-point functions can then be obtained by taking n − 2 derivatives of the two-point function with respect to J by repeated application of Eq. (17), cf. the equivalent approach of Ref. [11], before multiplying by ( ∆) −n to amputate the external two-point functions. Specifically, the connected n-point function (for n > 2) is φ n conn = ∂ ∂J n−2 − ∂ 2 W ∂J 2 = ∂ ∂J n−2 ( ∆),(18) and the amputated n-point vertex can be written as Γ (n>2) = − ( ∆) −n φ n conn = −∆ −n ∆ ∂ ∂φ − ∂ 2 Γ ∂φ∂∆ ∂ 2 Γ ∂∆ 2 −1 ∂ ∂∆ n−2 ∆.(19) The overall minus sign and factor of are such that Γ (n>2) coincides with the tree-level vertex to zeroth order in ; e.g., for the action in Eq. (3), Γ (4) = λ + O( ).(20) To order λ 2 , the explicit expression for the 2PI effective is (see Refs. [5,8] for the complete derivation) Γ(φ, ∆) = 1 2 φ 2 + λ 4! φ 4 + 2 ln ∆ −1 + 1 + λ 2 φ 2 ∆ − 1 + 2 λ 8 ∆ 2 − λ 2 12 φ 2 ∆ 3 − 3 λ 2 48 ∆ 4 .(21) Hence, ∂ 2 Γ ∂φ∂∆ = λ 2 φ + O(λ 2 ),(22a)∂ 2 Γ ∂∆ 2 = 2 ∆ −2 + O(λ),(22b) and so Γ (4) = ∆ −3 ∂ ∂φ − λφ∆ 2 ∂ ∂∆ λφ∆ 3 = λ − 3λ 2 φ 2 ∆ + O(λ 3 ).(23) We emphasise that φ and ∆ are independent variables, such that ∂∆/∂φ = 0 and ∂φ/∂∆ = 0. Notice that all of these expressions are obtained directly from the 2PI effective action in terms of the variables φ and ∆. For comparison, the equivalent procedure for the 1PI effective action involves taking only derivatives with respect to φ, i.e., Γ (n>2) 1PI = −∆ −n ∆ ∂ ∂φ n−2 ∆,(24) where ∆ −1 = ∂ 2 Γ 1PI ∂φ 2 .(25) Thus, the 4-point vertex has the simple expression Γ (4) 1PI = ∂ 4 Γ 1PI ∂φ 4 − 3Γ(3) 1PI ∆Γ 1PI . We stress that the variables φ and ∆ are not independent in the case of the 1PI effective action. In the 2PI case, the additional degree of freedom provided by K ensures that the variables φ and ∆ are independent. IV. VERTEX RG FLOWS We now turn our attention to the flow equations for the 2PI effective action of our zerodimensional theory. The main aim of this section (and indeed this work) is to go beyond Ref. [8] and to compare the flow equations for the quartic vertex, as obtained from the usual 1PI framework and the 2PI approach of Refs. [8,13], the latter of which we now describe. We proceed by promoting the source K → R k , which plays the role of the regulator in the 2PI approach for deriving exact flow equations with the parameter k emulating the RG scale. 4 The variation of the 2PI effective action with respect to the scale k is then ∂ k Γ(φ, ∆ k ) = 2 R k ∂ k ∆ k ,(27) where ∆ k is given by Eq. (10a) (see Refs. [8,13]). In order to derive the flow equations, we make the following Ansatz for the 2PI effective action: Γ(φ, ∆ k ) = α k (∆ k ) + 1 2 β k (∆ k )φ 2 + 1 4! γ k (∆ k )φ 4 .(28) Using (10a) and (5b), along with K → R k , this leads to the following expression for the inverse two-point function [8]: ∆ −1 k = β k − R k (φ, ∆ k ) + 1 2 γ k φ 2 − φ ∂β k ∂∆ k + 1 3! φ 3 ∂γ k ∂∆ k 2 ∂ 2 α k ∂∆ 2 k + 1 2 φ 2 ∂ 2 β k ∂∆ 2 k + 1 4! φ 4 ∂ 2 γ k ∂∆ 2 k −1 ,(29) where R k (φ, ∆ k ) = 2 ∂α k ∂∆ k + 1 2 φ 2 ∂β k ∂∆ k + 1 4! φ 4 ∂γ k ∂∆ k(30) and we have suppressed the arguments of the α k , β k and γ k for conciseness. Their flow equations can be extracted by taking derivatives of the flow equation with respect to φ and ∆ and evaluating at φ = 0, as described in Ref. [8]. If we intend to work to order λ 2 , this procedure leads to the following system of equations 4 We use a non-standard sign convention in the definition of the regulator R k . for the ∆-derivatives of α(∆), β(∆) and γ(∆) [8]: ∂α k (∆) ∂∆ = 2 R k (0, ∆), (31a) ∂ 2 α k (∆) ∂∆ 2 = 2 [β k (∆) − R k (0, ∆)] 2 + 2 4 γ k (∆) − 3 4 γ 2 k (∆) [β k (∆) − R k (0, ∆)] −2 , (31b) ∂ 3 α k (∆) ∂∆ 3 = − [β k (∆) − R k (0, ∆)] 3 − 3 2 γ 2 k (∆) [β k (∆) − R k (0, ∆)] −1 , (31c) ∂β k (∆) ∂∆ = 2 γ k (∆) − 2 2 γ 2 k (∆) [β k (∆) − R k (0, ∆)] −2 ,(31d)∂ 2 β k (∆) ∂∆ 2 = − 2 γ 2 k (∆) [β k (∆) − R k (0, ∆)] −1 , (31e) ∂γ k (∆) ∂∆ = O(γ 4 k ). (31f) Note that the regulator is evaluated at φ = 0 in the above expressions. We see that the "coupling parameter" γ k (∆) is fixed straightforwardly by our boundary condition, i.e., γ k (∆) = λ, at this order. A potentially striking observation (as made in Ref. [8]) is that γ k does not run until order λ 4 , when, under the assumption that γ k is related to the four-point vertex, we might expect it to run at order λ 2 . The remaining flow equations are [8] ∂ k α k (∆ k ) = 2 R k (0, ∆ k )∂ k [β k (∆) − R k (0, ∆)] −1 , (32a) ∂ k β k (∆ k ) = 2 λ − 2 2 λ 2 [β k (∆) − R k (0, ∆)] −2 ∂ k [β k (∆) − R k (0, ∆)] −1 . (32b) The solutions to order λ 2 are [8] β k (∆ k ) = β 0 + λ 2 1 1 − R k (0, ∆ k ) − 5 2 λ 2 12 1 [1 − R k (0, ∆ k )] 3 , (33a) α k (∆ k ) = α 0 + 2 1 1 − R k (0, ∆ k ) + 2 ln[1 − R k (0, ∆)] + 2 λ 8 1 − 3R k (0, ∆ k ) [1 − R k (0, ∆)] 3 − 3 λ 2 12 1 − 5R k (0, ∆ k ) [1 − R k (0, ∆ k )] 5 ,(33b) where α 0 and β 0 are integration constants. We fix β 0 = 1 by matching to the limit λ → 0. At this point, we are, in some sense, finished, since we have all that we need to reconstruct the 2PI effective action and all of the resulting n-point functions, correct to order λ 2 , as was done in Ref. [8]. However, it is instructive to obtain the four-point vertex and derive its flow equation, both evaluated at φ = 0, as we will now do. In particular, this will allow us to understand why the coefficient γ k of the φ 4 term in the Ansatz (28) does not appear to run at order λ 2 [see Eq. (31f) above]. We first use Eq. (19) with n = 3 to calculate the expression for the three-point vertex Γ (3) of the effective action in Eq. (28), which we need only to first order in φ (since we will subsequently take one more derivative with respect to φ, before setting φ = 0): Γ (3) k = ∆ −2 k ∂ 2 Γ k ∂φ∂∆ k ∂ 2 Γ k ∂∆ 2 k −1 = ∆ −2 k φ ∂β k ∂∆ k + φ 3 3! ∂γ k ∂∆ k ∂ 2 α k ∂∆ 2 k + 1 2 φ 2 ∂ 2 β k ∂∆ 2 k + 1 4! φ 4 ∂ 2 γ k ∂∆ 2 k −1 = ∆ −2 k φ ∂β k ∂∆ k ∂ 2 α k ∂∆ 2 k −1 + O(φ 3 ).(34) Multiplying this result by ( ∆ k ) 3 , differentiating with respect to J / via Eq. (17), and amputating four factors of ∆ k , only the φ derivative contributes at φ = 0, and we obtain Γ (4) k | φ=0 = ∆ −2 k ∂β k ∂∆ k ∂ 2 α k ∂∆ 2 k −1 φ=0 .(35) We need only the lowest-order terms in the expressions for ∂β k (∆ k ) ∂∆ k and ∂ 2 α k (∆ k ) ∂∆ 2 k from Eq. (31) along with ∆ k from Eq. (29), giving Γ (4) k | φ=0 = γ k (∆ k ) − 3 2 γ 2 k (∆ k ) 1 [β k (∆ k ) − R k (0, ∆ k )] 2 .(36) Since the coupling parameter γ k (∆ k ) does not flow until order λ 4 in the 2PI approach [8], the flow equation for Γ (4) k | φ=0 to order λ 2 is ∂ k Γ (4) k | φ=0 = −3 λ 2 ∂ k R k (0, ∆ k ) [1 − R k (0, ∆ k )] 3 ,(37) wherein we have used the fact that β k (∆ k ) = 1 + O(γ k ) and ∂ k β k (∆ k ) is order γ k (∆ k ). Most significantly, this agrees with the flow equation for the four-point vertex of the average 1PI approach, also at order λ 2 , as we describe below. The average 1PI effective action is given by [29] (for a review, see, e.g., Ref. [19]) Γ av 1PI (φ, R k ) = W(J , R k ) + J (φ)φ + 1 2 R k φ 2 .(38) It differs from the 2PI effective action in that there has been no Legendre transform with respect to the source R k . As a result, its natural variables are φ and R k , and it is for this reason that φ and ∆ are not independent variables in the 1PI case. The flow equation is [14][15][16] ∂ k Γ av 1PI = − 2 ∆ k ∂ k R k ,(39) and we draw attention to our non-standard choice of sign for the definition of the regulator. The two-point function is given in terms of derivatives of Γ av 1PI via ∆ −1 k = ∂ 2 Γ av 1PI ∂φ 2 − R k ,(40) cf. Eq. (10a), wherein the additional terms illustrate the difference in the degree of resummation provided by the 2PI approach. Proceeding by making the Ansatz Γ av 1PI (φ, R k ) =α k (R k ) + 1 2β k (R k )φ 2 + 1 4!γ k (R k )φ 4 ,(41) the flow equations are obtained by taking φ derivatives at φ = 0. Working to order λ 2 , we obtain [8] ∂ kαk = − 2 ∂ k R k β k − R k , (42a) ∂ kβk = 2γ k ∂ k R k [β k − R k ] 2 ,(42b)∂ kγk = −3 γ 2 k ∂ k R k [β k − R k ] 3 .(42c) The solutions strictly to order λ 2 are [8] α k =α 0 + 2 ln(1 − R k ) + 2 λ 8 1 (1 − R k ) 2 − 1 (1 − R k ) 4 ,(43a)β k = 1 + λ 2 1 1 − R k − 5 2 λ 2 12 1 (1 − R k ) 3 ,(43b)γ k = λ − 3 λ 2 2 1 (1 − R k ) 2 .(43c) Notice that, in the average 1PI approach, the four-point vertex function at φ = 0 coincides with the coefficientγ k (R k ) of φ 4 in the Ansatz (41) for the average 1PI effective action, i.e., Γ av (4) 1PI | φ=0 =γ k (R k ). This is not so for the 2PI case, because, in the 2PI resummation, the λ 2 correction to the four-point vertex is absorbed in α k (∆ k ) and β k (∆ k ), and not in the coefficient γ k (∆ k ) of φ 4 . It is for this reason that the coupling parameter γ k (∆ k ) does not run at order λ 2 , whereasγ k (R k ) of the 1PI approach does. With these observations, and comparing Eqs. (37) and (42c), we see that the flows of the four-point vertex at φ = 0, as obtained from the 2PI and average 1PI approaches, agree at order λ 2 . Due to the differing resummation implicit in the 2PI versus 1PI approaches, however, there is no reason why these flow equations should agree once we move away from the strictly fixed-order result. V. FIELD THEORY GENERALISATION In order to extend the previous results to non-zero dimensions, it is convenient to consider first the generalisation to N fields, i.e., φ → φ α and ∆ → ∆ αβ with α, β = 1, 2, . . . , N. In the latter case, the 2PI effective action becomes Γ({φ}, {∆}) = W({J }, {K}) + J α φ α + 1 2 K αβ (φ α φ β + ∆ αβ ).(44) A comprehensive discussion of how the convexity properties can be translated to the multifield case, along with the resulting identities (cf. Eq. (10)), is provided in Ref. [8]. Making use of those results, the derivative with respect to the source J α , generalising Eq. (17), becomes ∂ ∂J α = ∆ αβ ∂ ∂φ β − ∂ 2 Γ ∂φ β ∂∆ γδ ∂ 2 Γ ∂∆ γδ ∂∆ ρσ −1 ∂ ∂∆ ρσ .(45) The expression for the n-point vertex [Eq. (19)] is then promoted to Γ (n>2) α 1 α 2 ...αn = − n+1 ∆ −1 α 1 β 1 ∆ −1 α 2 β 2 · · · ∆ −1 αnβn φ β 1 φ β 2 · · · φ βn conn = −∆ −1 α 1 β 1 ∆ −1 α 2 β 2 · · · ∆ −1 αnβn ∂ ∂J β 1 ∂ ∂J β 2 . . . ∂ ∂J β n−2 ∆ β n−1 βn .(46) The generalisation from this multi-field case to the full field theory case in d > 0 dimensions is straightforward: we must simply interpret the field-space indices to include coordinate variables, which, when contracted, are integrated over with an appropriate spacetime measure, as per the conventions of the DeWitt notation. VI. CONCLUDING REMARKS We have outlined a procedure for extracting n-point vertex functions from the two-particle irreducible (2PI) effective action that exploits its convexity in the two-dimensional configuration space specified by the one and non-connected two-point functions. This result is expected to increase the utility of approaches based on the 2PI effective action, including alternative derivations (see Refs. [8,13]) of the exact flow equations for interacting quantum field theories that are complementary to long-standing approaches thanks to the differing ways that the resummation of loop corrections is organised in the 2PI versus 1PI frame- works. An explicit application of these results to the RG evolution of an interacting scalar field theory in d > 0 spacetime dimensions will be provided in a revised version of a previous unpublished work [32]. λ 2 12 AcknowledgmentsThe Authors thank the organisers and participants of the 11 th International Conference on the Exact Renormalization Group 2022 (ERG2022) and especially Urko Reinosa for helpful discussions. 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{'abstract': 'By exploiting the convexity of the two-particle-irreducible (2PI) effective action, we describe a procedure for extracting n-point vertex functions. This procedure is developed within the context of a zero-dimensional "quantum field theory" and subsequently extended to higher dimensions. These results extend the practicability and utility of a recent, alternative approach to the functional renormalization group programme [see Phys. Rev. D 104 (2021) 069906; J. Phys. A 54 (2021) 465401], and clarify the relationship between the flow equations for coupling parameters and vertices. This is an author-prepared post-print of J. Phys. A: Math. Theor. 55 (2022) 435402, published by IOP Publising under the terms of the CC BY 4.0 license. * Electronic address: peter.millington@manchester.ac.uk † Electronic address: paul.saffin@nottingham.ac.uk early applications of higher-order Legendre transforms in non-relativistic statistical mechanics, e.g., by De Dominicis and Martin [2, 3]. 2 The importance of identifying the true convex-conjugate variables, in terms of the non-connected functions, was emphasised earlier in Ref. [6]. 3 Recently, zero-dimensional quantum field theories have also been used to study the functional renormalization group in the context of O(N ) models [25-27].', 'arxivid': '2206.08865', 'author': ['Peter Millington ', 'Paul M Saffin ', '\nDepartment of Physics and Astronomy\nSchool of Physics and Astronomy\nUniversity of Manchester\nM13 9PLManchesterUnited Kingdom\n', '\nUniversity of Nottingham\nNG7 2RDNottinghamUnited Kingdom\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nSchool of Physics and Astronomy\nUniversity of Manchester\nM13 9PLManchesterUnited Kingdom', 'University of Nottingham\nNG7 2RDNottinghamUnited Kingdom'], 'corpusid': 253735239, 'doi': '10.1088/1751-8121/ac99ae', 'github_urls': [], 'n_tokens_mistral': 12407, 'n_tokens_neox': 10631, 'n_words': 5783, 'pdfsha': '3cb9cc2fdcafec7281bca11e99ab8b11d73787e7', 'pdfurls': ['https://export.arxiv.org/pdf/2206.08865v2.pdf'], 'title': ['Vertex functions and their flow equations from the 2PI effective action', 'Vertex functions and their flow equations from the 2PI effective action'], 'venue': []}

arxiv

Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images Snehal Singh snehal@smail.iitm.ac.in Indian Institute of Technology Madras Tomar A N Rajagopalan Indian Institute of Technology Madras Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images With the advent of an increasing number of Augmented and Virtual Reality applications that aim to perform meaningful and controlled style edits on images of human faces, the impetus for the task of parsing face images to produce accurate and fine-grained semantic segmentation maps is more than ever before. Few State of the Art (SOTA) methods which solve this problem, do so by incorporating priors with respect to facial structure or other face attributes such as expression and pose in their deep classifier architecture. Our endeavour in this work is to do away with the priors and complex pre-processing operations required by SOTA multi-class face segmentation models by reframing this operation as a downstream task post infusion of disentanglement with respect to facial semantic regions of interest (ROIs) in the latent space of a Generative Autoencoder model. We present results for our model's performance on the CelebAMask-HQ and HELEN datasets. The encoded latent space of our model achieves significantly higher disentanglement with respect to semantic ROIs than that of other SOTA works. Moreover, it achieves a 13% faster inference rate and comparable accuracy with respect to the publicly available SOTA for the downstream task of semantic segmentation of face images. Introduction Multi-class semantic segmentation of facial regions of interest is central to various AR and VR applications. Yet, there exists a paucity of publicly available pre-trained models which can perform this task with reasonable accuracy. The promise of deep learning for general semantic segmentation has been explored by several works ( [18], [3], [19], [16], [12], [1]) across a variety of scene settings. Face segmentation is a particularly challenging problem because of the irregular shapes, sizes, and textures of facial regions of interest. A category of literature ( [23], [10]) solves the problem of segmenting out one region of interest (hair in most cases) by incorporating priors unique to that region in their architecture and losses. Figure 1. Representative semantic segmentation results on test images sampled from the CelebAMask-HQ dataset. Columns 1, 3 contain input face images and Columns 2, 4 contain their corresponding output segmentation maps, respectively. The color coding used for semantic regions in the segmentation maps is given by; blue: hair, green: skin, red: nose, orange: eyes, and grey: lips + mouth. The SOTA when it comes to segmenting multiple regions of interest in face images are [11] and [22]. While, [11] relies on heavily pre-processed images, [22] incorporates relationships with facial expressions by learning graph representations. The computational overhead that preprocessing operations (warping) incur and the narrow scope of generalization of representation learning are key limitations of these works. These bottlenecks warrant the need for a pre-processing and prior independent approach that can generate fine-grained multi-class segmentation maps with a single forward pass over the input images. To this end, we propose the use of Generative Autoencoders (GAs) capable of producing regions of interest in a selective fashion, given an input image. GA models have shown promising results for tasks such as high fidelity reconstruction of images, style transfer, and style manipulation. However, the latent space of these models is heavily entangled and very high dimensional in nature. This limits their ability to enable spatially localised manipulation of images as a consequence of specific perturbations to their encoded latent space. The Swapping Autoencoder (SAE) [14] is an especially insightful work as it learns a latent representation with a neat distinction between structure and style information of the input image. Inspired by [14], we build upon its capabilities further in this work. Previously, generative models have been used extensively ( [4], [24], [13], [20], [21]) for semantic image synthesis, which is the task of generating photo-realistic images that match the structure of a semantic segmentation map given as input. However, the problem at hand has not seen significant attempts in the past. The contributions of this paper can be summarized as below: • We infuse a strong disentanglement with respect to the structure of semantic ROIs in re-generated images, in the latent space of SAE [14]. Ours is the first work which does so for any Generative Autoencoder model. We provide quantitative metrics for the extent of disentanglement achieved. • We harness the disentangled nature of our model's latent space to perform the challenging downstream task of semantic face segmentation. Results obtained for this task are close to the current SOTA. • Generating the segmentation map for a given ROI from an image belonging to a particular distribution using our model trained on a similar distribution amounts to a simple forward pass with appropriate masking (retention of a single non-zero slice corresponding to the ROI) applied to the latent space. This eliminates the need of any priors for semantic segmentation and underscores the applicability of our model to any generic semantic segmentation problem. Methodology The SAE [14] is a generative Autoencoder model which embeds the structure and style information present in input images (H × H × 3) into a structure tensor (S s , having dimensions H/16 × H/16 × 8) and texture vector (S t , having dimensions 1 × 1 × 2048). The latent space S = {S s , S t } serves as the point of initiation for our work. Our objective in this work, is to disentangle the tensor slices within S s such that they correspond to the structure information of individual regions of interest, namely: hair, skin, nose, eyes, and (lips + mouth) in the reconstructed image. This, in effect, entails that masking (setting to zero) all slices of S s except one should should produce an image containing only its corresponding semantic region of interest, when decoded together with S t . This observation is key to our work and the results and experiments that follow, are based on it. Network Architecture and Losses Figure 2 depicts our model which is a parallelized version of the SAE [14] wherein, disentanglement with respect to semantic regions of interest is infused in the latent space. We achieve this by slicing the structure tensor and enforcing faithful reconstruction of the ROI by the decoder, when given the texture vector and a masked structure tensor (having only one nonzero slice that should correspond to the chosen ROI) as input. The correspondences that we have sought to develop are; Slice 1 (Channels 1 and 2 of S s ): Lips + Mouth (R 5 ). We did not optimize the chosen slicing scheme on the basis of amount of disentanglement obtained. Thus, our results are independent of the chosen slicing scheme. We treat each Siamese decoder together with the encoder as a separate SAE (including all its components viz. the encoder, generator (decoder in our case), discriminator, and patch co-occurrence discriminator) while computing losses. The loss for each correspondence pair is defined as: Hair (R 1 ),l(Y i , Y i ) = L rec (Y i , Y i ) + 0.5L GAN, rec (Y i , Y i ) +0.5L GAN, swap (Y i , Y i ) + 0.5L CooccurGAN (Y i , Y i )(1) Here, Y i referes to the reconstruction obtained from the decoder for the latent space {S s f eature i , S t }, Y i refers to the ground-truth image containing R i only, L rec refers to the reconstruction(L1) loss, L GAN, rec refers to the non-saturating GAN loss, L GAN, swap refers to the non-saturating GAN loss for images generated post swapping, and L CooccurGAN refers to the patch co-occurrence discriminator loss as defined in [14]. For every batch of training data, parameters of the encoder and decoder (all Siamese decoders share the same parameters) are optimized using the following overall loss: L overall = 5 i=1 0.2 · l(Y i , Y i )(2) Training was initiated using the pre-trained weights provided by [14], post training on the FFHQ dataset [6]. Inferring segmentation maps from our model amounts to a simple forward pass over the network with appropriate masking applied to S s in the latent space, depending upon the ROI for which the segmentation has to be performed. Data Preparation We have used images (resized to 128 × 128 × 3 dimensions) from the CelebAMask-HQ [9] and HELEN [8] (contains images in the wild) datasets for training and evaluation. A batch of training data comprised of {X, Y 1 , ..., Y 5 } where X denotes a batch of input images and Y i denotes a batch of region specific images, R i . We sampled only those images from the CelebAMask-HQ dataset which had annotations for all the intended ROIs and split them into 27016 training and 862 test images. The HELEN dataset was used as made available. Experiments Since the objective of this work is to disentangle the latent space of a GA model so as to develop strong correspondences between latent slices and semantic regions of interest in order to harness the same for the downstream application of semantic segmentation of face images, we analyse our model's performance primarily with respect to two criteria, namely the Degree of Disentanglement achieved and the accuracy of predictions for the chosen downstream task. Degree of Disentanglement Degree of Disentanglement implies the extent to which changes made within a chosen slice of the latent representation (S s ) affect the pixel intensities in the desired semantic region of interest (R i ), without causing any deviation to those in other ROIs. We propose the use of an activeness measure inspired by [15] to quantify the degree to which a latent space conforms to this desired attribute. We define the activeness of a latent tensor masked in accordance with the i th slice of S s and denoted as S s f eature i , with respect to a particular semantic ROI (R j ) as: A ij = E n,x (σ 2 ((Decoder(S s f eature i , S t ) · M ask Rj ))) (3) where, Encoder(x) = {S s , S t } (4) S s · M ask Rj = S s f eature j(5) In essence, activeness of S s f eature i with respect to R j for a given input x, is the expectation of variance observed in R j due to addition of noise (n ∈ 0.01 · N (0, 1)) to S s f eature i , taken over all the added noise tensors. We also define the average activeness map (M ap) for a data distribution, such that M ap ij = E x (A ij ). Figure 3 depicts the average activeness map for our model obtained on test data sampled from the CelebAMask-HQ dataset. Since the map (matrix) is nearly diagonal, S s f eature i controls the pixel intensity levels in R j significantly, only if j = i. Thus, the claim made in section 2.1 regarding the correspondences that our model develops, stands validated. We define the ratio of sum of diagonal elements of the average activeness map to that of sum of all elements of the average activeness map as the Activeness Compaction Score (ACS). The ACS for our model with regard to the CelebAMask-HQ dataset [9] was found to be 0.8186 which suggests a nearly diagonal nature of the obtained M ap. Thus, it is evident that the slices of our model's Segmentation Accuracy Qualitative results have been presented in Figure 1 (refer to section 1) and Figure 4. We chose to disentangle two large slices (each containing 4 channels) of S s , with respect to Hair and Skin, respectively, while working with the HE-LEN dataset as the number of training images was lesser than that required to infuse disentanglement within several slices. We compare the accuracy of our model's predicted semantic labels for different ROIs and its rate of inference with SOTA, on the basis F1 scores and computation time per input image in Table 2. Our model is faster and comparable to SOTA for the chosen downstream task. From Table 2, we infer that certain entanglements in the latent space of our model are essential for near perfect re-generation of structure information by the decoder, since it inherits from the StyleGAN2 [7] architecture. Therefore, there is a tradeoff between the extent to which S s is disentangled, and the segmentation accuracy obtained. Since, this work focuses on disentanglement, we have optimized only the amount of disentanglement achieved, and will take up the characterization of this trade-off as a future work. We also observe that the accuracy of predicted segmentation maps and the number of classes for which segmentation is feasible, has a direct correlation with the amount of varied and wellannotated training data available. Conclusion In this work, we proposed and evaluated a method to disentangle the latent space of Generative Autoencoder models with respect to semantic ROIs. We illustrated its applicability to the downstream task of semantic segmentation of face images. Our model outperforms SOTA in terms of disentanglement and is faster while being comparably accurate in performing the downstream task. Our model shall find tremendous utility, especially in AR/VR applications that require selective control over semantic ROIs as a prerequisite since it is entirely prior-agnostic. Figure 2 . 2A schematic representation of our model's architecture and training pipeline. The schema used for slicing Ss and for formation of masked Ss f eature i have been annotated in the legend. l refers to the operation defined by Eq. 1. Slice 2 ( 2Channels 3 and 4 of S s ): Skin (R 2 ), Slice 3 (Channels 5 and 6 of S s ): Nose (R 3 ), Slice 4 (Channel 7 of S s ): Eyes (R 4 ), and Slice 5 (Channel 8 of S s ): Figure 3 . 3Our model's average activeness map obtained on test data sampled from the CelebAMask-HQ dataset. The near diagonal nature of the map suggests a high level of disentanglement among the slices of Ss with respect to the intended semantic ROIs. Figure 4 . 4Challenging instances from test images belonging to (a) HELEN (in the wild) and (b) CelebAMask-HQ dataset. Our model performed well despite the occlusions (regions not belonging to any semantic ROI), being present. The color coding used for the predicted segmentation maps is the same as that used forFigure 1. Table 1 . 1Comparative analysis of our model's latent space with that of SOTA Generative Models which either encode or project input images onto a structured latent space on the basis of ACS using test images from CelebAMask-HQ[9] dataset. Our model's latent space is the most disentangled and by a large margin, with respect to semantic ROIs. for these experiments were taken to be close to ours in order to maintain consistency. Our model's latent space has the most disentanglement (with respect to semantic ROIs) infused in it.Method Latent Dimensions Slicing Scheme ACS↑ [0:4, :] : Hair [4:8, :] : Skin Pixel2Style2Pixel 18 × 512 [8:12, :] : Nose 0.189 [17] [18, 512] [12:16, :] : Eyes [16:18, :] : Lips + Mouth [0:4, :] : Hair [4:8, :] : Skin StyleGAN2ADA 18 × 512 [8:12, :] : Nose 0.194 [5] [18, 512] [12:16, :] : Eyes [16:18, :] : Lips + Mouth [0:2, :, :] : Hair [2:4, :, :] : Skin Swapping Autoencoder 8 × 8 × 8 [4:6, :, :] : Nose 0.259 [14] [8, 8, 8] [6:7, :, :] : Eyes [7:8, :, :] : Lips + Mouth [0:2, :, :] : Hair [2:4, :, :] : Skin Latents2Segments 8 × 8 × 8 [4:6, :, :] : Nose 0.819 (Ours) [8, 8, 8] [6:7, :, :] : Eyes [7:8, :, :] : Lips + Mouth Method F1 Score ↑ Computation Time(ms) ↓ Hair Skin Nose Eyes Skin+Mouth Modified BiSeNet 0.9524 0.89 0.931 0.81 0.741 142.69 Ours 0.7803 0.8532 0.7605 0.578 0.6715 124.00 Table 2. F1 scores and time taken per input image for segmenta- tion with respect to different semantic ROIs obtained on test im- ages from the CelebAMask-HQ dataset. Our model's disentangled latent space yields performance comparable to publicly available pre-trained SOTA (Modified BiSeNet [2]) for most ROIs and the rate of inference is faster. latent space have a direct correspondence with pixel inten- sities in the intended semantic regions of interest only. We present a comparative analysis of our model's latent space with that of several SOTA works in Table 1. The slicing schema chosen Acknowledgement: Support from Institute of Eminence (IoE) project No. SB20210832EEMHRD005001 is gratefully acknowledged. Maxim Maximov, Cyrill Stachniss, Jens Behley, and Laura Leal-Taixe. 4d panoptic lidar segmentation. Mehmet Aygun, Aljosa Osep, Mark Weber, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)Mehmet Aygun, Aljosa Osep, Mark Weber, Maxim Maxi- mov, Cyrill Stachniss, Jens Behley, and Laura Leal-Taixe. 4d panoptic lidar segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5527-5537, June 2021. 1 Rethinking bisenet for real-time semantic segmentation. Mingyuan Fan, Shenqi Lai, Junshi Huang, Xiaoming Wei, Zhenhua Chai, Junfeng Luo, Xiaolin Wei, 2021Mingyuan Fan, Shenqi Lai, Junshi Huang, Xiaoming Wei, Zhenhua Chai, Junfeng Luo, and Xiaolin Wei. Rethink- ing bisenet for real-time semantic segmentation. In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 9711-9720, 2021. 4 Piotr Dollar, and Ross Girshick. Mask r-cnn. 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In Proceedings of European Conference on Computer Vision (ECCV), 2020. 3 Yolov3: An incremental improvement. Joseph Redmon, Ali Farhadi, abs/1804.02767CoRRJoseph Redmon and Ali Farhadi. Yolov3: An incremental improvement. CoRR, abs/1804.02767, 2018. 1 Encoding in style: a stylegan encoder for image-to-image translation. Elad Richardson, Yuval Alaluf, Or Patashnik, Yotam Nitzan, Yaniv Azar, Stav Shapiro, Daniel Cohen-Or, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Elad Richardson, Yuval Alaluf, Or Patashnik, Yotam Nitzan, Yaniv Azar, Stav Shapiro, and Daniel Cohen-Or. Encoding in style: a stylegan encoder for image-to-image translation. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2021. 4 U-net: Convolutional networks for biomedical image segmentation. Olaf Ronneberger, Philipp Fischer, Thomas Brox, abs/1505.04597CoRROlaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. CoRR, abs/1505.04597, 2015. 1 Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, Liang-Chieh Chen, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. Mobilenetv2: Inverted residuals and linear bottlenecksMark Sandler, Andrew Howard, Menglong Zhu, Andrey Zh- moginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In 2018 IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pages 4510-4520, 2018. 1 Diverse semantic image synthesis via probability distribution modeling. Zhentao Tan, Menglei Chai, Dongdong Chen, Jing Liao, Qi Chu, Bin Liu, Gang Hua, Nenghai Yu, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)Zhentao Tan, Menglei Chai, Dongdong Chen, Jing Liao, Qi Chu, Bin Liu, Gang Hua, and Nenghai Yu. Diverse se- mantic image synthesis via probability distribution model- ing. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition (CVPR), pages 7962- 7971, June 2021. 2 Semantic image synthesis via efficient class-adaptive normalization. Zhentao Tan, Dongdong Chen, Qi Chu, Menglei Chai, Jing Liao, Mingming He, Lu Yuan, Gang Hua, Nenghai Yu, abs/2012.04644CoRRZhentao Tan, Dongdong Chen, Qi Chu, Menglei Chai, Jing Liao, Mingming He, Lu Yuan, Gang Hua, and Nenghai Yu. Semantic image synthesis via efficient class-adaptive nor- malization. CoRR, abs/2012.04644, 2020. 2 Edgeaware graph representation learning and reasoning for face parsing. Gusi Te, Yinglu Liu, Wei Hu, Hailin Shi, Tao Mei, European Conference on Computer Vision. SpringerGusi Te, Yinglu Liu, Wei Hu, Hailin Shi, and Tao Mei. Edge- aware graph representation learning and reasoning for face parsing. In European Conference on Computer Vision, pages 258-274. Springer, 2020. 1 Real-time hair segmentation and recoloring on mobile gpus. Andrei Tkachenka, Gregory Karpiak, Andrey Vakunov, Yury Kartynnik, Artsiom Ablavatski, Valentin Bazarevsky, Siargey Pisarchyk, abs/1907.06740CoRRAndrei Tkachenka, Gregory Karpiak, Andrey Vakunov, Yury Kartynnik, Artsiom Ablavatski, Valentin Bazarevsky, and Siargey Pisarchyk. Real-time hair segmentation and recol- oring on mobile gpus. CoRR, abs/1907.06740, 2019. 1 Unpaired image-to-image translation using cycleconsistent adversarial networks. Jun-Yan Zhu, Taesung Park, Phillip Isola, Alexei A Efros, Proceedings of the IEEE International Conference on Computer Vision (ICCV). the IEEE International Conference on Computer Vision (ICCV)Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle- consistent adversarial networks. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), Oct 2017. 2

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{'abstract': "With the advent of an increasing number of Augmented and Virtual Reality applications that aim to perform meaningful and controlled style edits on images of human faces, the impetus for the task of parsing face images to produce accurate and fine-grained semantic segmentation maps is more than ever before. Few State of the Art (SOTA) methods which solve this problem, do so by incorporating priors with respect to facial structure or other face attributes such as expression and pose in their deep classifier architecture. Our endeavour in this work is to do away with the priors and complex pre-processing operations required by SOTA multi-class face segmentation models by reframing this operation as a downstream task post infusion of disentanglement with respect to facial semantic regions of interest (ROIs) in the latent space of a Generative Autoencoder model. We present results for our model's performance on the CelebAMask-HQ and HELEN datasets. The encoded latent space of our model achieves significantly higher disentanglement with respect to semantic ROIs than that of other SOTA works. Moreover, it achieves a 13% faster inference rate and comparable accuracy with respect to the publicly available SOTA for the downstream task of semantic segmentation of face images.", 'arxivid': '2207.01871', 'author': ['Snehal Singh snehal@smail.iitm.ac.in \nIndian Institute of Technology Madras\n\n', 'Tomar A N Rajagopalan \nIndian Institute of Technology Madras\n\n', 'Snehal Singh snehal@smail.iitm.ac.in \nIndian Institute of Technology Madras\n\n', 'Tomar A N Rajagopalan \nIndian Institute of Technology Madras\n\n'], 'authoraffiliation': ['Indian Institute of Technology Madras\n', 'Indian Institute of Technology Madras\n', 'Indian Institute of Technology Madras\n', 'Indian Institute of Technology Madras\n'], 'corpusid': 250280204, 'doi': '10.48550/arxiv.2207.01871', 'github_urls': [], 'n_tokens_mistral': 8172, 'n_tokens_neox': 7095, 'n_words': 4129, 'pdfsha': '0f98a9d5bd3e6f451d74f710ee189cc8614acfa5', 'pdfurls': ['https://arxiv.org/pdf/2207.01871v2.pdf'], 'title': ['Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images', 'Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images', 'Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images', 'Latents2Segments: Disentangling the Latent Space of Generative Models for Semantic Segmentation of Face Images'], 'venue': []}

arxiv

Task-Space Control of Robot Manipulators based on Visual SLAM 8 Feb 2023 Hamed Seyed Jouni Hashemi Mattila Task-Space Control of Robot Manipulators based on Visual SLAM IEEE TRANSACTIONS ON AUTOMATIC CONTROL XX18 Feb 2023Index Terms-Hybrid systemsrobot manipulatortask- space controlvisual SLAM This paper aims to address the open problem of designing a globally stable vision-based controller for robot manipulators. Accordingly, based on a hybrid mechanism, this paper proposes a novel task-space control law attained by taking the gradient of a potential function in SE(3). The key idea is to employ the Visual Simultaneous Localization and Mapping (VSLAM) algorithm to estimate a robot pose. The estimated robot pose is then used in the proposed hybrid controller as feedback information. Invoking Barbalat's lemma and Lyapunov's stability theorem, it is guaranteed that the resulting closed-loop system is globally asymptotically stable, which is the main accomplishment of the proposed structure. Simulation studies are conducted on a six degrees of freedom (6-DOF) robot manipulator to demonstrate the effectiveness and validate the performance of the proposed VSLAM-based control scheme. presented approach is applicable even in the case of singularity or redundancy. Slotine and Li in [6] introduced a feedback law which takes advantage of both a Proportional-Derivativetype (PD-type) controller and a full dynamics feed-forward compensator. This controller eliminates the need for feedback information on joint accelerations and inverse of the estimated inertia matrix. The work in [6] was revisited by [7], where it was proven that a system that tracks desired paths while keeping the regressor matrix exciting over time satisfies the necessary and sufficient conditions of uniform global asymptotic stability. Since the demand for robot manipulators has grown remarkably in different industrial applications, numerous recent studies have been devoted to designing various controllers in task-space, such as quaternion-based [8], dual quaternion-based [9], and inverse dynamics controllers [10]. The vision-based controller is another commonly used method for solving the robot control problem in Cartesian space. It utilizes camera measurements as feedback information. This is because robots endowed with cameras can provide more environmental information and are more flexible in dealing with unstructured environments [11]. Based on the class of feedback information provided by vision sensors, vision-based controllers can be categorized into three main types [12]: 1) image-based visual servoing (IBVS), 2) positionbased visual servoing (PBVS), and 3) hybrid visual servoing (HVS). For its advantages, such as ease of implementation, robustness against image noise, and camera calibration, 2D IBVS has received considerable attention. Nonetheless, IBVS still suffers from some challenges: 1) it can only ensure stability in a region close enough to the desired set, 2) it requires an exact interaction matrix, 3) it can get stuck in a local minima, and 4) it requires all features to be tracked in two consecutive frames [13]. Accordingly, PBVS has been widely researched since it provides 3D pose estimation, a broad field of view, and higher accuracy for motion control. Nevertheless, this method has its own disadvantages, such as sensitivity to camera calibration and a need for accurate robot and camera models [14]. Moreover, the main shortcoming of existing vision-based control techniques is that they still do not consider the effects of robot dynamics in stability proofs. In light of the aforementioned discussion, this paper introduces a new control scheme that exploits every advantage of the vision-based control strategy while trying to overcome its problems. To the best of the authors' knowledge, the stability of state-of-the-art vision-based control methods is only valid locally in confined task-space. Furthermore, the stability of these techniques depends on the number of visible features and camera calibration. Consequently, for the first time, this paper suggests utilizing a VSLAM observer to estimate robot pose because its stability is independent of camera and feature information. As a result, this paper proposes a new VSLAMbased control structure that, to improve upon existing taskspace control approaches, makes the following contributions: • A new hybrid feedback law is designed directly in taskspace to control end-effector pose. The proposed control law is a gradient-based controller obtained by taking the gradient of a positive-valued continuously differentiable function in SE(3). • A geometric VSLAM algorithm is derived directly from the Lie group of SLAM n (3). The introduced VSLAM observer estimates the robot pose, which is then used as taskspace feedback information for the proposed controller. • The main contribution of this paper is to show how a VSLAM-based control structure gains global asymptotic stability by incorporating robot dynamics. This paper consists of six sections, including the introduction. Preliminary mathematical definitions, a generic background on hybrid systems, SLAM kinematics, and the kinematic and dynamic equations of the robot manipulator are provided in section 2. A visual SLAM algorithm is described in section 3. Section 4 presents a design and stability analysis of the proposed hybrid feedback law. Section 5 demonstrates the performance of the proposed VSLAM-based control structure based on simulation results. By providing some concluding remarks, section 6 summarizes the paper. II. PRELIMINARIES A. Notation By R, this paper denotes the set of real numbers, and by N, it denotes the set of natural numbers. S n := {y ∈ R n+1 : y = 1} and B := {y ∈ R n : y ≤ 1}, respectively, represent a unit n-dimensional sphere and a closed unit ndimensional ball. R n denotes n-dimensional Euclidean space, where {e i } 1≤i≤n ⊂ R n is the canonical basis of R n . The Euclidean norm of a vector x ∈ R n is given by x = x, x , where x, y := x T y are the inner products and x A := min y∈A x − y . Given a matrix A ∈ R n×n , its trace, determinant, transpose, and skew-symmetric parts and = {X = Ψ(R, p, η) : R ∈ SO(3), p ∈ R 3 , η ∈ R 3×n } is the SLAM group and X −1 = Ψ(R T , −R T p, −R T η) is its inverse. The Lie algebra of SLAM n (3) is given by slam n (3) := {V(ω, v, ξ) = Γ(ω) v ξ 0 n+1×3 0 n+1×1 0 n+1×n : ω, v ∈ R 3 , ξ ∈ R 3×n }. Let M be a smooth manifold and T X M denote its tangent space, as in T X SLAM n (3) := {X V : X ∈ SLAM n (3) and V ∈ slam n (3)}. The map Ad X : SLAM n (3)×slam n (3) → slam n (3) is called the adjoint map for the SLAM group and its associated Lie algebra slam n (3), which transforms a tangent vector of one element into a tangent vector of another. The following maps are frequently utilized throughout the paper. Γ(y) =   0 −y 3 y 2 y 3 0 −y 1 −y 2 y 1 0   , ϕ(A) = 1 2   A (3,2) − A (2,3) A (1,3) − A (3,1) A (2,1) − A (1,2)   ,φ(A) = ϕ(A) 1 2 y , Ψ(R, p, η) =   R p η 0 1×3 1 0 1×n 0 n×3 0 n×1 I n×n   , Ψ(R, p) = R p 0 1×3 1 , A = A y c d , Υ(B) = Υ( A B 2 B T 3 B 4 ) = skew(A) B 2 0 n+1×3 0 n+1×n+1 , Υ(A) = Υ( skew(A) y 0 1×3 0 ), AD X = R 0 3×3 Γ(p)R R , ∀X =Ψ(R, p) y ∈ R 3×1 , A ∈ R 3×3 , c ∈ R 1×3 , d ∈ R (B 2 , B 3 ) ∈ R 3×n+1 , B 4 ∈ R n+1×n+1(1) Moreover, ∇ X m ∈ T X M represents the gradient of a differentiable smooth function m : M → R, which is determined throuh the following equation: dm.X V = ∇ X m, X V X = X −1 ∇ X m, V(2) In Equation (2), dm and ., . X stand for the differential of m and a Riemannian metric on the matrix Lie group, respectively, such that X V 1 , X V 2 X = V 1 , V 2 . A Rodrigues formula ℜ : R × S 2 → SO(3) defined by ℜ(θ, y) = I + sin(θ)Γ(y) + (1 − cos(θ))Γ 2 (y), or ℜ(θ, y) = exp(θΓ(y)),(3) is utilized to describe a rotation matrix R ∈ SO(3) in terms of its axis y ∈ S 2 and angle θ ∈ R of rotation. B. Hybrid System Framework The following equation describes the framework for hybrid dynamical systems H used throughout this paper [15]: H : ẋ = f (x, u), (x, u) ∈ C x + = g(x, u), (x, u) ∈ D(4) Here, x ∈ R n and u ∈ R m denote the state vector and the input of H, respectively. The flow map f : R n × R m → R n defines the continuous evolution of x when (x, u) belongs to the flow set C ⊂ R n ×R m . The jump map g : R n ×R m → R n describes the behavior of the systems during jumps when (x, u) belongs to the jump set D ⊂ R n × R m . C and D illustrate where continuous evolution and jumps are permitted, respectively. A solution to H is defined in a hybrid time domain E ⊂ R ≥0 × N, which is parameterized by the time variable t ∈ R ≥0 and jump variable j ∈ N. The subset E is a hybrid time domain if it can be written as E = I i=1 ([t i , t i+1 ] , i) for finite sequences of time 0 = t 0 ≤ t 1 · · · ≤ t I+1 . Lemma 1 [16]: For the hybrid system H, the closed set A ⊂ R n is judged locally exponentially stable if there exist (α 1 > α 2 , s 1 , s 2 , n) ∈ R ≥0 and a continuously differentiable function V : R n → R ≥0 such that the following inequalities hold: α 2 x n A ≤ V (x) ≤ α 1 x n A , ∀x ∈ (C ∪ D ∪ g(D)) ∩ (A + s 1 B) ∇V (x), f ≤ −s 2 V (x), ∀x ∈ C ∩ (A + s 1 B) V (g) ≤ exp(−s 2 )V (x), ∀x ∈ D ∩ (A + s 1 B).(5) Function V is defined in an open set containing the closure of C. When s 1 → ∞ and both s 2 → 0 and s 1 → ∞, set A is said to be globally exponentially stable and globally asymptotically stable, respectively. C. SLAM Kinematics The kinematic equations of motion of a rigid body and the i th landmark can be expressed aṡ R = RΓ(ω),(6)p = Rv,(7)η i = Rξ i , i = 1, . . . , n(8) where ξ i ∈ R 3 , ω ∈ R 3 , and v ∈ R 3 , respectively, denote the linear speed of the i th landmark, angular rate, and linear velocity of a rigid body with respect to the body-fixed frame B. Furthermore, p ∈ R 3 represents the position of a rigid body in the inertial frame I and η i ∈ R 3 represents the location of the i th landmark in I. The motion kinematics in (4-6) can be rephrased more compactly aṡ X = X V.(9) This paper focuses on stationary landmarks, which signify that ξ i = 0. The robot is equipped with sensors to measure its linear and angular velocity. The robot is also equipped with a camera that can measure ranges θ b = η i − p and bearings  = R T (η i − p)/θ b in relation to landmarks. Accordingly, β i denotes a camera measurement that contains both range and bearing measurements and is given by β i := X −1 r i =   R T (η i − p) 1 −e i   , r i =   0 3×1 1 −e i   . (10) D. Robot Manipulator Dynamics The dynamics of an n-link robot manipulator can be governed by the following so-called Euler-Lagrange equation [17]: M (q)q + C(q,q)q + G(q) + F (q) = τ,(11) where q = [q 1 , q 2 , · · · , q n ] T ∈ R n is the joint position anḋ q = [q 1 ,q 2 , · · · ,q n ] T ∈ R n is the joint velocity. Furthermore, M (q) ∈ R n×n is the nominal inertia matrix, C(q,q) ∈ R n×n represents the nominal Coriolis-centrifugal matrix, G(q) ∈ R n is the gravity vector, and F (q) ∈ R n contains the frictional force coefficients. Moreover, τ ∈ R n represents the applied control torque. The forward kinematics of a robotic manipulator provide a map between Cartesian space and joint space given by x end (t) = h(q(t)).(12) Here, x end denotes the attitude and position of a manipulator end-effector in task-space. Taking the derivative with respect to time on both sides of (12) yields the relation between the joint and Cartesian space velocities. x end (t) = ∂h(q) ∂qq (t) = Jq(t), or, Z = ω v = Jq(t) (13) Here, J is the Jacobian matrix of the forward kinematics, and ω, v represent the angular and linear velocity components of the end-effector velocity vector, respectively. III. VISUAL SLAM ALGORITHM This section presents the visual SLAM algorithm, which is used to estimate end-effector poses. There is an extensive literature full of various methods for solving the VSLAM problem. Consequently, existing VSLAM algorithms fall into three categories: optimization-based methods [18], geometrictype techniques [19], and Kalman-type algorithms [20]. All these techniques have their own advantages and disadvantages. For example, geometric-type algorithms can only guarantee almost global stability due to the existence of sets with Lebesgue measure zero in SO(3). Likewise, Kalman-type methods and optimization-based strategies suffer from performance dependency on the initialization and also cannot ensure stability. Consequently, this paper makes use of the VSLAM algorithm introduced by the authors in [21], where it was proven that the provided VSLAM method can guarantee global asymptotic stability and overcome problems associated with existing VSLAM algorithms. The first step in designing the proposed VSLAM algorithm is defining the potential function, U : SLAM n (3) → R, which is given by U(X ) = 1 2 tr((I − X )A(I − X ) T ), A := n i=1 k i r i r i T , k i ∈ R ≥0 .(14) The following identity is useful in determining the gradient of potential function ∇ X U, which is obtained with the aid of a Riemannian metric in SLAM n (3) and identities in the Appendix. dU.X V = X −1 ∇ X U, V , or, dU.X V = tr(−A(I − X ) T X V) = Υ(X −1 (X − I)A), V = Υ((I − X −1 )A), V(15) As a result, the gradient of potential function U with respect to X is determined as follows: ∇ X (U) = X Υ((I − X −1 )A)(16) Estimation error is often defined as the difference between the true state value X and estimated state valueX , i.e.,X = XX −1 withR = RR T ,p = p −Rp, andη = η −Rη. Hence, the following identities represent the potential function (14) and its gradient (16) in terms of estimation error: Υ( n i=1 k i (r i −X β i )r T i ) = Υ((I −X −1 )A), (a) n i=1 k i r i −X β i 2 = tr((I −X )A(I −X ) T ), (b)(17) The following equation describes the dynamics of the VS-LAM algorithm introduced by the authors in [21]: Ẋ =X (V − ∆),X ∈ Ċ q = 0,   X + = X q ,X ∈ D q + = arg min q∈Q U(X q ), C := {(U(X ) − miñ Xq∈Q U(X q ) ≤ δ), D := {(U(X ) − miñ Xq∈Q U(X q ) ≥ δ), X q = Ψ(ℜ(qθ, ℓ), 0, 0)Ψ(R,p,η), q ∈ N ∆ = −AdX −1 Υ( n i=1 k i (r i −X β i )r T i )K,(18) Here, K := k o I n+4×n+4 with k o ∈ R >0 is the observer gain, q ∈ N belongs to a compact set Q = {X q ∈ SLAM n (3) : q ∈ N, ℓ ∈ S 2 , θ ∈ R >0 }, andX q = X X −1 q . Moreover, (θ, δ) ∈ R >0 , and ℓ ∈ S 2 are arbitrary constants and an arbitrary fixed vector, respectively. Theorem 1: Consider the SLAM kinematics (9) evolving on SLAM n (3) along with bounded measurements (10). The VSLAM algorithm defined by (18) is a global asymptotic convergent observer, i.e., state estimation errorX globally asymptotically converges to I n+4×n+4 . Proof: The proof of Theorem 1 is omitted here; however, full details can be found in [21]. IV. PROPOSED HYBRID FEEDBACK LAW The proposed hybrid feedback law is developed in this section. In the past decade, hybrid controllers have frequently been used for stabilizing systems evolving in matrix Lie groups [22] since it was proven that continuous and discontinuous feedback laws cannot globally stabilize these systems in the desired set [23]. This is due to the non-contractibility of the configuration space of the attitude and existence of sets that have Lebesgue measure zero [24]. Nonetheless, to the best of the authors' knowledge, hybrid controllers have not yet been applied to a robot manipulator. Consequently, for the first time, this paper designs a new hybrid feedback law to control robot manipulators in task-space. The proposed hybrid feedback law is directly designed in SE(3) (end-effector configuration space) by constructing a potential function in SE(3) and taking its gradient. Accordingly, potential function U : SE(3) × R → R is defined as follows: U(X, h) = 1 2 tr((I − X h X)G(I − X h X) T ), X h = ℜ(θ h , ) 0 3×1 0 1×3 1(19) Here, G ∈ R 4×4 is a symmetric positive definite matrix, X ∈ SE(3) := {X =Ψ(R, p) : R ∈ SO(3), p ∈ R 3 },  ∈ S 2 is an arbitrary constant vector, and θ h ∈ R belongs to a compact set Ξ := {θ h ∈ R : |θ h | ≤ π/2}. By following the same procedure of gradient calculation for the previous potential function, the gradient of U(X, h) is determined as follows (for details, see [25]): ϕ(X −1 ∇ X U(X, h)) = AD −T X hφ ((I − (X h X) −1 )G) (20) Consequently, the proposed hybrid feedback control law is defined as τ * = N (q,q) − M (q)J −1 (Jq +φ(X −1 e ∇ Xe U(X e , h) + G d Y)), N (q,q) = C(q,q)q + G(q) + F (q).(21) In Equation (21), G d = g d I 6×6 with g d ∈ R >0 is the controller gain, and definitions of X e and Y are given in the proof of Theorem 2. The following hybrid mechanism calculates the switching variable h: ḣ = 0, (X e , h) ∈ C ′ h + = arg min h ′ ∈Ξ U(X e , h ′ ), (X e , h) ∈ D ′ C ′ := {(U(X e , h) − min h ′ ∈Ξ U(X e , h ′ ) ≤ δ), D ′ := {(U(X e , h) − min h ′ ∈Ξ U(X e , h ′ ) ≥ δ),(22) Theorem 2: Consider the robot dynamic equation (11) in a closed loop with the proposed hybrid feedback law (21) and observer (18). Then, the compact set A : = {X d ∈ SE(3),X ∈ SLAM n (3) : X d =Ψ(R d , p d ),X = I} is globally asymptotically stable for the resulting closed-loop systems. Proof: In accordance with Lemma 1, the proof of Theorem 2 fall into two phases. Step 1: The second condition of (5) is proven in this step. It can easily be shownẊ −1 = −X −1Ẋ X −1 by using the fact that X −1 X = I 4×4 . The end-effector motion is represented by the following kinematic model: X = X Γ(ω) v 0 0 = XW(23) To define the end-effector pose tracking error X e = X −1 d X, one can formulate the tracking error dynamics as follows. X e =Ẋ −1 d X + X −1 dẊ ⇒ X e = X e (W − Ad X −1 e W d ) = X e Y(24) Here, the definition of W d is the same as that of W with the desired constant angular velocity ω d and constant linear velocity v d . This paper employs the Lyapunov candidate function as follows: V (X e ,X , Y) = U(X e , h) + U(X ) + 1 2φ T (Y)φ(Y) (25) The time derivative of the proposed Lyapunov function is given bẏ V = ∇ Xe U, X e Y Xe + ∇X U,Ẋ X +φ T (Y)φ(Ẏ).(26) Therefore, by substituting Equations 24 and 27 [26] into Equation (26) and using the fact thatẆ d = 0, one getṡ V = −k o Υ((I −X −1 )A) 2 F + X −1 e ∇ Xe U, Y +φ T (Y)φ(Ẇ)(27) In Equation (27),φ(Ẇ) is achieved by taking the time derivative of Equation (13) sinceφ(W) = Z. Then, Z = Jq +Jq,(28) and replacingq by Equation (11), one getṡ Z = JM −1 (q)(τ − N (q,q)) +Jq.(29) The following equation is the result of applying the proposed hybrid feedback law (τ * ) to Equation (29): Z = −φ(X −1 e ∇ Xe U + G d Y)(30) Consequently, Equation (27) is simplified tȯ V = −k o Υ((I −X −1 )A) 2 F − g dφ T (Y)φ(Y) + X −1 e ∇ Xe U, Y −φ T (Y)φ(X −1 e ∇ Xe U), ⇒ V = −k o Υ((I −X −1 )A) 2 F − g dφ T (Y)φ(Y) ≤ 0.(31) As a result, it follows from (31) that both the tracking error and estimation error are globally bounded, hence;V is also globally bounded. By invoking Barbalat's lemma, it can be deduced that lim t→+∞V = 0; therefore, lim t→+∞ X → X d and lim t→+∞ X →X . Step 2: This step provides proof for the third condition of (5). Due to the existence of switching variables (h, q), it is necessary to test variation in V (X e ,X , Y) to guarantee that the Lyapunov function remains negative during jumps. Given the last condition of (5), the variation in V among jumps is defined by V + (X e ,X , Y) − V (X e ,X , Y) = U(X e , h + ) + U(X + ) − U(X e , h) − U(X ) = U(X q ) − U(X ) + U(X e , h + ) − U(X e , h)(32) From (22) and (18), one can determine that miñ Xq∈Q U(X q ) − U(X ) ≤ −δ, min h ′ ∈Ξ U(X e , h ′ ) − U(X e , h) ≤ −δ.(33) Accordingly, from Lemma 1, one can easily derive that set A is globally asymptotically stable. It is worth noting that the estimated poseΨ(R,p) can be substituted for the true poseΨ(R, p) in the proposed feedback law without invalidating the stability proof since lim t→+∞ X → X . The block diagram of the proposed VSLAM-based control structure is depicted in Figure (1), and the salient features of the proposed method are 1) its simple structure, 2) global stability, and 3) light computational burden. V. SIMULATION RESULTS In this section, a numerical simulation study is performed to assess the performance of the proposed VSLAM-based control structure given by (21). This study uses a 6-DOF manipulator with a long arm, illustrated in Figure (2). A detailed description of and supplementary material on this manipulator can be found in [27]. The observer gain and controller gain were determined as follows through trial and error until a satisfactory performance was obtained: 6-DOF manipulator arm utilized to validate the proposed structure. Fig. 3. End-effector trajectory, desired trajectory, and estimated trajectory in 3D space. K = 100I n×n , G = 200I 4×4 , G d = 5I 6×6 Here, the task of the robot is to draw a square with sides of 50 cm to demonstrate the robustness of the proposed observer against initial conditions. The initial position of the observer is chosen as [3.6 1 0], which is different from the true value. The simulation results in Figures (3-6) are achieved by applying the proposed VSLAM-based control scheme to the 6-DOF manipulator with the long arm. The desired trajectory, actual trajectory, and trajectory estimated by VSLAM are depicted in Figure ( Figures (4-5) that a tracking error of approximately 2% is obtained despite the existence of 1% error in the estimated position. The torque produced by the introduced VSLAM-based control structure is illustrated in Figure (6). This figure confirms that the torque produced by the proposed controller is realizable and applicable. VI. CONCLUSION This paper investigated the problem of designing a globally stable vision-based controller for robot manipulators. To address this problem, a novel control structure was introduced in which a visual SLAM observer was employed for estimating the robot pose. Based on feedback information provided by the VSLAM observer, a new hybrid feedback law was directly designed in SE(3). The proposed hybrid controller was derived by taking the gradient of a potential function defined in SE (3). The global asymptotic stability of the proposed VSLAM-based control structure was proven with the help of the Lyapunov stability theorem. Finally, the proposed control scheme was tested on a 6-DOF robot manipulator to demonstrate its accuracy and efficiency. APPENDIX I USEFUL PROPERTIES OF SLAM n (3) AND SE(3) This subsection presents some useful identities, properties, and maps related to the matrix Lie groups SLAM n (3) and Frobenius norm are defined as tr(A), det(A), A T , skew(A) = (A − A T )/2, and A F = A, A = tr(A T A), respectively. In this paper, matrix R belongs to the special orthogonal group of order three SO(3) := {R ∈ R 3×3 : R T R = RR T = I, det(R) = 1}, which denotes the attitude of a rigid body. The Lie algebra of SO(3) is denoted by so(3) = {A ∈ R 3×3 : A T = −A}. The matrix Lie group SLAM n (3) : Fig. 1 . 1Block diagram of the proposed scheme. 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{'abstract': "This paper aims to address the open problem of designing a globally stable vision-based controller for robot manipulators. Accordingly, based on a hybrid mechanism, this paper proposes a novel task-space control law attained by taking the gradient of a potential function in SE(3). The key idea is to employ the Visual Simultaneous Localization and Mapping (VSLAM) algorithm to estimate a robot pose. The estimated robot pose is then used in the proposed hybrid controller as feedback information. Invoking Barbalat's lemma and Lyapunov's stability theorem, it is guaranteed that the resulting closed-loop system is globally asymptotically stable, which is the main accomplishment of the proposed structure. Simulation studies are conducted on a six degrees of freedom (6-DOF) robot manipulator to demonstrate the effectiveness and validate the performance of the proposed VSLAM-based control scheme.", 'arxivid': '2302.04163', 'author': ['Hamed Seyed ', 'Jouni Hashemi ', 'Mattila '], 'authoraffiliation': [], 'corpusid': 256662697, 'doi': '10.48550/arxiv.2302.04163', 'github_urls': [], 'n_tokens_mistral': 11000, 'n_tokens_neox': 9688, 'n_words': 5766, 'pdfsha': '615c276a0bd35ef46780dde92621d1808d3723fb', 'pdfurls': ['https://export.arxiv.org/pdf/2302.04163v1.pdf'], 'title': ['Task-Space Control of Robot Manipulators based on Visual SLAM', 'Task-Space Control of Robot Manipulators based on Visual SLAM'], 'venue': ['IEEE TRANSACTIONS ON AUTOMATIC CONTROL']}

arxiv

Modélisation et manipulation de données historisées et archivées dans un entrepôt orienté objet Modelling and querying temporal and archive data in an object-oriented warehouse Franck Ravat Olivier Teste teste@irit.fr Modélisation et manipulation de données historisées et archivées dans un entrepôt orienté objet Modelling and querying temporal and archive data in an object-oriented warehouse Entrepôt de DonnéesBDOODonnées Temporelles et ArchivéesLangage de Manipulation Keywords : Data WarehouseOODBTemporal and Archive DataQuery Language Cet article aborde la modélisation et la manipulation des entrepôts objet intégrant des données historisées et archivées. Dans un premier temps, nous proposons un modèle décrivant l'entrepôt comme un référentiel centralisé de données complexes et temporelles. Notre modèle intègre les concepts d'objet entrepôt et d'environnement. Un tel objet est composé d'un état courant, de plusieurs états passés (modélisant les évolutions détaillées) et de plusieurs états archivés (modélisant les évolutions de manière résumée). Le concept d'environnement définit les parties temporelles dans le schéma de l'entrepôt avec une granularité pertinente (attribut, classe, graphe). Dans un second temps, nous définissons une algèbre de manipulation des données de l'entrepôt. Cette algèbre basée sur une extension des algèbres objet propose des opérateurs temporels et des opérateurs de manipulation d'ensembles d'états des objets entrepôt. Une contribution importante réside dans la proposition d'opérateurs spécifiques de restructuration en série temporelle et des opérateurs facilitant les traitements analytiques.AbstractThis paper deals with temporal and archive object-oriented data warehouse modelling and querying. In a first step, we define a data model describing warehouses as central repositories of complex and temporal data extracted from one information source. The model is based on the concepts of warehouse object and environment. A warehouse object is composed of one current state, several past states (modelling value changes) and several archive states (summarising some value changes). An environment defines temporal parts in a warehouse schema according to a relevant granularity (attribute, class or graph). In a second step, we provide a query algebra dedicated to data warehouses. This algebra, which is based on common object algebras, integrates temporal operators and operators for querying object states. An other important contribution concerns dedicated operators allowing users to transform warehouse objects in temporal series as well as operators facilitating analytical treatments. Introduction De nos jours, les entreprises ont recours à des systèmes décisionnels (OLAP), basés sur l'approche des entrepôts de données [28] pour exploiter d'importants volumes d'information à des fins d'analyse et d'aide à la décision. Un entrepôt [7,18,28] permet de stocker les données nécessaires à la prise de décision ; il est alimenté par des extractions de données portant sur des bases opérationnelles, appelées sources de données. Nos travaux de recherche [20,21,22,23,26] se placent dans le contexte des systèmes d'aide à la décision basés sur l'approche des entrepôts de données. Nos travaux s'intègrent dans le cadre du projet REANIMATIC 1 et visent plus particulièrement à développer des systèmes décisionnels aptes à supporter efficacement des analyses, afin d'améliorer la qualité des soins et le devenir des patients en réanimation. Notre approche se base sur une dichotomie entre deux espaces de stockage au sein du système décisionnel [3] : -L'entrepôt de données (data warehouse) est le lieu de stockage centralisé d'un extrait des bases de production. Cet extrait concerne les données pertinentes pour le support à la décision. Elles sont intégrées et historisées. L'organisation des données est faite selon un modèle qui facilite la gestion efficace des données et leur historisation. -Le magasin de données (data mart) est un extrait de l'entrepôt. Les données extraites sont adaptées à une classe de décideurs ou à un usage particulier (recherche de corrélation, logiciel de statistiques,...). L'organisation des données suit un modèle spécifique qui facilite les traitements décisionnels. L'objet de nos travaux est donc de spécifier des modèles de représentation et des langages de manipulation dédiés aux entrepôts et aux magasins de données complexes et évolutives. Peu de recherches traitent de cette double problématique dans le cadre des systèmes décisionnels : fournir des modèles de données complexes et évolutives, et spécifier les langages de manipulation associés. Nos premiers travaux ont permis de définir l'architecture fonctionnelle d'un système d'aide à la décision [26] distinguant les problématiques de recherche (cf. figure 1). -L'intégration se propose de résoudre les problèmes d'hétérogénéité (modèles, formats et sémantiques des données, systèmes,…) des différentes sources de données en intégrant celles-ci dans une source globale. Cette dernière est décrite au moyen du modèle de données orientées objet standard de l'ODMG [6]. Le choix du paradigme objet se justifie car il s'avère parfaitement adapté pour l'intégration de sources hétérogènes [4] couramment utilisées dans le milieu médical [18]. Cette source globale est virtuelle : les données utilisées pour la décision restent stockées dans les sources et sont extraites au moment des mises à jour de l'entrepôt. L'intégration s'appuie sur des techniques de bases de données fédérées [24] et/ou réparties [17]. -La construction consiste à extraire les données pertinentes pour la prise de décision, puis à les recopier dans l'entrepôt, tout en conservant, le cas échéant, les changements d'états des données. Par conséquent, l'entrepôt constitue une collection centralisée, de données matérialisées et historisées (conservation des évolutions), disponibles pour les applications décisionnelles. Le modèle de l'entrepôt doit supporter des structures complexes [18] et l'évolution des données au cours du temps [18,31]. -La structuration réorganise l'information dans des magasins de données afin de supporter efficacement les processus d'interrogation et d'analyse, tels que les applications OLAP (On-Line Analytical Processing) [8] et la fouille de données (data mining) [10] ; les industriels proposent de nombreux outils permettant une telle activité (Express, Warehouse Builder, Business Object, Impromptu,…). Pour ce faire, les données importées dans les magasins sont souvent organisées de manière multidimensionnelle [1,22]. Cette organisation des systèmes décisionnels offre deux visions : [11]. Des limites subsistent dans l'élaboration d'un système décisionnel basé sur la dualité entre un entrepôt et des magasins de données. Les travaux actuels se basent sur des modèles relationnels ou multidimensionnels qui n'intègrent ni des structures complexes ni une sémantique riche. Ceci oblige les concepteurs à un effort d'abstraction important afin de représenter le monde réel dans l'entrepôt. De nouvelles propositions vont dans ce sens ; notamment, [19] définit un modèle orienté objet multidimensionnel pour données complexes et temporelles. Néanmoins, ce modèle n'est pas adapté au niveau de notre entrepôt ; aucun langage de manipulation des données n'est proposé. L'approche multidimensionnelle est adaptée à l'interrogation et l'analyse des magasins de données, mais reste inadéquate pour maintenir et gérer efficacement les données d'un entrepôt sur de longues périodes de temps. Les propositions actuelles concernant la gestion du temps dans les entrepôts [16,31] n'offrent pas de mécanismes flexibles pour l'historisation des données ; le plus souvent, les données anciennes sont simplement supprimées de l'entrepôt. Or, aucune proposition ne fournit de mécanismes intermédiaires permettant, par exemple, d'archiver automatiquement les données détaillées à un niveau plus élevé exploitable par les décideurs. [16] fournit un modèle multidimensionnel et un langage temporel T-OLAP de manipulation des données. Ces travaux sont relatifs aux magasins de données (dans notre architecture) et ne paraissent pas adaptés pour un entrepôt dont le modèle de représentation des données ne suit pas une organisation multidimensionnelle. En outre, aucun mécanisme d'archivage n'est proposé. Modèle temporel orienté objet Dans cette section, nous définissons un modèle de données pour les entrepôts, basé sur le paradigme objet. Notre modèle subit l'influence du modèle objet standard de l'ODMG [6] qui est étendu pour prendre en compte les caractéristiques des entrepôts de données. Notamment, notre modèle intègre la dimension temporelle d'une manière flexible en permettant l'archivage des données temporelles. Objet entrepôt Chaque information extraite (objet, partie ou groupe d'objets source) est représentée dans l'entrepôt par un objet entrepôt qui conserve ses évolutions de valeur au cours du temps (tandis que la source de données ne contient que l'état courant [7], ou bien, ne conserve qu'une partie récente des évolutions, insuffisante pour la prise de décision [31]). Dans un entrepôt, l'administrateur peut décider de conserver : -l'image de l'information extraite c'est-à-dire l'état courant, ainsi que les états successifs que prend au cours du temps l'information extraite, c'est-à-dire ses états passés, -uniquement un résumé des états passés successifs, c'est-à-dire l'agrégation de certains états passés, appelée état archivé. Les états passés ainsi résumés sont supprimés de l'entrepôt afin de limiter l'accroissement du volume des données. Un objet entrepôt est donc défini par le quadruplet (oid, S 0 , EP, EA) où oid est l'identifiant interne, S 0 est l'état courant, EP = {S p1 , S Classe entrepôt Définition Les objets entrepôt qui ont la même structure et le même comportement, sont regroupés dans une classe. Pour prendre en compte les caractéristiques des objets entrepôt, nous définissons le concept de classe entrepôt c caractérisé par un n-uplet constitué -du nom de la classe propriétés temporelles de c (une propriété est temporelle lorsque ses évolutions sont conservées par des états passés). Le filtre temporel caractérise la structure des états passés des objets de la classe. -d'un filtre d'archives Archi c définissant l'ensemble des propriétés archivées de c (une propriété est archivée lorsque ses évolutions passées sont résumées dans des états archivés). Le filtre d'archives caractérise la structure des états archivés des objets de la classe. Mécanisme d'extraction Chaque classe entrepôt est partiellement définie par une fonction de construction Mapping c appliquée sur la source de données. Cette fonction est une composition de fonctions de base. Nous proposons une taxinomie des fonctions de base supportées par notre modèle d'entrepôt de données répondant aux différents problèmes posés : -les fonctions de structuration (FS) induisent la structure (attributs et relations) des classes entrepôt ; -les fonctions de peuplement (FP) induisent les objets source à partir desquels l'extension des classes entrepôt est calculée ; -les fonctions ensemblistes (FE) correspondent aux opérations ensemblistes classiques de l'algèbre objet de [25] Mécanisme d'historisation Le filtre temporel Tempo c ={(p 1 , f 1 ), (p 2 , f 2 ),…, (p t , f t )} Environnements, schéma de l'entrepôt, configurations Cette dualité (passé/archive) dans la conservation de l'évolution des données pose une difficulté relative à la définition du comportement temporel des classes. Il est indispensable de fournir des mécanismes permettant : -de définir des critères pour caractériser les états passés à archiver (seuil de déclenchement de l'archivage), -de garantir l'intégrité des relations sémantiques (associations, compositions) temporelles. En effet, conserver l'évolution d'une relation exige de conserver les états passés impliqués dans la relation. Remarquons que ce concept d'environnement aide l'administrateur à définir différentes parties temporelles dans l'entrepôt. Ceci permet de concevoir un entrepôt flexible qui s'adapte aux différentes exigences des décideurs. Un entrepôt se caractérise par son schéma S ED défini par un nom Nom ED , l'ensemble fini des classes de l'entrepôt C ED = {c 1 , c 2 ,…, c n }, l'ensemble fini des environnements Env ED = {Env 1 , Env 2 ,…, Env ne } et un ensemble de règles de configuration Config ED , visant à définir les différents paramètres de configuration globale de l'entrepôt (période de rafraîchissement,…). Remarquons que la clause if comporte soit une expression booléenne, soit une requête de sélection [29]. Dans notre exemple, la requête permet d'obtenir un ensemble d'états passés sur lequel sera appliqué l'action (si le résultat est vide, l'action n'est pas déclenchée). Résumé Opérateurs classiques C urrent(E ns) P ast(E ns) A rchive(E ns) Opérateurs temporels L'objet de cette section est d'offrir des mécanismes souples et puissants permettant de manipuler simplement la dimension temporelle inhérente aux objets entrepôt. Extension des relations de Allen aux domaines temporels Afin de faciliter la manipulation des domaines temporels, nous généralisons les treize relations définies par Allen [2] aux domaines temporels associés aux états des objets entrepôt. L'intérêt de cette généralisation est qu'elle offre la puissance nécessaire pour combiner simplement les domaines temporels, c'est-à-dire les valeurs temporelles associées aux états des objets entrepôt. Soient X et Y deux domaines temporels. On pose -X=<[TDeb X 1 , TFin X 1 ], [TDeb X 2 , TFin X 2 ],…, [TDeb X hx , TFin X hx ]> et -Y=<[TDeb Y 1 , Tfin Y 1 ], [TDeb Y 2 , Tfin Y 2 ],…, [TDeb Y hy , Tfin Y hy ]>. TAB. 2 -Extension des opérateurs de Allen aux domaines temporels. RELATION DEFINITION RECIPROQUE X Precedes Y TFin X hx<TDeb Y 1 Y Follows X X Meets Y TFin X hx=TDeb Y 1 Y IsMeeted X X Overlaps Y IntiX, IntjY, TDeb X i<TDeb Y j  TDeb Y j<TFin X i  TFin X i<TFin Y j Y IsOverlaped X X During Y IntiX, IntjY, TDeb X i>TDeb Y j  TFin X i<TFin Y j Y IsDuringX X Starts Y TDeb X 1=TDeb Y 1 Y IsStarted X X Ends Y TFin X hx=TFin Y hy Y IsFinished X X Equals Y i[1,h1], h1=h2, TDeb X i=TDeb Y i  TFin X i=TFin Y i Y Equals X Opération de restriction temporelle Opérations de jointure temporelles La jointure est étendue pour permettre de combiner les ensembles d'états. Cette adaptation de la jointure concerne le caractère temporel des états pour lequel deux possibilités sont offertes : -le résultat peut être calculé à partir de l'intersection des domaines temporels des états ; -le résultat peut être calculé à partir de l'union des domaines temporels des états. Formellement, nous définissons les jointures par intersection et par union des domaines temporels de la manière suivantes : -IJoin(Ens 1 , Ens 2 , P)={(DomT, V 1 , V 2 ) | S 1 Ens 1 , S 2 Ens 2 , DomT=DomT 1 DomT 2  V 1 S 1  V 2 S 2  P(V 1 , V 2 )} ; -UJoin(Ens 1 , Ens 2 , P)={(DomT, V 1 , V 2 ) | S 1 Ens 1 , S 2 Ens 2 , V 1 S 1  V 2 S 2  P(V 1 , V 2 )  ((DomTDomT 2  V=V 1  (DomT, V)Ens 1 )  (DomTDomT 1  V=V 2  (DomT, V)Ens 2 ))} EXEMPLE : Un utilisateur souhaite savoir quand le poids de Dupond Michel était inférieur à celui de Dulong Jeanne (on utilise uniquement les évolutions détaillées). Nous exprimons la requête en utilisant la jointure par union des domaines temporels. UJoin(h1 Flatten(Past( Select(p Patient, p.nom=«Dupond»  p.prénom=«Michel»))), h2 Flatten(Past( Select(p Patient, p.nom=«Dulong»  p.prénom=«Jeanne»))), h1.poids < h2.poids) Le résultat obtenu est constitué de domaines temporels formés par unions des domaines temporels initiaux. La partie structurelle est constituée du poids et d'un couple de valeurs pour la tension conformément à la définition du filtre temporel qui caractérise la structure des états passés (cf. section 3.2.3). Opérations de groupements temporels L'aspect temporel des objets entrepôt permet l'introduction de deux nouveaux opérateurs réalisant des regroupements en fonction de critères temporels : -un opérateur de groupement se base sur les unités temporelles en permettant de regrouper un ensemble d'états à une unité temporelle supérieure UGroup({S 1 , S 2 ,…, S n }, U)=Ens'. Ens' est un ensemble de n-uplets, dont la valeur est l'ensemble des valeurs structurelles des états Ens, regroupés à unité temporelle U. -un opérateur de groupement se base sur la durée en permettant de regrouper des états par durées successives depuis le temps d'origine de l'ensemble des états ; DGroup({S 1 ,…, S n }, D)=Ens'. Ens' est un ensemble de n-uplets, dont chaque valeur est l'ensemble des valeurs structurelles des états Ens, regroupés par période de temps (exprimée par une durée depuis le plus petit des instants des domaines temporels des états). Opérateurs de transformation et de traitements analytiques La prise de décision s'appuie très souvent sur les évolutions organisées chronologiquement [7] ; il s'agit de manipuler ces chronologies en appliquant systématiquement un traitement sur chaque valeur. Or, les opérateurs précédents ne permettent pas d'effectuer ce type de manipulation. Par exemple il est souvent utile d'appliquer une agrégation par accumulation sur des états (qui doivent être organisés chronologiquement) pour obtenir de nouveaux états dont la valeur est calculée en agrégeant les valeurs cumulées des états initiaux. De tels traitements n'ont de sens que sur une série d'états chronologiquement ordonnés. Opération de transformation Conclusion Les travaux de recherche exposés dans cet article se placent dans le contexte des systèmes décisionnels constitués d'entrepôts de données. Notre approche repose sur la dichotomie de deux espaces de stockage au sein du système décisionnel : -l'entrepôt de données centralise, stocke et historise l'information pertinente pour les décideurs ; -les magasins de données représentent un extrait de l'entrepôt sous une forme adaptée aux analyses et au support des processus décisionnels. Ainsi, l'entrepôt de données offre une vision informatique de l'information décisionnelle tandis que les magasins de données proposent une vision utilisateur. Cet article traite plus particulièrement de la vision informatique en proposant un modèle de représentation et une algèbre de manipulation des données de l'entrepôt. Dans un premier temps, nous avons défini un modèle d'entrepôt de données complexes et temporelles, basé sur le paradigme objet et reposant principalement sur trois concepts. -Le concept d'objet entrepôt modélise l'état courant d'une information extraite, ainsi que des états passés (représentant les évolutions de l'objet sous une forme détaillée) et des états archivés (correspondant aux évolutions de l'objet décrites sous une forme résumée). L'intérêt de cette modélisation est de conserver les données de l'entrepôt ainsi que leurs évolutions à un niveau de détail pertinent, limitant le stockage des évolutions. -Le concept de classe entrepôt intègre les caractéristiques de notre approche par une fonction de construction, un filtre temporel et un filtre d'archives. -Le concept d'environnement permet de définir simplement les parties temporelles homogènes (même période de rafraîchissement, même critère d'archivage,…). Dans un second temps, nous avons défini une algèbre de manipulation des données de l'entrepôt par extension des principales opérations proposées dans les langages de base de données objet. L'extension se situe essentiellement au niveau des opérateurs temporels et des opérateurs de manipulation des ensembles d'états présents dans les objets entrepôt. En outre, une contribution importante réside dans la proposition d'opérateurs spécifiques de restructuration en série temporelle ainsi que des opérateurs facilitant les traitements analytiques sur les données. L'intérêt majeur de notre algèbre est qu'elle facilite l'interrogation en offrant un cadre générique pour l'interrogation et la manipulation des données de l'entrepôt. Notre étude fait l'objet d'un développement au travers du prototype GEDOOH 2 [26], acronyme de Générateur d'Entrepôts de Données Orientées Objet et Historisées. Il comporte une interface (visualisant graphiquement la source globale et l'entrepôt de données) et un module générateur (permettant de créer automatiquement des entrepôts ainsi que les processus d'alimentation et de rafraîchissement). GEDOOH est opérationnel : il comprend 8000 lignes de code Java (jdk1.2). Les perspectives que nous envisageons de conduire sont les suivantes : -La première problématique que nous allons étudier concerne l'adaptation dynamique des systèmes décisionnels. Actuellement, les techniques utilisées se contentent de proposer une collection de schémas en étoile adaptés aux différents besoins décisionnels. Nous pensons que des profils utilisateurs pourraient être mis à profit pour améliorer l'adaptabilité des systèmes décisionnels en fonction des catégories de décideurs. -Une seconde problématique concerne la méthodologie de conception des systèmes décisionnels. A l'heure actuelle, aucune méthode de conception n'est disponible pour aider les administrateurs dans leur démarche de conception de ces systèmes. Nous pensons que cette méthode doit s'appuyer sur une étude préalable des méta-données dans les systèmes décisionnels. Ces dernières ont fait l'objet de travaux dans le cadre du projet européen DWQ. Cependant, ces recherches se sont focalisées sur la qualité des systèmes élaborés, sans tenir compte des aspects importants de la démarche et des contraintes de conception. Références Nom c , -d'un type Type c définissant la structure Structure c et le comportement Comportement c des objets entrepôt de c (à chaque classe entrepôt correspond un type), -d'un ensemble fini de super classes Super c (c i est une super classe de c, notée c≼c i si et seulement si, Type c Type ci et Extension c Extension ci ), -d'une extension Extension c = {o 1 , o 2 ,…, o x }, -d'une fonction de construction Mapping c qui permet de spécifier le processus d'extraction et de transformation mis en jeu pour créer la structure et le peuplement de la classe c à partir de la source globale, -d'un filtre temporel Tempo c définissant l'ensemble des EXEMPLE : La classe entrepôt PATIENT possède un filtre temporel et un filtre d'archivage afin de spécifier respectivement les propriétés temporelles et d'archives. Il reste cependant à spécifier le comportement temporel de la classe, et en particulier, il est nécessaire de définir un critère d'archivage qui caractérise les états passés qui doivent être archivés. Nous définissons l'environnement suivant : environment Evolution { PATIENT } Ensuite, nous spécifions une règle de configuration qui permet d'archiver tous les états passés plus ancien que juin 2000. rule critere_archive on Evolution when self.refresh() if select T from P in PATIENT, T in P.PastStates() where precedes(T.domT, Date('07-2000', 'mm-aaaa')) then T.archive() ; Nous proposons pour l'entrepôt de données un modèle de représentation orienté objet et intégrant la dimension temporelle de manière flexible au travers un mécanisme d'archivage automatique ; il s'agit d'une extension du modèle standard proposé par l'ODMG. L'originalité de notre modélisation repose sur les concepts d'objets entrepôt, de classe entrepôt et d'environnement.-L'objet entrepôt est une extension du concept d'objet avec l'intégration de valeurs courantes, mais également celle des valeurs passées sous une forme détaillée (états passés) ou résumée (états archivés). FIG. 4 - 4Opérateurs d'accès aux états. ... -La vision informatique, dédiée à l'entrepôt, permet une gestion et une historisation efficace des données utiles pour les décideurs. L'exploitation directe de l'information n'est réalisable que par des informaticiens. -La vision utilisateur, dédiée aux magasins, permet une représentation des données adaptée aux décideurs. Les analyses et les interrogations sont effectuées de manière indirecte au travers de cette vision multidimensionnelle des données. Il est important de remarquer que notre entrepôt de données n'est pas organisé de manière multidimensionnelle puisqu'il ne supporte pas directement les processus OLAP. Cette activité est réservée aux magasins de données qui améliorent les performances d'interrogation sans se soucier des redondances d'information ; chaque magasin stocke une partie de l'information disponible dans l'entrepôt afin de répondre à un objectif décisionnel précis ou à un groupe d'utilisateurs ayant les mêmes besoins.Dans cet article, nous nous focalisons sur l'entrepôt de données. La problématique abordée est double. -Nous souhaitons proposer un modèle de représentation de l'information intégrant des données complexes et évolutives. L'entrepôt doit stocker uniquement l'information utile pour les décideurs ; l'historisation des données doit être réalisée de manière flexible (sous forme détaillée ou résumée) afin de conserver les seules évolutions utiles. -Nous souhaitons également proposer un ensemble d'opérateurs de manipulation et d'interrogation des données entreposées. Ces opérateurs doivent permettre d'effectuer des manipulations sur les données courantes et passées. La section 2 présente les principaux travaux traitant de la modélisation des entrepôts complexes et temporels. La section 3 décrit le modèle de données que nous définissons pour les entrepôts. La section 4 définit l'algèbre de manipulation et d'interrogation de l'entrepôt. FIG. 1 -Principes d'exploitation du système décisionnel. Cependant, ces travaux ne sont pas suffisants et les aspects plus conceptuels restent peu étudiésI N T E G R A T I O N C O N S T R U C T I O N sou rces sou rce glob ale virtu elle en trep ôt d e d on n ées D écid eu rs S T R U C T U R A T I O N in form aticien s O L T P O L A P m agasin s d e d on n ées Interrogation indirecte Interrogation directe V IS IO N IN F O R M A T IQ U E V IS IO N U T IL IS A T E U R 2 Limites des approches actuelles Les travaux relatifs aux entrepôts de données abordent globalement deux problématiques. -La première traite essentiellement de l'organisation des données. Cette organisation dite multidimensionnelle [1, 13, 15, 19, 22] vise à supporter efficacement les analyses OLAP en offrant une vision des données adaptées et les temps de réponse sont accélérés en calculant de nombreux pré-agrégats. -La deuxième étudie principalement la sélection et la maintenance incrémentale des vues matérialisées [12, 27, 28, 30, 31, 32] afin de collecter les données utiles aux décideurs, de les stocker dans l'entrepôt et de les maintenir cohérentes avec les données source. Ces travaux se focalisent sur des aspects physiques ou logiques (vues, index,…). p2 ,…, S pn } est un ensemble fini contenant les états passés et EA = {S a1 , S a2 ,…, S am } est un ensemble fini contenant les états archivés. Un état S i d'un objet entrepôt est défini par le couple (v i , h i ) où v i est la valeur de l'objet pour les instants de h i et h i = <[td 1 , tf 1 [;…;[td h , tf h [> est le domaine temporel (ensemble ordonné d'intervalles disjoints deux à deux) définissant les instants durant lesquels la valeur de l'état S i est courante. La modélisation des domaines temporels s'effectue au travers d'un modèle temporel, linéaire, discret qui définit le temps par le biais d'unités temporelles ; l'espace continu du temps, représenté par une droite de réels, elle-même décomposée en une suite d'intervalles consécutifs disjoints [9]. Chaque partition correspond à une unité temporelle caractérisée par la taille des intervalles décomposant la droite du temps. Notre modèle gère un ensemble d'unités temporelles nommées (année, semestre, trimestre,…) muni d'une relation d'ordre partiel est-plus-fine permettant de comparer les unités. Nous définissons plusieurs types temporels de base : l'instant, l'intervalle ainsi que le domaine temporel. Ce dernier est un ensemble ordonné d'intervalles disjoints deux à deux et non contigus, noté h i = <[td 1 , tf 1 [; [td 2 , tf 2 [;…;[td h , tf h [> où chaque intervalle est non vide (k[1..h], td k tf k ) et possède une même unité temporelle (k[1..h],j[1..h], unit([td k ,tf k [)=unit([td j ,tf j [ où la fonction unit(Int) retourne l'unité temporelle de Int).EXEMPLE : Nous considérons deux objets entrepôt qui décrivent des patients admis dans un établissement thermal. Nous supposons que différents paramètres sont relevés pour chaque patient (poids, tension…) ; nous avons réduit le nombre de paramètres par rapport à la réalité (dans le projet REANIMATIC, environ 80 paramètres sont définis). Dans notre exemple d'illustration, la périodicité des relevés est fixée au mois (dans les services de réanimation du projet REANIMATIC, les paramètres sont relevés chaque 8h).FIG. 2 -Exemple d'objets entrepôt.C u rren t S tate : < [01-2001; n ow ]> nom : "D upond" prénom : "M ichel" poids : 75 tension : [m in : 8 ; m ax : 13] hém atocrite : 45 plaquettes : 220 urée : 10 P ast S tate : < [11-2000; 12-2000]> poids : 77 tension : [m in : 8 ; m ax : 15] P ast S tate : < [09-2000; 10-2000]; [07-2000; 07-2000]> poids : 80 tension : [m in : 10 ; m ax : 16] P ast S tate : < [08-2000; 08-2000]> poids : 79 tension : [m in : 10 ; m ax : 15] A rch ive S tate : < [01-2000; 06-2000]> poids : 82 nb_passé : 4 A rch ive S tate : < [07-1999; 12-1999]> poids : 90 nb_passé : 3 O ID 1 : P A T IE N T valeur courante valeurs passées (évolutions détaillées) valeurs archivées (évolutions résumées) C u rren t S tate : < [03-2001; n ow ]> nom : "D ulong" prénom : "Jeanne" poids : 65 tension : [m in : 6 ; m ax : 10] hém atocrite : 40 plaquettes : 180 urée : 8 P ast S tate : < [12-2000; 02-2001]> poids : 69 tension : [m in : 7 ; m ax : 10] P ast S tate : < [10-2000; 11-2000] > poids : 76 tension : [m in : 8 ; m ax : 14] P ast S tate : < [07-2000; 09-2000]> poids : 82 tension : [m in : 10 ; m ax : 16] A rch ive S tate : < [01-2000; 06-2000]> poids : 88 nb_passé : 3 A rch ive S tate : < [07-1999; 12-1999]> poids : 92 nb_passé : 3 O ID 2 : P A T IE N T Nous posons c i C ED , Mapping ci =f ci Par souci de simplification, nous ne détaillons pas les différentes fonctions d'extraction ; voir[26] pour une étude détaillée. En outre, nous avons étudié l'extraction du comportement des données dans[20].EXEMPLE :Nous considérons l'exemple précédent. Nous supposons que la source de données à partir de laquelle la classe entrepôt PATIENT a été définie, est composée des classes suivantes :attribute Integer plaquettes ; attribute Integer uree ; relationship Personnes patient inverse Personnes::parametres ; } A partir de cette source de données, décrite suivant l'ODMG, il est possible de définir la fonction de construction suivante : pp.nom, prénom : pp.prenoms[0], poids : pp.poids, tension : pp.tension, hématocrite : pp.hematocrite, plaquettes : pp.plaquettes, urée : pp.uree})La classe entrepôt produite est définie par :en offrant des mécanismes puissants pour combiner et transformer les classes afin de constituer des classes entrepôt adaptées aux besoins des décideurs ; -les fonctions de hiérarchisation (FH) organisent la hiérarchie d'héritage dans l'entrepôt en créant des super-classes et des sous-classes. 1 of ci 2 o…of ci m avec j[1,m], f ci j FS  f ci j FE  f ci j FP  f ci j FH. interface Personnes { attribute String nom ; attribute List<String> prenoms ; attribute Boolean sexe ; attribute Date naissance ; relationship Variables parametres inverse Variables::patient ; Integer age() ; } interface Variables { attribute Integer poids ; attribute Struct T_tension {Integer min, Integer max} tension ; attribute Integer hematocrite ; PROJECT(pp JOIN( p Personnes, v Variables, p.parametres=v ), {nom :interface PATIENT { attribute String nom ; attribute String prénom ; attribute Integer poids ; attribute Struct T_tension {Integer min, Integer max} tension ; attribute Integer hématocrite ; attribute Integer plaquettes ; attribute Integer urée ; } caractérise les propriétés temporelles d'une classe entrepôt. Il est constitué d'un ensemble de couples (p j , f j ) où -p j est une propriété temporelle et -f j est soit un attribut, soit une relation, soit une opération retournant un résultat (fonction). Les évolutions détaillées des propriétés temporelles sont conservées au travers d'états passés. S'il s'agit d'une opération, les évolutions de son résultat sont conservées à chaque point d'extraction (rafraîchissement de la classe). EXEMPLE : La définition de la classe entrepôt PATIENT est complétée de la manière suivante : interface PATIENT { attribute String nom ; attribute String prénom ; attribute Integer poids ; attribute Struct T_tension {Integer min, Integer max} tension ; attribute Integer hématocrite ; attribute Integer plaquettes ; attribute Integer urée ; } with temporal filter {(poids, poids), (tension, tension)} ; Suivant la définition, les évolutions détaillées du poids et de la tension des patients seront conservées. Le filtre d'archives Archi c ={(a 1 , f 1 ), (a 2 , f 2 ),…, (a s , f s )} caractérise les propriétés archivées de la classe entrepôt. Il est constitué d'un ensemble de couples (a j , f j ) où a j est un attribut et f j est une fonction d'agrégation. L'ensemble des attributs archivés est un sous-ensemble des attributs temporels. Chaque attribut archivé est associé à une fonction d'agrégation qui définit la manière dont sont résumées les valeurs temporelles. Les propriétés archivées sont associées à une fonction d'agrégation qui indique comment sont résumées les évolutions détaillées de la propriété temporelle correspondante. Notre modèle supporte plusieurs catégories de fonctions d'agrégation : -les fonctions d'agrégation forte (avg, sum, count, max, min) résument les états passés sélectionnés pour l'archivage dans un seul état archivé ; -les fonctions d'agrégation modérée (avg_t, sum_t, count_t, max_t, min_t) résument les états passés sélectionnés pour l'archivage avec plusieurs états archivés. Les états passés sélectionnés sont regroupés par grain de temps à une unité temporelle supérieure. FIG. 3 -Comparaison de l'archivage fort et modéré. Suivant la définition, les évolutions détaillées du poids seront archivées (résumées) par périodes de six mois. La mise en place d'un archivage, nécessite un seuil à partir duquel il est déclenché ( se reporter à la section 3.3).3.2.4 Mécanisme d'archivage état courant état passé état archivé L égen d e : juil. 1999 janv . 2000 juin. 2000 janv . 2001 A rch ivage fort : archive filter {poids:avg(poids)} juil. 1999 janv . 2000 juin. 2000 janv . 2001 A rch ivage m od éré : archive filter {poids:avg(poids)} by month(6) EXEMPLE : La définition de la classe entrepôt PATIENT est complétée de la manière suivante : interface PATIENT { attribute String nom ; attribute String prénom ; attribute Integer poids ; attribute Struct T_tension {Integer min, Integer max} tension ; attribute Integer hématocrite ; attribute Integer plaquettes ; attribute Integer urée ; } with temporal filter {(poids, poids), (tension, tension)}, archive filter {(poids, t_avg(poids))} by month(6) ; La section suivante se propose de fournir les moyens d'exploiter l'information de l'entrepôt de données.-Le concept d'objet entrepôt nécessite d'étendre les classes par le concept de classe entrepôt. En plus des caractéristiques standard définissant les classes, une classe entrepôt se caractérise par des filtres temporels et d'archives (pour définir les propriétés temporelles de la classe) et par une fonction de construction (pour définir le processus d'élaboration à partir de la source). -Le concept d'environnement décrit le regroupement de classes entrepôt ayant un même comportement temporel. L'atout des environnements est qu'ils permettent à l'administrateur de spécifier des parties temporelles homogènes et de taille adéquate aux exigences décisionnelles. 4 Algèbre pour la manipulation des données Cette section décrit l'algèbre associée au modèle de données préalablement défini. Cette algèbre s'inspire des principales algèbres temporelles objet [5, 25]. Nous reprenons les opérations algébriques des langages pour objets temporels que nous adaptons aux spécificités des objets entrepôt ; ces opérateurs doivent être étendus afin d'intégrer le concept d'état. Nous définissons également de nouveaux opérateurs, spécifiques à notre modélisation, pour manipuler les différentes catégories d'états qui composent les objets entrepôt. Enfin, nous proposons des mécanismes permettant de transformer les données afin d'offrir différentes perspectives sur les évolutions afin de faciliter les traitements analytiques. =<[TDeb i ;TFin i ]>, -les états sont ordonnés chronologiquement, i[1..s-1], domT i Precedes domT i+1 . Nous proposons un nouvel opérateur de transformation des objets entrepôt en série temporelle. Il s'agit d'un formatage des données sous la forme d'une série temporelle d'états. L'objectif principal de cette transformation est de rendre plus simple la compréhension de l'évolution dans le temps d'un objet entrepôt en permettant l'application de traitements analytiques. L'opérateur MakeSerie transforme un ensemble d'états en une série temporelle d'états. Il se définit de la manière suivante : -MakeSerie(Ens)=SR où Ens={S 1 , S 2 , S n } est un ensemble d'états et SR=<S s1 , S s2 ,…, S ss > est une série temporelle d'états. EXEMPLE : Un utilisateur veut obtenir la moyenne glissante (avec un décalage de deux mois) du poids de Dupond Michel. Il ne considère que les états passés. Le résultat obtenu est le suivant : Un traitement couramment utilisé lors des analyses est celui du changement de l'échelle d'observation, c'est-à-dire celui du changement de la granularité des domaines temporels des états. Cette transformation peut s'opérer dans deux sens. -Soit la granularité est augmentée, c'est-à-dire que l'on augmente l'unité temporelle des domaines temporels (cela revient à réduire le détail d'observation des évolutions). On parle de transformation "scale-up" ; ScaleUp(SR, U + , Archi)=SR'. -Soit la granularité est diminuée, c'est-à-dire que l'on diminue l'unité temporelle des domaines temporels (cela revient à augmenter le détail d'observation des évolutions). On parle de transformation "scale-down" ; ScaleDown(SR, U -, Archi)=SR'. EXEMPLE : Un utilisateur veut obtenir la moyenne du poids de Dupond Michel avec une échelle temporelle d'observation augmentée du mois au trimestre. Le langage de manipulation de l'entrepôt est une extension des algèbres objet prenant en compte les caractéristiques du modèle de représentation de l'entrepôt. L'extension se situe principalement au niveau des opérateurs temporels et des opérateurs de manipulation des ensembles d'états présents dans les objets entrepôt. Une contribution importante réside dans la proposition d'opérateurs spécifiques de restructuration en série temporelle ainsi que des opérateurs facilitant les traitements analytiques sur les données. L'intérêt de cette algèbre réside essentiellement dans la base formelle qu'elle offre. Celle-ci peut servir pour le développement futur d'interfaces et de langages graphiques performant permettant la manipulation directe de l'entrepôt dans sa globalité. TAB. 3 -Synthèse des opérateurs proposés. Opérations ensemblistes basées sur l'égalité de valeur Ex Opérations ensemblistes basées sur l'égalité d'identifiant Ex Les opérateurs proposés sont soit spécifique (Sp) à notre algèbre, soit correspondent à une extension (Ex) d'une opération issue d'algèbres temporelles objet.Nous définissons donc une série temporelle d'états à partir d'un ensemble d'états dont les domaines temporels sont réorganisés sous la forme d'un intervalle. L'ensemble des états est ordonné chronologiquement. Une série temporelle d'états est un ensemble ordonné d'états <S 1 , S 2 ,…, S s > tel que -les domaines temporels des états sont des intervalles, i[1..s], domT i EXEMPLE : Nous considérons l'objet entrepôt OID1. Les états passés de cet objet possèdent des domaines temporels complexes (ils ne peuvent être ordonnés chronologiquement pour simplifier l'analyse). L'utilisateur peut transformer les états passés en une série temporelle d'état ordonnés chronologiquement ; on notera dans la suite SR cette série temporelle. MakeSerie( Project( pp Flatten( Past( Select(p Patient, p.nom=«Dupond»  p.prénom=«Michel») ) ), {pp.poids, pp.domT}) ) <[poids=80 ; domT=<[07-2000;07-2000]>] ; [poids=79 ; domT=<[08-2000;08-2000]>] ; [poids=80 ; domT=<[09-2000;10-2000]>] ; [poids=77 ; domT=<[11-2000;12-2000]>]> 4.4.2 Opérations d'agrégation Nous adoptons différentes transformations pour les séries temporelles d'états : -L'opérateur d'agrégation Agreg transforme une série temporelle d'états en une valeur ; Agreg(SR, Archi)=V où Archi={(att 1 , f 1 ), (att 2 , f 2 ),…, (att t , f t )} est un filtre d'archivage, c'est-à-dire un ensemble d'attributs (des états de SR) associés à une fonction d'agrégation et V est la valeur résultat de l'agrégation des valeurs structurelles des états de Ens. EXEMPLE : Un utilisateur désire obtenir la moyenne du poids de Dupond Michel. Il ne considère que les états passés (la période considérées concerne donc le second semestre 2000). Il exprime la requête suivante : Agreg(SR, {(poids, avg(poids))}) Le résultat obtenu est le suivant : [poids=79] -L'opérateur d'agrégation cumulative consiste à cumuler les résultats d'une agrégation appliquée aux valeurs successives. L'opérateur d'agrégation cumulative ACum transforme une série temporelle d'états en une autre série d'états dont les valeurs sont le résultat d'une agrégation cumulée ; ACum(SR, Archi)=SR' où Archi={(att 1 , f 1 ), (att 2 , f 2 ),…, (att t , f t )} est un filtre d'archivage, c'est-à-dire un ensemble d'attributs (des états de SR) associés à une fonction d'agrégation et SR'=<S a1 , S a2 ,…, S aa > est une série temporelle d'états dont les valeurs structurelles sont le résultat d'une agrégation cumulée. EXEMPLE : Un utilisateur veut obtenir la moyenne cumulée du poids de Dupond Michel. Il ne considère que les états passés qui correspondent au second semestre 2000. ACum(SR, {(poids, avg(poids))}) Le résultat obtenu est le suivant : <[poids=80 ; domT=<[07-2000;07-2000]>] ; [poids=79,5 ; domT=<[07-2000;08-2000]>] ; [poids=79,6 ; domT=<[07-2000;09-2000]>] ; [poids=79,6 ; domT=<[07-2000;10-2000]>] ; [poids=79 ; domT=<[07-2000;11-2000]>] ; [poids=79 ; domT=<[07-2000;12-2000]>]> -L'opérateur d'agrégation dynamique permet d'appliquer des agrégations à des ensembles d'états sur une période de temps "qui se déplace". Les opérateurs d'agrégation cumulative dynamique AMove transforment un ensemble d'états en un autre ensemble d'états dont les valeurs sont le résultat d'une agrégation cumulée par glissements ; AMove(SR, Archi, D)=SR' où Archi={(att 1 , f 1 ), (att 2 , f 2 ),…, (att t , f t )} est un filtre d'archivage, c'est-à-dire un ensemble d'attributs (des états de SR) associés à une fonction d'agrégation, D est une durée de même unité temporelle que les états de Ens et SR=<S a1 , S a2 ,…, S aa > est une série temporel d'états dont les valeurs structurelles sont le résultat d'une agrégation cumulée. AMove(SR, {(poids, avg(poids))}, Duration(2, month)) <[poids=79,5 ;domT=<[07-2000;08-2000]>] ; [poids=80 ;domT=<[09-2000;10-2000]>] ; [poids=77 ;domT=<[11-2000;12-2000]>]> 4.4.3 Opérations de changement d'échelle temporelle ScaleUp(SR, 'trimestre', {(poids, avg(poids))}) Le résultat obtenu est le suivant : <[poids=79,6 ;domT=<[07-2000;09-2000]>] ; [poids=78,5 ;domT=<[10-2000;12-2000]>] > 4.5 Résumé OPERATEURS DESCRIPTIONS ORIGINE (*) VUnion, VIntersect, VDifference IUnion, IIntersect, IDifference Flatten Opération de déstructuration d'un ensemble d'ensembles Ex DupElim Opération de suppression des doubles Ex EmptyElim Opération de suppression des ensembles vides Ex Select Opération de sélection Ex Project Opération de projection Ex Join Opération de jointure Ex Nest, UnNest Opérations de groupement et dégroupement Ex Current Opération d'accès aux états courants Sp Past Opération d'accès aux états passés Sp Archive Opération d'accès aux états archivés Sp State Opération d'accès aux valeurs selon une fenêtre temporelle Sp UJoin Opération de jointure temporelle par union des domaines temporels Ex IJoin Opération de jointure temporelle par intersection des domaines temporels Ex UGroup Opération de groupement temporel à une unité temporelle de granularité supérieure Ex DGroup Opération de groupement temporel suivant une durée Ex MakeSerie Opération de restructuration en série temporelle Sp Agreg Opération d'agrégation des séries temporelles Sp ACum Opération d'agrégation cumulée des séries temporelles Sp AMove Opération d'agrégation glissante des séries temporelles Sp ScaleUp Opération de changement d'échelle à une unité temporelle de granularité supérieure Sp ScaleDown Opération de changement d'échelle à une unité temporelle de granularité inférieure Sp (*) http://www.irit.fr/SSI/ACTIVITES/EQ_SIG/gedooh.html R Agrawal, A Gupta, A Sarawagi, Modeling Multidimensional Databases, ICDE'97. 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Kansas City (Kansas, USARavat F., Teste O., Zurfluh G., Towards the Data Warehouse Design, ACM CIKM'99, Kansas City (Kansas, USA), Nov 1999. Database Architecture for Data Warehousing: An evolutionary Approach, DEXA'98. J Samos, F Saltor, J Sistrac, A Bardés, Vienna (AustriaSamos J., Saltor F., Sistrac J., Bardés A., Database Architecture for Data Warehousing: An evolutionary Approach, DEXA'98, Vienna (Austria), 1998. A Query Algebra for Object-Oriented Databases, ICDE'90. G M Shaw, S B Zdonik, Shaw G.M., Zdonik S.B., A Query Algebra for Object-Oriented Databases, ICDE'90, pp. 154-162, 1990. Modélisation et manipulation d'entrepôts de données complexes et historisées, Thèse de l'Université Paul Sabatier. O Teste, Toulouse (FranceTeste O., Modélisation et manipulation d'entrepôts de données complexes et historisées, Thèse de l'Université Paul Sabatier, 18 Décembre 2000, Toulouse (France). Incremental Design of a Data Warehouse. D Theodoratos, T Sellis, Journal of Intelligent Information Systems. 151Kluwer Academic PublishersTheodoratos D., Sellis T., Incremental Design of a Data Warehouse, Journal of Intelligent Information Systems, Kluwer Academic Publishers, 15(1) :pp. 7- 27, 2000. Research problems in data warehousing. J Widom, ACM CIKM'95. Widom J., Research problems in data warehousing, ACM CIKM'95, 1995. The Starburst Active Database Rule System. J Widom, Transaction on Knowledge and Data Engineering. 84Widom J., The Starburst Active Database Rule System, Transaction on Knowledge and Data Engineering, 8(4):pp. 583-595 (1996). Maintaining Temporal Views over Non-Temporal Information Sources For Data Warehousing. J Yang, J Widom, 98Valencia (SpainYang J., Widom J., Maintaining Temporal Views over Non-Temporal Information Sources For Data Warehousing, EDBT'98, Valencia (Spain), 1998. Temporal View Self-Maintenance in a Warehousing Environment. J Yang, J Widom, EDBT'00Konstanz (GermanyYang J., Widom J., Temporal View Self-Maintenance in a Warehousing Environment, EDBT'00, Konstanz (Germany), March 2000. Algorithms for materialized view design in data warehousing environment. J Yang, K Karlapalem, Q Li, 97Yang J., Karlapalem K., Li Q., Algorithms for materialized view design in data warehousing environment, VLDB'97, 136-145.

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{'abstract': "Cet article aborde la modélisation et la manipulation des entrepôts objet intégrant des données historisées et archivées. Dans un premier temps, nous proposons un modèle décrivant l'entrepôt comme un référentiel centralisé de données complexes et temporelles. Notre modèle intègre les concepts d'objet entrepôt et d'environnement. Un tel objet est composé d'un état courant, de plusieurs états passés (modélisant les évolutions détaillées) et de plusieurs états archivés (modélisant les évolutions de manière résumée). Le concept d'environnement définit les parties temporelles dans le schéma de l'entrepôt avec une granularité pertinente (attribut, classe, graphe). Dans un second temps, nous définissons une algèbre de manipulation des données de l'entrepôt. Cette algèbre basée sur une extension des algèbres objet propose des opérateurs temporels et des opérateurs de manipulation d'ensembles d'états des objets entrepôt. Une contribution importante réside dans la proposition d'opérateurs spécifiques de restructuration en série temporelle et des opérateurs facilitant les traitements analytiques.AbstractThis paper deals with temporal and archive object-oriented data warehouse modelling and querying. In a first step, we define a data model describing warehouses as central repositories of complex and temporal data extracted from one information source. The model is based on the concepts of warehouse object and environment. A warehouse object is composed of one current state, several past states (modelling value changes) and several archive states (summarising some value changes). An environment defines temporal parts in a warehouse schema according to a relevant granularity (attribute, class or graph). In a second step, we provide a query algebra dedicated to data warehouses. This algebra, which is based on common object algebras, integrates temporal operators and operators for querying object states. An other important contribution concerns dedicated operators allowing users to transform warehouse objects in temporal series as well as operators facilitating analytical treatments.", 'arxivid': '1005.0219', 'author': ['Franck Ravat ', 'Olivier Teste teste@irit.fr '], 'authoraffiliation': [], 'corpusid': 2734452, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18714, 'n_tokens_neox': 17712, 'n_words': 7672, 'pdfsha': '5918bf442832b50545cf2dc8d2bd65a3f881bca7', 'pdfurls': ['https://arxiv.org/pdf/1005.0219v1.pdf'], 'title': ['Modélisation et manipulation de données historisées et archivées dans un entrepôt orienté objet Modelling and querying temporal and archive data in an object-oriented warehouse', 'Modélisation et manipulation de données historisées et archivées dans un entrepôt orienté objet Modelling and querying temporal and archive data in an object-oriented warehouse'], 'venue': []}

arxiv

SRTR: SELF-REASONING TRANSFORMER WITH VISUAL-LINGUISTIC KNOWLEDGE FOR SCENE GRAPH GENERATION Yuxiang Zhang School of Information and Electronics Beijing Institute of Technology Zhenbo Liu Noah's Ark Lab Huawei Shuai Wang Department of Chemistry The University of Hong Kong SRTR: SELF-REASONING TRANSFORMER WITH VISUAL-LINGUISTIC KNOWLEDGE FOR SCENE GRAPH GENERATION Index Terms-Scene Graph GenerationOne- corresponding author: zyx829625@163com StageSelf-reasoningNatural Language Supervi- sionContrastive Learning Objects in a scene are not always related. The execution efficiency of the one-stage scene graph generation approaches are quite high, which infer the effective relation between entity pairs using sparse proposal sets and a few queries. However, they only focus on the relation between subject and object in triplet set <subject entity, predicate entity, object entity>, ignoring the relation between subject and predicate or predicate and object, and the model lacks self-reasoning ability. In addition, linguistic modality has been neglected in the one-stage method. It is necessary to mine linguistic modality knowledge to improve model reasoning ability. To address the above-mentioned shortcomings, a Self-reasoning Transformer with Visual-linguistic Knowledge (SrTR) is proposed to add flexible self-reasoning ability to the model. An encoder-decoder architecture is adopted in SrTR, and a self-reasoning decoder is developed to complete three inferences of the triplet set, s+o→p, s+p→o and p+o→s. Inspired by the largescale pre-training image-text foundation models, visual-linguistic prior knowledge is introduced and a visual-linguistic alignment strategy is designed to project visual representations into semantic spaces with prior knowledge to aid relational reasoning. Experiments on the Visual Genome dataset demonstrate the superiority and fast inference ability of the proposed method. I. INTRODUCTION Scene graph generation (SGG) is an advanced semantic understanding task based on object detection, which aims to provide a graphical representation of objects and their relationships in images. Vision and natural language are connected by a scene graph, which provides a local semantic representation of the visual scene and is considered useful in many visual tasks, such as visual question answering [1][2][3], image captioning [4,5], image generation [6,7], and image retrieval [8]. The scene graph is composed of a triplet set <subject entity, predicate, object entity>, in which the entity is the node and the predicate is the edge. The graph structure is used to model the scene and mine the possible relationships between all entities in the scene. The development of SGG has been divided into two stages and one stage from the method level. In the two-stage method, they use existing object detectors (e.g. Faster-RCNN [9]) to detect entity proposals, and then input the relational inference module to predict the predicate classes of these entity combinations. Although this strategy achieves high recalls in SGG tasks, their computational cost is quite high, because there is too much predicate proposals for the combination of all objects in the scene. Recently, one-stage object detectors have been widely concerned (e.g. Detection Transformer, DETR [10]). They regard object detection as an end-to-end set prediction task and propose set-based loss through bipartite matching. This prompted the development of the one-stage SGG. After obtaining the entity queries through DETR, a fixed number of relation queries are set and sent to the relation transformer for relational reasoning. In this strategy, there is no need to pair the objects to predict all relationships, which has great development potential due to its fast inference speed and low computational cost. Most one-stage techniques use custom relation queries and entity queries extracted by Convolutional Neural Networks (CNN) backbone to learn relations via relation transformer, which has limited model self-reasoning ability and flexibility. Self-reasoning refers to the ability to automatically infer the third element from two of the known elements in the triple, such as the model can autonomously infer the corresponding possible object when it knows subject and predicate. Furthermore, linguistic modality is rarely taken into account in the one-stage approach. Text has been shown in multimodal learning to be helpful for visual representation learning, and linguistic modality can well depict the relationship between objects. As a result, it is critical to understand how to leverage prior information in linguistic modality to guide relation transformer in learning the relationship between objects. In order to address the aforementioned problems, a straightforward multi-modal self-reasoning SGG framework, called Self-reasoning Transformer with Visual-linguistic Knowledge (SrTR), is proposed. SrTR is made up of three major modules: entity decoder, self-reasoning decoder, and visual-linguistic alignment. First, we introduce CNN and Deformable DETR for extracting the encoded multi-scale visual feature contexts. To obtain the entity representation, the Deformable DETR decoder is adopted as entity decoder and utilized to interactively encode entity queries and visual features. In addition, to implement the self-reasoning training of the model, we construct a self-reasoning decoder. <subject entity, predicate, object entity> is defined in the form of triplet query, and three inferences of the triplet set, s+o→p, s+p→o and p+o→s, are completed in the self-reasoning decoder. Finally, the linguistic features of the triplet set are generated using the large-scale trained CLIP [11], and the visual-linguistic alignment is designed to introduce visual-linguistic prior knowledge. Using supervised contrastive learning, the visual representation of the triplet set output by the self-reasoning encoder is projected into a prior semantic space, which strengthens the representation learning and assists the relational reasoning. The main contributions of this work are summarized as follows. • A relational self-reasoning module is proposed to learn the potential relationship among subject, object and predicate in the training phase. • Integrating multi-scale features and bounding box into triplet query for embedding visual and location information needed in the selfreasoning process. • Visual-linguistic prior information is introduced into the one-stage SGG framework, and visual-linguistic alignment is designed to the subject, object and predicate to aid the self-reasoning of model. The rest of the paper is organized as follows. Section II introduces relevant concepts of SGG. Section III elaborates on the proposed SrTR. The extensive experiments and analyses are presented in Section IV. Finally, conclusions are drawn in Section V. II. RELATED WORK II-A. Two-stage Scene Graph Generation Existing SGG works primarily concentrate on improving the context modeling structure [12][13][14][15][16] or solving the class imbalance problem [17][18][19][20][21] (i.e., long-tail distribution). Wang et al. [15] addressed the issue of annotation bias and sparse annotation in visual genome (VG) and suggested a new SGG training scheme with two relation classifiers, one of which offers less biased settings for the other. Based on the graph neural networkbased message passing, Lin et al. [13] proposed a Regularized Unrolling Network ( RU-Net ) to solve the problem that it is very sensitive to the correlation between spurious nodes. To lessen noise in context modeling, Li et al. [19] employed a relationship prediction confidencebased adaptive message transmission technique. Tang et al. [20] suggested an unbiased approach that eliminates the vision-agnostic bias with counterfactual causality. The above complex two-stage methods usually need to predict the relationship between densely connected entity pairs, resulting in high computational cost and poor representation learning flexibility, and also limits end-to-end optimization. II-B. One-stage Scene Graph Generation One-stage target detection due to its excellent performance, recent researches have begun to explore the one-stage SGG framework. These firstlevel methods use a fully convolutional network [22,23] or Transformer [24,25] architecture to detect relationships directly from visual features without pre-training the target detection, and are simple, fast, and easy to train. Liu et al. [22] proposed a fully convolutional scene graph generation (FCSGG), which is the first fully convolutional-based one-stage SGG framework. The object is encoded as the center point of the bounding box, and the relationship is encoded as a two-dimensional vector field. Cong et al. [24] presented an end-to-end SGG model Relation Transformer (RelTR) inspired by the advantages of DETR in object detection. The approach is made up of the encoder-decoder structure. The encoder infers the visual feature context, whereas the decoder infers a set of fixed-size subjectpredicate-object triples using various types of attention mechanisms with coupled subject and object queries. III. PROPOSED SELF-REASONING TRANSFORMER The proposed SrTR is broken into three parts: an entity decoder, a self-reasoning decoder, and a visual-linguistic alignment. The flowchart for each part is illustrated in Fig. 1. (1) Backbone and Entity Decoder: Given an image, a multi-scale visual feature context M e is encoded by a CNN and Deformable DETR encoder, which is then delivered to the Deformable DETR decoder, which interacts with entity query Q e to produce the entity representation H e and its matching bounding box B e . (2) Self-reasoning Decoder: In order to obtain multi-scale entity visual features Fea e , we map the bounding box B e back to multi-scale space, and initialize triplet query Q t through crossattention with entity visual features Fea e to embed visual information and location information. And then, triplet query Q t is split into subject query Q s , predicate query Q p and object query Q o , and input into a self-reasoning decoder for two-byone self-reasoning training. (3) Visual-linguistic Alignment: The class names of the subject and object, and the triplet they make with the predicate, construct a semantic space with visual-linguistic priors using a CLIP encoder. A visual-linguistic alignment strategy is designed to map the triplet representation to the semantic space to achieve semantic priori auxiliary relational reasoning. III-A. Backbone and Entity Decoder ResNet50 is adopted as CNN backbone to provide deep visual features for subsequent models, and multi-scale visual feature contexts M e is generated by Deformable DETR encoder. The decoder of Deformable DETR is used as the entity decoder to interactively decode with the learnable entity query Q e . The entity decoder is defined as a mapping function F df dec . The initial entity query Q e and the visual feature context M e are input to obtain the entity representation H e and its associated prediction bounding box B e , H e , B e = F df dec (M e , Q e )(1) The bounding box B e is mapped back to multiscale space to obtain the multi-scale entity visual features Fea e , Fea e = M ap (M e , B e )(2) III-B. Self-reasoning Decoder Similar to the entity query in entity decoder, using a fixed set of learnable query is a simple strategy for initializing predicate proposals. However, this triplet query ignores entity candidate visual information and location information. The resulting triplet representation has poor ability to capture structured information and multi-diversity relation reasoning. In order to solve this problem, the multi-scale entity visual features Fea e and bounding box B e are integrated into the process of initializing triplet query. Specifically, the com-bination of Fea e and B e builds keys and values, K init t = V init t = Fea e + Relu (F C (B e )) (3) where F C denotes the fully connected layer. Fig. 2. The first step of the self-reasoning decoder is to infer the predicate representation H p , i.e., s+o→p. To incorporate entity representation selfreasoning, we replace subject and object representation with entity representation H e , and enter H e into multi-layer multi-head self-attention via the skip-connected feed-forward network, F (Q, K, V) = F F N (M HA (Q, K, V)), Q l p = F p SrD Q l−1 p + E p , H e + E e , H e + E e (5) where the value of l denote as the number of F p SrD layers, l = 1, 2, 3. The predicate representation H p = Q 3 p . H p and Q s are then concatenated and passed through the FC to form the key of s+p, K sp = F C (cat (Q s , H p )). After that, K sp is sent to F o SrD to reason object representation H o , i.e., s+p→o, Q l o = F o SrD Q l−1 o + E o , K sp + E s + E p , K sp + E s + E p (6) The object representation H o = Q 3 o . Similarly, H o and Q p are then concatenated and form the key of p+o via FC, K po = F C (cat (Q p , H o )). The subject representation H s obtained by F s SrD , Q l s = F s SrD Q l−1 s + E s , K po + E p + E o , K po + E p + E o(7) III-C. Visual-linguistic Alignment Most of one-stage methods only focus on visual modality and ignore the importance of linguistic modality, because the triplet set <subject entity, predicate, object entity> can be well represented by linguistic features. We introduce a text encoder pre-trained by CLIP to obtain the linguistic features of triplet set, which correspond well to visual features and have visual-linguistic prior knowledge. In the form of "A photo of {}", we select triplet set from the label of each picture and send it to the text encoder to obtain linguistic features T = {T s , T p , T o }. As shown in Fig. 1, subject corresponds to "A photo of person", object corresponds to "A photo of motorcycle', predicate corresponds to "A photo of person riding motorcycle'. To achieve the alignment of visual features and linguistic features of triplet set by class, supervised contrastive learning is performed. Firstly, a supervised contrastive learning is defined as, L supcon = − N i=0 1 |K(i)| k∈K(i) log exp x T i x + k τ a∈A(i) exp x T i x − a τ (8) where for each embedding feature x i in minibatch, K(i) and A(i) are the positive and negative sample sets, |K(i)| is the number of positive samples, x + k and x − a are one of the positive and negative samples. The predicted triplet is matched to the appropriate ground truth using the Hungarian matching algorithm [26], and the visual features in the prediction and the linguistic features in the ground truth are chosen from t = {t s , t p , t o } and t = {t s ,t p ,t o } based on the matching index. The alignment losses L vla of visual to linguistic and linguistic to visual are calculated, L vla = − N i=0 1 |K(i)| k∈K l (i) log exp(t T t + k /τ) a∈A l (i) exp(t T t − a /τ) + k∈Kv(i) log exp(t Tt+ k /τ) a∈Av (i) exp(t Tt − a /τ) (9) where K v (i) and A v (i) are the positive and negative sample sets of visual feature, K l (i) and A l (i) are the positive and negative sample sets of linguistic feature, |K v (i)| = |K l (i)| = |K(i)|. In addition, the temperature parameter τ , which governs the range of the logits in the softmax, is explicitly optimized as a log-parameterized multiplicative scalar during training. When t is equal to t p , t = cat t p , t proto p ,t = cat t p ,t p , where t proto p is the prototype of this predicate and reflects the mean of all the features of the predicate of the corresponding index. IV. EXPERIMENTAL RESULTS AND DISCUSSION Experiments using the Visual Genome dataset are conducted to validate the proposed SrTR. For comparison algorithms, several state-of-theart SGG algorithms are used, including two-stage algorithms, RelDN [14], VCTree-TDE [20], G-RCNN [16], Motifs [27], GPS-Net [28], KERN [29], BGNN [19], IMP [12], CISC [30], onestage algorithms, FCSGG [22], RelTR [24]. The R@K and mR@K are employed to evaluate performance. IV-A. Experimental Setting Visual Genome contains a total of 108k images with 150 entity classes and 50 predicate classes. 70% of the images are used as training set, and the remaining 30% are used as test set. 5k images are extracted from the training set for validation. There are three evaluation settings: (1) Predicate classification (PredCLS/PredDET): Given the bounding box and object label, the model predicts possible predicates between objects; (2) Scene graph classification (SGCLS): Given the bounding box, the model predicts the object label and the predicate relation between objects; (3) SGDET: Bounding box, object label, and predicate between objects are directly predicted by the model. SrTR is implemented on the Pytorch platform. All the experiments are conducted on 8 Nvidia GTX 3090 GPU. The initial learning rates of ResNet50 backbone and Transformer are set to 10-5 and 10-4, respectively. The number of encoder and decoder layers in SrTR is set to 3, and multihead attention modules with 8 heads are used. The number of entity query and triplet query are set to 100 and 200. IV-B. Ablation Study The self-reasoning decoder (SrD) is the key components of SrTR, and visual-linguistic alignment (VLA) is the main strategy for introducing visual-linguistic knowledge. Ablation analyses are carried out by eliminating each component from the total framework in order to evaluate the contribution of important SrTR components. It is clear from Table I that both SrD and VLA improve the performance of baseline when using one of them, and SrD is more promising, with an increase of about 1.2 in mR@. When SrD and VLA are simultaneously added to SrTR, the mR@ increased by about 1.8. IV-C. Performance on Visual Genome The results for R@K and mR@K are given in Table II in the two-stage and one-stage methods, the best recall is represented in blue bold for the two-stage method and in red bold for the onestage method. The mR@K metric of the proposed SrTR in three evaluation settings is higher than that of the one-stage model FCSGG and RelTR, and the number of parameters is also the lowest. On PredCLS, SrTR reached mR@20=21.1, mR@50=22.3, 1.6 higher than RelTR. Since the performance of the entity detector in SrTR is only 23.1, its advantages in SGCLS and SGDET are not obvious, but there is a slight improvement on mR@20/mR@50. Compared with the onestage model, the two-stage model has better performance. The entity detection accuracy of BGNN is as high as 29.0, which is about 2 higher than SrTR on SGDET. However, they have many parameters, and even the IMP with Table II. Comparison with state-of-the-art scene graph generation methods on Visual Genome test set. These methods are divide into two-stage and one-stage. The best numbers in two-stage methods are shown in blue bold, and the best numbers in one-stage methods are shown in red bold. Method AP50 PredCLS SGCLS SGDET #params(M) R@20 R@50 mR@20 mR@50 R@20 R@50 mR@20 mR@50 R@20 R@50 mR@20 mR@50 two stage MOTIFS 20.0 58.5 65. 2 the lowest parameters is 3.8 times that of SrTR, which is not conducive to lightweight deployment in practical applications. In general, SrTR has strong competitiveness and can better capture the relationship between entities when there are entity classes and bounding boxes and only the relationship between entities is considered. Qualitative results for scene graph generation of Visual Genome dataset are illustrated in Fig. 3, where the blue box is the subject box and the orange box is the object box. We show 6 relationships with the highest confidence scores and the generated scene graphs. There is a significant deviation in the training data set in Visual Genome. As the most frequent predicate, "on" is the most easily predicted relationship, and the probability of "of" being mispredicted as "on" is the highest. As shown in Fig. 3, in most cases, "eye of man" is more appropriate than "eye on man". This indicates that SrTR predicts predicates that are more consistent with the subject and object semantic relationships. In addition, Qualitative results of examples of rational relational inference are shown in Fig. 4, which are selected from the top 10 relationships with the highest confidence scores. Among them, sitting on, above, near and walking on are low-frequency predicates in the training set, which can be inferred in SrTR with higher confidence scores. V. CONCLUSIONS From the perspective of self-reasoning, a Selfreasoning Transformer with Visual-linguistic Knowledge (SrTR) is proposed, which enhances the model to relation reasoning capacity. A selfreasoning decoder is designed to realize the reasoning between elements in <subject entity, predicate, object entity >. Furthermore, considering the importance of linguistic modality in representing the relationship between entities, we introduce the visual-linguistic prior information of the large-scale pre-training image-text foundation model CLIP, and design visual-linguistic alignment as triplet representation to embed prior knowledge in the training process. Experiments on the Visual Genome dataset show that SrTR has competitive performance compared to other state-of-the-art one-stage methods. Fig. 1 . 1Flowchart of the proposed SrTR. One image is encoded by CNN, Deformable DETR encoder and decoder to obtain multi-scale visual features and entity representation. The predicted bounding box and multi-scale visual features complete the initialization of the triplet queries. The decoupled triplet queries are input into the self-reasoning decoder for self-reasoning training. The linguistic features of the triplet set are obtained by using the CLIP encoder, and visual-linguistic priors are introduced to make visual-linguistic alignment with the visual embedding features obtained by the self-reasoning encoder. Fig. 2 . 2The flowchart of self-reasoning decoder, where s+o→p, s+p→o and p+o→s executed sequentially. The self-reasoning decoder uses {H e , Q 0 s , Q 0 p , Q 0 o } and the position encoding {E e , E s , E p , E o }, where E e is a fixed sine position encoding in Deformable DETR and E s , E p , E o are the learnable parametric position encoding. A schematic of the self-reasoning decoder is shown in The subject representation H s = Q 3 s . The predicted classes and bounding boxes are obtained by entering H s , H p and H o into the embedding layer. In addition, a visual embedding layer is set up to transform them into 512-dimension to complete the visual-linguistic alignment strategy, i.e.,T = {T s ,T p ,T o }. The embedding layers used here are all fully connected layers. Fig. 3 . 3Qualitative results for scene graph generation of Visual Genome dataset. Fig. 4 . 4Qualitative results of examples of rational relational inference. The blue box is the subject box and the orange box is the object box. Table I . ISelf-reasoning Decoder (SrD) and the Visual-linguistic Alignment (VLA) are isolated separately from the framework.Ablation Setting SGDET SrD VLA R@20 R@50 mR@20 mR@50 × × 18.7 21.9 4.5 6.3 √ × 20.1 23.8 5.6 7.7 × √ 19.5 22.3 5.0 6.9 √ √ 20.5 24.7 6.1 8.4 Q t = M HA Q init t , K init t , V init t(4)Here, the triplet query Q t has vision and location awareness, and subject query, predicate query and Inferring and executing programs for visual reasoning. J Johnson, B Hariharan, L Van Der Maaten, J Hoffman, L Fei-Fei, C L Zitnick, R Girshick, J. Johnson, B. Hariharan, L. van der Maaten, J. Hoffman, L. Fei-Fei, C. L. Zitnick, and R. 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Zagoruyko, G. Synnaeve, F. Massa, N. Carion, F. Massa, G. Synnaeve, N. Usunier, A. Kirillov, and S. Zagoruyko, "End-to-end object detection with transformers," european conference on computer vision, 2020. Learning transferable visual models from natural language supervision. A Radford, J W Kim, C Hallacy, A Ramesh, G Goh, S Agarwal, G Sastry, A Askell, P Mishkin, J Clark, G Krueger, I Sutskever, international conference on machine learningA. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, G. Krueger, and I. Sutskever, "Learning transferable vi- sual models from natural language supervi- sion," international conference on machine learning, 2021. Scene graph generation by iterative message passing," computer vision and pattern recognition. D Xu, Y Zhu, C Choy, L Fei-Fei, D. Xu, Y. Zhu, C. Choy, and L. Fei-Fei, "Scene graph generation by iterative message passing," computer vision and pattern recog- nition, 2017. Ru-net: Regularized unrolling network for scene graph generation. X Lin, C Ding, J Zhang, Y Zhan, D Tao, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern RecognitionX. Lin, C. Ding, J. Zhang, Y. Zhan, and D. Tao, "Ru-net: Regularized unrolling net- work for scene graph generation," in Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 19 457-19 466. Graphical contrastive losses for scene graph parsing. Z Ji, J S Kevin, E Ahmed, T Andrew, C Bryan, IEEE Conference Proceedings. Z. Ji, J. S. Kevin, E. Ahmed, T. Andrew, and C. Bryan, "Graphical contrastive losses for scene graph parsing," IEEE Conference Proceedings, 2019. Tackling the unannotated: Scene graph generation with bias-reduced models. T.-J J Wang, S Pehlivan, J Laaksonen, british machine vision conferenceT.-J. J. Wang, S. Pehlivan, and J. 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Nie, "Stacked hybrid-attention and group collaborative learning for unbiased scene graph generation," 2022. Fully convolutional scene graph generation," computer vision and pattern recognition. H Liu, N Yan, M Mortazavi, B Bhanu, H. Liu, N. Yan, M. Mortazavi, and B. Bhanu, "Fully convolutional scene graph genera- tion," computer vision and pattern recogni- tion, 2021. Structured sparse rcnn for direct scene graph generation. Y Teng, L Wang, arXiv: Computer Vision and Pattern Recognition. Y. Teng and L. Wang, "Structured sparse r- cnn for direct scene graph generation." arXiv: Computer Vision and Pattern Recognition, 2021. Reltr: Relation transformer for scene graph generation. Y Cong, M Y Yang, B Rosenhahn, 2022Y. Cong, M. Y. Yang, and B. Rosenhahn, "Reltr: Relation transformer for scene graph generation," 2022. Relation transformer network. R Koner, P Sinhamahapatra, V Tresp, arXiv: Computer Vision and Pattern Recognition. R. Koner, P. Sinhamahapatra, and V. Tresp, "Relation transformer network," arXiv: Com- puter Vision and Pattern Recognition, 2020. The hungarian method for the assignment problem. H W Kuhn, Naval Research Logistics Quarterly. H. W. Kuhn, "The hungarian method for the assignment problem," Naval Research Logistics Quarterly, 1955. Neural motifs: Scene graph parsing with global context. R Zellers, M Yatskar, S Thomson, Y Choi, R. Zellers, M. Yatskar, S. Thomson, and Y. Choi, "Neural motifs: Scene graph parsing with global context," computer vision and pattern recognition, 2017. Gps-net: Graph property sensing network for scene graph generation. X Lin, C Ding, J Zeng, D Tao, 2020X. Lin, C. Ding, J. Zeng, and D. Tao, "Gps-net: Graph property sensing network for scene graph generation," computer vision and pattern recognition, 2020. Knowledge-embedded routing network for scene graph generation. T Chen, W Yu, R Chen, L Lin, computer vision and pattern recognition. T. Chen, W. Yu, R. Chen, and L. Lin, "Knowledge-embedded routing network for scene graph generation." computer vision and pattern recognition, 2019. Exploring context and visual pattern of relationship for scene graph generation. W Wang, R Wang, S Shan, X Chen, computer vision and pattern recognition. W. Wang, R. Wang, S. Shan, and X. Chen, "Exploring context and visual pattern of re- lationship for scene graph generation," com- puter vision and pattern recognition, 2019.

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{'abstract': 'Objects in a scene are not always related. The execution efficiency of the one-stage scene graph generation approaches are quite high, which infer the effective relation between entity pairs using sparse proposal sets and a few queries. However, they only focus on the relation between subject and object in triplet set <subject entity, predicate entity, object entity>, ignoring the relation between subject and predicate or predicate and object, and the model lacks self-reasoning ability. In addition, linguistic modality has been neglected in the one-stage method. It is necessary to mine linguistic modality knowledge to improve model reasoning ability. To address the above-mentioned shortcomings, a Self-reasoning Transformer with Visual-linguistic Knowledge (SrTR) is proposed to add flexible self-reasoning ability to the model. An encoder-decoder architecture is adopted in SrTR, and a self-reasoning decoder is developed to complete three inferences of the triplet set, s+o→p, s+p→o and p+o→s. Inspired by the largescale pre-training image-text foundation models, visual-linguistic prior knowledge is introduced and a visual-linguistic alignment strategy is designed to project visual representations into semantic spaces with prior knowledge to aid relational reasoning. Experiments on the Visual Genome dataset demonstrate the superiority and fast inference ability of the proposed method.', 'arxivid': '2212.09329', 'author': ['Yuxiang Zhang \nSchool of Information and Electronics\nBeijing Institute of Technology\n\n', "Zhenbo Liu \nNoah's Ark Lab\nHuawei\n", 'Shuai Wang \nDepartment of Chemistry\nThe University of Hong Kong\n\n'], 'authoraffiliation': ['School of Information and Electronics\nBeijing Institute of Technology\n', "Noah's Ark Lab\nHuawei", 'Department of Chemistry\nThe University of Hong Kong\n'], 'corpusid': 254853831, 'doi': '10.48550/arxiv.2212.09329', 'github_urls': [], 'n_tokens_mistral': 9032, 'n_tokens_neox': 8222, 'n_words': 5111, 'pdfsha': 'dd43503ef714865251daa69feef23debb866ad0e', 'pdfurls': ['https://export.arxiv.org/pdf/2212.09329v1.pdf'], 'title': ['SRTR: SELF-REASONING TRANSFORMER WITH VISUAL-LINGUISTIC KNOWLEDGE FOR SCENE GRAPH GENERATION', 'SRTR: SELF-REASONING TRANSFORMER WITH VISUAL-LINGUISTIC KNOWLEDGE FOR SCENE GRAPH GENERATION'], 'venue': []}

arxiv

THE EIGEN-PROBLEM FOR SOME SPECIAL NEAR-TOEPLITZ CENTRO-SKEW TRIDIAGONAL MATRICES 30 Jan 2011 Date: February 1, 2011 Antonio Behn Department of Mathematics Mathematics Department Mathematics Department University of Chile Iowa State University Iowa State University Kenneth R Driessel Department of Mathematics Mathematics Department Mathematics Department University of Chile Iowa State University Iowa State University Irvin R Hentzel Department of Mathematics Mathematics Department Mathematics Department University of Chile Iowa State University Iowa State University THE EIGEN-PROBLEM FOR SOME SPECIAL NEAR-TOEPLITZ CENTRO-SKEW TRIDIAGONAL MATRICES 30 Jan 2011 Date: February 1, 2011Table of contents • Introduction • A Reduction • Tridiagonal Toeplitz Matrices • The Eigenvalues • Acknowledgements • References1991 Mathematics Subject Classification Primary: 15B05 Toeplitz, Cauchy, and related matricesSecondary: 15B35 Sign pattern matrices15A18 Eigenvalues, singular values, and eigenvectors Key words and phrases Tridiagonal, Toeplitz, eigenvalue, eigenvector, centro symmetric, centro skew symmetric, sign pattern Let n ≥ 2 be an integer. Let R n denote the n × n tridiagonal matrix with −1's on the sub-diagonal, 1's on the superdiagonal, −1 in the (1, 1) entry, 1 in the (n, n) entry and zeros elsewhere. We find the eigen-pairs of the matrices R n . Introduction We consider some special n-by-n, near-Toeplitz, tridiagonal matrices with entries from the set {0, 1, −1}. In particular, we consider tridiagonal matrices having the following form: R n := Tridiag (subdiag, diag, supdiag) where • subdiag := (−1, −1, . . . , −1), • diag := (−1, 0, 0, . . . , 0, 0, 1), and • supdiag := (1, 1, . . . , 1, 1). In other words, R n is the tridiagonal matrix that has all −1's on the subdiagonal, all 0's on the diagonal except for a −1 in the (1,1) entry and a 1 in the (n,n) entry, and has all 1's on the superdiagonal. For example, when n := 4 we have: R 4 :=     −1 1 0 0 −1 0 1 0 0 −1 0 1 0 0 −1 1     . We find the eigenvalue-vector pairs (or eigen-pairs for short) of the matrices R n . In particular, we prove the following result: Proposition 1. The eigenvalues of the matrix R n are 0 and 2i cos(jθ) for j = 1, . . . , n − 1 where θ := π/n. Remark: If n is even then, 0 is an eigenvalue with multiplicity 2. Let T n denote the set of n×n tridiagonal real matrices that satisfy the following conditions: the sub-diagonal is negative, the super-diagonal is positive, the (1, 1) entry is negative, the (n, n) entry is positive and all other entries are zero. Drew, et al(2000) conjectured that this sign pattern class contains matrices with arbitrary spectra. Note that the matrix R n is in this class. They provided evidence for the conjecture. (Elsner, et al(2003) provided further evidence.) We believe that understanding the properties of R n may be an important step toward understanding the arbitrary spectrum conjecture for the sign pattern class T n . Here is a summary of the contents. In the section with title "A Reduction", we show that R n is similar to a near skew-symmetric, tridiagonal, Toeplitz matrix. In the section with title, "Tridiagonal Toeplitz Matrices" we review the solution of the eigen-problem for such matrices; in particular, we determine the eigen-pairs for a skew-symmetric tridiagonal Toeplitz matrix. In the section with title "The Eigenvalues", we determine the eigenvalues of R n ; in other words, we prove the proposition given above. Aside: Let E be the n × n matrix defined by E(i, j) := δ(i + j, n + 1), for 1 ≤ i, j ≤ n, where δ is the Kronecker delta. This matrix is called the exchange matrix or the flip matrix. Let P be an n × n matrix. Then P is a centro-symmetric matrix if EP E = P and P is a centro-skew matrix if EP E = −P . Here is an older reference on centro-symmetry: Weaver(1985). Here is a more recent reference: Trench(2004). These papers contain further references. We shall not use the theory of centro-symmetric matrices in this paper. EndAside. A reduction Recall that we are considering special matrices R n where R n is the tridiagonal matrix which that has all −1's on the subdiagonal, has all 0's on the diagonal except for a −1 in the (1,1) entry and a 1 in the (n,n) entry, and has all 1's on the superdiagonal. We want to find the eigenvalues and eigenvectors of these matrices. Here is another description of R n . Let Z n denote the lower shift matrix which is the matrix that has 1's on the subdiagonal and 0's elsewhere. For example, when n = 4, we have Z 4 :=     0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0     . Aside: Here we follow notation and terminology used by T. Kailath. See, for example, Kailath, Kung and Morf(1979a), Kailath, Kung and Morf(1979b) or Kailath and Sayed(1999). EndAside. Note that R n = Z T n − Z n − e 1 e T 1 + e n e T n where e k denotes the column vector which has 1 in the kth coordinate and 0's elsewhere. Proposition 2. Reduction. Let S n := I n + Z n where I n is the n × n identity matrix. Then S −1 n R n S n = K n + e n e T n−1 where K n := Z T n − Z n . Remark: Here is a picture of S 4 : S 4 :=     1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1     Note that I n + Z n is invertible since Z n is nilpotent. In fact (I n + Z n ) −1 = I n − Z n + Z 2 n − · · · ± Z n−1 n . When n = 4, for example, the conclusion of the proposition is     1 0 0 0 −1 1 0 0 1 −1 1 0 −1 1 −1 1         −1 1 0 0 −1 0 1 0 0 −1 0 1 0 0 −1 1         1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1     =     0 1 0 0 −1 0 1 0 0 −1 0 1 0 0 0 0     . Proof. In order to reduce the number of symbols in this proof, we omit the subscript n on I, Z, R and S. Note that the equation in the conclusion of the proposition is equivalent to We also have Se n e T n−1 + (e 1 e T 1 − e n e T n )S = (I + Z)e n e T n−1 + (e 1 e T 1 − e n e T n )(I + Z) = e 1 e T 1 − e n e T n . Tridiagonal Toeplitz matrices In this section, we consider the skew symmetric tridiagonal matrices K n := Z T n − Z n . We determine the eigenvalues and eigenvectors of these matrices. Let T n (a, b, c, ) := Tridiag(subdiag, diag, supdiag) denote the n × n tridiagonal matrix determined by subdiag : = (a, a, . . . , a, a), diag := (b, b, . . . , b, b), and supdiag := (c, c, . . . , c, c). In other words, T n (a, b, c) is the tridiagonal matrix that has all a's on the subdiagonal, has all b's on the diagonal and has all c's on the superdiagonal. Recall that matrices with constant diagonals are called Toeplitz matrices. For example, when n := 4 we have: T 4 (a, b, c) :=     b c 0 0 a b c 0 0 a b c 0 0 a b     . We want to find the eigenvalues and eigenvectors of the matrices T n (a, b, c). The following result is well-known. Proposition 3. The matrix T n (a, b, c) is diagonally similar to the symmetric matrix T n ( √ ac, b, √ ac) provided ac = 0. In particular, DT n (a, b, c)D −1 = T n ( √ ac, b, √ ac) where D := Diag(1, d, d 2 , . . . , d n ) and d := c/a. Furthermore, T n (a, b, c)u = λu iff T n ( √ ac, b, √ ac)Du = λDu and hence the eigen-pairs of T n (a, b, c) are determined by the eigen-pairs of T n ( √ ac, b, √ ac). We call the transformation T n (a, b, c) → DT n (a, b, c)D −1 = T n ( √ ac, b,√ ac) the diagonal similarity symmetrizing transformation. Here is an example:   1 0 0 0 d 0 0 0 d 2     b c 0 a b c 0 a b     1 0 0 0 d −1 0 0 0 d −2   =   b c/d 0 da b c/d 0 da b   =   b √ ac 0 √ ac b √ ac 0 √ ac b   . The following result is well-known. (See, for example, Rayleigh(1894).) We include a simple proof for the reader's convenience. Proposition 4. Symmetric tridiagonal Toeplitz eigen-pairs. The eigenvalues λ j and corresponding eigenvectors u j of the n × n matrix T n (a, b, a) are, for j = 1, . . . , n, given by λ j := b + 2a cos(jθ) and u j := (sin(jθ), sin(2jθ), · · · , sin(njθ)) T , where θ := π/(n + 1). Proof. Recall that matrices A and bI + A have the same eigenvectors. Also recall that λ is an eigenvalue of A iff b + λ is an eigenvalue of bI + A. Recall that, for a = 0, the matrices A and aA have the same eigenvectors. Also recall that λ is an eigenvalue of A iff aλ is an eigenvlue of aA. Using these reductions, we see that we only need to verify the eigenpairs of T n (1, 0, 1). To reduce the number of symbols in this calculation we take n = 3. In this case, we have θ := π/4 and The last equality follows from the well-known 3 term recurrence relation for the sine function: sin(k + 1)α − 2 cos α sin(kα) + sin(k − 1)α = 0. Proposition 5. Skew-symmetric tridiagonal Toeplitz eigen-pairs. The eigenvalues λ j and corresponding eigenvectors u j of the n × n matrix K n := Z T n − Z n are, for j = 1, . . . , n, given by λ j := 2i cos(jθ) and u j := (i sin(jθ), i 2 sin(2jθ), i 3 sin(3jθ), · · · , i n sin(njθ)), where θ := π/(n + 1). Proof. Note that K n = Z T n − Z n = T n (−1, 0, 1). We apply the diagonal similarity symmetrizing transformation with D −1 := Diag(1, i, i 2 , . . . , i n ) to get DT n (−1, 0, 1)D −1 = T n (i, 0, i) = iT n (1, 0, 1). We know the eigen-pairs of T n (1, 0, 1) from above. Thus we can easily determine the eigen-pairs of K n . In particular, if T n (1, 0, 1)u = λu then DT n (−1, 0, 1)D −1 u = iT n (1, 0, 1) = iλu. Hence T n (−1, 0, −1)(D −1 u) = iλD −1 u. The Eigenvalues Proposition 6. The eigenvalues of the matrix R n are 0 and 2i cos(jθ) for j = 1, . . . , n − 1 where θ := π/n. Proof. It is clear that the vector (1, 1, . . . , 1) T is an eigenvector with 0 as corresponding eigenvalue. From the reduction proposition, we have S −1 n R n S n = K n + e n e T n−1 . Note that K n + e n e T n−1 = K n−1 e n−1 0 0 . Also note that if K n−1 u = λu then K n−1 e n−1 0 0 u 0 = K n−1 u = λu = λ u 0 . Remark: The eigenvectors of R n are determined in the proof of the proposition. ( * ) RS = SK + Se n e T n−1 . Also noteRS = (K − e 1 e T 1 + e n e T n )S = KS + (e n e T n − e 1 e T 1 )S = SK + [K, S] + (e n e T n − e 1 e T 1 )S where [X, Y ] := XY − Y X.Hence, to prove ( * ), we need only prove the followingClaim: [K, S] = Se n e T n−1 + (e 1 e T 1 − e n e T n )S We have [K, S] = [K, I + Z] = [K, Z] = [Z T − Z, Z] = [Z T , Z] = Z T Z − ZZ T = Diag(1, 1, . . . , 1, 0) − Diag(0, 1, . . . , 1, 1) = e 1 e T 1 − e n e T n . Table of contents of• Introduction • A Reduction • Tridiagonal Toeplitz Matrices • The Eigenvalues • Acknowledgements • References AcknowledgementsThis paper was written in October of 2010 while Behn was visiting the mathematics department at Iowa State university (under grant number FONDECYT 1100135).We thank Wayne Barrett (Brigham Young University) for his careful reading of this paper and his constructive comments about it.Driessel thanks Wolfgang Kliemann, chair of the Mathematics Department at Iowa State, for arranging his affiliation with that department. Spectrally arbitrary patterns. J H Drew, C R Johnson, D D Olesky, P Van Den Driesche, Linear Algebra and Appl. 308• Drew, J.H.; Johnson, C.R.; Olesky, D.D.; and van den Dri- esche, P. (2000) Spectrally arbitrary patterns, Linear Algebra and Appl. 308, 121-137 Low rank perturbations and the spectrum of a tri-diagonal sign pattern. L Elsner, D D Olesky, P Van Den Driesche, Linear Algebra and Appl. 308• Elsner, L.; Olesky, D.D.; and van den Driesche, P. (2003) Low rank perturbations and the spectrum of a tri-diagonal sign pat- tern, Linear Algebra and Appl. 308, 121-137 Displacement ranks of matrices and linear equations. T Kailath, S Y Kung, M Morf, J. Math. Anal. Appl. 68• Kailath, T.; Kung, S.Y.; and Morf, M. (1979a) Displacement ranks of matrices and linear equations, J. Math. Anal. Appl. 68, 395-407 Displacement ranks of a matrix. T Kailath, S Y Kung, M Morf, Bull. Amer. Math. Soc. 1• Kailath, T.; Kung, S.Y.; and Morf, M. (1979b) Displacement ranks of a matrix, Bull. Amer. Math. Soc. 1, 769-773 Fast Reliable Algorithms for Matrices with Structure. T Kailath, A H Sayed, SIAM• Kailath, T. and Sayed, A.H. (1999) Fast Reliable Algorithms for Matrices with Structure, SIAM The Theory of Sound. J W S • Rayleigh, MacmillanReprinted by Dover in 1945• Rayleigh, J.W.S. (1894) The Theory of Sound, Macmillan (Reprinted by Dover in 1945.) Characterization and properties of matrices with generalized symmetry or skew symmetry. W F Trench, Lin. Alg. Appl. 377• Trench, W.F. (2004) Characterization and properties of matri- ces with generalized symmetry or skew symmetry, Lin. Alg. Appl. 377, 207-218 Centrosymmetric (Cross-Symmetric) Matrices. J R Weaver, American Mathematical Monthly. 92• Weaver, J.R. (1985) Centrosymmetric (Cross-Symmetric) Ma- trices, American Mathematical Monthly 92, 711-717

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{'abstract': "Let n ≥ 2 be an integer. Let R n denote the n × n tridiagonal matrix with −1's on the sub-diagonal, 1's on the superdiagonal, −1 in the (1, 1) entry, 1 in the (n, n) entry and zeros elsewhere. We find the eigen-pairs of the matrices R n .", 'arxivid': '1101.5788', 'author': ['Antonio Behn \nDepartment of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n\n', 'Kenneth R Driessel \nDepartment of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n\n', 'Irvin R Hentzel \nDepartment of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n\n'], 'authoraffiliation': ['Department of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n', 'Department of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n', 'Department of Mathematics\nMathematics Department\nMathematics Department\nUniversity of Chile\nIowa State University\nIowa State University\n'], 'corpusid': 119127430, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5014, 'n_tokens_neox': 4281, 'n_words': 2493, 'pdfsha': '773abff913adf85819522a66fab60d02db4efea4', 'pdfurls': ['https://arxiv.org/pdf/1101.5788v1.pdf'], 'title': ['THE EIGEN-PROBLEM FOR SOME SPECIAL NEAR-TOEPLITZ CENTRO-SKEW TRIDIAGONAL MATRICES', 'THE EIGEN-PROBLEM FOR SOME SPECIAL NEAR-TOEPLITZ CENTRO-SKEW TRIDIAGONAL MATRICES'], 'venue': []}

arxiv

Special numbers, special quaternions and special symbol elements 5 Dec 2017 Diana Savin Special numbers, special quaternions and special symbol elements 5 Dec 2017(l, 1, p + 2q, q · l) − quaternions, (l, 1, p + 2q, q · l) − symbol elements. 2000 AMS Subject Classification: 15A24, 15A06, 16G30, 11R52, 11B39, 11R54.quaternion algebras; symbol algebrasFibonacci numbersLu- cas numbersFibonacci-Lucas quaternionsPell-Fibonacci-Lucas quaternions In this paper we define and we study properties of (l, 1, p + 2q, q · l) − numbers, (l, 1, p + 2q, q · l) − quaternions, (l, 1, p + 2q, q · l) − symbol elements. Finally, we obtain an algebraic structure with these elements. Introduction Quaternion algebras and of symbol algebras have applications in various branches of mathematics, but also in computer science, physics, signal theory. In this chapter we introduce special numbers, special quaternions, special symbol elements, and we present some of their properties and their applications in combinatorics, number theory and associative algebra theory. Let K be a field with char (K) = 2 and let α, β∈K\{0}. We recall that the generalized quaternion algebra H K (α, β) is an algebra over the field K with a basis {e 1 , e 2 , e 3 , e 4 } (where e 1 = 1 ) and the following multiplication: · 1 e 2 e 3 e 4 1 1 e 2 e 3 e 4 e 2 e 2 α e 4 αe 3 e 3 e 3 −e 4 β −βe 2 e 4 e 4 −αe 2 βe 2 −αβ Let x be element from H K (α, β) , x = x 1 · 1 + x 2 e 2 + x 3 e 3 + x 4 e 4 , where x i ∈ K, (∀)i ∈ {1, 2, 3, 4} and let x be the conjugate of x, x = x 1 · 1 − x 2 e 2 − x 3 e 3 − x 4 e 4 . The trace of x is t (x) = x + x = 2x 1 . The norm of x is n (x) = x · x = x 2 1 − αx 2 2 − βx 2 3 + αβx 2 4 . If K = R and α = β = −1, we obtain Hamilton quaternion algebra H R (−1, −1) , with the basis {1, i, j, k}. The generalization of a quaternion algebra is a symbol algebra. Let n be an arbitrary positive integer, n ≥ 3 and let K be a field with char(K), which does not divide n, containing ξ, where ξ is a primitive n-th root of unity. Let a, b ∈ K\{0}. The algebra A over K generated by elements x and y where x n = a, y n = b, yx = ξxy is called a symbol algebra and it is denoted by a, b K,ξ . Symbol algebras are also known as power norm residue algebras. For n = 2, we obtain the quaternion algebra over the field K. Quaternion algebras and symbol algebras are associative but non-commutative algebras, of dimension n 2 over K. Also, they are central simple over the field K (this means they are simple algebras and their centers are equal to K). Theoretical aspects about these algebras can be found in the books [Pi; 82], [La;04], [Mil], [Gi,Sz;06] [Ko], [Mi;71]. Several properties of these algebras and their applications in number theory, combinatorics, associative algebra, geometry, coding theory, mechanics can be found in the articles [Ak,Ko,To;14], [Fla;12], [Fl,Sa,Io;13], [Fl,Sa;14], [Fl,Sa;15], [Fl,Sa;15 (1)], [Fl,Sa;15 (2)], [Fl,Sa;17], [Fl,Sa;18], [Fl,Sh;13], [Fl,Sh;13 (1) [Fl,Sa;15], [Fl,Sa;18], [Fl,Sh;13],[Ha;12], [Sa;14], [Sw; 73], etc.). In this paper we define (l, 1, p + 2q, q · l) − numbers, (l, 1, p + 2q, q · l) − quaternions, (l, 1, p + 2q, q · l) − symbol elements. We also study properties and applications of these elements. This paper is organized as follow: section 2 is a preliminary section, containing theoretical notions which we will then use in our results. In Section 3 we introduce two special number sequences (namely (a n ) n≥0 , (b n ) n≥0 ), we obtain some interesting properties of these sequences and we also obtain some quaternion algebras which split or some division quaternion algebras. After these, in the same section, we introduce (l, 1, p + 2q, q · l) − numbers, (l, 1, p + 2q, q · l) − quaternions, (l, 1, p + 2q, q · l) − symbol elements and we obtain interesting properties and applications of them. Preliminaries First of all, we recall some results about prime integers, about diophantine equations or about the Fibonacci numbers, properties which will be necessary (in the next section) for to study some quaternion algebras. Proposition 2.1. ([Cu;06]). Let m be a fixed positive integer. The diophantine equation x 2 + my 2 = z 2 has an infinity of solutions: x = a 2 − mb 2 , y = 2mab, z = a 2 + mb 2 , a, b ∈ Z. Theorem 2.2. ( [Al,Go;99]). Let n be a positive integer. Then, there exist integers x, y such that n = x 2 + y 2 if and only if the exponent of any prime p ≡ 3 (mod 4) that divides n is even. Sa;14]). For each positive integer n, n ≡ 7 (mod 16), there exist integer numbers x, y so that, the Fibonacci number f n can be written as f n = x 2 + 9y 2 . Proposition 2.3. ([ Let K be a field with char (K) = 2, let α, β∈K\{0} and let the generalized quaternion algebra H K (α, β) . H K (α, β) is a division algebra if and only if for x ∈H K (α, β) we have n (x) = 0 only for x = 0. We recall that H K (α, β) is called split by K if it is isomorphic with a matrix algebra over K (see [Pi;82], [La;04], [Mil], [Gi,Sz;06]). It is known the following remark about the quaternion algebras. . Let K be a field with char K = 2 and let α, β∈K\{0}. Then, the quaternion algebra H K (α, β) is either split or a division algebra. In the book [Gi, Sz; 06] appears the following criterion to decide if a quaternion algebra splits. Proposition 2.5. ([Gi, Sz; 06]). Let K be a field with char K = 2 and let α, β∈K\{0}. The quaternion algebra H K (α, β) splits if and only if the conic C (α, β) : αx 2 +βy 2 = z 2 has a rational point over K(i.e. if there are x 0 , y 0 , z 0 ∈ K,not all zero such that αx 2 0 + βy 2 0 = z 2 0 ). Some properties of special quaternions Let l be a nonzero natural number. We consider the sequence (a n ) n≥0 a n = l · a n−1 + a n−2 , n ≥ 2, a 0 = 0, a 1 = 1 and let the sequence (b n ) n≥0 b n = l · b n−1 + b n−2 , n ≥ 2, b 0 = 2, b 1 = l. Let α = l+ √ l 2 +4 2 and β = l− √ l 2 +4 2 . It results immediately the following relations: Binet's formula for the sequence (a n ) n≥0 . a n = α n − β n α − β = α n − β n √ l 2 + 4 , (∀) n ∈ N. Binet's formula for the sequence (b n ) n≥0 . b n = α n + β n , (∀) n ∈ N. In the following, we show that the product of two elements belonging to the sequences (a n ) n≥0 , (b n ) n≥0 are transformed into sums of elements belonging to the same sequences. Also, we find another properties of these sequences. Proposition 3.1. Let (a n ) n≥0 , (b n ) n≥0 be the sequences previously defined. Then, the following equalities are true: i) b n b n+m = b 2n+m + (−1) n b m , (∀) n, m ∈ N; ii) a n b n+m = a 2n+m + (−1) n+1 a m , (∀) n, m ∈ N; iii) a n+m b n = a 2n+m + (−1) n a m , (∀) n, m ∈ N; iv) a n a n+m = 1 l 2 + 4 b 2n+m + (−1) n+1 b m , (∀) n, m ∈ N; v) b n + b n+2 = l 2 + 4 · a n+1 , (∀) n ∈ N; vi) a 2 n + a 2 n+1 = a 2n+1 , (∀) n ∈ N; vii) b 2 n + b 2 n+1 = l 2 + 4 · a 2n+1 , (∀) n ∈ N; Proof. Let n, m be two positive integers. Applying Binet's formulae, we have: i) b n b n+m = (α n + β n ) · α n+m + β n+m = = α 2n+m + β 2n+m + α n β n (α m + β m ) = b 2n+m + (−1) n b m . ii) a n b n+m = α n − β n α − β α n+m + β n+m = = α 2n+m − β 2n+m α − β − α n β n (α m − β m ) α − β = a 2n+m + (−1) n+1 a m . iii) a n+m b n = α n+m − β n+m α − β (α n + β n ) = = α 2n+m − β 2n+m α − β + α n β n (α m − β m ) α − β = a 2n+m + (−1) n a m . iv) a n a n+m = α n − β n α − β · α n+m − β n+m α − β = = 1 (α − β) 2 · α 2n+m + β 2n+m − α n β n (α m + β m ) = = 1 l 2 + 4 b 2n+m + (−1) n+1 b m . v) b n + b n+2 = α n + β n + α n+2 + β n+2 = = α n+1 · α + 1 α + β n+1 · β + 1 β = = α n+1 · l 2 + 4 − β n+1 · l 2 + 4 = l 2 + 4 · a n+1 . vi) Applying iv) for m = 0, we have: a 2 n + a 2 n+1 = 1 l 2 + 4 b 2n + (−1) n+1 b 0 + b 2n+2 + (−1) n+2 b 0 = = 1 l 2 + 4 [b 2n + b 2n+2 ] . Applying v) we obtain: a 2 n + a 2 n+1 = a 2n+1 . vii) Applying v) we have: b 2 n + b 2 n+1 = (α n + β n ) 2 + α n+1 + β n+1 2 = = α 2n + β 2n + 2 (−1) n + α 2n+2 + β 2n+2 + 2 (−1) n+1 = = b 2n + b 2n+2 = l 2 + 4 · a 2n+1 . Let (f n ) n≥0 be the Fibonacci sequence and let (l n ) n≥0 be the Lucas sequence. There are well known the Cassini's identities for Fibonacci and Lucas numbers: f n+1 f n−1 − f 2 n = (−1) n , (∀) n ∈ N * , and l n+1 l n−1 − l 2 n = 5 · (−1) n−1 , (∀) n ∈ N * , Now, we obtain similarly results for the the sequences (a n ) n≥0 , (b n ) n≥0 . Proposition 3.2. Let (a n ) n≥0 , (b n ) n≥0 be the sequences previously defined. Then, the following identities are true: i) a n+1 a n−1 − a 2 n = (−1) n , (∀) n ∈ N * ; ii) b n+1 b n−1 − b 2 n = (−1) n · l 2 + 4 , (∀) n ∈ N * . Proof. i) a n+1 a n−1 − a 2 n = α n+1 − β n+1 √ l 2 + 4 · α n−1 − β n−1 √ l 2 + 4 − (α n − β n ) 2 l 2 + 4 = = − (−1) n−1 · α 2 + β 2 + 2 (−1) n l 2 + 4 = (−1) n · (b 2 + 2) l 2 + 4 = (−1) n . ii) b n+1 b n−1 − b 2 n = α n+1 + β n+1 · α n−1 + β n−1 − (α n + β n ) 2 = = (−1) n−1 · α 2 + β 2 − 2 (−1) n = (−1) n−1 · (b 2 + 2) = (−1) n−1 · l 2 + 4 . Proposition 3.3. Let (b n ) n≥0 be the sequence previously defined. Then, the followings are true: i) if l is even, then b n is even (∀) n ∈ N; ii) if l is odd, then b n is even if and only if n ≡ 0 (mod 3); iii) if n ≡ 0 (mod 6), then b n−1 · b n+1 ≡3 (mod 4); iv) if n ≡ 3 (mod 6), then b n−1 · b n+1 ≡1 (mod 4). Proof. For i), ii), iii) and iv) the proof is immediate, using the principle of mathematics induction (after n ∈ N). Proposition 3.4. Let (a n ) n≥0 , (b n ) n≥0 be the sequences previously defined. Then, the followings are true: i) The quaternion algebraH Q (−1, f 2n+1 ) splits, (∀) n ∈ N * ; ii) The quaternion algebra H Q (−1, 5f 2n+1 ) splits, (∀) n ∈ N * ; iii) The quaternion algebra H Q (−1, a 2n+1 ) splits, (∀) n ∈ N * ; iv) The quaternion algebra H Q −1, l 2 + 4 · a 2n+1 splits, (∀) n ∈ N * ; v) The quaternion algebra H Q (−1, f 2n+1 f 2n−1 ) splits, (∀) n ∈ N * ; vi) The quaternion algebra H Q (−1, a 2n+1 a 2n−1 ) splits, (∀) n ∈ N * ; vii)The quaternion algebra H Q (−1, −b n+1 b n−1 ) is a division algebra, (∀) n ∈ N * ; viii) The quaternion algebra H Q (1, b n+1 b n−1 ) splits, (∀) n ∈ N * ; ix) If l is odd and n ≡ 0(mod 6), then the quaternion algebra H Q (−1, b n+1 b n−1 ) is a division algebra, (∀) n ∈ N * ; x) If 6 ∤ n and the exponent of any prime p ≡ 3(mod 4) that divides b n+1 b n−1 is even, then the quaternion algebra H Q (−1, b n+1 b n−1 ) splits, (∀) n ∈ N * ; xi) The quaternion algebraH Q (−9, f n ) splits, (∀) n ∈ N * , n ≡ 7 (mod 16). Proof. Since iii) is a generalization of i), we are proving directly iii). iii) If we consider the equation −x 2 + a 2n+1 · y 2 = z 2 , we apply Proposition 3.1 (vi) and we obtain that it has the following solution in Q × Q × Q : (x 0 , y 0 , z 0 ) = (a n , 1, a n+1 ) . According to Proposition 2.5, it results that the quaternion algebra H Q (−1, a 2n+1 ) splits, (∀) n ∈ N * . iv) Using Proposition 3.1 (vii), it results that the equation −x 2 + l 2 + 4 ·a 2n+1 · y 2 = z 2 has a solution in Q×Q×Q, namely (x 0 , y 0 , z 0 ) = (b n , 1, b n+1 ) . Applying Proposition 2.5, it results that the quaternion algebra H Q −1, l 2 + 4 · a 2n+1 splits, (∀) n ∈ N * . ii) This is a particular case of ii) (for l = 1). vi) This is a generalization of v), so we are proving only vi). Using Proposition 3.2 i), we find the following solution in Q × Q × Q for the equation −x 2 + a 2n+1 · a 2n−1 · y 2 = z 2 : (x 0 , y 0 , z 0 ) = ((−1) n , 1, a 2n ) . Applying Proposition 2.5, we obtain that the quaternion algebra H Q (−1, a 2n+1 a 2n−1 ) splits, (∀) n ∈ N * . vii) Let the quaternion algebra H Q (−1, −b n+1 b n−1 ) and let {1, e 2 , e 3 , e 4 } a basis in this algebra. Let x = x 1 ·1+x 2 ·e 2 +x 3 ·e 3 +x 4 ·e 4 ∈H Q (−1, −b 2n+1 b 2n−1 ) . The norm of x is n (x) = x · x = x 2 1 + x 2 2 + b n−1 b n+1 x 2 3 + b n−1 b n+1 x 2 4 . Since b n ∈N * , for (∀) n ∈ N * , it results that n (x) = 0 if and only if x = 0. So, H Q (−1, −b n+1 b n−1 ) is a division algebra for (∀) n ∈ N * . Similarly, it results immediately that H Q (−1, −f n+1 f n−1 ) , H Q (−1, −a n+1 a n−1 ) , H Q (−1, −l n+1 l n−1 ) are division algebras for (∀) n ∈ N * . viii) We study if the equation x 2 + b n+1 b n−1 · y 2 = z 2 has rational solutions. Applying Proposition 2.1, it results that the equation x 2 +b n+1 b n−1 ·y 2 = z 2 has solutions in integer numbers, so it has solutions in the set of rational numbers. Using Proposition 2.5, we obtain that the quaternion algebra H Q (1, b n+1 b n−1 ) splits. ix) If n ≡ 0(mod 6), according to Proposition 3.3, b n−1 · b n+1 ≡3 (mod 4). We study if the equation −x 2 + b n+1 b n−1 · y 2 = z 2 has integer solutions. We suppose that this equation has a solution (x 0 , y 0 , z 0 )∈Z × Z × Z \ {(0, 0, 0)} , g.c.d(x 0 , y 0 ) = g.c.d(y 0 , z 0 ) =g.c.d(y 0 , z 0 ) = 1. We have: b n+1 b n−1 · y 2 0 ≡0 or 3 (mod 4), but x 2 0 + z 2 0 ≡1 or 2 (mod 4), so we cannot have b n+1 b n−1 · y 2 0 = x 2 0 + z 2 0 . It results that the equation −x 2 + b n+1 b n−1 · y 2 = z 2 does not have integer solutions. We obtain immediately that the equation −x 2 + b n+1 b n−1 · y 2 = z 2 does not have solutions in the set of rational numbers, so the quaternion algebra H Q (−1, b n+1 b n−1 ) does not split. Applying Remark 2.4, we obtain that the quaternion algebra H Q (−1, b n+1 b n−1 ) is a division algebra, (∀) n ∈ N * . x) Case 1 : l is odd. If n ≡ 3 (mod 6), according to Proposition 3.3 (iv) b n−1 · b n+1 ≡1 (mod 4). If n ≡ 1 or 2 or 4 or 5 (mod 6), according to Proposition 3.3 (ii) b n−1 · b n+1 is even. Case 2 : l is even, according to Proposition 3.3 (i) b n−1 · b n+1 is even. In all these cases, it is possible to exist a prime p ≡ 3(mod 4) that divides b n+1 b n−1 . If the exponent of any prime p ≡ 3(mod 4) that divides b n+1 b n−1 is even, according to Theorem 2.2 there exist integers x 0 ; z 0 such that b n+1 b n−1 = x 2 0 + z 2 0 . This implies that (x 0 , 1, z 0 ) is a solution in integer numbers for the equation −x 2 + b n+1 b n−1 · y 2 = z 2 , so, according Proposition 2.5, the quaternion algebra H Q (−1, b n+1 b n−1 ) splits, (∀) n ∈ N * . xi) Let n be a positive integer number, n ≡ 7 (mod 16). Using Proposition 2.3 we obtain that there are x 0 , z 0 ∈Z such that (x 0 , 1, z 0 ) is a solution of the equation −9x 2 + f n · y 2 = z 2 . Applying Proposition 2.5, we obtain that the quaternion algebra H Q (−9, f n ) splits, (∀) n ∈ N * , n ≡ 7 (mod 16). Let p, q be two arbitrary integers and (a n ) n≥0 , (b n ) n≥0 are the sequences previously defined. If n∈N * , a −n = (−1) n+1 · a n . Let the sequence (u n ) n≥0 , u n+1 = pa n + qb n+1 , n ≥ 0. To avoid confusion, we will use the notation u p,q n for u n . We remark that u n = lu n−1 + u n−2 , (∀) n ∈ N, n ≥ 2, We calculate u 0 = pa −1 + qb 0 = p + 2q, u 1 = pa 0 + qb 1 = q · l. We call the elements of the sequence (u n ) n≥0 the (l, 1, p + 2q, q · l) −numbers. Remark 3.5. Let p, q be two arbitrary integers, and let (u p,q n ) n≥1 the sequence previously defined. Then, we have: pa n+1 + qb n = u p,q n + u pl,o n+1 , ∀ n ∈ N − {0}. Proof. We compute pa n+1 + qb n = pla n + pa n−1 + qb n = u p,q n + u pl,o n+1 . Let α, β∈Q * . We consider the generalized quaternion algebra H Q (α, β) with basis {1, e 1 , e 2 , e 3 }. We define the n-th (l, 1, p + 2q, q · l) − quaternion to be the element of the form U p,q n = u p,q n · 1 + u p,q n+1 · e 1 + u p,q n+2 · e 2 + u p,q n+3 · e 3 . Remark 3.6. Let U p,q n be the n-th (l, 1, p + 2q, q · l) − quaternion. Then, we have: U p,q n = 0 if and only if p = q = 0. Proof. " ⇐" It is trivial. " ⇒" If U p,q n = 0, using the fact that {1, e 1 , e 2 , e 3 } is a basis in quaternion algebra H Q (α, β) , we obtain that u p,q n = 0, u p,q n+1 = 0, u p,q n+2 = 0, u p,q n+3 = 0. From the recurrence relation of the sequence (u p,q n ) n≥1 , it results that u p,q n−1 = 0, u p,q n−2 = 0, ..., s p,q 1 = 0, u p,q 0 = 0. So, q = 0 and p = 0. About the generalized Fibonacci-Lucas quaternions (G p,q n ) n≥0 , in the paper [Fl,Sa;15] (Theorem 3.5), we proved that: i) The set M = n i=1 5G pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n ∪ {1} has a ring structure with quaternion addition and multiplication. ii) The set M is an order of the quaternion algebra H Q (α, β) . We generalized these results for (1, a, p + 2q, q)quaternions (S p,q n ) n≥0 ,, in the paper [Fl,Sa;17] (Proposition 5.4), namely: Let a be a nonzero natural number and let O be the set O = n i=1 (1 + 4a) S pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n ∪ {1} . Then O is an order of the quaternion algebra H Q (α, β) . Similarly, in the paper [Fl,Sa;18] we introduced the generalized Pell-Fibonacci-Lucas numbers (r p,q n ) n≥0 , the generalized Pell-Fibonacci-Lucas quaternions (R p,q n ) n≥0 , and we proved that (Proposition 3.7. from the paper [Fl, Sa; 18]) the set O = n i=1 8R pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n ∪ {1} is an order of the quaternion algebra H Q (α, β) . Here, we generalized these numbers and these quaternions: the sequence (a n ) n≥0 is the generalization for the Pell sequence (P n ) n≥0 and the sequence (b n ) n≥0 is the generalization for the Pell-Lucas sequence (Q n ) n≥0 . Also, the sequence (u p,q n ) n≥0 is the generalization for the sequence (r p,q n ) n≥0 and the sequence of the (l, 1, p + 2q, q · l) − quaternions (U p,q n ) n≥0 is the generalization for the sequence of the generalized Pell-Fibonacci-Lucas quaternions (R p,q n ) n≥0 . Let ǫ be a primitive root of the unity of order 3 and let K be a field with the property ǫ∈K. Let α 1 , α 2 ∈K * and let A = α1,α2 K,ǫ be the symbol algebra of degree 3. A has a K-basis x j1 y j2 |0 ≤ j 1 , j 2 < 3 , with x 3 = α 1 , y 3 = α 2 , yx = ǫxy. In the paper [Fl,Sa;14], we defined the n-th Fibonacci symbol element F n = f n · 1 + f n+1 · x + f n+2 · x 2 + f n+3 · y + f n+4 · xy+ +f n+5 · x 2 y + f n+6 · y 2 + f n+7 · xy 2 + f n+8 · x 2 y 2 . In the paper [Fl,Sa,Io;13] we defined the n-th Lucas symbol element L n = l n · 1 + l n+1 · x + l n+2 · x 2 + l n+3 · y + l n+4 · xy+ +l n+5 · x 2 y + l n+6 · y 2 + l n+7 · xy 2 + l n+8 · x 2 y 2 . Now, we define the n-th (l, 1, p + 2q, q · l) − symbol element to be the element of the form U p,q n = u p,q n · 1 + u p,q n+1 · x + u p,q n+2 · x 2 + u p,q n+3 · y+ +u p,q n+4 · xy + u p,q n+5 · x 2 y + u p,q n+6 · y 2 + u p,q n+7 · xy 2 + u p,q n+8 · x 2 y 2 . Remark 3.7. Let U p,q n be the n-th (l, 1, p + 2q, q · l) − symbol element. Then, we have: U p,q n = 0 if and only if p = q = 0. The proof of this remark is similar to the proof of Remark 3.6. With proof ideas similar to those in the Theorem 3.5 from the paper [Fl,Sa;15], Proposition 5.4 from the paper [Fl,Sa;17], Proposition 3.7. from the paper [Fl,Sa;18], we obtain the following results: Proposition 3.8. Let l be a nonzero natural number and let M 1 be the set M 1 = n i=1 l 2 + 4 U pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n ∪ {1} . Then M 1 is an order of the quaternion algebra H Q (α, β) . Proposition 3.9. Let l be a nonzero natural number and let M 2 be the set M 2 = n i=1 l 2 + 4 U pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n ∪ {1} . Then M 2 is an order of the symbol algebra A = α1,α2 K,ǫ . Since the proofs of Proposition 3.8 and Proposition 3.9 are similar, we only prove one of them. Proof of Proposition 3.9. We prove that M 2 is a free Z− submodule of rank 9 of the symbol algebra A = α1,α2 K,ǫ . According to Remark 3.7, U 0,0 n = 0∈O. Let n, m ∈ N * , p, q, p ′ , q ′ , c, d ∈ Z. We have: cu p,q n + du p So, M 2 is a free Z− submodule of rank 9 of the symbol algebra A. We consider the set M 3 = n i=1 l 2 + 4 u pi,qi ni |n ∈ N * , p i , q i ∈ Z, (∀)i = 1, n . We are proving that M 2 is a subring of the symbol algebra A. Taking into account the relation (5.1.) , it is enough to prove that l 2 + 4 U p,q n l 2 + 4 U p ′ ,q ′ m ∈M 2 . For this, it is enough to prove that l 2 + 4 u p,q n l 2 + 4 u p ′ ,q Remark 2.4. ([La;04], [Le; 05]) , [Le; 05], [Als, Ba; 04], [Vi; 80], [Vo; 10], ],[Ha; 12],[Ho; 63],[Ja, Ya;13],[Ka,Ha; 17], [Li; 12], [Ra; 15], [Sa, Fa, Ci; 09], [Sa; 14 (1)], [Sa; 16], [Sa; 16 (1)], [Sa; 17], [Sa; 17 (1)], [Sw; 73], [Ta; 13]. Many mathematicians studied the Fibonacci numbers, Lucas numbers, Pell numbers, Pell-Lucas numbers, generalized Fibonacci-Lucas numbers, the generalized Pell-Fibonacci-Lucas numbers, Fibonacci polynomials, Jacobsthal-Lucas polynomials, Fibonacci quaternions, the generalized Fibonacci-Lucas quaternions, the generalized Pell-Fibonacci-Lucas quaternions (see [Ho; 63], [Ca; 15], [Ca, Mo; 16], [Ca; 16], ′ m ∈M 3 . Let m, n be two integers, n < m. We calculate:Using Proposition 3.1, we have:Applying the definition of the sequence (u n ) n≥0 and Remark 3.5, we obtain:So, l 2 + 4 u p,q n · l 2 + 4 u p ′ ,q ′ m ∈M 3 . It results that M 2 is an order of the symbol algebra A = α1,α2 K,ǫ . Fibonacci generalized quaternions. Ak, ; M Ko, H H Akyigit, M Koksal, Tosun, Adv. Appl. Clifford Algebras. 243Ak, Ko, To; 14] M. Akyigit, HH. Koksal, M. Tosun, Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras 2014, 24(3), 631-641. Quaternion Orders, Quadratic Forms and Shimura Curves. Al, ; V Go, N M Alexandru, P Gosoniu ; M. Alsina, ; P Bayer, ; P Catarino, M L Catarino, Morgado, On Generalized Jacobsthal and Jacobsthal-Lucas polynomials. Als, Ba; Ca, MoAmerican Mathematical Society99publisher Bucharest UniversityChaos, Solitons and FractalsAl, Go; 99] V. Alexandru, N.M. Gosoniu, Elements of Number Theory (in Ro- manian), publisher Bucharest University, 1999. [Als, Ba; 04] M. Alsina, P. Bayer, Quaternion Orders, Quadratic Forms and Shimura Curves, CRM Monograph Series, 22, American Mathematical Society, 2004. [Ca; 15] P. Catarino, A note on h (x) Fibonacci quaternion polynomials, Chaos, Solitons and Fractals, 77(2015), 1-5. [Ca, Mo; 16] P. Catarino, M. L. 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Ciobanu, Some properties of the symbol algebras, Carpathian Journal of Mathematics , 25(2)(2009), 239-245 [Sa; 14] D. Savin, Fibonacci primes of special forms, Notes on Number Theory and Discrete Mathematics, vol. 20, 2014, no.2, 10-19. [Sa; 14 (1)] D. Savin, About some split central simple algebras, An. St. Univ. Ovidius Constanta, Mat. Ser., 22 (1) (2014), 263-272. D. Savin, Quaternion algebras and symbol algebras over algebraic number field K, with the degree [K : Q] even. D Savin, Carpathian Journal of Mathematics. 322Gulf Journal of Mathematics16] D. Savin, About division quaternion algebras and division symbol alge- bras, Carpathian Journal of Mathematics, vol. 32, No. 2 (2016), 233 -240. [Sa; 16 (1)] D. Savin, Quaternion algebras and symbol algebras over algebraic number field K, with the degree [K : Q] even, Gulf Journal of Mathematics, Vol 4, Issue 4 (2016), 16-21. About split quaternion algebras over quadratic fields and symbol algebras of degree n. D Savin, ; D Savin, About Special Elements in Quaternion Algebras Over Finite Fields. 27Advances in Applied Clifford Algebras17] D. Savin, About Special Elements in Quaternion Algebras Over Finite Fields, Advances in Applied Clifford Algebras, June 2017, Vol. 27, Issue 2, 1801-1813. [Sa; 17 (1)] D. Savin, About split quaternion algebras over quadratic fields and symbol algebras of degree n, Bull. Math. Soc. Sci. Math. Roumanie, Tome 60 (108) No. 3, 2017, 307-312. M N S Swamy, On generalized Fibonacci Quaternions, The Fibonacci Quaterly. 11M. N. S. Swamy, On generalized Fibonacci Quaternions, The Fibonacci Quaterly, 11(5)(1973), 547-549. A characterization of the quaternion group. M Tarnauceanu, An. StM. Tarnauceanu, A characterization of the quaternion group, An. St. Arithmetique des algebres de quaternions. M F Vigneras, J Voight, Lecture Notes in Math. 10800SpringerThe Arithmetic of Quaternion Algebras80] Vigneras, M.F., Arithmetique des algebres de quaternions, Lecture Notes in Math., no. 800, Springer, 1980. [Vo; 10] Voight, J., The Arithmetic of Quaternion Algebras, available on the au- thors website: http://www.math.dartmouth.edu/ jvoight/ crmquat/book/quat- modforms-041310.pdf, 2010. . Bd. Mamaia. 124Diana SAVIN Faculty of Mathematics and Computer Science, Ovidius UniversityDiana SAVIN Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA http://www.univ-ovidius.ro/math/ e-mail: savin.diana@univ-ovidius.ro, dianet72@yahoo.com

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{'abstract': 'In this paper we define and we study properties of (l, 1, p + 2q, q · l) − numbers, (l, 1, p + 2q, q · l) − quaternions, (l, 1, p + 2q, q · l) − symbol elements. Finally, we obtain an algebraic structure with these elements.', 'arxivid': '1712.01941', 'author': ['Diana Savin '], 'authoraffiliation': [], 'corpusid': 54624093, 'doi': '10.1007/978-3-030-00084-4_23', 'github_urls': [], 'n_tokens_mistral': 12981, 'n_tokens_neox': 10990, 'n_words': 5939, 'pdfsha': 'd9a15e988b0aead38bc9f32a3b0405d1bd5ec941', 'pdfurls': ['https://arxiv.org/pdf/1712.01941v1.pdf'], 'title': ['Special numbers, special quaternions and special symbol elements', 'Special numbers, special quaternions and special symbol elements'], 'venue': []}

arxiv

Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems 28 Aug 2014 J K Asbóth B Tarasinski P Delplace Institute for Solid State Physics and Optics Wigner Research Centre Instituut-Lorentz Hungarian Academy of Sciences P.O. Box 49H-1525BudapestHungary Laboratoire de Physique, Ecole Normale Superieure de Lyon Universiteit Leiden 47 allee dItalieP.O. Box 95062300 RA, 69007Leiden, LyonThe Netherlands, France Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems 28 Aug 2014(Dated: May 2014) In periodically driven lattice systems, the effective (Floquet) Hamiltonian can be engineered to be topological: then, the principle of bulk-boundary correspondence guarantees the existence of robust edge states. However, such setups can also host edge states not predicted by the Floquet Hamiltonian. The exploration of such edge states, and the corresponding unique bulk topological invariants, has only recently begun. In this work we calculate these invariants for chiral symmetric periodically driven one-dimensional systems. We find simple closed expressions for these invariants, as winding numbers of blocks of the unitary operator corresponding to a part of the time evolution. This gives a robust way to tune these invariants using sublattice shifts. We illustrate our ideas on the periodically driven Su-Schrieffer-Heeger model, which, as we show, can realize a discrete time quantum walk: this opens a useful connection between periodically driven lattice systems and discrete time quantum walks. Our work helps interpret the results of recent simulations where a large number of Floquet Majorana fermions in periodically driven superconductors have been found. arXiv:1405.1709v2 [cond-mat.mes-hall] Controlling the topological phases of matter is an important challenge in solid state physics. In the recent years, periodic driving has emerged as an important tool to meet this challenge. Topologically protected edge states, the hallmarks of topological phases, have been predicted and observed in periodically driven systems, such as materials irradiated by light 1-4 , in shaken optical lattices 5,6 , and in photonic crystals 7 . In the above cases, the principle of bulk-boundary correspondence 8 was applied to the effective (Floquet) Hamiltonian of the periodically driven system. The variety of topological phases that periodically driven systems can display, however, is much wider than those of their Floquet Hamiltonians, and the systematic exploration of these phases has only just begun 9 . An important example is the case of periodically driven onedimensional topological superconductors,where, the bulk Z 2 invariant is replaced by a pair of Z 2 invariants, whose calculation necessitates information beyond that represented by the Floquet Hamiltonian 10 . The edge states then are the Floquet Majorana fermions, with potential applications in quantum information processing 11 . Such states, not predicted by the bulk Floquet Hamiltonian, have also been observed in optical realization of a onedimensional quantum walk 12 . Simulations of one-dimensional periodically driven superconductors have shown that they can host a large number of Floquet Majorana fermions at their ends 13,14 . This can be explained by an extra chiral symmetry (CS) of the Floquet Hamiltonian, which prevents Majorana fermions on the same sublattice from recombining into complex fermions. Although this explanation is sufficient in some cases 13,14 , it cannot be general as it only relies on the Floquet Hamiltonian. Thus, the question is still open: what are the bulk topological invariants for periodically driven systems with CS? In this paper, we find the bulk-boundary correspondence for periodically driven one-dimensional quantum systems with chiral symmetry, building on the theory of CS in discrete-time quantum walks [15][16][17][18] . We show how CS can be ensured in a periodically driven system, whose time evolution in a period starts with a unitary operator F , by choosing an appropriate second part for the period. We show that the topological invariants predicting the number of 0 and π quasienergy end states are the winding numbers of the blocks of F in a canonical basis. Our formulas give a direct recipe to tune the topological invariants using a sublattice shift operation. We give an example of how to realize this operation in the simplest periodically driven one-dimensional Floquet insulator with CS, the periodically driven Su-Schrieffer-Heeger (PDSSH) model. We show how this model realizes a discrete-time quantum walk, and how this can be used to calculate the topological invariants of particle-hole symmetric quantum walks. Floquet formalism. We consider periodically driven single-particle lattice Hamiltonians, H(t + 1) = H(t). The long-time dynamics of H(t), i.e., over many periods, is governed by the time-evolution operator of one period, the Floquet operator U (τ ), U (τ ) = Te −i τ +1 τ H(t)dt ,(1) where T stands for time ordering. If at time τ the system is in an eigenstate |Ψ of the Floquet operator, U (τ )|Ψ = e −iε |Ψ , then at all times τ + n, for n ∈ Z, it will be in state e −inε |Ψ . In this sense, the periodically driven system acts as a stroboscopic simulator of the effective (Floquet) Hamiltonian H eff , H eff (τ ) = ilnU (τ ).(2) We fix the branch of the logarithm by restricting the eigenvalues ε of H eff , the quasienergies, to −π < ε ≤ π. The Floquet operator U (τ ), and thus also the effective Hamiltonian H eff (τ ), depend on the choice of the starting time of the period, τ . Changing τ amounts to a unitary transformation of the Floquet operator and the effective Hamiltonian (quasienergies are independent of τ ). Chiral symmetry of periodically driven systems. Ensuring CS of the periodically driven system amounts to ensuring that there is an initial time τ such that the corresponding effective Hamiltonian has CS, i.e., there is a unitary, Hermitian, and local (within a unit cell) operator Γ, that satisfies ΓH eff (τ )Γ = −H eff (τ ) ⇔ ΓU (τ )Γ = U −1 (τ ). (3) The effective Hamiltonian does not inherit CS from the instantaneous Hamiltonian, as is the case with particlehole symmetry 19 . However, CS of the periodically driven system is ensured if there is an intermediate time 0 < t 1 < 1 that splits the period into a first and second part in a special way. Let F denote the time evolution of the first part of the cycle, F = Te −i τ +t 1 τ H(t)dt .(4) The second part of the cycle has to fulfil ΓF † Γ = Te −i τ +1 τ +t 1 H(t)dt .(5) It is easy to check that in that case, not only U ≡ U (τ ), but also U ≡ U (τ ) have CS, where τ = τ + t 1 . These Floquet operators read U = ΓF † ΓF ; U = F ΓF † Γ.(6) Topological invariants of the effective Hamiltonians due to chiral symmetry. Consider a one-dimensional Floquet insulator: a long chain, with a translation invariant insulating bulk part, whose quasienergy spectrum has gaps around ε = 0 and π. If the system has CS, a local basis transformation can be performed that diagonalizes Γ, so that each lattice site has a sublattice index A or B, defined via the projectors Π A/B = (1 ± Γ)/2. We call such a basis a canonical basis. For the system to be a Floquet insulator, the number of A and B sites in each bulk unit cell has to be equal (or else the system would have flat bands at 0 or π quasienergy). We denote this number by N . In a canonical basis, the CS operator acts in each unit cell independently, as Γ = σ z ⊗ 1 N . The spectrum of an effective Hamiltonian with CS is symmetric: stationary states |Ψ of H eff with quasienergy ε = 0, π have chiral symmetric partners Γ|Ψ , that are also eigenstates with quasienergy −ε. Such states can be chosen to have equal support on both sublattices. The system can also host states |Ψ L/R with quasienergy ε = 0 or π, whose wavefunctions are expelled from the bulk to the left/right by the gaps in the bulk spectrum. These end states can be chosen to have support only on one sublattice. The effective Hamiltonians H eff and H eff have CS, as per Eqs. (6), and thus can be assigned topological invariants ν and ν . These are obtained by standard procedure 8 , whereby we first isolate the bulk part of H eff and H eff , by imposing periodic boundary conditions on the translation invariant central part of these Hamiltonians, and taking the thermodynamic limit. The bulk Hamiltonians are periodic functions of the quasimomentum k ∈ [−π, π), and, in the canonical basis, are block off-diagonal, H eff (k) = 0 h(k) h † (k) 0 .(7) Here, and later on, H eff refers to either of H eff or H eff , and similarly for U and h. The topological invariants are ν = ν[h ]; ν = ν[h ],(8) where the function ν[h] is a winding number, ν[h] = 1 2πi π −π dk d dk ln det h(k).(9) These integers cannot change under adiabatic deformation of the bulk Hamiltonians, and so are equal to the winding numbers of the flat band limits of these Hamiltonians, which are the topological invariants of Ryu et al 8 . They can be interpreted as the dimensionless bulk sublattice polarization 20 of the effective Hamiltonians, at times τ and τ . Topological invariants of the driven system. To derive the topological invariants of the periodically driven system, we start by adopting the results obtained for discrete-time quantum walks (DTQW) with CS 18 to periodically driven systems. The derivations follow very closely those of Ref. 18, and so we omit them here, but for completeness, we give details in Appendix A. As with DTQWs, also in periodically driven systems, the wavefunctions of quasienergy π end states switch sublattices as they evolve from time τ to τ , and so, neither ν , nor ν , on their own, give useful information about the number of end states (observations to the contrary in specific models 13,14 do not generalize). The winding numbers ν and ν must be combined to obtain the bulk topological invariants controlling the number of end states, ν 0 = ν + ν 2 ; ν π = ν − ν 2 .(10) We now proceed to simplify Eqs. (10), and express them using the blocks of F in the canonical basis: F (k) = a(k) b(k) c(k) d(k) .(11) Along the way, we will use simple properties of the function ν[A(k)] of Eq. (9): ν[AB] = ν[A] + ν[B] and ν[A † ] = −ν[A] , for arbitrary A(k) and B(k). There are two constraints on the winding numbers of the blocks of the Floquet operator F representing the first part of the drive cycle, both following from the unitarity of F . First, substituting Eqs. (11) directly into F (k)F (k) † = 1 gives ac † = −bd † . Taking the winding numbers of the two sides gives ν[c] − ν[a] = ν[d] − ν[b].(12) Second, F represents an operation on an open chain, terminated at its ends. Thus, the average displacement of a state in the bulk, with this average going over all possible states, has to be zero: Otherwise, unitarity of F would be violated in the end regions. This average displacement is given by the winding number of F itself 19 , which, since F is unitary, can be written as ν[F ] = 1 2πi dkTr F † (k) d dk F (k).(13) Inserting the decomposition of F in the canonical basis, Eq. (11), into ν[F ] = 0, gives ν[F ] = ν[a] + ν[c] + ν[b] + ν[d] = 0.(14) To use the relations derived above, we note, that U = e −iH eff = cos H eff − i sin H eff .(15) Because of the block off-diagonal structure of H eff , the first term in the sum above corresponds to the block diagonal and the second to the block off-diagonal parts of U . Now since sign(ε) = sign (sin ε) for ε ∈ [−π, π], the winding number of H eff is the same as that of sin H eff . Therefore, in Eq. (9) above, we can substitute the offdiagonal block of U in a canonical basis: h → iU 12 . For the topological invariants of the effective Hamiltonians H eff and H eff , using Eqs. (6), substituting the blocks of F , we obtain ν = ν[a † b − c † d] and ν = ν[−ac † + bd † ]. We can simplify these using the unitarity of F , whereby a † b + c † d = 0 and ac † + bd † = 0, and the fact that ν[αc] = ν[c] for any α ∈ C. We obtain ν = ν[b] − ν[a] = ν[d] − ν[c]; (16a) ν = ν[a] − ν[c] = ν[b] − ν[d].(16b) Inserting these equations into Eqs. (10), together with Eqs. (14) and (12), gives us ν 0 = ν[b]; ν π = ν[d].(17) These equations are the central result of our paper: In one-dimensional periodically driven systems with CS, the windings of the determinant of the off-diagonal and the diagonal blocks of the Floquet operator in a canonical basis fix the number of end states at quasienergy 0 and π, respectively. Eqs. (17) determine the topological invariant ν 0 (ν π ) even if the gap of H eff at quasienergy ε = π (ε = 0) is closed, a problem raised by Tong et al. 13 . Consider cos H eff = 1 − 2 c † c 0 0 b † b = 2 a † a 0 0 d † d − 1. (18) If there is a quasimomentum k where the gap of H eff closes around ε = 0, then cos H eff (k) has a doubly degenerate eigenvalue +1. At that k, using the first relation of Eq. (18), either c(k) or b(k) (or both) have an eigenvalue zero. This means ν 0 is not well defined, and neither are ν or ν . However, ν π of Eq. (17) is still well defined. Similarly, if at some k the gap of H eff around ε = π closes, then, using the second relation of Eq. (18), a(k) or d(k) must have an eigenvalue zero, and ν π is not well defined, but ν 0 is. Geometrical picture. In case of a two-band 1D Floquet insulator with CS, we can give a geometrical interpretation for the topological invariants ν 0 and ν π . We relegate details to Appendix B, and just summarize the results here. Disregarding an irrelevant global phase, the evolution operator for the first half of the period reads F (k) = e −i f (k) σ , with f (k) a three-dimensional real vector inside a unit sphere of radius π, all points on whose surface are identified with each other, and σ the vector of Pauli matrices. As k traverses the Brillouin zone [−π, π[, f (k) describes a directed, smooth, closed loop. If the gap around ε = 0 is open, the loop of f (k) cannot touch the z-axis or the surface of the sphere, and we find that the invariant ν 0 is given by the winding of the loop around the z axis. If the gap of H eff around ε = π is open, the path of f (k) cannot touch the circle in the xy plane of radius π/2. In that case, ν π is given by the winding of the loop around that circle. Tuning the invariants. Formulas (17) allow for a simple way to tune the topological invariants of a periodically driven system, using a unitary sublattice shift operation S(n), whose bulk part reads S(n, k) = exp(−inΓk). (19) In the bulk, S(n) displaces sites on sublattice A (B) to the right (left) by n sites. Therefore, at the left/right end, under the effect of S(n), n states must switch sublattices, transitioning B → A / A → B (if n is negative, vice versa). How this transition happens depends on the details of S(n) that have no influence on the topological invariants (nor on the number of end states). To tune the invariants of a periodically driven system, obeying Eq. (6), with some F = F (0) , insert extra sublattice shifts before and after F (0) , F (1) = S(m)F (0) S(n).(20) Substituting into Eqs. (17), we obtain directly the topological invariants of the modified driven system, Example: the periodically driven SSH model. We now illustrate the concepts introduced above on the PDSSH model, given by ν (1) 0 = ν (0) 0 + m − n; ν (1) π = ν (0) π − m − n.(21)H SSH (t) = M j=1 v(t)c 2j c † 2j−1 + w(t)c 2j+1 c † 2j + h.c.,(22) where c x annihilates the fermion on site x. For simplicity, we keep the intracell hopping amplitudes v(t) and the intercell hopping amplitudes w(t) real, homogeneous in space, and modulated periodically, with period 1. We fix open boundary conditions by identifying c 2M +1 = 0 (as opposed to periodic boundary conditions, which would require c 2M +1 = c 1 ). The sublattice shift operator S(n) can be realized 9 by the following drive sequence: a pulse of v of area π/2, followed by a pulse of w of area −π/2. This allows us to realize a discrete time quantum walk as a periodically driven lattice Hamiltonian. As a concrete example, we consider the PDSSH model on an open chain of 40 sites (M = 20 unit cells). The drive sequence, shown in Fig. 1(a), consists of a train of nine pulses, chosen to be Gaussian for numerical convenience, applied to v and w homogeneously. We ensure CS by way of Eq. (6), with t 1 = 0.5, by choosing both v(t) and w(t) to be even functions of time. We follow the recipe of Eq. (20), to realize ν 0 = −1, ν π = −2. The role of role of F (0) is played by the first half of the central Gaussian pulse, where w = 5v: thus, it is a short pulse e −iπ/2H1 , where H 1 is an SSH Hamiltonian in the topologically nontrivial phase. So, we have ν (0) 0 = 1, ν (0) π = 0. To test the robustness of the recipe, we realize the sublattice displacement S(n = 2) only approximately by allowing considerable overlaps between the π/2 area pulses of v and the −π/2 area pulses of w. We find that the bulk topological invariants and the end states agree perfectly with the theory above. The invariants are the winding numbers of the curve of Fig. 1 (b), which are ν 0 = −1, ν π = −2. Correspondingly, in the local density of states, Fig. 1 (c), at each end, we find 2 end states at ε = π, and 1 end state at ε = 0, exclusively localized on B/A sublattice at the left/right end. The time dependence of these end states, Fig. 1 (d-f), shows that that they indeed spread over both sublattices at intermediate times, but return to a single sublattice at t = 0.5. For the 0/π energy end states, this is the same/opposite sublattice as that occupied at t = 0. Since we restricted the hopping amplitudes v and w to be real, the instantaneous SSH Hamiltonian, Eq. (22), has particle-hole symmetry (PHS), represented by ΓK, where K denotes complex conjugation.The PDSSH model inherits this symmetry, and therefore, its the end states are analogous to 0 and π quasienergy Floquet Majorana fermions. If CS is violated, but PHS is maintained, only the parity of the number of the Floquet Majorana fermions at each edge and at each quasienergy 0,π is protected. There is a corresponding pair of bulk Z 2 topological invariants 10 . In the case of the PDSSH model, we can follow the construction of Jiang et al. 10 , and find that the Z 2 invariants can simply be obtained from the complete areas of the pulses of v and w. For details, see Appendix C. Outlook. The topologically protected states our theory predicts should have experimental signatures in different kinds of setups. Optical experiments, where edge states are routinely imaged directly 12,21 , are in the best position to test our predictions. Alternatively, in transport measurements, the end states should give rise to transmission resonances, similar to the ones predicted for Floquet Majorana fermions 22 . Our work leaves a couple of theoretical questions open. First, is the decomposition of the drive cycle U into F and ΓF † Γ, as per Eqs. (4-6), a necessary requirement for a periodically driven Hamiltonian to have CS? For previously studied cases 13,14 we can find such a decomposition, but if a counterexample were to be found, the theory we presented here would need to be expanded. Second, the bulk effective Hamiltonian H eff (τ, k) of a one-dimensional Floquet insulator (with or without CS) is periodic in both τ and k, and thus has a Chern number. In all the examples we examined numerically, we found this Chern number to be zero, but can it take on a nonzero value? If so, what is the physical interpretation of this number? Last, how can the topological invariants we found here be formulated in the frequency domain 9 ? This is especially an interesting question, as previous work on the PDSSH model using this approach ? has not detected the pair of topological invariants we found. We thank J. Dahlhaus, J. Li, A. Gábris and J. Edge, for useful discussions. PD acknowledges useful comments from A. Bernevig. This research was realized in the frames of TAMOP 4.2.4. A/1-11-1-2012-0001 "National Excellence Program -Elaborating and operating an in-land student and researcher personal support system", subsidized by the European Union and co-financed by the European Social Fund. This work was also supported by the Hungarian National Office for Research and Technology under the contract ERC HU 09 OPTOMECH and the Hungarian Academy of Sciences (Lendület Program, LP2011-016). This research was supported by the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientific Research (NWO/OCW). state at the other special time τ , |Ψ = F |Ψ . This is an eigenstate of U with the same quasienergy ε. This state is also on one sublattice only, because ΓF |Ψ = ΓF Γe iγ |Ψ = ΓF Γe i(γ−ε) ΓF −1 ΓF Ψ = e i(γ−ε) F |Ψ . So |Ψ is on the same (opposite) sublattice as |Ψ if ε = 0 (ε = π). This can be written succintly as n A,π − n B,π = n B,π − n A,π = 0; (A2a) n A,0 − n A,0 = n B,0 − n B,0 = 0. Using Eqs. (A2) to simplify ν + ν and ν − ν from Eqs. (A1), we obtain ν 0 = ν + ν 2 ; ν π = ν − ν 2 ,(A3) which are Eqs. (10) we set out to demonstrate. We find that for the PDSSH model, the invariant of Jiang et al. 10 can be given by simple closed formulas. At the momenta k = 0 and k = π, the Hamiltonians at different times all commute with each other, and therefore, all that matters is the total area under the v and w pulses, V = 1 0 v(t)dt; W = 1 0 w(t)dt.(C2) A short calculation gives Q 0 = sgn sin V + W 2 sin V − W 2 ; (C3) Q 0 Q π = sgn sin(V + W ) sin(V − W ) .(C4) FIG. 1 : 1Floquet eigenstates of a periodically driven SSH chain of 40 sites. (a) Time dependence of the intracell (continuous) and intercell (dotted) hopping amplitudes. (b) The curve f (k), which winds -1 times around the z axis (red) and -2 times around the circle of radius π/2 on the xy plane, showing that ν0 = −1 and νπ = −2. (c) Local Density of States of the effective Hamiltonian H eff (0). (d) Time evolution of the position distribution | Ψ(t)|x | 2 of the single end state with ε = 0, and (e,f) of two orthogonal end states with ε = π. FIG. 2 : 2Effect of breaking CS by time-shifting the pulse of the intracell hopping v(t) with respect to the other pulses. (a) In the PDSSH model, the extra PHS protects the end states at ε = 0. (b) In the PDRM model, there is no PHS, and all end state energies are affected by the time shift. L. N. H., G. Refael, and V. Galitski, Nature Physics 7, 490495 (2011). 2 B. Dóra, J. Cayssol, F. Simon, and R. Moessner, Phys. Rev. Lett. 108, 056602 (2012), URL http://link.aps. org/doi/10.1103/PhysRevLett.108.056602. 3 T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev. B 84, 235108 (2011), URL http://link.aps. org/doi/10.1103/PhysRevB.84.235108. 4 Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and Appendix A: Derivation of Eqs.(10)To derive Eqs.(10), we follow closely the line of thought of Ref.18. We consider an open, periodically driven chain with CS, which has one bulk and two ends. Let n A/B,0/π denote the number of end states at the left end on the A/B sublattice at quasienergy 0/π of the Hamiltonian H eff , and n A/B,0/π the corresponding quantities for H eff . The bulk-boundary correspondance for the effective Hamiltonians H eff and H eff reads ν = n A,0 − n B,0 + n A,π − n B,π ;(A1a) ν = n A,0 − n B,0 + n A,π − n B,π .Topologically protected end states of periodically driven one-dimensional lattices with CS can be divided to two classes: a), they have quasienergy 0 and are on the same sublattice at τ and τ , or b) have quasienergy π and are on opposite sublattices. Indeed, consider a topologically protected end state |Ψ , which is an eigenstate of U with eigenvalue e −iε , with ε ∈ {0, π}. It is only on a single sublattice: Γ|Ψ = e −iγ |Ψ , with γ = 0/π corresponding to sublattice A/B. Now consider the same end Appendix B: Geometrical picture For a two-band 1D Floquet insulator with CS, we can give a direct geometrical picture for the topological invariants ν 0 and ν π . Since the global phase cannot wind (F cannot have quasienergy winding), it can safely be disregarded, and the evolution operator for the first half of the period then reads F (k) = e −i f (k) σ . Here f is a 3-dimensional vector, of magnitude f ∈ [0, π] and σ the vector of Pauli matrices. The k-dependent vector f (k) is restricted inside a spherical ball of radius π, with all points on the surface identified with each other. The a, b, c, d in Eq.(11)are just complex number valued functions of k,using spherical coordinates. As k traverses the Brillouin zone, f (k) describes a directed, smooth, closed loop, that can at some k exit the ball at a point on the surface and reenter at the same k at the antipodal point. If the gap around ε = 0 is open, the loop of f (k) cannot touch the z-axis, nor the surface of the sphere. Thus, the loop has a well defined winding number around the z axis,Since both f (k), θ(k) ∈]0, π[ for all k, this is the same as the winding number ν 0 obtained by substituting (B1) into Eq.(17). The gap of H eff around ε = π closes when f (k) is on the circle on the n z = 0 plane of radius π/2 (n z = 0 and f = π/2). Thus, if the gap around ε = π is open, the loop of f (k) has a well defined winding number around that circle. To calculate this winding number, first discard the φ information, by setting φ = 0. This transforms the 3D closed path of f (k) into a 2D path in a semicircle, with the points on the circular boundary with the same x coordinate identified. We need the winding of this path around the single point, f = π/2, n z = 0. This is found by deforming the semicircle yet again, by the transformation (f sin θ, f cos θ) → (cos f, sin f cos θ), into circle, into whose origin the point f = π/2, n z = 0 is mapped. The winding number is thenwhich is the same as ν π obtained by substituting Eq. (B1) into Eq. (17b).The PDSSH model, Eq. (22), has particle-hole symmetry (PHS), represented by ΓK, where K stands for complex conjugation. This antiunitary symmetry is inherited by the effective Hamiltonian from the instantaneous Hamiltonian19.If we break CS in the PDSSH model, an end state can remain protected if it can have no PHS partner. This happens whenever the number of end states at a given energy and at a given end is odd: then, after breaking CS, a single end state is still protected by PHS. We illustrate this on the PDSSH model. If we break CS by delaying the intracell hopping amplitude v by δt with respect to the intercell hopping w pulses, as shown inFig. 2(a), the lone end state at ε = 0 is still topologically protected, while the pair of end states at ε = π hybridize and move away from the edge of the energy Brillouin zone (except for a time shift of 0.5, where the conditions for CS are again fulfilled). To break PHS, we can add a sublattice potential to the SSH model, obtaining the periodically driven Rice-Mele (PDRM) model,Now, CS still holds if in addition to v(t) and w(t) being even functions of time, u(t) is odd: u(t) = −u(−t). We choose u(t) = sin(2πt). This time, if we break CS by shifting the v(t) pulse in time with respect to the w(t) and u(t) pulses, as shown inFig. 2(b), all end states move away from their original energies (again except for the time shift of 0.5).The extra PHS of the PDSSH model brings with it an extra pair of bulk topological invariants, (Q 0 , Q π ) ∈ Z 2 ×Z 2 , which predict the number of end states protected by PHS at 0 and π energy. If we have CS, the invariants are just Q ε = ν ε mod 2; if CS is broken, however, they can only be obtained by a procedure involving analytic continuation based on the full cycle H(t), as found by Jiang et al.10. The PDSSH model, besides being the simplest periodically driven topological insulator, also gives a lattice realization of the discrete time split-step quantum walk. For the quantum walk, we need to define the basis states |R/L, x , for coin state predicting the next step right/left, and the walker at position x. These basis states are identified with states on the SSH chain asThe basic operations of the split-step walk are rotations of the internal state of the walker, R(θ) = e −iθσy , and shifts of the R/L internal state to the right/left, given by S ± = e −ik(σz±1) . One timestep of the split-step walk is defined asA pulse of v of area V followed by a pulse of w of area W , in the basis of Eq. D2, can be written as U = e −iW (cos kσy−sin kσx) e −iV σy , (D4) which reproduces the timestep of the split-step walk with the anglesThe above mapping is important as it allows us to apply results about the topological phases of periodically driven systems to quantum walks.As an example, consider the invariants due to CS, via Eqs.(17), for the simple quantum walk, given by U = S − S + e −iθσy . According to the mapping above, the winding numbers are ν 0 = ν[−i(s + ce ik ], ν π = ν[c − se −ik ], with c = cos(π/4 + θ/2), s = sin(π/4 + θ/2). 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{'abstract': 'In periodically driven lattice systems, the effective (Floquet) Hamiltonian can be engineered to be topological: then, the principle of bulk-boundary correspondence guarantees the existence of robust edge states. However, such setups can also host edge states not predicted by the Floquet Hamiltonian. The exploration of such edge states, and the corresponding unique bulk topological invariants, has only recently begun. In this work we calculate these invariants for chiral symmetric periodically driven one-dimensional systems. We find simple closed expressions for these invariants, as winding numbers of blocks of the unitary operator corresponding to a part of the time evolution. This gives a robust way to tune these invariants using sublattice shifts. We illustrate our ideas on the periodically driven Su-Schrieffer-Heeger model, which, as we show, can realize a discrete time quantum walk: this opens a useful connection between periodically driven lattice systems and discrete time quantum walks. Our work helps interpret the results of recent simulations where a large number of Floquet Majorana fermions in periodically driven superconductors have been found. arXiv:1405.1709v2 [cond-mat.mes-hall]', 'arxivid': '1405.1709', 'author': ['J K Asbóth ', 'B Tarasinski ', 'P Delplace ', '\nInstitute for Solid State Physics and Optics\nWigner Research Centre\nInstituut-Lorentz\nHungarian Academy of Sciences\nP.O. Box 49H-1525BudapestHungary\n', '\nLaboratoire de Physique, Ecole Normale Superieure de Lyon\nUniversiteit Leiden\n47 allee dItalieP.O. Box 95062300 RA, 69007Leiden, LyonThe Netherlands, France\n'], 'authoraffiliation': ['Institute for Solid State Physics and Optics\nWigner Research Centre\nInstituut-Lorentz\nHungarian Academy of Sciences\nP.O. Box 49H-1525BudapestHungary', 'Laboratoire de Physique, Ecole Normale Superieure de Lyon\nUniversiteit Leiden\n47 allee dItalieP.O. Box 95062300 RA, 69007Leiden, LyonThe Netherlands, France'], 'corpusid': 119230103, 'doi': '10.1103/physrevb.90.125143', 'github_urls': [], 'n_tokens_mistral': 11563, 'n_tokens_neox': 9917, 'n_words': 5913, 'pdfsha': '23c1d96ce3297a4328ba37fb25d27ab162d07501', 'pdfurls': ['https://arxiv.org/pdf/1405.1709v2.pdf'], 'title': ['Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems', 'Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems'], 'venue': []}

arxiv

A Modular Test Bed for Reinforcement Learning Incorporation into Industrial Applications ⋆ Simon Hirländer simon.hirlaender@plus.ac.at Paris Lodron University Salzburg SalzburgAustria Stefan Wegenkittl stefan.wegenkittl@fh-salzburg.ac.at Salzburg University of Applied Sciences SalzburgAustria A Modular Test Bed for Reinforcement Learning Incorporation into Industrial Applications ⋆ Reinforcement Learning · Industry 40 · OPC UA This application paper explores the potential of using reinforcement learning (RL) to address the demands of Industry 4.0, including shorter time-to-market, mass customization, and batch size one production. Specifically, we present a use case in which the task is to transport and assemble goods through a model factory following predefined rules. Each simulation run involves placing a specific number of goods of random color at the entry point. The objective is to transport the goods to the assembly station, where two rivets are installed in each product, connecting the upper part to the lower part. Following the installation of rivets, blue products must be transported to the exit, while green products are to be transported to storage. The study focuses on the application of reinforcement learning techniques to address this problem and improve the efficiency of the production process. Introduction Schäfer et al. stress how Reinforcement Learning (RL) can be used to overcome demands posed by the concepts of Industry 4.0, including shorter time-to-market, mass customization of products, and batch size one production proposing an Operational Technology (OT)-aware RL architecture in [8]. RL is an important machine learning paradigm for Industry 4.0 because it has the potential to surpass human level performance in various complex tasks [6] and does not require pre-generated data in advance. RL agents can learn optimal policies for executing control tasks, potentially leading to productivity maximization and cost reduction [3]. RL agents can also explore their environment to generate new data, which is particularly useful in environments where data is scarce. [9]. Based on a quantitative text analysis and a qualitative literature review, Hermann et al. [2] identified the following four Industry 4.0 design principles, which we are addressing with the created test bed: (i) Decentralized Decisions, (ii) Technical Assistance, (iii) Interconnection and (iv) Information Transparency. (i) and (ii) describe the interconnection of objects and people which allows for decentralized decision-making in Industry 4.0 enabled by Cyber-Physical Systems (CPSs). Meanwhile, humans' role is shifting towards strategic decision-making and problem-solving, supported by assistance systems and physical support by robots. Including RL into the aforementioned setting allows for incorporation of both principles. (iii) and (iv) address the increasing number of interconnected objects and people in the Internet of Everything (IoE) which enables collaborations and information transparency, but also requires common communication standards and cybersecurity, and relies on context-aware systems for appropriate decision-making based on real-time information provision. Combining Open Platform Communication Unified Architecture (OPC UA) [5] with the standard RL setting, as we are proposing in [8], enables the support for design principles (iii) and (iv). Test Bed The case study presented by [7] and extended by [1] aims to simulate a real production system using a model factory. The model factory, depicted in Fig. 1, comprises five modules: entry, rotary table, assembly station, storage, and exit. The entry storage of the model factory stores three different types of parts, namely the transport carriage, lower part of the product, and upper part of the product. The transport carriage is used for moving goods on the conveyor belts, while the lower and upper parts of the product are required for product assembly. The rotary table, which serves as a pivotal element in the model factory, can transport goods from the entry to the storage unit or from the entry to the assembly station. At the assembly station, rivets must be added to the product, and during the insertion process, the conveyor must remain stationary. The assembled products come in different randomly assigned colors. The factory is also equipped with several sensors that track the products on their carriages as they move through the facility, and can recognize the color and presence of rivets in a product. The model is designed to be representative of a real production system and includes several important aspects such as transportation of goods using conveyor belts and a rotary table, product modification through the assembly station, and storage of goods in both entry point and the main storage unit. Moreover, it represents a modular production plant consisting of multiple Programmable Logic Controllers (PLCs) of different manufacturers. OPC UA based RL-OT Integration Addressing the principles (iii) and (iv) introduced in Sect. 1, [8] proposes an OPC UA based architecture for RL in the context of industrial control systems. OPC UA allows communication between various industrial devices, including both real and simulated devices. The proposed architecture uses OPC UA nodes to extend the standard RL setting. The mapping of the RL action and state space with the OPC UA address space is performed by the RL mapper. The agent's action is turned into an OPC UA call using a function that sets the corresponding actuators using the client-server model. The environment is notified of each sensor change using the PubSub mechanism, mapping OPC UA sensor nodes to RL states. After each sensor change, a reward evaluation is triggered and a state space transition occurs. The paper also discusses how custom object types can be created for nodes accessible to the RL agent to automate the mapping between the RL action and state space with the OPC UA address space. This architecture allows for a seamless integration of different out-of-the-box implementations of RL agents into OT systems. Integration of Different RL Agents Kozlica et al. present a simulation environment for a production line in which an agent controlled by RL algorithms transports and assembles goods [4]. The goal is to test the performance of different RL algorithms and their ability to solve the given task. Two different reward functions are defined and compared. The first reward function only focuses on correctly assembling and sorting the products, with no penalty for a simple transition. The second reward function also considers the number of transitions needed for task completion. Negative rewards are assigned to collisions, incorrectly sorted products, and invalid transitions. A positive reward is assigned for completing the task. Moreover, two different RL algorithms are compared: Deep Q-Learning (DQN) and Proximal Policy Optimization (PPO). The results show that both agents can learn to solve the production line task, with the PPO agent generally outperforming the DQN agent in terms of task completion and reward. The authors conclude that the presented simulation environment is suitable for testing and comparing different RL algorithms for the production line task. Conclusion Our proposed test bed aims to demonstrate how RL can be used to address the demands posed by Industry 4.0, particularly in relation to the four design principles decentralized decisions, technical assistance, interconnection, and information transparency. The test bed consists of a modular production plant and is used for different research based scenarios. In this paper, the authors have shown how to fulfill the Industry 4.0 design principles by incorporating an OPC UA information model in the general RL setting. Additionally, it was shown, that different already available RL agent implementations can be used for solving the defined sorting task. Fig. 1 . 1Model factory placed in the Smart Factory Lab 3 . Additionally, RL agents can exploit their environment to detect unexpected behavior ⋆ Reuf Kozlica and Simon Hirländer are supported by the Lab for Intelligent Data Analytics Salzburg (IDA Lab) funded by Land Salzburg (WISS 2025) under project number 20102-F1901166-KZP. Georg Schäfer is supported by the JRC ISIA project funded by the Christian Doppler Research Association.arXiv:2306.01440v1 [cs.AI] 2 Jun 2023 early on, supporting the creation of more realistic digital representations of the environment. For more information on RL refer to Sutton & Barto https://its.fh-salzburg.ac.at/forschung/smart-factory-lab/ Strategies for developing a supervisory controller with deep reinforcement learning in a production context. J Harb, S Riedmann, S Wegenkittl, 10.1109/CCTA49430.2022.99660862022 IEEE Conference on Control Technology and Applications (CCTA). Harb, J., Riedmann, S., Wegenkittl, S.: Strategies for developing a supervisory con- troller with deep reinforcement learning in a production context. In: 2022 IEEE Conference on Control Technology and Applications (CCTA). pp. 869-874 (2022). https://doi.org/10.1109/CCTA49430.2022.9966086 Design principles for industrie 4.0 scenarios. M Hermann, T Pentek, B Otto, 10.1109/HICSS.2016.48849th Hawaii International Conference on System Sciences (HICSS). Hermann, M., Pentek, T., Otto, B.: Design principles for industrie 4.0 scenarios. In: 2016 49th Hawaii International Conference on System Sciences (HICSS). pp. 3928-3937 (2016). https://doi.org/10.1109/HICSS.2016.488 Reinforcement learning in robotics: A survey. J Kober, J A Bagnell, J Peters, The International Journal of Robotics Research. 3211Kober, J., Bagnell, J.A., Peters, J.: Reinforcement learning in robotics: A survey. The International Journal of Robotics Research 32(11), 1238-1274 (2013) Deep q-learning versus proximal policy optimization: Performance comparison in a material sorting task. R Kozlica, S Wegenkittl, S Hirländer, submitted to 32nd International Symposium on Industrial Electronics (ISIE). Kozlica, R., Wegenkittl, S., Hirländer, S.: Deep q-learning versus proximal policy optimization: Performance comparison in a material sorting task, submitted to 32nd International Symposium on Industrial Electronics (ISIE). OPC Unified Architecture. W Mahnke, S H Leitner, M Damm, Springer Science & Business MediaMahnke, W., Leitner, S.H., Damm, M.: OPC Unified Architecture. Springer Science & Business Media (2009) A review on reinforcement learning: Introduction and applications in industrial process control. R Nian, J Liu, B Huang, Computers & Chemical Engineering. 139106886Nian, R., Liu, J., Huang, B.: A review on reinforcement learning: Introduction and applications in industrial process control. Computers & Chemical Engineering 139, 106886 (04 2020) Timed coloured petri net simulation model for reinforcement learning in the context of production systems. S Riedmann, J Harb, S Hoher, B A Behrens, A Brosius, W G Drossel, W Hintze, S Ihlenfeldt, Nyhuis, P.Springer International PublishingChamProduction at the Leading Edge of TechnologyRiedmann, S., Harb, J., Hoher, S.: Timed coloured petri net simulation model for reinforcement learning in the context of production systems. In: Behrens, B.A., Brosius, A., Drossel, W.G., Hintze, W., Ihlenfeldt, S., Nyhuis, P. (eds.) Production at the Leading Edge of Technology. pp. 457-465. Springer International Publishing, Cham (2022) An architecture for deploying reinforcement learning in industrial environments. G Schäfer, R Kozlica, S Wegenkittl, S Huber, Computer Aided Systems Theory -EUROCAST 2022. Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A.ChamSpringer Nature SwitzerlandSchäfer, G., Kozlica, R., Wegenkittl, S., Huber, S.: An architecture for deploying reinforcement learning in industrial environments. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) Computer Aided Systems Theory -EUROCAST 2022. pp. 569-576. Springer Nature Switzerland, Cham (2022) Reinforcement learning: An introduction. R S Sutton, A G Barto, MIT press2nd edn.Sutton, R.S., Barto, A.G.: Reinforcement learning: An introduction. MIT press, 2nd edn. (2018)

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{'abstract': 'This application paper explores the potential of using reinforcement learning (RL) to address the demands of Industry 4.0, including shorter time-to-market, mass customization, and batch size one production. Specifically, we present a use case in which the task is to transport and assemble goods through a model factory following predefined rules. Each simulation run involves placing a specific number of goods of random color at the entry point. The objective is to transport the goods to the assembly station, where two rivets are installed in each product, connecting the upper part to the lower part. Following the installation of rivets, blue products must be transported to the exit, while green products are to be transported to storage. The study focuses on the application of reinforcement learning techniques to address this problem and improve the efficiency of the production process.', 'arxivid': '2306.01440', 'author': ['Simon Hirländer simon.hirlaender@plus.ac.at \nParis Lodron University Salzburg\nSalzburgAustria\n', 'Stefan Wegenkittl stefan.wegenkittl@fh-salzburg.ac.at \nSalzburg University of Applied Sciences\nSalzburgAustria\n'], 'authoraffiliation': ['Paris Lodron University Salzburg\nSalzburgAustria', 'Salzburg University of Applied Sciences\nSalzburgAustria'], 'corpusid': 259064274, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3160, 'n_tokens_neox': 2789, 'n_words': 1754, 'pdfsha': '5e79dcd3dee9ecff84797d8a4f5678e0b67ece0f', 'pdfurls': ['https://export.arxiv.org/pdf/2306.01440v1.pdf'], 'title': ['A Modular Test Bed for Reinforcement Learning Incorporation into Industrial Applications ⋆', 'A Modular Test Bed for Reinforcement Learning Incorporation into Industrial Applications ⋆'], 'venue': []}

arxiv

DIFFUSION-LIMITED AGGREGATION AS BRANCHED GROWTH Thomas C Halsey Department of Physics The James Franck Institute The University of Chicago 5640 South Ellis Avenue Chicago60637Illinois DIFFUSION-LIMITED AGGREGATION AS BRANCHED GROWTH arXiv:cond-mat/9401077v1 31 Jan 1994 1/19/946460A68700520 I present a first-principles theory of diffusion-limited aggregation in two dimensions. A renormalized mean-field approximation gives the form of the unstable manifold for branch competition, following the method of Halsey and Leibig [Phys. Rev. A 46, 7793 (1992)]. This leads to a result for the cluster dimensionality, D ≈ 1.66, which is close to numerically obtained values. In addition, the multifractal exponent τ (3) = D in this theory, in agreement with a proposed "electrostatic" scaling law. Diffusion-limited aggregation (DLA) is a model of pattern formation in which clusters grow by the accretion of successive random walkers. 1 Each random walker arrives from infinity, and sticks to the growing cluster at whichever surface point it first contacts. Only after the accretion of a walker does the next walker commence its approach to the cluster. The clusters thereby obtained are fractal in all dimensionalities d > 1, and are qualitatively and/or quantitatively similar to patterns observed in such diverse phenomena as colloidal aggregation, electrodeposition, viscous fingering, and dielectric breakdown. 2 At the heart of the problem of diffusion-limited aggregation is the following question: what is the relationship between the scale-invariance of the diffusive growth process and the hierarchical structure of the clusters generated by this process? 3 A preliminary, and incomplete, answer to this question was provided by this author in collaboration with M. Leibig. 4 In this work, it was hypothesized that the quantitative process by which one branch screens, i.e., takes growth probability from, a neighboring branch, has a specific form, independent of the length scale on which this process takes place. This assumption allows the development of a qualitatively correct theory, which yields multifractal scaling of growth probability, as well as agreement with a phenomonological scaling law, the "Turkevich-Scher" law, relating the scaling of the maximum growth probability over all sites on the cluster to the dimension of the cluster as a whole. 5 In this letter, I shall present a more complete and a priori theory of diffusionlimited aggregation in two dimensions based upon a specific mean-field calculation of the dynamics of branch competition. Because the mean-field approximation is implemented on all length scales, it is perhaps better to regard this theory as an ansatz solution in the case where certain types of fluctuations on all length scales are neglected, while others are included. This specific model allows verification of all qualitative aspects of branch competition that were advanced as (reasonable) hypotheses in Ref. 4. The result obtained for the dimensionality of the cluster, D = 1.66, is within 3% of the oft-quoted value D = 1.71 obtained from the scaling of the cluster radius-of-gyration in numerical studies. An additional scaling law (the "electrostatic scaling law"), relating the multifractal exponent τ (3) of the growth measure to the dimensionality D by D = τ (3), is seen to be exact within this theory. 6 In the growth process, each particle attaches itself to a unique "parent" particle in the pre-existing cluster. Furthermore, the cluster is observed to be a branched structure, with no loops and with each particle having asymptotically zero, one or two "children", i.e. particles to whom it stands as a parent. 7 Very rarely particles have more than two children; primarily for reasons of convenience I neglect this possibility. Consider a particle with two children. Each of the two children separately, with all particles descended from each, I term a "branch". Thus these two-child particles are parents of two branches, which occupy neighboring regions of space. The total number of particles in one branch I term n 1 , and the total in the other n 2 . The total number of descendants of the parent particle is thus n b ≡ n 1 + n 2 . Now consider the next particle to accrete to the cluster. I say that this particle has a total probability p 1 to stick anywhere on the first branch, and a total probability p 2 to stick anywhere on the second branch, yielding a total probability p b ≡ p 1 + p 2 . Let us now consider the normalized quantities x = p 1 /p b and y = n 1 /n b . Clearly dn 1 /dn = p 1 , where n is the total number of particles in the cluster, and we are neglecting fluctuations of O( √ n b ). Thus y obeys the following equation of motion: dy d ln n b = x − y.(1) The right-hand side of this equation is a function only of x and y. Now x will obey an equation of the form dx d ln n b = G(x, y; n; {φ i }),(2) where {φ i } is some parameterization of all of the variables describing the structure of the cluster. In ref. that there will be a stable and an unstable direction; the eigenvalue corresponding to the latter direction we define to be ν. When a pair of branches is first created by a tip-splitting event, its initial growth up to the stage at which n b ≫ 1 is determined by complicated microscopic dynamics, which do not recognize the existence of the unstable fixed point. Thus we expect the probability that a newly created branch pair will be a distance ǫ ν from the unstable fixed point will be ρ(ǫ ≪ 1)dǫ ∝ ǫ ν−1 dǫ; we are assuming a constant probability density of branch creation near the unstable fixed point. This assumption has been specifically verified by numerical study in ref. 4. The choice of ǫ ν for this initial distance insures that position along the unstable manifold in the x − y plane can be parameterized by the variable ǫn b . It is possible to relate the eigenvalue ν to the cluster dimensionality D by the following argument. 4 Consider the strongest branch in the cluster, that obtained by always following the stronger child (with the larger values of x,y) at each branching. The total number of side-branches (or branch points) from such a branch is ∼ r, where r is the cluster radius. In order that the cluster have a dimension D > 1, a number ∼ 1 of these side branches must have a total number of particles ∼ n, the total number in the cluster. A side branch obeying this criterion must have ǫn ∼ 1, so that at that branching, both descendant branches are roughly equal in size. The probability of this happening at any particular branching is n −1 dǫ ρ(ǫ) ∝ n ν , and there are ∼ r different sidebranchings at which this might occur. Thus rn ν ∼ 1, or D = 1/ν. In order to determine g(x, y), we turn to an explicit description of the growth process. 6,8 Suppose that we parameterize the accessible surface of the cluster by arc-length s. If a particle attaches at the surface point s ′ , it thereby reduces the growth probability at all points s for which |s − s ′ | > a, where a is the particle size. This is because a certain number of the random walks that would have reached s previously are now obstructed by the new particle at s ′ . If the probability that a particle lands at s ′ is p(s ′ ), and the probability that a random walker goes from s ′ to s without contacting the surface is H(s, s ′ ), this implies that dp(s) dn = ds ′ (H(s, s ′ ) − h(s)δ(s − s ′ )) p 2 (s ′ ),(3) where we have modelled effects on the scale |s − s ′ | < a by the δ-function, the coefficient of which, h(s), is set by the conservation of the total growth probability, dsp(s) = 1. Note that in Eq. (3), two factors of p(s ′ ) appear-one corresponds to the original probability that a particle lands at p(s ′ ), the other to the potential trajectories arriving at s that are blocked by such a particle. For a ≪ |s − s ′ | ≪ an, conformal transformation shows that the function H(s, s ′ ) is given in two dimensions by the simple form 9 H(s, s ′ ) = p(s)p(s ′ ) s ′ s ds ′′ p(s ′′ ) 2 ,(4) where the integral in the denominator is the total growth probability between the points s and s ′ . It is convenient to parameterize the interface by this quantity, the "growth probability" distance between points z(s), defined by z(s ′ ) − z(s) = s ′ s ds ′′ p(s ′′ ). Then our fundamental equation becomes dp(z) dn = p(z) dz ′ 1 (z − z ′ ) 2 −h(z)δ(z − z ′ ) p 2 (z ′ ),(5) where a serves as an ultra-violet cutoff to prevent divergence of the integral, and h(z) is related to h(s) and to the function z(s); its precise form is of no interest to us. I wish to use this equation to determine the function dx/d ln n b = g(x, y). Repeated application of the chain rule yields dx d ln n b = n b p 2 b (1 − x) dp 1 dn − x dp 2 dn . (6) Consider a branch with probability p ′ and a number of particles n ′ . We suppose that this branch extends from z = 0 to z = p ′ . Eqs. (5) and (6) imply that if we can write p 2 (z) on this branch (and by extension, all other branches) as p 2 (z) = (p ′ ) 2 n ′ f (z/p ′ ),(7) where f (z) is a universal function that depends neither upon p ′ nor upon n ′ , then we will be able to write dx/d ln n b = g(x, y), with the right-hand side a function of x and y alone. Equation (7) is motivated by the fact that p 2 (z) must be proportional to (p ′ ) 2 ; the dependence on n ′ is specifically chosen to lead to an n ′ -independent g(x, y). Only if we can find a method of computing an n ′ -independent f (z) will this ansatz be justified. Thus the crux of the problem is this "branch envelope" function f (z), which represents, with the appropriate normalization, the distribution of growth probability in different regions of a branch. Now in our picture, each branch can be divided into two distinct sub-branches, which compete according to the dynamics established by g(x, y). Our central mean-field assumption is that we can compute f (z) by averaging the envelope functions f (z) of these sub-branches over the stochastic parameter ǫ appropriate to the competition of these two sub-branches. In this way we obtain the following equation: f (z) = ∞ −∞ dǫρ(ǫ) x 2 (ǫn b ) y(ǫn b ) f z x(ǫn b ) + (1 − x(ǫn b )) 2 (1 − y(ǫn b )) f 1 − z 1 − x(ǫn b ) ,(8) where x(ǫn b ) and y(ǫn b ) give the values of x and y along the unstable manifold as functions of n b and the stochastic parameter ǫ. For convenience, we are defining ρ(ǫ) for negative values of ǫ as ρ(−ǫ) = ρ(ǫ), with x(−η) = 1 − x(η), y(−η) = 1 − y(η). This leads to the relatively compact expression of Eq. (8). For large n b , this equation has a solution independent of n b , which is determined by ∞ −∞ dη|η| ν−1 x 2 (η) y(η) f z x(η) + (1 − x(η)) 2 (1 − y(η)) f 1 − z 1 − x(η) − f (z) = 0. (9) Since the integrand goes to zero as η → ∞, we are justified in taking the small ǫ form for ρ(ǫ). Of course, in order to perform this integral, we must have the form of the unstable manifold, and thus we must already know g(x, y). We can determine g(x, y) from f (z) by simply integrating Eq. (5) over the appropriate intervals. We do not integrate over regions exterior to the two competing branches, but only investigate the influence of the two branches on one another. Skipping some tedious algebra, we may express the result as follows. Defining a function ψ(u) by ψ(u) = 1 0 dz 1 z − 1 z + u f (z),(10) we can write g(x, y) =x(1 − x) 2 x y ψ(∞) − (1 − x) 2 (1 − y)x ψ x 1 − x − 2 1 − x 1 − y ψ(∞) − x 2 y(1 − x) ψ 1 − x x .(11) The reader should note that we have a circular procedure, because g(x, y) is determined as a function of f (z) by Eqs. (10) and (11) In addition, this theory automatically agrees with the electrostatic scaling law, which states that ds p(s) 3 ∝ n −1 ,(12) where the integral is over the entire cluster surface. This is equivalent to the more usual statement that τ (3) = D. In ref. 4, we demonstrated that the multifractal exponents σ(q) defined by ds p(s) q ∝ n −σ(q) can be obtained from the integral condition 11 ∞ 0 dη η ν−1 x(η) q y(η) σ(q) + (1 − x(η)) q (1 − y(η)) σ(q) − 1 = 0.(13) By integrating Eq. (9) from z = 0 to z = 1, one obtains precisely this criterion, with q = 3 and σ(q) = 1, in agreement with the electrostatic scaling law. Though the electrostatic scaling law thus appears in a natural way in this theory, one should not say that it is predicted by this theory unless the solution obtained to Eq. (9) is stable. It may be that it is necessary to impose the electrostatic scaling law as a constraint to insure this stability. 10 From Figure 1, it is clear that although in some sense the unstable manifold that we have calculated is an acceptable average trajectory, the numerically 4 we assumed that by averaging the right-hand side of this equation over these parameters {φ i }, one obtains dx/d ln n b = g(x, y), where the right-hand side is now only a function of x and y. Given this function g(x, y), one has a closed system of equations describing the evolution of x and y as functions of ln n b .By symmetry, g(x, y) = −g(1 − x, 1 − y), so (x, y) = (1/2, 1/2) must be a fixed point of this process of competition between the two branches. In ref. 4, we explored the consequences of assuming that this fixed point is hyperbolic, with the unstable manifold emerging from the fixed point terminating in two stable fixed points at (x, y) = (0, 0) or (x, y) = (1, 1), these latter representing the situation in which one branch has been completely screened by the other. This assumption will be explicitly verified in the calculation below.If the central fixed point at (x, y) = (1/2, 1/2) is hyperbolic, then branch pairs which commence their existence (with n b ∼ 1) near the unstable fixed point will be quickly drawn onto the unstable manifold. Linearizing the system of equations for d(x, y)/d ln n b about the central fixed point, the hyperbolic assumption implies Figure 1 . 1, while f (z) is determined as a function of g(x, y), and in particular by the unstable manifold in the x − y plane as determined by g(x, y), by Eq.(9). Thus in practice we are looking for a solution of Eq. (9) where the functions x(η) and y(η) are implicitly determined by f (z).I have numerically obtained the unique solution to Eq. (9) under these conditions, which is displayed in the inset toFigure 1. 10 This validates our assumption regarding the scaling with n ′ in Eq. (7). The function g(x, y) determined from this function has all of the necessary qualitative features; in particular, the fixed point at (x, y) = (1/2, 1/2) is unstable and hyperbolic, and the unstable manifold leads from this point to stable fixed points at (x, y) = (0, 0) and (1, 1), as illustrated inFigure 1also shows numerical results for branch competition. The value of the unstable eigenvalue ν is ν ≈ .6020, implying that D = 1/ν ≈ 1.661, which is within 3% of the standard numerical result D ≈ 1.71. obtained trajectories do exhibit some dispersion about this average. This has significant results. The Makarov scaling law predicts that dσ(q)/dq| q=1 = 1/D 0 , 12 where D 0 is the surface fractal dimension (which according to some studies is significantly less than the radius-of-gyration dimension D.)13 My result, from Eq.(13),is dσ(q)/dq| q=1 ≈ 0.71,which is significantly different from the Makarov result. In practice, this quantity is quite sensitive to the way in which the unstable manifold approaches the stable fixed points at (x, y) = (0, 0) and (1, 1); since the numerical trajectories are quite dispersed in this region, I do not expect a good result for the Makarov scaling from a one-trajectory theory. However, the theory outlined in this letter can be easily generalized to account for the possibilty of trajectory dispersion, which may lead to better agreement with the Makarov result. Figure Caption Figure Caption 1 . 1Trajectories of branch competion in the x−y plane. The light solid trajectories are numerical results from ref. 4 for specific branch pairs in growing DLA clusters. The heavy solid line represents the unstable manifold predicted by this letter, which is quite close to the "average" numerical trajectory. The inset shows the computed branch envelope function f (z). AcknowledgementsI would like to acknowledge a stimulating discussion with L.P. Kadanoff, as well as conversations with R. Blumenfeld on a closely related topic. I am very grateful to A. Libchaber for encouragement at an early stage in this project. This work was supported by the National Science Foundation through a Presidential Young Investigator award, Grant DMR-9057156. . T A Witten, Jr , L M Sander, Phys. Rev. Lett. 471400T.A. Witten, Jr. and L.M. Sander, Phys. Rev. Lett. 47, 1400 (1981); . P Meakin, Phys. Rev. A. 271495P. Meakin, Phys. Rev. A 27, 1495 (1983). . R Brady, R C Ball, Nature. 309225R. Brady and R.C. Ball, Nature (London) 309, 225 (1984); . L Niemeyer, L Pietronero, H J Wiesmann, Phys. Rev. Lett. 521033L. Niemeyer, L. Pietronero, and H.J. Wiesmann, Phys. Rev. Lett. 52, 1033 (1984); . J Nittmann, G Daccord, H E Stanley, Nature. 314141J. Nittmann, G. Daccord, and H.E. Stanley, Nature (London) 314, 141 (1985). This question is also the focus of real-space studies such as. L Pietronero, A Erzan, C Evertsz, Phys. Rev. Lett. 61861This question is also the focus of real-space studies such as L. Pietronero, A. Erzan, and C. Evertsz, Phys. Rev. Lett. 61, 861 (1988); . X R Wang, Y Shapir, M Rubenstein, Phys. Rev. A. 1515974Physica APhysica A 151, 207 (1988), and X.R. Wang, Y. Shapir and M. Rubenstein, Phys. Rev. A 39, 5974 (1989); . J. Phys. A. 22507J. Phys. A 22, L507 (1989). . T C Halsey, M Leibig, Phys. Rev. A. 467793T.C. Halsey and M. Leibig, Phys. Rev. A 46, 7793 (1992); . T C Halsey, K Honda, unpublishedT.C. Halsey and K. Honda, unpublished. . L Turkevich, H Scher, Phys. Rev. Lett. 551026L. Turkevich and H. Scher, Phys. Rev. Lett. 55, 1026 (1985); . R Ball, R Brady, G Rossi, B R C Thompson ; T, P Halsey, I Meakin, Procaccia, Phys. Rev. Lett. 33854Phys. Rev. Lett.Phys. Rev. A 33, 786 (1986); see also R. Ball, R. Brady, G. Rossi, and B.R. Thompson, Phys. Rev. Lett. 55, 1406 (1985), and T.C. Halsey, P. Meakin, and I. Procaccia, Phys. Rev. Lett. 56, 854 (1986). . T C Halsey, Phys. Rev. Lett. 592067T.C. Halsey, Phys. Rev. Lett. 59, 2067 (1987); . Phys. Rev. A. 384749Phys. Rev. A 38, 4749 (1988). The fact that there are no loops follows from the fact that every particle has a unique parent. which is true in off-lattice versions of DLAThe fact that there are no loops follows from the fact that every particle has a unique parent, which is true in off-lattice versions of DLA. . B Shraiman, D Bensimon, Phys. Rev. A. 302840B. Shraiman and D. Bensimon, Phys. Rev. A 30, 2840 (1984); . R C Ball, M Blunt, Phys. Rev. A. 393591R.C. Ball and M. Blunt, Phys. Rev. A 39, 3591 (1989). . T C Halsey, Phys. Rev. A. 353512T.C. Halsey, Phys. Rev. A 35, 3512 (1987). The stability of this solution is a more difficult question. There appears numerically to be a single instability of the solution. which can be eliminated if one applies the electrostatic scaling law as a constraintThe stability of this solution is a more difficult question. There appears nu- merically to be a single instability of the solution, which can be eliminated if one applies the electrostatic scaling law as a constraint. For a general discussion of multifractality, see T. Vicsek, Fractal Growth Phenomena. World ScientificSingapore2nd ed.For a general discussion of multifractality, see T. Vicsek, Fractal Growth Phe- nomena, 2nd ed. (World Scientific, Singapore, 1992). . N G Makarov, Proc. London Math. Soc. 51369N.G. Makarov, Proc. London Math. Soc. 51, 369 (1985). . F Argoul, A Arneodo, G Grasseau, H Swinney, Phys. Rev. Lett. 612558In particular, D 0 ≈ 1.60 isIn particular, D 0 ≈ 1.60 is claimed by F. Argoul, A. Arneodo, G. Grasseau, and H. Swinney, Phys. Rev. Lett. 61, 2558 (1988).

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{'abstract': 'I present a first-principles theory of diffusion-limited aggregation in two dimensions. A renormalized mean-field approximation gives the form of the unstable manifold for branch competition, following the method of Halsey and Leibig [Phys. Rev. A 46, 7793 (1992)]. This leads to a result for the cluster dimensionality, D ≈ 1.66, which is close to numerically obtained values. In addition, the multifractal exponent τ (3) = D in this theory, in agreement with a proposed "electrostatic" scaling law.', 'arxivid': 'cond-mat/9401077', 'author': ['Thomas C Halsey \nDepartment of Physics\nThe James Franck Institute\nThe University of Chicago\n5640 South Ellis Avenue Chicago60637Illinois\n'], 'authoraffiliation': ['Department of Physics\nThe James Franck Institute\nThe University of Chicago\n5640 South Ellis Avenue Chicago60637Illinois'], 'corpusid': 34381487, 'doi': '10.1103/physrevlett.72.1228', 'github_urls': [], 'n_tokens_mistral': 6201, 'n_tokens_neox': 5515, 'n_words': 3530, 'pdfsha': 'd7eaaa0f066d26226f20a74ae79731c4da4516e0', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/9401077v1.pdf'], 'title': ['DIFFUSION-LIMITED AGGREGATION AS BRANCHED GROWTH', 'DIFFUSION-LIMITED AGGREGATION AS BRANCHED GROWTH'], 'venue': []}

arxiv

TEMPERATURE FLUCTUATION AND AN EXPECTED LIMIT OF HUBBLE PARAMETER IN THE SELF-CONSISTENT MODEL arXiv:gr-qc/0412036v1 8 Dec 2004 A B Morcos morcos@frcu.eun.eg Department of Astronomy National Research Institute Of Astronomy and Geophysics Helwan, CairoEgypt TEMPERATURE FLUCTUATION AND AN EXPECTED LIMIT OF HUBBLE PARAMETER IN THE SELF-CONSISTENT MODEL arXiv:gr-qc/0412036v1 8 Dec 2004 The temperature gradient of microwave background radiation (CMBR) is calculated in the Self Consistent Model. An expected values for Hubble parameter have been presented in two different cases. In the first case the temperature is treated as a function of time only ,while in the other one the temperature depends on relaxation of isotropy condition in the self-consistent model and the assumption that the universe expands adiabatically. The COBE's or WMAP's fluctuations in temperature of CMBR may be used to predict a value for Hubble parameter. Introduction The cosmic micro wave background radiation (CMBR) temperature is one of the important parameters of any cosmological model. The three characteristics of this radiation are its spectrum, spatial anisotropy and polarization. The COBE Far-Infrared Absolute Spectrophotometer (FIRAS) has determined the black body temperature of (CMBR) to be 2.728 ± 0.004 K (Keating et al. (1998)), and the COBE Differential Microwave Radiometer (DMR) experiment has detected spatial anisotropy of the (CMBR) on 10 o scales of △T T ≃ 1.1 × 10 −5 K. Ground and balloon-based experiments have detected anisotropy at smaller scales (cf. Silk, and White(1995)). Many authors referred these anisotropies to simple linear or nonlinear processing in the primordial fluctuations (cf. Hu et al. (1997), Challinor and Lasenby(1999), Melek(2002)). Recently Wilkinson Microwave Anisotropy Probe (WMAP) satellite, which is designed for precision measurement of the CMBR anisotropy on the angular scales ranging from the full sky down to several arc minutes. This ongoing mission has already provided a sharp record of the conditions in the universe from the epoch of last scattering to the present. WMAP results despite the absence of a direct dark energy interaction with our baryonic world (Rebort Caldwell and Michael Doran(2003)). Whenever Joshue et al.(2003) show that the Planck CMBR mission can be significant. In general the observational limit of the temperature fluctuations ∆T T becomes lower and lower and it reached almost 10 −6 (Keating et al. (1998)). In the present work ,the COBR or WMAP results for the temperature gradient is used to expect a value for the Hubble constant using the self-consistent model. In the next section a brief review of the self-consistent model will be given. In Section 3 timetemperature relation in the model is calculated. In Section 4 the theoretical technique for calculating the gradient of any scalar cosmic field is described. In Section 5 a lower limit for Hubble parameter is calculated. In Section 6 discussion and concluding remarks are given. 2 The Self-Consistent Model (SCM) Wanas(1989), has constructed the (SCM), a cosmological model in the frame work of Generalized Field Theory (GFT) (Mikhail and Wanas (1977)). This theory is constructed in a 4-dimensional Absolute Parallelism (Ap)-Geometry. In (1986) Wanas suggested a set of conditions to be satisfied by any geometric structure in (AP-Geometry) to be suitable for cosmological applications. This set of conditions, if satisfied, would guarantee that a geometric structure would represent, a homogeneous, isotropic, electrically neutral and non-empty universe. Wanas (1989) has used one of the AP-structures, constructed by Robertson(1932), satisfying the conditions mentioned above, to construct (SCM). The geometric structure used in that model is given in the spherical polar coordinates by, λ i µ =         √ −1 0 0 0 0 L + sinθcosφ 4R (L − cosθcosφ−4K 1 2 rsinφ) 4rR −(L − sinφ+4k 1 2 rcosθcosφ) 4rRsinθ 0 L + sinθsinφ 4R (L − cosθsinφ−4K 1 2 rcosφ) 4rR (L − cosφ−4k 1 2 rcosθsinφ) 4rRsinθ 0 L + cosθ 4R −L − sinθ 4rR K 1 2 4R         .(1) where L ± = 4 ± kr 2 , k is the curvature of the space and R(t) is an unknown function of (t) only. It is to be considered that the Riemannian space, associated with(1), is given by dS 2 =ĝ µν dx µ dx ν .(2) with the metric tensor given by,ĝ µν = i e i λ i µ λ i ν , g µν def. = i e i λ i µ λ i ν ,(3) where e i (= 1, −1, −1, −1) is Levi-Civita's indicator. Wanas(1989) got the following set of the differential equations: Ṙ 2 R 2 + 4k R 2 = 0,(4)2R R +Ṙ 2 R 2 + 4k R 2 = 0,(5) where the dots represents differentiation with respect to time (t). Integration of (4), gives immediately R =R ± 2(−k) 1 2 t,(6) whereR is a constant of integration, giving the value of the scale factor at t=0. If k takes the value zero , the SCM model will be a static empty one,and when k=+1 it will give an imaginary scale factor. So we must take k=-1 for non-static, non-empty model,and the solution (6) will take the form, R =R + 2t,(7) with k = -1. It is worth of mention that the SCM is a cosmological model fixing the the curvature constant to be -1. It is also satisfying the weak and strong energy conditions, and it is free of particle horizon( for more details see Wanas(1989), (2003)). The model is consistent with recent Supernovae observations Riess et al. (2004). The negative curvature, uniquely fixed by the model, is among recent discussed reasons of the WMAP low multi-pole anomaly (cf. Gurzadyan et al. (2003)). So, this model deserves further examination. The Time-Temperature Relation in SCM In what follows we are going to find the relation between time and temperature in SCM, we are going to assume that in the early stages of the universe, the radiations behaves as if it is coming from a black body with temperature T given by the will known relation(cf. Narlikar (1983) ) B 0 0 = a T 4 ,(8) where B µ ν is the phenomenological energy-momentum tensor, and a is the radiation constant. But the energy momentum tensor in the SCM is a geometric one, say S µ ν , that has the non-vanishing values: S 0 0 = 9k R 2 , S 1 1 = S 2 2 = S 3 3 = 3k R 2 .(9) We can assume that the geometric energy momentum tensor is related to the phenomenological one via the relation, S µ ν = HB µ ν ,(10) where H is a conversion constant equal to 8 Π G c 2 , G is the gravitational constant and c is the speed of light. If we use (7), (8), and (9), we get T = ( 9 4aH ) 1/4 ( 2 R + 2t ) 1/2 .(11) But it is well known that the relation between temperature and time depends on the type of particles filling the model and the kind of interaction between them at a certain temperature range. Thus it is more convenient to rewrite the relation(11) in the form, T = ( 9 aH γ (R + 2t) 2 ) 1/4 ,(12) where γ is a parameter depending on types of particles and their interactions. The relation (12) may be used to determine the parameterR , if all other constants are known. If it is assumed that at time t=0 the temperature of the universe is 10 12 o K, as it is usually used in the thermal literature (cf. Narlikar(1983)), the value of the parameterR is obtained from the relation (12) to be 3.7 × 10 −4 sec . Relation (12) then takes the form T = ( 9 aH γ (3.7 × 10 −4 + 2t) 2 ) 1/4 , where γ = 1.45 (cf. Narlikar (1983)). The Gradient of any Cosmic Scalar Field Melek(1992) generalized a procedure, used in meteorology, in studying the temperature gradient in the Earth's atmosphere, to study the matter density and temperature gradients in the universe. For any cosmic measurable scalar field S which can be related to the energy momentum tensor B µν , he defined the function F g , in a curved space-time with metric g µν , as : F g def. = dG dτ ,(14) where G def. = (g µν S µ S ν ) 1/2 ,(15)and S µ = ∂S ∂x µ ,(16) where S µ is a time-like covariant vector, τ is the cosmic time and µ = 0, 1, 2, 3. Melek has shown that the function F g has the form: F g = 1 G g µν S µ;σ S ν u σ ,(17) where S µ;σ is the usual covariant derivative with respect to x σ and u σ def. = dx σ dτ . The second derivative of the absolute value of the gradient of any cosmic scalar field S, with respect to the cosmic time τ , is given by: d 2 G dτ 2 = dF g dτ = 1 G {g µν u σ u α [ S µ;σα S ν + S µ;σ S ν;α ] − F 2 g }.(18) Melek(1995) applied this procedure to suggest an expression for the function F g in a spatially perturbed Fredman-Robertson-Walker cosmological model(FRW). He put a lower limit on the Hubble parameter. Melek (2000) used the same technique for (FRW) to study limits on cosmic time scale variations of gravitational and cosmological constants. Melek(2002)used the same procedure to find the primordial angular gradients in the temperature of the microwave background radiation and the density functions in the same cosmological model. In what follows we are going to use the same technique to find the gradient of microwave back ground radiation's temperature in SCM . Also, we are going to get a relation between this gradient and the value of Hubble parameter. CMBR Temperature Gradient in the SCM and Expected Limit of Hubble Parameter The metric of the Riemannian space, associated with the AP-space (1), can be written using, equations (2) and (3), as dS 2 = dt 2 − 16 R 2 L + 2 [dr 2 + r 2 dθ 2 + r 2 sinθ dφ],(19) where L + = 4 + r 2 k. Now if we follow the coordinate transformation, dt = R(t)dτ ,(20) in the metric (19),then we can write dS 2 = R 2 (t){dτ 2 − 16 L + 2 [dr 2 + r 2 dθ 2 + r 2 sinθ dφ]},(21) where R(t) is the scale factor and τ is the cosmic time. If we assume that the microwave background radiation temperature (T(t)), is our scalar field and this field varies with time only, then following (16), we can write T µ def. = d T d x µ ,(22)T 0 = d T d t d t d τ = R . T ,(23) where . T = d T d τ . Then, T µ ; ν = T µ , ν − ρ µν T ρ .(24) Now by taking into consideration that the temperature is a function of time only and using (24),we can write T 0 ; 0 = R(R .. T − . R . T ).(25) Using equations (14), (15), (16), (17), (18)and (25),taking into consideration that the CMBR is independent of the radial coordinate at any fixed cosmic time and the motion in the Universe is only due to its expansion, then we get after some straight forward calculations , F = ( .. T − . R R . T ).(26) Since the SCM has been assumed to be homogenous and isotropic then F = 0 i.e, .. T − . R R . T = 0.(27) Equation (27) T . T = H .(29) It is clear from the last equation that all the quantities on its left hand side are unmeasurable quantities till now, so if these quantities are measured by COBE, WMAP or any other satellite, the Hubble parameter is determined completely and at that moment can be fixed. As it is mentioned above, the most recently detected value of the anisotropy in the temperature of the CMBR is determined by COBE and WMAP for each 10 o . This means that it is more suitable to relax the condition of isotropy in the cosmological model used. To satisfy this aim we are going to use the spatially perturbed form of the metric of the SCM in the spherical polar coordinates. The metric (21) will take the form: dS 2 = R 2 (t){dτ 2 − 16(1 + h 1 ) L + 2 dr 2 − h 2 r dr dΩ − r 2 (1 + h 3 ) dΩ 2 },(30) where Ω is the solid angle defined in terms of θ and φ as dΩ 2 = dθ 2 + sin 2 θ dφ 2 and h 1 , h 2 and h 3 are small spatial perturbations. If we use the metric (30) taking into our consideration that the homogeneity is valid (i.e ∂ T ∂ r = 0 , and ∂ 2 T ∂ t ∂ r = 0 ), and assuming that the expansion is the only motion in the universe,then this expansion affects the temperature. If we assume now that the temperature of the CMBR is a function of the cosmic time and direction i.e T (t, Ω) and one follows the same procedure as before, then equation (17) takes the form F = ( 1 G R(t) )( . T )( .. T − . R R . T ) − ( 1 − h 2 r 2 )(T ′ )( ∂T ′ ∂ t − . R R T ′ ),(31) where T ′ def. = ∂ T ∂ Ω . If we write now the metric of the SCM , which is homogenous and isotropic,in the form dS 2 = R 2 (t){dτ 2 − 16 L + 2 dr 2 − r 2 dΩ 2 },(32) then, after some straight forward calculations, we can find the temporal variation of the magnitude of the gradient of T as F SCM = ( 1 G R(t) )( . T )( .. T − . R R . T ) − ( 1 r 2 )(T ′ )( ∂T ′ ∂ t − . R R T ′ ).(33) This gradient will be zero if the model is homogeneous and isotropic. From equations (31) and (33), assuming that the universe expands adiabatically, we get ( ∂T ′ ∂ t − . R R T ′ ) = 0.(34) Since H = . R R , then the Hubble parameter can be written as H = ( ∂T ′ ∂ t ) / T ′ .(35) Since COBE, WMAP and other space and ground based measurements have detected and confirmed anisotropy in the temperature of the CMBR, this means that the right hand side quantities of the equation (35), can be measured easily fixing the value of the Hubble parameter. Discussion and Concluding Remarks Using theAP-structure (1), Wanas(1989) has got a unique pure geometric world model . This model is non-empty and has no particle horizons. This model fixes a value for k(= −1) i.e. it has no flatness problem, and as it is clear from equation (7), it has no singularity at t=0. A further advantage of using pure geometric theories is that one did not need to impose any condition from outside the geometry used (e.g. equation of state) in order to solve the field equations (Wanas (1986(Wanas ( ) , (1989). In the present work the generalized procedure for studying gradients, which has been used by Melek (1992), is used to find the temperature gradient in the SCM. It is shown that when it is assumed that the CMBR temperature is a function of time only, the Hubble parameter (H) is given by (22). But all the quantities on the right hand side are nonmeasurable, so this relation can not determine the numerical value of H except for a satellite or a ground based observation arises the gradient of temperature and its rate of change with respect to time. When the isotropy condition in the self-consistent model is relaxed and the universe is assumed to expand adiabatically, the Hubble parameter is given by the relation (35). The quantity in the denominator of the right hand side of (35) may be determined by COBE or WMAP while the quantity in the numerator can not be determined at time being. It can be calculated after the accumulation of further data, and then the Hubble parameter can be determined. It is clear also from the relation (34) that the value of the Hubble parameter decreases as the temperature gradient decreases. This result is in agreement with Bellini(2001) results. It is worth of mention that the gradient relation my give the same form for many of cosmological models but each result depends essentially on the value of the scale factor fixed by the model under consideration, i.e this procedure is model dependent. Acknowledgement The author would like to express his deep thankfulness to Professor M.I.Wanas for his useful discussions.Also he is indebted to the late Dr. M.Melek for his previous guiding points. Part of this work has a been accomplished during the author's visit to the High Energy Section of the Abdos Sallam ICTP. So, the author would like to thank Professor S. Randjbar-Daemi the Head of High Energy Section, ICTP, Italy, for inviting him to visit and use the facilities of the ICTP, during the period from 25 July to 3 August 2004. leads directly to the following result.. T . T = . R R . (28) Noting that . R R = H , as usually done, then we get .. . M Bellini, A Challinor, A Lasenby, V G Gurzadyan, astro-ph/0312305Ap. J. 331GRGBellini, M. (2001) GRG, 33, 2081. Challinor, A. and Lasenby, A.(1999)Ap. J., 512, 1. Gurzadyan V.G. et al. (2003), astro-ph/0312305. . W Hu, N Sugiyama, J Silk, A F Joshue, H Dragan, V L Eric, S T Michael, astro-ph/0208100v2Nature. 386Hu,W., Sugiyama, N. and Silk, J.(1997) Nature, 386. Joshue , A. F., Dragan H., Eric, V. L. , Michael,S. T.(2003) astro-ph/0208100v2. . B Keating, P Timbie, A Polnarev, J Steinberger, Ap. J. 495580Keating, B., Timbie, P., Polnarev, A. and Steinberger, J.(1998) Ap. J., 495, 580. . M Melek, IC/92/95Melek, M.(1992) ICTP print no. IC/92/95. . M Melek, Astrophys. Space. Sci. 228743Astrophys. Space. Sci.Melek, M.(1995) Astrophys. Space. Sci., 228, 327. Melek, M.(2000) Astrophys. Space. Sci., 272, 417. Melek, M.(2002) Astrophys. Space. Sci., 281, 743. Narlikar (1983), Introduction to Cosmology. F I Mikhail, M I Wanas, Proc. Roy. Soc. Lond. A. 356Jones and Bartlett Publishers, IncMikhail, F.I. and Wanas, M.I. (1977) Proc. Roy. Soc. Lond. A,356,471. Narlikar (1983), Introduction to Cosmology, Jones and Bartlett Publishers, Inc. . R C Rebort, M Doran, H P Robertson, A G Riess, astro-ph/0305334v1Chaos, Solitons and Fractals. Wanas, M. I.332621Astrophys. Space. Sci.Rebort, R. C. and Doran, M. (2003), astro-ph/0305334v1. Robertson, H. P. (1932) Ann. Math. Princeton(2),33,496. Riess, A. G. , et al. (2004)Ap. J., 607, 665. Wanas, M. I. (1986) Astrophys. Space. Sci., 127, 21. Wanas, M. I. (1989) Astro. Space Sci., 154, 165. Wanas, M. I. (2003) Chaos, Solitons and Fractals, 16, 621.

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{'abstract': "The temperature gradient of microwave background radiation (CMBR) is calculated in the Self Consistent Model. An expected values for Hubble parameter have been presented in two different cases. In the first case the temperature is treated as a function of time only ,while in the other one the temperature depends on relaxation of isotropy condition in the self-consistent model and the assumption that the universe expands adiabatically. The COBE's or WMAP's fluctuations in temperature of CMBR may be used to predict a value for Hubble parameter.", 'arxivid': 'gr-qc/0412036', 'author': ['A B Morcos morcos@frcu.eun.eg \nDepartment of Astronomy\nNational Research Institute Of Astronomy and Geophysics\nHelwan, CairoEgypt\n'], 'authoraffiliation': ['Department of Astronomy\nNational Research Institute Of Astronomy and Geophysics\nHelwan, CairoEgypt'], 'corpusid': 119368473, 'doi': '10.5303/jkas.2006.39.4.081', 'github_urls': [], 'n_tokens_mistral': 5829, 'n_tokens_neox': 4950, 'n_words': 3052, 'pdfsha': '7c75bb4fbf5de9550d82f4a1980d0d4a0a3e48d7', 'pdfurls': ['https://export.arxiv.org/pdf/gr-qc/0412036v1.pdf'], 'title': ['TEMPERATURE FLUCTUATION AND AN EXPECTED LIMIT OF HUBBLE PARAMETER IN THE SELF-CONSISTENT MODEL', 'TEMPERATURE FLUCTUATION AND AN EXPECTED LIMIT OF HUBBLE PARAMETER IN THE SELF-CONSISTENT MODEL'], 'venue': []}

arxiv

A new route for enantio-sensitive structure determination by photoelectron scattering on molecules in the gas phase Kilian Fehre Nikolay M Novikovskiy Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Institute of Physics Southern Federal University 344090Rostov-on-DonRussia Sven Grundmann Gregor Kastirke Sebastian Eckart Florian Trinter Molecular Physics Fritz-Haber-Institut Max-Planck-Gesellschaft Faradayweg 4-614195BerlinGermany Jonas Rist Alexander Hartung Daniel Trabert Christian Janke Martin Pitzer Stefan Zeller Florian Wiegandt Miriam Weller Max Kircher Giammarco Nalin Max Hofmann Lothar H Ph Schmidt André Knie Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Andreas Hans Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Ltaief Ben Ltaief Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Arno Ehresmann Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Robert Berger Department of Chemistry Philipps-Universität Marburg Hans-Meerwein-Straße 435032MarburgGermany Hironobu Fukuzawa Institute of Multidisciplinary Research for Advanced Materials Tohoku University 980-8577SendaiJapan Kiyoshi Ueda Institute of Multidisciplinary Research for Advanced Materials Tohoku University 980-8577SendaiJapan Horst Schmidt-Böcking Joshua B Williams Department of Physics University of Nevada 89557RenoNevadaUnited States European XFEL GmbH Holzkoppel 422869SchenefeldGermany Reinhard Dörner Philipp V Demekhin demekhin@physik.uni-kassel.de Institut für Physik und CINSaT Universität Kassel Heinrich-Plett-Straße 4034132KasselGermany Markus S Schöffler schoeffler@atom.uni-frankfurt.de2 Institut für Kernphysik Goethe-Universität Frankfurt Max-von-Laue-Straße 160438Frankfurt am MainGermany A new route for enantio-sensitive structure determination by photoelectron scattering on molecules in the gas phase 1 X-ray as well as electron diffraction are powerful tools for structure determination of molecules. Studies on randomly oriented molecules in the gas-phase address cases in which molecular crystals cannot be generated or the interaction-free molecular structure is to be addressed. Such studies usually yield partial geometrical information, such as interatomic distances. Here, we present a complementary approach, which allows obtaining insight to the structure, handedness and even detailed geometrical features of molecules in the gas phase. Our approach combines Coulomb explosion imaging, the information that is encoded in the molecular frame diffraction pattern of core-shell photoelectrons and ab initio computations. Using a loop-like analysis scheme we are able to deduce specific molecular coordinates with sensitivity even to the handedness of chiral molecules and the positions of individual atoms, as, e.g., protons. Introduction During the last decade, the determination of the three-dimensional structure of molecules using electron crystallography developed into a complementary analysis method to the well-established X-ray crystallography (1)(2)(3). In particular, for the structural investigation of micro-and nano-crystalline materials, where sufficiently large single crystals for X-ray diffraction cannot be obtained (4) or crystalline sponge approaches for X-ray diffraction (5,6) fail, electron diffraction is the method of choice (7). It yields Ångstrom-resolution even when applied to large systems (8) or systems involving weak scatterers such as hydrogen atoms (9). For cases in which molecular crystals cannot be obtained or the interaction-free molecular structure is to be addressed, corresponding electron scattering approaches on gas-phase molecules exist (10,11). Such electron diffraction studies on randomly oriented molecules, however, can only provide information on interatomic distances, which is, in addition, challenging to extract in case of overlapping distance parameters. Furthermore, they do not reveal the handedness of chiral systems (12). In order to overcome the drawback of a random orientation of the molecules under investigation, several experiments have been performed utilizing electron (13,14) and x-ray diffraction (15) in combination with sophisticated 2-dim. and 3-dim. laser alignment schemes. An alternative approach employs electrons that are created from within the molecule as a probe. In these experiments, high-energetic single photons or strong-field laser pulses ionize isolated molecules in the gas phase. The emitted photoelectron is scattered by the molecular potential, yielding a very complex interference pattern, in which the structural information is encoded. Using table-top laser systems, laser-induced electron diffraction (LIED) has, for example, proven its capability determining internuclear distances with high accuracy for small molecules (16)(17)(18). In general, photoelectron diffraction by molecules in the gas phase has been successfully applied for determining molecular constituents (19), mapping bond lengths (20,21) and simple chemical reactions (22) on ultrafast timescales (17). Until now, however, corresponding studies were restricted to linear (17,19,22) or mostly symmetric molecules (23-25) such as, e.g., CO, CO2, H2O, or CH4. Apart from measuring the electron diffraction pattern (in terms of an electron angular emission distribution), the key to electron diffraction experiments on molecules is the knowledge of each individual molecule's orientation in space (26). A possible approach is to adsorb the molecule to a surface (13) or (as indicated above) to utilize weak laser pulses in various schemes (and in 2d or 3d arrangements) to orient (27,28) or align (15,(27)(28)(29)(30)(31) the molecule. Single cycle pulses in the THz were also used in the past to orient/align molecules (32). An alternative approach for detecting the molecule's spatial orientation is Coulomb explosion imaging (33), which, in addition, provides structural information, as well. Here, molecular ions or molecules are rapidly charged up by foil-induced electron stripping (34), multiple ionization by a short and strong laser pulse, or by photoionization and subsequent Auger decay (cascades) (34). After the charge-up, the ionic fragments are driven apart rapidly by Coulomb repulsion. Intriguingly, if more than three molecular fragments are generated in the Coulomb explosion, the triple product of three of their momentum vectors allows for identifying whether a chiral molecule was right-or lefthanded (35). However, despite the absolute configuration of chiral molecules can be determined in principle using this method (35,36), it has been restricted, so far, to small molecules with only a few atoms. So far, the largest molecule investigated using this approach was halothane consisting of eight atoms (37), but just recently iodopyridine (11 atoms) has been addressed in an experiment (38). Seribal et al. have shown in a simulation, that Coulomb Explosion Imaging in combination with a spatial orientation of the gas-phase target substance allows for retrieving the molecules' handedness even for system as large as camphor (39). In detail, however, gathering structural information of larger molecules with the help of Coulomb explosion techniques faces yet multiple technical challenges. These are, for example, the initial generation of high charge states, the rapidly declining detection efficiency for the coincident detection of multiple molecular fragments (40), and uncertainties in their correct m/q-assignment. In addition, the inversion of the measured momentum space information to position space is far from trivial as soon as the charge-up of the molecule and/or its fragmentation is governed by nuclear dynamics. Already when examining small molecules as H2O, support from sophisticated theory is required for the interpretation of Coulomb explosion data, which in turn provides valuable details on the fragmentation dynamics and processes (41). In this article, we demonstrate in a proof-of-principle study, how to overcome these obstacles. With a combination of the concepts of Coulomb explosion imaging, photoelectron diffraction imaging and support from ab initio modelling, we developed a method for addressing isolated molecules in the gas phase to determine their structure and their handedness. Our scheme is applicable without the need for advanced laseralignment schemes or elaborate abilities for detecting a multitude of ionic fragments in coincidence. We will show, furthermore, that our approach allows to determine even tiny details, as, for example, a slight displacement of a hydrogen atom in a methyloxirane molecule. Experimental method In our study we target methyloxirane molecule and examine its ionic fragments (occurring after photoionization and subsequent Auger decay) and the angular distributions of the emitted photoelectrons. The measurements were performed employing Cold Target Recoil Ion Spectroscopy (COLTRIMS) which is a multi-coincidence momentum imaging technique (42,43). In brief, ions and electrons created in the interaction of single photons from the Soleil synchrotron with single methyloxirane molecules were guided by electric and magnetic fields onto two time-and position-sensitive multichannel plate detectors. From the particles' positions of impact and times of flight the individual trajectories inside the COLTRIMS-spectrometer were reconstructed in an offline analysis of the data. This information yielded the particles' momenta and accordingly all derived observables as emission directions and kinetic energies. As a coincidence measurement has been performed, relative quantities are retrieved, as well, as for example relative emission angles. The experimental data were recorded at the same beam time as the data from a previous publication. Accordingly, the identical experimental setup was used and further details on the exact parameters of the COLTRIMS reaction microscope can found there (44). The employed photon energy of 550 eV addressed the O 1s-shell of the methyloxirane molecule leading to a photoelectron energy of about 11.5 eV and we restricted our analysis to electrons with kinetic energy of 11.5 ± 1.5 eV in order to suppress possible background. Furthermore, we employed the aforementioned photon energy as photoelectrons of this specific kinetic energy showed a large chiral response and this electron energy is amenable for accurate calculations (44). Several fragmentation pathways occur after the O-K-ionization of the molecule and subsequent Auger decay(s). As detailed below, we employ for our study cases where the molecule fragmented into at least three parts of which two are charged. The data presented in this study consists of a combination of the fragmentation channels C3H6O → C2H3 + + CH2 + + OH 0 + 2eand C3H6O → C2H2 + + CH2 + + H2O 0 + 2e -. About 3 • 10 events were recorded for both enantiomers of the molecule and both light helicities. Two other breakup channels, namely C3H6O → C2H3 + + CH3 + + O 0 + 2eand C3H6O → C2H2 + + CH3 + + OH 0 + 2e -, cannot be used because of the following reasons: It turns out that for the latter fragmentation channel the CH3 + group stems from the methyl group of methyloxirane. This different fragmentation dynamics manifests itself in the fact that the measured momenta of the ions define a completely different molecular coordinate system. In addition, the interference patterns for these fragmentation channels are washed out. We suspect that this is due to a weaker correlation between measured ionic momenta and the molecular orientation at the instant of ionization due to complex fragmentation dynamics. Theoretical method In order to extract information on the molecular geometry from the experimental data we performed a modeling of the electron diffraction pattern, i.e., of the molecular-frame angular emission distributions of the photoelectrons. The ionization transition amplitudes for the emission of O(1s) photoelectrons of the methyloxirane enantiomers were computed by using the single-center method and code (45,46) in the relaxed-core Hartree-Fock approximation, as described in our previous work on this molecule (44) (please see the supplementary information document of this reference for more details, as well). The averaging of the molecular frame photoelectron angular emission distributions over all incident directions of the ionizing light, required for the present study, was performed analytically. The average differential cross section reads: ( , ) Ω = * ( , ) with = 1 3 ℓ ℓ (−1) ℓ × ℓ ℓ × (2ℓ + 1)(2ℓ + 1)(2 + 1) 4 ℓ ℓ 0 0 0 ℓ ℓ − ℓ ℓ * Here, and are the photoelectron emission angles in the frame of molecular reference, are spherical harmonics, and ℓ are the dipole transition amplitudes for the emission of the partial photoelectron waves with the angular momentum quantum numbers ℓ and via the absorption of a photon of polarization , as defined in the frame of the molecule. Because of the mutual orthogonality of the Wigner rotational matrices (which transform the ionizing light of a given polarization from the laboratory to the molecular frame), the average molecular frame photoelectron angular distribution is independent of the polarization of the ionizing light that is used in the experiment. Results and Discussion As mentioned above, we use synchrotron light to ionize the molecule by emission of a core electron. The emerging photoelectron is diffracted by the molecular potential and serves as a messenger providing the molecular structure information in its angular emission pattern. Molecules are in most cases unstable after the emission of a core electron. Typically, at least one more additional electron is released in an Auger decay process and subsequently the molecule fragments into charged and neutral pieces. It turns out, that the detection of the fragmentation direction (i.e., the momentum vectors) of two charged fragments of a breakup of the molecule into at least three pieces is sufficient to gather the information on the spatial orientation of the molecule, which is needed for evaluating the photoelectron interference pattern. Dictated by conservation of linear momentum, the momentum vectors of three molecular fragments lie within a plane (turquoise arrows in Fig. 1A). They can be employed to form a fragment coordinate frame (X,Y,Z) in Fig. 1A, which was built as follows: ⃗ ( ⃗ respectively) points in the direction of the Y-axis, ⃗ × ⃗ ( ⃗ × ⃗ ) in the direction of the Z-axis, and ⃗ = ⃗ × ⃗ . The actual spatial orientation of the molecule within this fragment coordinate frame at the instant of photoionization remains, however, unknown. In larger systems, the fragments' emission directions are typically only loosely connected to the direction of the molecular bonds prior to the fragmentation, in particular if only few fragments are generated. Thus, the fragment frame (X,Y,Z) deduced from the ion direction measurement and a desired molecular coordinate frame (x',y',z') linked to its structure are typically skewed by some unknown angles ( , and , i. e., the rotation angles with respect to the X, Y, and Z axis). In addition, the measured ion momentum vectors alone do not provide any information on the handedness of the ionized molecule, as they define a plane and thus leave the sign of the Z-axis open, as depicted in Figs. 1B and C. Both, this information and the information on the skew-angles are, however, encoded in the electron diffraction pattern. Figure 1: Three-dimensional interference pattern of the scattered electron wave in the molecular frame of reference. A Spherical representation and definition of the fragment (X,Y,Z) and molecular (x',y',z') coordinate frames. For larger molecules, the fragment emission directions (i.e., their momenta after Coulomb explosion) do typically not coincide with molecular features such as bonds. As a result, the molecular coordinate frame (x',y',z') at the instant of photoionization is rotated against the fragment frame (X,Y,Z). The panel depicts the methyloxirane molecule employed in our studies and the turquoise arrows show the directions of the measured momentum vectors of the fragments (CH2 + , C2H3 + , and OH 0 ), which were used to generate the fragment (X,Y,Z) coordinate system, as discussed in the text. The electron wave employed for probing the molecular structure has been emitted from the oxygen 1s orbital. The surrounding-colored sphere shows the resulting three-dimensional probability distribution of the emission direction of the 11.5 eV photoelectron. The emission distributions are averaged over all incident directions of the ionizing light. B Same data as in A in a color-map representation. C is as B for the R-enantiomer. The mirror symmetry regarding the enantiomers is highlighted by the horizontal line to guide the eye at = 0 in B and C. A visualization of the two enantiomers and their orientation in the molecular frame are given above B and C. To extract the structural information from the experimental data we use the procedure that is outlined in Fig. 2. We start with an initial guess for the molecular structure and compute the photoelectron interference pattern in a guessed molecular frame (x',y',z'), which is assumed to coincide with the fragment frame. Then, we compare this pattern to the pattern obtained in our experiment, which is provided in the fragment frame (X,Y,Z). To quantify the agreement, we introduce the distance parameter between the renormalized experimental and computed interference pattern. This parameter depends, as well, on the relative rotation between (x',y',z') and (X,Y,Z) quantified by the rotation angles , and and the guess of the handedness. = ∬ (φ, cos(ϴ)) − (φ, cos(ϴ), , , ) φ cos ϴ We now determine the skew between the coordinate frames (x',y',z') and (X,Y,Z) varying the three rotation angles in order to obtain the smallest distance parameter (Fig. 2C). The minimized value of (i.e., after applying the rotation) is then used to quantify the overall agreement between the measured and the computed interference pattern for the initially hypothesized molecular structure and handedness. This procedure is then repeated with a slightly adjusted molecular structure as an input in order to further minimize d2. The model structure, which provides the smallest distance parameter d2 is assumed to be responsible for the measured interference pattern, thus providing the molecular structure and coordinate system at the instant of ionization. Figure 2: Sketch of the optimization procedure for obtaining the molecular structure from the measured electron interference patterns. Starting with an initial guess (A) the interference pattern of the photoelectron in a chosen molecular frame (x',y',z') is computed (B). This computed pattern is then compared to the measured interference pattern in the fragment frame (X,Y,Z). Typically, the fragment frame and molecular frame do not coincide. The skew between the two systems (given by the three rotation angles , and ) is determined by finding the minimum value of distance parameter for the computed molecular structure (C). D shows the computed interference pattern from B in the rotated molecular frame (x',y',z'). After the rotation, the minimized value of is used to quantify the agreement between measured and computed interference pattern for a specific hypothesized molecular structure. The molecular structure is slightly modified, and the interference pattern is recomputed (E). The molecular structure and coordinate system at the instant of ionization are obtained for the best agreement between measured and computed interference patterns, i.e., for lowest d2. In more detail, in order to actually calculate the distance parameter , we apply the following procedure: The minimum value occurring in the interference pattern is first subtracted from the pattern and then the pattern's integral is normalized to one. For each calculated molecular structure, the molecular system in coordinate space used in the calculation must be connected to the measured fragment momentum vectors. To do this, we rotate the measured and calculated interference pattern with respect to each other, applying the X-Y-Z convention (roll, pitch and yaw angle: , and ). We determine the transformation that connects the fragment system defined by the measured ionic momenta and a molecular system used in the computation by searching for the minimum in in a scan over the yaw, pitch and roll angles (see Fig. 2). We scan all angles in steps of one degree. This step size is small enough to ensure that the residual error in the molecular frame does not influence the result presented in the following. In order to test our approach, we employ the two enantiomers of methyloxirane as benchmark systems. As outlined in the experimental methods section, we are using a photoelectron of 11.5 eV kinetic energy emitted from the oxygen K-shell for the diffraction imaging and a naturally occurring subsequent Auger decay for the generation of two ionic and one neutral fragments. As shown in Figs. 1B and 1C, the observed interference pattern is vastly different for the two enantiomers making chiral discrimination straight forward. A comparison to the modelled pattern shown in Fig. 2D provides the information on the absolute configuration. As the geometry of methyloxirane is well known in the literature (47), we employ the algorithm described above (and illustrate the high sensitivity of our approach) in order to extract the exact location of distinct atoms inside the molecule. A corresponding table of the atomic coordinates at its equilibrium can be found in the supplemental material of (44). As a first example, we show in Figs. 3A-3F the effect of a modification of the * − distance in the oxirane ring around the equilibrium structure. The resulting variation of is shown in Fig. 2F and implies, that we are sensitive to a change of 5 % of the geometry-optimized * − distance. For comparison, a similar relative accuracy of a few percent for a bond length measurement has been demonstrated recently employing LIED examining OCS molecules (48). Particularly challenging for other methods of structure determination is the assignment of the location of hydrogen atoms. Electron scattering is known to be sensitive also to such weak scatterers. Accordingly, in a second demonstration, we investigate the sensitivity of our approach to a change of the * − bond length. The corresponding results are depicted in Fig. 3G which confirms, that within 5 % discrepancy, the correct bond length between the chiral carbon atom and the adjacent proton attached to it, has been found via the smallest value of . The experimental statistical error is smaller than the plotted dot size; however, different sources of systematic errors might alter the exact value of . The quality of the reconstruction as well as the achievable resolution depends on multiple factors, which cannot be easily quantified. For example, the recorded interference pattern has a finite experimental resolution. It is however not easy to estimate how resolving fine details of the interference pattern effects the reconstruction of geometrical features in the end. First estimates on the presented data suggest, that the experimental resolution is not limiting the accuracy of the geometrical reconstruction. Furthermore, background from other fragmentation channels or a complex interplay between areas of reduced detection efficiency on the electron and ion detectors might affect the geometry-reconstruction, as well. Yet, we expect that the statistic of the recorded datasets has a stronger impact on the reconstruction quality in our present study. An additional source of errors is connected to the approximation that is made in the ab initio computations. Each molecular configuration was calculated only for fixed internuclear distances (i.e., a single fixed molecular geometry) and only for a single photoelectron energy. In the experiment, however, the signal is averaged over a certain distribution of photoelectron energies and real-life molecules exhibit vibrational motion. In the case of a non-linear relationship between the influencing parameters and the observed variable, the mean value of the observed variable generally does not correspond to the mean value of the influencing factors. Therefore, an estimate of our resolution when determining the three-dimensional position of the atoms is provided by considering how close the minimum in comes to the result of the geometry-optimized structure when scanning across different molecular structures. Thus, the resolution is estimated to be of similar magnitude as our chosen step size of ~6 • 10 m. Please note, that with our technique the spatial resolution is not limited by the photoelectron's wavelength of 3.6 Å. Figure 3: Determination of the molecular structure via the best agreement between measured and computed interference pattern of the photoelectron. A-E Interference patterns from a scan in which the * − distance in the oxirane ring is set to 95, 100, 105, 110 and 120 % of the optimized structure. Our structure retrieval algorithm leads within 5 % accuracy to the energyoptimized structure highlighted in green (F). G A corresponding scan of the * − bond length demonstrates the sensitivity of the interference pattern to weak scatterers such as hydrogen. The smallest distance in leads, again, within 5 % accuracy to the * − bond length of the energyoptimized structure. Conclusion Combining partial Coulomb explosion imaging of a large molecule with the measurement of the photoelectron diffraction pattern in the molecular frame and quantum chemical computation allows for precise structural analysis and chiral discrimination of molecules in the gas phase. Unlike established X-ray or several of the electron diffraction techniques, our approach does not require a molecular crystal. Contrary to traditional Coulomb Explosion Imaging, it is scalable, so that larger molecules can be examined, as well. The only requirement is that the molecule breaks sufficiently fast into at least three fragments (of which at least two are charged), so that the measured ion momentum vectors are linked to the molecular orientation at the time of ionization. In addition, by adjusting the photon energy, distinct atoms of the molecule can be addressed and the emission source of the probing electron wave inside the molecule can be selected. By applying pump-probe schemes, the method will allow for tracking changes in the molecular structure on a femtosecond time scale (49,50) in the future. The described approach is in principle general and can be extended to larger molecules. If necessary, required calculations can rely on density function theory (DFT) based methods which are usually not limited by molecular size, like e. g. the TDDFT B-spline LCAO formalism (51). Data availability All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors. Correspondence and requests for materials should be addressed to K.F. (fehre@atom.uni-frankfurt.de), P.V.D. (demekhin@physik.uni-kassel.de), or M.S.S. (schoeffler@atom.uni-frankfurt.de). Author Contributions Conflicts of interest The authors declare that they have no competing interests. The experiment was conceived by M.S.S. and R.D. The experiment was prepared and carried out by S.G., G.K., S.E., F.T., J.R., A.H., D.T., C.J., M.P., S.Z., F.W., M.W., M.K., M.H, L.Ph.H.S., A.K., A.H., L.B.L., A.E., R.B., H.F., K.U., T.J., J.B.W., and M.S.S. Data analysis was performed by K.F. and M.S.S. Theoretical calculations were performed by P.V.D and N.M.N. All authors discussed the results and commented on the manuscript. K.F., P.V.D., R.D., T.J, and M.S.S. wrote the paper. K.F. and A.H. acknowledge support by the German National Merit Foundation. M.S.S. thanks the Adolf-Messer Foundation for financial support. 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{'abstract': 'X-ray as well as electron diffraction are powerful tools for structure determination of molecules. Studies on randomly oriented molecules in the gas-phase address cases in which molecular crystals cannot be generated or the interaction-free molecular structure is to be addressed. Such studies usually yield partial geometrical information, such as interatomic distances. Here, we present a complementary approach, which allows obtaining insight to the structure, handedness and even detailed geometrical features of molecules in the gas phase. Our approach combines Coulomb explosion imaging, the information that is encoded in the molecular frame diffraction pattern of core-shell photoelectrons and ab initio computations. Using a loop-like analysis scheme we are able to deduce specific molecular coordinates with sensitivity even to the handedness of chiral molecules and the positions of individual atoms, as, e.g., protons.', 'arxivid': '2101.03375', 'author': ['Kilian Fehre ', 'Nikolay M Novikovskiy \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n\nInstitute of Physics\nSouthern Federal University\n344090Rostov-on-DonRussia\n', 'Sven Grundmann ', 'Gregor Kastirke ', 'Sebastian Eckart ', 'Florian Trinter \nMolecular Physics\nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany\n', 'Jonas Rist ', 'Alexander Hartung ', 'Daniel Trabert ', 'Christian Janke ', 'Martin Pitzer ', 'Stefan Zeller ', 'Florian Wiegandt ', 'Miriam Weller ', 'Max Kircher ', 'Giammarco Nalin ', 'Max Hofmann ', 'Lothar H Ph ', 'Schmidt ', 'André Knie \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n', 'Andreas Hans \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n', 'Ltaief Ben Ltaief \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n', 'Arno Ehresmann \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n', 'Robert Berger \nDepartment of Chemistry\nPhilipps-Universität Marburg\nHans-Meerwein-Straße 435032MarburgGermany\n', 'Hironobu Fukuzawa \nInstitute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan\n', 'Kiyoshi Ueda \nInstitute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan\n', 'Horst Schmidt-Böcking ', 'Joshua B Williams \nDepartment of Physics\nUniversity of Nevada\n89557RenoNevadaUnited States\n\nEuropean XFEL GmbH\nHolzkoppel 422869SchenefeldGermany\n', 'Reinhard Dörner ', 'Philipp V Demekhin demekhin@physik.uni-kassel.de \nInstitut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany\n', 'Markus S Schöffler schoeffler@atom.uni-frankfurt.de2 ', '\nInstitut für Kernphysik\nGoethe-Universität Frankfurt\nMax-von-Laue-Straße 160438Frankfurt am MainGermany\n'], 'authoraffiliation': ['Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Institute of Physics\nSouthern Federal University\n344090Rostov-on-DonRussia', 'Molecular Physics\nFritz-Haber-Institut\nMax-Planck-Gesellschaft\nFaradayweg 4-614195BerlinGermany', 'Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Department of Chemistry\nPhilipps-Universität Marburg\nHans-Meerwein-Straße 435032MarburgGermany', 'Institute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan', 'Institute of Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577SendaiJapan', 'Department of Physics\nUniversity of Nevada\n89557RenoNevadaUnited States', 'European XFEL GmbH\nHolzkoppel 422869SchenefeldGermany', 'Institut für Physik und CINSaT\nUniversität Kassel\nHeinrich-Plett-Straße 4034132KasselGermany', 'Institut für Kernphysik\nGoethe-Universität Frankfurt\nMax-von-Laue-Straße 160438Frankfurt am MainGermany'], 'corpusid': 253183411, 'doi': '10.1039/d2cp03090j', 'github_urls': [], 'n_tokens_mistral': 13013, 'n_tokens_neox': 11017, 'n_words': 6514, 'pdfsha': '9bd95317c9607469d0a7ac1b663b2c2349681c9b', 'pdfurls': ['https://export.arxiv.org/pdf/2101.03375v2.pdf'], 'title': ['A new route for enantio-sensitive structure determination by photoelectron scattering on molecules in the gas phase', 'A new route for enantio-sensitive structure determination by photoelectron scattering on molecules in the gas phase'], 'venue': []}

arxiv

December 20, 2022 December 20, 2022 E C Herenz eherenz@eso.org European Southern Observatory Av. Alonso de Córdova 3107763 0355Vitacura, SantiagoChile Leiden Observatory Leiden University Niels Bohrweg 22333 CALeidenNLThe Netherlands J Inoue Department of Physics & Astronomy, Macalester College 1600 Grand Avenue55105Saint PaulMNUSA H Salas Leibniz-Institut für Astrophysik Potsdam (AIP) An der Sternware 1614482PotsdamGermany B Koenigs Department of Physics & Astronomy, Macalester College 1600 Grand Avenue55105Saint PaulMNUSA C Moya-Sierralta Pontificia Universidad Católica de Chile -Instituto de Astrofísica Av. Vicuña Mackenna 4860897, 0117Macul, SantiagoChile J M Cannon Department of Physics & Astronomy, Macalester College 1600 Grand Avenue55105Saint PaulMNUSA M Hayes Department of Astronomy Stockholm University AlbaNova University Centre, SE-106 91StockholmSweden P Papaderos Instituto de Astrofísica e Ciências do Espaço -Centro de Astrofísica da Universidade do Porto Rua das Estrelas4150-762PortoPortugal G Östlin Department of Astronomy Stockholm University AlbaNova University Centre, SE-106 91StockholmSweden A Bik Department of Astronomy Stockholm University AlbaNova University Centre, SE-106 91StockholmSweden A Le Reste Department of Astronomy Stockholm University AlbaNova University Centre, SE-106 91StockholmSweden H Kusakabe Observatoire de Genève Université de Genève Chemin Pegasi 511290VersoixSwitzerland A Monreal-Ibero Leiden Observatory Leiden University Niels Bohrweg 22333 CALeidenNLThe Netherlands J Puschnig Argelander-Institut für Astronomie Universität Bonn Auf dem Hügel 71D-53121BonnGermany December 20, 2022 December 20, 2022Astronomy & Astrophysics manuscript no. sbs0335_musevla_revision_leGalaxies: starburst -Galaxies: haloes -Galaxies: individual: SBS 0335-052E A ∼15 kpc outflow cone piercing through the halo of the blue compact metal-poor galaxy SBS 0335-052E ABSTRACT Context. Outflows from low-mass star-forming galaxies are a fundamental ingredient for models of galaxy evolution and cosmology. Despite seemingly favourable conditions for outflow formation in compact starbursting galaxies, convincing observational evidence for kiloparsec-scale outflows in such systems is scarce. Aims. The onset of kiloparsec-scale ionised filaments in the halo of the metal-poor compact dwarf SBS 0335-052E was previously not linked to an outflow. In this paper we investigate whether these filaments provide evidence for an outflow. Methods. We obtained new VLT/MUSE WFM and deep NRAO/VLA B-configuration 21cm data of the galaxy. The MUSE data provide morphology, kinematics, and emission line ratios of Hβ/Hα and [O iii]λ5007/Hα of the low surface-brightness filaments, while the VLA data deliver morphology and kinematics of the neutral gas in and around the system. Both datasets are used in concert for comparisons between the ionised and the neutral phase. Results. We report the prolongation of a lacy filamentary ionised structure up to a projected distance of 16 kpc at SB Hα = 1.5 × 10 −18 erg s cm −2 arcsec −2 . The filaments exhibit unusual low Hα/Hβ ≈ 2.4 and low [O iii]/Hα ∼ 0.4 − 0.6 typical of diffuse ionised gas. They are spectrally narrow (∼ 20 km s −1 ) and exhibit no velocity sub-structure. The filaments extend outwards from the elongated H i halo. On small scales, the N HI peak is offset from the main star-forming sites. The morphology and kinematics of H i and H ii reveal how star-formation-driven feedback interacts differently with the ionised and the neutral phase. Conclusions. We reason that the filaments are a large-scale manifestation of star-formation-driven feedback, namely limb-brightened edges of a giant outflow cone that protrudes through the halo of this gas-rich system. A simple toy model of such a conical structure is found to be commensurable with the observations. Introduction The response of interstellar gas to energy and momentum deposition from supernovae and stellar winds is the growth of a hot bubble surrounded by a dense shell. Star-forming populations can inject energy and momentum long enough to sweep the shell up and out to lower density gas, where Rayleigh-Taylor instabilities eventually deform it before breaking it up. The hot gas then vents into the circum-galactic halo of the galaxy as a wind that also entrains warm and cold gas. If the wind is powerful enough, the gas may escape the gravitational potential of a galaxy and thereby enrich the intergalactic medium (IGM) with metals. The gas from less powerful outflows then rains back on the galaxy and is available to form the next generation of stars. Outflows and winds -next to inflows of fresh gas -are a cornerstone of galaxy formation models and are vital for regulating cosmic chemical evolution (see reviews by Veilleux et al. 2005Veilleux et al. , 2020Collins & Read 2022, as well as Schneider & Robertson 2018, Nelson et al. 2019, Mitchell et al. 2020, Schneider et al. 2020, and Pandya et al. 2021 for state-of-the-art computer models). Observationally, signatures of galactic outflows in starforming galaxies are ubiquitous, both in the nearby (e.g. Heckman et al. 2011;Chisholm et al. 2016) and in the high-redshift universe (review by Erb 2015). Galaxies in the local Universe allow for detailed panchromatic mappings of outflow phenomena, from the highest energies (cosmic rays and hot gas) to the longest wavelengths possible (cold and molecular gas) -especially if the outflow occurs 'edge-on'. The most detailed observational studies that analyse multiple phases of outflows simultaneously are performed on nearby, more evolved, and massive systems. These observations nourish our understanding and provide robust con-Article number, page 1 of 22 arXiv:2212.01239v2 [astro-ph.GA] 16 Dec 2022 A&A proofs: manuscript no. sbs0335_musevla_revision_le straints of how mass, energy, and momentum from star formation couple with the different phases of the interstellar-medium (ISM) to drive outflows 1 . However, from a cosmological perspective, similar constraints are especially needed for the abundant and, in the early universe, dominating population of low-mass starforming galaxies. Currently, cosmological models have to implement those processes in the form of phenomenological 'sub-grid physics' recipes that are tuned to match observed galaxy population statistics, especially luminosity-and stellar-mass functions (Somerville & Davé 2015). Understanding the effects of feedback and winds in low-mass systems is especially required to grasp the large-scale physics of the universe during the cosmological phase transition known as the Epoch of Reionisation at z 6. This is because dense shells in the pre-fragmentation stage are more opaque to hydrogen ionising radiation than perforated bubbles at a later stage (e.g. Fujita et al. 2003). Theoretical modelling requires sustained momentum and energy injection to drive a wind and thereby significant Lyman continuum (LyC) photons out of low-mass galaxies (Kimm & Cen 2014;Paardekooper et al. 2015). Detailed 'zoomin' simulations of individual dwarf star-forming galaxies suggest that LyC escape is highly variable and highly anisotropic (Trebitsch et al. 2017). The angular dependence is modulated by high optical depth sight-lines along cold-flow accretion filaments (Park et al. 2021) and low optical depth sight-lines that follow the direction of the outflow phenomena (Trebitsch et al. 2017). Such anisotropic beamed LyC may have consequences for the mechanisms by which H ii zones percolate during reionisation (Paardekooper et al. 2015;Furlanetto & Oh 2016). While a pan-chromatic approach is required for a detailed understanding of the multiple phases of the outflowing material, narrow band imaging and kinematic analysis of the classical principal emission lines can expose and characterise feedback, outflow, and wind phenomena just from the T ∼ 10 4 K phase (e.g. Calzetti et al. 2004;Westmoquette et al. 2008;Zastrow et al. 2011;Lee et al. 2016;McQuinn et al. 2019). The increased sensitivity for low surface-brightness line emission offered by modern integral-field spectrographs (IFSs; Bacon & Monnet 2017) allows for unparalleled discoveries in this respect. Observational evidence for the T ∼ 10 4 K phase of feedbackdriven winds in nearby dwarf galaxies and potential earlyuniverse analogues have been amassed for decades from such imaging and spectral analyses (e.g. Marlowe et al. 1995;Mc-Quinn et al. 2015, 2018Collins & Read 2022). Combined H i -H ii morpho-kinematical investigations of those targets are feasible as well. The studies by van Eymeren et al. (2009avan Eymeren et al. ( ,b, 2010 are noteworthy as they compare 21cm HI and HII kinematics from Hα in five nearby irregular dwarf galaxies (NGC 4861, NGC 2366, NGC 5408, and IC 4622). In these galaxies, multiple localised velocity offsets between the ionised and the neutral phase are found and are interpreted as being driven feedback (see also Bomans 2001;Bomans et al. 2007 for reviews and X-ray follow-up). This idea is corroborated by filamentary features seen in Hα that extend towards regions of lower neutral columns, as well as the detection of partially broken super-shells in Hα. However, in the van Eymeren et al. studies, feedback was never found to be strong enough to drive outflows into the IGM. A similar conclusion was reached by Westmoquette et al. (2008) for the dwarf starburst NGC 1569, where the hot wind fluid is still presumed to be confined by super shells that extend up to 1 kpc outwards from the super-star clusters in this system. Roychowdhury et al. (2012) compared H i with H ii in three compact dwarf galaxies but found only tentative evidence for outflows. More recently, McQuinn et al. (2019) analysed Hα and 21cm morphologies simultaneously in a sample of 12 nearby dwarfs. They traced potential galactic winds via low surface-brightness Hα emission isophotes ( 3 × 10 −18 erg s −1 cm −2 arcsec −2 ) that extend beyond regions of their limiting neutral column (N HI ∼ 10 20 cm −2 or 1 M pc −2 ) gas; however, even the most convincing winds found in this study were not deemed strong enough to have material escaping into the IGM. More compelling early universe analogues for such observational studies are small, low-mass, and low-metallicity galaxies that exhibit high specific star-formation rates -so-called blue compact dwarf galaxies (Gil de Paz et al. 2003, see also the review by Thuan 2008 and the recent, more general review on dwarf galaxies by Henkel et al. 2022). Their shallow gravitational potential and the spatially and temporally concentrated energy and momentum injection from the starbursting population make them prime candidates for powering large-scale outflows (Mac Low & Ferrara 1999). And indeed, the unprecedented sensitivity for mapping low-surface brightness emission lines offered by the Multi Unit Spectroscopic Explorer (MUSE) at ESOs Very Large Telescope (VLT) UT4 (Bacon et al. 2014) unveiled spectacular outflows around the blue compact dwarfs Haro 11 (Menacho et al. 2019), Haro 14 (Cairós et al. 2022), and ESO 0338-IG04 (Bik et al. 2015(Bik et al. , 2018, as well as for the nearby HII galaxy Heinze 2-10 ( Cresci et al. 2017). In this paper we report new observational results on H i -H ii morphology and kinematics from the interstellar and circumgalactic medium of the observationally well-studied blue compact dwarf galaxy SBS 0335-052E (e.g. Izotov et al. 1990;Thuan et al. 1997;Izotov et al. 2001;Papaderos et al. 1998;Johnson et al. 2009;Adamo et al. 2010;Hunt et al. 2014;Kehrig et al. 2018;Wofford et al. 2021, and references therein). Numerous properties render this galaxy a unique analogue of early universe galaxies. With an absolute UV magnitude of M UV = −16.8, it is significantly fainter than the characteristic absolute UV magnitude of the z 6 luminosity functions (M UV < −20.6; Bouwens et al. 2015). With 12 + log(O/H) ∼ 7.25 (Izotov et al. 2009), it is one of the most metal-poor galaxies in the local Universe. This low-mass galaxy (M = 6 × 10 6 M ; Reines et al. 2008) is also extremely compact; the starburst is taking place within 500 pc, where six super-star clusters with diameters 60 pc form stars at ∼0.7 M yr −1 or at specific star-formation rates up to 20 M yr −1 kpc −2 . Our previous analysis of VLT/MUSE integral-field spectroscopic data from this galaxy unveiled two prominent ionised filaments in its halo (Herenz et al. 2017, hereafter H17). Here we present an analysis of new MUSE data in concert with new Karl G. Jansky Very Large Array (VLA) B-configuration observations that show that these filaments are likely related to an unprecedented large conical outflow structure that pinches deep into the halo of this compact starburst galaxy (Sect. 4). Prior to this, we summarise observations and data reduction in Sect. 2; also, the morphological and kinematic analyses of neutral and ionised gas are detailed in Sect. 3. In Sect. 5 we summarise and conclude. For distance conversions we assume a flat Λ cold dark matter Universe with Ω m = 0.3 and H 0 = 70 km s −1 Mpc −1 , and we adopt z = 0.01352 as the cosmological redshift for SBS 0335-052E (Moiseev et al. 2010). This translates to a luminosity distance of 58.42 Mpc and an angular scale of 276 pc/ . Observations and data reduction VLT/MUSE For the present analysis we combine data from two ESO/VLT MUSE programmes: Programme 096.B-0690 (PI: Hayes) with data taken on November 16th and 17th in 2015 (clear to photometric sky, DIMM seeing ≈0.7 -0.9 ) and Programme 0104.B-0834 (PI: Herenz) with data taken on October 2nd and 3rd in 2020 (photometric sky, DIMM seeing 1.1 -1.4 ). Both programmes were observed in wide-field mode (1 ×1 field of view) with the blue cut-off filter removed (extended mode, wavelength range 465 nm -930 nm) and without adaptive optics. Programme 096.B-0690 centred the field of view on SBS 0335-052E (RA: 03 h 37 m 44.075 s Dec: −05 • 02 39.5 ). These data lead to the discovery of ionised filaments in the halo of SBS 0335-052E (H17), and were also analysed by Kehrig et al. (2018) and Wofford et al. (2021) to understand the properties of the ionising sources of this galaxy. In order to map the extent of the filament we obtained new MUSE data in programme 0104.B-0834 with a pointing centred north-west of the galaxy (RA: 03 h 37 m 41.9167 s Dec: −05 • 01 57.2 '). This pointing overlaps partly with the discovery data. The total open shutter times are 5680 s (8 exposures a 710 s) and 5576 s (8 exposures a 697 s) for the central-and offset pointing, respectively. We reduced the data with version 2.8.3 of the MUSE data processing pipeline (Weilbacher et al. 2020). The reduction of the 2015 data was already detailed in Wofford et al. (2021, Sect. 3.2) and we apply here the same recipes to our new 2020 data. These recipes provide us first with sky-subtracted, flux calibrated, and astrometrically roughly calibrated pixel tables for each exposure. As explained in H17, the spatially extended line emission in the data requires a modification of the default pipeline sky subtraction procedure; significant flux from the extended principal H ii emission around SBS 0335-052E would be removed without this modification. The most prominent oversubtracted emission shows up as spikes at the observed wavelengths of Hα and [O iii] λ5007 in the sky continuum spectra that the pipeline determines after the best-fit sky emission line spectra have been subtracted (Weilbacher et al. 2020, Sect. 3.9.1). We thus altered the output sky continuum spectra from the initial run of the sky subtraction routine by replacing the values around principal emission lines via linear interpolation from the surrounding spectral bins. The corrected sky continuum spectra and the unaltered sky emission line spectra from the first run were then used as input in a second run of the sky subtraction procedure. This removed the visible oversubtraction artefacts at Hα and [O iii] λ5007 in the initial reduction, and to be secure we also repeated the procedure for the lines [O iii] λ4363, He ii λ4686, He i λ5876, [S ii] λλ6717,6731, and [O i] λ6300, even if such strong spikes were apparent in the continuum sky spectrum from the initial run of the sky subtraction. Lastly, the individual exposure pixel tables are resampled onto a 3-dimensional 473×532×3802 grid (datacube); the first two axes sample the spatial domain with 0.2 ×0.2 spectral pixels (spaxels) parallel to RA and Dec and the third axis samples wavelength linearly from 4599.96Å to 9351.21Å with 1.25Å bins. The final coordinate transformation between datacube spaxels and (RA,Dec)-coordinates is registered by cross-correlating the Pan-STARRS (Chambers et al. 2016;Magnier et al. 2020) rband image against a synthetic r-band image created from the datacube. VLA We obtained new HI 21cm observations with the Karl G. Jansky Very Large Array (VLA) in its B-configuration. The observations (project number 17B-234; PI: Herenz) made use of the L-Band receiver centred on the redshifted 21cm frequency of the galaxy. The programme was executed in 10×3h blocks from Sep 30th to Oct 20th, 2017. All raw VLA data are calibrated following standard prescriptions using the National Radio Astronomy Observatory (NRAO) casa software package (version 5.6.1-8;McMullin et al. 2007;Emonts et al. 2020). Radio-frequency interference is manually identified and excised from the raw data. These data are then calibrated, and once a satisfactory calibration is achieved, the data are continuum subtracted and weighted using CASA's uvcontsub and statwt functions. "Satisfactory" calibration constitutes an RMS noise value in source-free channels of the cube that is within 10% of the theoretically optimal RMS value given the duration of the observation, the configuration of the VLA, and the spectral properties of the datacube. Using tclean and the masking algorithm Auto-Multithresh (AMT; Kepley et al. 2020), we generate an intermediate image cube using casa version 6.3.0.48. From this intermediate cube, we extract the AMT-generated mask. We chose AMT for mask generation as it produced high quality data products with reproducibility and speed. All image cubes are created using Briggs weighting with a robust parameter of 0.5 for a compromise between angular resolution and sensitivity. We set the AMT noise threshold to 3σ. In doing so, AMT finds and masks all emission associated with our target sources, in addition to noise peaks that rise above this threshold. To address this, we then examine the AMT-generated mask interactively and extract only the regions that include source emission. Through this process of automated mask generation and interactive selection, we end up with a final, robustly generated mask that solely and fully masks the target sources. With this mask in hand, we then initiate a final tclean with the same parameters as used for the mask-generation clean, but now providing the source-only mask generated in the previous step. We clean down to 0.5σ, as this removes the need for residual flux rescaling. The resulting final image cube covers an area of 22.8 arcmin 2 sampled at 1 /pixel, and the channel width is 10.26 km s −1 . The typical rms-noise per channel is 2.5×10 −4 Jy beam −1 . The beam is 6.1 ×4.6 and it is oriented 23.6 • west of north. The positional accuracy of the VLA HI data, which is based on milli-arcsecond precision positions of quasars in the VLA calibrator manual 2 , can be assumed to be typically 10% of the synthesised beam size. We also create another datacube from the B-configuration data, that provides us with a different spatial resolution. This cube is created by applying a 7kλ UV-taper in the imaging process. This UV-tapering downweights the longest baselines using a Gaussian function in the uv-plane (with the FWHM measured in units of wavelengths), producing a cube at a spatial resolution that is roughly equivalent to VLA C-configuration observations (beam 15.9 ×14.7 at 66.4 • east of north). The datacube differs from the untapered cube by an adjusted spatial sampling of 3 /pixel and the RMS-noise per channel is 4×10 −4 Jy beam −1 . The 7kλ-tapered data are less sensitive than the untapered data, but this is compensated by the larger beam that enables the mapping of lower columns over larger areas. Analysis and results Detection and morphology of extended H ii and H i emission Ionised gas To optimally detect extended line emission in our MUSE data, we processed the reduced datacube with routines from the LSDCat software (Herenz & Wisotzki 2017). We started by subtracting a running median in the spectral direction. The bandwidth of this filter (which effectively enables the removal of low-frequency continua) is 151 px (188.75 Å). We then crosscorrelated the continuum-subtracted datacube with a 3D Gaussian template. The result of this procedure is a unit-less datacube, that we dub in LSDCat as "S/N cube" in reference to the signal-to-noise ratio maximising characteristic of matched filtering. However, we here use LSDCat only as a simple linear filter to suppresses high-frequency noise and enhance the emission line signal in the datacube. By experimentation we found that the parameters σ v = 150 km s −1 and σ G = 2 optimally augmented the spectrally narrow and spatially extended emission line features in the circumgalactic medium around SBS 0335-052E. We visually examined the resulting S/N cube with the QFitsView software (Ott 2012) in layers with strong nebular line detections in the integrated spectrum (cf. Fig. 3 in Wofford et al. 2021). We here report significant detections of extended filamentary emission in Hα, [O iii] λ5007, and Hβ around SBS 0335-052E ( Fig. 1). Hβ was not reported in H17, but the increased exposure time in the region where the discovery data overlaps with our new observations renders the detection of this line significant. The filaments remain undetected in other emission lines. The Hα narrowband image shown in Fig. 1 was synthesised by summing over the three layers in the median-filter subtracted datacube (6651.21 ± 1.25Å observed frame). Only these layers contain appreciable Hα emission (see also the line-width analysis in Sect. 3.3.1). The maximum S/N-images in Fig. 1 were created by taking the maximum values from the LSDCat crosscorrelated datacube over a few layers around the emission lines of interest. We define SN LSD Hα > 8 as a conservative criterion for detected emission line signal. Spaxels in the S/N cube that fulfil this criterion are surrounded by a white dotted contour in Fig. 1. As can be appreciated from Fig. 1, the new MUSE observations reveal a continuation of the filamentary structure first reported by H17. This Hα emitting complex extends up to 15 kpc in projection towards the north-west from the galaxy's main stellar body. While the pronounced high-SN emission stems from the two filamentary threads, Hα is also detected throughout a contiguous region in between the filaments. The To make the discussion of morphological features of the extended ionised structure text more accessible, we display in Fig. 2 Hα surface brightness contours with annotations. The two filaments that protrude towards the north (northern filament; PA ≈ 0 • ) and north-west (western filament; PA ≈ 315 • ) are characterised by SB Hα ≈ 3×10 −18 erg s −1 cm −2 . The northern filament fragments after ∼ 19 (∼5 kpc) into a lacier prolongation. This threadlike extension continues outwards up to ∼ 58 (∼16 kpc). The western filament, on the other hand, brightens noticeably after ∼ 25 (∼7 kpc) to SB Hα ≈ 5 × 10 −18 erg s −1 cm −2 (Hα "hot-spot"). After this brightening the western filament fans out into two short branches, one to the west and one to the north-east. The accented filamentary features are surrounded by diffuse emission at SB Hα ≈ 7 × 10 −19 erg s −1 cm −2 . Two shorter and fainter filaments are found towards the north-east of the galaxy; these are labelled as "eastern foreshortened filaments" in Fig. 2. They may trace a similar structure as the northern and the western filaments, but appear shorter due to a different alignment along our sight-line. While Hα in the northern circumgalactic halo shows complex morphological features, the southern halo displays a rather smooth and slowly decaying light profile. At our sensitivity we detect diffuse Hα emission up to 25 away from the main stellar body of the system. To provide a quantitative assessment of this emission we extract a radial surface-brightness profile, SB Hα (r), under the assumption of circular symmetry. Therefore, we first determine the photometric centre (x, y) = ( xy I xy x/ xy I xy , xy I xy y/ xy I xy ), where I xy is the Hα flux value of the pixel at coordinates (x, y). We find (x, y) = (153.7, 162.7) which is located at (RA, Dec) = (3 h 37 m 43.99 s , -5 • 02 39.2 ). We then extract SB Hα (r) in semicircular annuli of 0.6 width south y and centred on (x, y). The result of this procedure is shown in Fig. 3. The measured Hα SB profile of SBS 0335-052E is characterised by a central high-SB region ("core") which drops rapidly from 2 × 10 −14 to 2 × 10 −16 erg s −1 cm −2 arcsec −2 within the first 5 and then kinks into a more slowly decaying ("halo") profile. Further out, at ∼ 13 and ∼ 10 −17 erg s −1 cm −2 arcsec −2 , another kink towards an even flatter profile is found. As can be seen from Fig. 3, the outer profile can be described adequately by a sum of two exponentials with scale lengths r 1 = 1.62 (447 pc) and r 2 = 8.94 (2.47 kpc). We caution that given the extreme intensity contrast -four orders of magnitude between core and outer halo region -diffuse scattered light may have brightened the observed profile at the faintest isophotes. This observational effect of scattered excess light was studied extensively by Sandin (2014) and Sandin (2015). These papers caution against the physical interpretation of faint halo scale lengths around bright compact galaxies in the absence of an accurate extended point-spread function model for the observations. In particular the here measured flattening at larger radii was put forward as a tale-tale sign of diffuse scattered light (see especially Sect. 5 and Appendix E in Sandin 2015). The "core" high-SB region (SB Hα 2 × 10 −16 erg s −1 cm −2 ), denoted as "central circumgalactic zone" in Fig. 2, is characterised by an intricate morphology. This region is detected in a plethora of other emission lines in the MUSE IFS data (see Figures in Kehrig et al. 2018). To visualise the morphology of this region adequately we display in the left panel of Fig. 4 a zoomedin view of the Hα narrow band image from Fig. 1; we now use a different cyclic asinh-stretch that is appropriate to trace the brighter morphological features relevant at this magnification. To better visualise the size difference between circum-galactic ionised gas structures and main stellar body of the galaxy, we inset the Hubble Space Telescope (HST) F550M image into the brightest central (SB Hα > 10 −15 erg s −1 cm −2 ) region. This band is devoid of strong emission lines and thereby traces only the morpholgy of the stellar continuum. We show the HST FR656N image in the right panel of Fig. 4. This resolves morphological features in the central region in greater detail and nicely demonstrates the gain in depth of our MUSE data with respect to HST. Figure 4 reveals that the high-SB Hα region exhibits multiple nested arcs or loops, as well as outward pointing filaments or spurs. Generally loops seen in Hα at such large scales trace m 1 (r) m 2 (r) m 1 (r) + m 2 (r) core : SB Hα − (m 1 (r) + m 2 (r)) the limb-brightened edges of so-called super-bubbles or supergiant shells (e.g. Chu 1995;Bomans 2001;Bomans et al. 2007). Disconnected loops are believed to point towards regions where the shells are broken up and where the pressurised interior can vent out. The interaction of the hot wind (> 10 5 -10 7 K) with the colder material in the surroundings and warm gas (∼ 10 4 K) entrained in these bubble-blow outs, as well as the merging of super-bubbles, are believed to manifest in outward pointing filamentary spurs (e.g. Cooper et al. 2008;Tanner et al. 2016). Here we see that the two brightest inner arcs towards the northwest resemble closed loops, but variations in surface brightness point at inhomogeneities that indicate cracks in the shells caused by Rayleigh-Taylor instabilities. Further outwards towards the north-west, only barely detected in the FR656N image but clearly visible in the MUSE data, we see a flaring of spurlike features. These structures are very extended, up to ∼ 10 or 2.7 kpc from the main star-forming regions. These features may be interpreted as outflowing warm gas that is entrained in the hot wind that vents through the cracks in the shell. Neutral gas For our mapping of the signals from the neutral phase we processed the VLA datacubes with the software SoFiA (Source Finding Application; Serra et al. 2015). We used SoFiA's smooth+clip source detection algorithm that is most commonly employed for searches of extragalactic HI emission. Here the radio datacube is smoothed with a set of top-hat kernels of varying sizes and the final detected source mask is built from the union of the binary masks that result from thresholding each individually smoothed cube. We used the recommended parameters for detection of extragalactic HI signal, i.e. a set of 12 top-hat kernels of zero, three, seven, and 15 channels width and spatial dimensions of 0×0 (i.e. no spatial smoothing), 3 × 3, and 6 × 6 pixels (1 pixel = 1 arcsec 2 ), and a detection threshold of 5. The SoFia runs provide us with moment-0 maps, S ν [Jy km s −1 beam −1 ], that are of interest here and moment-1 maps that are used for the kinematic analysis in Sect. 3.3.2. The moment-0 maps are converted into maps of neutral column, N HI [cm −2 ], via N H = 1.1 × 10 24 (1 + z) 2 S ν /(ab) ,(1) where a [ ] and b [ ] are the FWHM of the major-and minor axis of the synthesised beam, respectively, and z is the redshift of the galaxy (Eq. 78 in Meyer et al. 2017). The N HI map of SBS 0335-052E from this procedure is displayed in Fig. 6 and the top panel of Fig. 7 shows the moment-0 map encompassing both galaxies SBS 0335-052E & W. The individual layers that contribute to the spatially resolved H i signal are assembled in Appendix B in the form of channel maps. Translating the detection threshold of the smooth-and-clip algorithm into an N HI sensitivity limit needs to account for the different smoothing kernels that contributed in building up the 3D source mask. We proceeded empirically by inspecting the final binary-mask cube from SoFiA to determine that the faintest pixels in the outskirts of the moment-0 map are mostly single channel detections. As stated in Sect. 2.2, and verified by the noise cube produced by SoFia, the typical rms-noise per channel in the B-configuration data is 2.5 × 10 −4 Jy beam −1 . For a 10 km s −1 wide-channel this translates via Eq. (1) into a 1-σ column-density limit of σ N HI = 1 × 10 20 cm −2 . We finally used the contouring algorithm of ds9 (Joye & Mandel 2003) without smoothing on the N HI map (Fig. 6) to produce contours at N HI = {2.5 (2.5σ), 5, 10, 20, 30, 40} × 10 20 cm −2 . The closing 2.5σ contour associated with SBS 0335-052E is shown as a Thompson et al. (2009); N HI contours as in Fig. 4. The viewport, indicated as a violet dashed square in Fig. 4 white-dashed line only in Fig. 1; all other contours are displayed in Fig. 1, Fig. 4, and Fig. 5. From Fig. 4 it becomes apparent that the peak of N HI is slightly offset to the west from the Hα peak and from the starclusters. The offset with respect to the star-clusters is made more apparent in Fig. 5, where we overlay the N HI contours onto the emission-line free HST/ACS F550M image. Such offsets are observed frequently in dwarf starbursts and it is hypothesised that that feedback from starbursting SSCs will mechanically disrupt and ionise H i in their vicinity (e.g. van Eymeren et al. 2010;Cannon et al. 2016;Teich et al. 2016;Jaiswal & Omar 2020). At lower column densities the resolved H i structures appear to relate spatially with Hα morphological features. Towards the north we find an extended tail at N HI = 10 21 cm −2 that bends towards the east at a slightly lower columns ( Fig. 1 and Fig. 4). The northern tail is almost co-spatial with a northern Hα spur before it curves around the Hα spur towards the east. At even lower neutral columns, close to the detection limit of our observations (2.5σ = 2.5×10 20 cm −2 ; white dashed contour in Fig. 1), we trace an extended tail that aligns with the foreshortened filaments towards the east. Towards the north-west and more prominently towards the south-east, two "ears" emerge at the lowest column density, both of which appear also co-spatial with spurs pointing in the same direction. Given the observed spatial correlations between HI and Hα we suspect that the low-column HI emission stems from cold gas that is interacting with the ionised phase. The entangled interactions between star-forming H ii regions and H i phase are observationally well studied on small ( 10 2 pc) scales in more nearby Article number, page 7 of 22 A&A proofs: manuscript no. sbs0335_musevla_revision_le systems (e.g. Egorov et al. 2014, Cannon et al. 2016, Egorov et al. 2018, and review by Veilleux et al. 2020). These processes are complex, observationally challenging to tackle, and far from being quantitatively fully understood. Here, in the extreme environment of the starburst, we may now have resolved such interactions on kiloparsec scales. To study the interaction between the ionised and neutral phase on larger scales, we use the VLA B-configuration data tapered at 7kλ (Sect. 2.2). The N HI moment-0 maps and the 1-σ detection threshold, 2.2×10 19 cm −2 , were computed in the same way as described above for the untapered dataset. The tapered data reveal H i within and around the pair SBS 0335-052E & W. We juxtapose the moment-0 maps for the two different beams in Fig. 7. We compare the large-scale HI contours from tapered data with our MUSE Hα image in Fig. 8. There we use Pan-STARRS imaging (Chambers et al. 2016;Magnier et al. 2020) outside the MUSE FoV. In Pan-STARRS the fainter companion SBS 0335-052W is prominently visible as a blue cometary object close to the position of the western N HI peak. The juxtaposition of the moment-0 maps in Fig. 7 illustrates how different beams are sensitive to different scales of the neutral gas distribution in and around the galaxies. The untapered data resolves the morphology of higher-column H i within the galaxies and in their immediate vicinity. The measured peak N HI = 4.1 × 10 21 cm −2 for SBS 0335-052E is only slightly lower than the inferred N COS H from Lyα absorption with HST/COS spectroscopy: N COS HI = (5.0 ± 0.5) × 10 21 cm −2 (James et al. 2014). However, this does not indicate that the gas seen by COS in absorption is spatially resolved with the VLA. Especially the spatial offset between the N HI peak and the starclusters discussed above (cf. Fig. 5) precludes such a conclusion. This rather indicates substantial substructure below our resolution limit in the neutral phase. The larger beam of the 7kλtapered data smooths out almost all resolved features in the untapered data (Fig. 7, central panel). This is evidenced by the peak column (9.5 × 10 20 cm −2 ) being only slightly higher than the average column (9×10 20 cm −2 ) in the untapered map. However, the tapered data reveals the diffuse lower-column gas that is spread over larger areas in the outskirts of the system. Here this gas forms a bridge between the galaxy pair and also possible tidal features towards the east and west. The N HI morphology from this reduction appears broadly consistent with the significant detected H i in the VLA C-and D-configuration data presented in Pustilnik et al. (2001). We find a similar level of agreement between our maps and the GMRT 21-cm observations presented by Ekta et al. (2009). As described both by Ekta et al. (2009) and Pustilnik et al. (2001) in detail, the morphological features (and also the kinematics) of the H i halo gas on larger scales strongly indicate an early stage of a merger between both galaxies. The onset of this merger is hypothesised as having triggered the current star-formation episodes in both galaxies. Comparing the morphology of the H i envelope and the extended Hα emission, it is evident that the filaments extend into regions that are characterised by lower neutral column densities or, at greater radial distance, are even devoid of detected H i; This is especially so for the eastern filament. However, the western filament appears to overlap more with the low-column gas of the bridge. Especially in Fig. 8 it appears as if the extended Hα structure starts to interact with the extended H i halo of the western galaxy. Interestingly, this boundary coincides with the fanning out of the western filament (Fig. 2). The elongation of the H i halo surrounding the eastern galaxy is oriented almost perpendicular to the direction of the filaments. Emission line ratios along the filaments Our detections of Hβ and [O iii] λ5007, in concert with Hα, enable us to make limited inferences on the physical state of the emitting plasma in the filaments. We show in Fig. 9 the continuum-subtracted narrowband images in [O iii] λ5007 and Hβ. The images were synthesised by summing over the datacube layers at 4927.46 ± 1.25Å and 5074.96 ± 1.25Å observed frame for Hβ and [O iii], respectively. These narrow windows are adequate, since the line emission in the filaments is barely resolved (see kinematic analysis in Sect. 3.3.1). We analyse the emission line ratios Hβ/Hα (Sect. 3.2.1) and [O iii]/Hα (Sect. 3.2.2) within two rectangular regions that cover each filament of an area where all three lines are detected; these slits are indicated in the bottom left panel of Fig. 1. Both slits cover 6 across each filament, and the length of the eastern and western slits are 27.6 and 19.2 , respectively. The flux distribution of Hα, Hβ, and [O iii] along (∆l) and across (∆w) the filaments is shown in the top three panels of Fig. 10. These flux distributions were extracted directly from the synthesised narrowband images. Corresponding variance maps in all lines were created by summing the respective layers in the variance datacube. We measured the individual emission line fluxes by summing the emission across the filaments in bins of 1.2 along the direction of the slit. We then calculate the ratios for each bin after correcting the individual measurements for galactic foreground emission using the Cardelli et al. (1989) extinction curve for a given A V = 0.123 from the Galactic dust map of Schlafly & Finkbeiner (2011). Inverse Balmer decrement Hβ/Hα Traditionally the Balmer decrement, Hα/Hβ, is used to measure interstellar extinction. We here use the inverse of the Balmer decrement since the statistics for the ratios of measurements for the low surface-brightness emission close to the detection limit are only well defined if the denominator is chosen to have the smaller error-bar. Then the standard 1-σ error on the ratio, where Var(Hα) and Var(Hβ) are the variances on the extracted fluxes Hα and Hβ, respectively. Equation (2) is only valid as long as Var(Hβ) 0.25 Hβ (e.g. Dunlap & Silver 1986), which is the case here. To aid the interpretation of this non-standard ratio, we plot in Fig. 11 its behaviour as a function of temperature for a recombining plasma under Case-A and Case-B recombination scenarios. Both in Fig. 10 and Fig. 11 we also include ticks on the right ordinate that allow reading off the conventional Balmer decrement. From an integrated spectrum covering the star-forming sites in the galaxy (depicted in Fig. 3 in Wofford et al. 2021) we find an inverse Balmer decrement of 0.34 (or Hα/Hβ = 2.92). As can be seen in Fig. 10, both the eastern and the western filament are predominantly characterised by higher Hβ/Hα ratios (lower Balmer decrements), that is Hβ/Hα 0.35 (Hα/Hβ 2.86). Interesting features in the radial ratio profiles are the dips before and after the Hα "hot-spot" in the western filament, for which the higher ratios appear closer to the galaxy; in contrast, the eastern filament shows the highest values at its end. The mean (median) of Hβ/Hα along the eastern and western filament are 0.4 (0.38) and 0.41 (0.39), respectively. Some of the here observed (high) low (inverse) Balmer decrements in the filaments are not compatible with standard Case-B or even Case-A recombination values (Fig. 11). The ansatz of the Case-B recombination scenario is infinite optical depth in all Lyman series lines (τ Lyn = ∞), but zero optical depth in all other transitions, whereas for Case-A also τ Lyn = 0. Case-B delivers realistic values for Hydrogen line ratios in the interstellar medium, since the absorption cross section for Lyman series photons is very high. Both in Case-B and in Case-A Hβ/Hα increases with increasing temperature, but the increase is more rapid for Case-A. However, even for T ≈ 3 × 10 4 K, the highest temperature provided in the calculations by Storey & Hummer (1995), the resulting Hβ/Hα = 0.385 for Case-A (red dashed line in Fig. 10) and Hβ/Hα = 0.370 for Case-B are well below the measured ratios, 0.4 Hβ/Hα 0.5, at the beginning (end) of the western (eastern) filament. Since the low-surface brightness filaments are observed in the low-density outskirts of the galaxy it is qualitatively conceivable that the filaments originate in gas where neither the A&A proofs: manuscript no. sbs0335_musevla_revision_le (2). For Hβ/Hα the Case-A recombination expectations from Storey & Hummer (1995), are indicated as dashed lines for three different temperatures (T = {1, 2, 3} × 10 4 K in green, orange, and red); cf. Fig. 11. For Hβ/Hα the corresponding conventional Balmer decrement values are measured on the right ordinate. τ Lyn = ∞ nor the τ Lyn = 0 approximation can be justified. Radiative transfer effects thus may alter the population levels, and neither the Case-A or Case-B approximation are reasonable anymore. Calculations for this intermediate τ Lyn → 0 regime were performed by Capriotti (1966) and Cox & Mathews (1969); cf. Chapter 4.5 of Osterbrock & Ferland (2006). Especially at low optical depths in Lyα (τ Lyα < 10 3 ) Balmer decrements compatible with the range of the here measured ratios in the filaments, 2 < Hα/Hβ < 2.5 (0.4 ≤ Hβ/Hα ≤ 0.5), are possible ( Fig. 4.3 in Osterbrock & Ferland 2006). Deviations from equilibrium occur in media that are ionised of fast-radiative shocks, but model calculations in low-metallicity environments favour higher Balmer decrements ≈ 3 (Alarie & Morisset 2019). [O iii]/Hα The (Cairós et al. 2022). From Fig. 9 it could be already anticipated that the western filament is significantly dimmer in [O iii] (see also Fig. 1 in H17). The ratio in the eastern filament is relatively flat, and only at the end (∆l 17 ) of it does the [O iii] flux decrease more rapidly than Hα. For the western filament the ratio appears to show an overall decrease with increasing distance from the galaxy, albeit with significant fluctuations. There appears to be no strong correlation between trends seen in Hβ/Hα and [O iii]/Hα. Interpreting the [O iii]/Hα ratio is complicated, as it is regulated by multiple parameters that characterise the physical conditions of the emitting plasma. At low-densities (n 1000 cm −3 ) the emissivity of [O iii] λ5007 depends only on T e (Luridiana et al. 2015;Morisset et al. 2020), but the ratio of O 2+ to H + ions is influenced both by the oxygen abundance and the ionisation mechanism. More specifically, if O 2+ /H + is held fixed then the important parameters regulating [O iii]/Hα are the hardness of the ionising radiation field and the ionisation parameter (ratio between density of ionising photons to density of H atoms) for photo-ionisation, whereas in fast radiative shocks the shockvelocity, density, and the strength of the magnetic field are influential (e.g. Alarie & Morisset 2019). We recall that the western filament appears to interact with the low neutral column gas that belongs to the tidal bridge that connects the western with the eastern halo, whereas the eastern filament appears to protrude more freely out of the neutral envelope (Sect. 3.1.2 and Fig. 8). Thus, a plausible interaction with the more pristine halo gas in the western filament might lower the oxygen abundance and thereby also attenuate the [O iii] line. On the other hand, this interaction could also lead to a higher neutral fraction within the western filament that in turn lowers the ionisation parameter and thereby also attenuates [O iii] line. Kinematic analysis H ii kinematics In Fig. 12 we display the line-of-sight velocity field (v los , top panel) and the velocity dispersion (σ v , bottom panel) of the line emitting plasma. We created these maps from fits of Gaussian profiles to Hα, Hβ, and [O iii] λλ4959,5007 in a Voronoi tessellated 3 continuum-subtracted datacube; in the outer filaments only Hα is contributing to the kinematic signal. By experimenting we found that a S /N threshold of 8.5 and a maximum binsize of 4 square arcseconds provided useful results, that is to say those parameters preserve the morphological appearance of the filaments while also providing a large number of high-SN bins throughout the low-SB regions; all bins in the high-SB region 3 The Voronoi tessellation was computed with the algorithm of Diehl & Statler (2006), which is an extension of the Cappellari & Copin (2003) binning scheme. consist of single spaxels. Despite the morphological complexity, the emission lines are all well modelled by a single Gaussian. We show in Appendix A the line profiles and model along the main filaments. Only for a few spaxels in the central regions a weaker secondary component could be added to better fit the observed skew in some profiles. As zero-point velocity we set v 0 = c · z = 4053 km s −1 and σ v is corrected for the instrumental line spread function using the prescription provided in the MUSE user manual. Low S/N σ v values obtained from Gaussian fits are known to be biased high, since increasing the noise relative to the line flux artificially broadens the fits. Therefore we show in the bottom panel of Fig. 12 only σ v 's where SB Hα > 1.5 × 10 −18 erg s −1 cm −2 arcsec −2 . Moreover, we only include bins in the v LOS map where the measurement uncertainty ∆v < 15 km s −1 . In the low-SB filaments typical values for the 1-σ uncertainties on velocity and velocity dispersion are ∆v ∼ 10 km s −1 and ∆σ v ∼ 20 km s −1 , respectively. Further, it is known for MUSE data, that velocity fields derived from narrow emission lines exhibit a characteristic striping pattern on small scales (see Sect. 3.4 in Weilbacher et al. 2015); this is due to the LSF being undersampled by the MUSE spectrograph. For the display in Fig. 12 we removed the most prominent effects of this striping pattern by slightly degrading the spatial resolution of the velocity fields. This is achieved by smoothing the computed v LOS map with a circular top-hat kernel of 0.4 (2 pixels) radius. The H ii v LOS map in the central high-SB region appears quite chaotic at first glance. Nevertheless, Moiseev et al. (2010), using SAO/SCORPIO Fabry-Perot spectroscopy of Hα in the central region and using VLA C-and D-configuration 21 cm data from Pustilnik et al. (2001), showed that the velocity field can be characterised by a disk-like north-west to south-east gradient that exhibits a distinct perturbation in the centre. Moiseev et al. modelled this central disturbance, which is shaped somewhat like a hook, as the effect of an expanding super-bubble disturbing a disk-like velocity field. The best-fitting disk was found to be aligned at a position angle of 53 • east of north. We plot this kinematic axis and the orthogonal axis forced through the photometric centre (Sect. 3.1.1) in Fig. 12. It can be appreciated how the orthogonal axis appears almost symmetric in between the two filaments. We do not observe velocity gradients along the direction of the extending filaments. As already noted in H17, the filaments seemingly inherit the velocity that is found at their base where they connect the high-surface brightness region. The northern filament appears to inherit the blue-shifts from the central perturbation, whereas the western filament appears slightly more redshifted. A conspicuous kinematic feature is the blue-shift (≈ −35 km s −1 ) at the western sub-branch at the tip of the westernfilament. As analysed in Sect. 3.1.2, this is where the 21 cm maps indicate an overlap between Hα emitting gas and the H i bridge that connects to the western galaxy. Excluding this sub-branch, the velocity difference between the average velocities in the filament is ∼ 3 − 5 km s −1 . The overall shear of the complete velocity field is v shear = 1/2 × (v 95 − v 5 ) = 28.5 km s −1 , with v 95 and v 5 being the upper and lower fifth-percentile of the velocity map. By limiting the calculation of v shear only to the high-SB region (SB Hα > 12.5 × 10 −18 erg s −1 cm −2 arcsec −2 ) we find v >12.5 shear = 22.9 km s −1 . This can be compared to the disk model by Moiseev et al. (2010), that is characterised by a maximal rotation velocity of v max = 28.2 km s −1 and an inclination of i = 37 • (see also Moiseev et al. 2015), that is the projected maximum amplitude of this disk is v max · sin i = 17 km s −1 < v >12. tributed this discrepancy between disk-like and observed motions in the high-SB region to non-circular motions due to the expanding shell near the centre. For the large-scale H ii velocity field the discrepancy is even more pronounced. That neither a large velocity offset between the filaments, nor a strong velocity gradient along the filaments, nor abrupt discontinuities with respect to the central velocity field are observed may be caused by projection effects (i.e. because of material moving mostly perpendicular to our sightline). However, it may also indicate the absence of strong flows of ionised gas along the filaments. The eastern "foreshortened" filaments (cf. Fig. 2) are not characterised by an abrupt break in line-of-sight velocities. Ostensibly they just continue the velocity gradient towards positive velocities along the north-eastern direction. However, the v shear > v >12.5 shear behaviour is caused by the redshifts of the foreshortened filaments and by the blue-shift in the western subbranch of the western filament. Hence, The velocities observed in those extended regions are not compatible with an idealised disk and thus the motions of the ionised filamentary halo gas are not predominantly driven by the gravitational potential of the galaxy. The velocity dispersion map in Fig. 12 (bottom panel) shows that the filaments are characterised by relatively narrow emission, with average values of ≈ 30 km s −1 . This is similar to the galaxy's luminosity weighted average σ 0 = x,y NB Hα x,y σ x,y / x,y NB Hα x,y = 29 km s −1 ; here σ x,y denote the dis- (Fig. 1). This value is in exact agreement with σ 0 derived from high-spectral resolution (R ∼ 86000) Fabry-Perot spectroscopy (Moiseev et al. 2010(Moiseev et al. , 2015. We note, however, that in the filaments σ v /∆σ v 2, that is there the Hα line is barely resolved with MUSE. The ratios of large-scale ordered motions, quantified by v max or v shear , to localised unordered motions, quantified by σ 0 , are a useful quantity to characterise the kinematical state of a galaxy (see review by Glazebrook 2013). We confirm v max /σ 0 ≈ 1 for the disk model of Moiseev et al. (2010), and we note v shear /σ 0 ∼ v max /σ 0 . The v/σ-ratio for SBS 0335-052E is lower that what is observed for typical disk-galaxies (v max /σ 0 4). A large fraction of star-forming galaxies at high-z show also v max /σ 0 1 (e.g. Turner et al. 2017;Wisnioski et al. 2019). Star-formationdriven feedback is deemed to be an important driver of those so-called dispersion dominated kinematics. The zone of narrow lines at the photometric centre is surrounded by zones of elevated velocity dispersions (45 -55 km s −1 , towards the north and south). These zones appear as double-peaked Hα profiles in high spectral resolution (R ∼ 10000) VLT/GIRAFFE ARGUS data (Izotov et al. 2006, their Fig. 7a), but at our lower spectral resolution we here only observe a broadening. These double peaked profiles were interpreted as expanding shells. Moreover, Izotov et al. (2006) conjectured the launch of an outflow perpendicular to the disk from the intricate line profiles seen in the small ARGUS field of view (see especially Fig. 7 and Fig. 8). This is consistent with our interpretation of the filaments being large-scale effects of this outflow (see discussion in Sect. 4). H i kinematics In Fig. 13 we display the intensity weighted line-of-sight velocity field of the 21 cm signal from SBS 0335-052E & W, both for the untapered and the 7kλ-tapered data products. The velocity maps in km s −1 were computed from the moment-1 maps produced by the source finding software SoFiA (Serra et al. 2015). The relevant input parameters for our SoFiA runs were stated in Sect. 3.1.2. The velocity map from the 7kλ-tapered datacube appears quantitatively consistent with the maps from GMRT data (Ekta et al. 2009) and with the maps from VLA C-and D-configuration data (Pustilnik et al. 2001). The H i envelope encompassing both systems may be characterised by an overall shear from east to west, whose gradient is, however, not perfectly smooth. Both Ekta et al. (2009) andPustilnik et al. (2001) argued for the envelope being comprised of two disk-like systems. Judging from our maps, but also from the maps presented in the aforementioned publications (Fig. 6 in Pustilnik et al. 2001 and Fig. 1 in Ekta et al. 2009), this reading appears overly simplistic. Focusing here on the eastern galaxy, another gradient from redshifts in the south-east to blue-shifts the north-west is perceivable. The alignment of this gradient along the directions of the filaments is suggestive of causal relationships between ionised and neutral phase. At the higher spatial resolution of the untapered datacube the velocity fields exhibit an even higher complexity, and simple disk-like gradients appear indiscernible. In the top-left panel of Fig. 14 we show a magnified view of the velocity field of the eastern galaxy; an overall north-west to south-east shear may be envisioned, but this velocity gradient is significantly warped. As already noted in Sect. 3.1.2, the faintest H i signals from the outskirts are often only single channel detections. This can also be appreciated from the channel maps in Appendix B. While our data does not allow for robust higher-order measurements, these channel maps supplement the view of high kinematic complexity in the H i on kpc scales in the system. From H i morphology and kinematics alone it is not possible ascribe tidal-or feedback effects as the source of this complexity. We thus proceed by comparing H i and H ii kinematics quantitatively in the next section. Comparison between H i and H ii kinematics We compare the ionised gas velocity field derived from the MUSE data (Sect. 3.3.1) to the H i velocity field from the VLA observations (Sect. 3.3.2). For this comparison we need to account for the lower spatial resolution and the coarser spatial sampling of the H i data products. To this aim we follow the method used by van Eymeren et al. (2009a,b) and convolve the H ii v los map (Fig. 12) with a 2D Gaussian, whose position angle, major-, and minor-axis are matched to the beam of the respective 21 cm data. We next re-bin this convolved velocity field to the native resolution of the VLA datacubes. The complete process is illustrated in Fig. 14, with the top right panel displaying the H ii velocity field as observed, with the bottom left panel displaying it after convolution, and with the bottom right panel displaying this convolved field after re-binning. For the 7kλ-tapered dataset we omit the intermediate step and show only the resulting resampled velocity field (bottom panel of Fig. 15). Qualitatively, both the velocity fields from the untapered and the 7kλ-tapered H i data show a high level of congruence with the degraded and resampled H ii velocity fields (Fig. 14 and Fig. 15). Our method ignored the effect of intensity variations on the resulting resolution-degraded and re-binned H ii velocity fields. If we instead convolve each MUSE datacube layer with the beamimitating 2D Gaussian prior to the kinematic fitting, then the resulting degraded velocity fields show almost no spatial variation, as they mostly trace H ii velocities stemming from the brightest regions. Hence, the unknown intrinsic 21-cm signal from the neutral phase does not mimic the extreme intensity variations seen in the H ii emitting plasma. The degraded H ii velocity fields appear more smoothly varying compared to the H i measurements, which is partly related to the low-SN of the H i data. We analyse the quantitative differences between H i velocity fields, v HI , and the resolution degraded and re-sampled H ii velocity fields, v rep HII , in Fig. 16. There we display ∆v = v HI − v rep HII maps both for the untapered and the 7kλ-tapered datasets. These ∆v-maps exhibit mostly an amplitude of absolute differences 10 km s −1 and thus confirm the qualitative impression of overall congruence already anticipated from Figs. 14 and 15. The similarity between both velocity fields manifests in small mean (median) absolute differences of 0.6 km s −1 (0.2 km s −1 ) and 1.1 km s −1 (0.6 km s −1 ) for the untapered and 7kλ-tapered velocity fields, respectively. There are, nevertheless, some notable differences. In the untapered data we find ∆v 15 km s −1 near the centre. This v HI > v rep HII zone is in close vicinity to the most prominent "hook-like" disturbance of the H ii map. A causal connection between H i-H ii velocity offsets and the expansion thus suggests itself. In the 7kλ-tapered data we find a prominent rim v HI > v rep HII along the south-west, parallel to the minor axis. The origin of this feature is not obvious. Both high-∆v zones in both datasets are related to the most prominent complexities of the H i velocity fields that inhibited the identification of simple disk-like velocity gradients in the first place (Sect. 3.3.2). Discussion The presented observational evidence in the previous section leads us to explain the observed features in the circumgalactic halo of SBS 0335-052E as limb-brightened edges of a conical outflow phenomenon. We briefly summarise the main clues towards the suggested interpretation before making an attempt to quantitatively illustrate this scenario with a didactic toy model. VLT/MUSE reveals two faint ionised filaments of delicate morphology that extent up to 15 kpc to the north-west of the starburst (Fig. 1). The orientation of a perceived symmetry axis that separates the rays of the filaments is perpendicular to the shearing direction of the large-scale velocity field (Fig. 12). This large-scale shear lines up with the major axis of the disk-model derived from SAO/SCORPIO Fabry-Perot observations in the central high-SB Hα region (Moiseev et al. 2010(Moiseev et al. , 2015; therefore, the symmetry axis of the bifurcating filaments is nearly parallel with the minor axis of this model (Fig. 12). The ionised filaments protrude away from the elongation axis of the H i envelope (Fig. 8). This envelope also exhibits overall disk-like shearing motions (Fig. 15). These observational facts are reminiscent to prototypical conical outflows in disk-galaxies that launch such cones from their central starburst outwards from the disk plane (Veilleux et al. 2005;Bland-Hawthorn et al. 2007;Nelson et al. 2019). We explore the possibility of this scenario by setting up a heavily idealised astrophysical structure with parameters suggested by the observations. Our toy model geometry is a simple cone of opening angle θ and height h. The apex of this cone is fixed at the centre of a disk at position angle ϑ and inclination i. Thus the observed opening angle, θ P , is a projection of the true cone opening θ = 2 × arctan[sin(i) × tan(θ P /2)]. We assume that the Hα filaments are confining this cone. We measure θ P = 34 • as the angle between the filaments from the Hα NB (Fig. 1) and the disk-model of Moiseev et al. (2010) which fits i = 43 • , hence θ = 27 • . We note, however, that conical winds can be tilted and asymmetric with respect to the centre; our value for θ thus represents an approximate working hypothesis. We also adopt ϑ = 52 • from Moiseev et al. (2010). We fix l p = 10 kpc for the projected length of the filaments from Fig. 1, where we ignore for now the threadlike extension. The height of the inclined cone is then h = l p · cos(θ/2)/ sin(i) = 16.2 kpc. Our estimate of the height h may be translated into a velocity requirement that the hypothesised wind from a stellar-population of age t * must have to blow out such a structure. The oldest starclusters in the north (SSC 5 & SSC 6;Fig. 5) have ages t * 15 Myr (Reines et al. 2008;Adamo et al. 2010). Accounting for the delay after which supernovae start to inject kinetic energy, t SN ≈ 5 Myr, we have v wind = h/(t * − t SN ) = 10 kpc/10 Myr × cos(θ/2)/ sin(i) ≈ 1000 km s −1 × cos(θ/2)/ sin(i) = 1620 km s −1 . This rough limit, which neglects variations in energy input and flow-speed, is broadly consistent with predictions for velocities in the tenuous hot phase (T > 10 6 −10 7 K, n ∼ 10 −3 −10 −4 cm −3 ) of galactic winds. This "wind-fluid" is believed to carry most of the momentum and energy of an outflow, and at larger radii even most of the mass (Schneider et al. 2020). If the volume of the cone, V hot = π/3 · tan 2 (θ/2) · h 3 = 256 kpc 3 , would be filled uniformly with this fluid, then its mass would comprise M hot ∼ 6 × 10 6 -6 × 10 5 M . This requires loading rates, Λ hot = (M hot )/(Ṁ SFR · ∆t), of order unity if all the mass is asserted to be loaded into the wind via star-formation over the last ∆t = 10 Myr given the determinedṀ SFR 1 M yr −1 in SBS 0335-052E (Reines et al. 2008;Thompson et al. 2009). This rough limit for Λ hot is higher than the expected average mass-return from stellar winds and supernovae in stellar populations, Λ = 0.1 (Leitherer et al. 1999), that is also used as the injected mass fraction for the hot phase in computer simulations of wind phenomena (e.g. Schneider & Robertson 2018;Schneider et al. 2020). However, a significant fraction of the fluid will consist of shock-heated ambient ISM, hence the required mass loaded into the interior of our cone does not appear unphysical. The hot fluid, which does not emit in optical recombination lines, is assumed to be surrounded by the warm ionised medium that fills the wall of the cone and leads to the observed structure in Hα. Denoting the thickness of the walls as t and writing ∆h = 2 · t · cos(θ/2) the volume inhabited by the diffuse H ii gas is V HII = π/3 · tan 2 (θ/2) · (h + ∆h) 3 − h 3 − ∆h 3 . Figuratively, V HII is the volume between two identical cones pushed into each other (∝ (h + ∆h) 3 − h 3 ), but ignoring the volume between the apexes of the inner and the outer cone (∝ ∆h 3 ). From the Hα NB we estimate t = 1.5 kpc and hence V HII = 142 kpc 3 . In the spirit of a toy-model we assume the plasma filling V HII is in recombination equilibrium, fully aware that deviations from equilibrium are expected in shocked gas (e.g. Allen et al. 2008;Morisset et al. 2020) T filling a volume V emits Hα emission at luminosity L Hα = Hα (T ) · n 2 · V (3) in recombination equilibrium, where (T ) denotes the Hα equilibrium emissivity that follows from atomic data. For T = {1.0, 1.5, 2.0} × 10 4 K in case-A recombination A Hα (T ) = {2.3, 1.5, 1.1} × 10 −25 erg s −1 cm 3 (Storey & Hummer 1995); the small density dependence on can be neglected for our purposes. Using a hand-drawn aperture that encompasses the filamentary structure while not covering inner high-SB spurs and shell-like structures (thin dotted line in Fig. 2), we estimate F Hα ≈ 2.6 × 10 −15 erg s −1 cm −2 as the flux from the supposed cone. From Eq. (3) then n A = {3, 4, 5} × 10 −2 cm −3 follows for T = {1, 1.5, 2} × 10 4 K in case-A. We recall from Sect. 3.2.1 that Case-A recombination appears more compatible with the observed Balmer decrement in the filaments than case-B recombination, noting here that B Hα (T ) < A Hα (T ) and that B Hα / B Hα is only mildly temperature dependent. Therefore we can approximate B Hα ≈ 0.65 A Hα and hence n B ≈ 1.24 · n A would be required for case-B. The required densities in the walls of the cone are only a factor of two to three lower than the canonical n = 0.1 cm −3 for diffuse warm ionised gas in the circumgalactic medium. They are consistent with the expectations for the T ∼ 10 4 K phase of galactic winds. The correspondence between the derived parameters in our idealised setup and the physical characteristics expected in outflows appears encouraging and supportive of the idea that we indeed observe a 15 kpc outflow cone emanating from a tiny starburst galaxy. We remark that the mass of ionised hydrogen in the toy-model, M HII = 1.5 − 1.7 × 10 8 M , requires a mass-loading factor Λ HII 10 if this mass has been loaded into the wind over the last 10 Myr. Neglecting the cold phase, which is expected to provide only a minor contribution to the total mass of outflowing material, we arrive at a total mass loading factor of Λ = (M hot + M HII )/(Ṁ SFR · ∆t) 10. Given the low-stellar mass of SBS 0335-052E, M ≈ 6 × 10 6 M (Reines et al. 2008), the required Λ is in line with theoretical expectations (Pandya et al. 2021) and observed trends for the other dwarf starbursts (Collins & Read 2022, their Fig. 4). Interestingly, the ratios Λ hot /Λ ≈ 0.1 and Λ HII /Λ ≈ 1 are in line with the expectations of the wind models by Pandya et al. (2021). We caution that our results have to be taken cum grano salis, as we assumed that the diffuse Hα emission in the cone is caused by pure Case-A recombination. As explained above, slightly higher densities would be required for the Case-B scenario, thus the mass in the warm phase would be slightly lower. However, we have shown in Sect. 3.2.1 that the Balmer-decrement in the filaments is suggestive of neither Case-A or Case-B being appropriate. Ultimately photo-ionisation models are required to obtain more stringent constraints on the observed mass in the warm phase. We now explore the observable Hα morphology of our toymodel cone-wall structure numerically. To this aim we compute the emission on a cubic 3D lattice with 500 3 volume cells that cover a 25 3 kpc 3 volume, that is each cell has a volume of V cell = 50 3 pc 3 . Then all grid cells C are set to emit Hα according to Eq. (3) with V cell when their centre-coordinates are within the walls of the cone. Formally, this is the set of cells C = {x , y , z : f (x , y , z |h, θ) ≤ 0 ⊕ f (x , y , z |h + ∆h, θ) ≤ 0}, where ⊕ denotes the exclusive disjunction and f (x , y , z |h, θ) = x 2 + y 2 − z 2 tan 2 (θ/2) is the implicit parametric equation for the surface of a single cone and x , y , and z ∈ [0, h] are the coordinates in a rotated frame that is tilted along ϑ and inclined along i using standard intrinsic Euler rotations. We set up the model grid with cell-coordinates x, y, z such that the y-axis is parallel to the sight-line. We project along this axis by summation after dividing by the surface area of a cell (6V 2/3 cell ). This provides a map of surface luminosities (erg s −1 kpc −2 ). This map is transformed into observable fluxes in erg s −1 cm −2 on a grid of 0.2 MUSE-like spaxels for the given distance to galaxy. Prior to this re-binning the received fluxes are convolved with a Gaussian of 1.2 FWHM, which was the average DIMM seeing in our observations (Sect. 2.1). Finally, we add noise to the re-binned image consistent with the noise determined in the outskirts of the Hα NB (σ Hα = 1.8 × 10 −19 erg s −2 cm −2 ). We show the result of this exercise for the observationally constrained parameters from above (θ = 27 • , i = 37 • , ϑ = 52 • , h = 16.2 kpc, and t = 1.5 kpc) and (n, T ) = (0.05 cm −3 , 2 × 10 4 K) in Fig. 17. To provide visual intuition regarding the projected morphology for different geometrical parameters we study the effects of varying θ, ϑ, and i in Appendix C; we note that the projection of the model is the same if the cone is inclined towards us or away from us. Furthermore we examine the effects of varying T and n on the overall observability in Appendix C. We also verified the consistency between numerical computation and the analytical results presented beforehand. We overlay in Fig. 17 the two faintest Hα SB isocontours from Fig. 1 to visualise morphological similarities and differences between calculation and observation. These contours were aligned by eye to maximise congruity. The resemblance between the MUSE mock data of the cone-wall model and observed reality is readily apparent. An interesting feature of the model is the brightening along the walls with increasing distance from the apex. This radial brightening is caused by more wall material being intersected perpendicular to the sightline closer to the base of the cone. In reality we observe a brightening with increasing distance only in the western filament (Hα "hot-spot" in Fig. 2); however, compared to the smooth radial gradient in the model this feature emerges more abruptly. In fact, the observed filaments brighten towards the central high-SB zone. Density or temperature variations, as well as variations of the ionisation mechanisms, may be invoked to explain the differences between model and observations and to produce localised features such as the Hα hot-spot. The observations reveal further morphological sub-structure that is not reproducible with our simple purely geometrical setup. Most prominently, the cone-wall structure cannot explain the threadlike extension and the fanning out of the western filament (Fig. 2). Parts of these filamentary sub-structures coincide spatially with the region where only one wall of the cone would be intersected by the sight-lines. Zones of higher density or lower temperature in the wall section or lumps of gas that have been advected by the hot fluid could lead to brighter Hα patches. Alternatively, these features may be interpreted as the terminating cloud-wind interface where dense gas has been compressed and pushed up by the hot wind fluid and now cools radiatively. State-of-the-art numerical models of starburst-driven winds (Schneider & Robertson 2018;Schneider et al. 2020) produce small-scale density and temperature variations in wind structures that all have been neglected in our didactic scenario. Substructure within the hot wind fluid arises due to interplay between the different gas-phases; cold and warm material swept up from the ISM may survive within the hot flow whereas density variations in the hot phase might lead to condensation and radiative cooling. Substructures in the HII emitting phase produce turbulent mixing layers between the fast moving tenuous fluid and the slower-moving denser warm-phase. The emergence of a conical structure appears, however, a common feature of wind phenomena. This "beaming" is believed to be a purely hydrodynamical collimation effect due to the intrinsic density distribution in the gaseous disk surrounding the energy and momentum injecting star-formation sites (Nelson et al. 2019). In this respect it appears interesting to note that Micheva et al. (2019) advocate to interpret θ as the opening angle of a Mach cone, that is the surfaces of the cone are created by a shock-front from the overlapping sound waves in a supersonic flow. The opening angle is then related to the Mach number M, that expresses the velocity of the flow as multiples of the speed of sound via M = 1/ sin(θ/2). For our cone we have M 4, that is to say in this interpretation the wind fluid is required to be super-sonic. Intriguingly, M 4 flows are again consistent with sophisticated model expectations for the tenuous hot wind fluid. Obviously our toy-model is static and neglects kinematics. A kinematic model requires many assumptions and is beyond the scope of our current analysis. Still, some important kinematical features need to be discussed qualitatively, especially since the absence of a strong velocity gradient and the narrow line widths appear incompatible with the idea that the H ii phase is being accelerated in an outflow. Projection affects surely may influence the observed line-of-sight velocities and velocity dispersions given that we look at the cone from the side. However, under the assumption of isotropic random motions the combination of ordered motions and inclination can only broaden the line. This effect is known to as "mixing term" in observational studies of velocity dispersions in disk galaxies (see, e.g. Sect. 2.1 in Bouché et al. 2015). In this scenario then the narrowness of the lines would indeed reflect an intrinsic narrow velocity dispersion. A projection effect is thus only expected if the distribution of velocities is larger parallel to the walls of the cone. Similarly, the lack of an observed velocity gradient along the walls of the cone may be an projection effect. Nevertheless, the large-scale wind simulations by Schneider et al. (2020) show that the H ii emitting gas at larger radii is moving at fairly constant velocities. Interestingly, we do not observe strong velocity differences between H i and H ii at the base of the cone in the 7kλ-tapered data ( Fig. 15 and right panel of Fig. 16). We thus speculate, that the walls of the cone are H i halo gas that is advected into the flow of the hot fluid. During this advection there might only be a short window of time before the dense warm medium, which is likely in the form of clumps, is being transformed into a more dilute wind fluid. If this is the case, then we expect that here observed emission of warm phase inherits mostly the velocity dispersion of the neutral gas. In this scenario the narrow lines appear consistent with the detection of H i emission in only 2-3 channels (see Fig. B.2), which corresponds to ∼ 20 − 30 km s −1 . Summary and conclusions We have presented new MUSE and VLA B-configuration 21cm observations of the extremely low-metallicity blue compact starburst galaxy SBS 0335-052E. Our observations reveal the continuation of two ionised filaments in Hα emission towards the north-west of the galaxy's main stellar body (Sect. 3.1.1 and Fig. 1). The onset of this structure was observed with MUSE observations centred on the galaxy and described in H17. The morphology of the newly detected prolongation can be described as thread-like lacy fragments. These somewhat detached portions of extended Hα emission are of lower SB (SB Hα ≈ 1.5 × 10 −18 erg s −1 cm −2 arcsec −2 ) than the inner straight portions of the filaments (SB Hα ≈ 3 × 10 −18 erg s −1 cm −2 arcsec −2 ). The rays of the inner filaments are aligned at a projected angle of θ P = 43 • relative to each other. Diffuse ionised gas with a smoothly declining brightness profile is found up to ∼ 25 towards the south of the galaxy. The inner parts of the filaments are detected in Hα, [O iii] λ5007, and Hβ. The flux ratios between those three lines are different compared to typical ISM ratios in the vicinity of star formation. The inferred average Balmer decrement is Hα/Hβ = 2.5 for both filaments, whereas [O iii]/Hα = 0.65 and [O iii]/Hα = 0.26 for the eastern and western filament, respectively. The unusual low Balmer decrement, which does not agree with standard recombination cascade expectations, is deemed to be caused by radiative transfer effects in the low-density halo gas that alter the Sect. 4 discusses the ionised cone-wall structure for parameters that are derived from the observations. However, it appears instructive to visualise projections of this structures for different paramters. In Fig. C.1 we fix the opening angle θ = 27 from Sect. 4, but vary ϑ and i. In Fig. C.2 we leave i = 37 • and ϑ = 52 • as in Sect. 4, but vary the opening angle. While for most of the parameters the cone geometry is apparent, projections at low inclination or large opening angles appear more disk-like. From Eq. (3) it is apparent that there is a degeneracy in observable flux between T and n. To obtain a feel for the effects of those parameters on the observable morphology, we explore variations of T at fixed n in Fig. C.3 while in Fig. C.4 we fix T but vary n. As in Sect. 4 we use case-A recombination emissivities for Hydrogen computed by Storey & Hummer (1995). alongside the fitted Gaussian models (centre panels) and the residuals (data -model, bottom panels). The left panels show the results for the eastern filament, whereas the right panels show the result for the western filament and the horizontal extend is the same as in Fig. 10. The x-axis is the cross dispersion axis in units of MUSE spaxels (1 spaxel = 0.2 ) and the y-axis is the dispersion axis (1 pixel = 1.25Å). The colour bar on the left corresponds to the two top panels, whereas the colour bar on the right corresponds to the bottom panel. Article number, page 20 of 22 Each channel is shown with a linear stretch from 0 to 6σ, where σ = 2.5 × 10 −4 Jy beam −1 is determined from fitting the negative distribution of pixels in each channel. Contours are drawn at 2σ (dashed), 4σ, and 6σ. The bottom right panel shows the same viewport (46 / 12.7 kpc along each axis), but with the Hα NB image and the total N HI contours from Fig. 1. [O iii] and Hβ detections, on the other hand, are confined exclusively to the filaments. The northern filament shows quite a strong detection in [O iii] (SN 25), whereas the north-western filament is significantly weaker in [O iii] (SN 10). An analysis of the observed line ratios from the filaments is presented in Sect. 3.2. Fig. 1 . 1H ii and H i emission as seen with MUSE and VLA around SBS 0335-052E. Top panel: Continuum-subtracted Hα narrowband image. Fluxes are encoded via an asinh-scaled cyclic colour map from 0 to 10 −18 erg s −1 cm −2 in the low surface-brightness, and from 10 −18 erg s −1 cm −2 to 10 −16 erg s −1 cm −2 in the central high surface-brightness region. The cyan solid contours demarcate Hα isophotes at SB Hα = {1.5, 2.5, 5, 12.5} × 10 −18 erg s −1 cm −2 arcsec −2 while the white dotted contour demarcates the limiting surface-brightness of SB Hα = 7.5×10 −19 erg s −1 cm −2 arcsec −2 . The image has been smoothed with a Gaussian of 0.4 FWHM. Subdued grey contours indicate HI column densities N HI = {5, 10, 20, 30, 40}×10 20 cm −2 from our VLA-B configuration observations; these contours are displayed more prominently inFig. 4. The dashed white contour demarcates the 2.5σ = 2.5 × 10 20 cm −2 detection limit. The VLA-B configuration beam, 6.1 ×4.6 oriented 23.6 • west of north, is indicated via a grey shaded ellipse in the top left. The displayed field of view is 1 35 × 1 46 corresponding to 26.2 kpc × 29.2 kpc in projection. The dashed violet square in the centre indicates the viewport used inFig. 4. North is up and east is to the left. Bottom panels: Signal-to-noise of the three principal emission lines(Hα, Hβ, and [O iii] λ5007 in the left, centre, and right panel, respectively) in which extended ionised gas is detected (see text for details). The maps are displayed with a linear cyclic colour map from 0 to 90 and from 90 to 900. White dotted contours mark out regions with SN > 8. The chartreuse dashed rectangles in the Hα panel outline the rectangular regions used for the line ratio analysis in Sect. 3.2.Article number, page 5 of 22 Fig. 3 . 3Hα surface brightness profile in the south, i.e. not affected by the filaments in the north. The points show the observed profile in circular annuli and error-bars are 2σ. The black line indicates the double exponential profile fitted to the outer (r > 5 ) profile, while dot-dashed and dashed lines indicate the individual components with scale-lengths r 1 = 1.62 (447 pc) and r 2 = 8.94 (2.47 kpc). The grey line indicates the light profile of the central (r ≤ 5 ) residual component that is obtained after subtracting the double exponential. Fig. 4 .Fig. 5 . 45Left panel: Zoomed view (22 ×22 / 6 kpc × 6 kpc) of the Hα narrow band image from the top panel ofFig. 1with N HI contours from the VLA B-configuration 21 cm observations (N HI = {5, 10, 20, 30, 40} × 10 20 cm −2 , rose lines); these contours are also shown inFig. 1as subdued white lines. The cyclic colour map encodes Hα flux in a asinh-scale from 0 to 10 −17 erg s −1 cm −2 to 10 −15 erg s −1 cm −2 . In the brightest central region the archival HST F550M image is inset, with an arbitrary asinh-scaling to highlight the compact super-star clusters that comprise the main stellar body of this system. The violet dashed square indicates the region that is displayed in the right panel. Right panel: Inner high-SB Hα region as seen in the archival HST FR656N image (13 ×13 / 3.5 kpc × 3.5 kpc) displayed here with a cyclic asinh-scale from 0 to 5 × 10 −15 erg s −1 cm −2 to 10 13 erg s −1 cm −2 . Here the violet dashed square indicates the region that is displayed inFig. 5. HST archival F550M image with super-star clusters labelled according to Fig. 6 . 621 cm moment-0 map of SBS 0335-052E with the colour-bar encoding the neutral column N HI in units of 10 20 cm −2 using an asinhscaling. The magenta ellipse indicates the dimension and orientation of the VLA B-configuration beam. This map has been used to draw the N HI contours inFig. 1, Fig. 4, and Fig. 5. Fig. 7 . 7Moment-0 maps from VLA B-configutration 21cm observations. The colour mapping encodes N HI in 10 20 cm −2 and the beam for each dataset is indicated as a black ellipse. Top panel: VLA-B configuration data. Bottm panel: VLA-B configuration observations, tapered at 7kλ; this map has been used to draw the N HI contours inFig. 8. FFig. 9 . 9Hβ Top panel: Continuum subtracted [O iii] λ5007 narrowband; contours indicate SB [OIII] = {5, 12.5, 25} × 10 −19 erg s −1 cm −2 arcsec −2 . The cyclic colour map encodes flux densities from 0 to 5 × 10 −19 erg s −1 cm −2 (colour bar) to 5 × 10 −19 erg s −1 cm −2 . The image has been smoothed with a Gaussian of 0.4 FWHM. Bottom panel: Continuum subtracted Hβ narrowband. The cyclic colour map encodes flux from 0 to 3 × 10 −19 erg s −1 cm −2 to 3 × 10 −17 erg s −1 cm −2 . The image has been smoothed with a Gaussian of 0.95 FWHM to enhance the visibility of the low-SB Hβ emission. ∆(Hβ/Hα), can be calculated via ∆(Hβ/Hα) = Var(Hα) + Var(Hβ)(Hβ/Hα) 2 Hβ , Fig. 10 . 10Analysis of emission line ratios along the eastern (left) and western (right) filament. In the top three panels the emission in Hα, Hβ, and [O iii] λ5007 is shown, now aligned such that the x-axis runs along the direction of the filament (∆l) and that the y-axis runs accros the filament (∆w); the correpsonding areas have also been outlined in the bottom left panel ofFig. 1. The two bottom panels show the Hβ/Hα and [O iii]/Hα ratio in bins of 1.2 along ∆l. These ratios are obtained after integrating the emission in each line along ∆w. The 1-σ errors on the ratios have been computed with Eq. Fig. 12 . 125 shear .Moiseev et al. at- Line-of-sight velocity map (top) and velocity dispersion map (bottom) of the ionised gas; subdued contours mark various Hα surface brightness levels as inFig. 1. As zero-point velocity v = c · z = 4053 km s −1(Moiseev et al. 2010) was adopted. The kinematic major and minor axis afterMoiseev et al. (2010) are indicated with a dash-dotted and dotted line, respectively. The viewport of this figure is 1 35 ×1 46 as inFig. 1. The blue dashed square indicates the 22 ×22 viewport centred on the central circumgalactic medium(Fig. 2)shown in the left panel ofFig. 4. Fig. 14 . 14Comparison between H i and H ii kinematics on small scales. The H i velocity field (top left panel) is a zoomed-in view of the velocity field shown in the top panel Fig. 13. The H ii velocity field (top right panel) is the same as in Fig. 12, but regions outside of detected H i have been subdued. In the bottom left panel we show the H ii velocity field convolved with a 2D Gaussian (6.1 ×4.6 , position angle -23.6 • ) to mimic the VLA B-configuration observations beam smearing (see text). The bottom right panel shows the smoothed Hα velocity field resampled onto the native grid of the VLA B-configuration cube. persion measures in each pixel of the dispersion map and NB Hα x,y are the flux values recorded in each pixel of the Hα narrow band image Fig. 15 .Fig. 16 . 1516Comparison between H i and H ii kinematics on larger scales; we here display a zoomed-in view of the 7kλ tapered 21cm data shown in the bottom panel of Fig. 13. The top panel shows the first moment from the H i data, the middle panel shows the ionised gas velocity field, with regions outside of H i detections being subdued, and the bottom panel shows the H ii velocity field convolved with a 2D Gaussian (15.9 ×14.7 , position angle 66.4 • ) to mimic the beam smearing and resampled onto the native grid of the H i observations. Maps showing ∆v = v HI − v rep HII , where v HI are the H i line-of-sight velocities and v rep HII are the resolution matched and re-projected H ii line-of-sight velocities. The left panel shows the comparison with the untapered VLA-B configuration data, i.e. the subtraction of the velocity field in the lower-right panel of Fig. 14 from the upper-left panel ofFig. 14, while the right panel shows the comparison for the 7kλ-tapered data, i.e. the subtraction of the velocity field in the lower panel ofFig. 15from the upper panel ofFig. 15. Fig. 17 . 17or due to radiative transfer effects (Sect. 3.2.1). An ionised hydrogen plasma of density n (≡ n e = n p , i.e. no electrons from other species) and temperature Article number, page 15 Toy model to explain the NW filaments as limb-brightened edges of an outflow cone. We assume a cone filled with a hot wind fluid that is surrounded by walls filled with ionised hydrogen in recombination equilibrium (T = 2 × 10 4 K, n = 0.05 cm −3 ). Details on the setup and the model parameters that are derived from the observations are given in the text. The left panel shows a projection of the resulting Hα emission (as surface-luminosity in erg s −1 kpc −1 ) from this model along the Y-axis of the model box; this axis is chosen to be parallel to the line of sight. The right panel shows how the Hα emission from this model would appear in our MUSE data under the assumption that the model is at the distance of SBS 03352-052E. Here we overlay some Hα contours fromFig. 1(dotted white lines) to indicate the degree of congruence between the simulated observations of this simple toy model and reality. Fig . A.1. Hα emission line profiles extracted along the pseudo slits shown in the bottom left panel of Figure 1 (top panels) Fig. B. 1 . 1Channel maps from the untapered VLA-B configuration datacube. Fig. B. 2 .. 1 .. 2 . 212Channel maps from the 7kλ-tapered datacube, similar to Fig. B.1; here σ = 4 × 10 −4 Jy beam −1 . The bottom right panel shows the viewport (210 ×114 / 58.1 kpc×31.5 kpc) over the Pan-STARRS false-colour image with the N HI contours from Fig. Surface-luminosity projections of the cone-wall structure from Sect. 4 for various position angles ϑ and inclinations i as indicated in the panels. Surface-luminosity projections of the cone-wall structure from Sect. 4 for various opening angles θ as indicated in the panels. Fig. C. 3 . 3Simulated MUSE Hα observation of the cone-wall structure from Sect. 4 at the distance of SBS 0335-052E for fixed n = 0.05 cm −2 but varying T as indicated in the panels. Fig. C. 4 . 4Simulated MUSE Hα observation of the cone-wall structure from Sect. 4 at the distance of SBS 0335-052E for fixed T = 20000 K but varying n as indicated in the panels. 700} × 10 −18 erg s −1 cm −2 arcsec −2 ). In the background the Hα narrowband is shown with a log-stretch to 5 × 10 −16 erg s −1 cm −2 . The thin-dotted line surrounding the filaments to the north-west demarcates the aperture that is used to estimate the total Hα flux from that structure (Sect. 4).A&A proofs: manuscript no. sbs0335_musevla_revision_le 3 h 37 m 46 s 44 s 42 s 1'30" 2'00" 30" 3'00" eastern foreshortened filaments northern filament threadlike extension western filament fanning out of western filament Hα "hot-spot" central circum-galactic zone Fig. 2. Annotated Hα surface brightness contours (contour levels SB Hα = {1.5, 2.5, 5, 12.5, 100, 0 5 10 15 20 25 r [arcsec] 10 -18 10 -17 10 -16 10 -15 10 -14 SB Hα [erg s −1 cm −2 arcsec −2 ] SBS 0335 − 052E SB Hα (south) , is 7 ×7 or 1.88 kpc×1.88 kpc. line ratio [O iii]/Hβ is typically used to map the excitation of the plasma. Given the limited coverage of detected Hβ emission we here substitute Hβ with Hα in the denominator. The standard This needs to be kept in mind, since the unusual low Balmer decrements discovered in the previous section preclude us from making definitive claims on the presence of dust within the filaments.From the integrated spectrum covering the central starforming sites (seeFig. 3in Wofford et al. 2021) we measure [O iii]/Hα = 1.13. For both filaments the ratio is significantly lower, with an mean (median) of 0.63 (0.65) and 0.26 (0.23) for the eastern and western filament, respectively. Such low [O iii]/Hα ratios have been reported recently in the outskirts of the blue compact dwarf galaxy Haro 14error on [O iii]/Hα, ∆([O iii]/Hα), can then be calculated in an analogous way to Eq. (2). In principle, [O iii]/Hα must be under- stood as a lower bound to the excitation, since extinction lowers the ratio. Proto-typical example galaxies with well-mapped outflows are M 82 (e.g.,Bland & Tully 1988;Leroy et al. 2015;Lokhorst et al. 2022) and Arp 220 (e.g.Perna et al. 2020); seeVeilleux et al. (2005) andVeilleux et al. (2020) for more examples. We also mention in passing that the centre of our galaxy also powers an outflow that can be studied in exquisite detail (e.g.Hsieh et al. 2016;Ponti et al. 2021). https://science.nrao.edu/facilities/vla/observing/ callist Article number, page 3 of 22 A&A proofs: manuscript no. sbs0335_musevla_revision_le Article number, page 4 of 22 E. C. Herenz et al.: Outflow cone of SBS 0335-052E Acknowledgements. We thank the anonymous referee for their valuable contribution to the manuscript. We express our gratitude to the ESO Office for Science in Santiago for funding research internships for H. Salas and C. Moya-Sierralta to contribute to this project. J.M. CannonA&A proofs: manuscript no. sbs0335_musevla_revision_le populations of the contributing energy levels compared to the classical calculations (Sect. 3.2.1;Osterbrock & Ferland 2006).Our VLA B-configuration observations enable us to resolve H i at unprecedented spatial resolution within and in the vicinity of the starburst. The peak of the 21 cm signal is spatially offset by ∼ 1 to the west from the main super-star-cluster complexes. At lower N HI (5−10×10 20 cm −2 ), we observe morphological correspondences between neutral and ionised phases. In particular, we observe a bent tail that lines up in the direction of the eastern foreshortened filaments. Our observations are thus mapping the entanglement and interactions between the neutral and ionised phases in feedback-driven gas.Tapering the VLA B-configuration data at 7kλ in the UV plane in the imaging process enabled us to map the neutral halo around the interacting galaxy pair SBS 0335-052E and SBS 0335-052W. Our results are consistent with the previous analysis of 21 cm data from VLA C-and D-configuration(Pustilnik et al. 2001) and GMRT(Ekta et al. 2009) observations. The Hα emitting filaments extend outwards from the east to west elongated neutral halo around SBS 0335-052E. However, there is significant overlap between halo gas and the western filament. Especially the coincidence of the Hα hot spot and the fanning-out of the western filament with the extended tidal halo gas from the western galaxy appears intriguing.Our kinematical analysis of the ionised gas shows that the line emission from the filaments is very narrow, σ v ≈ 20 − 30 km s −1 , and in fact barely spectrally resolved in the MUSE data. We do not map velocity gradients along the filaments. Zones nearer to the star clusters appear in the MUSE data at high σ v (45 − 50 km s −1 ), but they have previously been resolved into double-peaked profiles with VLT/GIRAFFE(Izotov et al. 2006). The seemingly chaotic appearance of the H ii line-of-sight velocity map near the centre has also been previously described as a kinematic disturbance of an expanding shell superimposed onto a disk-like velocity field(Moiseev et al. 2010). The position angle of the Moiseev et al. model velocity field, ϑ = 52 • , is parallel to the shearing velocities mapped at lower SB levels and in H i. Interestingly, this gradient is aligned perpendicular to the symmetry axis in between the bifurcating filaments.The line-of-sight kinematics of the neutral phase exhibit a significant degree of complexity, both in the untapered and in the 7kλ-tapered data products. The complexity on large scales is likely due to tidal effects from the encounter of the two galaxies. However, an interaction with the large-scale wind may also cause deviations from simple disk-like motions in and around SBS 0335-052E. On smaller scales, the velocity gradient appears significantly warped. We find that H i and H ii kinematics are broadly in agreement on large and small scales. Nevertheless, there appear to be zones where the bulk motions of the neutral phase are redshifted by ∼ 15 − 20 km s −1 with respect to the velocities of the ionised phase. These velocity differences are deemed to be caused by different responses of the neutral and ionised phases to injected energy and momentum from stellar winds and supernova explosions.The observational evidence assembled in our analysis merits an explanation as to the unprecedented detection of more than 10 kpc long Hα-emitting filaments as limb-brightened edges of a conical outflow; this new hypothesis supersedes the initial attempt at explaining the morphology in H17. A toy model of a cone structure, with geometrical parameters read off the MUSE Hα narrow band, is commensurable with the expected physical reality of a starburst-driven wind. Our setup is described by an inner cone that is filled with a hot wind fluid and a surrounding wall filled with ionised plasma. In order to be compatible with the dimensions of the cone, the hot wind fluid would be required to flow at radial velocities ∼ 1600 km s −1 . This hot wind fluid could be loaded easily with mass-loading factors Λ Hot 0.1. While there is some degeneracy between density and temperature in order to match the flux of the ionised phase, we generally require very low densities of n 0.05 cm −3 . If the ionised phase is purely gas expelled from the starburst, then mass-loading factors Λ ≈ Λ HII 10 would be required. A mock MUSE observation of the toy model clearly shows the limb-brightening effect at surface brightness levels similar to the observations. Obviously, our didactic setup fails to reproduce the morphological substructure observed in reality, especially the thread-like lacy prolongation at the largest projected distances.Given the small size of SBS 0335-052E's main stellar body, the dimensions of its outflow structure revealed in the ionised phase appear extreme. Similar-sized ∼ 10 − 20 kpc outflows are known from more massive and larger galaxies. The finding reported here was only possible because of the unprecedented sensitivity of MUSE for low-SB line emission. Currently it is unknown whether such structures are common around other metal-poor compact starbursts. One unique characteristic of the SBS 0335-052 system is its large neutral gas reservoir, and we may speculate here that the size of the ionised structure is connected to the size of the halo, since a significant fraction of the Hα-emitting gas may have been ionised in situ.In our analysis and discussion, we made no attempt to constrain the actual ionisation mechanism. Photo-ionisation from the starburst and the X-ray emitting wind fluid, as well as radiative shocks between wind fluid and neutral halo gas are expected to contribute. While the three emission lines in this study do not provide discriminating power with respect to these mechanisms, they can be used in connection with photo-ionisation or shock models to design deeper observations that may constrain the ionisation mechanisms. We believe that this is an intriguing avenue for future investigations.Appendix A: Emission line profiles along the filamentsIn Sect. 3.3.1 we stated that the emission lines in the outer low-SB regions are adequately modelled by a single Gaussian profile. To back this claim we here visualise the line profile in the main filaments. To this aim we extract 2D spectra along the two pseudo-slits shown in the bottom-left panel ofFigure 1. From these extractions we show inFigure A.1, the Hα profiles alongside the fitted Gaussian models and the residuals. It can be appreciated from this Figure in that the single Gaussian is an adequate model and that the line is extremely narrow. 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{'abstract': 'A ∼15 kpc outflow cone piercing through the halo of the blue compact metal-poor galaxy SBS 0335-052E ABSTRACT Context. Outflows from low-mass star-forming galaxies are a fundamental ingredient for models of galaxy evolution and cosmology. Despite seemingly favourable conditions for outflow formation in compact starbursting galaxies, convincing observational evidence for kiloparsec-scale outflows in such systems is scarce. Aims. The onset of kiloparsec-scale ionised filaments in the halo of the metal-poor compact dwarf SBS 0335-052E was previously not linked to an outflow. In this paper we investigate whether these filaments provide evidence for an outflow. Methods. We obtained new VLT/MUSE WFM and deep NRAO/VLA B-configuration 21cm data of the galaxy. The MUSE data provide morphology, kinematics, and emission line ratios of Hβ/Hα and [O iii]λ5007/Hα of the low surface-brightness filaments, while the VLA data deliver morphology and kinematics of the neutral gas in and around the system. Both datasets are used in concert for comparisons between the ionised and the neutral phase. Results. We report the prolongation of a lacy filamentary ionised structure up to a projected distance of 16 kpc at SB Hα = 1.5 × 10 −18 erg s cm −2 arcsec −2 . The filaments exhibit unusual low Hα/Hβ ≈ 2.4 and low [O iii]/Hα ∼ 0.4 − 0.6 typical of diffuse ionised gas. They are spectrally narrow (∼ 20 km s −1 ) and exhibit no velocity sub-structure. The filaments extend outwards from the elongated H i halo. On small scales, the N HI peak is offset from the main star-forming sites. The morphology and kinematics of H i and H ii reveal how star-formation-driven feedback interacts differently with the ionised and the neutral phase. Conclusions. We reason that the filaments are a large-scale manifestation of star-formation-driven feedback, namely limb-brightened edges of a giant outflow cone that protrudes through the halo of this gas-rich system. A simple toy model of such a conical structure is found to be commensurable with the observations.', 'arxivid': '2212.01239', 'author': ['E C Herenz eherenz@eso.org \nEuropean Southern Observatory\nAv. Alonso de Córdova 3107763 0355Vitacura, SantiagoChile\n\nLeiden Observatory\nLeiden University\nNiels Bohrweg 22333 CALeidenNLThe Netherlands\n', 'J Inoue \nDepartment of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA\n', 'H Salas \nLeibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternware 1614482PotsdamGermany\n', 'B Koenigs \nDepartment of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA\n', 'C Moya-Sierralta \nPontificia Universidad Católica de Chile -Instituto de Astrofísica\nAv. Vicuña Mackenna 4860897, 0117Macul, SantiagoChile\n', 'J M Cannon \nDepartment of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA\n', 'M Hayes \nDepartment of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden\n', 'P Papaderos \nInstituto de Astrofísica e Ciências do Espaço -Centro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal\n', 'G Östlin \nDepartment of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden\n', 'A Bik \nDepartment of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden\n', 'A Le Reste \nDepartment of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden\n', 'H Kusakabe \nObservatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland\n', 'A Monreal-Ibero \nLeiden Observatory\nLeiden University\nNiels Bohrweg 22333 CALeidenNLThe Netherlands\n', 'J Puschnig \nArgelander-Institut für Astronomie\nUniversität Bonn\nAuf dem Hügel 71D-53121BonnGermany\n'], 'authoraffiliation': ['European Southern Observatory\nAv. Alonso de Córdova 3107763 0355Vitacura, SantiagoChile', 'Leiden Observatory\nLeiden University\nNiels Bohrweg 22333 CALeidenNLThe Netherlands', 'Department of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA', 'Leibniz-Institut für Astrophysik Potsdam (AIP)\nAn der Sternware 1614482PotsdamGermany', 'Department of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA', 'Pontificia Universidad Católica de Chile -Instituto de Astrofísica\nAv. Vicuña Mackenna 4860897, 0117Macul, SantiagoChile', 'Department of Physics & Astronomy, Macalester College\n1600 Grand Avenue55105Saint PaulMNUSA', 'Department of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden', 'Instituto de Astrofísica e Ciências do Espaço -Centro de Astrofísica da Universidade do Porto\nRua das Estrelas4150-762PortoPortugal', 'Department of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden', 'Department of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden', 'Department of Astronomy\nStockholm University\nAlbaNova University Centre, SE-106 91StockholmSweden', 'Observatoire de Genève\nUniversité de Genève\nChemin Pegasi 511290VersoixSwitzerland', 'Leiden Observatory\nLeiden University\nNiels Bohrweg 22333 CALeidenNLThe Netherlands', 'Argelander-Institut für Astronomie\nUniversität Bonn\nAuf dem Hügel 71D-53121BonnGermany'], 'corpusid': 254221229, 'doi': '10.1051/0004-6361/202244930', 'github_urls': [], 'n_tokens_mistral': 37562, 'n_tokens_neox': 31209, 'n_words': 19280, 'pdfsha': '6c798f8a3ed682cf8c714802e31aff0e98ce107a', 'pdfurls': ['https://export.arxiv.org/pdf/2212.01239v2.pdf'], 'title': [], 'venue': []}

arxiv

Multicell Random Beamforming with CDF-based Scheduling: Exact Rate and Scaling Laws 8 Jun 2013 Yichao Huang Department of Electrical and Computer Engineering University of California San Diego La Jolla92093CA Bhaskar D Rao brao@ece.ucsd.edu Department of Electrical and Computer Engineering University of California San Diego La Jolla92093CA Multicell Random Beamforming with CDF-based Scheduling: Exact Rate and Scaling Laws 8 Jun 2013 In a multicell multiuser MIMO downlink employing random beamforming as the transmission scheme, the heterogeneous large scale channel effects of intercell and intracell interference complicate analysis of distributed scheduling based systems. In this paper, we extend the analysis in [1] and [2] to study the aforementioned challenging scenario. The cumulative distribution function (CDF)-based scheduling policy utilized in [1] and [2] is leveraged to maintain fairness among users and simultaneously obtain multiuser diversity gain. The closed form expression of the individual sum rate for each user is derived under the CDFbased scheduling policy. More importantly, with this distributed scheduling policy, we conduct asymptotic (in users) analysis to determine the limiting distribution of the signal-to-interferenceplus-noise ratio, and establish the individual scaling laws for each user. I. INTRODUCTION With the emerging heterogeneous cellular structure [3] and the ever shrinking cell size, achieving high capacity with low design complexity in a multicell multiuser MIMO downlink has drawn considerable interest in recent years, e.g, see [4] and the references therein. Distributed scheduling policies are often favored due to operational scalability and affordable complexity incurred by the limited capacity of the backhaul. Under the employed distributed scheduling policy, analysis on the multicell network builds upon the extensive studies and insights drawn from the single cell network. For the single cell network without intercell interference, capacity boosting scheme relies on the independent varying channels across users, i.e., the well known multiuser diversity gain [5]. To further harness this gain with multiple antennas, the notion of opportunistic beamforming is proposed [6], which is later extended to the notion of random beamforming [7] to have same sum capacity growth as nonlinear precoding schemes [8], [9] with reduced feedback requirement [10]. Multiuser diversity depends heavily on the scheduling policy, and it is important to guarantee scheduling fairness while achieving this gain in consideration of the heterogeneous large scale channel effects. This issue is tackled in [2] (the closed form sum rate in a homogeneous setup is derived in [11]), by leveraging the cumulative distribution function (CDF)-based scheduling policy [12] to satisfy the two desired This research was supported by Ericsson endowed chair funds, the Center for Wireless Communications, UC Discovery grant com09R-156561 and NSF grant CCF-1115645. features: multiuser diversity and scheduling fairness. According to the CDF-based scheduling policy [12], each user can be equivalently viewed as competing with other users with the same CDF, and thus each user's rate is independent of the statistics of other users. This interesting property enables a "micro" understanding of each user's rate performance compared to the conventional "macro" understanding of the sum rate performance. Due to this property, the notion of individual sum rate and individual scaling laws are proposed in [2] to further understand both the exact and the asymptotic performance of random beamforming in a heterogeneous setup (and with different selective feedback schemes). In a multicell network, the heterogeneous channel effects come naturally from the different experienced intercell interference across users, even for a SISO setup. This issue is investigated in [1] for a generic single antenna based heterogeneous multicell OFDMA network, with both exact rate expression and asymptotic rate approximation derived. In [13], the rate of convergence and the individual scaling laws are established for the SISO multicell setup. Both [1] and [13] employ the CDF-based scheduling to maintain the two aforementioned scheduling features. In [14], the rate scaling for a power controlled network is examined with additional distance-based random variables. Since the time variations for the large scale and small scale channel effects are vastly different [15], both [1] and [13] concentrate on the randomness of the small scale channel effects. In [16], a normalized form of transformation is applied in a multicell network to achieve fairness. The main difference of using CDF-based scheduling [12] is the inherent nonlinear functional transformation to strictly guarantee user fairness. In this work, we extend the analysis in [1] and [2] to a generic multicell multiuser MIMO setup. The random beamforming is utilized as the multi-antenna transmission scheme to reduce feedback need (for part of the literature survey regarding random beamforming, please refer to [2]). With spatial multiplexing in each cell, both intercell interference and intracell interference exist and users would experience heterogeneous interference. Under the CDF-based scheduling policy, we firstly derive the individual sum rate for each user from the exact analysis perspective. We further prove the type of convergence and the rate of convergence to the limiting distribution to establish the individual scaling laws. II. SYSTEM MODEL Consider the downlink of a generic multicell multiuser MIMO network. We assume a narrowband model and the established analysis in this work can be extended to the wideband model such as OFDMA using the techniques developed in [2]. Full spectrum reuse is assumed, and the process of cell association is assumed to be performed in advance. Without loss of generality, one base station B 0 equipped with M antennas from the base station set B and its associated single antenna users K 0 = {1, . . . , k, . . . , K 0 } with |K 0 | = K 0 are considered. The random beamforming strategy at base station B 0 employs M random orthonormal vectors φ (0) m ∈ C M×1 for m = 1, . . . , M , where the φ (0) i are drawn from an isotropic distribution independently every T (denoting the channel coherence interval for the block fading model) channel uses [7]. Denoting s (0) m (t) as the mth transmission symbol at time t, the transmitted vector s (0) (t) from base station B 0 at time t, is given as: s (0) (t) = M m=1 φ (0) m (t)s (0) m (t), t = 1, . . . , T.(1) The received signal y (0) k of user k (the time variable t is dropped for notational convenience) is represented by y (0) k = M m=1 G (0) k h (0) k φ (0) m s (0) m + J k b=1 M m=1 G (b) k h (b) k φ (b) m s (b) m + v (0) k , k ∈ K 0 ,(2) where the superscript indicates the base station with (0) being the cell of interest and (b = 0) being the interfering cells. J k denotes the number of interfering cells for user k, and v (0) k denotes the additive white noise distributed with CN (0, σ 2 k ). s (0) and s (b) are the transmitted symbols by the serving cell and the interfering cell B b with E |s (0) | 2 = p 0 and E |s (b) | 2 = p b . h (0) k ∈ C 1×M and h (b) k ∈ C 1×M , which are assumed to be independent across users, denote the small scale channel gains between the serving cell and user k, and between the interfering cell B b and user k. G k represent the large scale channel gains between the serving cell and user k, and between the interfering cell B b and user k respectively. Based on the aforementioned assumption, denoting Z (0) k,m as the SINR of user k for beam m, it can be expressed as: Z (0) k,m = G (0) k p0|h (0) k φ (0) m | 2 M i =m G (0) k p0|h (0) k φ (0) i | 2 + J k b=1 G (b) k p b M i=1 |h (b) k φ (b) i | 2 + M σ 2 k = ρ (0) k |h (0) k φ (0) m | 2 M i =m ρ (0) k |h (0) k φ (0) i | 2 + J k b=1 ρ (b) k M i=1 |h (b) k φ (b) i | 2 + 1 ,(3) where ρ (0) k G (0) k p0 Mσ 2 k , ρ (b) k G (b) k p b Mσ 2 k . Now we examine the statistics of Z k,m . Note that the time variations for the large scale and small scale channel effects are vastly different [1]. The variation of the small scale channel gain h occurs on the order of milliseconds; whereas the large scale channel gain G which may consist of path loss, antenna gain, and shadowing, varies usually on the order of seconds. Therefore, G is assumed to be known in advance by the system, through infrequent feedback; and the elements of h are modeled as complex Gaussian with zero mean and unit variance. Since for a given user k, the Z (0) k,m 's are identically distributed and correlated, the beam index m can be dropped in the expression of the CDF, which is derived in the following lemma. Lemma 1. The CDF of Z (0) k can be expressed as F Z (0) k (x) = 1 − e − x ρ (0) k J k b=1 ρ (0) k ρ (b) k M (x + 1) M−1 J k b=1 x + ρ (0) k ρ (b) k M , x ≥ 0.(4) Proof: The main technique relies on the use of the moment-generating function (MGF) [17]. Denote ϑ (0) k ρ (0) k |h (0) k φ (0) m | 2 , and ζ (0) k M i =m ρ (0) k |h (0) k φ (0) i | 2 + J k b=1 ρ (b) k M i=1 |h (b) k φ (b) i | 2 . Then the CDF of Z (0) k can be derived using the following procedure: F Z (0) k (x) = P ϑ (0) k ζ (0) k + 1 ≤ x = ∞ 0 P ϑ (0) k ≤ x(ζ (0) k + 1) f ζ (0) k (y)dy (a) = 1 − e − x ρ (0) k ∞ 0 e − xζ (0) k ρ (0) k f ζ (0) k (y)dy,(5) where (a) follows from the fact that P ϑ (0) k ≤ x(ζ (0) k + 1) corresponds to the CDF of the exponential distribution at x(ζ (0) k + 1). From (5), we note that the expression inside the integral corresponds to the MGF of ζ (0) k , denoted as Ψ ζ (0) k (τ ), at − x ρ (0) k . Due to the additive effect reflected in ζ (0) k , its MGF can be obtained below: Ψ ζ (0) k (τ ) = 1 (1 − ρ (0) k τ ) M−1 J k b=1 1 (1 − ρ (b) k τ ) M .(6) Combing (5) and (6) yields (4). F Z (0) k expressed in The SINR will be fed back 1 and used for scheduling, which is pursued next. III. CDF-BASED SCHEDULING POLICY AND INDIVIDUAL SUM RATE After receiving the SINR (0) k,m from user k for beam m, the scheduler is ready to perform scheduling. In order to guarantee scheduling fairness and obtain multiuser diversity gain, we employ the CDF-based scheduling policy [12] as ψ (0) k,j = 1 ((M − 1)(ℓ + 1) − j))! d (M−1)(ℓ+1)−j dx (M−1)(ℓ+1)−j    1 J k b=1 (x + ρ (0) k ρ (b) k ) M(ℓ+1)    x=−1 ψ (b) k,j = 1 (M (ℓ + 1) − j))! d M(ℓ+1)−j dx M(ℓ+1)−j    1 (x + 1) (M−1)(ℓ+1) J k q =b (x + ρ (0) k ρ (q) k ) M(ℓ+1)    x=− ρ (0) k ρ (q) k the opportunistic scheduling policy 2 . According to this policy, the scheduler will utilize F Z (0) k , and performs the following functional transformation: Z (0) k,m = F Z (0) k Z (0) k,m .(7) The transformed random variableZ (0) k,m is uniformly distributed ranging from 0 to 1, and can be regarded as the virtual received SINR of user k for beam m. The transformed random variablesZ (0) k,m 's are i.i.d. across users for a given beam, which enables the maximization at the scheduler side to perform scheduling in a fair manner. Denoting k * m as the random variable representing the selected user for beam m, then: k * m = arg max k∈K0Z (0) k,m .(8) After user k * m is selected per (8), the scheduler utilizes the corresponding Z (0) k * m ,m for rate matching of the selected user. Let X (0) m be the SINR of the selected user for beam m, and now consider the sum rate for a given base station B 0 defined as follows: R (0) = E M m=1 log 1 + X (0) m .(9) From the aforementioned analysis, the sum rate can be formulated as: R (0) (a) ≃ M E k * m 1 0 log 1 + F −1 Z (0) k * m ,m (x) dx K0 (b) = M K 0 K0 k=1 ∞ 0 log(1 + t)d(F Z (0) k (t)) K0 ,(10) where (a) follows from the sufficient small probability that multiple beams are assigned to the same user;(b) follows from the change of variable x = F Z (0) k * m ,m (t) and the fairness property of the CDF-based scheduling policy. The CDF-based scheduling enables a "micro" level understanding of each user's rate performance, from both exact and asymptotic perspective. The individual sum rate for user k is defined in [2] as the the individual user rate multiplied by the number of users in cell B 0 , namely: R (0) k K 0 R (0) k = M ∞ 0 log(1 + x)d(F Z (0) k (x)) K0 .(11) 2 For detailed motivation of the CDF-based scheduling as well as the rationale behind the notion of individual sum rate and individual scaling laws, please refer to [1] and [2]. Now employing the steps described below, the closed form expression for the individual sum rateR (0) k is derived. Step 1 (PDF Decomposition): This step, utilized similarly in [1] and [2], is the essential step to decompose d(F Z (0) k (x)) K0 into the following amenable input for Step 2. d(F Z (0) k (x)) K0 = K 0 K0−1 ℓ=0 K 0 − 1 ℓ (−1) ℓ ℓ + 1 × d     1 −     e − x ρ (0) k ( ρ (0) k ρ (b) k ) M (x + 1) M−1 J k b=1 (x + ρ (0) k ρ (b) k ) M     ℓ+1     .(12) Step 2 (Partial Fraction Expansion [18]): This step manipulates part of (12) for further integration. 1 (x + 1) (M−1)(ℓ+1) J k b=1 x + ρ (0) k ρ (b) k M(ℓ+1) = (M−1)(ℓ+1) j=1 ψ (0) k,j (x + 1) j + J k b=1 M(ℓ+1) j=1 ψ (b) k,j x + ρ (0) k ρ (b) k j ,(13) where the expressions for ψ k,j are given on top of this page. Combining the outcomes of the two aforementioned steps, we can derive the closed form expression forR (0) k in the following procedure: R (0) k = M K0 ln 2 K 0 −1 ℓ=0 K0 − 1 ℓ (−1) ℓ ℓ + 1 × ∞ 0 ln(1 + x)d      1 −     e − x ρ (0) k ( ρ (0) k ρ (b) k ) M (x + 1) M −1 J k b=1 (x + ρ (0) k ρ (b) k ) M     ℓ+1      = M K0 ln 2 K 0 −1 ℓ=0 K0 − 1 ℓ (−1) ℓ ℓ + 1 J k b=1 ρ (0) k ρ (b) k M × J k b=1 M (ℓ+1) j=1 ψ (b) k,j I1 ℓ + 1 ρ (0) k , ρ (0) k ρ (b) k , j + (M −1)(ℓ+1) j=1 ψ (0) k,j I2 ℓ + 1 ρ (0) k , 1, j + 1 ,(14) where I 1 (α, β, γ) ∞ 0 e −αx (1+x)(β+x) γ dx, and I 2 (α, β, γ) ∞ 0 e −αx (β+x) γ dx. The calculation for I 1 (α, β, γ) and I 2 (α, β, γ) has been discussed in [1], and their closed form expressions can be found in [1, (42)] and [1, (43)]. Up to now, we have performed exact analysis and derived the closed form results for the individual sum rate for an arbitrary selected user in a given base station. The derived results extend the exact analysis in [1] (multicell SISO setup) and [2] (single cell random beamforming), and can serve as a theoretical reference for evaluating the system performance under the CDF-based scheduling policy. In the next section, we will perform asymptotic analysis to evaluate the rate scaling laws forR (0) k , which helps in understanding the asymptotic behavior of an individual user. IV. INDIVIDUAL SCALING LAWS This section is devoted to the asymptotic analysis. Section IV-A shows the type of convergence of Z (0) k . In Section IV-B, the convergence rate to the limiting distribution is studied to establish the individual rate scaling laws. A. Type of Convergence Firstly, we need to examine the tail behavior of the statistics of Z (0) k , which has the form presented in (4). Tools from extreme value theory [19] are to be utilized. The following lemma describes the tail behavior of F Z (0) it must be shown that lim x→∞ d dx 1−F Z (0) k (x) f Z (0) k (x) = 0 [19]. The equivalent condition is: lim x→∞ F Z (0) k (x)−1 f ′ Z (0) k (x) (f Z (0) k (x)) 2 = 1. Since similar methodologies in proving [1, Corollary 1] can be employed, we omit the detailed proof. B. Rate of Convergence Knowing the type of convergence can lead to asymptotic approximation for the individual sum rate, which is investigated in [1]. Herein, dealing with higher order moments of the extreme order statistics and the rate of convergence is of interest. To establish the convergence rate to the limiting distribution for an individual user, the following definition of the so called growth function [19] is needed: g Z (0) k (x) 1 − F Z (0) k (x) f Z (0) k (x) .(15) One important step in proving the rate of convergence is to solve for a suitable coefficient sequence w k:K0 by solving the following equation: 1 − F Z (0) k (w k:K0 ) = 1 K 0 .(16) Due to the complicated form of F Z (0) k in (4), we need to find upper and lower bound for w k:K0 by constructive methods, namely, w LB k:K0 ≤ w k:K0 ≤ w UB F UB Z (0) k (x) = 1 − e − x ρ (0) k 1 + ρ (b min k ) k ρ (0) k x (J k +1)M−1 , x ≥ 0, (17) F LB Z (0) k (x) = 1 − e − x ρ (0) k 1 + ρ (b max k ) k ρ (0) k x (J k +1)M−1 , x ≥ 0. (18) Proof: (Sketch) The upper and lower bound can be constructed by examining the large scale channel effects of intercell interference for user k. Herein, we only provide an intuitive explanation. F UB Z (0) k can be obtained by assuming that the intercell and intracell interference has the same large scale channel effects ρ (b min k ) k . The F LB Z (0) k can be found by using a similar line of argument. Employing Lemma 3, the upper and lower bound for w k:K0 can be derived and are provided in the following corollary. Corollary 1. w UB k:K 0 = ρ (0) k log K0 − ρ (0) k ((J k + 1)M − 1) log ρ (b max k ) k log K0 + O(log log log K0).(19)w LB k:K 0 = ρ (0) k log K0 − ρ (0) k ((J k + 1)M − 1) log ρ (b min k ) k log K0 + O(log log log K0). (20) Proof: The w UB k:K0 can be obtained via solving 1 − F LB Z (0) k (w UB k:K0 ) = 1 K0 . Substituting the expression of F LB Z (0) k and taking the log operator of both sides yields: w UB k:K0 ρ (0) k + ((J k + 1)M − 1) log 1 + ρ (b max k ) k ρ (0) k w UB k:K0 = log K 0 . (21) Since w UB k:K0 → ∞ as K 0 → ∞, w UB k:K 0 ρ (0) k dominates(d UB k:K 0 = −ρ (0) k ((J k + 1)M − 1) log ρ (b max k ) k log K0 − ρ (0) k ((J k + 1)M − 1) log 1 + ρ (0) k + ρ (b max k ) k d UB k:K 0 ρ (0) k ρ (b max k ) k log K0 .(22) Therefore, w UB k:K0 exhibits a scaling performance as expressed in (19). The corresponding analysis for w LB k:K0 can be conducted following the same line of arguments. Using the results from Corollary 1 and observing the fact that log ρ (b max k ) k or log ρ (b min k ) k are inconsequential when K 0 goes large, we have the following expression for w k:K0 : w k:K0 = ρ (0) k log K 0 − ρ (0) k ((J k + 1)M − 1) log log K 0 + O(log log log K 0 ).(23) Once the expression of w k:K0 is obtained, we can have the following inequality for the selected user's SINR for beam m (denoted by X (0) m in Section III) by employing [7, Corollary A.1], as follows: P ρ (0) k log K 0 − ρ (0) k (J k + 1)M log log K 0 + O(log log log K 0 ) ≤ X (0) m ≤ ρ (0) k log K 0 − ρ (0) k ((J k + 1)M − 2) log log K 0 + O(log log log K 0 ) ≥ 1 − O 1 log K 0 .(24) Now we can state the following theorem for the rate scaling performance ofR (0) k . Corollary 2. lim K0→∞R (0) k M log log K 0 = 1.(25) Proof: (Sketch) R (0) k ≤ M P X (0) m ≤ w k:K0 + ρ (0) k log log K 0 × log 1 + w k:K0 + ρ (0) k log log K 0 + M P X (0) m ≥ w k:K0 + ρ (0) k log log K 0 log(1 + ρ (0) k K 0 ) ≤ M log 1 + w k:K0 + ρ (0) k log log K 0 + O(1). (26) A lower bound forR (0) k can be obtained similarly. Thus the individual scaling law forR (0) k exhibits a M log log K 0 growth in the large user regime. Remark: Corollary 2 informs us that after we bring in the notion of individual scaling law corresponding to the individual sum rate for an arbitrary selected user, it exhibits a M log log K 0 growth. This property is desirable from the perspective of opportunistic scheduling (e.g., greedy scheduling). In addition to this property, with the nonlinear functional transformation, the CDF-based scheduling policy can guarantee scheduling fairness in terms of long term user fairness (i.e., each user is equiprobable to be scheduled irrespective of their intercell and intracell interference). V. CONCLUSION The analytical impact of CDF-based scheduling policy in two special scenarios (multicell multiuser SISO and single cell multiuser MIMO) has been investigated in [1] and [2]. This work extends and generalizes our previous works by addressing the rate performance in a generic multicell multiuser MIMO downlink, with random beamforming as the signal transmission scheme. The most challenging part of this generalization lies in the existence of both intercell and intracell interference, as well as their accompanying heterogeneous large scale channel effects. The CDF-based scheduling helps us to deal with this challenging scenario, and enables a "micro" understanding of the rate performance for any selected user. The closed form individual sum rate is derived employing the MGF and the PDF decomposition. With the constructed bounding technique, we also establish the individual scaling laws to show that CDF-based scheduling exhibits the same scaling performance as opportunistic scheduling (but achieves scheduling fairness additionally). the domain of attraction of the Gumbel distribution, i.e., F Z (0) k ∈ D(G 3 ).Proof: (Sketch) In order to prove that F Z K0 ∈ o(log K 0 ). The expression for d UB k:K0 can be derived by substituting the formulation of w UB k:K0 into (21), which solves d UB k:K0 as21) and so we have w UB k:K0 ∼ ρ (0) k log K 0 . Then the sequence of w UB k:K0 can be further written as w UB k:K0 = ρ (0) k log K 0 + d UB k:K0 , where d UB k: Full feedback wherein each user feeds back the SINR for M beams is assumed. 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Wireless packet scheduling based on the cumulative distribution function of user transmission rates. D Park, H Seo, H Kwon, B G Lee, IEEE Trans. Commun. 5311D. Park, H. Seo, H. Kwon, and B. G. Lee, "Wireless packet scheduling based on the cumulative distribution function of user transmission rates," IEEE Trans. Commun., vol. 53, no. 11, pp. 1919-1929, Nov. 2005. On rate scaling laws with CDF-based distributed scheduling in multicell networks. Y Huang, B D Rao, Proc. IEEE International Symposium on Information Theory and its Applications (ISITA). IEEE International Symposium on Information Theory and its Applications (ISITA)Y. Huang and B. D. Rao, "On rate scaling laws with CDF-based dis- tributed scheduling in multicell networks," in Proc. IEEE International Symposium on Information Theory and its Applications (ISITA), Oct. 2012, pp. 511-515. Rate scaling laws in multicell networks under distributed power control and user scheduling. D Gesbert, M Kountouris, IEEE Trans. Inf. Theory. 571D. Gesbert and M. Kountouris, "Rate scaling laws in multicell networks under distributed power control and user scheduling," IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 234-244, Jan. 2011. The capacity gain from intercell scheduling in multi-antenna systems. W Choi, J G Andrews, IEEE Trans. Wireless Commun. 72W. Choi and J. G. Andrews, "The capacity gain from intercell scheduling in multi-antenna systems," IEEE Trans. Wireless Commun., vol. 7, no. 2, pp. 714-725, Feb. 2008. Information exchange limits in cooperative MIMO networks. A Tajer, X Wang, IEEE Trans. Signal Process. 596A. Tajer and X. Wang, "Information exchange limits in cooperative MIMO networks," IEEE Trans. Signal Process., vol. 59, no. 6, pp. 2927- 2942, Jun. 2011. Proakis, Digital Communications. J , McGraw HillJ. G. Proakis, Digital Communications. McGraw Hill, 2001. I S Gradshteyn, I M Ryzhik, Tables of Integrals, Series and Products. ., D. Zwillinger and A. JeffreyAcademic Press7th edI. S. 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{'abstract': 'In a multicell multiuser MIMO downlink employing random beamforming as the transmission scheme, the heterogeneous large scale channel effects of intercell and intracell interference complicate analysis of distributed scheduling based systems. In this paper, we extend the analysis in [1] and [2] to study the aforementioned challenging scenario. The cumulative distribution function (CDF)-based scheduling policy utilized in [1] and [2] is leveraged to maintain fairness among users and simultaneously obtain multiuser diversity gain. The closed form expression of the individual sum rate for each user is derived under the CDFbased scheduling policy. More importantly, with this distributed scheduling policy, we conduct asymptotic (in users) analysis to determine the limiting distribution of the signal-to-interferenceplus-noise ratio, and establish the individual scaling laws for each user.', 'arxivid': '1303.2587', 'author': ['Yichao Huang \nDepartment of Electrical and Computer Engineering\nUniversity of California San Diego\nLa Jolla92093CA\n', 'Bhaskar D Rao brao@ece.ucsd.edu \nDepartment of Electrical and Computer Engineering\nUniversity of California San Diego\nLa Jolla92093CA\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nUniversity of California San Diego\nLa Jolla92093CA', 'Department of Electrical and Computer Engineering\nUniversity of California San Diego\nLa Jolla92093CA'], 'corpusid': 12645936, 'doi': '10.1109/vtcfall.2013.6692293', 'github_urls': [], 'n_tokens_mistral': 9265, 'n_tokens_neox': 8299, 'n_words': 5051, 'pdfsha': 'a5f64a3070fb75160f818331a108f7e6ebcd1aa5', 'pdfurls': ['https://arxiv.org/pdf/1303.2587v2.pdf'], 'title': ['Multicell Random Beamforming with CDF-based Scheduling: Exact Rate and Scaling Laws', 'Multicell Random Beamforming with CDF-based Scheduling: Exact Rate and Scaling Laws'], 'venue': []}

arxiv

Temporal extensivity of Tsallis' entropy and the bound on entropy production rate 21 Mar 2006 Sumiyoshi Abe Institute of Physics University of Tsukuba 305-8571IbarakiJapan Yutaka Nakada Institute of Physics University of Tsukuba 305-8571IbarakiJapan Temporal extensivity of Tsallis' entropy and the bound on entropy production rate 21 Mar 2006arXiv:cond-mat/0603550 1 The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs entropy, plays a central role in nonextensive statistical mechanics of complex systems. A lot of efforts have recently been made on establishing a dynamical foundation for the Tsallis entropy. They are primarily concerned with nonlinear dynamical systems at the edge of chaos. Here, it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by Carati [A. Carati, Physica A 348, 110 (2005)]that, whenever relevant, the Tsallis entropy indexed by q is temporally extensive: linear growth in time, i.e., finite entropy production rate. Then, the universal bound on the entropy production rate is shown to be 11 /| | −q . The property of the associated probabilistic process, i.e., the sojourn time distribution, determining randomness of motion in phase space is also analyzed. PACS number(s): 65.40.Gr, 05.20.-y, 05.90.+m, 02.50.-r _ I. INTRODUCTION The Boltzmann-Gibbs entropy and ordinary statistical mechanics based on it require a system under consideration to be in the strongly chaotic regime. It may be possible to view this point in Boltzmann's Stosszahlansatz (i.e., molecular chaos hypothesis), which, combined with the H-theorem, enables one to obtain the thermal equilibrium distribution in the long-time limit. That is, there exists no correlation between colliding particles. Over a century after Boltzmann's period, physicists and mathematicians have made a lot of efforts to build a bridge between statistical mechanics and microscopic dynamics. As known today, one way to satisfy the above requirement is to demand that the system possesses at least one positive Lyapunov exponent [1,2]. The Kolmogorov-Sinai entropy [1,2], which is the nonlinear dynamics counterpart of the Boltzmann-Gibbs entropy, quantifies the strength of dynamical chaos. A point of crucial importance is its change due to dynamical evolution: it grows linearly in time, that is, the entropy production rate is constant. Then, the Pesin identity [1,2] tells that the rate is given by the sum of positive Lyapunov exponents. In analogy between statistical mechanics and chaotic dynamics, this linear growth of the Kolmogorov-Sinai entropy corresponds to thermodynamic extensivity of the Boltzmann-Gibbs entropy [1], i.e., linear scaling with respect to the number of particles. And, it is considered that extensivity of entropy is an indispensable requirement, with which thermodynamics can be constructed. This may be the case even if system energy is nonextensive [3]. It should be noted that, in general, it is sufficient to realize the linear growth of the Kolmogorov-Sinai entropy and thermodynamic extensivity of the Boltzmann-Gibbs entropy only in the long-time limit and the thermodynamic limit, respectively. A physical quantity is said to be temporally extensive if it grows linearly in time. Thus, for example, the Kolmogorov-Sinai entropy possesses temporal extensivity for chaotic dynamical systems. Now, the situation becomes different for complex dynamical systems prepared at the edge of chaos, in which their maximum Lyapunov exponents strictly vanish. In this case, the standard Pesin identity becomes trivial ( 00 =) , offering no information, and the Kolmogorov-Sinai entropy fails to be temporally extensive. Then, the so-called generalized Lyapunov exponents or, q-Lyapunov exponents [4][5][6][7][8][9][10] and the Tsallis entropy [11] may become physically relevant. The authors of Refs. [4][5][6][7][8][9][10] have studied several kinds of nonlinear dynamical systems at the edge of chaos and have found that the generalized Pesin identity with the q-Lyapunov exponents holds and the Tsallis entropy with the entropic index q different from unity (i.e., different from the Kolmogorov-Sinai limit) is temporally extensive. Most of these works are numerical, but remarkably Ref. [8] rigorously shows these results for the logistic map by using exact renormalization group analysis. Generalized statistical mechanics based on the Tsallis entropy, termed nonextensive statistical mechanics [12], is considered to be a consistent and unified framework for describing complex statistical systems and is currently under vital investigation. Yet, there is another approach to thermostatistics, which is based on time averages. This method proposed by Carati [13,14] allows us to establish a connection between a probabilistic process in a phase space and definition of entropy. Since it directly treats dynamics as a time series, it may shed new light on a long-standing problem of statistics and dynamics. In this paper, we show temporal extensivity of the Tsallis entropy, without recourse to employing any specific dynamical system. This is executed by generalizing the above-mentioned formulation of thermostatistics based on time averages. Then, we derive the universal bound on the Tsallis-entropy production rate. We shall also see what probabilistic process in a phase space underlies nonextensive statistical mechanics. The paper is organized as follows. In Sec. II, the formulation of thermostatistics based on time averages presented in Refs. [13,14] is reviewed with some modifications. In Sec. III, such a formulation is generalized, and then it is shown that the Tsallis entropy is temporally extensive. In Sec. IV, the universal bound on the Tsallis-entropy production rate is presented. Analysis of the probabilistic process associated with the Tsallis entropy and nonextensive statistical mechanics is performed in Sec. V. Finally, Sec. VI is devoted to concluding remarks. Throughout this paper, the Boltzmann constant, k B , is set equal to unity for the sake of simplicity. II. THERMOSTATISTICS BASED ON TIMES AVERAGES In this section, we wish to review Carati's formulation of thermostatistics based on time averages, which is equivalent to Boltzmann-Gibbs theory. But, at the same time, we shall make some modifications of it. Consider a system in a phase space M. Its dynamics yields a sequence {} ,, x nn = ⋅⋅⋅ 12 . This sequence is generated by a map, φ : MM →, defining the iteration xx nn + = 1 φ () . T h e average of a certain physical quantity A over a fixed long time interval 1 ≤≤ nN ( N >> 1) is given by Ax N Ax n n N () () 0 1 1 ≡ = ∑ .( 1 ) T h i s quantity is random, depending on the initial data, x 0 . A procedure of coarse graining is to divide M into a lot of cells: LL L K 12 ,,, ⋅⋅⋅ (K >> 1). Let A i and n i be the representative value of A in the ith cell L i and the number of times the system visits L i , respectively. Then, Eq. (1) is well approximated as follows: Ax A n N i i K i () 0 1 ≅ = ∑ .( 2 ) In this representation, {} ,, , n ii K =⋅ ⋅ ⋅ 12 is random, depending on x 0 . Next, let Pn n Fn ii () ( ) ≤≡ be the cumulative probability that the cell L i is visited nn i () ≤ times by the system. It essentially describes the sojourn time characterizing system dynamics in the phase space [15]. Then, the average value of A to be compared with observation is given by <> = <> = ∑ A N An i i K i 1 1 ,( 3 ) where <> = = = = = ∏ ∫ ∏ ∫ n dF n n dF n i ji j K nN j j K nN j j () () 1 1 Σ Σ ,( 4 ) provided that the integral should be understood as the Lebesgue-Stieljes integral. Eq. (3) can be rewritten as <> = − ∂ ∂ = A N Z 1 0 λ λ λ ln ( )(5) with the generating function Zd F n e j j K An nN ii i j () ( ) λ λ = ∑ = − = ∏ ∫ 1 Σ = ∑ − = − ∏ ∫ dF n e N n j j K An ii ii i () ( ) 1 λ δ Σ .( 6 ) If the constraint is imposed on the energy U N n ii i K = = ∑ 1 1 ε (7) with ε i the representative value of the system energy in the cell L i , Eq. (6) should be replaced by Zd F n e N n U n N Uj j K An ii iii ii i () ( ) ( ) ( / ) λδ δ ε λ = ∑ −− = − ∏ ∫ 1 ΣΣ = −∞ ∞ −∞ ∞ = + ∫ ∫ ∏ ∫ N dk dk dF n e j j K ik UN ik N () () 2 2 12 1 12 π × −+ + = ∏ e n A ik ik i K ii i () λε 12 1 .( 8 ) Accordingly, the average value in Eq. (5) is modified as <> = − ∂ ∂ = A N Z UU 1 0 λ λ λ ln ( ) .( 9 ) I t is convenient to define the function, χζ () , as follows: dF n e e n () () − ∫ = ζ χζ .(10) Then, Eq. (8) becomes 12 1 2 Z N dk dk e U ik UN ik N A ik ik i ii () () () λ π χλ ε = ∑ −∞ ∞ −∞ ∞ ++ + + ∫ ∫ 2 2 12 . ( 1 1 ) In the large-N limit, the integrals are evaluated by the method of steepest descents. The steepest-descent condition leads to U N ik ik i i K i =− + = ∑ 1 1 12 εχ ε '( ),(12)Ni k i k i K i =− + = ∑ 1 12 χε '( ),(13) in the limit λ → 0 , where χ ' is the derivative of χ with respect to the argument. As can be seen from Eq. (9), the relation <> = − + = ∑ A N Ai k i k Ui i K i 1 1 12 χε '( )(14) holds in the leading order of N. (It is discussed in Ref. [13] how fluctuations around the steepest descents are small.) Eqs. (12)- (14) imply that −+ χθ ε α '( ) i is the average number of times that the cell L i is visited by the system, where the analytic continuations, ik 1 =θ and ik 2 =α, have been made. Therefore, defining the sojourn time probability p N i i =− + χθ ε α '( ) ,(15)= ∞ − − ∑ = − () ! 0 ζ ζ ,( 17) showing χζ ζ () =− − Npe Np,(18) where p is a positive constant to be determined later. In anticipation of the relation between χζ () and the free energy, consider the following Legendre transformation: s ii i i () () νν ζχ ζ =+ 0 ,( 19)νχ ζ ii =− '( ) 0 ,( 20) with ζθ εα ii =+ . In these equations, we are using only the relevant part of χζ () i in Eq. (18) χζ ζ 0 () i Npe i = − ,(22) since the term, −Np, is nothing but a constant shift of the free energy. From Eq. (18), ζ i and ν i are calculated to be ζ ii pp =−ln ( / ),(23)νχ ζ ζ ii i Npe Np i == = − 0 () ,( 24) where p i is given in Eq. (15). Then, the entropy is found to be given by the Boltzmann- Gibbs entropy Ss i i K ii i i K ==+ == ∑∑ () [ () ] νν ζ χ ζ 1 0 1 =− = ∑ Np p i i K i 1 ln ,(25) provided that p has been fixed as p e = 1 .(26) The thermodynamic entropy, S th , may be defined by S S N th = .(27) From Eqs. (12), (20), and (27), it follows that ∂ ∂ = S U th θ .(28) Thus, θ is found to be the inverse temperature, θβ =≡ 1/T ( k B ≡ 1). Now, using Eqs. (15) and (22), and recalling N i i =− + ∑ χθ ε α '( ), one obtains p e Z i i = −βε β () , Ze i i K () β βε = − = ∑ 1 ,(29) which is precisely the canonical distribution in Boltzmann-Gibbs statistical mechanics. Note that S in Eq. (25) is temporally extensive, linearly scaling with respect to N, and the thermodynamic entropy in Eq. (27) is actually the entropy production rate. III. TEMPORAL EXTENSIVITY OF TSALLIS ENTROY We wish to construct the Tsallis entropy [11] and nonextensive statistical mechanics in the spirit of the discussion in the preceding section. Although this issue has already been addressed in Ref. [13], several points remain to be clarified and some generalizations are needed. We shall fully develop nonextensive statistical mechanics based on time averages. First of all, we note that, as shown in Ref. [16], what has to be used in nonextensive statistical mechanics is not the ordinary expectation value in Eq. (16) but the normalized q-expectation value <>= = ∑ AA P Uq i i K i q , () 1 ,( 30) where P i q () is the escort distribution [1] associated with the original distribution, p i : P p p i q i q j q j () () () = ∑ .( 31) Since the time average should be identified with the normalized q-expectation value, Eq. (15) has to be replaced by P N i q i () '( ) =− + χθ ε α .( 32) However, the Legendre transformation should remain form invariant as in Eqs. (19) and (20). That is, the thermodynamic formalism is kept unchanged. = − −       = ∑ N q p i q i K 1 1 1 () ,( 42) provided that r q is chosen to be And, at this level, the entropic index, q, is taken to be positive. Clearly, Eq. (42) becomes Eq. (25) in the limit q → 1. We note that the solution in Eq. (43), and therefore the Tsallis entropy in Eq. (42), can exist if and only if c q < 2.(44) The marginal case, c q →− 20 , corresponds to divergently large r q . The consistency condition in Eq. (44) leads to an important result, which will be discussed in the next section. Thus, we conclude that the Tsallis entropy necessarily has temporal extensivity if the time average is given in terms of the escort distribution in Eq. (31) with p Z e i q qi =− + 1 () (( ) ) β θε α , Ze qq i i K () (( ) ) βθ ε α =− + = ∑ 1 .( 45) Finally, we point out that, as in Eqs. (27) and (28), ∂ ∂ = S U q th, θ(46) implying that θ is the inverse temperature, 1/T. Therefore, Eq. (44) is precisely the qexponential distribution in nonextensive statistical mechanics [12]. Closing this section, we wish to emphasize that temporal extensivity of the Tsallis entropy plays a crucial role in Eq. (46). Were it not temporally extensive, could temperature not be defined. In this respect, we recall that, also in recent papers [14,17], a discussion has been developed about necessity of extensivity of the Tsallis entropy for temperature to be definable. IV. BOUND ON TSALLIS-ENTROPY PRODUCTION RATE S Nq q < − 1 1 ( q > 1 ).(48) The situation becomes nontrivial in the case when 01 << q . T h e maximum value of c q is realized by the equiprobability, pK i = 1/ (iK =⋅ ⋅ ⋅ 12 ,, , ) , giving cK q q max = − 1 . Therefore, the consistency condition in Eq. (44) yields 1 2 1 −< < ln ln K q .(49) Since K is very large, this essentially means the Boltzmann-Gibbs limit, q → 1. However, it is not a physical state considered in nonextensive statistical mechanics of complex systems. The phase space of a complex systems at the edge of chaos has a highly nontrivial structure: only a small number of the cells are visited by the system. Therefore, the equiprobability is not realized. This leads to the conclusion that the phase-space configuration is realized in such a way that Eq. (44) is satisfied for each system-specific value of q ∈(,) 01. T hus, we find that the Tsallis-entropy production rate has the bound S Nq q < − 1 1 ( 01 << q ).(50) From Eqs. (48) and (50), we conclude that the Tsallis-entropy production rate, which is actually the thermodynamic entropy, has the following universal bound: where˜( ) () fz e = χζ ,( 53)ze = ζ .( 54) Therefore, ˜( ) fn is the z-transformation [18] of fn () . T h e inverse z-transformation is given by [18] fn i dz f z z n C ()() = − ∫ 1 2 1 π ,( 55) where C is a circle centered at z = 0 (in the complex z-plane) surrounding all the poles of ˜( ) fn. Upon applying the above inversion formula to Eq. (33), an analytic expression is needed for χζ () , since z (and therefore ζ ) is complex. Accordingly, we have to employ the following expression: where a is the logarithm of the radius of the circle C. Unfortunately, it does not seem to be possible to calculate Eq. 1 1 − − − −               << − −       >          .( 63) In particular, Eq. (60) tends to the asymptotic form of the Poisson distribution in the limit q → 1 fn e n nn n Np q Np ( ) exp ln → − →  −       1 2 π( 64) with rp e q →= 1/ ( q → 1) in Eq. (36). VI. CONCLUDING REMARKS We have shown by generalizing the formulation of thermostatistics based on time averages that the Tsallis entropy is temporally extensive. We have also shown that, as Boltzmann-Gibbs statistical mechanics, nonextensive statistical mechanics can also be consistently constructed based on time averages. Then, we have presented the universal bound on the Tsallis-entropy production rate. In addition, we have analyzed the probabilistic process (i.e., the sojourn time distribution in the cells) in the phase space and have determined its asymptotic property. In the present work, we have been concerned with temporal extensivity of entropy. It is considered [1] that temporal extensivity in a fully chaotic dynamical system corresponds to thermodynamic extensivity (i.e., linear scaling with respect to the number of particles) in a statistical system. From the combined viewpoints of statistics and dynamics, extensivity of entropy seems to be an indispensable premise for temperature to be definable [3,14,17]. Recently, it has been shown [19] that the Tsallis entropy behaves as an extensive quantity if a system contains strong correlation of a specific type. It is necessary to understand such correlation from dynamics at the edge of chaos. On the other hand, upon proving temporal extensivity of the Tsallis entropy, we have used the probabilistic process in the phase space. Thus, it is equally necessary to clarify if dynamics at the edge of chaos can certainly yield such a process (and, simultaneously, nonextensive statistical mechanics). There is yet another interesting issue to be addressed. Strictly speaking, it is sufficient to realize temporal extensivity of entropy only in the large-N limit, i.e., the long-time limit. In recent works [20,21], it has numerically been shown that not only the Tsallis entropy but also the quantum-group entropy [22] and the κ -entropy [23] possess temporal extensivity in the long-time limit for several dynamical systems at the edge of chaos. These generalized entropies are concave and Lesche-stable [24,25], as the Tsallis entropy is [26]. These facts indicate that there may be a certain level of diversity in microscopic description of thermostatistics of complex systems. It is desirable to be possible to formulate theories for those entropies based on time averages and to show them temporally extensive, as done in the present work. These remaining issues are to be clarified and solved for deeper understanding of statistical mechanics of complex systems. Now, examine the following deformation of the exponential factor in Eq. deformation-free limit q → 1, this function converges to the ordinary exponential function, e x . So, Eq. (33) tends to Eq. the preceding section, we can ignore the constant shift, −Nr q , in Eq. here is the value of the energy in the cell L i of a nonextensive complex system, which different from an ordinary simple system considered in the preceding section. in fact converges to pe = 1/ in Eq. (26) in the limit q → 1, as required in Eq. (36). of interest to examine this bound for an analytic examples known in the literature. In Ref.[8], the logistic map, xx tt + , with the initial condition,x 0 = 0,is analytically discussed at the edge of chaos µ = ⋅⋅⋅ 1 4011 . . [This system is dissipative, and therefore the generalized canonical distribution in Eq. (45) is irrelevant and only the property of the Tsallis entropy may be mentioned.] The value of the entropic index of such a system is calculated to be q = ⋅⋅⋅ 0 2445 . . On the other hand, the Tsallis-entropy production rate, which is the q-Lyapunov exponent, λ q , is a marginal case of Eq. (51). In Ref. [9], the conservative triangle map on a torus at the edge of chaos is numerically analyzed with an ensemble of various initial conditions, and the values of q and λ q are found to be about zero and unity, respectively. Again, it is a marginal case. V. PROBABILISTIC PROCESS FOR NONEXTENSIVE STATISTICAL MECHANICS In Sec. III, we have seen that the deformation of χζ () from Eq. (18) to Eq. (33) uniquely leads to the Tsallis entropy and nonextensive statistical mechanics. (It is in fact unique since the discussion is based on the Legendre transformation and the thermodynamic formalism.) In the case of Eq. (18), the underlying probabilistic processin the phase space is Poissonian, as in Eq.(17). What is the corresponding one for χζ () in Eq. (33)? This is the issue to be addressed in this section.Let fn () be a distribution function of a discrete random variable, (57) analytically in terms of known functions. Here, we make the asymptotic evaluation of fn () for large values of n by the method of steepest descents. Let us rewrite Eq. (57) with Eq. for large values of n, fn () asymptotically behaves as follows: An interesting discovery of Carati is that Boltzmann-Gibbs statistical mechanics can be reproduced if Fn () is Poissonian, i.e., the completely random process. In this case,we have <> = = ∑ AA p Ui i K i 1 . ( 16) Eq. (10) becomes e Np n ee Np n n n Npe Np − As noted in Sec. III, the Tsallis entropy and associated nonextensive statistical mechanics can consistently be constructed within the time-average formulation if andonly if Eq. (44) is satisfied. The consistency condition in Eq. (44) is always satisfied when q > 1, since, in this case, cp qi i q ≡< ∑ () 1 . T h i s immediately gives rise to the bound on the Tsallis- entropy production rate C Beck, F Schlögl, Thermodynamics of Chaotic Systems: An Introduction. CambridgeCambridge University PressC. Beck and F. Schlögl, Thermodynamics of Chaotic Systems: An Introduction (Cambridge University Press, Cambridge, 1993). R C Hilborn, Chaos and Nonlinear Dynamics. OxfordOxford University Press2nd ed.R. C. Hilborn, Chaos and Nonlinear Dynamics, 2nd ed. (Oxford University Press, Oxford, 2000). . S Abe, A K , Phys. Lett. A. 337292S. Abe and A. K. Rajagopal, Phys. Lett. A 337, 292 (2005). . C Tsallis, A R Plastino, W.-M Zheng, Chaos, Solitons & Fractals. 8885C. Tsallis, A. R. Plastino, and W.-M. Zheng, Chaos, Solitons & Fractals 8, 885 (1997). . U Tirnakli, G F J Ananos, C Tsallis, Phys. Lett. A. 28951U. Tirnakli, G. F. J. Ananos, and C. Tsallis, Phys. Lett. A 289, 51 (2001). . U Tirnakli, C Tsallis, M L Lyra, Phys. Rev. E. 6536207U. Tirnakli, C. Tsallis, and M. L. Lyra, Phys. Rev. E 65, 036207 (2002). . E P Borges, C Tsallis, G F J Añaños, P M C De Oliveira, Phys. Rev. Lett. 89254103E. P. Borges, C. Tsallis, G. F. J. Añaños, and P. M. C. de Oliveira, Phys. Rev. Lett. 89, 254103 (2002). . F Baldovin, A Robledo, Phys. Rev. E. 6945202F. Baldovin and A. Robledo, Phys. Rev. E 69, 045202(R) (2004). . A See Also, Robledo, Physica D. 193153See also, A. Robledo, Physica D 193, 153 (2004); . H Hernández-Saldaña, A Robledo, cond-mat/0507624H. Hernández-Saldaña and A. Robledo, e-print cond-mat/0507624. . G Casati, C Tsallis, F Baldovin, Europhys. Lett. 72355G. Casati, C. Tsallis, and F. Baldovin, Europhys. Lett. 72, 355 (2005). . U Tirnakli, C Tsallis, cond-mat/0502607e-printto appear in Phys. Rev. EU. Tirnakli and C. Tsallis, e-print cond-mat/0502607, to appear in Phys. Rev. E. . C Tsallis, J. Stat. Phys. 52479C. Tsallis, J. Stat. Phys. 52, 479 (1988). A comprehensive list of references about nonextensive statistical mechanics and its applications can be. Abe , Y Okamoto, Nonextensive Statistical Mechanics and Its Applications. HeidelbergSpringer-VerlagNonextensive Statistical Mechanics and Its Applications, edited by S. Abe and Y. Okamoto (Springer-Verlag, Heidelberg, 2001). A comprehensive list of references about nonextensive statistical mechanics and its applications can be found at URL: http://tsallis.cat.cbpf.br/TEMUCO.pdf . A Carati, Physica A. 348110A. Carati, Physica A 348, 110 (2005). . A Carati, cond-mat/0501588to appear in Physica AA. Carati, e-print cond-mat/0501588, to appear in Physica A. . E G D Cohen, Physica A. 30519E. G. D. Cohen, Physica A 305, 19 (2002). . S Abe, G B Bagci, Phys. Rev. E. 7116139S. Abe and G. B. Bagci, Phys. Rev. E 71, 016139 (2005). . S Abe, cond-mat/0504036to appear in Physica AS. Abe, cond-mat/0504036, to appear in Physica A. Theory and Application of The z-Transform Method. E I See, Jury, WileyNew YorkSee, for example, E. I. Jury, Theory and Application of The z-Transform Method (Wiley, New York, 1964). C Tsallis, M Gell-Mann, Y Sato, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA10215377C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. USA 102, 15377 (2005); . Europhys. News. 36186Europhys. News 36, 186 (2005). . R Tonelli, G Mezzorani, F Meloni, M Lissia, M Coraddu, Prog. Theor. Phys. 11523R. Tonelli, G. Mezzorani, F. Meloni, M. Lissia, and M. Coraddu, Prog. Theor. Phys. 115, 23 (2006). . A Celikoglu, U Tirnakli, cond-mat/0603245e-printA. Celikoglu and U. Tirnakli, e-print cond-mat/0603245. . S Abe, ibid. 244Phys. Lett. A. 224229S. Abe, Phys. Lett. A 224, 326 (1997); ibid. 244, 229 (1998). . G Kaniadakis, ibid. 72Phys. Rev. E. 6636108G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); ibid. 72, 036108 (2005). . B Lesche, J. Stat. Phys. 27419B. Lesche, J. Stat. Phys. 27, 419 (1982); . Phys. Rev. E. 7017102Phys. Rev. E 70, 017102 (2004). . S Abe, G Kaniadakis, A M Scarfone, J. Phys. A. 3710513S. Abe, G. Kaniadakis, A.M. Scarfone, J. Phys. A 37 (2004) 10513. . S Abe, Phys. Rev. E. 6646134S. Abe, Phys. Rev. E 66, 046134 (2002).

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{'abstract': 'The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs entropy, plays a central role in nonextensive statistical mechanics of complex systems. A lot of efforts have recently been made on establishing a dynamical foundation for the Tsallis entropy. They are primarily concerned with nonlinear dynamical systems at the edge of chaos. Here, it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by Carati [A. Carati, Physica A 348, 110 (2005)]that, whenever relevant, the Tsallis entropy indexed by q is temporally extensive: linear growth in time, i.e., finite entropy production rate. Then, the universal bound on the entropy production rate is shown to be 11 /| | −q . The property of the associated probabilistic process, i.e., the sojourn time distribution, determining randomness of motion in phase space is also analyzed. PACS number(s): 65.40.Gr, 05.20.-y, 05.90.+m, 02.50.-r ________________________________', 'arxivid': 'cond-mat/0603550', 'author': ['Sumiyoshi Abe \nInstitute of Physics\nUniversity of Tsukuba\n305-8571IbarakiJapan\n', 'Yutaka Nakada \nInstitute of Physics\nUniversity of Tsukuba\n305-8571IbarakiJapan\n'], 'authoraffiliation': ['Institute of Physics\nUniversity of Tsukuba\n305-8571IbarakiJapan', 'Institute of Physics\nUniversity of Tsukuba\n305-8571IbarakiJapan'], 'corpusid': 6679226, 'doi': '10.1103/physreve.74.021120', 'github_urls': [], 'n_tokens_mistral': 8533, 'n_tokens_neox': 7290, 'n_words': 4291, 'pdfsha': 'ec687d6e1c3851a05152fb2eaecd7d7ee390ef36', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0603550v1.pdf'], 'title': ["Temporal extensivity of Tsallis' entropy and the bound on entropy production rate", "Temporal extensivity of Tsallis' entropy and the bound on entropy production rate"], 'venue': []}

arxiv

Experimental demonstration of fully contextual quantum correlations on an NMR quantum information processor Dileep Singh Jaskaran Singh Kavita Dorai Arvind Department of Physical Sciences Indian Institute of Science Education & Research Mohali Sector 81 SAS Nagar 140306Manauli, PunjabPOIndia Experimental demonstration of fully contextual quantum correlations on an NMR quantum information processor numbers: 0365Ud0367Lx0367Mn The existence of contextuality in quantum mechanics is a fundamental departure from the classical description of the world. Currently, the quest to identify scenarios which cannot be more contextual than quantum theory is at the forefront of research in quantum contextuality. In this work, we experimentally test two inequalities, which are capable of revealing fully contextual quantum correlations, on a Hilbert space of dimension eight and four respectively, on an NMR quantum information processor. The projectors associated with the contextuality inequalities are first reformulated in terms of Pauli operators, which can be determined in an NMR experiment. We also analyze the behavior of each inequality under rotation of the underlying quantum state, which unitarily transforms it to another pure state. I. INTRODUCTION Non-contextual hidden variable (NCHV) theories in which outcomes of measurements do not depend on other compatible measurements, have been shown not to reproduce quantum correlations [1,2]. Quantum mechanics (QM) exhibits the property of contextuality [3][4][5] which implies that measurement results of observables depend upon other commuting observables which are within the same measurement test. Much recent research is going on in the direction of guessing the physical principle responsible for this form of contextuality [6]. The pertinent questions that arise include whether there is any theory more contextual than quantum mechanics and whether the simplest scenario in which more general theories cannot be more contextual than quantum mechanics can be identified [7][8][9][10]. Contextuality tests correspond to the violation of certain inequalities involving expectation values, and the first such test was proposed by Kochen and Specker [2] by using a single qutrit system (the KS theorem), and a modified KS scheme was constructed by Peres [11]. State-independent [12][13][14] tests use the set of observables such that for any quantum state there is no probability distribution which can describe the outcome of measurement of these observables on that state, hence these tests are able to reveal the contextual behavior of any state of the quantum system. On the other hand, the state-dependent [15][16][17] tests typically use fewer observables to show that no joint probability distribution can describe the measurement outcomes on a certain subset of states of the quantum system. The smallest indivisible physical system exhibiting quantum contextu-ality for repeatable measurements is a qutrit (a threelevel quantum system) [1]. The simplest state-dependent non-contextual inequality which is commonly referred to as the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality [15], for a qutrit requires five experiments, each of them involving two compatible yes-no tests [7]. Several experimental tests of quantum contextuality have been demonstrated by different groups using photons [18][19][20][21][22], ions [23,24], neutrons [25] and nuclear spins [26,27]. In this paper, we experimentally demonstrate fully contextual quantum correlations via two different inequalities, on an NMR quantum information processor. The first inequality as proposed by Cabello [7], utilizes ten projectors and requires five measurements on a state in a Hilbert space of dimension at least six. We demonstrate this inequality by realizing the six-dimensional subspace on states in an eight-dimensional Hilbert space. The second inequality as proposed by Nagali et. al [21], uses ten projectors and ten measurements which we implement on states in a four-dimensional Hilbert space. For experimental verification of both the inequalities, we decompose all the projectors involved in terms of Pauli operators. The advantage is two-fold: first, it reduces the need of performing quantum state tomography which is a resource-intensive procedure and second, the inequalities can be tested by using a fewer number of observables. The eight-dimensional and four-dimensional Hilbert spaces are physically realized using three and two NMR qubits, respectively. Violation of the inequalities as observed experimentally match well with theoretical predictions and have an experimental fidelity ≥ 0.96. We also study the behavior of both the inequalities when the underlying quantum state undergoes a rotation. Our results imply that the violation of both inequalities follows a nonlinear trend with respect to the rotation angle of the underlying state. We also find that fully contextual quantum correlations on an eight-dimensional Hilbert space are more robust against state rotation, as compared to the ones on the four-dimensional Hilbert space, allowing a greater angle for violation. arXiv:1903.09912v1 [quant-ph] 24 Mar 2019 The material in this paper is arranged as follows: Section II describes the fully contextual quantum correlations, the quantum state and the yes/no tests required to reveal correlations with zero non-contextual content and their experimental implementation on an eightdimensional quantum system using three NMR qubits. Section III describes fully contextual quantum correlations in a four-dimensional Hilbert space, and its experimental implementation using two NMR qubits. Section IV contains a few concluding remarks. II. FULLY CONTEXTUAL QUANTUM CORRELATIONS IN AN EIGHT-DIMENSIONAL HILBERT SPACE In this section, we first review a contextuality inequality which is capable of revealing fully contextual quantum correlations as developed by Cabello [7], which requires a Hilbert space dimensionality of at least 6. We then design a modified version of the inequality via decomposition of the projectors into Pauli matrices, for ease of experimental implementation. We experimentally test the inequality on an eight-level quantum system, physically realized via three NMR qubits. The simplest test of quantum contextuality requires the measurement of five different projectors {Π i }, i ∈ {0, 1, 2, 3, 4} and Π i = |v i v i |, where |v i are unit vectors [15]. These projectors follow the exclusivity relation P (Π i = 1) + P (Π i⊕1 = 1) = 1, where P (Π i = 1) represents the probability of obtaining the outcome Π i , and addition is taken modulo five. For projective measurements, this relationship implies that only one of Π i or Π i⊕1 can be obtained in a joint measurement of both. Pauli operators Pauli operators A0 = 1 ⊗ 1 ⊗ σx A18 = σx ⊗ σz ⊗ σx A1 =1 ⊗ 1 ⊗ σz A19 = σx ⊗ σz ⊗ σz A2 = 1 ⊗ σx ⊗ 1 A20 = σy ⊗ 1 ⊗ σy A3 = 1 ⊗ σx ⊗ σx A21 = σy ⊗ σx ⊗ σy A4 = 1 ⊗ σx ⊗ σz A22 = σy ⊗ σy ⊗ 1 A5 = 1 ⊗ σy ⊗ σy A23 = σy ⊗ σy ⊗ σx A6 = 1 ⊗ σz ⊗ 1 A24 = σy ⊗ σy ⊗ σz A7 = 1 ⊗ σz ⊗ σx A25 = σy ⊗ σz ⊗ σy A8 = 1 ⊗ σz ⊗ σz A26 = σz ⊗ 1 ⊗ 1 A9 = σx ⊗ 1 ⊗ 1 A27 = σz ⊗ 1 ⊗ σx A10 = σx ⊗ 1 ⊗ σx A28 = σz ⊗ 1 ⊗ σz A11 = σx ⊗ 1 ⊗ σz A29 = σz ⊗ σx ⊗ 1 A12 = σx ⊗ σx ⊗ 1 A30 = σz ⊗ σx ⊗ σx A13 = σx ⊗ σx ⊗ σx A31 = σz ⊗ σx ⊗ σz A14 = σx ⊗ σx ⊗ σz A32 = σz ⊗ σy ⊗ σy A15 = σx ⊗ σy ⊗ σx A33 = σz ⊗ σz ⊗ 1 A16 = σx ⊗ σy ⊗ σy A34 = σz ⊗ σz ⊗ σx A17 = σx ⊗ σz ⊗ 1 A35 = σz ⊗ σz ⊗ σz The corresponding test, termed as KCBS inequality [7] is of the form 1 2 4 i=0 P (Π i + Π i⊕1 = 1) NCHV ≤ 2 QM ≤ √ 5 GP ≤ 5 2 ,(1) where the inequalities correspond to the maximum value achievable for non-contextual hidden variable (NCHV) theories, quantum mechanics (QM) and generalized probabilistic (GP) theories. As is evident from Eqn. (1), the maximum violation that can be achieved in quantum mechanics is less than what can be attained if an underlying GP model is considered. Therefore, for the KCBS scenario, quantum correlations are not fully contextual. Recently, it has been shown that there exist tests of contextuality for which quantum correlations saturate the bound as imposed by GP models [28]. For these scenarios, quantum correlations are either non-contextual or fully contextual. The simplest test of contextuality, capable of revealing fully contextual quantum correlations again requires only five measurements, but of ten different projectors {Π i } and is of the form, K = 1 2 4 i=0 P (Π i +Π i+1 +Π i+5 +Π i+7 = 1) NCHV ≤ 2 QM, GP ≤ 5 2 , (2) where the sum in the indices is defined such that 4+1 = 0 and 3 + 7 = 5. Since both the KCBS and the aforementioned inequality (Eqn. 2) require only five different measurements, the above scenario is termed as a twin inequality of KCBS, with the only difference that it is capable of revealing fully contextual quantum correlations and requires quantum systems having Hilbert space dimension at least six. We will henceforth refer to this inequality as the "KCBS-twin" inequality. The scenario corresponding to the KCBS-twin inequality (Eqn. 2) can be represented by an exclusivity graph as shown in Fig. 1. In this graph, each vertex corresponds to a unit vector |v i used to construct the projectors Π i , and two vertices are connected by an edge if and only if they are exclusive. From the graph it is possible to identify five different measurements M i which are defined as M i = {Π i , Π i+1 , Π i+5 , Π i+7 }, ∀i ∈ {0, 1, ..., 9}. (3) These measurements can be identified from the graph in Fig. 1 by five sets of four interconnected vertices, each represented by a different line style. ω H (ppm) ω C (ppm) (a) 1 H 13 C 19 F (b) (c) 1 H 19 F 13 C |11 |01 |10 |00 |01 |00 |10 |11 |01 |00 |10 |11 |00 |00ω H (ppm) ω F (ppm) ω C (ppm) FIG. 2. (a) Molecular structure of 13 C-labeled diethyl fluoromalonate used to physically realize three qubits. NMR spectra of (b) the thermal equilibrium state and (c) the pseudopure state |000 . Each peak is labeled with the logical state of the qubit which is passive during the transition. The horizontal scale represents the chemical shifts in ppm. An explicit form of the KCBS-twin inequality (Eqn. 2) which saturates the QM and GP bound can be obtained if we consider the unit vectors |v i defined as: v 0 | ≡ 1 √ 8 ( √ 2, − √ 2, 0, 0, 2, 0, 0, 0), (4a) v 1 | ≡ 1 √ 8 ( √ 2, 0, 0, √ 2, −1, √ 3, 0, 0), (4b) v 2 | ≡ 1 2 (1, −1, −1, −1, 0, 0, 0, 0), (4c) v 3 | ≡ 1 2 (1, −1, 1, 1, 0, 0, 0, 0), (4d) v 4 | ≡ 1 √ 8 ( √ 2, 0, 0, − √ 2, −1, √ 3, 0, 0), (4e) v 5 | ≡ 1 √ 8 ( √ 2, 0, − √ 2, 0, −1, − √ 3, 0, 0), (4f) v 6 | ≡ 1 √ 8 ( √ 2, 0, √ 2, 0, −1, − √ 3, 0, 0), (4g) v 7 | ≡ 1 2 (1, 1, 1, −1, 0, 0, 0, 0), (4h) v 8 | ≡ 1 √ 8 ( √ 2, √ 2, 0, 0, 2, 0, 0, 0), (4i) v 9 | ≡ 1 2 (1, 1, −1, 1, 0, 0, 0, 0).(4j) The state |ψ on which the measurements M i will be performed is chosen as ψ| ≡ (1, 0, 0, 0, 0, 0, 0, 0),(5)so that v i |ψ = 1 2 ∀ i ∈ {0, 1, ..., 9} which subsequently ensures the exclusivity relation P (Π i + Π i+1 + Π i+5 + Π i+7 = 1) = 1, i = 0, 1, ..., 4. In order to evaluate the KCBS-twin inequality experimentally, we first decompose the projectors involved in terms of Pauli operators, σ j , j ∈ {x, y, z} for three qubits given by: Π 0 = 1 16 − A 0 + A 1 + 2A 6 − A 7 + A 8 + √ 2(A 9 − A 10 + A 11 + A 17 − A 18 + A 19 − A 20 − A 25 ) − A 27 − A 28 − A 34 − A 35 + 21 ,(6a)Π 1 = 1 32         − √ 3A 0 − A 1 + 2A 3 − 2A 5 + 2A 6 − √ 3A 7 + A 8 + √ 2 −A 9 + √ 6A 10 − A 11 + √ 6A 12 − A 13 − √ 6A 14 + A 15 + A 16 − A 17 + √ 6A 18 + √ 2 −A 19 − √ 6A 20 − A 21 + √ 6A 22 − A 23 − √ 6A 24 − √ 6A 25 + √ 3A 27 + A 28 + 2A 30 − 2A 32 − 2A 33 + √ 3A 34 + 3A 35 + 41         ,(6b)Π 2 = 1 8 −A 4 + A 5 − A 7 + A 26 − A 31 + A 32 − A 34 + 1 ,(6c)Π 3 = 1 8 (A 4 − A 5 − A 7 + A 26 + A 31 − A 32 − A 34 + 1) ,(6d)Π 4 = 1 32         − √ 3A 0 − A 1 − 2A 3 + 2A 5 + 2A 6 − √ 3A 7 + A 8 + √ 2 −A 9 + √ 6A 10 − A 11 − √ 6A 12 + A 13 + √ 6A 14 − A 15 − A 16 − A 17 + √ 6A 18 + √ 2 −A 19 − √ 6A 20 + A 21 − √ 6A 22 + A 23 + √ 6A 24 − √ 6A 25 + √ 3A 27 + A 28 − 2A 30 + 2A 32 − 2A 33 + √ 3A 34 + 3A 35 + 41         ,(6e)Π 5 = 1 32         √ 3A 0 + A 1 − 2A 2 − 2A 4 + 2A 6 + √ 3A 7 − A 8 + √ 2 −A 9 − √ 6A 10 − A 11 + A 12 + √ 6A 13 + A 14 + √ 6A 16 − A 17 − √ 6A 18 + √ 2 A 19 + √ 6A 20 − √ 6A 21 + A 22 + √ 6A 23 + A 24 + √ 6A 25 − √ 3A 27 + 3A 28 − 2A 29 − 2A 31 − 2A 33 − √ 3A 34 + A 35 + 41         ,(6f)Π 6 = 1 32         √ 3A 0 + A 1 + 2A 2 + 2A 4 + 2A 6 + √ 3A 7 − A 8 + √ 2 −A 9 − √ 6A 10 − A 11 − A 12 − √ 6A 13 − A 14 − √ 6A 16 − A 17 − √ 6A 18 + √ 2 −A 19 + √ 6A 20 + √ 6A 21 − A 22 − √ 6A 23 − A 24 + √ 6A 25 − √ 3A 27 + 3A 28 + 2A 29 + 2A 31 − 2A 33 − √ 3A 34 + A 35 + 41         ,(6g)Π 7 = 1 8 (A 4 + A 5 + A 7 + A 26 + A 31 + A 32 + A 34 + 1) ,(6h)Π 8 = 1 16    + A 0 + A 1 + 2A 6 + A 7 + A 8 + √ 2 (A 9 + A 10 + A 11 + A 17 + A 18 + A 19 + A 20 + A 25 ) + A 27 − A 28 + A 34 − A 35 + 21    ,(6i)Π 9 = 1 8 (−A 4 − A 5 + A 7 + A 26 − A 31 − A 32 + A 34 + 1) . (6j) where A i s are given in [27,29,30] given by the observables {A i }. (a) R y (θ) |000 000| State Initialisation (b) 1 H 19 F 13 C 1 H 19 F 13 C G x π 4 π 4 −y x Using the decomposition given in Eqn. (6), the inequality K (given in Eqn. 2) can be re-written as: K = 1 8 Tr [A · ρ] NCHV ≤ 2 QM, GP ≤ 5 2 ,(7) where ρ = |ψ ψ| and A = A 1 + 4A 6 + A 8 + 4A 26 + A 28 − 2A 33 + A 35 + 101.(8) By experimentally measuring the expectation value of each observable A i for the state ρ, the value of the inequality K can be estimated. The explicit mapping of expectation value of the observables onto Pauli z operators for three qubits is given in Table II. The underlying state |ψ is unitarily rotated by an angle θ as: |ψ(θ) = U θ ⊗ 1 ⊗ 1|ψ ,(9) where, To experimentally implement the KCBS-twin inequality capable of revealing fully contextual quantum correlations for an eight-dimensional quantum system, we used the molecule of 13 C -labeled diethyl fluoromalonate dissolved in acetone-D6, with the 1 H, 19 F and 13 C spin-1/2 nuclei being encoded as 'qubit one', 'qubit two' and 'qubit three', respectively (see Fig 2 for the molecular structure and corresponding NMR spectrum of the PPS state, and Table III for details of the experimental NMR parameters). The NMR Hamiltonian for a three-qubit system is given by [29] U θ = cos θ 2 − sin θ 2 sin θ 2 cos θ 2(10)H = − 3 i=1 v i I i z + 3 i>j,i=1 J ij I i z I j z(11) where the indices i, j = 1, 2, or 3 label the qubit, ν i is the chemical shift of the ith qubit in the rotating frame, J ij is the scalar coupling interaction strength, and I i z is z-component of the spin angular momentum operator of the i th qubit. The system was initialized in a pseudopure state (PPS), i.e., |000 , using the spatial averaging technique [31] with the density operator given by ρ 000 = 1 − 2 3 I 8 + |000 000|(12) where is proportional to the spin polarization and I 8 is the 8×8 identity operator. The fidelity of the experimentally prepared PPS state was computed to be 0.96 using the fidelity measure [32]. Full quantum state tomography (QST) [33,34] was performed to experimentally reconstruct the density operator via a set of preparatory pulses {III, IIY, IY Y, Y II, XY X, XXY, XXX}, where I implies no operation, and X(Y ) denotes a qubit-selective rf pulse of flip angle 90 • of phase x(y). Experiments were performed at room temperature (294 K) on a Bruker Avance III 600-MHz FT-NMR spectrometer equipped with a QXI probe. Local unitary operations were achieved by using highly accurate and calibrated spin selective transverse rf pulses of suitable amplitude, phase, and duration. Nonlocal unitary operations were achieved by free evolution under the system Hamiltonian, of suitable duration under the desired scalar coupling with the help of embedded π refocusing pulses. The durations of the π 2 pulses for 1 H, 19 F, and 13 C nuclei were 9.36 µs at 18.14 W power level, 23. The quantum circuit to construct the states required to test fully contextual quantum correlations is shown in Fig. 3(a) and the corresponding NMR pulse sequence is shown in Fig 3(b). Different states can be prepared by varying the value of the flip angle θ of the rf pulse. We prepared seven different states by varying the flip angle θ to attain a range of values: 180 • , 120 • , 90 • , 60 • , 45 • , 36 • , 0 • . The state prepared with θ = 180 • gives the minimum value of K, while the state prepared without applying any rf pulse (θ = 0 • ) gives the maximum value. All the states required to demonstrate the KCBS-twin inequality on an 8-dimensional Hilbert space which are capable of revealing the transformation from classical correlations to fully contextual Fig. 4. For each of the initial states, the contextuality test was repeated three times. The mean values and the corresponding error bars were computed and the result is shown in Fig 5, where the inequality values are plotted for different values of the parameter θ. The maximum of the sum of probabilities using classical theory is 2 and the maximum of sum of probabilities using quantum theory is 2.5, which are depicted by dotted and dashed lines respectively in Fig 5. The theoretically computed and experimentally obtained values of the inequality for different values of the θ parameter are tabulated in Table IV. The theoretical and experimental values match well, within the limits of experimental errors. From Fig. 5 it is also seen that the violation observed for the KCBS-twin inequality decreases as the original state |ψ is rotated through an angle θ, with no violation when the transformed state is orthogonal to the original state. Furthermore, the plot is nonlinear, indicating that smaller rotations lead to minor changes in violation, while larger rotations may also lead to observing no violation at all. III. FULLY CONTEXTUAL QUANTUM CORRELATIONS IN A FOUR-DIMENSIONAL HILBERT SPACE In this section, we first review a contextuality inequality which is capable of revealing fully contextual quantum correlations as developed by Nagali et. al [21] which utilizes states in a Hilbert space of dimension at least four. We provide a modified version of the inequality by decomposition into Pauli matrices which we experimentally test on a four-level quantum system using two NMR qubits. Fully contextual quantum correlations can also be achieved for scenarios other than KCBS. As shown in Reference [21], one such scenario entails measurements corresponding to ten different projectors Π j = |u j u j |, j = {0, 1, ..., 9}. In this particular scenario, the projec- tors follow exclusivity relationships as depicted in Fig. 6, where each vertex represents a projector Π i and two projectors are connected by an edge if and only if they are exclusive. The corresponding test of contextuality is then given by the inequality: The scenario is reminiscent of the KCBS-twin inequality discussed in the previous section, however this test requires ten different measurements rather than five and is capable of revealing fully contextual quantum correlations in a much smaller Hilbert space (of minimum dimension four). The inequality can be explicitly tested if we consider the unit vectors |u i as follows: 1, 1, 1), (14e) C = 9 i=0 P (Π i = 1) NCHV ≤ 3 QM, GP ≤ 7 2 .(13)u 0 | ≡ 1 √ 2 (0, 0, 1, 1), (14a) u 1 | ≡ 1 2 (1, −1, 1, −1),(14b)u 2 | ≡ 1 2 (1, −1, −1, 1),(14c)u 3 | ≡ 1 √ 2 (1, 0, 0, −1),(14d)u 4 | ≡ 1 2 (1,u 5 | ≡ 1 √ 2 (0, 1, 0, −1),(14f)u 6 | ≡ 1 2 (−1, 1, 1, 1),(14g)u 7 | ≡ 1 √ 2 (1, 0, 0, 1),(14h)u 8 | ≡ 1 2 (1, 1, 1, −1),(14i)u 9 | ≡ 1 2 (1, 1, −1, 1). (14j) The corresponding projective measurements are of the form M j = {Π j , 1 − Π j } ∀ j ∈ {0, 1, ..., 9},(15) which are performed on the state φ| ≡ (0, 0, 0, 1). For the experimental implementation of the inequality, we again decompose the projectors {Π j } in terms of Pauli operators : Π 0 = 1 4 (−σ z ⊗ 1 − σ z ⊗ σ x + 1 ⊗ σ x + 1 ⊗ 1),(17a)Π 1 = 1 4 (σ x ⊗ 1 − σ x ⊗ σ x − 1 ⊗ σ x + 1 ⊗ 1),(17b)Π 2 = 1 4 (−σ x ⊗ 1 + σ x ⊗ σ x − 1 ⊗ σ x + 1 ⊗ 1),(17c)Π 3 = 1 4 (−σ x ⊗ σ x + σ y ⊗ σ y + σ z ⊗ σ z + 1 ⊗ 1), (17d) Π 4 = 1 4 (σ x ⊗ 1 + σ x ⊗ σ x + 1 ⊗ σ x + 1 ⊗ 1),(17e)Π 5 = 1 4 (−σ x ⊗ 1 + σ x ⊗ σ z − 1 ⊗ σ z + 1 ⊗ 1),(17f)Π 6 = 1 4 (−σ x ⊗ σ z + σ y ⊗ σ y − σ z ⊗ σ x + 1 ⊗ 1), (17g) Π 7 = 1 4 (σ x ⊗ σ x − σ y ⊗ σ y + σ z ⊗ σ z + 1 ⊗ 1), (17h) Π 8 = 1 4 (σ x ⊗ σ z + σ y ⊗ σ y + σ z ⊗ σ x + 1 ⊗ 1),(17i)Π 9 = 1 4 (−σ x ⊗ σ z − σ y ⊗ σ y + σ z ⊗ σ x + 1 ⊗ 1). (17j) Using Eqn. (13) and Eqn. (17), the inequality C can be re-written as C = 1 4 Tr [B · ρ ] NCHV ≤ 3 QM, GP ≤ 7 2 ,(18) where ρ = |φ φ| and B = B 0 + B 1 − B 2 + 2B 3 − B 4 + 101,(19) with B 0 = σ x ⊗ σ x , B 1 = σ y ⊗ σ y , B 2 = σ z ⊗ 1, B 3 = σ z ⊗ σ z , B 4 = 1 ⊗ σ z .(20) The underlying state |φ is unitarily rotated by an angle θ as: |φ(θ) = U θ ⊗ 1|φ ,(21) where U θ has been defined in Eqn. (10). 7. (a) Molecular structure of 13 C-labeled chloroform used as a two-qubit quantum system. NMR spectra of (b) the thermal equilibrium state and (c) the pseudopure state |00 . Each peak is labeled with the logical state of the qubit which is passive during the transition. Horizontal scale represents the chemical shifts in ppm. ω H (ppm) ω F (ppm) ω C (ppm) FIG. By experimentally evaluating the expectation value of the observables B j , the value of the inequality C can be estimated. To implement the non-contextual inequality capable of revealing fully contextual quantum correlations on a four-dimensional quantum system, the molecule of 13 C-enriched chloroform dissolved in acetone-D6 was used, with the 1 H and 13 C spins being labeled as 'qubit one' and 'qubit two', respectively (see Fig. 7 and Table VI for details of the experimental parameters). The Hamiltonian for a two-qubit system is given by [30] (a) |0 |0 R y (π) with the flip angle θ = 180 • and state fidelity 0.99 is depicted in Fig. 9. For each of these eight different initial states, the contextuality test was repeated three times. The mean values and the corresponding error bars were calculated and result is shown in Fig. 10 of probabilities using classical theory is 3 and the maximum of sum of probabilities using quantum theory is 3.5, which are shown by dotted and dashed lines respectively in Fig 10. As seen from the values tabulated in Table VII, the theoretically computed and experimentally measured values of the inequality agree well to within experimental errors. From Fig. 10 it is seen that the violation for the inequality C decreases as the original state |φ is rotated through an angle θ. It is seen that no violation is observed for angle θ > 70 • , which is in contrast with the inequality K, which exhibits violation for a larger range of θ. However, certain similarities remain, most notably the nonlinear nature of violation with respect to rotation. It is again observed that smaller rotations lead to minor changes in the violation, while larger rotations may lead to a situation where no violation is observed. R y (π) R y (θ) π π π π 4 π 4 β θ (b) 1 H 13 C G x x x −x −y y y x −x y τ12 IV. CONCLUDING REMARKS In this paper we experimentally demonstrated fully contextual quantum correlations on an NMR quantum information processor. We studied two distinct inequalities capable of revealing such correlations: the first inequality used five measurements on an eight-dimensional Hilbert space, while the second inequality used ten measurements on a four-dimensional Hilbert space to reveal the contextuality of the state. However, both the inequalities involved the same number of projectors. For an experimental demonstration of each inequality, every projector was decomposed in terms of the Pauli basis, and the corresponding inequality recast in terms of Pauli operators, thereby reducing the need for resource-intensive full state tomography. Both the inequalities K and C were experimentally implemented with a fidelity of ≥ 0.96 by measuring the expectation values of only seven and five Pauli operators for the state which maximizes the violation, respectively. In addition to demonstration of fully contextual quantum correlations, we analyzed the behavior of each inequality under rotation of the underlying state, which unitarily transforms it to another pure state. The experiments were repeated for various states rotated through an angle θ and were in good agreement with theoretical results. It was seen that both the inequalities follow a nonlinear trend, while the inequality K offers a greater range of violation than the inequality C with respect to the parameter θ. An experimental implementation of fully contextual quantum correlations is an important step towards achieving information processing tasks, for which no post-quantum theory can do better. While the inequality C has been experimentally observed on optical systems, an experimental demonstration of the inequality K is difficult owing to the high dimensionality of the Hilbert space required. Our work asserts the fact that NMR is an optimal test bed for such scenarios. FIG. 1 . 1Orthogonality graph corresponding to the KCBStwin inequality K. Vertices correspond to projectors, while edges represent the orthogonality relationship between two vertices. Five sets of four interconnected vertices correspond to measurements involved in testing K and are differentiated by different edge line styles. FIG. 3 . 3(a) Quantum circuit for state preparation; the parameter θ in the unitary Ry(θ) is used to generate different quantum states. (b) Corresponding NMR pulse sequence for the quantum circuit. The sequence of pulses before the first dashed black line achieves initialization of the state into the pseudopure |000 state. The value of the flip angle α is kept fixed at 57.87 • , while the flip angle θ is varied over a range of values. The broad unfilled rectangles denote π pulses, and the flip angle and phases of the other pulses written below each pulse. The time intervals τ12, τ13, τ23 are set to 1 2J HF , 1 2J HC , 1 2J F C , respectively. FIG. 4 . 4Real (left) and imaginary (right) parts of the theoretically expected and the experimentally reconstructed tomographs of the ψ1| = (0, 0, 0, 0, 1, 0, 0, 0) state in the eightdimensional quantum system, with an experimental state fidelity of 0.97. FIG. 5 . 525 µs at a power level of 42.27 W, and 15.81 µs at a power level of 179.47 W, respectively. Graph representing quantum correlations corresponding to the inequality K for various states rotated by angle θ from the initial state |ψ . FIG. 6 . 6Orthogonality graph corresponding to the inequality C. Vertices correspond to projectors and two vertices are connected by an edge if they are orthogonal. FIG. 8 . 8(a) Quantum circuit for the required state, generated randomly generated with the different flip angles. (b) NMR pulse sequence for the corresponding quantum circuit. The sequence of pulses before the first dashed black line achieves state initialization into the |00 state. The value of the flip angle β is kept fixed at 59.69 • , while the pulse rf flip angle θ is varied. The time interval τ12 is set to 1 2J HC . FIG. 9 . 9Real (left) and imaginary (right) parts of the theoretical and experimental tomographs of the φ1| = (0, −1, 0, 0) state in the four-dimensional Hilbert space, prepared with an experimental state fidelity of 0.99. FIG. 10 . 10, where the inequality values are plotted for different θ values. The maximum of sum Quantum correlations corresponding to the inequality C for various states plotted for different initial states |φ , as a function of the θ parameter. TABLE I . IPauli operators for three qubits, used to decompose the corresponding projectors for the experimental demonstration of the inequality K. TABLE II . IIProduct operators for a three-qubit system, mapped to the Pauli z operators via the initial state transformation ρ → ρi = Ui.ρ.U † i .Observable Expectation Unitary Operator σ3z = Tr[ρ1.σ3z] U1=Identity σ2z = Tr[ρ2.σ2z] U2=Identity σ2zσ3z = Tr[ρ3.σ3z] U3 = CNOT23 σ1z = Tr[ρ4.σ1z] U4=Identity σ1zσ3z = Tr[ρ5.σ3z] U5 = CNOT13 σ1zσ2z = Tr[ρ6.σ2z] U6 = CNOT12 σ1zσ2zσ3z = Tr[ρ7.σ3z] U7 = CNOT23 .CNOT12 TABLE III . IIINMR parameters for the three-qubit 13 C-labeled diethyl fluoromalonate system.Qubit ν (Hz) J (Hz) T1 (sec) T2 (sec) 1 H 3334.24 JHF = 47.5 3.4 1.6 19 F − 110999.94 JHC = 161.6 3.7 1.5 13 C 12889.53 JF C = −191.5 3.6 1.3 Table I . IIn NMR, the observed z magnetization of a nuclear spin in a quantum state is proportional to the expectation value of the σ z operator of the spin in that state. The time-domain NMR signal, i.e., the free-induction decay with appropriate phase gives Lorentzian peaks when Fourier transformed. Thesenormalized experimental intensities give an estimate of the expectation value of σ z of the quantum state. The observables of interest are A 1 , A 6 , A 8 , A 26 , A 28 , A 33 , A 35 for the eight-dimensional Hilbert space being considered. The task of experimentally demonstrating the inequality K (given in Eqn. 2) on an NMR quantum information processor becomes particularly accessible while dealing with the Pauli basis, since the NMR signal is propor- tional to ensemble average of the σ z operator. Thus mea- surement of the expectation value of the projectors {Π i } involved becomes simplified when they are decomposed into Pauli operators TABLE IV . IVTheoretically computed and experimentally measured values of quantum correlations corresponding to the inequality K for various states, rotated by angle θ, from the initial state |ψ .θ Theoretical Experimental 180 • 1.500 1.522±0.042 120 • 1.750 1.785±0.035 90 • 2.000 2.016±0.031 60 • 2.250 2.239±0.030 45 • 2.353 2.330±0.033 36 • 2.404 2.385±0.045 0 • 2.500 2.449±0.046 correlations, were experimentally prepared with state fi- delities of ≥ 0.96. The tomograph of one such exper- imentally reconstructed state with flip angle θ = 180 • with state fidelity 0.97 is depicted in TABLE V . VProduct operators for a two-qubit system mapped onto the Pauli z operators via the initial state ρ → ρi = Ui.ρ.U † i . Observable Expectation Unitary Operator σ1xσ2x = Tr[ρ1.σ2z] U1 = CNOT12 Y2Y1 σ1yσ2y = Tr[ρ2.σ2z] U2 = CNOT12 X2X1 σ1z = Tr[ρ3.σ1z]. U3=Identity σ1zσ2z = Tr[ρ4.σ2z] U4 = CNOT12 σ2z = Tr[ρ5.σ2z] U5=Identity TABLE VII . VIITheoretically computed and experimentally measured values of quantum correlations corresponding to the inequality C for different quantum states parameterized by the angle θ.θ Theoretical Experimental 180 • 2.000 2.024±0.025 120 • 2.375 2.433±0.031 90 • 2.750 2.754±0.029 69.23 • 3.016 2.989±0.040 60 • 3.125 3.171±0.034 45 • 3.280 3.334±0.035 30 • 3.399 3.434±0.040 0 • 3.500 3.501±0.032 where ν H , ν C are the chemical shifts, I H z , I C z are the zcomponents of the spin angular momentum operators of the 1 H and 13 C spins respectively, and J HC is the scalar coupling constant. The system was initialized in the pseudopure state (PPS) |00 , using the spatial averaging technique[35,36]with the density operator given bywhere I 4 is the 4 × 4 identity operator, is proportional to the spin polarization and can be evaluated from the ratio of magnetic and thermal energies of an ensemble of magnetic moments µ in a magnetic field B at temperature T ; ∼ µB k B T and at room temperature and for a B ≈ 10 Tesla, ≈ 10 −5 . The state fidelity of the experimentally prepared PPS was computed to be 0.99. For the experimental reconstruction of density operator full quantum state tomography (QST) was performed using a set of preparatory pulses {II, IX, IY, XX}. Most of the experimental details are the same as for the three-qubit case. The durations of π 2 pulses for 1 H, 13 C nuclei were 9.56 µs at power level 18.14 W and 16.15 µs at a power level of 179.47 W, respectively.Let π i be the observables (projectors) whose expectation value is to be measured in a state ρ = |ψ ψ|. Instead of measuring π , the state ρ can be mapped to ρ i by using ρ i = U i .ρ.U † i followed by a z-magnetization measurement of one of the qubits[30].Table Vdetails the mapping of Pauli basis operators (used in this paper) to the single-qubit Pauli z operator, where X, X, Y and Y represent the π 2 rotations with phases x, −x, y and −y, respectively. The observables of interest are B 0 , B 1 , B 2 , B 3 , B 4 for the four-dimensional Hilbert space under consideration.The quantum circuit to achieve the required states to test the inequality C on a four-dimensional quantum system is shown inFig. 8(a)and the corresponding NMR pulse sequence is shown inFig. 8(b). Eight different states were generated by varying the flip angle θ over a range of values: 180 • , 120 • , 90 • , 69.23 • , 60 • , 45 • , 30 • , 0 • . 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{'abstract': 'The existence of contextuality in quantum mechanics is a fundamental departure from the classical description of the world. Currently, the quest to identify scenarios which cannot be more contextual than quantum theory is at the forefront of research in quantum contextuality. In this work, we experimentally test two inequalities, which are capable of revealing fully contextual quantum correlations, on a Hilbert space of dimension eight and four respectively, on an NMR quantum information processor. The projectors associated with the contextuality inequalities are first reformulated in terms of Pauli operators, which can be determined in an NMR experiment. We also analyze the behavior of each inequality under rotation of the underlying quantum state, which unitarily transforms it to another pure state.', 'arxivid': '1903.09912', 'author': ['Dileep Singh ', 'Jaskaran Singh ', 'Kavita Dorai ', 'Arvind ', '\nDepartment of Physical Sciences\nIndian Institute of Science Education & Research Mohali\nSector 81\n', '\nSAS Nagar\n140306Manauli, PunjabPOIndia\n'], 'authoraffiliation': ['Department of Physical Sciences\nIndian Institute of Science Education & Research Mohali\nSector 81', 'SAS Nagar\n140306Manauli, PunjabPOIndia'], 'corpusid': 85498029, 'doi': '10.1103/physreva.100.022109', 'github_urls': [], 'n_tokens_mistral': 15952, 'n_tokens_neox': 12660, 'n_words': 7167, 'pdfsha': '29be51726a87c041efcdd2a3228a69dfd6f3df69', 'pdfurls': ['https://arxiv.org/pdf/1903.09912v1.pdf'], 'title': ['Experimental demonstration of fully contextual quantum correlations on an NMR quantum information processor', 'Experimental demonstration of fully contextual quantum correlations on an NMR quantum information processor'], 'venue': []}

arxiv

Spectroscopic metallicities of Vega-like stars 26 May 2008 May 26, 2008 C Saffe csaffe@casleo.gov.ar Complejo Astronómico El Leoncito CC 4675400San JuanArgentina M Gómez Observatorio Astronómico de Córdoba Laprida 8545000CórdobaArgentina O Pintado opintado@tucbbs.com.ar Instituto Superior de Correlación Geológica (INSUGEO) 4000TucumánArgentina E González Facultad de Ciencias Exactas, Físicas y Naturales (UNSJ) 5400San JuanArgentina Spectroscopic metallicities of Vega-like stars 26 May 2008 May 26, 2008Received September 15, 1996; accepted March 16, 1997Astronomy & Astrophysics manuscript no. carlosTechniques: spectroscopic -Stars: abundances -Stars: late-type Aims.To determine the metallicities of 113 Southern Hemisphere Vega-like candidate stars in relation to the Exoplanet host group and field stars.Methods. We applied two spectroscopic methods of abundance determinations: equivalent width measurements together with the ATLAS9 (Kurucz 1993) model atmospheres and the WIDTH9 program, and a comparison of observed spectra with the grid of synthetic spectra of Munari et al.(2005).Results. For the Vega-like group, the metallicities are indistinguishable from those of field stars not known to be associated with planets or disks. This result is quite different from the metallicities of Exoplanet host stars which are metal-rich in comparison to field stars. Introduction It is well established that Exoplanet host stars are, on average, metal-rich in comparison to stars that do not harbor Doppler detected planets (see, for example, Santos et al. 2004). Two hypotheses have been put forward to explain this peculiarity of the Exoplanet host stars a) a primordial origin and b) a pollution of the convective zone of the star. In the first case the "excess" of metallicity was already present in the parent cloud from which the star bearing planet/s was formed (see, for example, Santos et al. 2001). In the pollution scenario the convective zone of the star is contaminated by the infall or accretion of planets and/or planetesimals (see, for example, González et al. 2001). Santos et al. (2004) found a lack of correlation between the thickness of the convective zone and the metallicity for a sample of FG dwarfs with planets. As the convective zone acts as a diluting medium, for a given amount of accreted material, F dwarfs with thinner convective zones should exhibit a greater degree of pollution than G dwarfs with thicker zones. On average, F and G dwarfs ⋆ On a fellowship from CONICET, Argentina. exhibit similar metallicities and the pollution hypothesis is not favored by these observations. The primordial origin of the "excess" remains an alternative to explain the relatively high metallicity of stars with planets with respect to field stars. However, Pasquini et al. (2007) compared the metallicities of giant and dwarf stars with planets and found that the first group has, on average, lower metallicities than the dwarfs. The smaller mass of the convective zone of the dwarfs with respect to the giants provides a plausible explanation for this difference. The diluting effect of the convective zone is efficient for the giants and tends to lower the metallicity to its primordial value. In this case, the pollution scenario is favored (over the primordial origin) since it can explain the observed difference in metallicities between dwarfs and giants with planets. Even when the origin or the cause of the "excess" of metallicity of stars with planets is not well understood, the Exoplanet host stars are metal-rich and this is a feature that distinguishes this group among stars with similar physical properties and no giant planets detected. Vega-like stars are a group of objects that show infrared excesses in their spectral energy distributions that can be attributed to the presence of dust in circumstellar disks. The first members or candidate members of the class were selected by IRAS and had mainly A-F spectral types (Aumann et al. 1984;Gillett 1986;Backman & Paresce 1993;Sylvester et al. 1996;Mannings & Barlow 1998;Fajardo-Acosta et al. 1999;Sylvester & Mannings 2000;Habing et al. 2001;Laureijs et al. 2002;Sheret et al. 2004). Vega (α Lyr) is one of the four prototypes of the group or the "fabulous four" (Vega, β Pictoris, Fomalhaut = α PsA and ǫ Eridanis; Gillett 1986) and has given the name to the class. More recently, Spitzer has contributed with the detection of G dwarfs with infrared excesses (Meyer et al. 2004;Rieke et al. 2005;Kim et al. 2005;Chen et al. 2005;Uzpen et al. 2005;Beichman et al. 2005Bryden et al. 2006;Silverstone et al. 2006;Su et al. 2006;Trilling et al. 2008). Since the excesses come from distances similar to the Kuiper-Belt to the Sun, these stars have also received the designation of Kuiper-Belt analogs or Kuiper-Belt-like stars. In this contribution we adopt the term "Vega-like stars" to refer to both IRAS and Spitzer detections. The metallicity of Vega-like stars has previously been investigated by Greaves et al. (2006) and Chavero et al. (2006), deriving nearly solar values. However these works analyzed relatively small samples of objects. Greaves et al. (2006) studied a group of 18 FGK Vega-like stars whereas Chavero et al. (2006) included 42 FG dwarfs with infrared excesses in their metallicity determination. In addition these previous works do not include stars of A spectral type which represent the bulge of IRAS detections. Greaves et al. (2006) derived their sample from the Doppler searches for planets that in general include solar type stars. Chavero et al. (2006) used the Strömgren photometry to determine the metallicity. These authors were also restricted to late spectral types. Both the stars with planets and the Vega-like stars have evidence of the presence of circumstellar material, in the form of planet/s, in the first case, or dust in a circumstellar disk, in the second. As mentioned before, the Exoplanet hosts are metal-rich. This fact may have facilitated the formation of planets (Pollack et al. 1996). In this contribution we determine spectroscopic metallicities of a large sample of Vega-like stars to compare with the Exoplanet host group. We include objects of B-K spectral types, observable from the Southern Hemisphere. The sample We compiled a total of 113 Southern Hemisphere Vega-like candidate stars from the literature, based on their infrared or submillimetric excess emissions (Backman & Paresce 1993;Sylvester et al. 1996;Mannings & Barlow 1998;Fajardo-Acosta et al. 1999;Sylvester & Mannings 2000;Habing et al. 2001;Laureijs et al. 2002;Sheret et al. 2004). This compilation also includes G dwarfs with infrared excess recently detected by Spitzer Bryden et al. 2006;Su et al. 2006;Trilling et al. 2008). Specifically the list comprises objects with BAFGK spectral types (22, 38, 28, 17 and 8, respectively). All the stars are luminosity class V (Hipparcos catalogue) and have distances between 5 and 300 pc. Table 1 lists the observed objects. Table 1 includes a sub-sample of stars that were originally selected by IRAS as candidate Vegalike stars. However, when observed by Spitzer the infrared excesses were deemed to be of little significance. These objects are: HD 10800, HD 20794, HD 38393, HD 41700, HD 68456, HD 160691, HD 169830, HD 203608, andHD 216437 (Beichman et al. 2005, 2006;Bryden et al. 2006;Hillenbrand et al. 2008;Trilling et al. 2008). For example, Bryden et al. (2006) Observations and data reduction The stellar spectra were obtained at the Complejo Astronomico El Leoncito (CASLEO), using the Jorge Sahade 2.15-m telescope equipped with a REOSC echelle spectrograph 1 and a TEK 1024×1024 CCD detector. The REOSC spectrograph uses gratings as cross dispensers. We used a grating with 400 lines mm −1 , covering the spectral range λλ3500-6500, giving a resolving power The spectra were reduced using IRAF 2 standard procedures for echelle spectra. We applied bias and flat corrections and then normalized order by order with the continuum task, using 7-9 order Chebyshev polynomials. We also corrected by the scattered light in the spectrograph (apscatter task). We fitted the background with a linear function on both sides of the echelle apertures, using the task apall. The resolution of the reduced spectra is 0.17 Å/pix. Metallicity determinations We used two different methods of abundance determination: 1) Fe lines equivalent width measurements together with the ATLAS9 (Kurucz 1993) model atmosphere corresponding to a given star and the WIDTH9 3 program. 2) A comparison of the observed and synthetic spectra using the Downhill method (Gray et al. 2001 Munari et al. (2005). This method offers the advantage that there is no need to identify and measure the equivalent widths of many Fe lines as with the WIDTH9 program. Metallicity determinations using the WIDTH program To determine abundances by this method it is necessary to estimate the stellar parameters T eff and Log g, by means of the Strömgren photometry, for example. With these quantities we adopt the Kurucz (1993)'s model atmosphere appropriated to each star. The model that initially is chosen has solar metallicity. Finally the Kurucz's model together with the measured equivalent widths are used by the WIDTH9 program (Kurucz 1992(Kurucz , 1993 to derive the metallicity. To obtain T eff and Log g, we have used the uvbyβ mean colors of Hauck & Mermilliod (1998) with two different calibrations: Napiwotzki et al. (1993) and Castelli et al. (1997) and Castelli (1998) (hereafter N93 and C97, respectively), with the TEMPLOGG code (Rogers et al. 1995). This program has been used in the COROT mission preparation (see, for example, Lastennet et al. 2001;Guillon & Magain 2006) and includes reddening corrections, according to Domingo & Figueras (1999) for stars in the range A3-F0, and to Nissen (1988) for spectral types F0-G2. We have compared the temperatures and gravities derived using both calibrations (N93 and C97) and noticed some differences particularly in the later parameter. For this reason we initially determined metallicities using values derived from both calibrations and later on considered if they significantly affect the final metallicity values. We have also confronted the obtained T eff with those published by Nordström et al. (2004). We found a good agreement in particular with the N93 calibration. With the values of T eff and Log g derived for each object, we have chosen the corresponding model atmosphere using the Kurucz ATLAS9 (Kurucz 1993) code. The stellar lines were identified using the general references of A multiplet table of astrophysical interest (Moore 1945) and Wavelengths and transition probabilities for atoms and atomic ions -Part 1: Wavelengths (Reader et al. 1980), as well as more specialized references for the Fe ii lines (Johansson 1978). The equivalent widths were measured by fitting Gaussian profiles through the stellar metallic lines using the IRAS splot task. There is no more than a 15% difference among the equivalent widths of the same lines, measured in different spectra. We have excluded from our abundance determinations seriously blended lines. To determine the abundances we need an initial estimation of the microturbulent velocity (ξ). For this estimation we have used the standard method. We computed the abundances from the Fe lines for a range of possible values of ξ satisfying two conditions: a) that the abundances of Fe lines were not dependent on the equivalent widths and b) that the rms errors were minima. To achieve the first condition the slope in the plot abundance vs ξ must be zero. We tried different ξ values to fulfill this requirement. In this sense the abundance and microturbulent velocity determinations are recursive and simultaneous. Once a ξ value has been fixed the corresponding abundances to all chemical species measured are determined using the WIDTH9 code. The WIDTH9 code requires the model atmosphere calculated by the ATLAS9 program, the equivalent width of each line as well as atomic constants such as oscillator strength (Log gf) values, excitation potentials, damping constants, etc. In particular for the Log gf we used Fuhr et al. (1988) and Kurucz (1992). This code calculates the theoretical equivalent widths for an initial C. Saffe et al.: Spectroscopic metallicities of Vega-like stars 5 input abundance and compares these values with the measured equivalent widths. Then the code modifies the abundance to achieve a difference between theoretical and measured equivalent widths < 0.01 mÅ. The final values of the metallicities corresponding to the N93 and C97 calibrations, are listed in Table 2. We have included the number of lines used in each determination as well as the rms of the average. To estimate errors for our WIDTH metallicities we consider the following facts. The most significant contribution to the final uncertainties, probably, comes from the equivalent width measurements. We assume a 5% error due to the continuum level determination. This translates into 20% maximum uncertainties in the metallicity estimation. The atomic constants may also have uncertainties. In particular we estimate that the oscillator strength values may cause differences of about 10% in the calculated metallicity. Finally to provide an estimation of "typical" errors introduced by the WIDTH method we increased the T eff by 150 K and the Log gf by 0.15, and recalculated the metallicity value for each star. We derived a median difference of 0.20 dex. The largest difference corresponds to HD 28978 (0.55 dex). Metallicity derivations from synthetic spectra: The Downhill method The WIDTH method is not practical when the number of stars is large. For each object, we need to identify and measure many spectral lines. An alternative would be to compare the observed spectra with a grid of synthetic ones corresponding to different values of the metallicites and choose from the grid the spectrum that better reproduces the observed data (Gray et al. 2001). This comparison has the advantage that the complete profiles of the lines and not only the equivalent widths are used in the metallicity determinations. In general synthetic spectra depend on four parameters: T eff , surface gravity (Log g), metallicity ([Fe/H]) and microturbulent velocity (ξ). Following Gray et al. (2001), we applied a multidimensional Downhill Simplex technique, in which the observed spectrum is compared to a grid of synthetic spectra. The "final" synthetic spectrum is an interpolation of spectra, rather than a single point in the grid. As we are working with four variables (T eff , Log g, [Fe/H] and ξ) the interpolation is done in 4d, minimizing the square differences in each wavelength (i.e., the χ 2 statistics). The stellar parameters are determined with a higher accuracy than the steps in the grid since they correspond to interpolated values. The grid of synthetic spectra was taken from Munari et al. (2005). The parameters range covered by the grid is the following: Synthetic spectral lines were convolved with the instrumental line profile corresponding to the REOSC/CASLEO. Finally they were also convolved with a Gaussian profile corresponding to the rotational velocities of the sample stars, taken from the literature (Glebocki et al. 2000;Mora et al. 2001;Yudin 2001;Royer et al. 2002;Cutispoto et al. 2002Cutispoto et al. , 2003Pizzolato et al. 2003;Strom et al. 2005;Reiners 2006). We weighted the synthetic spectra by the blaze function of each of the REOSC spectrograph order. Finally we normalized and re-sampled our data to compare them with Munari et al. (2005)'s grid. The spectral sampling of the synthetic spectra is 0.02 Å. 3500 K < T eff < 40000 K, We have implemented the Downhill method (Gray et al. 2001) by means of a Fortran program. From the stellar spectral type or the Strömgren photometry it is possible to estimate "a starting point" in the 4d grid. The Downhill method provides a searching algorithm within the 4d grid and finds the best match, minimizing the χ 2 . In our case, the final spectrum is obtained by an interpolation of 16 spectra of Munari et al. (2005)'s grid. In general it takes 15 -20 min for each star (50 -60 iterations) in a Pentium IV 2.0 GHz to find the best interpolated spectrum. Table 3 lists the metallicities obtained with the Downhill method for our sample of Vega-like stars. To estimate the uncertainties in the metallicities obtained by the Downhill method, we carried out a few tests. We first applied this method to 30 synthetic spectra of known metallicities. The median difference between the derived and known metallicities is 0.2 dex. The internal consistency of the method has been checked, by fixing one of the four variables and comparing the resultant metallicities. Fixed values for each variable were obtained, for example, from an adopted calibration: a) T eff was taken from the N93 calibration, b) Log g was adopted from the N93 calibration, c) ξ was fixed at 2.9 km/s, the solar value. The median difference, calculated by fixing 3 of the 4 variables with respect to the "standard" procedure (i.e., with 4 variable), was 0.05 dex. Considering this value and the median difference derived from the comparison with 30 synthetic spectra of known metallicities (0.2 dex), we estimate a "typical" uncertainty of 0.06 dex for the metallicities derived by the Downhill method. We have also compared the Downhill method derived metallicities with those obtained by Nordström et al. (2004) and Fischer & Valenti (2005). We first noticed a systematic difference of ∼ 0.09 dex between these two determinations. With these limitations in mind, we consider that the external consistency of the Downhill method derived metallicities is acceptable. We finally mention two parameters taken as fixed by the Downhill method, the radial and the rotational velocities. Radial velocities are initially determined, minimizing the χ 2 with an accuracy of 0.1 km/s or a median value of 0.03 dex in metallicity. Rotational velocities (v sin i) from Table 4). The KS-test (Press 1992) indicates that these distributions are similar and represent the same parent population. Comparison of metallicity determinations by the WIDTH and the Downhill methods We finally adopt the metallicities calculated with the Downhill method for the sample of Vegalike stars, as these determinations use the complete line profiles and not only the equivalent widths. In addition "typical" uncertainties are smaller than those estimated for the WIDTH method. Discussion of the results The metallicity of the Solar Neighborhood is usually represented by a control sample of stars, which should exclude, in our case, known Vega-like stars. (Napiwotzki et al. 1993) and C97 (Castelli et al. 1997;Castelli 1998) Table 4). pared with the two groups, are 0.13 and 0.226 dex, respectively. In other words, the "excess" is real, but the amount depends on the control sample used. The two control sets contain different classes of stars. The magnitude-limited sample includes more massive and metal-rich stars than the volume-limited set. The metallicity distribution of Exoplanet host stars is usually compared with a volume-limited group of solar neighborhood stars (González 1998(González , 1999González et al. 2001;Santos et al. 2000Santos et al. , 2003Santos et al. 2004;Sadakane et al. 2002;Laws et al. 2003). We compared the metallicity distribution of our Vega-like sample with a volume-limited sample of 71 stars, without Doppler detected Exoplanets (Santos et al. 2001;Gilli et al. 2006) and with 98 Exoplanets host stars (Santos et al. 2004). Metallicity values for these two comparison samples were obtained from Nordström et al. (2004). As discussed in Section 4.2, the agreement between our metallicities and those obtained by these authors is acceptable. Figure 2 shows these distributions. degrees. The vertical lines indicate the medians of each distribution: stars without planets, Vegalike stars, and exoplanet host stars, respectively (see Table 5). relation is also applicable to A stars (the bulge of IRAS detected Vega-like stars), the low median value of the metallicity for the Vega-like group (−0.11 dex, see Table 5) indicates that the probably for these stars to host a planet/s of the type detected by radial velocity surveys is also low. We note, however, that the dispersion of metallicities within the Vega-like stars is also significant (0.26 dex) and at least a fraction of these stars has metallicities high enough to host giant planets, assuming the "excess" of metallicity/presence of a giant planet/s holds for A spectral type stars. In addition it is worthwhile to mention that Doppler searches do not achieve the required precision to detect planets in A stars as metal lines practically disappear. We also compared the metallicity distribution of Vega-like stars, with a sample of 115 stars without excess at 24 or 70 µm, observed by Spitzer Bryden et al. 2006;Su et al. 2006). Figure 3 shows these distributions. Vega-like stars are indicated by the empty histogram whereas the stars without excess at 24 or 70 µm are shown by the histogram shaded at 45 degrees. The KS test shows no significant difference between the two distributions. Table 5 lists the medians and the dispersions of the four samples compared in Figures 2 & 3. The results in Table 5 indicate that, on average, the Vega-like group has metallicities similar to the stars in the Solar Neighborhood without detected planets or disks, in contrast to the Exoplanet host stars group. This result confirms and extends previous works by Greaves et al. (2006) and Chavero et al. (2006), based on relatively small numbers of FG Vega-like stars. In Figure 4 we analyze the metallicity distribution of Vega-like stars of different spectral types. The number of objects corresponding to each spectral type is indicated between brackets. The vertical bars are the dispersions within the spectral types. A-spectral-type stars still dominate the Vega-like group although Spitzer has significantly contributed with F and G stars during the last few years Bryden et al. 2006;Su et al. 2006). Figure 4 shows no trend of the metallicity with the spectral type for the Vega-like group. As suggested by Greaves et al. (2006) the relatively high metallicity of Exoplanet host stars as well as the solar metallicity value for the Vega-like stars can be understood within the core accumulation model of Pollack et al. (1996). The high metal content of the disk favors the fast formation of giant planets, which needs to accrete an atmosphere and migrate inward before the gas is dissipated from the disk. On the contrary, for Vega-like objects no giant planet needs to be formed and/or migrate inward. The gas may dissipate and still the planetesimal in the external part of the disk may produce dust by collisions. We tentatively analyzed two small sub-sets of Vega-like objects: the Vega-like stars with planets and the Vega-like group with no Doppler detected planets. The first group is composed of 7 stars: 6 with 70 µm excess detected by Spitzer (HD 33636, HD 50554, HD 52265, HD 82943, HD 128311 and HD 117176;) and ǫ Eri with infrared and submillimieter excesses Zuckerman 2001). In the second group we include 5 stars without Exoplanets detected by the Doppler technique (Santos et al. 2004;Gilli et al. 2006) and showing infrared excess in 24 or 70 µm (HD 7570, HD 38858, HD 69830, HD 76151 and HD 115617;Bryden et al. 2006). The median metallicity of Vega-like stars with planets is +0.07 dex and the dispersion is 0.16 dex. For the Vega-like objects without planets these values are: −0.08 and 0.18 dex, respectively. It seems that when a Vega-like star has a planet the metallicity increases slightly. However the small number of objects available as well as the dispersions prevent us from giving any statistical significance to this initial trend. Greaves et al. (2007) proposed that the solid-mass (i.e., metals) content in primordial disks, called M S , is the fundamental parameter that regulates the planet/disk formation. If M S is small, the star will form a Vega-like disk, while if M S is larger, a giant planet may be formed. Summary and Conclusions We derived spectroscopic metallicities for a group of 113 Southern Hemisphere Vega-like stars. We applied two methods to determine metallicities: the "classical" WIDTH method and a comparison with the grid of synthetic spectra of Munari et al. (2005) by means of the Downhill algorithm. The later method offers the advantage that the complete profile of the line is used in the metallicity derivation and not only the equivalent width. In addition we estimated smaller uncertainties in the metallicities derived by the Downhill method (0.1 dex) than with the WIDTH code (0.2 dex). Vega-like stars have metallicities similar to Solar Neighborhood stars without planets or disks and significantly different from the Exoplanet host stars. This result confirms previous estimations by Greaves et al. (2006) and Chavero et al. (2006), based on comparatively smaller samples. The low metallicities for the Vega-like group (median = −11 dex) in relation to the Exoplanet host stars (median = + 0.17, see for example , Fischer & Valenti 2005), may indicate that the probability for these stars to host a planet/s of the type detected by radial velocity surveys is also low. However the dispersion of metallicities within the Vega-like stars is also significant (0.26 dex) and thus a fraction of these objects may have metallicities high enough to form giant planets. We caution that Exoplanet host stars are mainly of FGK spectral types whereas the bulge of IRAS detected Vega-like stars has A spectral type which are, in general, excluded from radial velocity searches since high precisions are not feasible. In this we are assuming that the probability of a A star to be associated with a giant planet depends on the metallicity as is the case for FGK stars. We find no trend in the metallicities of Vega-like objects with the spectral type. Greaves et al. (2006) suggestion make compatible the relative high metallicity of Exoplanet host stars and the solar Neighborhood value for Vega-like stars with the core accumulation model of Pollack et al. (1996). Analyzing two relatively small sub-samples, we find that Vega-like stars with a Doppler detected planet have slightly higher metallicities than Vega-like stars known not to harbor such a planet. However this must be considered only as an initial trend that needs to be confirmed by increasing both samples to achieve a statistical significant result. of ∼ 12500 . 12500Three individual spectra for each object were obtained in four observing runs: August 05-08 2005, August 18-22 2005, February 18-25 2006 and May 04-07 2007 and have S/N ratio of about 300. < Log g < 5.0 dex, with steps of 0.5 dex, −2.5 dex < [Fe/H] < 0.5 dex, with steps of 0.5 dex, and ξ values of 0, 1, 2, and 4 km/s. In addition to these parameters, the synthetic spectra are calculated for 15 different rotation velocities, ranging 0 -500 km/s. In all, Munari et al. (2005)'s library contains 625000 different spectra. These authors calculated the complete synthetic spectral library for four resolving powers: 20000, 11500 (GAIA), 8500 (RAVE) and 2000 (SLOAN). To our request, Dr. U. Munari kindly provided a grid corresponding to the REOSC/CASLEO resolving power ( Fischer & Valenti (2005)'s determinations are, on average, larger than those from Nordström et al. (2004)'s. Our Downhill method derived metallicities show a better agreement with Nordström et al. (2004)'s value than with Fischer & Valenti (2005)'s. However this later comparison is based on a relatively small number of common stars.In the work ofNordström et al. (2004)'s the metallicities are derived as a secondary parameter obtained photometrically. In the case ofFischer & Valenti (2005), the metallicities are obtained by a comparison with synthetic spectra but using only a small range of wavelengths (6000 -6200 Å). Figure 1 1compares the metallicity distributions calculated with the WIDTH method plus the N93 calibration (histogram shaded at 0 degree) and the C97 calibrations (histogram shaded at 45 degrees), respectively. The empty histogram shows the distribution derived with the Downhill method for the Vega-like sample. Vertical lines indicate the medians of each distribution. The left line corresponds to the WIDTH+N93 median, and the right line shows (superimposed) the WIDTH+C97 and Downhill medians (see calibrations, respectively. The empty histogram shows the metallicity distribution derived by the Downhill method. The vertical lines indicate the medians of each distribution. The left line corresponds to the WIDTH+N93 median, and the right line shows (superimposed) the WIDTH+C97 and Downhill medians (see Fig. 3 . 3Metallicity distributions for the Vega-like sample, empty histogram, and for stars without excess at 24 or 70 µmBryden et al. 2006;Su et al. 2006). The vertical lines (almost superimposed) indicate the medians of each distribution. Fig. 4 . 4Metallicity of Vega-like stars of different spectral types. Between brackets is indicated the number of objects in each spectral type bin. The vertical lines are the corresponding dispersions. Table 4 . 4Medians and dispersions of the metallicities for the Vega-like sampleMethod Median Dispersion N [Fe/H] [Fe/H] WIDTH+N93 −0.14 0.28 113 WIDTH+C97 −0.11 0.26 113 Downhill −0.11 0.27 113 Note -N93: Napiwotzki et al. (1993)'s calibration; C97: Castelli et al. (1997) and Castelli (1998)'s calibration. the literature have "typical" dispersions of 5 -10%, corresponding to an error of about 10% in metallicity. In summary, we have estimated an internal uncertainty of 0.06 dex for metallicities derived from the Downhill method. A more conservative estimation would indicate a value of 0.1 dex. This corresponds to half of the uncertainty calculated for the WIDTH method (0.2 dex). In this manner, the Downhill method allows a more precise determination of the metallicities for our sample of Vega-like objects. Table 4 4lists the medians and the dispersions of the metallicities derived by applying the WIDTH and the Downhill methods for the Vega-like group. In the case of the WIDTH method we present the results corresponding to the two calibrations used (N93 and C97). The derived median values are practically indistinguishable. Vega-like stars are represented by the empty histogram, stars with planets by the histogram shaded at 0 degree and stars known not to harbor planets detected by the Doppler technique, by the histogram shaded at 45 degrees. The KS test shows no significant difference between the metallicities distributions of the Vega-like stars and stars without planets. On the other hand, the Vega-like stars metallicitydistribution is different from the metallicity distribution for stars with planets with a high level of confidence. Fischer & Valenti (2005) obtained that the probability that a FGK star harbors a giant planet/s increases as P(Z) ∝ (10 Z ) 2 , where Z is the stellar metallicity (see also, Wyatt et al. 2007b). If this Fig. 2. Metallicity distributions for the Vega-like sample, empty histogram, for stars with planets, histogram shaded at 0 degree, and for stars known not to harbor planets, histogram shaded at 45 Table 5 . 5Medians and dispersions of the Vega-like sample and three comparison groupsSample Median Dispersion N [Fe/H] [Fe/H] Vega-like stars −0.11 0.27 113 Exoplanet host stars +0.17 0.22 98 Volume-limited sample without planets −0.16 0.25 71 Stars without excess at 24 or 70 µm −0.12 0.24 115 Table 1 of 1without planets agree withGreaves et al. (2007)'sTable 1. However this can only be considered as an initial trend that needs to be confirmed by increasing the number of Vega-like objects with planets as well as objects known not to harbor Doppler detected planetary mass objects.Greaves et al. (2007) shows the range of metallicity and the final configurations (planet+debris, debris, etc.) derived by these authors. The medians of the metallicities of Vega-like stars with and Table 1 . 1Sample of Vega-like stars observed at the CASLEOTable 1. Continued.Star Distance V Spectral Reference [pc] Type HD 105 40 7.51 G0V DEC03 HILL08 HD 142 26 5.70 G1V BE05 TR08 HD 2623 365 7.93 K2 SB91 HD 3003 46 5.07 A0V MB98 O92 SB91 WY07 HD 9672 61 5.62 A1V MB98 O92 SB91 PW91 WW88 SN86 WY07 HD 10647 17 5.52 F8V MB98 O92 SB91 DEC03 TR08 HD 10700 4 3.49 G8V MB98 HDJL01 DEC03 HD 10800 27 5.88 G2V MB98 BE06 BR06 HD 17206 14 4.47 F5V O92 SB91 HD 17848 51 5.25 A2V MB98 HD 18978 26 4.08 A4V SCBS01 HD 20010 14 3.80 F8V O92 WW88 HD 20794 6 4.26 G8V DEC03 BE06 HD 21563 182 6.14 A4V MB98 HD 22049 3 3.72 K2V SB91 WW88 HDJL01 DEC03 HD 22484 14 4.29 F9V DEC03 TR08 HD 23362 309 7.91 K2 SB91 HD 25457 19 5.38 F5V DEC03 PAS06 HILL08 HD 28375 118 5.53 B3V O92 SB91 TR08 HD 28978 125 5.67 A2V BP93 HD 30495 13 5.49 G3V HDJL01 DEC03 TR08 HD 31295 37 4.64 A0V SN86 WY07 HD 33262 12 4.71 F7V BR06 TR08 HD 33636 29 7.00 G0 BE05 TR08 HD 33949 172 4.36 B7V MB98 O92 SB91 PW91 SN86 HD 35850 27 6.30 F7V DEC03 PAS06 AP08 HD 36267 88 4.20 B5V BP93 HD 37484 60 7.26 F3V PAS06 HILL08 HD 38206 69 5.73 A0V MB98 DEC03 WY07 HD 38385 53 6.25 F3V MB98 HD 38393 9 3.59 F7V MB98 HDJL01 SH03 BE06 HD 38678 22 3.55 A2V MB98 O92 PW91 AP91 C87 HDJL01 DEC03 SU06 WY07 HD 39014 44 4.34 A7V O92 SB91 C87 JU04 HD 39060 19 3.85 A3V MB98 O92 C92 SB91 PW91 AP91 WW88 C87 JJE86 HDJL01 DEC03 WY07 HD 40136 15 3.71 F1V MB98 BE06b HD 41700 27 6.35 G0V DEC03 HILL08 HD 41742 27 5.93 F4V MB98 HD 43955 305 5.51 B3V MB98 Table 2 . 2Metallicities, dispersion (δ) and number of lines (N) used with the WIDTH9 program, applying the N93 and C97 calibrations for the Vega-like sample Star [Fe/H] δ[Fe/H] [Fe/H] δ[Fe/H]N93 N93 C97 C97 N HD 105 −0.37 0.26 −0.33 0.26 15 HD 142 −0.45 0.27 −0.27 0.25 20 HD 2623 −0.20 0.26 0.09 0.20 23 HD 3003 0.17 0.22 0.07 0.31 17 HD 9672 −0.32 0.26 −0.31 0.21 24 HD 10647 0.12 0.22 −0.07 0.28 29 HD 10700 −0.73 0.29 −0.67 0.23 17 HD 10800 0.16 0.27 0.12 0.26 30 HD 17206 −0.22 0.27 0.03 0.21 22 HD 17848 −0.02 0.28 −0.17 0.20 21 HD 18978 −0.39 0.25 −0.11 0.23 22 HD 20010 −0.64 0.20 −0.62 0.30 27 HD 20794 −0.17 0.29 −0.58 0.24 17 HD 21563 −0.41 0.30 −0.10 0.26 19 HD 22049 −0.08 0.25 −0.13 0.27 22 HD 22484 −0.22 0.28 −0.19 0.26 21 HD 23362 −0.07 0.25 −0.47 0.29 25 HD 25457 0.18 0.21 −0.22 0.22 29 HD 28375 0.10 0.29 −0.19 0.26 23 HD 28978 0.20 0.22 0.33 0.24 18 HD 30495 0.11 0.29 0.13 0.24 23 HD 31295 −0.68 0.24 −0.76 0.25 27 HD 33262 0.07 0.25 −0.09 0.24 27 HD 33636 0.03 0.20 −0.09 0.27 16 HD 33949 0.00 0.21 −0.23 0.27 19 HD 35850 −0.23 0.27 −0.12 0.24 26 HD 36267 −0.23 0.23 −0.02 0.23 21 HD 37484 −0.17 0.31 −0.25 0.30 28 HD 38206 −0.06 0.21 0.32 0.23 23 HD 38385 0.09 0.25 0.12 0.30 24 HD 38393 0.30 0.21 0.21 0.27 26 HD 38678 −0.13 0.21 −0.35 0.28 29 HD 39014 −0.41 0.26 −0.39 0.30 23 HD 39060 0.00 0.29 0.17 0.21 16 HD 40136 −0.27 0.30 −0.33 0.27 29 HD 41700 −0.14 0.22 −0.41 0.22 23 HD 41742 −0.31 0.28 −0.30 0.30 16 Acknowledgements. 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{'abstract': 'Aims.To determine the metallicities of 113 Southern Hemisphere Vega-like candidate stars in relation to the Exoplanet host group and field stars.Methods. We applied two spectroscopic methods of abundance determinations: equivalent width measurements together with the ATLAS9 (Kurucz 1993) model atmospheres and the WIDTH9 program, and a comparison of observed spectra with the grid of synthetic spectra of Munari et al.(2005).Results. For the Vega-like group, the metallicities are indistinguishable from those of field stars not known to be associated with planets or disks. This result is quite different from the metallicities of Exoplanet host stars which are metal-rich in comparison to field stars.', 'arxivid': '0805.3936', 'author': ['C Saffe csaffe@casleo.gov.ar \nComplejo Astronómico El Leoncito\nCC 4675400San JuanArgentina\n', 'M Gómez \nObservatorio Astronómico de Córdoba\nLaprida 8545000CórdobaArgentina\n', 'O Pintado opintado@tucbbs.com.ar \nInstituto Superior de Correlación Geológica (INSUGEO)\n4000TucumánArgentina\n', 'E González \nFacultad de Ciencias Exactas, Físicas y Naturales (UNSJ)\n5400San JuanArgentina\n'], 'authoraffiliation': ['Complejo Astronómico El Leoncito\nCC 4675400San JuanArgentina', 'Observatorio Astronómico de Córdoba\nLaprida 8545000CórdobaArgentina', 'Instituto Superior de Correlación Geológica (INSUGEO)\n4000TucumánArgentina', 'Facultad de Ciencias Exactas, Físicas y Naturales (UNSJ)\n5400San JuanArgentina'], 'corpusid': 15059920, 'doi': '10.1051/0004-6361:200810260', 'github_urls': [], 'n_tokens_mistral': 22471, 'n_tokens_neox': 18256, 'n_words': 9245, 'pdfsha': '8d2121081cf80d0553438ec96d312ef46406fce0', 'pdfurls': ['https://arxiv.org/pdf/0805.3936v1.pdf'], 'title': ['Spectroscopic metallicities of Vega-like stars', 'Spectroscopic metallicities of Vega-like stars'], 'venue': []}

arxiv

A NEW METHOD FOR RECOGNISING SUZUKI GROUPS 18 Jun 2017 John N Henrik Bäärnhielm A NEW METHOD FOR RECOGNISING SUZUKI GROUPS 18 Jun 2017arXiv:1706.05697v1 [math.GR] We present a new algorithm for constructive recognition of the Suzuki groups in their natural representations. The algorithm runs in Las Vegas polynomial time given a discrete logarithm oracle. An implementation is available in the Magma computer algebra system. Introduction In [1] and [2], algorithms for constructive recognition of the Suzuki groups in the natural representation are presented. They depend on a technical conjecture, which is still open, although supported by substantial experimental evidence. Here we present a new algorithm for this problem, which does not depend on any such conjectures, and which is also more efficient. We shall use the notation of [2], but for completeness we state the important points here. The ground finite field is F q where q = 2 2m+1 for some m > 0, and we define t = 2 m+1 so that x t 2 = x 2 for every x ∈ F q . For a, b ∈ F q and λ ∈ F × q , define the following matrices. U (a, b) =     1 0 0 0 a 1 0 0 a t+1 + b a t 1 0 a t+2 + ab + b t b a 1     ,(1)M ′ (λ) =     λ t+1 0 0 0 0 λ 0 0 0 0 λ −1 0 0 0 0 λ −t−1     ,(2) If ω ∈ F q is a primitive element, then Sz(q) = U (1, 0), M ′ (ω), T . This is our standard copy of Sz(q), denoted Σ. This group acts on the Suzuki ovoid, which is O = {(1 : 0 : 0 : 0)} ∪ (a t+2 + ab + b t : b : a : 1) | a, b ∈ F q .(4)Let F = { U (a, b) | a, b ∈ F q } and H = M ′ (λ) | λ ∈ F × q . Then F H = HF is the stabiliser of (1 : 0 : 0 : 0) ∈ O, a maximal subgroup of Sz(q) and F H = U (1, 0), M ′ (ω) ∼ = F q .F q .F × q . The group Sz(q) is partitioned into two sets as Sz(q) = F H ∪ F HT F = HF ∪ HF T F . If G is a conjugate of Sz(q), so that G c = Sz(q) for some c ∈ GL(4, q), we say that the ordered triple of elements α, h, γ ∈ G are rewriting generators for G with respect to c if • α c ∈ F , h c ∈ F H, γ c = T , • α has order 4 and h has odd order not dividing r − 1 for any r such that q is a non-trivial power of r. 1 Note that these conditions imply that G = α, h, γ . The main results are Theorem 8 and 9. The following result is a consequence: Theorem 1. Given a random element oracle for subgroups of GL(4, q) and an oracle for the discrete logarithm problem in F q : • There exists a Las Vegas algorithm that, given a conjugate G = X of Σ, constructs g ∈ GL(4, q) such that X g = Σ and also constructs rewriting generators for G with respect to g as SLPs in X. The algorithm has expected time complexity O log(q) log log(q) field operations. • Let h ∈ GL(4, F ). Then: (1) We can determine whether h ∈ G in time O 1 field operations. (2) Given some preprocessing that is independent of h, and given that h ∈ G, we can construct h as an SLP in X in time O log(q) field operations. The preprocessing, which only has to be done once per constructive recognition of G, has complexity O (log(q)) 2 . Overview The group G = Sz(q) acts doubly transitively on O. In [2,Corollary 4.8], an algorithm for constructing a generating set of a stabiliser of a given point P ∈ O is described. The generators are constructed as SLPs in a given generating set X for Sz(q). This is then used in [2,Theorem 5.2] to obtain generating sets for the stabilisers of the two points P ∞ = (1 : 0 : 0 : 0) and P 0 = (0 : 0 : 0 : 1), which consist of lower and upper triangular matrices, respectively. These stabilisers are used in constructive membership testing to convert any given g ∈ Sz(q) to an SLP in X, given as [2,Algorithm 2]. In the natural representation, the constructive recognition problem reduces to the construction of a conjugating matrix, and [2, Theorem 5.2] is also used. The new algorithm presented here replaces the central task of constructing a point stabiliser with a new method. A difference compared to the old method is that the stabilised point of O is not part of the input, instead a random point stabiliser is constructed. We therefore obtain two random point stabilisers of the input group, but it turns out that we can construct a matrix that conjugates the input group to G and also the two point stabilisers to G P∞ and G P0 . This is done using a simplified version of [2,Lemma 7.4]. After that we perform constructive membership testing with a method similar to [2,Theorem 5.2], but here we present a deterministic algorithm instead of a probabilistic one. Constructing a point stabiliser The algorithm consists of two parts. The second part is an adaptation of [7, Theorem 17] for the natural representation. This is given as Theorem 7. The first part is the algorithm that constructs an element of order 4, to be used as input to the algorithm in Theorem 7. This is presented as Algorithm 1. We use Mat n (F ) to denote the algebra of n by n matrices over the field F . To calculate the pseudo-order of a matrix, we use [9] (see also [10,Section 2.2]). Note that we do not need to compute precise orders, hence large integer factorisation can be avoided. On the other hand, it turns out that one can use integer factorisation to avoid some potential discrete logarithm calculations. We make use of a discrete logarithm oracle DiscreteLog(λ, a), which returns k 0 such that λ k = a, or −1 if no such number exists, that is if a / ∈ λ F × q . We also make use of a random element oracle Random(G) that returns independent random elements of a group G = X as SLPs in X. This is polynomial time by [4]; in practice we use the product replacement algorithm [8], which is also polynomial time by [11]. We construct an element of order 4 by constructing elements of trace 0. In G, the elements of trace 0 have orders 1, 2 or 4. Algorithm 1: Order4Elt(G) Input: Group G = X where X ⊆ GL(4, q) such that G ∼ = Sz(q). Output: Element f ∈ G as SLP in X, such that |f | = 4. 1 g := Random(G) 2 if |g| | q − 1 and g = 1 then 3 u, c, λ := Diagonalise(g) 4 ✄ Now u = g c where u = diag(λ t+1 , λ, λ −1 , λ −t−1 ) ∈ G and c ∈ GL(4, q) 5 h := Random(G) 6 B := h c 7 if u B = u −1 then 8 A := diag(xy, x, x −1 , x −1 y −1 ) ∈ Mat 4 (F q (x, y)) 9 eqns := {Tr(AB) = 0, Tr(AB) t = 0} 10 ✄ eqns determines 2 polynomial equations in x and y (with y used for x t ) 11 if eqns has a root (r, r t ) for (x, y) where r ∈ F × q then 12f := diag(r t+1 , r, r −1 , r −t−1 )B 13 if f = 4 then 14 i := DiscreteLog(λ, r) 15 ✄ This will return −1 if r / ∈ λ 16 if i 0 then 17 ✄ Now λ i = r 18 f := g i h 19 ✄ Now f is an SLP in X andf = f c 20 return f end end end end end 21 return fail Lemma 2. With u and B as in Algorithm 1, there is a diagonal matrix δ ∈ GL(4, q) such that u δ = u and B δ lie in Σ. Proof. Let ∆ be the subgroup of diagonal matrices of GL(4, q). We observe that λ t+1 , λ, λ −1 , λ −t−1 are distinct elements of F × q , since λ = 1 (enforced by g = 1). For example, if λ t+1 = λ −1 , we obtain λ t+2 = 1, and raising to the power of 1 − (t/2) gives us λ = 1. Thus C GL(4,q) (u) = ∆. The normaliser of u in GL(4, q) must permute the eigenspaces of u, and thus any element therein must raise u to one of the powers ±1, ±(t+ 1). We observe that T inverts u, while u ±(t+1) has eigenvalues λ ±(t+1) , λ ±(t+1) 2 . Therefore, if u ±(t+1) is conjugate in GL(4, q) to u or u −1 , then λ (t+1) 2 = λ or λ −1 . But (t + 1) 2 = 2q + 2t+ 1 ≡ 2t+ 3 mod q − 1, and so we get λ 2t+3 = λ or λ −1 . Now 2t+ 2 and 2t+ 4 are invertible powers when applied to F × q , and so λ = 1. This contradiction implies that u is not conjugate in GL(4, q) to u ±(t+1) . Thus N GL(4,q) ( u ) = ∆ T = T ∆. Now it is well-known that all copies of Sz(q) in GL(4, q) are conjugate therein. Notice also that (u, B) = (g, h) c . Thus u, B γ1 Σ for some γ 1 ∈ GL(4, q). Moreover, there is γ 2 ∈ Σ such that (u γ1 ) γ2 is a diagonal matrix diag(µ t+1 , µ, µ −1 , µ −(t+1) ). Naturally, the eigenvalues of this diagonal matrix coincide with the eigenvalues of u, and the argument of the previous paragraph now shows that µ = λ ±1 . Therefore u γ1γ2 = u ±1 . So γ 1 γ 2 normalises u , and so γ 1 γ 2 = δζ, where δ ∈ ∆ and ζ = ζ −1 = T j for some j ∈ {0, 1}. Thus u, B δ = ( u, B γ1 ) γ2ζ Σ γ2ζ = Σ, since γ 2 , ζ ∈ Σ, and this establishes our claim. The next theorem asserts that the equations we have to solve in Algorithm 1 have few solutions (with one exception) and also provides a way of reducing the problem of their solution to finding roots of a univariate polynomial of degree at most 4 over F q . To summarise the next theorem, let the diagonal entries of B be a, b, c, d. Line 11 is then done as follows: • Line 7 excludes the case a = b = c = d = 0. Proof. In the general case, we establish that y is the root of a non-zero univariate polynomial of degree at most 4. Exceptional cases are dealt with explicitly. The analysis of one of the cases becomes simpler if B is in our standard copy Σ. By Lemma 2, there is a diagonal matrix δ such that B δ is in Σ. Note that A δ = A, and so Tr(AB δ ) = Tr(A δ B δ ) = Tr((AB) δ ) = Tr(AB). In fact, B and B δ have the same diagonal entries. The upshot is that Tr(AB ′ ) = Tr(AB ′ ) t = 0 gives the same equations for both B ′ = B and B ′ = B δ . From now on, we presume that B (the old value of B δ ) has been standardised to lie in Σ. We have Tr(AB) = axy + bx + cx −1 + dx −1 y −1 , where a = B 11 , b = B 22 , c = B 33 and d = B 44 . Multiplying through by xy and also considering Tr(AB) t gives us: ax 2 y 2 + bx 2 y + cy + d = 0,(6)a t x 4 y 2 + b t x 2 y 2 + c t x 2 + d t = 0.(7) We now try to solve these equations. Note that x and y determine each other via y = x t and x = y t/2 . We now let p 1 = (aa t y 3 + a t by 2 )x 2 + ab t y 3 + bb t y 2 + a t cy 2 + ac t y + a t dy + bc t and p 2 = a 2 y 3 + b 2 y. Evaluating p 1 × (6) + p 2 × (7) gives us ab t cy 4 + (a 2 d t + ab t d + a t c 2 + bb t c)y 3 + (acc t + bb t d)y 2 +(ac t d + a t d 2 + b 2 d t + bcc t )y + bc t d = 0.(8) (The values of p 1 and p 2 may be obtained using polynomial division to divide Tr(AB) t x t y t by Tr(AB)xy, treating x 2 as the variable and everything else as a constant.) Clearly, (8) reduces to the trivial equation 0 = 0 whenever a = b = 0 or c = d = 0. We now show that it is non-trivial in all other cases, in which case y is a root of a non-trivial polynomial equation of degree at most 4 (over F q ), and thus can have at most four values. (Note that a solution to (8) does not necessarily yield a solution to the original equations (6) and (7).) By inspecting the coefficients of y 4 and 1 in (8), we see that this equation can only be trivial if ab t c = bc t d = 0, putting us in one of the three cases a = d = 0, b = 0 or c = 0, which we now analyse separately below. M-I. If a = d = 0, (8) reduces to bb t cy 3 + bcc t y = 0 (so bb t cy 2 + bcc t = 0). If b = 0 or c = 0, then we are in one of the excluded cases a = b = 0 or c = d = 0, and if b, c = 0 we get b t y 2 + c t = 0, with unique solution y = (c/b) t/2 . M-II. If b = 0, (8) reduces to y(a 2 d t y 2 +a t c 2 y 2 +acc t y +ac t d+a t d 2 ) = 0. Because of the excluded case a = b = 0, we may assume that a = 0. So for this equation to be identically zero, c must be 0 (coefficient of y 2 ), giving us y(a 2 d t y 2 + a t d 2 ) = 0, and thus forcing a 2 d t = a t d 2 = 0. Since a = 0 this forces d 2 = 0, whence d = 0. But c = d = 0 is also an excluded case. M-III. If c = 0, (8) reduces to y(a 2 d t y 2 + ab t dy 2 + bb t dy + a t d 2 + b 2 d t ) = 0. We have d = 0, since we are excluding c = d = 0, so for a degenerate equation we need b = 0 (from the coefficient of y 2 ). This also implies that we need a 2 d t = a t d 2 = 0. So a = 0 (as d = 0), putting us in another excluded case, namely a = b = 0. We now deal with the cases when (8) is identically zero, namely a = b = 0 and c = d = 0. With the exception of the case a = b = c = d = 0, the original equations (6) and (7) are not identically zero. S (6), as x 2 y = 0. S-III. Exactly three of a, b, c, d are 0. Now (6) reduces to one of ax 2 y 2 = 0, bx 2 y = 0, cy = 0 or d = 0, which have no solutions, since we want x, y = 0. -I. a = b = 0, c, d = 0. In this case y = d/c, from (6). S-II. c = d = 0, a, b = 0. In this case y = b/a, from S-IV. a = b = c = d = 0. This does occur. The subspace generated by the first row of B is a point of O, so it is P ∞ or (e 2 e t + ef + f t : f : e : 1) for some e, f ∈ F q . The requirement that B 11 = 0 obviously rules out the P ∞ case, and imposes the requirement that e 2 e t +ef +f t = 0 in the other case. This actually forces e = f = 0. So B = M ′ (k)U (e 1 , f 1 )T for some k, e 1 , f 1 ∈ F q (with k = 0). This element has B 11 = B 22 = 0, B 33 = e t 1 /k and B 44 = (e 2 1 e t 1 + e 1 f 1 + f t 1 )/(kk t ), which imposes the relations e 1 = f 1 = 0. So B = M ′ (k)T is an element of order 2 in the copy of D 2(q−1) that normalises the C q−1 containing u. (We solve e 2 e t + ef + f t = 0 as follows. Taking t-th powers gives us e 2 e 2t + e t f t + f 2 = 0; take away e t times the original equation to leave us with f 2 + ee t f = 0. This gives us f = 0 or f = ee t , and substituting these into the original equation gives us e 2 e t = 0 (so e = 0) in the case f = 0 or e 2 e t +eee t +(ee t ) t = 0 (so e t e 2 = 0) in the case when f = ee t . In either case, the only solution is e = f = 0. Note that (z t+1 ) t−1 = z t 2 −1 = z 2−1 = z and z t+2 = (z t+1 ) t for all z ∈ F q , so that t + 1 and t + 2 are both invertible powers when applied to F q .) Using the algorithms of [3], we can usually construct a copy of Sz(q) as a subgroup of GL(4, q). However, the above algorithm, lemma, theorem and proofs hold with obvious minor modifications even when G is a group isomorphic to Sz(q) embedded in GL(4, F ) for any field F (including infinite ones) containing F q . Furthermore, the diagonal entries of the semi-standardised matrix B must lie in F q , even though the off-diagonal entries need not do so. Proof. In total in G, there are q(q − 1)(q 2 + 1) elements of order 4, and q 2 (q 2 + 1) cosets of g . By Theorem 3 there are at most four elements of order 4 in each coset. Hence there are at least q(q − 1)(q 2 + 1)/4 cosets containing an element of order 4, and the proportion of such cosets is 1/4 − 1/(4q) 7/32, since q 8. Theorem 5. Algorithm 1 is a Las Vegas algorithm, with success probability at least 7/(384 log log(q)) − O 1/q . When the algorithm succeeds it returns an element as an SLP in X. Proof. It is clear that g and h are constructed as SLPs, hence f will have an SLP as well. There are (q − 2)q 2 (q 2 + 1)/2 non-trivial elements of order dividing q − 1, hence the test at line 2 succeeds with probability (q/2 − 1)/(q − 1), which is at least 3/7. From now on, we can (and shall) assume that g is uniformly distributed among the non-trivial elements of G having order dividing q − 1. Since C G (G) ∼ = C q−1 , any two distinct subgroups of G isomorphic to C q−1 have trivial intersection. So we can, for the purpose of this proof, choose g by randomly picking a copy of C q−1 , and let g be a random non-trivial element therein, in both cases using a uniform distribution. We define C := C G (g) ∼ = C q−1 , and D := N G ( g ) = N G (C) ∼ = D 2(q−1) . The (right-)cosets of C in G partition G into sets all of the same size. So we can choose h by first choosing a random coset S (of C), and letting h be a random element of S, again with all distributions uniform. There are just q − 1 elements that invert g, namely those elements that lie in the coset D \ C of C, and all of them are involutions, so the test at line 7 succeeds with probability 1 − O 1/q . (Note that the pairs (u, B) and (g, h) are conjugate in GL(4, q).) So now we can assume that S is uniformly distributed among the cosets of C other than D \ C. We use Theorem 3 to reduce the equations on line 11 to a univariate polynomial, and we can then compute its roots using [13, Algorithm 14.10]. Using 3 log 2 (q) applications of [13,Algorithm 14.8] in the latter, the probability of success is 1 − O 1/q . The solutions of these equations are in one-to-one correspondence with the elements h 1 , . . . , h s of S having order lying in {1, 2, 4}. By Corollary 4, these equations have a solution with probability at least 7/32, and Algorithm 1 produces an element f = cf c −1 which is one of h 1 , . . . , h s . Unfortunately, given g, the elements f produced are not uniformly distributed among the elements of orders 1, 2, 4 not lying in D \ C. For if h 1 , . . . , h s do occur with the same probability, then f has a probability of [s(q 4 − q + 1)] −1 , and s can vary, depending on the coset S. But G has just (q 2 + 1)(q − 1) involutions, and 1 identity, while Corollary 4 implies that at least 7(q 4 + q 2 − 1)/32 cosets S( = D \ C) will have an element of order 4. Thus h 1 , . . . , h s will all have order 4 with probability 1 − O 1/q , and so the test at line 13 succeeds with probability 1 − O 1/q . If |g| = q − 1 then the test at line 16 always succeeds, and by [2, Lemma 2.5] this happens with probability at least 1/(12 log log(q)). (Note that in general, if we choose a solution at line 11 uniformly at random, then f will be a random element of h 1 , . . . , h s , chosen uniformly, depending only on the coset S, and independent of h. Then the discrete log test at line 16 succeeds if and only if h ∈ g f , which happens with probability | g f | / |S| = |g| /(q − 1).) We see that the success probabilities of lines 2, 7, 11, 13 and 16 combine to at least 7/(384 log log(q)) − O 1/q . We remark that successful runs of Algorithm 1 produce elements of order 4 in G uniformly at random, provided that the solutions found at line 11 are chosen uniformly at random; the fact that g is random smooths out the outputs obtained when g is fixed. By [9], line 2 has expected time complexity O log(q) field operations. At lines 3 and 11 we find the roots of a polynomial over F q of degree at most 4. The expected time complexity is O log(q) field operations, by [13,Corollary 14.11]. In practice, when implementing Algorithm 1, in order to avoid unnecessary work, and excessive calls to Random(G), on rerunning the algorithm we only execute those steps that are necessary. Thus if Algorithm 1 fails because g has the wrong order or DiscreteLog does not succeed, we should rerun the algorithm with a new value of g (we can retain h if we choose). In other cases of failure, we should retain g and use a new value of h. If the algorithm reaches the DiscreteLog stage, it will succeed there with probability |g| /(q − 1). Taking account of this small probability of success is awkward, and suppressed analysis shows that the success probability for this stage does not exceed 1.03φ(q − 1)/(q − 2). This means that the expected number of executions of Dis-creteLog before a successful run of Algorithm 1 is at least 0.97(q − 2)/φ(q − 1). Formally, this quantity has magnitude O log log(q) . In practice, while it is the case that n/φ(n) > log log(n) for infinitely many values of n, we do not know the behaviour when n is q − 1 for q = 2 2m+1 . We can also modify Algorithm 1 and introduce exactly one call to integer factorisation (to factor q − 1), hence only having to call DiscreteLog exactly once. In this case, line 2 now tests whether g has order exactly q − 1. This takes time O log q(log t + 1) field operations by [9], where t is the number of distinct prime factors of q − 1, and so time O log(q) log log(q) in the worst case. We now run lines 1 and 2 an expected 2(q − 1)/φ(q − 1) times until we have produced an element g of order q − 1, and do not run this part of the algorithm again. We then execute lines 3 and 4, and repeat the rest of the algorithm (lines 5-21) until success occurs. The tests on lines 7, 11 and 13 succeed with the same probabilities as before, but now, when executed, the discrete log test on line 16 always succeeds (and in this case the entire algorithm succeeds). In this variant the success probability of lines 5-21 is at least 7/32 − O 1/q . Hence, the expected time complexity until this variant of the algorithm succeeds (ignoring the single call to DiscreteLog) is O log(q)(log log(q)) 2 field operations, which is dominated by finding g. Theorem 7. Given a random element oracle for subgroups of GL(4, q), there exists a Las Vegas algorithm that, given G = X GL(4, q) such that G ∼ = Sz(q), and an element f ∈ G of order 4, constructs Y = {f, h} ⊆ GL(4, q) such that O 2 (G P ) < Y G P , where P is the unique point of the G-ovoid fixed by f . The algorithm has expected time complexity O log(q) log log(q) field operations. (If G = Σ g , where Σ is our standard Sz(q), then the G-ovoid is O g .) Proof. The algorithm proceeds as follows: (1) Use the algorithm of [6] to construct g ∈ C G (f 2 ) ∼ = F q .F q . Thus we pick a random c ∈ G such that [f 2 , c] has odd order 2k + 1 for some k ∈ N, and we calculate g := c[f 2 , c] k (see note (a) below about this). If [f 2 , c] has even order then start again. If |g| = 4 then let j := g 2 , otherwise j := g. Repeat this step until j / ∈ {1, f 2 }. This requires expected time O log(q) field operations, and the element j produced is uniformly distributed among the set I of q − 2 involutions of G P \ {f 2 }. See note (b) below for further details about this step. (2) Construct random c ∈ G such that j c / ∈ C G (f 2 ) (see note (c) below). An equivalent requirement is that c / ∈ G P , and so this step requires O 1 field operations. The product of two involutions has odd order if they lie in distinct point stabilisers, so f 2 j c = 2k + 1 for some k ∈ N. Some notes on aspects of the above algorithm are given below. (a) In order to prove that g ∈ G ∼ = Sz(q) has odd order, it suffices to show that g = 1 or g 4 = 1. And if g has odd order 2k + 1, but we only know an odd multiple (2k + 1)(2l + 1) = 2(2kl + k + l) + 1 of this order, then we can still compute g k , since g 2kl+k+l = g l(2k+1)+k = g k . In our case, (q 2 + 1)(q − 1) is always a multiple of the order of any odd order element. (b) The probability of success at each iteration is 1 − O 1/q . Note that [f 2 , c] has odd order whenever c / ∈ G P , and also when c ∈ O 2 (G P ), and that the possible g obtained this way are uniformly distributed throughout all of C G (f 2 ) = O 2 (G P ). Among the q 2 possibilities for g, precisely 2 + q of them produce a forbidden value of j. The other q 2 − q − 2 = (q − 2)(q + 1) of them produce a j uniformly distributed among the q − 2 elements of I. (If [f 2 , c] has even order, necessarily 2, then we can set j := g = [f 2 , c] (not quite in line with [6]). The element j depends solely on the coset O 2 (G P )c, always lies in I, and has uniform distribution therein.) (c) In Step 2, letting c run through X until success occurs, as it must, may be the best way to execute this step, especially if X is small, which will typically be the case. In theory, f is independent of X, and so the probability of success for each choice of c ∈ X is 1 − O 1/q . Note that restricting c to lie in X does not change the success probability of subsequent steps, notably Step 4. This is because all elements conjugating j to f 2 lie in the coset O 2 (G P )h, they all have the same order, so the order of h depends only on the pair (f 2 , j), which is fully constructed by the end of Step 1. Hence it is not wise to use c ∈ X when executing Step 1, as this determines j, and Step 4 could fail for all choices of c ∈ X. (d) If we have used the integer factorisation oracle earlier (to factor q − 1), then it seems neater to generate the whole of G P here, and in that case, at Step 4, we test if h has order q − 1. This test takes time O log(q) log log(q) field operations, and succeeds with probability φ(q − 1)/(q − 2). This variant of the algorithm has expected time complexity O log(q)(log log(q)) 2 field operations. To finish off the proof, we must calculate the probability that Step 4 succeeds, and show that correct output is returned. But the possible elements h correspond to non-trivial cosets of O 2 (G P ) in G P (any element of O 2 (G P )h conjugates j to f 2 ), Moreover, h and O 2 (G P )h have the same order, and the possible elements h are uniformly distributed, and so the order distribution of h is the same as that for the non-trivial elements of a C q−1 . Each order test carried out in Step 4 excludes at most 3 √ q − 2 of these, and there are O log(q) such tests. Thus Step 4 succeeds with probability 1 − O 1/ √ q . Now h acts on O 2 (G P )/Φ(O 2 (G P )) ∼ = (F q , +) in a manner corresponding to the F 2 -action of λ (or λ t ) for some λ ∈ F q \ {0, 1}, and the F 2 h -module O 2 (G P )/Φ(O 2 (G P )) is irreducible if and only if λ does not belong to a proper subfield. Our order restrictions on h rule out that case, giving an irreducible action. Therefore O 2 (G P ) f, h , since f ∈ O 2 (G P ) \ Φ(O 2 (G p )). Constructive recognition The new constructive recognition algorithm for the natural representation is given as the following two results. The first of these describes how to conjugate an arbitrary copy X of Sz(q) to the standard copy, and the second uses the first to provide deterministic constructive membership testing of an element of GL(4, q) inside X . Theorem 8. Given a random element oracle for subgroups of GL(4, q) and an oracle for the discrete logarithm problem in F q , there exists a Las Vegas algorithm that, for each conjugate X of Σ, constructs g ∈ GL(4, q) such that X g = Σ and rewriting generators α 1 , h 1 , γ of X with respect to g as SLPs in X. The algorithm has expected time complexity O log(q) log log(q) field operations. The discrete logarithm oracle is only needed in the initial phase, in order to obtain an element of order 4, where it is used, at worst, O log log(q) times. Proof. Let G = X . The algorithm proceeds as follows. (1) Use Algorithm 1 to construct an element α 1 of order 4. By Theorem 5, the expected number of invocations of the algorithm is O log log(q) , so by Theorem 6, this step requires expected time O log(q) log log(q) field operations. (2) Use Theorem 7 to construct a set of matrices Y 1 = {α 1 , h 1 } such that O 2 (G P ) < Y 1 G P for some P in our G-ovoid. Use α 1 for this. This requires expected time O log(q) log log(q) field operations. Theorem 7, Step 4 forces h 1 to have the required order. (3) There are just three non-trivial proper submodules of Y 1 (and even of α 1 ), namely V P 1 < V P 2 < V P 3 , where dim V P i = i for i = 1, 2, 3. For each i, we can obtain V P i as the nullspace of (α 1 − 1) i . This requires expected time O 1 field operations. (4) Choose random β ∈ G that does not fix P (the subspace V P 1 constructed in Step 3), and let γ = (α 2 1 ) β and Q = P γ. Then Q = P and hence Y 1 γ = Y 2 G Q , where Y 2 = {α 2 , h 2 } = {α γ 1 , h γ 1 }. Also Y γ 2 = Y 1 , since γ is an involution. This requires expected time O 1 field operations. (It is probably sensible to take β ∈ X here.) Note that G is generated by any two of its distinct Sylow 2-subgroups. Thus G = O 2 (G p ), O 2 (G Q ) = α 1 , h 1 , α 2 , h 2 = α 1 , h 1 , γ . (5) Use the method of Step 3 to construct the non-trivial proper submodules V Q 1 < V Q 2 < V Q 3 of Y 2 (so V Q i is the nullspace of (α 2 − 1) i ). (6) Define U 1 = V P 1 , U 2 = V P 2 ∩V Q 3 , U 3 = V P 3 ∩V Q 2 , U 4 = V Q 1 . For i = 1, 2, 3, 4, we have dim U i = 1, and also γ swaps U 1 with U 4 and U 2 with U 3 . We choose nonzero u 1 ∈ U 1 and u 2 ∈ U 2 , and define u 3 = u 2 γ ∈ U 3 and u 4 = u 1 γ ∈ U 4 . Let k be the inverse of the matrix whose i-th row is u i . Then by the proof of [2,Lemma 7.4], there is a diagonal matrix d ∈ GL(4, q) such that (G k ) d = Σ, our standard copy of Sz(q). This requires expected time O 1 field operations. (We have corrected the definitions of U 2 and U 3 here.) (7) Let J = antidiag (1, 1, 1, 1 ), and d = diag(d 1 , d 2 , d 3 , d 4 ), where (G k ) d = Σ. Then G k preserves the form dJd T = antidiag( d 1 d 4 , d 2 d 3 , d 2 d 3 , d 1 d 4 ) , a form that is unique up to scalars, since G acts absolutely irreducibly on its natural module. Since G k contains γ k = antidiag(1, 1, 1, 1), Σ con- tains d −1 (γ k )d = antidiag(d −1 1 d 4 , d −1 2 d 3 , d −1 3 d 2 , d −1 4 d 1 ) , and so this must be M ′ (κ)T for some κ ∈ F × q , and conjugating this by the diagonal matrix M ′ ( √ κ) ∈ Σ gives us T (= J = γ k ). Therefore, we may assume that d centralises γ k , and this forces d = diag(d 1 , d 2 , d 2 , d 1 ). But conjugating by scalars has no effect, and so we can take d 1 = 1. Therefore d = diag(1, d 2 , d 2 , 1) for some d 2 , and the form preserved by G k is K := antidiag(1, d 2 2 , d 2 2 , 1). The equations hKh T = K for h ∈ α k 1 , h k 1 give us many linear equations for d 2 2 , at least some of which are non-trivial. (Note that γ k automatically preserves the form K.) This requires O log q field operations (to square-root d 2 2 ). Note that using γ to partially standardise G k simplifies this step compared to [2,Lemma 7.3]. 1), hence Σ P ∞ Y 1 g and Σ P0 Y 2 g . Most of the output criteria and complexity issues have been dealt with as we went along. To finish off, we note that γ g = (γ k ) d = T d = T . Theorem 9. Let X, α 1 , h 1 , γ, g be as in Theorem 8. Thus G = X = α 1 , h 1 , γ ∼ = Sz(q) is a subgroup of GL(4, q) and G g = Σ. Let h ∈ GL(4, F ). Then: (1) We can determine whether h ∈ G in O 1 field operations. (2) Given some preprocessing that is independent of h, and given that h ∈ G, we can construct h as an SLP in α 1 , h 1 , γ in time O log(q) field operations. The SLP has length O log(q) . Thus we also get h as an SLP in X. The preprocessing, which only has to be done once for any (X, α 1 , h 1 , γ, g), has complexity at most O (log(q)) 2 , the true value being dependent on the complexity of matrix inversion. (We are counting log 2 (q) bit operations as being 1 field operation, as one field operation over F q must take at least log 2 (q) bit operations.) Proof. We note that h ∈ GL(4, q) belongs to G if and only if h g ∈ G g = Σ, which we solve as follows. We make use of the partitioning from (5). (1) Consider the first row of h g . If it is µ(1, 0, 0, 0) for some µ = 0 let k 1 = Id 4 . If it is c(a t+2 +ab+b t , b, a, 1) for some a, b, µ with µ = 0 let k 1 = (T U (a, b)) −1 . In any other case, h g does not preserve O, so return fail. In successful cases, let k 0 = (h g )k −1 1 . (2) Now the first row of k 0 has the form µ(1, 0, 0, 0) for some µ = 0. So in the case when h g ∈ Σ, we have k 0 ∈ Σ, and so k 0 = M ′ (λ)U (c, d) for some c, d, λ with λ = 0. Look at the (2, 2)-entry of k 0 . This should be λ, so if it is 0 then return fail. In the other cases, define k 3 = M ′ (λ), and k 2 = k −1 3 k 0 . If k 2 is U (c, d) for some c, d then we succeed, otherwise we return fail. All the above clearly requires just O 1 field operations. In the successful cases, we wish to write h as an SLP in X. First, we show how to write each of the elements T , M ′ (λ) and U (a, b) in terms of f = α g 1 , e = h g 1 and z = γ g . The easy one is that T = z. We also have M ′ (λ) = zU (0, λ 1+t/2 )zU (λ −t/2 , λ −1−t/2 )zU (λ t/2 , 0), which just defers the problem. Last, but not least, we consider U (a, b). We have f = U (a 1 , b 1 ) and e = M ′ (µ)U (a 2 , b 2 ) for some a 1 , b 1 , a 2 , b 2 , µ with a 1 , µ = 0. Now O 2 ( e, f ) = O 2 (G P∞ ) has generating set L := { f e i , (f 2 ) e i | 0 i 2m }. In terms of matrices f e i = U (µ it a 1 , * ) and (f 2 ) e i = U (0, µ (t+1)it a t+1 1 ) = U (0, µ (t+2)i a t+1 1 ), where we have not calculated the starred entry. By construction of h 1 , µ does not lie in a subfield of F q , and hence the elements µ it a 1 and µ (t+2)i a t+1 1 form vector space bases for F q over F 2 . Now solve a linear system over F 2 and calculate n 0 , . . . , n 2m ∈ {0, 1} such that a = a 1 (n 0 + n 1 µ t + · · · + n 2m µ 2mt ). We set j 1 = 2m i=0 (f e i ) ni = f n0 e −1 f n1 e −1 . . . e −1 f n2m e 2m , and note that its matrix has form U (a, * ). We have U (a, b) = j 1 U (0, β) for some β, and solve another linear system to obtain β = a t+1 1 (p 0 + p 1 µ t+2 + · · ·+ p 2m µ 2m(t+2) ) for some p 0 , . . . , p 2m ∈ {0, 1}. This gives us j 2 = 2m i=0 ((f 2 ) ei ) pi = f 2p0 e −1 f 2p1 e −1 . . . e −1 f 2p2m e 2m = U (0, β), and so U (a, b) = j 1 j 2 writes U (a, b) as a word or SLP of length O log(q) in {e, f }. Note that the matrices, and their inverses, used in the linear system solving do not depend on h, and hence we precompute them. (This leads to Θ(log(q)) space complexity of the algorithm, which may be unavoidable in any case.) We then obtain n i and p i by multiplication with these inverse matrices, which requires O (log(q)) 2 bit operations, and thus O log(q) field (F q ) operations. Therefore, writing U (a, b) as an SLP in e and f also requires O log(q) field operations. The precomputation requires us to invert two degree log 2 (q) matrices over F 2 , for which the classical algorithm uses O (log(q)) 3 bit operations and thus O (log(q)) 2 field operations. (It is known that asymptotically faster matrix inversion algorithms exist. It is not known whether it is possible for Gaussian elimination to be asymptotically faster than matrix inversion.) Having shown how to write each of the elements T , M ′ (λ) and U (a, b) as SLPs in α g 1 , h g 1 , γ g , we can now easily obtain h g as an SLP in α g 1 , h g 1 , γ g since we have already noted that h g = M ′ (λ)U (c, d) or M ′ (λ)U (c, d)T U (a, b) for some a, b, c, d, λ. The same SLP gives h in terms of α 1 , h 1 , γ, and since these three elements have known SLPs in terms of X, so now does h. As we have seen, writing h as an SLP in {α 1 , h 1 , γ} requires at most 5 invocations of the above method that writes an element of F as an SLP in {e, f }. Therefore this requires time complexity O log(q) field operations, and produces an SLP for h having length O log(q) in {α 1 , h 1 , γ}. Elements of order 4 Following Corollary 4, we conjecture that the actual proportion of cosets of g possessing an element of order 4 is 5q 3 − 3q 2 + 14q − 16 8q(q 2 + 1) , a value we have checked for q = 2, 8, 32, 128, 512. We associate the vector (v 1 , v 2 , v 4 ) to a coset of g , where v i is the number of elements of order i in that coset. We have proved that there are just nine possibilities for this vector. (Theorem 3 restricts the possible vectors, but some other arguments are needed too.) The possible vectors, and the number of cosets (of g ) in Sz(q) that we conjecture have this vector, are tabulated below. vector #cosets vector #cosets (0, 0, 0) 1 8 (q − 1)(3q 3 + 2q 2 − 8q + 16) (0, 1, 0) 1 2 (q − 1)q(q + 2) (0, 0, 1) 1 6 (q − 1)q(2q 2 + q + 20) (0, 1, 2) 1 2 (q − 1)q(q − 2) (0, 0, 2) 1 4 (q − 1)q 2 (q − 2) (0, q − 1, 0) 1 (0, 0, 3) 1 2 (q − 1)q(q − 2) (1, 0, 0) 1 (0, 0, 4) 1 24 (q − 1)q(q − 2)(q − 8) Again, we have checked these values for q = 2, 8, 32, 128, 512. For the case q = 2 and vector (0, 1, 0) one should sum the values listed for the cases (0, 1, 0) and (0, q−1, 0). Implementation and performance An implementation of the algorithms described here is available in Magma [5], as part of the CompositionTree package [3]. The implementation uses the existing Magma implementations of the algorithms described in [8], [9] and [13,Corollary 14.10]. A benchmark of the algorithm in Theorem 8, for field sizes q = 2 2m+1 , with m = 1, . . . , 100, is given in Figure 1. For each field size, 100 random conjugates of Sz(q) were recognised, and the average running time for each call, as well as the average time spent in discrete logarithm calculations, is displayed. As expected, the running time is completely dominated by the time to compute discrete logarithms, and the two plots in Figure 1 are almost indistinguishable. The timings jump up and down due to the cost of discrete logarithms and due to the way that Magma handles finite field computations: it uses Zech logarithms for finite fields up to a certain size, and for larger fields it tries to find a subfield smaller than this size. Hence field arithmetic speed depends on the prime divisors of 2m + 1. We have also benchmarked the algorithm in Theorem 9 in a similar way. For each field size, SLPs of 100 random elements were calculated and the average running time for each call is displayed in Figure 2. The time to precomputate the matrices used in the linear system solving is not included in the running time. In all benchmarks, we used a generating set of size 2. We can always switch to a generating set of this size by choosing random elements of the input group. The probability that 2 uniformly random elements generate the group is high, and we can detect this using [2, Theorem 6.2]. The benchmark was carried out using Magma V2.22-7, Intel64 CUDA 5.5 flavour, on a PC with an Intel Core i5-4690 CPU running at 3.5 GHz. We used the software package R [12] to produce the figure. • If exactly three of a, b, c, d are 0, then the equations at line 9 have no solutions of the required type. • If a = b = 0 and c, d = 0 then y = d/c. • If c = d = 0 and a, b = 0 then y = b/a. • In all other cases, equation (8) is a non-degenerate univariate polynomial equation in y, and we consider all non-zero roots of this. The roots can be found using [13, Algorithm 14.10]. For each value of y obtained above, we set x = y t/2 and check if equation(6)holds. This provides the values r at line 11. (There is no need to check equation(7)since it will automatically hold if (6) does.) Theorem 3. The equations at line 11 in Algorithm 1 determine at most four solutions for x, except when B is an involution that inverts u. Corollary 4 . 4Let G ∼ = Sz(q), where q = 2 2m+1 for some m > 0, and let g ∈ G have order q − 1. At least 7/32 of the cosets of g have at least one element of order 4. Theorem 6 . 6Given an oracle for the discrete logarithm problem in F q and a random element oracle, Algorithm 1 has expected time complexity O log(q) field operations. Proof. All matrix arithmetic can be done in O 1 field operations, except exponentiation which can be done in O log(q) field operations, for example by [10, Lemma 10.1]. Computation of characteristic polynomial and eigenspaces also requires O 1 field operations. ( 3 ) 3Let h = c(f 2 j c ) k (see note (a) below). This requires O log(q) field operations. Then j h = f 2 , so h must fix the point fixed by f 2 and j, and hence f, h lies in a point stabiliser.(4) Now |h| = 2 or 4 or |h| | q − 1, since f, h lies in a point stabiliser, and h / ∈ C G (f 2 ) now forces |h| = 1, 2, 4. For each prime divisor p of 2m + 1, set a p = 2 (2m+1)/p − 1, and verify that h ap = 1. Each test takes time O log(q) field operations, and we have to do at most O log log(q) of them, for an overall time complexity of O log(q) log log(q) . (We have no divide-and-conquer strategy of [9] to speed things up.) Otherwise (if h ap = 1 for some p) return to Step 1. (5) Set Y = {f, h}, and return Y . Note that Y is the whole of the point stabiliser G P if and only if |h| = q − 1. ( 8 ) 8Let g = kd, where k is as constructed in Step 6 and d is as constructed in Step 7. Then G g = Σ, the standard copy of Sz(q). It is clear from the proof of [2, Lemmas 7.3 & 7.4] that P g = P ∞ = (1 : 0 : 0 : 0) and Qg = P 0 = (0 : 0 : 0 : ( 3 ) 3In the cases of success, we have also written h g in the form k 3 k 2 k 1 = M ′ (λ)U (c, d).1 or M ′ (λ)U (c, d)T U (a, b) for some a, b, c, d, λ with λ = 0. Figure 1 . 1Benchmark of recognition Algorithmic problems in twisted groups of Lie type. Henrik Bäärnhielm, Queen Mary, University of LondonPh.D. thesisHenrik Bäärnhielm, Algorithmic problems in twisted groups of Lie type, Ph.D. thesis, Queen Mary, University of London, 2007. Recognising the Suzuki groups in their natural representations. Henrik Bäärnhielm, J. Algebra. 3001Henrik Bäärnhielm, Recognising the Suzuki groups in their natural representations, J. Algebra 300 (2006), no. 1, 171-198. A practical model for computation with matrix groups. Henrik Bäärnhielm, Derek Holt, C R Leedham-Green, E A O&apos;brien, J. Symbolic Comput. 68Henrik Bäärnhielm, Derek Holt, C.R. Leedham-Green, and E.A. O'Brien, A practical model for computation with matrix groups, J. Symbolic Comput. 68 (2015), 27-60. Local expansion of vertex-transitive graphs and random generation in finite groups. László Babai, STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of computing. New York, NY, USAACM PressLászló Babai, Local expansion of vertex-transitive graphs and random generation in finite groups, STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of computing (New York, NY, USA), ACM Press, 1991, pp. 164-174. The Magma algebra system. I. The user language. Wieb Bosma, John Cannon, Catherine Playoust, Computational algebra and number theory. London24MR MR1484478Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265, Computational algebra and number theory (London, 1993). MR MR1484478 An improved method for generating the centralizer of an involution. John N Bray, Arch. Math. (Basel). 744John N. Bray, An improved method for generating the centralizer of an involution, Arch. Math. (Basel) 74 (2000), no. 4, 241-245. Standard generators for the Suzuki groups. John N Bray, Henrik Bäärnhielm, preprintJohn N. Bray and Henrik Bäärnhielm, Standard generators for the Suzuki groups, preprint (2015). Generating random elements of a finite group. Frank Celler, Charles R Leedham-Green, Scott H Murray, Alice C Niemeyer, E A O&apos;brien, Comm. Algebra. 2313Frank Celler, Charles R. Leedham-Green, Scott H. Murray, Alice C. Niemeyer, and E.A. O'Brien, Generating random elements of a finite group, Comm. Algebra 23 (1995), no. 13, 4931-4948. Calculating the order of an invertible matrix, Groups and computation. Frank Celler, C R Leedham-Green, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 28Amer. Math. SocIIFrank Celler and C.R. Leedham-Green, Calculating the order of an invertible matrix, Groups and computation, II (New Brunswick, NJ, 1995), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 28, Amer. Math. Soc., Providence, RI, 1997, pp. 55-60. Constructive recognition of classical groups in odd characteristic. C R Leedham-Green, E A O&apos;brien, J. Algebra. 3223C.R. Leedham-Green and E.A. O'Brien, Constructive recognition of classical groups in odd characteristic, J. Algebra 322 (2009), no. 3, 833-881. Igor Pak, The product replacement algorithm is polynomial, FOCS '00: Proceedings of the 41st Annual Symposium on Foundations of Computer Science. Washington, DC, USAIEEE Computer SocietyIgor Pak, The product replacement algorithm is polynomial, FOCS '00: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (Washington, DC, USA), IEEE Computer Society, 2000, pp. 476-485. R: A language and environment for statistical computing, R Foundation for Statistical Computing. Vienna, AustriaR Development Core TeamR Development Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2005, 3-900051-07-0. Joachim Von Zur Gathen, Jürgen Gerhard, Modern computer algebra. CambridgeCambridge University Presssecond ed.Joachim von zur Gathen and Jürgen Gerhard, Modern computer algebra, second ed., Cam- bridge University Press, Cambridge, 2003. E-mail address: j.n.bray@qmul.ac.uk

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{'abstract': 'We present a new algorithm for constructive recognition of the Suzuki groups in their natural representations. The algorithm runs in Las Vegas polynomial time given a discrete logarithm oracle. An implementation is available in the Magma computer algebra system.', 'arxivid': '1706.05697', 'author': ['John N ', 'Henrik Bäärnhielm '], 'authoraffiliation': [], 'corpusid': 38488527, 'doi': '10.1016/j.jalgebra.2017.05.040', 'github_urls': [], 'n_tokens_mistral': 15282, 'n_tokens_neox': 13443, 'n_words': 8971, 'pdfsha': 'bc0e2b80810215147dcf8066a8db9d3d71d5933a', 'pdfurls': ['https://arxiv.org/pdf/1706.05697v1.pdf'], 'title': ['A NEW METHOD FOR RECOGNISING SUZUKI GROUPS', 'A NEW METHOD FOR RECOGNISING SUZUKI GROUPS'], 'venue': []}

arxiv

Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts Eugenio Aulisa Jonathon Loftin Department of Mathematics and Statistics Department of Mathematics and Computer Science Texas Tech University Southern79410LubbockTXUSA Arkansas University Magnolia, AR71753USA Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts Subdomain IntegrationEmbedded Interface IntegrationEquivalent PolynomialPolylogarithm 2020 MSC: 65D3211G5565M60 The implementation of discontinuous functions occurs in many of today's state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when integrating functions whose discontinuity falls in the element's interior are for low order degree polynomials, not easily extended to higher order degree polynomials, and cover a restricted set of geometries. Many approaches to this issue have been developed in recent years. Among them one of the most elegant and versatile is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, allowing integration to occur over the entire domain rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its problems. The exact subdomain integration requires a machinery that quickly grows in complexity when increasing the polynomial degree and the geometry dimension, restricting its applicability to lower order degree finite element families. The current work eliminates this issue. We provide algebraic expressions to exactly evaluate the subdomain integral of any degree polynomial on parent finite element shapes cut by a planar interface. These formulas also apply to the exact evaluation of the embedded interface integral. We provide recursive algorithms that avoid overflow in computer arithmetic for standard finite element geometries: triangle, square, cube, tetrahedron, and prism, along with a hypercube of arbitrary dimensions. Introduction Partial differential equation (PDE) solvers are ubiquitous among many engineering and applied mathematics practitioners. Today, many PDE solvers employ discontinuous functions, especially in fluid dynamics problems. These methods use discontinuous functions to distinguish different subdomains and ensure no extrinsic contributions incur while utilizing an arbitrary discontinuity. A few particular extensions of the Finite Element Method (FEM) using discontinuous functions are CutFEM or Extended FEM (XFEM), generalized FEM (GFEM), and nonlocal FEM. In XFEM and GFEM, an enrichment function, e.g., the Heaviside function, is employed to distinguish different domains defined by a common interface, alleviating cumbersome remeshing techniques [1,2]. The nonlocal FEM implements a kernel function, represented by the step function, that ensures nonlocal contributions are zero outside of some specified region [3,4,5]. Venturing outside of the FEM, an example of a method that also employs discontinuous functions is the Volume of Fluid (VOF) method. The VOF method uses the characteristic function to determine what portion of a cell is occupied by a fluid [6,7,8]. From the methods above, one can see the critical role of discontinuous functions in many of today's PDE solvers, all of which benefit from an accurate and efficient way of dealing with the integration of a discontinuous function. Discontinuous function integration can be cast as integration over several disjoint subdomains involving continuous functions, i.e., the region over which the integration occurs can be broken up into multiple subdomains where only continuous functions are defined. However, the boundary defining the subdomains is rarely trivial, and traditional integration schemes are not practical. Even invoking the divergence theorem in such cases has proven to be intractable for even simple geometries and discontinuities, as seen in [9,10], since integration over the subdomain boundaries must be performed. There has also been working devoted to moment fitting approaches, such as in [9], which also rely on the divergence theorem. Another approach, which depends on the convexity of the region of integration, is presented in [11]. Although in general, it cannot be expected that the region of integration is convex. Exact quadrature rules exist only for triangle and tetrahedral geometries, with planar cuts, and for low degree polynomials, specifically for quadratic polynomials, [12]. In there, the authors suggest they can extend their work to a higher degree polynomial integration. Still, the automation seems challenging since every time a higher degree polynomial is considered, one must recompute a new set of quadrature points and rules. The most common approach to discontinuous function integration is an adaptive algorithm, i.e., an algorithm that uses a grid refinement technique to capture the discontinuity better and produce a more accurate approximation to the integral. However, adaptive methods still require extensive information about the boundaries of the subdomains and typically lead to high computational costs, ultimately slowing down the numerical PDE scheme. There have been several recent developments that deal with the issue of discontinuous function integration, avoiding expensive adaptive methods. Among these are the use of equivalent polynomials [13,14,15,10], more specifically, a polynomial that replaces the discontinuous function in the integrand and yields an equivalent integral. Equivalent polynomial methods allow for integration of continuous functions over an entire region without the difficulty of discontinuous functions and, for line/plane discontinuities, the high computation cost of adaptive quadrature methods. The equivalent polynomial method was first introduced by Ventura in [10], where equivalent polynomials 2 were found analytically for simple geometries and discontinuities [16,17,18]. The ideas presented in [10] were limited to lower order elements, e.g., linear triangles and tetrahedra, and bi-linear quadrilaterals, as a consequence of using the divergence theorem to analytically calculate the coefficients of the equivalent polynomial. The difficulty behind this method is introduced when analytical integration is applied to a generic discontinuity since integration must be carried out on two subdomains. This causes severe restrictions when the dimension increases, resulting in impractical discontinuity considerations, even when the discontinuity is a hyperplane. Moreover, the automation becomes difficult when the degree of the polynomial and/or the dimension increases. In [19], the authors implemented the work of Ventura up to 4 th degree polynomials in 3D for the integration of fluid-structure Nitsche-type cutFEM coupling terms. However, the extension to higher degree polynomial integration can be numerically difficult. The work in [15], by Ventura and Benvenuti, builds on the core idea presented in [10], i.e., the idea of replacing a discontinuous function with an equivalent polynomial. The work's limitation in [10] is overcome by using a regularized Heaviside function, which approaches the Heaviside function in the limit, in place of the Heaviside function. This regularized Heaviside function is continuous and differentiable for any value of the regularization parameter ρ. The regularized Heaviside function allows one to perform analytical integration over the entire domain and then take the limit of the resulting expression when deriving the equivalent polynomial coefficients. The extended work in [15] creates a more robust method by eliminating the need for analytical integration over arbitrary subdomains created by the discontinuity. This method requires equality between the integral of the regularized Heaviside function multiplied by some monomial and the integral of the equivalent polynomial multiplied by the same monomial. The highest degree of the monomial and the dimension dictate the size of the linear system that needs to be solved to recover the coefficients of the equivalent polynomial. Since the equation for the discontinuity appears in the regularized Heaviside function, the equivalent polynomial coefficients will be dependent upon the discontinuity and the regularization parameter. Large values of the regularization parameter can then be taken to approximate the Heaviside function. Automation of this method relies on numerical libraries to calculate the polylogarithm function, which naturally arises from integrating the regularized Heaviside function. Using the regularized Heaviside function, one is left with expressions that involve a linear combination of polylogarithm functions of various orders. The two sources of error arising from the use of equivalent polynomials, as mentioned in [15], are the numerical evaluation of the polylogarithm and round-off error introduced by large values of the regularization parameter ρ. In [13] the concept of equivalent polynomials was extended to incorporate Legendre polynomials, which give rise to very beneficial properties. The main idea is to represent the equivalent and element shape polynomials with Legendre polynomials. The properties of Legendre polynomials are then utilized to allow for analytical integration over specified squares in 2-D or cubes in 3-D. Hence the error incurred from this method is produced by a spacetree refinement algorithm for complex discontinuities. It is stated in [13] that the analytical integration results are the same as those in [15] for a line or plane discontinuity; however, the implementation of the equivalent Legendre polynomials for the specified discontinuities lacks the ease of algorithmic automation for the analytical integration. In this work, we overcome these difficulties and provide closed-form recursive algebraic formulas to exactly evaluate the subdomain integral for any degree polynomial on parent finite element shapes cut by a planar interface. The proposed method completes the equivalent polynomial technique in [15]. Similar to their work, ours eliminates the need to integrate over a specified, often intractable, subdomain Ω i by integrating on the whole domain and using the Heaviside function as a weight. Again the regularized Heaviside function is substituted by the polylogarithm function. Still, here, we take advantage of its derivative and limiting properties, yielding a formulation that ultimately eliminates the need for evaluating the polylogarithm. As a result, both sources of error introduced by the polylogarithm in [15] are removed. By utilizing the derivative properties of the polylogarithm and the relationship the derivative of the Heaviside function shares with the Dirac delta distribution, we derive the same type of formulas to exactly evaluate the embedded interface integral for any degree polynomial. We provide the pseudo-codes for the subdomain and embedded interface integrals on hypercubes, triangles, tetrahedra, and prisms cut by planar surfaces. Much attention has been given to algorithms that avoid overflow in computer arithmetic, using, when needed, alternative formulas derived to eliminate round-off errors. The recursive nature of the algorithms allows for full automation. All expressions are algebraic and easy to implement. The outline of this paper is as follows. In Section 2, we discuss the properties of the polylogarithm, which are implemented in the paper. In Section 3, the closed-form expressions for the different elements are derived. The pseudo-codes for the n-dimensional cube, triangle, tetrahedron, and wedge can also be found in this section. Lastly, in Section 4, we provide some useful notes on the practical implementation of the equivalent polynomials. First we deal with the ill-conditioned Gram mass matrix and then we show that our algorithm can be executed offline, while new quadrature rules can be computed online at very little cost and to any accuracy using simple interpolation. In the Conclusion section, we refer to a new result we obtained for curved cuts and for now only available in the Ph.D. thesis of the second author [20]. All the algorithms developed in this article are implemented in FEMuS [21], an in-house open-source finite element C++ library built on top of PETSc [22] and publicly available on GitHub. Preamble The polylogarithm, Li s (z) where s, z ∈ C with |z|< 1, can be defined as Li s (z) = ∞ k=1 z k k s , or in integral form as Li s (z) = z Γ(s) ∞ 0 x s−1 e x − z dx, by analytic continuation, where Γ(s) is the gamma function. The integral representation of Li s (z) is analytic for z ∈ C \ [1, ∞) and (s) > 0 [23,24]. When the above integral is replaced with an appropriate complex contour integral we can consider s ∈ Z − ∪ {0} [24]. For the purpose of this paper we will only consider polylogarithms of the form Li s (w), where s ∈ {−1, 0, 1, ...} and w ∈ R. The polylogarithm can be defined in closed form for s = 1, 0, −1, ... . Specifically, Li 0 (w) = w 1 − w . All identities in this paper are used when Li s (w) is well defined. Two useful properties used throughout this paper are Li [23,25]. Two cases are of particular importance: s = 0 and s = −1. For s = 0, the polylogarithm Li 0 is used to represent the Heaviside function U, and for s = −1, the polylogarithm Li −1 is used to represent the Dirac delta distribution δ. Namely, for any smooth level set function G(x) s (−e w ) = −F s−1 (w), where F s−1 (w) is the Complete Fermi-Dirac integral, and d Li s (−e µ ) dµ = Li s−1 (−e µ )(1)U(G(x)) = − lim t→∞ Li 0 (− exp(G(x)t)),(2) where U is the Heaviside function with half-maximum convention U(G(x)) =          1 for G(x) > 0 0.5 for G(x) = 0 0 for G(x) < 0 ,(3) and for any differentiable function f (x) D f (x) δ(G(x)) ∇G dx = − lim t→∞ t D f (x) Li −1 (− exp(G(x)t)) ∇G dx.(4) Equality (4) is the weak convergence of −t Li −1 (− exp(G(x)t)) ∇G to the the Dirac delta distribution δ(G(x)) ∇G [26,27]. The proof is quite technical and it is given in Appendix A. For a hyperplane level set function G(x) = n · x + d, with unit normal n, we have ∇G = n, ∇G = 1, and Eq. (4) further simplifies to D f (x) δ(n · x + d)dx = − lim t→∞ t D f (x) Li −1 (− exp((x · n + d)t))dx.(5) The results of this paper should be implemented with consideration given to the aforementioned properties. The following proposition, which applies to a general domain, relates subdomain and embedded interface integrals to the limits of certain domain integrals of the polylogarithm functions Li 0 and Li −1 . This result is obtained by first connecting the subdomain and interface integrals to corresponding domain integrals, 5 using the Heaviside function U and the Dirac delta distribution δ as weights, and then by using equality (2) between U and the limit of Li 0 and equality (5) between the integral of δ and the limit of the integral of Li −1 , respectively. D1 P m (x)dx = − lim t→∞ D P m (x) Li 0 (− exp((n · x + d)t))dx (6) = lim t→∞ D P m (x) (1 + Li 0 (− exp(−(n · x + d)t))) dx.(7) Moreover, if Γ is tangential to ∂D at most on a set of measure zero and n = 1, then Γ P m (x)dµ = − lim t→∞ D P m (x) t Li −1 (− exp((n · x + d)t))dx (8) = − lim t→∞ D P m (x) t Li −1 (− exp(−(n · x + d)t))dx.(9) Proof. Equality (6) follows from the integral equality D1 P m (x)dx = D P m (x) U(n · x + d)dx, Eq. (2) and the dominated convergence theorem [28], i.e., D P m (x) U(n · x + d)dx = D P m (x) − lim t→∞ Li 0 (− exp((n · x + d)t)) dx = − lim t→∞ D P m (x) Li 0 (− exp((n · x + d)t))dx. 6 Eq. (7) follows from Eq. (6) and the following integral equality D1 P m (x)dx = D P m (x) (1 − U(−(n · x + d))) dx. Equality (8) follows from the integral equality Γ P m (x)dµ = D P m (x)δ(n · x + d)dx, and from Eq. (5) D P m (x)δ(n · x + d)dx = − lim t→∞ D P m (x) t Li −1 (− exp((n · x + d)t))dx. Eq. (9) follows from Eq. (8) and the following integral equality D P m (x)δ(n · x + d)dx = D P m (x)δ(−(n · x + d))dx. Remark 2.1. In proving (8), we assumed that the interface Γ is tangential to the boundary of D only on a set of measure zero. Such distinction is needed, since otherwise the domain of the Dirac delta distribution, centered on ∂D and aligned with the normal direction, would be only half contained within D, thus contributing only for half to the interface integral. In all the applications we are going to consider next, D will only be a convex domain with piece-wise flat boundaries. In doing so, Γ is either completely tangential or never tangential to ∂D. This allows us to compute the interface integral also in the tangential case (the boundary integral) by doubling the value of the computed integral in (8). In the remaining part of this section we will build the needed tools to exactly evaluate integrals as the ones in Eqs. (6)-(9) for the one dimensional case, where the domain D is the interval [0, 1] and level set function G(x) = ax + d. The resulting formula will hold for all integer s ≥ −1. The following two propositions are a direct consequence of the properties found in [23,25,24]. Proposition 2.2. For s = 0, 1, 2, . . . , limLi s (a) := lim t→∞ Li s (− exp(at)) t s =              −0.5 if s = 0 and a = 0 − a s s! if a > 0 0 otherwise . Proof. Case 1: s = a = 0. In this case we have limLi 0 (0) = Li 0 (−1) = −1 1 + 1 = − 1 2 . 7 Case 2: By induction on s, with a > 0. For s = 1 we have limLi 1 (a) = lim t→∞ Li 1 (− exp(at)) t = lim t→∞ − ln (1 + e at ) t = −a. For s = k − 1: assume lim t→∞ Li k−1 (−e at ) t k−1 = − a k−1 (k − 1)! . Then for s = k we have lim t→∞ Li k (−e at ) t k = lim t→∞ a k Li k−1 (−e at ) t k−1 = − a k k! , where we used the derivative property in Eq. (1). Case 3: For any other case, i.e., a ≤ 0, s = 0, the terms of the appropriate series expansion for the Fermi-Dirac integral vanish when the limit is taken inside the series, where the series converges uniformly [23]. In the following proposition we show that, for any integer s ≥ −1, it is possible to derive close form expressions for the antiderivative of the polylogarithm function Li s (− exp(ax + d)t) multiplied by any monomial. x m Li s (− exp(ax + d)t) dx = m+1 i=1 m! (−1) i−1 (m + 1 − i)! x m+1−i Li s+i (− exp(ax + d)t) (at) i + C. Proof. In the domain of interest, where the polylogarithm function converges uniformly, we use the identity Li s (−e µ ) = Li s−1 (−e µ )dµ + C, which follows from the derivative property in Eq. (1). For ease of notation we drop the constant in the proof. For m = 0, we get Li s (− exp((ax + d)t))dx = Li s+1 (− exp((ax + d)t)) at = 1 i=1 (−1) i+1 0! x 0+1−i Li s+i (− exp((ax + d)t)) (0 − i + 1)! (ta) i i s+1 . For m = k assume x k Li s+1 (− exp((ax + d)t))dx = k+1 i=1 k! (−1) i+1 (k + 1 − i)! x k+1−i Li s+1+i (− exp((ax + d)t) (at) i . 8 Then for m = k + 1, from integration by parts, we have x k+1 Li s (− exp((ax + d)t))dx = x k+1 Li s (− exp((ax + d)t))dx − (k + 1) x k Li s+1 (− exp((ax + d)t) at dx = x k+1 at Li s+1 (− exp((ax + d)t)) − (k + 1) k+1 i=1 k! (−1) i+1 (k + 1 − i)! x k+1−i Li s+i+1 (− exp((ax + d)t) (at) i+1 = k+2 i=1 (k + 1)! (−1) i+1 (k + 2 − i)! x k+2−i Li s+i (− exp((ax + d)t) (at) i . This completes the proof. Integrals, as the ones in Proposition 2.1, require the evaluation of the limit for t → ∞ of the definite integral of polynomials multiplied by polylogarithm functions. Using the results in Propositions 2.2 and 2.3 we are now capable of evaluating such limits for one-dimensional integrals. Later, in Section 3, we will extend these results to limits of integrals in higher dimensions. Proposition 2.4. For , m = 0, 1, 2, . . . and s = −1, 0, 1, . . . , the limit of the following definite integral is given by I 1 = − lim t→∞ 1 t s 1 0 x m Li s (− exp((ax + d)t) dx = m+1 i=1 m! (m + 1 − i)! 1 (−a) i limLi s+i (a + d) − m! (−a) m+1 limLi s+m+1 (d).(10) Proof. The proof follows directly from combining Propositions 2.2 and 2.3. Definition 2.1. For a = 0, let I 2 = s i=0 (−a) s−i (a + d) i i! (m + 1 + s − i)! .(11) Remark 2.2. In the following proposition, we will show that Eq. limLi s+i (O(a)) a i = O(a) s+i a i = O(a) s for all i. Thus, all terms in Eq. (10) would have comparable size and, since a does not appear in the denominator, it no longer contributes to overflows for a → 0. Proposition 2.5. For s ≥ 0, a = 0, d > 0, and a + d > 0 I 1 = I 2 . Proof. First note that the conditions a = 0, d > 0 and a + d > 0 are equivalent to −d a ∈ [0, 1]. In proving the proposition one simply needs to apply integration by parts and utilize Proposition 2.2. For a fixed s ≥ 0 and a = 0 we have − lim t→∞ 1 t s 1 0 x m Li s (− exp((ax + d)t) dx = − lim t→∞ 1 t s x m+1 m + 1 Li s (− exp((ax + d)t) 1 0 − at m + 1 1 0 x m+1 Li s−1 (− exp((ax + d)t)) dx = (a + d) s s! (m + 1) − a m + 1 − lim t→∞ 1 t s−1 1 0 x m+1 Li s−1 (− exp((ax + d)t)) dx = (a + d) s m! s! (m + 1)! − a(a + d) s−1 m! (s − 1)! (m + 2)! − a 2 (m + 1)(m + 2) − lim t→∞ 1 t s−2 1 0 x m+2 Li s−2 (− exp((ax + d)t)) dx = . . . = s i=0 m! (−a) s−i (a + d) i i! (m + 1 + s − i)! − (−a) s+1 m! (m + 2 + s) − lim t→∞ 1 0 x m+2+s t Li −1 (− exp((ax + d)t)) dx = s i=0 m! (−a) s−i (a + d) i i! (m + 1 + s − i)! , where we have used the weak convergence of t Li −1 (− exp((ax + d)t)) to the the Dirac distribution δ with −d a ∈ [0, 1]. Polynomial basis integration In this section, we provide closed-form algebraic expressions for the integration of selected polynomial bases for several standard FEM shapes. For a particular element, the polynomial basis implemented results in a monomial integrand, after an appropriate transformation. For each element, several integration strategies are provided in order to avoid overflow in computer arithmetic. Each integration is then summarized into a detailed algorithm. LSI: Line Segment Integration on [0, 1], with a = 0 For a fixed s = −1, 0, 1, . . . , we want to evaluate integrals in the form LSI m s (a, d) = − lim t→∞ 1 t s 1 0 x m Li s (− exp((ax + d)t)dx. From Eq. (10) LSI m s (a, d) := m+1 i=1 m! (m + 1 − i)! 1 (−a) i limLi s+i (a + d) − m! (−a) m+1 limLi s+m+1 (d).(12) For all a, d ∈ R, with |a|> 0, we have the subdomain integral LSI m 0 (a, d) = 1 0 x m U(ax + d)dx,(13) and, for a 2 = 1, the interface integral LSI m −1 (a, d) = 1 0 x m δ(ax + d)dx.(14) For s = −1 and |a|> 0 we also have the explicit point evaluation formula LSI m −1 (a, d) =                  1 |a| − d a m if 0 < − d a < 1 1 2|a| − d a m if − d a = 0 or − d a = 1 0 elsewhere ,(15) with the assumption that 0 0 = 1. That is the case for m = 0 and d = 0. Although equivalent to Eq. (12), for s = −1, Eq. (15) is generally faster to compute and does not suffer from overflow in computer arithmetic. Also, for s ≥ 0, d > 0, and a + d > 0 we replace Eq. (12) with the equivalent Eq. (11) to avoid overflow. The pseudo-code for the line segment integration is given in Algorithm 1. LSI m −1 (a, d) =        1 |a| − d a m if 0 ≤ − d a ≤ 1 0 elsewhere ,(16) again with the assumption that 0 0 = 1. if s = −1 then 3: return LSI m −1 (a, d) from Eq. (15) 4: else 5: if d ≤ 0 or a + d ≤ 0 then 6: return m+1 i=1 m! (m + 1 − i)! 1 (−a) i limLi s+i (a + d) − m! (−a) m+1 limLi s+m+1 (d) 7: else 8: return s i=0 m! (−a) s−i (a + d) i i! (m + 1 + s − i)! 9: end if 10: end if 11: end function 3.2. SQI: Square Integration on [0, 1] 2 , with a 2 + b 2 > 0 Fix s = −1, 0, 1, . . . , we want to evaluate integrals in the form SQI mn s (a, b, d) = − lim t→∞ 1 t s 1 0 1 0 x m y n Li s (− exp((ax + by + d)t)dy dx. We will first consider the case when the interface Γ is parallel to either the square sides, and then all the remaining cases. If a = 0 the iterated integral can be split in the product of two integrals SQI mn s (0, b, d) = 1 0 x m dx − lim t→∞ 1 t s 1 0 y n Li s (− exp((by + d)t)dy = 1 m + 1 LSI n s (b, d) . Similarly, if b = 0 SQI mn s (a, 0, d) = 1 n + 1 LSI m s (a, d). If both a and b are different from zero, after the integration of the inner integral we get SQI mn s (a, b, d) = − lim t→∞ 1 0 x m   − n+1 j=1 n! (n + 1 − j)! 1 (−b) j Li s+j (− exp((ax + b + d)t) t s+j + n! (−b) n+1 Li s+n+1 (− exp((ax + d)t) t s+n+1 dx = − n+1 j=1 n! (n + 1 − j)! 1 (−b) j LSI m s+j (a, b + d) + n! (−b) n+1 LSI m s+n+1 (a, d),(17) Then, for all a, b, d ∈ R , such that a 2 + b 2 > 0, we have the subdomain integral SQI mn 0 (a, b, d) = 1 0 1 0 x m y n U(ax + by + d) dy dx, and, for a 2 + b 2 = 1, the interface integral SQI mn −1 (a, b, d) = 1 0 1 0 x m y n δ(ax + by + d) dy dx. These formulas are general and versatile: they work regardless of where the line ax + by + d = 0 intersects the square domain, and the orientation of the Heaviside function follows the orientation of the normal a, b . Remark 3.2. In the special cases SQI mn −1 (a, 0, d) (or SQI mn −1 (0, b, d)), with −d/a = 0 or 1 (or −b/d = 0 or 1) , the corresponding line ax + d = 0 (or by + d = 0) overlaps with one of the sides of the square. Depending on which definition is used for LSI m −1 , either Eq. (15) or Eq. (16), one is left with half the boundary integral or the entire boundary integral, respectively, over the specified side of the square. This is also the case for the cube and the hypercube we are going to consider next. CBI: Cube integration on [0, 1] 3 , with a 2 + b 2 + c 2 > 0.1 n + 1 LSI o s (c, d).(18) For all other cases, after integrating in z we get CBI mno s (a, b, c, d) = − lim t→∞ 1 0 1 0 x m y n − o+1 k=1 o! (o + 1 − k)! 1 (−c) k Li s+k (− exp((ax + by + c + d)t) t s+k + o! (−c) o+1 Li s+o+1 (− exp((ax + by + d)t) t s+k dy dx = − o+1 k=1 o! (o + 1 − k)! 1 (−c) k SQI mn s+k (a, b, c + d) + o! (−c) o+1 SQI mn s+o+1 (a, b, d).(19) The cases a = 0 or b = 0 are handled by the square integrals as described in the previous section. 13 Then, for all a, b, c, d ∈ R, with a 2 + b 2 + c 2 > 0, we have the subdomain integral We are seeking integrals in the form Then, dropping the superscript, the problem reduces to evaluating integrals in the form HCI m s,dim (n, d) = − lim t→∞ 1 t s [0,1] dim dim i=1 x mi i Li s (− exp((n · x + d)t)dx,HCI A m s,dim (n, d) = − lim t→∞ 1 t s HC dim dim i=1 x mi i Li s (− exp((n · x + d)t)dx, with |a i |≤ |a i+1 | and a 1 = 0. Following the same integration strategy used for the square and the cube, with m = m dim and a = a dim , we obtain the following recursive formula HCI B m s,dim (n, d) = − m+1 i=1 m! (m + 1 − i)! 1 (−a) i HCI A m − s+i,dim−1 (n − , a + d) + m! (−a) m+1 HCI A m − s+m+1,dim−1 (n − , d),(20) where Similarly to Remarks 2.2 and 2.3, the HCI B formula also suffers from overflow in computer arithmetic when dim i=1 a i + d >> |a dim |. To overcome these difficulties, for this case only, we introduce the alternative formula HCI C m s,dim (n, d) = s i=0 m! (m + 1 + i)! (−a) i HCI A m − s−i,dim−1 (n − , a + d) + m! (m + s + 1)! (−a) s+1 HCI A m * −1,dim (n, d),(21) where m * = m 1 , m 2 , . . . , m dim−1 , m+s+1 . This formula is obtained from the derivative identity in Eq. (1) and by recursive integration by parts, increasing the monomial power and reducing the polylogarithm order s until it reaches −1. More specifically, HCI C follows the idea in Proposition 2.5, where an equivalent closed-form expression is given in which a does not appear in the denominator. Note that in HCI C HCI A m * −1,dim (n, d) = HCI A m * −1,dim (−n, −d), which not only permits choosing the optimal sign for the normal n, but also satisfies dim i=1 (−a i ) − d << −|a dim |. The pseudo-code for general dimension dim ≥ 1 is given in Algorithms 2 and 3. In Algorithm 2, the contributions of each component with a zero coefficient a i are handled first. Algorithm 3 is then called to 15 compute the contributions from all the remaining components. The recursive nature of the algorithm follows from the patterns developed in the HCI A , HCI B , and HCI C formulas. Note that Algorithm 2 also handles the case n = 0. Although this case was excluded here, it will be needed later when integrating on the prism. if dim > 0 then 12: Sort n, and accordingly m, from the smallest to the largest coefficient in magnitude 13: return HCI * Hypercube Integration A(dim, n, d, m, s) Then, for all a, b, d ∈ R such that a 2 + b 2 > 0, the subdomain integral is given by TRI mn 0 (a, b, d) = Tri (1 − x) m y n U(ax + by + d) dA, and, for a 2 + b 2 = 1, the interface integral is given by TRI mn −1 (a, b, d) = Tri (1 − x) m y n δ(ax + by + d) dA. In Eq. (22), changing variables and renaming constants as follows x = 1 − x, y = y, a = −a, b = b, d = d + a, First, we will consider the three separate cases where the interface Γ is parallel to one of the triangle edges. t→∞ 1 t s 1 0 n! n+1 i=1 (−1) i−1 (n + 1 − i)! x m+n+1−i Li s+i (− exp((ax − ax + d)t) (−at) i −(−1) n x m Li s+n+1 (− exp(ax + d)t) (−at) n+1 dx = n! n+1 i=1 1 a i limLi s+i (d) (n + 1 − i)! 1 0 x m+n+1−i dx + 1 a n+1 LSI m s+n+1 (a, d) = n! n+1 i=1 1 a i limLi s+i (d) (n + 1 − i)! (m + n + 2 − i) + 1 a n+1 LSI m s+n+1 (a, d) .(26) Next, we consider the remaining cases where the interface Γ is not parallel to one of the triangle edges. = − lim t→∞ 1 t s 1 0 n!   n+1 j=1 (−1) j−1 (n + 1 − j)! x m+n+1−j Li s+j (− exp(((a + b)x + d)t) (bt) j −(−1) n x m Li s+n+1 (− exp(ax + d)t) (bt) n+1 dx = − n+1 j=1 n! (−b) j (n + 1 − j)! LSI m+n+1−j s+j (a + b, d) + n! (−b) n+1 LSI m s+n+1 (a, d).(27) Alternatively, the same integral could be evaluated by reversing the order of integration. Specifically, TRI B mn s (a, b, d) = − lim t→∞ 1 t s 1 0 1 y x m y n Li s (− exp((ax + by + d)t) dx dy = − lim t→∞ 1 t s 1 0 m!   m+1 j=1 (−1) j−1 (m + 1 − j)! y n Li s+j (− exp(by + a + d)t) (at) j − (−1) j−1 (m + 1 − j)! y m+n+1−j Li s+j (− exp(((a + b)y + d)t) (at) j dx = m+1 j=1 m! (m + 1 − j)! −1 (−a) j LSI m+n+1−j s+j (a + b, d) − LSI n s+j (b, a + d) .(28)TRI BR mn s (a, b, d) =n!   limLi s+m+n+2 (d) (−(a + b)) m+n+2 n+1 j=1 (m + n + 1 − j)! (n + 1 − j)! a + b b j + LSI m s+n+1 (a, d) (−b) n+1 ,(29) and TRI BR mn s (a, b, d) = m!   − limLi s+m+n+2 (d) (−(a + b)) m+n+2 m+1 j=1 (m + n + 1 − j)! (m + 1 − j)! a + b a j + n! (−b) n+1 m+1 j=1 1 (m + 1 − j)! limLi s+n+j+1 (a + d) (−a) j   ,(30) which are less expensive to compute. For s = −1 and a + b + d > 0, we can still take advantage of this reduction by changing the sign of the normal and utilizing Proposition 2.1. Namely, TRI B mn −1 (a, b, d) = TRI BR mn −1 (−a, −b, −d). A similar reasoning can be extended to the cases TRI B TRI BR mn s (a, 0, d) = 1 n + 1 − limLi s+1 (a + d) a + (m + n + 1)! (−1) m+n+1 limLi s+m+n+2 (d) a m+n+2 ,(31)TRI BR mn s (0, b, d) = 1 m + 1 n! (−1) n limLi s+n+1 (d) b n+1 − (m + n + 1)! (−1) m+n+1 limLi s+m+n+2 (d) b m+n+2 ,(32)TRI BR mn s (a, −a, d) = limLi s+1 (d) (m + n + 1)a + n! m! m+1 i=1 (−1) i limLi s+n+1+i (a + d) (m + 1 − i)! a n+1+i ,(33) which hold for a + b + d ≤ 0 and s ≥ −1. We also include the two alternative formulas below. These are obtained from the derivative identity in Eq. (1) and by recursive integration by parts, increasing the monomial power and reducing the polylogarithm order s until it reaches −1. Namely, for s ≥ 0 and b = 0, TRI C m,n s (a, b, d) = n! (n + s + 1)! (−b) s+1 TRI A m,n+s+1 −1 (a, b, d) + s i=0 n! (n + i + 1)! (−b) i LSI m+n+i+1 s−i (a + b, d),(34) and, for s ≥ 0 and a = 0, TRI C m,n s (a, b, d) = m! (m + s + 1)! (−a) s+1 TRI m+s+1,n −1 (a, b, d) + s i=0 m! (m + i + 1)! (−a) i LSI n s−i (b, d + a) − LSI m+n+i+1 s−i (a + b, d) .(35) For a + b + d > max(|a|, |b|), the combination of Remark 3. return Triangle Integration A(−a, b, d + a, m, n, s) 3: end function 3.6. TTI: Tetrahedron Integration on          0 ≤ x ≤ 1 0 ≤ y ≤ 1 − x 0 ≤ z ≤ 1 − x − y , with a 2 + b 2 + c 2 > 0 To ease the computation in the case of the tetrahedron, we choose different polynomial bases depending on the magnitude of the coefficients a, b, and c. . Details of computation are given below. We make the following change of variables and constant renaming • for Eq. (37), x = x + y + z, y = y + z, z = z, a = a, b = b − a, c = c − b, d = d, • for Eq. (38), x = y + z + x, y = z + x, z = x, a = b, b = c − b, c = a − c, d = d, The case max(|b|, |c|) = max(m 1 , m 2 ) = 0 was already considered in Eq.(39). This corresponds to TTI A mno s (a, 0, 0, d), whose integration is straightforward. Below, we consider only max(|b|, |c|) = max(m 1 , m 2 ) > 0. For |b|≤ |c|, after integrating in z TTI B mno s (a, b, c, d) = − lim t→∞ 1 t s 1 0 x 0 x m y n o+1 i=1 o! (−1) i−1 (o + 1 − i)! y o+1−i Li s+i (− exp((ax + by + cy + d)t) (ct) i − o! (−1) o Li s+o+1 (− exp((ax + by + d)t) (ct) o+1 dy dx. = − lim t→∞ 1 t s 1 0 x 0 o+1 i=1 o! (−1) i−1 (o + 1 − i)! x m y n+o+1−i Li s+i (− exp((ax + (b + c)y + d)t) (ct) i − o! (−1) o x m y n Li s+o+1 (− exp((ax + by + d)t) (ct) o+1 dy dx.(41) Simplifying and using the triangle integration formula yields TTI B mno s (a, b, c, d) = − o+1 i=1 o! (o + 1 − i)! 1 (−c) i TRI A m, n+o+1−i s+i (a, b + c, d) + o! (−c) o+1 TRI A mn s+o+1 (a, b, d).(42) For |c|< |b|, we reverse the order of integration and after simplification get TTI B s mno (a, b, c, d) = − lim t→∞ 1 t s 1 0 x 0 x z x m y n z o Li s (− exp((ax + by + cz + d)t) dy dz dx = n+1 i=1 n! (n + 1 − i)! 1 (−b) i TRI A m, n+o+1−i s+i (a, b + c, d) − TRI A m+n+1−i,o s+i (a + b, c, d) .(43) All limiting cases, are left to be handled by the triangle integration formula as described in the previous section. For s ≥ 0 and a + b + c + d > max(|b|, |c|), we also include the alternative formulas below. These are obtained from the derivative identity in Eq. + s i=0 o! (o + i + 1)! (−c) i TRI A m,n+o+i+1 s−i (a, b + c, d),(44) otherwise TTI C s m,n,o (a, b, c, d) = n! (n + s + 1)! (−b) s+1 TTI A m,n+s+1,o −1 (a, b, c, d) + s i=0 n! (n + i + 1)! (−b) i TRI A m+n+i+1,o s−i (a + b, c, d) − TRI A m,n+o+i+1 s−i (a, b + c, d) .(45) The pseudo-code for the integration over the tetrahedron is given in Algorithms 6 and 7. Every time the triangle integration formula TRI A is needed, the function Triangle Integration A in Algorithm 5 is called.          0 ≤ x ≤ 1 0 ≤ y ≤ 1 − x −1 ≤ z ≤ 1 , with a 2 + b 2 + c 2 > 0 The implementation of a polynomial basis, whose elements are given by (1 − x) m y n z o , allows for computational simplicity when considering integrals in the form PRI = − lim t→∞ 1 t s Pri (1 − x) m y n 1 + z 2 o Li s (− exp((ax + by + cz + d)t) dV 2 .(46) By using the following transformation For |c|≥ max(|a|, |b|), after integrating in z we get if |b|≤ |c| then return 3: x = 1 − x, y = y, z = 1 + z 2 , a = −a, b = b, c = 2c, d = d + a − cPRI B mno s (a, b, c, d) = − lim t→∞ 1 0 x 0 x m y n − o+1 i=1 o! (o + 1 − i)! 1 (−c) i Li s+i (− exp((ax + by + c + d)t) t s+i + o! (−c) o+1 Li s+o+1 (− exp((ax + by + d)t) t s+k dy dx = − o+1 i=1 o! (o + 1 − i)! 1 (−c) i TRI A mn s+i (a, b, c + d) + o! (−c) o+1 TRI A mn s+o+1 (a, b, d).(47)s i=0 o! (o + i + 1)! (−c) i TRI A m,n+o+i+1 s−i (a, b + c, d) 4: + o! (o + s + 1)! (−c) s+1 Tetrahedron Integration A(a, bn! (n + i + 1)! (−b) i TRI A m+n+i+1,o s−i (a + b, c, d) − TRI A m,n+o+i+1 s−i (a, b + c, d)= − n+1 i=1 n! (n + 1 − i)! 1 (−b) i HCI m+n+1−i,o s+i,2 ( a + b, c , d) + n! (−b) n+1 HCI m,o s+n+1,2 ( a, c , d).(48) Lastly, for all other cases, after integrating first in x and simplifying we obtain PRI B mno s (a, b, c, d) = − lim t→∞ 1 t s 1 0 1 0 1 y x m y n z o Li s (− exp((ax + by + cz + d)t) dx dy dz = m+1 i=1 m! (m + 1 − i)! 1 (−a) i − HCI n,o s+i,2 ( b, c , a + d) + HCI m+n+1−i,o s+i,2 ( a + b, c , d) .(49) All limiting cases are left to be handled by the triangle and the hypercube integration formulas previously described. For s ≥ 0 and a + b + c + d > max(|a|, |b|, |c|), we also include the alternative formulas below. These are obtained from the derivative identity in Eq. (1) and by recursive integration by parts, increasing the monomial power and reducing the polylogarithm order s until it reaches −1. Namely, for |c|≥ max(|a|, |b|), we utilize the formula PRI C s m,n,o (a, b, c, d) = o! (o + s + 1)! (−c) s+1 PRI A m,n,o+s+1 −1 (a, b, c, d) + s i=0 o! (o + i + 1)! (−c) i TRI A m,n s−i (a, b, c + d),(50) and for |b|≥ |a| we implement PRI C s m,n,o (a, b, c, d) = n! (n + s + 1)! (−b) s+1 PRI A m,n+s+1,o −1 (a, b, c, d) + s i=0 n! (n + i + 1)! (−b) i HCI m+n+i+1,o s−i,2 ( a + b, c , d).(51) For any other case we employ PRI C s m,n,o (a, b, c, d) = m! (m + s + 1)! (−b) s+1 PRI A m+s+1,n,o −1 (a, b, c, d) + s i=0 m! (m + i + 1)! (−a) i HCI n,o s−i,2 ( b, c , a + d) − HCI m+n+i+1,o s−i,2 ( a + b, c , d) .(52) The pseudo-code for integration over the prism is given in Algorithms 8 and 9. Every time the triangle integration formula TRI A and the hypercube integration formula HCI are used, the functions Triangle Integration A in Algorithm 5 and HyperCube Integration in Algorithm 2 are called. if |c|≥ max(|a|, |b|) then return 3: The equivalent polynomial problem can be stated as follows: find the equivalent polynomial coefficients s i=0 o! (o + i + 1)! (−c) i TRI A m,n s−i (a, b, c + d) 4: + o! (o + s + 1)! (−c) s+1 Prism Integration A(a, bc o , such that M c o = f o , where f o = − lim t→∞ t −s            Ω b o,0 (x) Li s (− exp((n · x + d)t)) dx Ω b o,1 (x) Li s (− exp((n · x + d)t)) dx . . . Ω b o,L (x) Li s (− exp((n · x + d)t)) dx            and M =            Ω b o,0 (x) b o,0 (x) dx Ω b o,1 (x) b o,0 (x) dx · · · Ω b o,L (x) b o,0 (x) dx Ω b o,0 (x) b o,1 (x) dx Ω b o,1 (x) b o,1 (x) dx · · · Ω b o,L (x) b o,1 (x) dx . . . . . . . . . . . . Ω b o,0 (x) b o,L (x) dx Ω b o,1 (x) b o,L (x) dx · · · Ω b o,L (x) b o,L (x) dx            , with s = −1 or 0. Here b o is the basis of the polynomial space. Then, the equivalent polynomial is given by p(x) = c T o · b o . In order to avoid an ill-conditioned Gram matrix M , we implement orthogonal polynomials, via Grahm-Schmidt orthogonalization, using the L 2 inner product [29,30]. This yields the following relation Rather than storing and interpolating the equivalent polynomial coefficients, we store and interpolate the values of the equivalent polynomial evaluated at the quadrature points, for a given quadrature rule. This becomes particularly useful in the case of the tetrahedron where we used two different bases in the parent element depending of the values of the normal n. While the coefficients c n would differ for the two bases, the equivalent polynomial p(x) = c n b n remains the same. Thus interpolation is still possible even when using interpolants evaluated with different bases. Conclusion The many closed-form algebraic expressions provided in the current work can easily be implemented into numerous PDE solvers when discontinuous functions are implemented. We have eliminated the need to consider complicated subdomains while simultaneously eliminating any error produced by a regularization 30 parameter and polylogarithm approximation. We provide exact formulas for cumbersome subdomain and interface integrals, along with the associated algorithms. These closed-forms were designed with floating point arithmetic in mind. The results of this work provide one with the tools to eliminate many of the problems posed by discontinuous function integration. In this work, the discontinuities we considered were points, lines, and planes. Analytical integration on subdomains bounded by curved surfaces is currently being investigated. We have shown that analytical integration is still possible for elements cut by surfaces as complex as P n (x) + y(ax + b) + cz + d = 0, for any degree P n polynomial. A preliminary version of this result is already available in the PhD thesis of the second author [20] and will be analyzed in details in a forthcoming paper. Funding This work was supported by the National Science Foundation (NSF) Division of Mathematical Sciences (DMS) program, project 1912902. Figure 1 : 1Domain D cut by the planar interface n · x + d = 0. Proposition 2.1. With respect to Figure 1, let the region D, with boundary ∂D, be cut by the plane n·x+d = 0 into two subregions D 1 and D 2 , with normal n, pointing from D 2 to D 1 . Let Γ be the embedded interface between D 2 and D 1 . Let P m (x) be a polynomial of degree m in D. Then, the following integral equalities hold Proposition 2. 3 . 3Let a = 0, m = 0, 1, 2, . . . and s = −1, 0, 1, . . . , then ( 10 ) 10is equivalent to Eq. (11) for s ≥ 0 and positive arguments of the limLi functions. In computer arithmetic Eq. (10) suffers from overflow for d |a|> 0 and a → 0, because of the presence of the 1 a i terms in the sums. Proposition 2.5 will show all these terms actually simplify after expanding the definition of limLi for positive argument. Remark 2 . 3 . 23In Eq. (10), for d ≤ 0 or a + d ≤ 0, and a → 0, either the arguments of the polylogarithm functions are non positive, or, if positive, they are of the same order of a. In the first case the contribution 9 of their limits is zero. In the second case using the definition of limLi with positive argument one would get Remark 3. 1 . 1The formula for LSI m −1 (a, d) halves the value of the interface integral if the point −d/a is one of the two boundary points. This happens because half of the domain of the Dirac distribution falls outside the line segment, thus it does not contribute to the integral value. If this is not the desired behavior, and the boundary integral should account for the whole value, the definition of LSI m −1 (a, d) should be replaced by Algorithm 1 1Pseudo-code for integration on the line segment [0, 1] with a = 0 and s = −1, 0, 1, . . . . For s = −1 and |a|= 1 it corresponds to the interface integral. For s Fix s = −1, 0, 1, . . . , we want to evaluate integrals in the y n z o Li s (− exp((ax + by + cz + d)t)dz dy dx. For c = 0 0the above integral reduces to the square case, i.e. = 1 1y n z o U(ax + by + cz + d) dz dy dx, and, for a 2 + b 2 + c 2 y n z o δ(ax + by + cz + d) dz dy dx.It is remarkable how such simple formulas can handle all possible intersections between the cube and the plane. Moreover, they can be easily extended to evaluate corresponding integrals on hypercubes cut by hyperplanes for any dimension.3.4. HCI: Hypercube Integration on [0, 1] dim , with n = a 1 , a 2 , . . . , a dim , n > 0 and m = m 1 , m 2 , . . . , m dim . where we assume |a i |≤ |a i+1 |. However, if this is not the case, one can perform a reordering of the normal coefficients due to the symmetry of the domain and the integrand. Define dim 0 ∈ N 0 with dim 0 ≤ dim such that dim 0 is an upper bound for the indices corresponding to all the a i = 0 ∀i < dim 0 . Define dim := dim − dim 0 , m := m dim0+1 , . . . , m dim and n := a dim0+1 , . . . , a dim . Then HCI m s,dim (n, HCI m s,dim (n , d). m − = m 1 , . . . , m dim−1 and n − = a 1 , . . . , a dim−1 . This formula is recursively applied until dimension 1, where the the line segment integration formula, LSI, is used. At each level of integration two 14 contributions occur, one that involves a sum and a single term. The most expensive terms to compute are the ones involving a summation, with each one of them requiring the k ≥ dim. It is then desirable to have dim i a i + d < 0, so that all the limLi contributions vanish. From Proposition 2.1, changing the sign of the normal without any contribution is only allowed for s = −1, hence HCI B m −1,dim (n, d) = HCI B m −1,dim (−n, −d). Algorithm 2 2Pseudo-code for the integration on the hypercube [0, 1] dim cut by the hyperplane n · x + d = 0 with n = a 1 , a 2 , . . . , a dim , m = m 1 , m 2 , . . . , m dim and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. 1: function Hypercube Integration(dim, n, d, m, ,( 1 1with a 2 + b 2 > 0To ease the computation we choose a non-standard polynomial basis, namely (1 − x) m y n . − x) m y n Li s (− exp((ax + by + d)t) dA. Algorithm 3 3Pseudo-code for the integration on the hypercube [0, 1] dim cut by the hyperplane n · x + d = 0 with n = a 1 , a 2 , . . . , a dim , a 1 = 0 and |a i |≤ |a i+1 | for all i = 1, ..., dim − 1, m = m 1 , m 2 , . . . , m dim , and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral.1: function Hypercube Integration A(dim, n, d, m, a) i Hypercube Integration A(dim − 1, n, a + d, m, s + i) + m! (−a) m+1 Hypercube Integration A(dim − 1, n, d, m, s + m + 1) 5: end function 1: function Hypercube Integration C(dim, n, d, m, s) 2: m = m dim ; a = a dim ; m dim = m dim + s a) s+1 Hypercube Integration A(dim, n, d, m, Hypercube Integration A(dim − 1, n, a + d, m, s − i) 6: return HCI 7: end function Li s (− exp((a x + b y + d )t))dy dx .Dropping the superscript, for a fixed s = −1, 0, 1, . . . , the problem reduces to evaluating integrals y n Li s (− exp((ax + by + d)t)dy dx. y n Li s (− exp((by + d)t) dx dy n − y n+m+1 ) Li s (− exp((by + d)y n Li s (− exp((ax − ay + d)t) dy dx = − lim y n Li s (− exp((ax + by + d)t) dy dx In the limit for b → 0, with |a|> M > 0, Eq. (27) may suffer from overflow. Similarly, in the limit for a → 0, with |b|> M > 0, Eq. (28) may suffer from overflow. The choice of which formula to use, Eq. (27) or Eq. (28), should take into consideration the magnitude of a and b.Remark 3.3. In Eq. (27), for a + b + d ≤ 0 the summation within the LSI m+n+1−j s+j (a + b, d) terms vanishes. This is due to limLi s+1+i (x) = 0, with s ≥ −1, i ∈ Z + , and non positive argument x. Specifically, for a + b + d ≤ 0, Eqs. (27) and (28) reduce to b, d) and TRI B mn s (a, −a, 0), when a + d ≤ 0, b + d ≤ 0, and d ≤ 0, respectively. However, special attention should be used if s = −1 and a + b + d = 0, since for this case the first terms in the "supposedly vanishing" sums would be limLi 0 (0) = −0.5 = 0. Rewriting the three reduced formulas in a conservative way, always including the first term in the sum, leads to 3 and Eqs. (34)-(35) yields a formulation which protects against overflow for a → 0 and/or b → 0. In particular, the calls to the TRI A m,n+s+1 −1(a, b, d) and TRI A we include the degenerate case when both a = 0 and b = 0 for s ≥ 0, which was excluded because of the constraint a 2 + b 2 > 1. This case is needed for external calls made by higher dimensional objects, such as the tetrahedron and prism, for which the normal n = a, b, c could take the form n = 0, 0, y n Li s (− exp(dt))dy dx = − limLi mn s (d) (n + 1)(m + n + 2).(36)The pseudo-code for the triangle integration is given inAlgorithms 4 and 5. Algorithm 4 evaluates the integral in Eq. (22) on the triangle {(x, y) : x ∈ [0, 1], y ∈ [0, 1 − x]}. It calls the function Triangle Integration A in Algorithm 5, which evaluates the transformed integral in Eq. (23) on the triangle {(x, y) : x ∈ [0, 1], y ∈ [0, x]}. Triangle Integration A handles the degenerate case a = b = 0 and sorts the different s−cases. For each case it ensures that the reduced integration function, Triangle Integration BR, is called only for a + b + d ≤ 0. For 0 < a + b + d ≤ max(|a|, |b|), the function Triangle Integration B is called, otherwise the alternative function Triangle Integration C is used. The recursive calls follow from the patterns developed in Eqs. (34) and (35). Every time the line segment integration formula, LSI, is needed the function Line Segment Integration in Algorithm 1 is called. Algorithm 4 4Pseudo-code for the integration of Eq. (22) on the triangle {(x, y) : x ∈ [0, 1], y ∈ [0, 1 − x]} cut by the line a x + b y + d = 0 with n = a, b , n > 0, m = m, n , and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. 1: function Triangle Integration(a, Let m 1 1= max(|b − a|, |c − b|) and m 2 = |a − c|. For m 1 ≥ m 2 , we evaluate integrals in the form TTI mno s (a, b, c, d) = − lim t→∞ 1 t s Tet (x + y + z) m (y + z) n z o Li s (− exp((ax + by + cz + d)t) dV, (37) Algorithm 5 Pseudo-code for the integration of Eq. (23) on the triangle {(x, y) : x ∈ [0, 1], y ∈ [0, x]} cut by the line a x + b y + d = 0 with n = a, b , m = m, n and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 and n > 0 it corresponds to the subdomain integral. a + b + d ≤ 0 then return Triangle Integration BR(a, b, d, m, n, a + b + d ≤ 0 then return Triangle Integration BR(a, b, d, m, n, s) 10: else if a + b + d ≤ max(|a|, |b|) then return Triangle Integration B(a, a + b = 0 then return TRI B mn (a, −a, d) from Eq. a + b = 0 then return TRI BR mn (a, −a, d) from Eq. + i + 1)! (−a) i LSI n s−i (b, d + a) − LSI m+n+i+1 s−i (a + b, d) 7: + m! (m + s + 1)! (−a) s+1 Triangle Integration A(a, b, d, m + s + 1, n, z + x) m (z + x) n x o Li s (− exp((ax + by + cz + d)t) dV. (38)For a = b = c (or max(m1, m2) = 0), integral (37) is considered and after integration we Li s (− exp((a x + b y + c z + d )t) dz dy dx .Dropping the superscript, for a fixed s = −1, 0, 1, . . . , the problem reduces to find integrals in the y n z o Li s (− exp((ax + by + cz + d)t) dz dy dx. ( 1 ) 1and by recursive integration by parts,, increasing the monomial power and reducing the polylogarithm order s until it reaches −1. Namely, For |b|≤ |c| we use Algorithm 6 6Pseudo-code for the integration of Eqs. (37)-(38) on the tetrahedron {(x, y, z) : x ∈ [0, 1], y ∈ [0, 1 − x], z ∈ [0, 1 − x − y]} cut by the plane a x + b y + c z + d = 0 with n = a, b, c , n > 0, m = m, n, o and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. = max(|a − b|, |c − b|), m 2 = |a − c| Li s (− exp((a x + b y + c z + d )t))dz dy dx Dropping the superscript, for a fixed s = −1, 0, 1, . . . , the problem reduces to integrals in the y n z o Li s (− exp((ax + by + cz + d)t) dz dy dx. Algorithm 7 7Pseudo-code for the integration of Eq. (40) on the tetrahedron {(x, y, z) : x ∈ [0, 1], y ∈ [0, x], z ∈ [0, y]} cut by the plane a x + b y + c z + d = 0 with n = a, b, c , either b = 0 or c = 0, m = m, n, o and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. 1: function Tetrahedron Integration A(a, |b|≤ |c| then return TTI B from Eq. (42) 3: else return TTI B from Eq. function Tetrahedron Integration C(dim, n, d, m, s) 2: y n z o Li s (− exp((ax + by + cz + d)t) dy dz dx Algorithm 8 return 8Pseudo-code for the integration of Eq. (46) on the prism {(x, y, z) : x ∈ [0, 1], y ∈ [0, 1 − x], z ∈ [−1, 1]} cut by the plane a x + b y + c z + d = 0 with n = a, b, c , n > 0, m = m, n, o and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. Prism Integration A(−a, b, 2c, d + a − c, m, n, o, s) 3: end function Algorithm 9 Pseudo-code for the integration on the prism {(x, y, z) : x ∈ [0, 1], y ∈ [0, x], z ∈ [0, 1]} cut by the plane a x + b y + c z + d = 0 with n = a, b, c , n > 0, m = m, n, o and s = −1, 0, 1, . . . . For s = −1 and n = 1 it corresponds to the interface integral. For s = 0 it corresponds to the subdomain integral. |c|≥ max(|a|, |b|) then return PRI B from Eq. (47) 3: else if |b|> |a| then return PRI B from Eq. (48) 4:else return PRI B from Eq. for basis elements: b n = Ab o , where the components in the new basis, b n , are a linear combination of the components in the old basis, b o . The matrix A is an (L + 1) × (L + 1) lower triangular matrix, where L is the dimension of the space spanned by the basis vector b o .The implementation of equivalent polynomial using an orthonormal basis yieldsIc n = f n = Af o , resulting in p(x) = (c n ) T b n = f T o A T Ab o (x).Note that the term A T Ab o (x) is independent of the hyperplane cut and can be evaluated off-line. Insteadf o changes and has to be recalculated for every new cut. To this end, the continuous dependence of f o with respect to the coefficients of the cut planes n and d is of great help. Namely, for each considered element, we can explicitly evaluate ∂f o ∂n and ∂f o ∂d , 29 and prove differentiability almost everywhere of f 0 with respect to these parameters. This implies that for each element a given set of quadrature rules can be evaluated and stored off-line, and a new quadrature integration rule can be reconstructed on-line by interpolation at very little cost and to any accuracy, making this technique far superior to any other existing method. In 2D for a given line ax + by + d = 0 we use the two parameter family given by the polar angle θ = atan2(b, a) and the x−intercept between the given line and the lines x = y, if θ is in the 1 st or 3 rd quadrant, or 1 − x = y, if θ is in the 2 nd or 4 th quadrant, respectively. In 3D for a given plane ax + by + cz + d = 0 we use the three parameter family given by the polar angle θ = atan2(b, a), the azimuthal angle φ = acos(c/ √ a 2 + b 2 + c 2 ), and the x−intercept between the given plane and the lines x = y = z, if θ and φ are in the 1 st or 7 th octant, or 1 − x = y = z, if θ and φ are in the 2 nd or 8 th octant, or x = y = 1 − z, if θ and φ are in the 3 rd or 5 th octant, or x = 1 − y = z, if θ and φ are in the 4 th or 6 th octant, respectively. These choices assure that for lines or planes cutting any of the considered elements the x−intercept is always in the interval [0, 1]. Then, for each quadrant or octant, as above, we construct off-line matrices of coefficients spanning the whole range of parameters, and use on-line Lagrange interpolation to reconstruct the values of the coefficients for any (x, θ) ∈ [0, 1] × [−π, π] in 2D, or (x, θ, φ) ∈ [0, 1] × [−π, π] × [0, π] in 3D. , c, d, m, n, o + s + 1, −1)5: else return 6: s i=0 , c, d, m, n, o + s + 1, −1) s+1 Prism Integration A(a, b, c, d, m, n + s + 1, o, −1) s+1 Prism Integration A(a, b, c, d, m + s + 1, n, o, −1) end if 12: end function 4. Note on the equivalent polynomial5: else if |b|≥ |a| then return 6: s i=0 n! (n + i + 1)! (−b) i HCI m+n+i+1,o s−i,2 ( a + b, c , d) 7: + n! (n + s + 1)! (−b) 8: else return 9: s i=0 m! (m + i + 1)! (−a) i HCI n,o s−i,2 ( b, c , a + d) − HCI m+n+i+1,o s−i,2 ( a + b, c , d) 10: + m! (m + s + 1)! (−a) 11: The authors have no conflicts of interest to declare that are relevant to the content of this article.Data availabilityData sharing is not applicable to this article as no datasets were generated during the current study.Appendix A Lemma 5.1. Let D be a bounded connected domain with smooth boundary ∂D. Let G(x) be a smooth level set function. Let Γ = {x ∈ D : G(x) = 0} be a continuous smooth embedded interface, that separates D in the two subregions D 1 and D 2 , such that G(x) > 0 for all x ∈ D 1 and G(x) < 0 for all x ∈ D 2 . Assume theThe ∇G term in both sides is needed since the level set G(x) only approximates the required condition, ∇d = 1, for a true distance d(x), see Appendix in[12].Proof. Let n on ∂D be the outer unit normal vector to D. Let n = − ∇G ∇G be defined everywhere on D. n is the unit vector orthogonal to the the level curves G(x) = const, pointing in the direction of maximum decrease. On the interface Γ, n is the unit outer normal to D 1 . 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L , Journal of Mathematical Analysis and Applications. 3221L. Onural, Impulse functions over curves and surfaces and their applications to diffraction, Journal of Mathematical Analysis and Applications 322 (1) (2006) 18-27. . X Wang, Lecture Notes in Real Analysis. SpringerX. Wang, Lecture Notes in Real Analysis, Springer, 2018. On the condition number of some Gram matrices arising from least squares approximation in the complex plane. Y Saad, Numerische Mathematik. 483Y. Saad, On the condition number of some Gram matrices arising from least squares approximation in the complex plane, Numerische Mathematik 48 (3) (1986) 337-347. On multivariate orthogonal polynomials. Y Xu, SIAM Journal on Mathematical Analysis. 243Y. Xu, On multivariate orthogonal polynomials, SIAM Journal on Mathematical Analysis 24 (3) (1993) 783-794.

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{'abstract': "The implementation of discontinuous functions occurs in many of today's state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when integrating functions whose discontinuity falls in the element's interior are for low order degree polynomials, not easily extended to higher order degree polynomials, and cover a restricted set of geometries. Many approaches to this issue have been developed in recent years. Among them one of the most elegant and versatile is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, allowing integration to occur over the entire domain rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its problems. The exact subdomain integration requires a machinery that quickly grows in complexity when increasing the polynomial degree and the geometry dimension, restricting its applicability to lower order degree finite element families. The current work eliminates this issue. We provide algebraic expressions to exactly evaluate the subdomain integral of any degree polynomial on parent finite element shapes cut by a planar interface. These formulas also apply to the exact evaluation of the embedded interface integral. We provide recursive algorithms that avoid overflow in computer arithmetic for standard finite element geometries: triangle, square, cube, tetrahedron, and prism, along with a hypercube of arbitrary dimensions.", 'arxivid': '2110.00642', 'author': ['Eugenio Aulisa ', 'Jonathon Loftin ', '\nDepartment of Mathematics and Statistics\nDepartment of Mathematics and Computer Science\nTexas Tech University\nSouthern79410LubbockTXUSA\n', '\nArkansas University\nMagnolia, AR71753USA\n'], 'authoraffiliation': ['Department of Mathematics and Statistics\nDepartment of Mathematics and Computer Science\nTexas Tech University\nSouthern79410LubbockTXUSA', 'Arkansas University\nMagnolia, AR71753USA'], 'corpusid': 253384191, 'doi': '10.1007/s11075-023-01502-3', 'github_urls': ['https://github.com/eaulisa/MyFEMuS.'], 'n_tokens_mistral': 23652, 'n_tokens_neox': 21055, 'n_words': 12227, 'pdfsha': '614f1609b400c91ac0e3bdb8d0a33c9634d0318c', 'pdfurls': ['https://export.arxiv.org/pdf/2110.00642v2.pdf'], 'title': ['Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts', 'Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts'], 'venue': []}

arxiv

Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology Florian Biersack florian.biersack@mathematik.uni-wuerzburg.de University of Würzburg Chair for Complex Analysis Emil-Fischer-Strasse 4097074Würzburg, BavariaGermany Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology We study the class HQ(D), the set of harmonic quasiconformal automorphisms of the unit disk D in the complex plane, endowed with the topology of uniform convergence. Several important topological properties of this space of mappings are investigated, such as separability, compactness, path-connectedness and completeness. Introduction The idea for investigating the harmonic quasiconformal automorphisms of D := z ∈ C |z| < 1 was on the one hand inspired from a topic that has drawn much attention in recent years: The harmonic quasiconformal mappings. Initiated by Martio in 1968 (see [10, p. 238] and [14, p. 366]), this particular class of homeomorphisms attracted much interest in the recent past, see [3,Introduction], [10], [11], [14], [15,Section 10.3] and the references therein, to name only a few. In particular, Kalaj and Pavlović worked intensively in this area and achieved numerous results, among others several characterization statements for harmonic quasiconformal automorphisms of the unit disk (see [10,Theorem A,p. 239] and Proposition 3.8 below). On the other hand, in [2], the authors studied the quasiconformal automorphism groups of simply connected domains in the complex plane. For this class of domains, the unit disk in C can be regarded as the reference element, not least due to the classical Riemann Mapping Theorem (RMT) and its quasiconformal counterpart, the Measurable RMT (see e.g. [13,Mapping Theorem,p. 194]). In view of these circumstances and by the Theorem of Radó-Kneser-Choquet (see Proposition 2.2), a similarly striking result on harmonic mappings, the following discussion will focus on the special case of harmonic quasiconformal automorphisms of the unit disk in C. The topology used in this paper is the uniform topology, induced by the supremum metric Definition and basic properties In [2], the authors studied the following space of mappings: That is, the mappings in HQ(D) are the harmonic quasiconformal automorphisms of D. Here and henceforth, a complex-valued mapping f = u + iv defined on a domain is called harmonic if both, its real and imaginary parts, are real-valued harmonic mappings, which in turn are defined via the Laplace equation ∆u = ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0, the differential polynomial ∆ := ∂ 2 ∂x 2 + ∂ 2 ∂y 2 being the Laplace operator. Harmonic mappings possess numerous important properties, such as the mean-value property and the maximum principle ([6, p. 12]), which are in turn deeply connected with holomorphic functions by well-known results from Complex Analysis. An immediate conclusion to be drawn is Σ(D) ⊆ HQ(D), where Σ(D) := {f ∈ Q(D) f is conformal} denotes the subset of conformal automorphisms of D. In particular, it is id D ∈ HQ(D) and therefore HQ(D) = ∅, where id D is the identity on D. An important fact about harmonic mappings in C and their representation is given by the following result due to Radó, Kneser Proposition 2.2 (Radó-Kneser-Choquet). Let G C be a convex Jordan domain and γ : ∂D −→ ∂G be a weak homeomorphism, i.e. a continuous mapping of ∂D onto ∂G such that the preimage γ −1 (ξ) of each ξ ∈ ∂G is either a point or a closed subarc of ∂D. Then the harmonic extension P[γ](z) := 1 2π 2π 0 1 − r 2 1 − 2r cos(t − φ) + r 2 γ(e it ) dt, z = re iφ ∈ D,(2) defines an injective harmonic mapping of D onto G; moreover, P[γ] is unique. Conversely, if G C is a strictly convex Jordan domain and f : D −→ G is an injective harmonic mapping, then f has a continuous extension to D which defines a weak homeomorphism of ∂D onto ∂G. Moreover, if f ∈ C(D) is harmonic in D, then f | D can be written in the form (2). For a (Jordan) domain G ⊆ C, let H * (∂D, ∂G) denote the set of all weak homeomorphisms of ∂D onto ∂G and in the special case G = D define H * (∂D) := H * (∂D, ∂D). Consequently, let H + (∂D, ∂G) and H + (∂D) denote the corresponding subsets of all orientation-preserving homeomorphisms, respectively. Remark 2.3. The harmonic extension P[γ] defined by (2) is also called the Poisson transformation of γ ∈ H * (∂D, ∂G), and the corresponding integral kernel 1 − r 2 1 − 2r cos(t) + r 2 is called the Poisson kernel, see [6, p. 12] and [15, pp. 5-6]. From the Radó-Kneser-Choquet Theorem 2.2, one obtains the following (see also [11, pp. 337-338]) Corollary 2.4. HQ(D) = Q(D) P[γ] γ ∈ H + (∂D) Corollary 2.4 also makes sense when recalling that every quasiconformal automorphism of a Jordan domain admits a homeomorphic boundary extension (see [12, p. 13]). In particular, the induced boundary mapping is injective, hence an element of H + (∂D). A concrete harmonic automorphism of the unit disk is visualized in Example 2.5. For x ∈ [0, 1], consider the piecewise-defined function φ(x) =      2x, x ∈ [0, 1 3 ] 2 3 , x ∈ [ 1 3 , 3 4 ] 4 3 x − 1 3 , x ∈ [ 3 4 , 1] for e it ∈ ∂D defines a weak homeomorphism of ∂D onto itself, i.e. γ ∈ H * (∂D). The corresponding harmonic extension provided by Proposition 2.2 therefore yields a harmonic homeomorphism P[γ] of D onto itself. Figure 1 shows the (approximated) mapping behaviour of this harmonic extension, visualized by concentric circles around the origin, radial rays and an Euclidean grid. However, the mapping P[γ] is not quasiconformal due to the fact that its boundary function -which equals γ by construction -is not injective, but this would necessarily follow. Remark 2.6. In particular, the harmonic extension P[γ] discussed in Example 2.5, with γ given by (3), provides a concrete example of a sense-preserving homeomorphism of the unit disk that is not quasiconformal, i.e. P[γ] ∈ H + (D)\Q(D). Another example of such a mapping will be presented in Proposition 4.3. A basic fact in the theory of harmonic mappings is that the composition of two such mappings is not necessarily harmonic again (see [6, p. 2]). In the same manner, the inverse mapping of an injective harmonic mapping is also not harmonic in general, except for special situations, as stated in (see [6, Theorem, pp. 145-148]) Proposition 2.7 (Choquet-Deny). Suppose f is an orientation-preserving injective harmonic mapping defined on a simply connected domain G ⊆ C, and suppose that f is neither analytic nor affine. Then the inverse mapping f −1 is harmonic if and only if f has the form f (z) = α βz + 2i Arg(γ − e −βz ) + δ,(4) where α, β, γ, δ ∈ C are constants with αβγ = 0 and e −βz < |γ| for all z ∈ G. This result and the previously stated facts immediately imply (see also [ Topological properties of HQ(D) This section is intended to study some central topological properties of the space HQ(D). Since this situation is settled in the context of metric spaces, many of these topological notions can be expressed in terms of convergent sequences in the space HQ(D). Thus, certain convergence results for uniformly convergent sequences of harmonic mappings will prove valuable, as stated in An elementary persistence property in the interplay between harmonic and holomorphic mappings is that the post-composition of a holomorphic function with a harmonic one remains harmonic (see [6, p. 2]). This fact is utilized in order to prove Another property to be studied is the path-connectedness of HQ(D). In this context, the following integral operator will be of crucial importance (see [14, p. 367] and [15, p. 305]): Definition 3.5 (Hilbert transformation). For periodic ϕ ∈ L 1 ([0, 2π]) and x ∈ R, the expression H(ϕ)(x) := − 1 π lim →0 + π ϕ(x + t) − ϕ(x − t) 2 tan(t/2) d t (5) is called the (periodic) Hilbert transformation of ϕ. Remark 3.6. (i) In Fourier theory and trigonometric series, the Hilbert transformation plays a prominent role. However, the definition of the operator H is not completely consistent in the vast literature about this topic. For example, a different formulation is given by H(ϕ)(x) = − 1 π lim →0 + π ϕ(x + t) − ϕ(x − t) t d t, which is -at least for existence questions -equivalent to (5) due to 2 tan(t/2) − t = 0 for t −→ 0 (see [15, p. 306] and [17, Vol. I, p. 52]). (ii) The notion of Hilbert transformation is also present in further mathematical areas, for example in the classical theory of quasiconformal mappings in C (see [13, pp. 156-160]) and Teichmüller spaces (see [9, pp. 319-320]). However, the circumstance that the definitions are in parts considerably different from each other is also present in these contexts. Due to the presence of the tangent function in the integrand's denominator in (5), the question for existence of H raises, partially answered in (see [14, p. 367 (b) ϕ is strictly increasing and bi-Lipschitz; (c) the Hilbert transformation of ϕ is an element of L ∞ (R). A mapping g : X −→ Y between metric spaces (X, d X ) and (Y, d Y ) is called bi-Lipschitz if there exists a constant L ∈ [1, +∞) such that 1 L d X (x 1 , x 2 ) ≤ d Y (g(x 1 ), g(x 2 )) ≤ Ld X (x 1 , x 2 ) for all x 1 , x 2 ∈ X, thus sharpening the classical notion of a Lipschitz-continuous mapping. In view of Corollary 2.4 and Proposition 3.8, the following characterization for the elements of the space HQ(D) is valid: A harmonic (orientation-preserving) homeomorphism P[e iϕ ] of D onto itself is quasiconformal if and only if the corresponding mapping ϕ is an element of H + qc := ϕ ∈ C([0, 2π]) ϕ is strictly increasing and bi-Lipschitz, ϕ(2π) − ϕ(0) = 2π, H( ϕ ) ∈ L ∞ (R) . Here, ϕ denotes the canonical extension of ϕ ∈ H + qc to all of R via ϕ(t + 2kπ) := ϕ(t) + 2kπ for all t ∈ [0, 2π] and every k ∈ Z. By the requirement of strict increasing monotonicity, every mapping ϕ ∈ H + qc is differentiable almost everywhere in (the interior of) [0, 2π]. Consequently, each extended mapping ϕ ∈ C(R) is differentiable almost everywhere in R with ϕ being 2π-periodic by construction. Furthermore, the assumption that ϕ is bi-Lipschitz yields ϕ ∈ L 1 ([0, 2π]) (see [16,Theorem 10,p. 124]). Therefore, the condition H( ϕ ) ∈ L ∞ (R) is reasonable. In view of the path-connectedness of HQ(D), the first important observation to be made here is + (1 − λ)ϕ 2 . (i) Monotonicity: For t, t ∈ [0, 2π] with t < t , it is λϕ 1 (t) + (1 − λ)ϕ 2 (t) < λϕ 1 (t ) + (1 − λ)ϕ 2 (t ) due to λ, (1 − λ) ≥ 0, hence λϕ 1 + (1 − λ)ϕ 2 is strictly increasing. (ii) Bi-Lipschitz property: Let t, t ∈ [0, 2π] and L := max{L 1 , L 2 } with L j denoting the bi-Lipschitz constant of ϕ j , j = 1, 2. Then on the one hand, by means of the triangle inequality, it is |λϕ 1 (t) + (1 − λ)ϕ 2 (t) − λϕ 1 (t ) − (1 − λ)ϕ 2 (t )| ≤ λ |ϕ 1 (t) − ϕ 1 (t )| + (1 − λ) |ϕ 2 (t) − ϕ 2 (t )| ≤ λL|t − t | + (1 − λ)L|t − t | = L|t − t |. Hence λϕ 1 + (1 − λ)ϕ 2 is Lipschitz-continuous with Lipschitz constant L. Without loss of generality, assume t > t , then on the other hand, it is (recall that λϕ 1 + (1 − λ)ϕ 2 is strictly increasing by (i)) λϕ 1 (t) + (1 − λ)ϕ 2 (t) − λϕ 1 (t ) − (1 − λ)ϕ 2 (t ) = λ(ϕ 1 (t) − ϕ 1 (t )) + (1 − λ)(ϕ 2 (t) − ϕ 2 (t )) ≥ λ 1 L (t − t ) + (1 − λ) 1 L (t − t ) = 1 L (t − t ). Finally, switching the roles of t and t shows that λϕ 1 + (1 − λ)ϕ 2 is bi-Lipschitz continuous on [0, 2π] with bi-Lipschitz constant L. (iii) Image interval has length 2π: It is λϕ 1 (2π) + (1 − λ)ϕ 2 (2π) − (λϕ 1 (0) + (1 − λ)ϕ 2 (0)) = λ(ϕ 1 (2π) − ϕ 1 (0)) + (1 − λ)(ϕ 2 (2π) − ϕ 2 (0)) = λ2π + (1 − λ)2π = 2π. (iv) Hilbert transformation: First of all, λ ϕ 1 + (1 − λ) ϕ 2 is differentiable almost everywhere in R by (i) with (λ ϕ 1 + (1 − λ) ϕ 2 ) = λ ϕ 1 + (1 − λ) ϕ 2 . The function λϕ 1 + (1 − λ)ϕ 2 is contained in L 1 ([0, 2π]) as the linear combination of such elements. Following Definition 3.5, the Hilbert transformation of λ ϕ 1 + (1 − λ) ϕ 1 is given by H(λ ϕ 1 + (1 − λ) ϕ 2 )(x) = − 1 π π 0 + λ ϕ 1 (x + t) + (1 − λ) ϕ 2 (x + t) − λ ϕ 1 (x − t) − (1 − λ) ϕ 2 (x − t) 2 tan(t/2) dt = − 1 π π 0 + λ( ϕ 1 (x + t) − ϕ 1 (x − t)) + (1 − λ)( ϕ 2 (x + t) − ϕ 2 (x − t)) 2 tan(t/2) dt. Since ϕ 1 , ϕ 2 ∈ H + qc , it is H( ϕ 1 ), H( ϕ 2 ) ∈ L ∞ (R) by definition of the set H + qc , thus using the linearity of (improper) integrals the previous equation can be rewritten as H(λ ϕ 1 + (1 − λ) ϕ 2 )(x) = − λ π π 0 + ϕ 1 (x + t) − ϕ 2 (x − t) 2 tan(t/2) dt − 1 − λ π π 0 + ϕ 2 (x + t) − ϕ 2 (x − t) 2 tan(t/2) dt = λH( ϕ 1 )(x) + (1 − λ)H( ϕ 2 )(x). Since L ∞ (R) is a R-vector space, the previous equation yields H(λ ϕ 1 + (1 − λ) ϕ 2 ) ∈ L ∞ (R). All in all, the mapping λϕ 1 + (1 − λ)ϕ 2 is contained in H + qc for every λ, hence H + qc is convex. Thus, as a subset of the normed vector space C([0, 2π]), H + qc is also path-connected. Continuing the investigation, the set H + qc now gives rise to consider the mapping (6) is continuous and surjective. Λ : H + qc −→ HQ(D), ϕ −→ D z −→ Λ(ϕ)(z) := P[e iϕ ](z) .(6) Proof. The fact that Λ is surjective was already mentioned above. Hence, let (ϕ n ) n∈N converge in H + qc to ϕ ∈ H + qc . The characterization of elements in HQ(D) stated in Proposition 3.8 implies that (Λ(ϕ n )) n∈N is a sequence in HQ(D) and Λ(ϕ) ∈ HQ(D). In particular, Λ(ϕ n ) and Λ(ϕ) are harmonic quasiconformal automorphisms of D, continuous on D and coincide with e iϕn and e iϕ on ∂D, respectively (see also [6, p. 12]). Therefore, since Λ(ϕ n ) − Λ(ϕ) is harmonic as well, the Maximum Principle for harmonic mappings applies, concluding in sup z∈D |Λ(ϕ n )(z) − Λ(ϕ)(z)| = sup z∈∂D |Λ(ϕ n )(z) − Λ(ϕ)(z)| = sup t∈[0,2π] e iϕn(t) − e iϕ(t) ≤ sup t∈[0,2π] |ϕ n (t) − ϕ(t)| = d sup (ϕ n , ϕ). In the estimate, the elementary inequality |e ix − e iy | ≤ |x − y| for x, y ∈ R was used. The last expression tends to zero for n → ∞, proving the continuity of Λ. Finally, combining the statements of Lemma 3.9 and Theorem 3.10 yields In order to prove this claim, some helpful results are collected in the following. The principal idea of the proof of Theorem 4.1 is to construct a sequence of homeomorphic mappings of the interval [0, 1] onto itself converging uniformly to the Cantor function C : [0, 1] −→ [0, 1]; for basic information on this function, see [5] and [16,Section 2.7,. A result of Božin and Mateljević shows that, via the Poisson transformation, an appropriately modified variant of the mapping C induces a harmonic homeomorphism of the unit disk D onto itself which is not quasiconformal (see Proposition 4.3). However, this harmonic homeomorphism will be seen to arise as the uniform limit of harmonic quasiconformal automorphisms of D, thus implying that HQ(D) cannot be complete. First of all, an approximation procedure for the Cantor function C in terms of a certain recursively defined sequence, which will be of central importance, is stated (see [5,Proposition 4.2,p. 9]): C(x) =      1 2 C(3x), 0 ≤ x ≤ 1 3 1 2 , 1 3 < x < 2 3 1 2 + 1 2 C(3x − 2), 2 3 ≤ x ≤ 1. Moreover, for arbitrary ψ 0 ∈ B([0, 1]), the sequence (ψ n ) n∈N0 defined by ψ n+1 (x) :=      1 2 ψ n (3x), 0 ≤ x ≤ 1 3 1 2 , 1 3 < x < 2 3 1 2 + 1 2 ψ n (3x − 2), 2 3 ≤ x ≤ 1(7) for n ∈ N 0 converges uniformly on [0, 1] to C. An approximation of the Cantor function using the recursively defined sequence given by (7) is shown in Figure 2. Basically, the principal idea of the approximation procedure and the mappings ψ n is that the initial mapping ψ 0 is "copied" and gets "duplicated in a scaled fashion", being added to the graph of ψ n more and more times as the index increases. This is visualized by the right-hand picture in Figure 2: In the first step (in blue), the scaled initial mapping ψ 0 can be seen two times, namely on the intervals [0, 1 3 ] and [ 2 3 , 1]. After the second iteration (in orange), the mapping ψ 0 appears four times in a scaled manner. Finally, in the third step (in yellow), the appropriately scaled version of ψ 0 is present eight times. In particular, it becomes obvious that all continuity and differentiability questions regarding ψ n depend solely on the behaviour of the initial mapping ψ 0 (and eventually existing derivatives) at the boundary points x = 0 and x = 1 of the starting interval. Furthermore, in Lemma 4.2, the stated approximation part and the related uniqueness of C is based on Banach's Contraction Principle (see [16, p. 216]). The following Proposition contains the mentioned result of Božin/Mateljević concerning a harmonic homeomorphism of D which fails to be quasiconformal (see [3, Furthermore, ψ 0 is strictly increasing on (0, 1) and leaves the boundary points fixed -in other words, ψ 0 maps [0, 1] homeomorphically onto itself. Lemma 4.2 implies that the corresponding sequence (ψ n ) n∈N0 defined via (7) converges uniformly on [0, 1] to the Cantor function C, and by construction, it is ψ n ∈ C 2 ([0, 1]) for every n ∈ N 0 due to (8). Transferring the ψ n to the interval [0, 2π] via ϕ n (t) := π ψ n t 2π Figure 2: Left: An approximation of the Cantor function C using the recursively defined sequence (ψ n ) n described in Lemma 4.2. The initial function is given by ψ 0 (x) = 6x 5 − 15x 4 + 10x 3 for x ∈ [0, 1], and for the approximation, the index value n = 15 was chosen. Right: The first three function in the approximation sequence (ψ n ) n : ψ 1 in blue, ψ 2 in orange and ψ 3 in yellow. + t 2π , t ∈ [0, 2π],(9) yields a sequence (ϕ n ) n of C 2 -homeomorphism of [0, 2π] onto itself. Accordingly, this sequence (ϕ n ) n clearly converges uniformly on [0, 2π] to the mapping ϕ C defined in Proposition 4.3. As a next step, the mappings ϕ n and ϕ C are extended to all of R by setting ϕ n (t + 2kπ) := ϕ n (t) + 2kπ (10) for k ∈ Z and t ∈ [0, 2π], yielding a sequence (ϕ n ) n∈N0 ⊆ C 2 (R); likewise, the mappings ψ n and C are extended in the same manner (the extended mappings are denoted by the same letter). In particular, the ϕ n are differentiable with ϕ n (t + 2kπ) = ϕ n (t) for all t ∈ R by construction, i.e. the ϕ n (and thus the ϕ n as well) are continuous 2π-periodic mappings. Lifting these mappings to the unit circle by γ n (e it ) := e iϕn(t) for t ∈ [0, 2π] and each n ∈ N yields orientation-preserving homeomorphisms of ∂D onto itself, hence the harmonic extensions P[γ n ] by means of the Radó-Kneser-Choquet Theorem 2.2 are (orientation-preserving) harmonic homeomorphisms of D onto itself. In order to visualize this procedure and to illustrate a concrete mapping of the described type, the mapping behaviour of the harmonic unit disk automorphism P[γ 4 ] is visualized in Figure 3. Moreover, by Pavlović's characterization result stated in Proposition 3.8, the mappings P[γ n ] in fact define quasiconformal automorphisms of D, which can be seen as follows: It is ϕ n ∈ C 2 (R) strictly increasing with ϕ n (t + 2π) = ϕ n (t) + 2π for all t ∈ R by construction, see (9) and (10). Furthermore, as C 2 -homeomorphisms, each mapping ϕ n is Lipschitz-continuous, and the corresponding inverse mappings ϕ −1 n are also C 2 by construction due to (8), thus also Lipschitz-continuous. In consequence, the mappings ϕ n are bi-Lipschitz. Hence, in view of Proposition 3.8(ii), the Hilbert transformation condition (c) needs to be verified. Therefore, let x ∈ R, then it is |ϕ n (x + t) − ϕ n (x − t)| ≤ L n · |x + t − (x − t)| = 2L n |t|, since ϕ n ∈ C 2 (R), thus ϕ n is Lipschitz-continuous on R with Lipschitz constant L n ∈ R + . This yields π 0 + ϕ n (x + t) − ϕ n (x − t) t dt ≤ π 0 + |ϕ n (x + t) − ϕ n (x − t)| t dt ≤ π 0 + 2L n t t dt = 2πL n < +∞, and now Remark 3.6(i) implies that H(ϕ n ) is (essentially) bounded for ϕ n , n ∈ N 0 (note that the conclusion could also have been drawn from Lemma 3.7 since ϕ n and ϕ n are periodic and continuous on R). Thus Proposition 3.8 shows that the mappings P[γ n ] are quasiconformal automorphisms of D. Finally, it will be shown that the mappings P[γ n ] converge uniformly on D to the non-quasiconformal mapping h C in question (from |ϕ n (t) − ϕ C (t)| . Since ϕ n converges uniformly on [0, 2π] to ϕ, the claim follows: The sequence (P[γ n ]) n∈N0 in HQ(D) converges uniformly to h C ∈ HQ(D), showing that the space HQ(D) is incomplete. d sup (f, g) := sup z∈D |f (z) − g(z)| for (bounded) mappings f, g : D −→ C. Definition 2 . 1 . 21Let G C be a bounded, simply connected domain, then Q(G) := f : G −→ G f is a quasiconformal mapping of G onto G Several central topological properties of Q(G) in the topology of uniform convergence induced by d sup were studied in [2]. In the unit disk D, a particularly specialized subclass of such mappings arises by demanding the additional property of harmonicity, i.e. by considering HQ(D) := f ∈ Q(D) f is harmonic ( Figure 1 : 1Preimage (left) and image (right) of concentric circles and radial rays (top) as well as of an Euclidean grid (bottom) in D under the harmonic extension of γ defined by (3). which is easily seen to map the interval [0, 1] continuously, but not injectively onto itself while keeping the endpoints x = 0 and x = 1 fixed. Transferring φ to the interval [0, 2π] by conjugating it via the mapping x −→ t = 2πx yields a function ψ ∈ C([0, 2π]) with the same properties. Consequently, the mapping γ(e it ) = e i ϕ(t) HQ(D) is no semigroup with respect to composition of mappings. In particular, HQ(D) is no subgroup of Q(D). Proposition 3. 1 . 1Let (f n ) n∈N be a sequence of harmonic mappings on a domain G ⊆ C. (i) If (f n ) n converges locally uniformly on G to some function f , then f is harmonic (Weierstraß-type Theorem, see [1, Theorem 1.23, p. 16]). (ii) If additionally, all f n are injective and the sequence (f n ) n converges locally uniformly on G to f , then f is either injective, a constant mapping, or f (G) lies on straight line (Hurwitz-type Theorem, see [4, Theorem 1.5]). The first result concerning certain topological aspects of HQ(D) is given in Theorem 3.2. The space HQ(D) is separable and non-compact. As a subspace, it is closed in Q(D). Proof. If (f n ) n∈N ⊆ HQ(D) converges uniformly on D to f ∈ Q(D), then f is harmonic by Proposition 3.1(i), hence f ∈ HQ(D). Therefore, HQ(D) is closed in Q(D). As for the separability of HQ(D), it suffices to observe that the ambient metric space Q(D) is separable by [2, Theorem 6, p. 5]. The claimed separability of HQ(D) is then implied by the fact that subspaces of separable metric spaces are also separable. In order to see that HQ(D) is a non-compact space, suppose the contrary, i.e. HQ(D) is compact in the uniform topology. Due to the completeness of Σ(D) (see [8, Satz 1, p. 229]), the space Σ(D) is closed in the ambient space HQ(D). However, this yields that Σ(D) would also be compact as a closed subspace of the compact space HQ(D), contradicting the non-compactness of Σ(D) (see [8, Satz 1, p. 229]). Hence HQ(D) is not compact. The space HQ(D) is dense-in-itself, i.e. it does not contain any isolated points. Proof. The space Q(D) is a topological group (see [2, Theorem 3, p. 3]) and not discrete ([2, Theorem 12, p. 8]); in particular, Σ(D) is not discrete (as already noticed in [8, p. 230]). Hence, let h ∈ HQ(D) be arbitrary and choose a sequence (f n ) n∈N in Σ(D)\{id D } converging to id D . Then, for each n ∈ N, the mapping g n := h•f n is harmonic and quasiconformal, thus (g n ) n is a sequence in HQ(D). The continuity of left multiplication in the topological group Q(D) yields d sup (g n , h) = d sup (h • f n , h) n→∞ −→ 0 due to d sup (f n , id D ) The space HQ(D) is perfect, i.e. it is closed in Q(D) and contains no isolated points. ] and [ 17 , 17Vol. I, p. 52]) Lemma 3.7.For periodic ϕ ∈ L 1 ([0, 2π]), the Hilbert transformation H(ϕ)(x) exists for almost every x ∈ R. Furthermore, H(ϕ)(x) exists if ϕ (x) exists and is finite at x ∈ R. Now the connection between the Hilbert transformation H and HQ(D) will be clarified. By the Radó-Kneser-Choquet Theorem 2.2, every mapping γ = e iϕ ∈ H + (∂D) defines a harmonic automorphism of D by means of the Poisson transformation P[e iϕ ] (this statement remains true even for γ ∈ H * (∂D), see also[11, (1.3), p. 338]). The question for whether this harmonic extension is quasiconformal has been answered in a characterizing manner by Pavlović in[14], and is stated in (see[15, Theorem 10.18, p. 305]) Proposition 3.8. Let f : D −→ D be an orientation-preserving harmonic homeomorphism of the unit disk onto itself. Then the following conditions are equivalent:(i) f is quasiconformal, i.e. f ∈ HQ(D);(ii) f = P[e iϕ ], where the function ϕ has the following properties: (a) ϕ(t + 2π ) 2π− ϕ(t) = 2π for all t ∈ R; 2π]) is convex. In particular, H + qc is path-connected in the Banach space C([0, 2π]). Proof. Let ϕ 1 , ϕ 2 ∈ H + qc , λ ∈ [0, 1] and consider the mapping λϕ 1 The space HQ(D) is path-connected. 4 Incompleteness of HQ(D): Statement, auxiliary results and proofThis section is concerned with the proof of the following statement: The space HQ(D) is incomplete. Let B([0, 1]) denote the Banach space of bounded real-valued functions on [0, 1]. The Cantor function C is the unique element of B([0, 1]) for which For t ∈ [0, 2π], define ϕ C (t) := π(C( t 2π ) + t 2π ) and γ C (t) := e iϕ C (t) . Then the function h C := P[γ C ] is a harmonic homeomorphism of D onto itself that is not quasiconformal.Now all preparations are made in order to prove the claim of Theorem 4.1: Proof of Theorem 4.1. Consider the polynomial functionψ 0 : [0, 1] −→ R, x −→ ψ 0 (x) := 6x 5 − 15x 4 + 10x 3whose first and second derivatives satisfy ψ 0 (0) = ψ 0 (1) = 0 = ψ 0 (0) = ψ 0 (1). Figure 3 : 3Left-hand side: Preimage of concentric circles and radial rays (top) and of an Euclidean grid (bottom) in the unit disk. Right-hand side: Image of the concentric circles, radial rays and the Euclidean grid in D under the mapping P[γ 4 ]. Proposition 4.3), which is essentially based on the same idea as the proof of Theorem 3.10: Applying the Maximum Principle for harmonic functions to P[γ n ] − h C yields sup z∈D |P[γ n ](z) − h C (z)| = max z∈∂D |P[γ n ](z) − P[γ C ](z)| = max t∈[0,2π] e iϕn(t) − e iϕ C (t) ≤ max t∈[0,2π] Harmonic Function Theory, 2. edition. S Axler, P Bourdon, W Ramey, Graduate Texts in Mathematics. 137Springer-VerlagAXLER, S., BOURDON, P., RAMEY, W.: Harmonic Function Theory, 2. edition. Graduate Texts in Mathematics, vol. 137. Springer-Verlag New York, 2001. F Biersack, W Lauf, Topological Properties of Quasiconformal Automorphism Groups, The Journal of Analysis. to appearBIERSACK, F., LAUF, W.: Topological Properties of Quasiconformal Automorphism Groups, The Jour- nal of Analysis, to appear. Some counterexamples related to the theory of HQC mappings. V Božin, M Mateljević, Filomat. 244BOŽIN, V., MATELJEVIĆ, M.: Some counterexamples related to the theory of HQC mappings. Filomat, 24(4), pp. 25-34 (2010). Univalent Harmonic Mappings in the Plane. D Bshouty, W Hengartner, Annales Universitatis Mariae Curie-Sklodowska, Sectio A -Mathematica. 483BSHOUTY, D., HENGARTNER, W.: Univalent Harmonic Mappings in the Plane. Annales Universitatis Mariae Curie-Sklodowska, Sectio A -Mathematica, 48(3), pp. 12-42 (1994). The Cantor function. O Dovgoshey, O Martio, V Ryazanov, M Vuorinen, Expositiones Mathematicae. 241DOVGOSHEY, O., MARTIO, O., RYAZANOV, V., VUORINEN, M.: The Cantor function. Expositiones Mathematicae, 24(1), pp. 1-37 (2006). Harmonic Mappings in the Plane. P Duren, Cambridge Tracts in Mathematics. 156Cambridge University PressDUREN, P.: Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics, no. 156. Cambridge University Press, 2004. A Variational Method for Harmonic Mappings onto Convex Regions. P Duren, G Schober, Complex Variables. 9DUREN, P., SCHOBER, G.: A Variational Method for Harmonic Mappings onto Convex Regions. Com- plex Variables, 9(2-3), pp. 153-168 (1987). Über Räume konformer Selbstabbildungen ebener Gebiete. D Gaier, Mathematische Zeitschrift. 1872GAIER, D.:Über Räume konformer Selbstabbildungen ebener Gebiete. Mathematische Zeitschrift, 187(2), pp. 227-257 (1984). . F P Gardiner, N Lakic, Quasiconformal Teichmüller Theory. Mathematical Surveys and Monographs. 76American Mathematical SocietyGARDINER, F.P., LAKIC, N.: Quasiconformal Teichmüller Theory. Mathematical Surveys and Mono- graphs, vol. 76. American Mathematical Society, 2000. Quasiconformal and harmonic mappings between Jordan domains. D Kalaj, Mathematische Zeitschrift. 2602KALAJ, D.: Quasiconformal and harmonic mappings between Jordan domains. Mathematische Zeitschrift, 260(2), pp. 237-252 (2008). Harmonic automorphisms of the unit disk. J G Krzyz, M Nowak, Journal of Computational and Applied Mathematics. 1051-2KRZYZ, J.G., NOWAK, M.: Harmonic automorphisms of the unit disk. Journal of Computational and Applied Mathematics, 105(1-2), pp. 337-346 (1999). Univalent Functions and Teichmüller Spaces. O Lehto, Graduate Texts in Mathematics. 109Springer-VerlagLEHTO, O.: Univalent Functions and Teichmüller Spaces. Graduate Texts in Mathematics, vol. 109. Springer-Verlag New York, 1987. Quasiconformal Mappings in the Plane, 2. edition. Die Grundlehren der mathematischen Wissenschaften. O Lehto, K I Virtanen, Springer-Verlag126Berlin Heidelberg New YorkLEHTO, O., VIRTANEN, K.I.: Quasiconformal Mappings in the Plane, 2. edition. Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer-Verlag Berlin Heidelberg New York, 1973. Boundary Correspondence under Harmonic Quasiconformal Homeomorphisms of the Unit Disk. M Pavlović, Annales Academiae Scientiarum Fennicae. Mathematica. 27PAVLOVIĆ, M.: Boundary Correspondence under Harmonic Quasiconformal Homeomorphisms of the Unit Disk. Annales Academiae Scientiarum Fennicae. Mathematica. vol. 27, pp. 365-372 (2002). Function Classes on the Unit Disk: An Introduction. M Pavlović, De Gruyter Studies in Mathematics. 52Walter de GruyterPAVLOVIĆ, M.: Function Classes on the Unit Disk: An Introduction. De Gruyter Studies in Mathe- matics, vol. 52. Walter de Gruyter, Berlin/Boston, 2014. . H L Royden, P M Fitzpatrick, Prentice HallReal Analysis, 4. editionROYDEN, H.L., FITZPATRICK, P.M.: Real Analysis, 4. edition. Prentice Hall, 2010. A Zygmund, Trigonometric Series, 3. edition, Volumes I & II combined, with a foreword by R. Fefferman. Cambridge Mathematical Librar Series. Cambridge University PressZYGMUND, A.: Trigonometric Series, 3. edition, Volumes I & II combined, with a foreword by R. Fefferman. Cambridge Mathematical Librar Series. Cambridge University Press, 2002.

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{'abstract': 'We study the class HQ(D), the set of harmonic quasiconformal automorphisms of the unit disk D in the complex plane, endowed with the topology of uniform convergence. Several important topological properties of this space of mappings are investigated, such as separability, compactness, path-connectedness and completeness.', 'arxivid': '2304.03993', 'author': ['Florian Biersack florian.biersack@mathematik.uni-wuerzburg.de \nUniversity of Würzburg\nChair for Complex Analysis\nEmil-Fischer-Strasse 4097074Würzburg, BavariaGermany\n'], 'authoraffiliation': ['University of Würzburg\nChair for Complex Analysis\nEmil-Fischer-Strasse 4097074Würzburg, BavariaGermany'], 'corpusid': 258049289, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11128, 'n_tokens_neox': 9535, 'n_words': 5468, 'pdfsha': '3f0b57e8b44092031d615ae4cbe8ede43a5f74f2', 'pdfurls': ['https://export.arxiv.org/pdf/2304.03993v1.pdf'], 'title': ['Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology', 'Topological Characteristics of Harmonic Quasiconformal Unit Disk Automorphisms in the Uniform Topology'], 'venue': []}

arxiv

Radial Oscillations in Neutron Stars with Delta Baryons Ishfaq A Rather Centro de Astrofísica e Gravitação-CENTRA Instituto Superior Técnico Universidade de Lisboa 1049-001LisboaPortugal Kauan D Marquez Departamento de Física -CFM Universidade Federal de Santa Catarina CEP 88.040-900FlorianópolisSCBrazil; Grigoris Panotopoulos Departamento de Ciencias Físicas Universidad de la Frontera Casilla 54-D4811186TemucoChile Ilídio Lopes Centro de Astrofísica e Gravitação-CENTRA Instituto Superior Técnico Universidade de Lisboa 1049-001LisboaPortugal Radial Oscillations in Neutron Stars with Delta Baryons We investigate the effect of ∆ baryons on the radial oscillations of neutron and hyperon stars, employing a density-dependent relativistic mean-field model. The spin-3/2 baryons are described by the Rarita-Schwinger Lagrangian density. The baryon-meson coupling constants for the spin-3/2 decuplet and the spin-1/2 baryonic octet are calculated using a unified approach relying on the fact that the Yukawa couplings present in the Lagrangian density of the mean-field models must be invariant under the SU(3) and SU(6) group transformations. We calculate the 20 lowest eigenfrequencies and corresponding oscillation functions of ∆-inclusive nuclear (N+∆) and hyperonic matter (N+H+∆) by solving the Sturm-Liouville boundary value problem and also verifying its validity. We see that the lowest mode frequencies for N+∆ and N+H EoSs are higher as compared to the pure nucleonic matter because of the deltas and hyperons present. Furthermore, the separation between consecutive modes increases with the addition of hyperons and ∆s. I. INTRODUCTION The densest observed stars in the universe, neutron stars (NSs), serve as natural laboratories for the investigation of cold dense nuclear matter. The equation of state (EoS) of nuclear matter is the decisive factor that, theoretically, governs the structure and properties of NSs. To ensure the stability of nuclear matter inside NSs, they contain some amount of protons as well, apart from the neutrons. Because of the strong interaction's nonperturbative nature, we still know relatively little about the EoS of dense nuclear matter, especially at densities considerably higher than the nuclear saturation density (ρ 0 ), where exotic degrees of freedom are likely to exist. Nearly all theoretical descriptions of these objects encompass the entire spin-1/2 baryon octet i.e, nucleons and hyperons [1]. These investigations gave rise to the widely discussed "hyperon puzzle". Hyperons soften the EoS, leading to a lower maximum mass on the mass-radius curve of stars. [2]. Within the relativistic mean-field approach, Glendenning [3] considered several exotic degrees of freedom such as hyperons, kaons, and delta baryons (∆) in the NS matter. With the coupling parameters chosen, he found that the ∆ baryons could be present only at densities ≈ 10 ρ 0 inside the NSs. However, recent studies have shown that with the proper couplings between ∆ baryons and nucleons constrained by several experiment measurements, they might be present inside the NSs [4][5][6][7][8][9] and that they could in fact make up a large fraction of the baryons in NS matter and have a significant effect on the properties of NSs. Also, since ∆ baryons are approximately 30% heavier than the nucleons (m ∆ = 1232 MeV) and even lighter than the heaviest spin-1/2 baryons of the octet (Ξ hyperons), it is reasonable to expect the ∆ baryons to exist inside NSs at almost the same density range as the hyperons (≈ 2-3ρ 0 ). We have so far been capable of investigating the properties of dense matter under extremely difficult circumstances thanks to the recent accomplishment of gravitational wave (GW) detection by LIGO and Virgo Collaborations (LVC) of a binary neutron star (BNS) merger GW170817 event [10,11]. GWs produced by the coexistence of BNS merger events provide enough information to significantly constrain the EoS and the internal composition of NSs [10][11][12][13][14]. The oscillating NSs also emit GWs with several frequency modes that can be used to investigate the internal constituents and hence various properties of the star [15,16]. Following their formation in the supernovae, oscillating NSs emit a range of frequencies depending on the restoring force and there are numerous mechanisms that could be the diverse causes of these oscillations [17][18][19]. Overall, oscillations can be divided into two categories: radial and non-radial. In a pioneering work, Chandrasekhar [20,21] investigated the radial oscillations of stellar models. Importantly, radial oscillation characteristics can reveal details regarding the stability and EoS of compact stars. As radial oscillations cannot produce GWs on their own, their detection is rather difficult. They are linked to non-radial oscillations, which amplify GWs and improve the likelihood of detecting them [22,23]. However, Chirenti et al. [24] observed that in the post-merger event of BNS, a hyper-massive NS is created along with the emission of a short gamma-ray burst (SGRB), which could be influenced by radial oscillations. The high-frequency oscillations of the hyper-massive NS in the range of 1-4 kHz could be observed. Similar to different families of modes arising from different physical origins as described in Ref. [25], the radial oscillation modes can also be categorized into two families that are largely independent of one another. One family resides mostly in the neutron star's high-density core, whereas the other resides primarily in its low-density en-velope [26]. The significant shift in the matter's stiffness at the neutron drip point causes a "wall" in the adiabatic index that separates the two regions. Given that it is related to the neutron drip point, which is a part of the low-pressure regime and is the same for all EoSs, this wall effect is present for any realistic EoS. We investigate various radial oscillations of NSs with different matter compositions in this work. Several studies on the investigation of various radial oscillations of NSs with different exotic phases such as dark matter and deconfined quark matter have already been carried out [25,[27][28][29][30][31]. But the radial oscillation of NSs with ∆ baryons (∆-inclusive nucleonic stars) and hyperon stars with ∆ baryons (∆-inclusive hyperonic stars) is being studied for the first time. The neutron star EoS at supra-nuclear densities has been constructed using a variety of models with a range of parameterizations. The saturation properties of highly dense nuclear matter have been extensively studied using density functional theories (DFT), in which the nucleonnucleon interaction is effectively defined by fitting ground state properties of finite nuclei [32][33][34][35][36][37]. From many-body theories, the nuclear matter EoS at saturation density is well constrained. The properties of neutron stars are interpreted by extrapolating these EoSs to densities several times the nuclear saturation density. The density-dependent relativistic mean-field (DD-RMF) model [38] is a widely used and successful model with the advantage that the self-and cross-coupling of various mesons in the RMF model are replaced by the densitydependent nucleon-meson coupling constants. The results produced by the density-dependent coupling constants are comparable to those of other models and allow for consistent measurement of NS properties. It takes into account the Dirac-Brueckner model's characteristics and uses microscopic interactions at varying densities as an input. DD-RMF parameter sets such as DD-ME1 [39], DD-ME2 [40], and DD-MEX [41] generate a very stiff EoS and hence predict a massive NS with a maximum mass in the range of 2.3-2.5M ⊙ [42,43]. Our work is organized as follows: in Section (II.1), the EoS for the DD-RMF model along with the addition of ∆ baryons and the couplings used is discussed. The Sturm-Liouville eigenvalue equations for the internal structure and radial oscillations of NSs are introduced in Section (II.2). In Section (III), the EoS and the Mass-Radius profile for different compositions of the matter are discussed in Section (III.1). Section (III.2) describes the numerical results obtained for NSs and hyperon stars with ∆ baryons. The summary and concluding remarks are finally given in Section (IV). II. THEORETICAL FRAMEWORK AND FORMALISM II.1. Equation-of-state In this study, the hadronic matter composing the neutron stars is described within a relativistic mean-field approach with density-dependent couplings (DD-RMF). This type of model is shown to be very consistent in the description of nuclear matter experimental properties and also holds when astrophysical constraints are imposed [44][45][46]. The interaction is described considering nucleons (and other hadrons) interacting through the exchange of virtual mesons, and the DD-RMF model adopted here considers the scalar meson σ, the vector mesons ω and ϕ (that carries hidden strangeness), isoscalars, and the isovector-vector meson ⃗ ρ. The Lagrangian density is the basic ansatz of any RMF theory and contains the contributions from free baryons and mesons together with the terms describing the interaction between them. The lagrangian of the relativistic model in the mean field approximation used to describe the hadronic interactions here is given by L RMF = b∈Hψ b iγ µ ∂ µ − γ 0 g ωb ω 0 + g ϕb ϕ 0 + g ρb I 3b ρ 03 − (m b − g σb σ 0 ) ψ b − i 2 b∈∆ψ bµ ε µνρλ γ 5 γ ν ∂ ρ − γ 0 (g ωb ω 0 + g ρb I 3b ρ 03 ) − (m b − g σb σ 0 ) ς µλ ψ bν + λψ λ (iγ µ ∂ µ − m λ ) ψ λ − 1 2 m 2 σ σ 2 0 + 1 2 m 2 ω ω 2 0 + 1 2 m 2 ϕ ϕ 2 0 + 1 2 m 2 ρ ρ 2 03(1) where the first sum represents the Dirac-type interacting Lagrangian for the spin-1/2 baryon octet (H = {n, p, Λ, Σ − , Σ 0 , Σ + , Ξ − , Ξ 0 }) and the second sum represents the Rarita-Schwinger-type interacting Lagrangian for the particles of the spin-3/2 baryon decuplet (∆ = ∆ − , ∆ 0 , ∆ + , ∆ ++ }), where ε µνρλ is the Levi-Cicita symbol, γ 5 = iγ 0 γ 1 γ 2 γ 3 and ς µλ = i 2 γ µ , γ λ . We point to the fact that spin-3/2 baryons are described by the Rarita-Schwinger Lagrangian density and that their vector-valued spinor has additional components when compared to the four components in the spin-1/2 Dirac spinors, but, as shown in [47], spin-3/2 equations of motion can be written compactly as the spin-1/2 ones in the RMF regime. The last sum describes the leptons admixed in the hadronic matter as a free non-interacting fermion gas (λ = {e, µ}), as their inclusion is necessary in order to ensure the β-equilibrium and charge neutrality essential to stellar matter. The remaining terms account for the purely mesonic part of the Lagrangian. In DD-RMF models, the coupling constants can be either dependent on the scalar density n s or the vector density n B , but usually, the vector density parameterizations are considered which influences only the self-energy instead of the total energy [38]. In this study, we use the DD-RMF parametrization known as DDME2 [48], where the meson couplings are scaled with the baryonic density factor η = n B /n 0 obeying the function g ib (n B ) = g ib (n 0 ) a i + b i (η + d i ) 2 a i + c i (η + d i ) 2(2) for i = σ, ω, ϕ and g ρb (n B ) = g ib (n 0 ) exp −a ρ η − 1 ,(3) for i = ρ. The model parameters are fitted from experimental constraints of nuclear matter at or around the saturation density n 0 , namely the binding energy B/A, compressibility modulus K 0 , symmetry energy S 0 , and its slope L 0 , shown in Table I [44,48]. The model-free parameters are fitted considering pure nucleonic (protons and neutrons only) matter. In order to determine the meson couplings to other hadronic species we define the ratio of the baryon coupling to the nucleon one as χ ib = g ib /g iN , with i = {σ, ω, ϕ, ρ}. In this work, we consider hyperons and/or deltas inclusive in the nucleonic matter and follow the proposal of [50] to determine their respective χ ib ratios. It is made through a unified approach relying on symmetry arguments such as the fact that the Yukawa couplings terms present in the Lagrangian density of the DD-RMF models must be invariant under SU(3) and SU(6) group transformations. Hence, the couplings can be fixed to reproduce the potentials U Λ = −28 MeV, U Σ = 30 MeV, U Ξ = −4 MeV and U ∆ ≈ −98 MeV in terms of a single free parameter α v . Our choice of α v = 1.0 for the baryon-meson coupling scheme corresponds to an unbroken SU(6) symmetry, and the values of χ ib are shown in Table II taking into account the isospin projections in the lagrangian terms [51]. [50]. b χ ωb χ σb I 3b χ ρb χ ϕb Λ 2/3 0.611 0 0.471 Σ − ,Σ 0 , Σ + 2/3 0.467 −1, 0, 1 -0.471 Ξ − , Ξ 0 1/3 0.284 −1/2, 1/2 -0.314 ∆ − , ∆ 0 1 1.053 −3/2, −1/2, 1/2, 3/2 0 From the Lagrangian, thermodynamic quantities can be calculated in the standard way for RMF models. The baryonic and scalar densities of a baryon of the species b are given, respectively, by n b = λ b 2π 2 k F b 0 dk k 2 = λ b 6π 2 k F 3 b ,(4) and n s b = λ b 2π 2 k F b 0 dk k 2 m * b k 2 + m * b 2 ,(5) with k F denoting the Fermi momentum, since we assume the stellar matter to be at zero temperature, and λ b is the spin degeneracy factor (2 for the baryon octet and 4 for the deltas). The effective masses are m * b = m b − g σb σ 0 .(6) The energy density is given by ε B = b γ b 2π 2 k F b 0 dkk 2 k 2 + m * b 2 + λ 1 π 2 k F λ 0 dkk 2 k 2 + m 2 λ + m 2 σ 2 σ 2 0 + m 2 ω 2 ω 2 0 + m 2 ϕ 2 ϕ 2 0 + m 2 ρ 2 ρ 2 03 . (7) The effective chemical potentials read µ * b = µ b − g ωb ω 0 − g ρb I 3b ρ 03 − g ϕb ϕ 0 − Σ r ,(8) where Σ r is the rearrangement term due to the density-dependent couplings Σ r = b ∂g ωb ∂n b ω 0 n b + ∂g ρb ∂n b ρ 03 I 3b n b + ∂g ϕb ∂n b ϕ 0 n b − ∂g σb ∂n b σ 0 n s b ,(9) and the µ b are determined by the chemical equilibrium condition µ b = µ n − q b µ e ,(10) in terms of the chemical potential of the neutron and the electron, with µ µ = µ e . The particle populations of each individual species are determined by Eq. (10) together with the charge neutrality condition i n i q i = 0, where q i is the charge of the baryon or lepton i. The pressure, finally, is given by P = i µ i n i − ϵ + n B Σ r ,(11) which receives a correction from the rearrangement term to guarantee thermodynamic consistency and energymomentum conservation [52,53]. II.2. Radial oscillations Einstein's equations of General Relativity govern the structure and dynamical evolution of NSs because of their intense gravitational field. Moreover, the static equilibrium structure-based Einstein field equation can be used to calculate the radial oscillation properties [54]. Consider a spherically symmetric system with only radial motion, where the metric is now time-dependent. For radial displacement ∆r with ∆P as the perturbation of the pressure, the small perturbation of the equations governing the dimensionless quantities ξ = ∆r/r and η = ∆P/P are defined as [26,55] ξ ′ (r) = − 1 r 3ξ + η γ − P ′ (r) P + E ξ(r),(12)η ′ (r) = ξ ω 2 r(1 + E/P )e λ−ν − 4P ′ (r) P − 8π(P + E)re λ + r(P ′ (r)) 2 P (P + E) + η − EP ′ (r) P (P + E) − 4π(P + E)re λ ,(13) where ω is the frequency oscillation mode and γ is the adiabatic relativistic index defined as γ = 1 + E P c 2 s ,(14) where c 2 s is the speed of sound squared c 2 s = dP dE c 2 .(15) The two coupled differential equations Eqs. 12 and 13 are supplemented with two additional boundary conditions, one at the center where r = 0, and another at the surface where r = R. The boundary condition at the center requires that η = −3γξ (16) must be satisfied. The equation Eq. (13) must be finite at the surface and hence η = ξ −4 + (1 − 2M/R) −1 − M R − ω 2 R 3 M(17) must be satisfied where M and R correspond to the mass and radius of the star, respectively. The frequencies are computed by ν =ω 2π (kHz),(18) whereω = ωt 0 is the dimensionless quantity computed at t 0 = 1 ms. These equations represent the Sturm-Liouville eigenvalue equations for ω. The solutions provide the discrete eigenvalues ω 2 n and can be ordered as ω 2 0 < ω 2 1 < ... < ω 2 n , where n is the number of nodes for a given NS. For a real value of ω, the star will be stable and for an imaginary frequency, it will become unstable. Also, since the eigenvalues are arranged in above defined manner, it is important to know the fundamental f -mode frequency (n = 0) to determine the stability of the star. Fig. 1 shows the variation of pressure with energy density (EoS) for an NS in beta-equilibrium and charge-neutral conditions. The pure nucleonic matter produces stiff EoS in the high-density region. The EoS softens when ∆s are added to the nuclear matter. This is because the presence of more degrees of freedom distributes the Fermi pressure among the many particles as a result of the inclusion of new particles, softening the EoS. However, we must point out the fact that only ∆ 0 and ∆ − baryons are considered in the ∆-inclusive nuclear and ∆-inclusive hyperonic matter because the inclusion of ∆ + and ∆ ++ baryons allows the nucleon effective mass to drop to zero for very low densities and hence precluding the neutron stars from achieving densities high enough to describe the maximum mass star. A more detailed explanation of such behavior of ∆ baryons is explained in Ref. [7], where the authors show that the increase of the exotic particle abundance adds to the negatively contributing term of the effective nucleon mass, through the scalar density dependence of the σ field. This issue was already known for some hyperon matter models, but the fact that the SU(6) coupling scheme enhances very strongly the abundance of resonances makes this behavior very sensible when ∆ baryons are present. As ∆ + and ∆ ++ baryons ought to be unfavored in the low and intermediate densities due to the charge neutrality condition, excluding them altogether is a possible workaround to that problem. While the hyperons further soften the EoS, the addition of ∆s in the hyperonic matter, N+H+∆, is more complex. As seen from Fig. 1, at lower densities, the N+H+∆ is softer than the N+H composition. With the increase in the density, the EoS with the N+H+∆ composition becomes stiffer than the N+H composition. III. NUMERICAL RESULTS AND DISCUSSION III.1. EoS and MR Profile The explanation for this is that the appearance of ∆ − baryon replaces a neutron-electron pair at the top of their Fermi seas which are favored over the light baryons because of the attractive potential. The electric charge-neutral particles, Λ 0 and ∆ 0 , appear later. The inset in Fig. 1 shows the number density vs pressure variation for different matter compositions. The joint constraints from the heavy-ion collision (HIC) experiments and multi-messenger astrophysics (Astro), orange (68%), and green (95%) credible ranges are also shown [56]. As we can see, the EoSs nearly satisfy the joint constraints at a lower density. The appearance of delta baryons ensures that all the EoSs satisfy these constraints. For the unified EoS, the Baym-Pethick-Sutherland (BPS) EoS [57] is used for the outer crust part. For the inner crust, the EoS in the non-uniform matter is generated by using the DD-ME2 parameter set in Thomas-Fermi approximation [58][59][60]. density for different matter compositions. For pure nucleonic matter, the γ curve increases to a peak value at low energy density and then drops smoothly. The presence of hyperons, especially Λ 0 , softens the EoS and the value of γ drops at around ≈ 350 MeV/fm 3 , and each following peak can be associated with the onset of a new particle species. For nuclear and hyperonic matter with ∆ baryons, the value of γ drops sharply at around ≈ 250 MeV/fm 3 due to the onset of ∆ − baryons. But as the density increases, the γ also increases and becomes larger than the pure nucleonic matter. This large increase in the behavior of γ is not seen in the hyperonic matter. For ∆-inclusive hyperonic matter, we see a huge drop in the value γ due to the ∆ − threshold followed by a quick increase and then a new drop due to the onset of Λ 0 hyperon. 3 depicts the behavior of the speed of sound squared as a function of energy density for different compositions of the matter studied in this work. The speed of sound is an important quantity that conveys information about shear viscosity, tidal deformability, and gravitational wave signatures [61,62]. It is defined as the derivative of pressure with respect to energy density with its square defined by Eq. (15). It can also be interpreted as a measurement of the stiffness of the EoS, with a higher speed yielding a higher pressure at a given energy density and allowing a larger star mass for a given radius. Thermodynamic stability ensures that c 2 s > 0 and causality implies an absolute bound c 2 s ≤ 1. For very high densities, perturbative QCD findings anticipate an upper limit of c 2 s = 1/3 [63]. The two solar mass requirements, according to several studies [63][64][65], necessitates a speed of sound squared that exceeds the conformal limit (c 2 s = 1/3), revealing that the matter inside of NS is a highly interacting system. From Fig. 3, we can see a very large value of the c 2 s for the pure nucleonic matter. When different particle compositions are considered, one can see the kinks corresponding to the onset of a new particle species at the same point as the ones in the adiabatic index curves. The conformal limit is violated in the case of pure nucleonic and ∆-inclusive nuclear matter. Also, the curve for N+H+∆ composition predicts a higher value of the speed of sound squared at intermediate densities because of the early appearance of ∆ − particles, as explained earlier. With the solutions from the TOV equations for different EoSs, Fig. 4 displays the mass-radius profile for different compositions of the matter. For pure nucleonic matter, a maximum mass of 2.46 M ⊙ is achieved at a radius of 12.05 km. With the ∆ baryons present in the star, the maximum mass and the corresponding radius decrease to a value of 2.24 M ⊙ and 11.87 km, respectively. The decrease in the mass and the radius of ∆-inclusive nuclear matter depends upon the value of α v . The higher the value of α v , the lower the maximum mass, corresponding radius, and the radius at the canonical mass. In our case with α v = 1.0, the radius at the canonical mass decrease from 13.29 km for the pure nucleonic matter to 12.82 km for ∆-inclusive nuclear matter. The presence of hyperons softens the EoS and hence the maximum mass decrease to a value of 1.87 M ⊙ with the corresponding radius of 12.09 km. For the ∆-inclusive hyperonic matter at α v =1.0, The stiffness of the EoS predicts a maximum mass of 1.90 M ⊙ with a radius of 11.90 km. The constraints on the mass and the radius from various measurements [66][67][68][69] are very well satisfied by the N and N+∆ profile, while the N+H and N+H+∆ profiles nearly satisfy the description of the PSR J0740+6620 for mass and radius [66,67]. The joint constraints from heavy-ion collision (HIC) experiments and multi-messenger astrophysics are very well satisfied by all the mass-radius profiles. Since the maximum mass for N+H and N+H+∆ EoSs is lower than the mass of the super heavy pulsar PSR J0740+6620, these EoSs can be ruled out. However, for the comparison, we keep the results and calculate the radial profiles with N+H and N+H+∆ EoSs also to see the effect of delta The dotted, dashed, and dashdotted lines represent the MR profile for ∆-inclusive nuclear matter, hyperons, and ∆-inclusive hyperonic matter, respectively. The 68% (violet) and 95% (turquoise) credible regions for mass and radius are inferred from the analysis of PSR J0740+6620 [66,67]. For PSR J0030+0451, the indigo dotted region is for 68% credibility while the cyan dotted region is for 95% credibility [68]. The grey upper (brown lower) shaded region corresponds to the higher (smaller) component of the GW170817 event [69]. The joint constraints from HIC experiments and multi-messenger astrophysics with 68% (orange) and 95% (green) credible ranges are also shown [56]. baryons on the hyperonic matter. III.2. Radial Profiles The radial displacement perturbation profile ξ(r) and pressure perturbation profile η(r) as a function of dimensionless radius distance r/R is plotted in Figs. 5 and 6, respectively. These profiles are plotted for different particle compositions, pure nucleonic (upper left), ∆-inclusive nucleonic (upper right), hyperonic (lower left), and ∆inclusive hyperonic (lower right) matter at the corresponding maximum masses (with different central densities). Only the f -mode (n = 0), lower order p-modes (n = 1, 2, 3), intermediate (n = 9, 10, 11), and high excited modes (n = 17,18,19) are shown. In the region 0 < r < R, exactly n nodes are obtained for the nth mode both for ξ and η profiles, thereby following the Sturm-Liouville system. From Fig. 5, one can see that the amplitude of ξ n (r) for each frequency mode ν n is larger near the center and small at the surface. The lower modes show a smooth drop in their profiles while the higher modes depict small oscillations which would become large for higher modes. For ∆-inclusive nucleonic matter, we see a small kink at around r/R = 0.8 for n = 0 mode. For hyperonic matter, the kink at the same node is large and present at around r/R = 0.3. These kinks in ξ(r) represent the emergence of new exotic particles which provides a discontinuity in the adiabatic index, that appears in Eq. (12) explicitly. One can see a rapid sign change near the center of the star which along with the amplitude decrease as one moves toward the surface of the star. From Fig. 6, it's observed that the amplitude of η n (r) is larger near the center and also at the surface of the star. Although the η oscillations are directly proportional to the Lagrangian pressure variation ∆P , the amplitude of η n (r) for consecutive n have large amplitudes near the surface, and hence the contribution from η n+1 -η n cancels out because of the opposite signs, thereby satisfying the condition that P (r = R) = 0. This implies that η n+1 -η n and also ξ n+1 -ξ n are more sensitive to the star's core. As a result, the measurement of ∆ν n = ν n+1 -ν n is an observational imprint of this star's innermost layers. Table III displays the frequencies, ν in kHz, of the first 20 nodes for pure nucleonic matter, ∆-inclusive nucleonic matter, hyperonic matter, and ∆-inclusive hyperonic matter, respectively. All these frequencies are obtained at the corresponding maximum masses of the EoSs. The node n = 0 corresponds to the f -mode frequency while the others correspond to the lower and highly excited p-modes. The frequency of the f -mode for pure nucleonic EoS is lower as compared to the other EoSs with deltas and hyperons. Fig. 7 shows the frequencies of radially oscillating NS with different matter compositions, as a function of central energy density for lower radial modes, n = 0, 1, 2, and 3. It is clear from the figure that for the same core density, stellar models with softer EoSs exhibit higher f -mode frequencies than those of stiffer EoSs. The stellar models of softer EoSs are typically linked to larger average densities and more centrally compressed stars. The star is getting closer to its stability limit as the center density rises and the f -mode frequency (n = 0) begins to shift toward zero at the same moment. An eigenmode with zero frequency is a characteristic of the stability limit itself. The f -mode frequency of N+∆ and N+H EoSs is higher as compared to the pure nucleonic matter because of the delta and hyperonic that make the EoS softer. Since the N+H+∆ is stiffer than the N+H EoS, the corresponding f -mode frequency is lower. Higher modes oscillate more frequently than lower stable modes do, and for all modes, this frequency appears to decrease as the center energy density approaches the minimum value of the specific star model. The explanation for this comes from the fact that the NSs at very high densities can be approximated as being homogeneous and thus the angular frequency ω 2 0 follows the relation ω 2 0 = ρ(4 − 3γ) [27,70]. From Fig. 7, it is observed that with ∆s in the pure nucleonic and hyperonic matter, the higher modes show small kinks. This illustrates an essential observation that leads to a series of "avoided crossings" between the various modes: the frequencies of two subsequent modes from different families reject each other as they approach one another [25,26]. This "avoided crossing" is a characteristic of a realistic EoS [25] and is present in all four cases at lower densities. Fig. 8 displays the frequency difference ∆ν n = ν n+1ν n vs ν n in kHz for pure nucleonic (N), ∆-inclusive nucleonic (N+∆), hyperonic (N+H), and ∆-inclusive hyperonic matter (N+H+∆) matter. For pure nucleonic EoS, the separation between the modes is almost the same and there are no fluctuations at lower modes (n = 0). While the frequencies in the ∆-inclusive nucleonic matter are higher than pure nucleonic one, as one would expect because of the soft N∆ EoS than N, the difference between consecutive modes is also the same with minor fluctuations. With the hyperonic and ∆-inclusive hyperonic matter, they oscillate with higher frequencies, and the magnitude of ∆ν n is higher. This shows that the decrease in the central baryon density of the star, and, hence, of its mass leads to a large separation ∆ν n . We also observe the erratic fluctuations present in ∆ν n for N+H and N+H+∆ cases. These fluctuations arise from the significant variation of the speed of sound squared c 2 s or the relativistic adiabatic index γ on the transition layer separating the inner and outer core of the NS, which has an amplitude proportionate to the magnitude of the discontinuity. This is also due to the fact that we have considered a unified EoS in the present study. Although the radial oscillation for the lowest order mode (n = 0) is not highly impacted by the crust because it typically accounts for less than 10% of the stellar radius and the oscillation nodes are situated far inside the NS core. But other high oscillation modes are present in the crust of the star and hence the eigenfrequencies are modified (characterized by the peaks in the ∆ν n ) [28]. For a given EoS without crust, the variation in the frequency, ∆ν n , is smooth as discussed in Refs. [29][30][31]. IV. SUMMARY AND CONCLUSION In this work, we studied the radial oscillations of ∆inclusive neutron and hyperon stars employing the DD-RMF model with the DD-ME2 parameter set. The spin-3/2 baryons (∆s) are described using the Rarita-Schwinger Lagrangian density. For the spin-3/2 decuplet and the spin-1/2 baryonic octet, the baryon-meson coupling constants are calculated using the Clebsch-Gordan coefficients of the SU(3) group. The coupling constants of the scalar meson are fixed to replicate the known potential depth using a QHD model that essentially satisfies all requirements at the saturation density, thus allowing a unified approach to the coupling constants of hyperons and delta resonances. We studied the 20 lowest eigenfrequencies and corresponding oscillation functions of ∆-inclusive nu-clear (N+∆) and hyperonic matter (N+H+∆) by solving the Sturm-Liouville boundary value problem and also verifying its validity. For the hydrostatic equilibrium, we numerically solved the structural equations to obtain the mass-radius relationship of ∆-inclusive neutron and hyperon stars. The Sturm-Liouville equations were then solved for the perturbations imposing the necessary boundary conditions in order to examine radial oscillations of pulsating stars. This allowed us to calculate the frequencies of the modes as well as the related wave functions. 19 excited p-modes and the fundamental f -mode have been calculated. The addition of hyperons softens the EoS, decreasing the maximum mass and hence increasing the corresponding frequencies of the pulsating star. While the addition of ∆ baryons to nucleonic matter softens the EoS, it gets stiffer for the hyperonic matter. Compared to hyperonic matter, the adiabatic index γ for the ∆inclusive matter exhibits far more complex behavior. Due to the onset of the ∆ − , we observe a significant decrease in the value of the parameter followed by a rapid increase. This increases even more at intermediate densities than it does for the pure nucleonic case, a behavior not seen in the hyperonic case. We investigated the radial displacement perturbation profile ξ(r) and pressure perturbation profile η(r) with ∆-inclusive matter and found that they oscillate with exactly n nodes for the n th mode for all cases. The lowest modes show a smooth drop in their profiles while the higher modes depict lower oscillations which become large for higher modes. For ∆-inclusive nucleonic matter, small kinks are present for n = 0 mode. For hyperonic matter, the kink at the same node is found to be large and present at a small radius. These kinks in ξ(r) correspond to the emergence of new exotic particles which provides a discontinuity in the adiabatic index γ. We see that the lowest mode frequencies for N+∆ and N+H EoSs are higher as compared to the pure nucleonic matter because of the deltas and hyperons. Furthermore, the separation between consecutive modes increases with the addition of hyperons and ∆s. One of the main reasons for an abrupt change in the oscillatory property of the star as one undertakes a small variation in the star's stellar configuration is the precisely defined division of the star's inner core and outer crust. These two regions have different EoS and hence different oscillation properties. The stellar configurations such as the star's mass or central energy density essentially determine the oscillation frequencies of the crust and core pulsations. For the crust part, while the oscillation functions are expanding pretty steeply, they either decline or essentially remain constant for the NS core. The oscillatory properties of the star could be due to pulsations in the core or crust of the NS. The mass of the star affects the frequency spectrum. For moderately massive NSs, the oscillatory properties of the lower-order radial modes are determined by the core pulsations. If the stellar mass is low enough or the frequency is high enough, the star may be affected by the crust pulsations. The avoided crossing phenomena are closely related to the changes in matter's compressibility all across the star, which are modeled by the adiabatic index γ. The maximum of γ occurs close to the boundary between the core and crust as the stiffness of the matter increases outward. Unlike non-radial oscillations, radial oscillations do not possess a gravitational wave counterpart, making them a distinctive means to directly discern the influence of the EoS on the structure of the neutron star. This allows for a more straightforward analysis without the added complexities associated with gravitational wave measurements. Observing multiple radial modes, including the fundamental mode (f -mode) and pressure modes (p-modes), offers a precise means of measuring the radius of compact stars. This methodology has proven successful in other branches of Asteroseismology, demonstrating its reliability and applicability. The future observation of multiple radial oscillation modes, particularly through the computation of the large separation, holds the potential to identify the presence of delta baryons or hyperons within neutron stars. This would contribute to validating the existence or absence of these species in different regions of the NS. As a result, it is possible to probe the exotic degrees of freedom existing inside the NS using the high sensitivity of the enormous separation to the interior structure of the star. The properties of more realistic environments, such as temperature, rotation, and magnetic field, should also be incorporated in order to study radial oscillations in newborn NSs following supernova explosions or the merger of NSs. We leave these studies for future work. FIG . 1. (color online) Energy density and pressure variation for the given DD-ME2 parameter set. The solid line represents the pure nucleonic matter (N) while dotted, dashed, and dashdotted lines represent the EoS for ∆-inclusive nuclear matter (N+∆), hyperons (N+H), and ∆-inclusive hyperonic matter (N+H+∆), respectively, for αv = 1.0. The inset plot shows the number density vs pressure variation for different matter compositions. The orange (68%) and green (95%) shaded regions show the joint constraints from the heavy-ion collision (HIC) experiments and multi-messenger astrophysics (Astro)[56]. Fig. 2 FIG. 2 . 22(Color online) Adiabatic index as a function of energy density for the DD-ME2 parameter set with the pure nuclear matter, ∆-inclusive nuclear matter, hyperonic matter, and ∆-inclusive hyperonic matter. FIG . 3. (Color online) Speed of sound squared as a function of energy density for the DD-ME2 parameter set with the pure nuclear matter, ∆-inclusive nuclear matter, hyperonic matter, and ∆-inclusive hyperonic matter. The solid orange line represents the conformal limit c 2 s = 1/3. Fig. Fig. 3 depicts the behavior of the speed of sound squared as a function of energy density for different compositions of the matter studied in this work. The speed of sound is an important quantity that conveys information about shear viscosity, tidal deformability, and gravitational wave signatures [61, 62]. It is defined as the derivative of pressure with respect to energy density with its square defined by Eq. (15). It can also be interpreted as a measurement of the stiffness of the EoS, with a higher speed yielding a higher pressure at a given energy density and allowing a larger star mass for a given radius. Thermodynamic stability ensures that c 2 s > 0 and causality implies an absolute bound c 2 s ≤ 1. For very high densities, perturbative QCD findings anticipate an upper limit of c 2 s = 1/3 [63]. The two solar mass requirements, according to several studies [63-65], necessitates a speed of sound squared that exceeds the conformal limit (c 2 s = 1/3), revealing that the matter inside of NS is a highly interacting system. From Fig. 3, we can see a very large value of the c 2 s for the pure nucleonic matter. When different particle compositions are considered, one can see the kinks corresponding to the onset of a new particle species at the same point as the ones in the adiabatic index curves. The conformal limit is violated in the case of pure nucleonic and ∆-inclusive nuclear matter. Also, the curve for N+H+∆ composition predicts a higher value of the speed of sound squared at intermediate densities because of the early appearance of ∆ − particles, as explained earlier. FIG . 4. (Color online) Mass-Radius profile for DD-ME2 parameter set with different compositions of ∆ baryons and hyperons. The solid (dashed) lines represent the MR plot for the pure nucleonic matter. FIG . 5. (Color online) The radial displacement perturbation ξ(r) = ∆r/r as a function of dimensionless radius distance r/R for lower f -mode (n = 0), lower order p-modes (n = 1, 2, 3), intermediate p-modes (n = 9, 10, 11), and high excited modes (n = 17, 18, 19). The upper left (right) panel represents the result for pure nucleonic (∆-inclusive nucleonic) matter, while the lower left (right) panel represents the result for hyperonic (∆-inclusive hyperonic) matter. online) The radial pressure perturbation η(r) = ∆r/r as a function of dimensionless radius distance r/R for lower f -mode (n = 0), lower order p-modes (n = 1, 2, 3), intermediate p-modes (n = 9, 10, 11), and high excited modes (n =17,18,19). The upper left (right) panel represents the result for pure nucleonic (∆-inclusive nucleonic) matter, while the lower left (right) panel represents the result for hyperonic (∆-inclusive hyperonic) matter. FIG. 7 .FIG 7(Color online) Frequencies of radially oscillating NS as a function of central energy density for a) pure nucleonic matter (N), b) ∆-inclusive nucleonic matter (N+∆), c) hyperonic matter (N+H), and d) ∆-inclusive hyperonic matter (N+H+∆). The frequencies for lower radial modes (. 8. (Color online) Frequency difference ∆νn = νn+1 -νn vs νn in kHz for pure nucleonic (N), ∆-inclusive nucleonic (N+∆), hyperonic (N+H), and ∆-inclusive hyperonic matter (N+H+∆) matter. TABLE I . IDDME2 parameters (top) and its predictions to the nuclear matter at saturation density (bottom). i mi(MeV) ai bi ci di giN (n0) σ 550.1238 1.3881 1.0943 1.7057 0.4421 10.5396 ω 783 1.3892 0.9240 1.4620 0.4775 13.0189 ρ 763 0.5647 - - - 7.3672 Quantity Constraints [44, 49] This model n0 (f m −3 ) 0.148-0.170 0.152 −B/A (MeV) 15.8-16.5 16.4 K0 (MeV) 220-260 252 S0 (MeV) 31.2-35.0 32.3 L0 (MeV) 38-67 51 TABLE II . IIBaryon-meson coupling constants χ ib TABLE III . III20 lowest order radial oscillation frequencies, ν in (kHz) for different EoSs considered. For each EoS, the frequencies are calculated at the maximum mass of the corresponding star.Nodes EoS N N+∆ N+H N+H+∆ 0 0.571 1.578 1.977 1.865 1 5.173 5.891 6.582 6.241 2 8.163 9.118 10.497 9.656 3 11.060 12.379 14.231 13.390 4 13.960 15.674 17.907 16.652 5 16.878 18.984 21.527 19.986 6 19.812 22.316 25.064 23.422 7 22.758 25.649 28.758 27.116 8 25.713 29.011 32.317 30.754 9 28.674 32.358 35.633 34.492 10 31.639 35.732 39.177 38.135 11 34.609 39.089 42.800 41.779 12 37.580 42.457 46.347 45.505 13 40.556 45.822 50.016 49.174 14 43.535 49.189 53.464 52.622 15 46.518 52.556 57.117 56.275 16 49.500 55.933 60.527 59.685 17 52.483 59.308 64.180 63.380 18 55.469 62.688 67.796 67.054 19 58.457 66.061 71.603 70.618 V. ACKNOWLEDGEMENTWe would like to thank the anonymous reviewer for valuable comments and suggestions. N K Glendenning, Compact stars: Nuclear physics, particle physics, and general relativity. N. K. 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{'abstract': 'We investigate the effect of ∆ baryons on the radial oscillations of neutron and hyperon stars, employing a density-dependent relativistic mean-field model. The spin-3/2 baryons are described by the Rarita-Schwinger Lagrangian density. The baryon-meson coupling constants for the spin-3/2 decuplet and the spin-1/2 baryonic octet are calculated using a unified approach relying on the fact that the Yukawa couplings present in the Lagrangian density of the mean-field models must be invariant under the SU(3) and SU(6) group transformations. We calculate the 20 lowest eigenfrequencies and corresponding oscillation functions of ∆-inclusive nuclear (N+∆) and hyperonic matter (N+H+∆) by solving the Sturm-Liouville boundary value problem and also verifying its validity. We see that the lowest mode frequencies for N+∆ and N+H EoSs are higher as compared to the pure nucleonic matter because of the deltas and hyperons present. Furthermore, the separation between consecutive modes increases with the addition of hyperons and ∆s.', 'arxivid': '2303.11006', 'author': ['Ishfaq A Rather \nCentro de Astrofísica e Gravitação-CENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\n1049-001LisboaPortugal\n', 'Kauan D Marquez \nDepartamento de Física -CFM\nUniversidade Federal de Santa Catarina\nCEP 88.040-900FlorianópolisSCBrazil;\n', 'Grigoris Panotopoulos \nDepartamento de Ciencias Físicas\nUniversidad de la Frontera\nCasilla 54-D4811186TemucoChile\n', 'Ilídio Lopes \nCentro de Astrofísica e Gravitação-CENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\n1049-001LisboaPortugal\n'], 'authoraffiliation': ['Centro de Astrofísica e Gravitação-CENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\n1049-001LisboaPortugal', 'Departamento de Física -CFM\nUniversidade Federal de Santa Catarina\nCEP 88.040-900FlorianópolisSCBrazil;', 'Departamento de Ciencias Físicas\nUniversidad de la Frontera\nCasilla 54-D4811186TemucoChile', 'Centro de Astrofísica e Gravitação-CENTRA\nInstituto Superior Técnico\nUniversidade de Lisboa\n1049-001LisboaPortugal'], 'corpusid': 257632169, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19438, 'n_tokens_neox': 15812, 'n_words': 8803, 'pdfsha': '6912827a7f7319ea9e4049cca2942d9bcb3ac557', 'pdfurls': ['https://export.arxiv.org/pdf/2303.11006v2.pdf'], 'title': ['Radial Oscillations in Neutron Stars with Delta Baryons', 'Radial Oscillations in Neutron Stars with Delta Baryons'], 'venue': []}

arxiv

Reanalysis of critical exponents for the O(N ) model via a hydrodynamic approach to the Functional Renormalization Group Fabrizio Murgana Department of Physics and Astronomy University of Catania Via S. Sofia 64I-95125CataniaItaly INFN-Sezione di Catania Via S. Sofia 64I-95123CataniaItaly Institut für Theoretische Physik Goethe-Universität Max-von-Laue-Straße 1D-60438Frankfurt am MainGermany Adrian Koenigstein Institut für Theoretische Physik Goethe-Universität Max-von-Laue-Straße 1D-60438Frankfurt am MainGermany Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Dirk H Rischke Institut für Theoretische Physik Goethe-Universität Max-von-Laue-Straße 1D-60438Frankfurt am MainGermany Helmholtz Research Academy Hesse for FAIR Campus Riedberg, Max-von-Laue-Straße 12D-60438Frankfurt am MainGermany Reanalysis of critical exponents for the O(N ) model via a hydrodynamic approach to the Functional Renormalization Group We compute the critical exponents of the O(N ) model within the Functional Renormalization Group (FRG) approach. We use recent advances which are based on the observation that the FRG flow equation can be put into the form of an advection-diffusion equation. This allows to employ well-tested hydrodynamical algorithms for its solution. In this study we work in the local potential approximation (LPA) for the effective average action and put special emphasis on estimating the various sources of errors. Our results complement previous results for the critical exponents obtained within the FRG approach in LPA. Despite the limitations imposed by restricting the discussion to the LPA, the results compare favorably with those obtained via other methods. I. INTRODUCTION The understanding of the behavior of systems where microscopic degrees of freedom are strongly interacting is the main goal of many areas of physics, ranging from condensed matter [1,2] to elementary-particle theories [3,4], and extending even to quantum gravity [5,6]. However, first-principle calculations for these systems are often very difficult and demand the use of powerful tools. Calculations are in particular challenging when a system undergoes a phase transition, since new degrees of freedom may arise or become relevant. In this case the underlying theory must consistently relate the two phases and thus describe the transition from one set of degrees of freedom to the other. For second-order transitions, the behavior of a system at all length scales is determined by a finite set of so-called critical exponents. A widely exploited theoretical method to treat this problem is the perturbative Renormalization Group (RG) introduced by Callan, Symanzik, and others. This technique describes systems where many interacting degrees of freedom are present which interact via small effective couplings [7]. Unfortunately, this approach cannot be used in systems where such a small coupling, or in general a small parameter in which one can perform perturbative calculations, simply does not exist or is hard to identify. Furthermore, the perturbative series does not converge in general and one has to exploit resummation techniques or other methods. Another approach to describe phase transitions are computer simulations employing Monte Carlo (MC) methods, which have been successfully used to obtain high-precision estimates of the critical exponents, see, e.g., Refs. [8,9]. However, one of the major issues with these methods is the extremely large amount of computer time needed to obtain a reliable infinite-volume and continuum limit. Yet another method which can address critical phenomena is the conformal-field theory approach [10][11][12], which for example leads to a high precision for the critical exponents of the Ising model, originally in 2D and later in 3D thanks to the Conformal Bootstrap (CB) technique. In order to overcome the difficulties of the perturbative RG method and numerical simulations, a different, but similarly powerful approach can be used: the Functional Renormalization Group (FRG) [13][14][15][16][17], also referred to as Exact RG or Non-perturbative RG, which is ultimately based on seminal ideas by Wilson [18][19][20][21][22] and others, see, e.g., Refs. [23,24]. The central object of the FRG approach is a flow equation, which describes the evolution of correlation functions or, equivalently, their generating functional under the influence of fluctuations. It connects a well-defined initial quantity, e.g., the microscopic ultraviolet (UV) action, in an exact manner with the desired full infrared (IR) effective action, where all fluctuations are integrated out. Hence, solving the flow equation corresponds to solving the full theory and is therefore equivalent to a direct computation of the generating functional. Thanks to the fact that it is non-perturbative and connects degrees of freedom at different scales, the FRG approach is well suited to address the issue of describing systems approaching criticality and phase transitions, and thus a tool for the computation of critical exponents, cf. Refs. [25][26][27][28][29][30][31][32][33]. Even though the FRG has a solid theoretical foundation, it is very hard to find analytical solutions to the flow equation. For specific problems, it can be converted into an infinite number of coupled partial differential equations (PDEs) and/or ordinary differential equations (ODEs), which, if suitably truncated, can be subjected to a numerical treatment in order to obtain a solution. In fact, as a truncation scheme one often employs a derivative expansion of the effective action and solves separate FRG flow equations for the coefficients of this expansion. The leading-order truncation in this expansion only accounts for the flow of the local effective potential, and is thus called local potential approximation (LPA). In this work, we briefly introduce the FRG approach and apply it to the O(N ) model, for which we compute the critical exponents in the LPA. The critical exponents are just one example of physical quantities that can be computed by the FRG method. However, they are relevant since they can provide a good benchmark to test the convergence of this method, because other techniques, such as CB and MC, yield a very precise determination of these quantities. The critical exponents of the O(N ) model have already been studied within the FRG approach some time ago, see, e.g., Refs. [27,30,31,33,34]. The reason why we decided to repeat such an investigation is that recently a novel method to solve the FRG flow equations has been proposed [35]. This method relies on the observation that the FRG flow equation for the effective potential can be cast into the form of a hydrodynamic advection-diffusion equation. This suggests to use numerical techniques well-known from hydrodynamics to solve that equation, see also Refs. [36][37][38][39][40][41][42] as well as Refs. [43][44][45][46][47][48][49] for some early developments. In particular, hydrodynamic conservation laws in general allow for the formation of discontinuities or, more general, non-analyticities in the solution. Therefore, the applied numerical scheme has to be able to handle discontinuities in an appropriate manner. While Ref. [35] used a discontinuous Galerkin method to solve the FRG flow equation, here we will exploit a well-established finite-volume central scheme, the Kurganov-Tadmor (KT) algorithm, which is designed to have a high-order accuracy, preserving stability and introducing negligible dissipation, while allowing to treat discontinuities in the solution (see Ref. [50] for more details). Applying this method, we perform a more systematic study of various sources of errors in determining the critical exponents of the O(N ) model as compared to previous attempts using the FRG approach in the LPA (without working with the fixed-point equations directly). 1 This work is organized as follows. In Sec. II we give a brief introduction to the FRG approach and discuss the FRG flow equation. In Sec. III we derive the FRG flow equation for the O(N ) model in the LPA and rewrite it in the form of an advection-diffusion equation in order to solve it with the KT scheme. Numerical results for the critical exponents are presented in Sec. IV. Finally, we conclude this work in Sec. V with a summary and an outlook. A detailed discussion of error estimates is delegated to App. A. II. FUNCTIONAL RENORMALIZATION GROUP APPROACH In this section, we briefly recapitulate the FRG approach in the formulation given by Wetterich et al. [51][52][53][54]. For further details, see Refs. [13][14][15][16][17]. The central object of this approach is the effective action Γ[Φ], which is the generating functional of 1PI vertex functions. In order to compute Γ[Φ], one introduces the so-called effective average actionΓ k [Φ]. This quantity depends on the parameter k, which is a coarse-graining scale with physical dimension of a momentum. The FRG flow equation describes the evolution ofΓ k [Φ] as k runs from the UV, k → ∞, to the IR, k → 0. The effective average action interpolates between the bare classical action S bare [Φ] in the UV and the full quantum effective action Γ[Φ] in the IR, i.e., Γ k→∞ [Φ] = S bare [Φ] ,Γ k→0 [Φ] = Γ[Φ] .(1) In practice, one may not always be able to send k → ∞. Therefore, one usually introduces a UV scale Λ as initial scale for the FRG flow equation, which is chosen to be sufficiently large, i.e., much larger than any other physical scale of the theory, and assumes that the bare classical action describes the underlying theory at this scale. However, using a finite cutoff is an approximation and one has to ensure that the results are independent of the choice of Λ, e.g., by respecting RG consistency [55]. The equation that describes the RG-scale evolution ofΓ k [Φ], i.e., how the effective average action varies as one integrates out fluctuations with increasingly smaller momenta, is the so-called Wetterich equation, or Exact Renormalization Group flow equation, or simply FRG flow equation [51,56,57], ∂ kΓk [Φ] = Tr 1 2 ∂ k R k Γ (2) k [Φ] + R k −1 = ,(2) where the trace indicates an integral over momenta and a sum over all internal degrees of freedom. The equation has one-loop structure, which is indicated by the Feynman-diagram representation on the right-hand side. Here, the black line represents the full propagator G k [Φ] = (Γ (2) k [Φ] + R k ) −1 and the crossed circle stands for 1 2 ∂ k R k , where R k in Eq. (2) is the so-called regulator. The latter must be chosen such thatΓ k [Φ] smoothly interpolates between the bare classical action S bare [Φ] at k = Λ and the full effective action Γ[Φ] at k = 0. This requirement imposes certain conditions on R k . In particular, (i) it should act like a mass term for momenta smaller than k, regulating the full propagator G k [Φ] = (Γ (2) k [Φ] + R k ) −1 in the IR, (ii) it has to vanish when k → 0, and (iii) it has to diverge when k → ∞, such that the functional integral definingΓ k [Φ] is dominated by the bare classical action. The choice of the specific shape of the regulator is a delicate issue and can be optimized depending on the problem, see, e.g., Refs. [58][59][60][61] for details. Despite its deceptively simple form, the Wetterich equation is a functional integro-differential equation, and as such cannot be solved exactly for an arbitraryΓ k [Φ]. Thus, it is clear that some approximation has to be made. Two common approximation schemes used in the literature are the vertex expansion [53,62] and the derivative expansion [17,63]. The former expandsΓ k [Φ] in powers of the fields Φ, where the vertex functionsΓ (n) k (x 1 , . . . , x n ) = δ nΓ k [Φ]/δΦ(x 1 ) · · · δΦ(x n ) Φ=Φ0 of the theory are the expansion coefficients. On the other hand, in the derivative expansion one expandsΓ k [Φ] in terms of all composite operators constructed from space-time derivatives (in principle of arbitrary order) of the fields. These composite operators must be compatible with the symmetries of the theory. In both cases, one obtains an infinite system of coupled integro-differential equations which needs to be truncated, either at a certain order of the vertex functions or at a certain order of space-time derivatives of the fields. In this work we will focus on the latter, since the vertex expansion assumes regularity ofΓ k [Φ], which is violated near phase transitions, where the effective action develops discontinuities or points of non-analyticity during the FRG flow [35,36,40]. Since in this paper we are interested in critical exponents of second-order phase transitions, we have to employ a method which allows to treat discontinuities and non-analyticities in the FRG flow of the effective action. The derivative expansion fulfills this requirement, even in the most simple truncation, the aforementioned local potential approximation (LPA), which will be discussed in the next section. S bare [ φ ] = d d x (∂ µ φ a ) 2 + V (ρ) ,(3) where V (ρ) is the potential parametrizing the form the N scalar fields interact with each other. The O(N ) symmetry requires that V (ρ) is solely a function of the variable ρ = 1 2 φ a φ a ,(4) where a sum over a from 1 to N is implied. Despite its apparent simplicity, the O(N ) model can describe a large variety of physical systems at different energy scales. The reason for the wide range of applicability of such a simple model is the well-known universal behavior of physical systems close to criticality, when the microscopic degrees of freedom are not important and only the main features, like the symmetry of the model, have to be taken into account in the description of the physical behavior of the system. For this reason, the O(N ) model can be considered as a prototype model to understand relevant mechanisms that govern a phase transition. For example, for N = 4 the model can describe the chiral phase transition in Quantum Chromodynamics (QCD) with two quark flavors [64]. For N = 3 it belongs to the universality class of the Heisenberg model, describing a ferromagnetic phase transition [65]. The N = 2 case can be used to describe the XY-model [66] and N = 1 belongs to the Ising universality class [67]. B. Derivation of the flow equation in LPA In order to proceed with the quantitative study of the O(N ) model, the first step will be to derive the flow equation that describes the k-evolution of the effective average action. As already mentioned in the previous section, we need to truncate the effective action, i.e., we need to choose an ansatz. We use the derivative expansion, expanding the effective action in terms of powers of gradients of the field. In particular we consider the lowest order of the expansion, the so-called LPA, where the only space-time derivatives of the fields appear in the kinetic term and the effective potential V (ρ) is solely a function of the fields but not of their space-time derivatives. The advantage of the LPA is that it leads to a simple expression for the FRG flow equation, while still capturing many non-trivial features of the theory. Furthermore, this approximation becomes exact in the limit N → ∞ [68]. In LPA the effective average action readsΓ k [ ϕ ] = d d x 1 2 (∂ µ ϕ a ) 2 + V k ( ) ,(5) where V k ( ) is the effective potential which depends on the FRG scale parameter k and the O(N )-invariant = 1 2 ϕ a ϕ a , i.e., the quantity ρ defined in Eq. (4), evaluated for the mean fields ϕ a . The next order in the derivative expansion would be the so-called LPA , which considers non-trivial wave-function renormalization corrections, i.e., (∂ µ ϕ a ) 2 → Z k (∂ µ ϕ a ) 2 , with Z k being a non-trivial function of the FRG flow parameter k. For even better truncations in similar models, see, e.g., Refs. [69][70][71][72]. Once the ansatz (5) for the effective action is specified, the Wetterich equation (2) turns into a PDE for the effective potential V k ( ). For a concrete result, we have to choose a regulator which satisfies the conditions we described in the previous section. It is possible to prove [58] that, in the case of the LPA, an optimal choice for the regulator in terms of stability of the FRG flow equation is the so-called Litim regulator R k (q, p) = (2π) d δ (d) (q + p) p 2 r k (p) , r k (p) = k 2 p 2 − 1 Θ k 2 p 2 − 1 ,(6) where r k (p) is the corresponding regulator shape function. The next step is the computation of the regularized propagator, or equivalently the two-point vertex function. To this end, we choose a simple background field ϕ = (0, · · · , 0, 2 ) , ⇐⇒ ϕ a = 2 δ aN .(7) In this way we obtain for the two-point vertex function in momentum spacē Γ (2) k,ab (q, p; ) = (2π) d δ (d) (q + p) q 2 + V k ( ) δ ab + 2 V k ( ) δ aN δ bN ,(8) where V k ( ) = ∂ V k ( ). Inserting Eq. (8) into Eq. (2) and performing the trace one obtains ∂ k V k ( ) = d d q (2π) d q 2 1 2 ∂ k r k (q) N − 1 q 2 [1 + r k (q)] + V k ( ) + 1 q 2 [1 + r k (q)] + V k ( ) + 2 V k ( ) .(9) Inserting Eq. (6) into Eq. (9), the integration becomes trivial and employing Eq. (16) we arrive at the following well-known flow equation for the effective potential, ∂ k V k ( ) = A d k d+1 N − 1 k 2 + V k ( ) + 1 k 2 + V k ( ) + 2 V k ( ) ,(10) with A d = Ω d d(2π) d , Ω d = 2π d/2 Γ(d/2) .(11) Here, Ω d is the volume of the d − 1 dimensional unit sphere and Γ is the Gamma function. C. FRG flow equation in terms of the field σ and boundary conditions Up to now the PDE (10) for the effective potential is written in terms of the O(N )-invariant . However, when solving the PDE with a numerical scheme that relies on a discretization in -direction, we face the following problem. Usually, such a numerical scheme requires a stencil of points in the vicinity of any given point in the domain where V k ( ) is defined. In particular, this stencil is then also required at the boundary = 0, i.e., we would have to specify V k ( ) for some negative value of , which does not exist. A solution is to reformulate the equations above in terms of the field expectation value σ = √ 2 , i.e., we consider V k (σ), ∂ σ V k (σ), and ∂ 2 σ V k (σ) instead of V k ( ), ∂ V k ( ), and ∂ 2 V k ( ), respectively. We then rewrite Eq. (10) as follows: ∂ k V k (σ) = A d k d+1 N − 1 k 2 + 1 σ ∂ σ V k (σ) + 1 k 2 + ∂ 2 σ V k (σ) .(12) This equation will be the reference point for our further investigations of the FRG flow in LPA. The boundary condition at σ = 0 (including points in its vicinity) is now specified exploiting the residual Z 2 symmetry of the potential, V k (σ) = V k (−σ) ,(13) which translates into a Z 2 antisymmetry for ∂ σ V k (σ), ∂ σ V k (σ) = −∂ σ V k (−σ) ,(14) and again a Z 2 symmetry for ∂ 2 σ V k (σ), ∂ 2 σ V k (σ) = ∂ 2 σ V k (−σ) .(15) The PDE (12) cannot be solved numerically on an infinite domain σ ∈ [0, ∞). Therefore, we have to choose a sufficiently large value of σ for the upper boundary, say σ max , and we also need to specify V k (σ max ). In the UV, at k → ∞, the effective potential V k (σ) is given by the potential term in the bare action S bare , which is usually a polynomial in powers of σ. For a Z 2 -symmetric potential, these powers must be even. For an interacting theory, the smallest power of σ must then be at least four. This means that for large σ the right-hand side of the FRG flow equation (12) is at least of order ∼ σ −2 . Choosing a sufficiently large σ max ensures that the right-hand side of Eq. (12) can be neglected, which means that the effective potential V k (σ max ) does not change under the FRG flow and stays at its UV value. For a more detailed discussion on the boundary conditions and how to implement them we refer to Ref. [37]. D. Advection-diffusion formulation of the FRG flow equation For the sake of convenience, let us introduce the "RG-time" parameter t via t = − ln k Λ , ∂ ∂t = −k ∂ ∂k .(16) Note that we choose the opposite sign as in most of the standard FRG literature, because we deem it more natural for a time to flow from t = 0 (at k = Λ) to t = ∞ (at k = 0). We will now show that it is possible to cast Eq. (12) into the form of an advection-diffusion equation well-known from hydrodynamics. This is possible because Eq. (12) is independent of the potential itself and thus one can rewrite it in terms of a PDE for u(t, σ) = ∂ σ V (t, σ) .(17) Analogously we will denote u (t, σ) = ∂ σ u(t, σ) = ∂ 2 σ V (t, σ) .(18) Here, we have defined V (t, σ) = V k (σ), i.e., we replaced the k-dependence of V k (σ) by an equivalent dependence on the variable t. We also considered V (t, σ) as a function of the two continuous variables t and σ. In the interpretation of the FRG flow equation as an advection-diffusion equation, t will retain its role as a time variable for the FRG flow, while σ will assume the role of a spatial variable. The main advantage of considering the FRG flow equation as an advection-diffusion equation is that it allows us to exploit the powerful toolbox of numerical methods that have been developed to solve hydrodynamical equations. We now introduce the function f (t, u, σ) = (N − 1) A d (Λe −t ) d+2 (Λe −t ) 2 + 1 σ u(t, σ) ,(19) which corresponds to the non-linear advection flux in the hydrodynamical interpretation, as well as the function g(t, u ) = −A d (Λe −t ) d+2 (Λe −t ) 2 + u (t, σ) ,(20) which corresponds to a non-linear diffusion flux. Taking the derivative of Eq. (10) with respect to σ we then obtain the equation for u(t, σ) as ∂ t u(t, σ) + ∂ σ f (t, u, σ) = ∂ σ g(t, u ) ,(21) which has exactly the form of a non-linear advection-diffusion equation for some fluid field u. For more details on the properties of this equation in the FRG framework we refer to Refs. [35][36][37][38][39][40][41][42]. IV. CRITICAL BEHAVIOR In order to study the critical behavior of the O(N ) model, we first have to set an initial condition for the FRG flow in the UV, i.e., at t = 0 or k = Λ. The potential in the UV is chosen to have the well-known φ 4 -form, V k=Λ ( ) = λ 4 ( − 0 | t=0 ) 2 ,(22) where 0 | t=0 is the minimum of the potential at k = Λ. For 0 | t=0 > 0, the system is in the broken phase, while for 0 | t=0 = 0 it is in the symmetric phase. In particular, in the case where 0 | t=0 > 0, one component of the N -dimensional scalar field φ develops a nonvanishing expectation value, see Eq. (7), and the O(N )-symmetry group is spontaneously broken to O(N − 1). According to Goldstone's theorem, this leads to N − 1 massless modes, the so-called Nambu-Goldstone bosons. The remaining component of the field ϕ is the radial sigma mode discussed above in Sec. III C, which develops a nonvanishing mass proportional to the minimum 0 | t=0 [73]. For example, considering the model at finite temperatures, the O(N ) symmetry is restored via a second-order phase transition and all modes become degenerate in mass when the temperature is increased above a critical value. It seems clear that, due to the non-perturbative nature of the phase transition, a simple perturbative approach fails to describe this phenomenon. On the other hand, the non-perturbative FRG approach allows us to go beyond any finite order in perturbation theory, and thus to correctly treat this kind of problem. Via this approach we will determine several critical exponents in the LPA. A. Critical scaling In order to compute the critical exponents we need the generalization of the flow equation (12) to finite temperature. However, we can circumvent this step by exploiting the well-known phenomenon of dimensional reduction [74][75][76][77]: in the limit of high temperatures, or more precisely, when the temperature is much larger than the FRG flow parameter, T k, the non-zero Matsubara modes with energy ∼ 2πnT , n = 0, decouple from the evolution, leaving only a three-dimensional effective theory involving the zero Matsubara mode. From a different perspective, if the correlation length, which is the scale associated with the system, dominates over the inverse temperature, then it is not possible to resolve the compactified Euclidean time dimension, leading to a dimensionally reduced effective theory. Thus we do not need to use a finite-temperature flow equation in order to investigate the critical region of the phase transition. Instead, it is sufficient to employ the d = 3-dimensional zero-temperature equation (12). According to universality-class arguments [78][79][80][81], there are two relevant parameters that can be tuned in order to bring the system to the critical point. This can be immediately understood considering the ferromagnetic Ising model, which has a discrete Z 2 = O(1) symmetry, as an example: here the parameters that drive the system towards the phase transition are the temperature and the external magnetic field. Since we are considering the O(N ) model without external fields, only one relevant parameter is left in our case: the temperature. However, since we exploit the dimensional-reduction phenomenon and work in a zero-temperature field theory, the temperature does not explicitly enter the description and we need to identify a relevant variable which assumes its role. Obviously, such a variable decides whether the system ends up in the symmetric or broken phase when t → ∞. This leads us to take the UV minimum 0 | t=0 of the potential as the variable replacing the temperature. Indeed, if 0 | t=0 is larger than a critical value c 0 | t=0 , the system will end up in the broken phase in the IR, i.e., this is equivalent to working at temperatures T < T c . Analogously, if 0 | t=0 < c 0 | t=0 , the system will end up in the symmetric phase in the IR, which is equivalent to working at temperatures T > T c . Due to the convexity of the effective potential, see for example Ref. [82], in the broken phase the potential V IR (σ) is flat for σ ≤ σ 0 | IR , where σ 0 | IR = 2 0 | IR > 0 is the minimum of the potential in the IR, i.e., for t → ∞. On the other hand, in the symmetric phase σ 0 | IR = 0. This allows us to identify σ 0 | IR as the order parameter of the phase transition, since σ 0 | IR > 0 in the broken phase and σ 0 | IR = 0 in the symmetric one. This qualitative discussion is confirmed by solving the FRG flow equation, cf. Fig. 1, where we plot the effective potential V k (σ) (right panel) and its derivative u(t, σ) (left panel) for different values of σ 0 | t=0 at the FRG time t = 3, which is sufficiently far in the IR such that the flow does not change the minimum of the potential and the curvature mass at the minimum by an appreciable amount (for a detailed discussion of the numerical errors, see App. A). One observes that, for small values of σ 0 | t=0 the system ends up in the symmetric phase in the IR (σ 0 | IR = 0), while for larger values of σ 0 | t=0 the symmetry remains broken in the IR (σ 0 | IR > 0) and the potential exhibits a plateau. Since we are dealing with a second-order phase transition, we expect an IR fixed point of the FRG flow, close to which the theory is scale-invariant. Thus the critical behavior has to be described by a solution of the FRG flow equation which is scale-independent for sufficiently small (large) values of k (t). Since we only have one relevant variable ( 0 | t=0 ), we are allowed to set λ, which is an irrelevant variable from the RG perspective, to an arbitrary value. Here, we choose λ = 0.5. We then tune 0 | t=0 in order to find the so-called scaling solution or critical trajectory, i.e., a solution of the FRG flow equation which becomes t-independent for sufficiently large values of t. In particular, the closer 0 | t=0 is to c 0 | t=0 , the closer the solution is to the critical trajectory when approaching the IR. This implies that properly rescaled dimensionless quantities will exhibit a constant behavior at sufficiently large t. In particular, the dimensionless minimum˜ 0,k = k 2−d 0,k(23) tends to a constant (fixed-point) value as 0 | t=0 approaches c 0 | t=0 . This qualitative behavior is quantitatively confirmed by an explicit solution of the FRG flow equation, as shown in Fig. 2, cf. Ref. [31]. Here we plot the evolution of the dimensionless minimum (23) with t for different initial values 0 | t=0 . One observes that, during the FRG flow,˜ 0,k approaches the critical trajectory, the asymptotic fixedpoint value of which is shown by the red horizontal line. However, eventually it deviates upwards (for initial values 0 | t=0 > c 0 | t=0 ) or downwards (for initial values 0 | t=0 < c 0 | t=0 ). This can be easily explained by considering that, in d = 3 dimensions, • in the broken phase ( 0 | t=0 > c 0 | t=0 ) the minimum 0,k of the potential tends to a constant value 0 | IR > 0 when k → 0, which means that˜ 0,k = 0,k /k → +∞, Fig. 1). The green curves correspond to initial values 0|t=0 > c 0 |t=0, indicating that the system is in the broken phase, while the purple curves correspond to initial values 0|t=0 < c 0 |t=0, i.e., the system is in the symmetric phase. The red horizontal line indicates the fixed-point value of the scaling solution in the IR. • in the symmetric phase ( 0 | t=0 < c 0 | t=0 ) the minimum 0,k of the potential goes to 0 already at a non-zero value of k > 0, which means that˜ 0,k = 0,k /k → 0. The closer the initial value 0 | t=0 is to the critical value c 0 | t=0 , the larger is the value of t where the deviation from the scaling solution occurs. The critical trajectory, ending in the fixed-point value as t → ∞, can actually not be reached in practice, as it would require infinite numerical precision and infinite spatial resolution in the domain of the PDE. Of course not only˜ 0,k exhibits a scaling behavior, but all other appropriately rescaled dimensionless quantities first approach and then deviate from the critical trajectory at certain RG times which depend on the initial value 0 | t=0 . Here, we choose˜ 0,k since the minimum of the potential is the order parameter characterizing the phase transition. B. Critical exponents It is a well-known fact [78][79][80][81] that some physical quantities derived from the partition function or, equivalently, from the free energy of the system, may diverge when approaching a phase transition, i.e., for 0 | t=0 → c 0 | t=0 . In particular they exhibit a power-law behavior characterized by so-called critical exponents, which depend on the dimension d and the symmetries of the system, and which define the universality class of the theory. There exist six critical exponents: α, β, γ, δ, ν, η. However, only two of them are independent since they are constrained by the following so-called scaling laws or scaling relations: α = 2 − dν , Josephson law ,(24)γ = ν(2 − η) , Fisher law ,(25)γ = β(δ − 1) , Widom law ,(26)2 = α + 2β + γ , Rushbrooke law .(27) In the following we will consider β and ν as independent and discuss in more detail how to extract them from the FRG flow equation. 1. Exponent for the order parameter: β Close to criticality, the order parameter of the phase transition σ 0 | IR is described by the following behavior σ 0 | IR = 0 , 0 | t=0 < c 0 | t=0 , σ 0 | IR ∼ ( 0 | t=0 − c 0 | t=0 ) β , 0 | t=0 > c 0 | t=0 .(28) In Fig. 3, we show ln(σ 0 | IR ) as a function of ln( 0 | t=0 − c 0 | t=0 ) for N = 3 and d = 3 in the LPA. The exponent β is then read off from the slope of this function. One observes that, for small values of ln( 0 | t=0 − c 0 | t=0 ), ln(σ 0 | IR ) deviates from the scaling behavior given by the second line of Eq. (28). This deviation is caused by the finite numerical precision with which we can determine c 0 | t=0 . The closer the initial 0 | t=0 is to the critical value, the more sensitive one is to deviations from the actual critical value caused by the finite numerical precision, and thus one simply does not follow the critical behavior anymore. Note that, if we work in the finite-temperature theory, σ 0 | IR would be a function of temperature and close to criticality would be proportional to (T c − T ) β . Thus c 0 | t=0 determines the critical temperature in three dimensions. Exponent for the correlation length: ν For large spatio-temporal distances, the correlator exhibits an exponential decay, G(x − y) = φ(x) φ(y) − φ(x) φ(y) ∼ e −|x−y|/ξ for |x − y| ξ ,(29) where ξ is the so-called correlation length. Close to criticality it behaves as ξ(T ) ∼ (T − T c ) −ν ,(30) which, in the dimensionally reduced theory, becomes ξ( 0 | t=0 ) ∼ 0 | t=0 − c 0 | t=0 −ν .(31) In order to compute ν, we exploit the fact that the correlation length is proportional to the inverse of the renormalized mass m, ξ ∼ m −1 . Note that in the LPA the pole, curvature, and screening masses are identical for zero-temperature calculations [83]. In the symmetric phase, the square of the renormalized mass is given by m 2 = lim t→∞ u (t, σ = 0) = lim t→∞ ∂ 2 σ V (t, σ = 0) .(32) Thus we compute m 2 and then exploit In this way we can obtain ν from the slope of ln m 2 as a function of ln 0 | t=0 − c 0 | t=0 , see Fig. 4. We note that one has to make sure to reach the symmetric phase before stopping the t-evolution, because only this ensures that m 2 > 0 at σ = 0. m 2 ∼ 0 | t=0 − c 0 | t=0 2ν .(33) We remark that the deviation of ln m 2 from the scaling solution at small values of ln 0 | t=0 − c 0 | t=0 has the same origin as that observed in Fig. 3. On the other hand, the deviation observed for large values of ln 0 | t=0 − c 0 | t=0 is due to the fact that one is simply too far away from the critical region, such that the scaling behavior of Eq. (29) does no longer apply. In Tabs. I and II we show our results for β and ν, respectively, obtained within the LPA for d = 3 and various values of N in comparison to results obtained within similar as well as other frameworks. Results obtained within the same FRG framework but including a non-trivial RG-scale dependent wave-function renormalization [31] are denoted as "RG' ". Results obtained from MC simulations [8,9] are listed under "MC", those from perturbative RG are denoted as "PT" [84], while those from the ε-expansion at order ε 6 are shown under "ε-exp." [85]. Results from CB are denoted as "CB" [86][87][88], and finally those from a derivative expansion up to fourth order as "DE 4 " [30]. We observe that the results obtained in this work are overall in good agreement with the ones obtained with other approaches. As compared to the results of RG', which is from a technical perspective closest to our approach, we improve the precision by one order of magnitude and in addition provide an error estimate, see App. A for details. Although RG' includes a non-trivial wave-function renormalization, while our approach does not, our results for the values of the critical exponents are not significantly worse than RG'. Furthermore, we have checked that the values obtained for finite N tend to converge to the ones in the large-N limit, shown in Tab. III, since in that case the LPA becomes exact [53]. V. CONCLUSION AND OUTLOOK In this work we have applied the Functional Renormalization Group (FRG) approach to compute the critical exponents of the O(N ) model. We exploited the recent realization that the FRG flow equation for the effective potential can be put into the form of an advection-diffusion equation [35], allowing us to employ widely used and welltested hydrodynamic algorithms to solve it. We demonstrated the feasibility of this approach for the computation of the O(N ) critical exponents by solving the FRG flow equation in the local potential approximation (LPA). As hydrodynamic algorithm, we used a finite-volume method, the so-called KT scheme [50]. The critical exponents of the O(N ) model have been studied previously within the FRG approach in LPA and our discussion closely follows Ref. [31]. However, here we demonstrated that the novel hydrodynamic approach to solve the FRG flow equations allows a better control of statistical errors. With this method, these errors can be more precisely determined and are of order 10 −2 to 10 −3 . They are thus comparable to the errors of other well-established methods like lattice calculations, perturbation theory, the ε-expansion, or CB. The range of applicability for the novel hydrodynamic method to solve FRG flow equations is very wide. Studies in the framework of the O(N ) model were already carried out including higher-order terms in gradients, such as wave-function renormalization factors [30,31,33,34]. Combining these high-precision studies with our developments is certainly a worthwhile future project. Furthermore, it is possible to study the system at finite temperature, without exploiting dimensional reduction. In particular, we expect that the non-zero Matsubara modes do not influence the exponents related to the singular part of free energy (the effective action), that is α, γ, η, ν, while in principle they could give contributions to the order parameter, thus modifying β, and to the critical isotherm, changing the value of δ. Another possibility is to extend the model to include fermions. From a mathematical point of view, the FRG flow equations is still an advection-diffusion equation, but now with a source term [36,40,89], which can also be solved using the approach described above. General fit error As discussed in Sec. IV, a linear fit to the data is performed, the slope of which gives the critical exponents. This procedure inevitably leads to some error due to the fact that, as mentioned in Sec. IV, some points do not follow the critical scaling and thus deviate from the linear fit. Therefore, we need a criterion which enables us to select which points we should take into account for the fit. This is an important issue, since, as can be seen from Figs. 6 and 7, the value of a particular critical exponent and the relative fit error may change a lot if only a single data point is added to (or subtracted from) the fit. Since our goal is to extract the critical exponents from the scaling region, where we expect a linear behaviour of the observables (the logarithm of the IR minimum and the logarithm of the curvature mass) as a function of ln 0 | t=0 − c 0 | t=0 , it seems reasonable to use those consecutive points for the fit which exhibit the highest degree of collinearity. As a criterion for collinearity, we choose the Pearson correlation coefficient. However, the number of points which are taken into consideration still have a significant impact on the value of the critical exponents. This can be clearly seen from Figs. 6 and 7, where the critical exponents are shown as a function of the number of consecutive most aligned points. This is due to the fact that, as shown in Fig. 5, either the region where the consecutive most aligned points are contained moves while modifying the number of points, or it includes points in the fit which are increasingly further away from a straight fitting line. Thus, a criterion is still needed according to which one should select the number of consecutive collinear points. It is worth noting that the behavior seen in Figs. 6 and 7 is not affected by varying the point density (number of points per unit interval of the fitting region), indicating that the values of the critical exponents depend on the size of the fitting region rather than on the number of points contained in it. Ideally, if all points belong to the scaling region, all of them would exhibit perfect collinearity and thus the critical exponents would be completely independent of the fitting region. Our criterion is thus to identify the scaling region by searching for the range on the abscissa in Figs. 6 and 7 where the critical exponent is least dependent on the number of consecutive collinear points and thus on the fitting region. Once this range is identified, we assume that the actual value of the critical exponent lies somewhere between the highest and the lowest value of the critical exponent in that range. This gives a contribution to the total error on the critical exponent which is half the difference of the highest and the lowest value in that range. Finding such a range of consecutive collinear points is straightforward in the case of β: if the number of points in the range considered is too small, the range moves a lot along the abscissa when it is extended. Therefore, we discard the leftmost points in Fig. 7 (left panel). From a certain size onward the range does not move anymore, but merely grows within the critical region. However, if the range is too large, also points from outside the critical region are included, which is clearly seen by the jump in Fig. 7 (left panel) at N p 130. Hence, this clearly restricts the size of the critical region. On the other hand, in the case of ν, if the fitting region is too large, the exponent starts to change more rapidly, indicating that the corresponding fitting region includes points which do not follow a straight line in Fig. 4. Thus, we restricted the fit to a maximum value of N p 180. However, from Fig. 6 it is clearly visible that it is much harder, if not almost impossible, to identify a critical region, because ν strongly depends on the fitting interval. We conclude this subsection by first noticing that the error originating from the choice of the fitting region is much larger than the error of the fit itself, i.e., the one extracted from the maximum deviation of the points in the fitting region from a straight line. Indeed, this error is at least one order of magnitude smaller than the error from the choice of the fitting region. Second, as we discuss next, the error from choosing the correct fitting region is also much larger than the numerical error coming from the KT scheme and the determination of the curvature mass, which is required for extracting ν. For β, however, the discretization error which results in an uncertainty on the position of the minimum is again comparable to the error arising from the determination of the size of the critical region. Error on m 2 In this subsection we analyze the impact of the discretization in field space, i.e., of the grid spacing ∆σ, on the value of the renormalized mass m 2 and on the corresponding error. In Fig. 8, we show the curvature mass squared m 2 (left panel) and its relative deviation (right panel) ∆m 2 rel (∆σ) = 1 − m 2 (∆σ) m 2 (∆σ min ) (A1) from the value m 2 (∆σ min ) calculated for the smallest cell size ∆σ min (which in our case is 10 −4 ) as a function of ∆σ. Note that each point in these figures is the result of an FRG flow evolution, obtained for σ 0 | t=0 = 0.3 in the symmetric phase (σ IR 0 = 0) at final time t IR = 25 (k IR ∼ 10 −11 ). The aim is to identify the value of m 2 from a plateau region, i.e., when its value does not change anymore upon changing the value of ∆σ. From Fig. 8 we observe that this happens for values of ∆σ < ∼ 10 −2.5 . Thus, in order to get a reasonable compromise between computational resources and needed precision, we use ∆σ = 0.0005 (corresponding to the fourth point from the left in Fig. 8 (left panel)). As one observes from the right panel of Fig. 8, this choice leads to a relative error in the determination of m 2 of the order of ∼ e −7 10 −3 . As anticipated, the right panel provides an estimate of the order of magnitude of the relative errors, which is especially useful for those points which belong to the plateau in the left panel. The fact that these relative differences scale as ∆σ α , with α > 2, is a direct consequence of the numerical scheme used, the KT scheme, which has second-order precision in the spatial resolution. This means that it is possible to significantly reduce the error in the determination of m 2 by choosing a sufficiently small ∆σ. We should also mention that there is no additional error from using a finite difference stencil in Eq. (32) to extract the curvature mass at σ = 0, for details see Refs. [37,40]. One final remark is related to the extrapolation of the value of m 2 in the IR. As final value of t in the IR we have chosen t = 25, since at this point the FRG flow is effectively frozen in, i.e., no quantity changes anymore with RG time. We convinced ourselves that this is true by computing m 2 for t = 50 and t = 100. The values of m 2 at these times were the same as at t = 25 up to machine precision. Thus, in the symmetric phase it is possible to reach arbitrarily small momentum scales to obtain IR quantities. 3. Error on σ IR 0 Several aspects have to be taken into account regarding the determination of the value of σ IR 0 . First of all, in the broken phase, u(t, σ) is negative for σ < σ 0 and approaches zero from below as t → ∞. As a consequence, the denominator in Eq. (19) becomes very small when approaching the IR, which leads to a stiff problem when solving the FRG flow equation numerically, i.e., the time step ∆t has to become increasingly smaller when t → ∞ in order to avoid numerical instabilities. In practice, this prevents us to go arbitrarily far into the IR within our setup. For recent advances on this issue, we refer to Ref. [90]. This problem is solved by determining the position of the minimum σ IR 0 via extrapolation. Here, we decided to perform an exponential extrapolation of σ 0 as a function of k via a linear extrapolation of the values of ln σ 0 ln σ 0 = a k + b , ⇐⇒ σ 0 = e b e ak ,(A2) as can be seen from Fig. 9. This functional form of the extrapolation is motivated by the behavior of σ 0 in the large-N limit, where such an exponential scaling is exact, as can be shown by solving the FRG flow equation analytically via the method of characteristics (see Refs. [35,91]). This extrapolation is the reason for the small oscillations of ln σ IR 0 observed in Fig. 3. Even if the extrapolation procedure were to give the correct value, the uncertainty on σ IR 0 is still limited from below by the grid spacing ∆σ. Assuming that the correct value lies within a cell, one can still determine whether σ 0 lies to the left or the right of the cell center, such that the uncertainty is ∆σ IR 0 < ∼ ∆σ/2. Indeed, it can be seen from Fig. 10 that, starting with the same fixed σ UV 0 , the position of the minimum is independent of the cell size ∆σ for larger momentum scales and becomes dependent on ∆σ as k decreases (the shifts of the curves in the right panel, which are performed to increase the visibility of the ∆σ bands, are smaller than the actual differences between the curves). However, we observe that, for any given ∆σ, for small k the fluctuations of σ 0 as a function of k stay within a band given by ±∆σ/2, confirming the aforementioned assumption for the error on σ IR 0 . In addition, these fluctuations seem to decrease as ∆σ is reduced. Thus, assuming ∆σ/2 as the error on every value of σ IR 0 , each point in Fig. 3 has an error given by δ ln σ IR 0 = δσ IR 0 σ IR 0 = ∆σ 2σ IR 0 . (A3) Note that this error depends on σ IR 0 itself. We then consider the value of β extracted from the line which passes through the points ( 0 | t=0 − c 0 | t=0 ) lef t , ln σ IR,lef t are the errors of the leftmost and rightmost points of the selected fit interval. The difference of this value of β and the one extracted from the slope of the straight line in Fig. 3 gives our estimate for δβ, which then reads δβ = δ ln σ IR,lef t 0 + δ ln σ IR,right 0 ∆ f it ,(A4) where ∆ f it is the range in ln( 0 | t=0 − c 0 | t=0 ) where the linear fit is performed. Using typical values encountered during the calculations, i.e., ∆ f it 3.5, ∆σ = 0.0005 one finds δβ 0.006 . (A5) We will assume this as error on the critical exponent β. It is worth noticing that this contribution is significantly larger than the one coming from the fit error, i.e., the one extracted from the maximum deviation of the points from the straight-line fit, which is usually of order δβ f it ∼ 0.0002 .(A6) If one wants to have an upper and lower bound for the error, one can use the minimum and the maximum value for σ IR 0 in the fit region in Eq. (A4). One gets δβ > = 0.024 δβ < = 0.007 . Thus, our estimate of the error seems a good compromise between the highest and the lowest possible errors. III. APPLICATION: O(N ) MODEL IN LPAIn this section we will briefly introduce the O(N ) model and then apply the FRG approach in LPA to derive the respective flow equation. A. The O(N ) model Let us consider the O(N ) model in d-dimensional Euclidean space-time. The model describes the dynamics of N scalar fields φ a (x), where a = 1, . . . , N , with the bare action FIG. 1 . 1The effective potential V k (σ) (right panel) and its derivative u(t, σ) (left panel) in LPA at RG time t = 3, for different values of σ0|t=0 = 2 0|t=0 and for N = 3. The UV cut-off scale is Λ = 1.0, the grid size is [0, σmax] = [0, 2.0], while the grid spacing is ∆σ = 0.005 (corresponding to 400 grid points). FIG. 2 . 2Scaling of˜ 0,k as a function of RG-time t in the LPA for N = 3 and d = 3. The calculations are performed for Λ = 1.0, σmax = 2.0, and grid spacing ∆σ = 0.0005 (i.e., an order of magnitude smaller than for FIG. 3 . 3Double-logarithmic plot of Eq. (28) (violet dots) in the LPA case for N = 3 and d = 3. The green dots mark the boundaries of the error band. The slope of the blue line gives the estimate for the critical exponent β. Other parameters of the calculation are the same as for Fig. 2. FIG. 4 . 4Double-logarithmic plot of Eq. (33) (red dots) in the LPA case for N = 3 and d = 3. The slope of the blue line gives twice the value of the critical exponent ν. Other parameters of the calculation are the same as for Fig. 2. Large N 0.5 1.0 0 5 −1.0 2.0 TABLE III. Critical exponents in the large-N case. FIG. 5 .FIG. 6 .FIG. 7 . 567Different fitting regions corresponding to different numbers of most aligned consecutive points, for the extraction of the critical exponents β (left panel) and ν (right panel). Here, N = 3 and d = 3. Other parameters of the calculation are the same as for Fig. 2. Critical exponent ν as a function of the number of the most aligned points Np taken into consideration in the fit, for the case N = 3 and d = 3. Other parameters of the calculation are the same as for Fig. 2. Critical exponent β as a function of the number of the most aligned points Np taken into consideration in the fit (left panel). The right panel is zooming in on the region where the value of the critical exponent is least dependent on Np. In both plots N = 3 and d = 3. Other parameters of the calculation are the same as for Fig. 2. FIG. 8 . 8Curvature mass squared in the IR, m 2 (left panel), and relative deviation of m 2 from the value calculated for the smallest cell size (right panel) as functions of the cell size ∆σ. Here, N = 3, d = 3, Λ = 1.0, and σmax = 2.0. FIG. 9 . 9Logarithm of the minimum σ0 of the effective potential as a function of the momentum scale k, for N = 3 and d = 3. The right panel shows the range 0.1 ≥ k ≥ 0 from the left panel in higher resolution, with the corresponding linear extrapolation to determine the minimum σ IR 0 at k = 0. Other parameters of the calculation are the same as for Fig. σ=0.00133)+0.0000 (Δ σ=0.00100)+0.0005 (Δ σ=0.00067)+0.0010 (Δ σ=0.00050)+0.0015 FIG. 10. Position of the minimum of the effective potential σ0 as a function of the momentum scale k for different values of the cell size ∆σ, for N = 3 and d = 3. The right panel shows the range 0.0001 ≥ k ≥ 0 from the left panel in higher resolution, where the fluctuations in the value of σ0 can be seen. Each band around a line has a width ∆σ. In order to avoid overlapping bands, we have performed a global shift of σ0 for the cases ∆σ = 0.001, 0.00067, and 0.0005 by 0.0005, 0.001, and 0.0015, respectively. Other parameters of the calculation are the same as for Fig. 2. 0 − 0δ ln σ IR,lef t 0 and ( 0 | t=0 − c 0 | t=0 ) right , ln σ IR,right 0 + δ ln σ IR,right 0 ,where δ ln σ IR,lef t 0 and δ ln σ IR,right 0 TABLE II . IICorrelation length exponent ν. As already mentioned, there are more advanced truncations available for high-precision calculations of the critical exponents of O(N ) models via the FRG[30,31,33,34], and we do not attempt to compete with the respective results. 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{'abstract': 'We compute the critical exponents of the O(N ) model within the Functional Renormalization Group (FRG) approach. We use recent advances which are based on the observation that the FRG flow equation can be put into the form of an advection-diffusion equation. This allows to employ well-tested hydrodynamical algorithms for its solution. In this study we work in the local potential approximation (LPA) for the effective average action and put special emphasis on estimating the various sources of errors. Our results complement previous results for the critical exponents obtained within the FRG approach in LPA. Despite the limitations imposed by restricting the discussion to the LPA, the results compare favorably with those obtained via other methods.', 'arxivid': '2303.16838', 'author': ['Fabrizio Murgana \nDepartment of Physics and Astronomy\nUniversity of Catania\nVia S. Sofia 64I-95125CataniaItaly\n\nINFN-Sezione di Catania\nVia S. Sofia 64I-95123CataniaItaly\n\nInstitut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany\n', 'Adrian Koenigstein \nInstitut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany\n\nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n', 'Dirk H Rischke \nInstitut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany\n\nHelmholtz Research Academy Hesse for FAIR\nCampus Riedberg, Max-von-Laue-Straße 12D-60438Frankfurt am MainGermany\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUniversity of Catania\nVia S. Sofia 64I-95125CataniaItaly', 'INFN-Sezione di Catania\nVia S. Sofia 64I-95123CataniaItaly', 'Institut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany', 'Institut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany', 'Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany', 'Institut für Theoretische Physik\nGoethe-Universität\nMax-von-Laue-Straße 1D-60438Frankfurt am MainGermany', 'Helmholtz Research Academy Hesse for FAIR\nCampus Riedberg, Max-von-Laue-Straße 12D-60438Frankfurt am MainGermany'], 'corpusid': 257804763, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 27795, 'n_tokens_neox': 22891, 'n_words': 12947, 'pdfsha': 'dbe7192b6973d3a06bd9317631af7beca6982805', 'pdfurls': ['https://export.arxiv.org/pdf/2303.16838v2.pdf'], 'title': ['Reanalysis of critical exponents for the O(N ) model via a hydrodynamic approach to the Functional Renormalization Group', 'Reanalysis of critical exponents for the O(N ) model via a hydrodynamic approach to the Functional Renormalization Group'], 'venue': []}

arxiv

Physics-informed neural network applied to surface-tension-driven liquid film flows Yo Nakamura Graduate School of Integrative Science and Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Suguru Shiratori Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Ryota Takagi Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Michihiro Sutoh Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Iori Sugihara Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Hideaki Nagano Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Kenjiro Shimano Department of Mechanical Systems Engineering Tokyo City University Setagaya-ku158-8557TokyoJapan Physics-informed neural network applied to surface-tension-driven liquid film flows machine learninglubrication equationlong-wave approximationLaplace pressurethickness variation A physics-informed neural network (PINN), which has been recently proposed by Raissi et al. [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time evolution of the thickness distribution h(x, t) owing to the Laplace pressure, which involves 4th-order spatial derivative and 4th-order nonlinear term. Even for such a PDE, it is confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some improvements are needed in training convergence and accuracy of the solutions. The precision of floating-point numbers is a critical issue for the present PDE. When the calculation is executed with a single precision floating-point number, the optimization is terminated due to the loss of significant digits. Calculation of the automatic differentiation (AD) dominates the computational time required for training and becomes exponentially longer with increasing order of derivatives. By splitting the original 4th-order single PDE into lower-order coupled PDEs, the computational time for each training iteration is greatly reduced. The sampling density of training data also significantly affects training convergence. For the problem considered in this study, improved convergence was obtained by allowing the sampling density of training data to be greater in earlier time ranges, where the rapid flattening of the thickness occurs. Introduction Thin liquid films are ubiquitous in nature and technology [1,2]. The coating process of photoresist films plays important roles in industrial micro-fabrication techniques. For semiconductor devices, coated photoresist films are removed in the middle of the fabrication process, whereas for MEMS devices, anti-reflection films for optical lenses, or color filters of image sensors, some parts of photoresist films are used as final structures of the devices. Therefore, the design of devices determines the film thickness, which often exceeds the order of 10 µm, and its uniformity is desired at a high level. However, liquid film suffers from various thickness undulations due to physical phenomena during the coating process. The better-known thickness undulations are "edge-bead" and "striation". Edge-bead is a thick ridge that appears along the substrate periphery. The substrate region eroded by edge beads is not usable, thus they result in the loss of product yield. Shiratori and Kubokawa investigated edge-bead generation for the case where the bead had a double-peaked shape in the direction away from the substrate periphery, and proposed a simple explanation for the mechanism of the double peak [3]. Striation is another typical thickness undulation, which appears as radial spoke-like patterns all over the film [4,5]. The mechanism of striation has been investigated and, Marangoni-Bénard instability is considered to play a fundamental role [4,5,6,7,8,9,10,11,12]. Shiratori et al. investigated the formation process of striations by measuring spatiotemporal thickness variations during a spin coating for mixtures of epoxy resin and xylene solvent [13]. They found that short-wavelength thickness variation may suddenly vanish in the middle of the process. They also performed a numerical simulation to predict the transient Marangoni number during spin-coating, and concluded that the sudden change in the thickness spectrum found in the experiment is associated with the decrease in the Marangoni number to a sub-critical value for a specific bifurcation of the flow regimes. Such thickness variations must be avoided or suppressed in industry. To this end, numerical simulations are often used to find the optimal coating conditions, which allows thickness undulations to be minimized. In previous research on thin liquid films, many numerical methods have been developed [14,15,16,17,18,19,20]. However, it has been difficult to solve the inverse problem of finding the optimal coating conditions, because of the computational cost of the time integration of the governing equations. In the general optimization procedure, it is necessary to evaluate the objective function and its gradient with respect to the parameters to be optimized. Both require considerable computational cost in conventional numerical methods. A fast-growing machine learning approach can be considered to resolve such problems. Previously, machine learning approaches have been applied to the field of fluid dynamics [21,22,23,24,25]. There are many potential applications including not only a surrogate model for high-fidelity simulation, but also the reduced-order modeling, closure modeling, or a flow control. The recent trend of the machine-learned fluid dynamics are summarized by, e.g. Brunton et al. [26], Fukami et al. [27] and Duraisamy [28]. In general, machine learning has been successful in fields such as image recognition and natural language processing, where it is easy to obtain a sufficiently large training dataset. The above-mentioned applications of machine learning to fluid dynamics primarily involve turbulence, for which abundant computational results have been accumulated. In fields where sufficient training data is not readily available, obtaining training data for numerical simulations is typically very expensive. This applies to research into liquid film flows. In this study, we focus on a physics-informed neural network (PINN), as recently proposed by Raissi et al. [29]. The PINN learns the solutions of a partial differential equation (PDE) for a given dataset. In the training process for a PINN, a loss function is defined as the mean square error of the predicted solutions of the PDE. To evaluate the loss function, the temporal and spatial derivatives of the unknowns are calculated by automatic differentiation (AD), which is implemented in the neural network (NN) framework. Once a PINN has been trained, the solutions for any time instance can be calculated directly without time integration by forward computation by the NN. In addition, the gradient of the solution with respect to the input variable can be calculated efficiently using AD. The methodologies of the PINN are based on supervised learning; nevertheless, it does not require supervisor data, because the supervisor is assigned to the governing equation which must equate to zero. Therefore, the PINN is expected to perform well with limited training data. However, the application and validation of a PINN has only been carried out for several simple problems. In particular, there is no prior research regarding applications of a PINN to liquid film flows. The typical governing equation for liquid film flows involves a fourth-order spatial derivative and fourth-order nonlinearity in the Laplace pressure term. For higherorder derivatives, the AD may require greater computational time. The Laplace pressure term behaves as a diffusion of the curvature with the cube of the thickness h 3 being the diffusion coefficient. Thus, if h is unexpectedly predicted as negative, the Laplace pressure term acts as an anti-diffusion, which leads to numerical instability. For equations in which this occurs, the applicability of a PINN has not been investigated with respect to accuracy and computational cost. The goal of the present study is to resolve this problem. We aim to clarify whether a PINN can predict the solution of the governing equation with sufficient accuracy and acceptable computational time. For validation, the solution predicted by the PINN is compared with that calculated by the conventional finite difference method (FDM). We also investigate the computational cost for the automatic differentiation of high-order derivatives. If excessive computational time is required, it may be preferable to divide the equation into multiple sub-equations of low-order derivatives, by introducing intermediate unknowns. After the problem formulation given in Sec. 2, the detailed architecture and numerical methods are formulated in Sec. 3. Detailed methods in the FDM, which is executed for comparisons, are described in Sec. 4. The validity and efficiency of the PINN for liquid film flows are confirmed in Sec. 5 by comparing the results obtained in this study with those obtained using the FDM. In Sec. 5, in addition to the representative case, several aspects are investigated to improve the performance of the PINN. Problem formulation Basic form We consider a liquid film of an incompressible Newtonian fluid of viscosity µ and surface tension σ, which is coated on the substrate with average thickness h 0 . As shown in Fig. 1, the film is initially bumped according to the following function: h ini (x) = (h 1 − h 0 ) 1 − x λ 2 exp − x 2 2λ 2 + h 0 ,(1) where h 1 is the maximum thickness, and λ is the parameter which determines the extent of the bump. Such a deformed interface has a curvature distribution, which causes a gradient in the Laplace pressure. Owing to the flow driven by the pressure gradient, the interface deformation decays with time. The problem addressed in this study may seem rather simple. In the practical liquid film flows, there may occur some physics other than the Laplace pressure, such as the advection and leveling effects of the gravity, the Marangoni convection, a wetting phenomena, or evaporation and so on. These physics can be expressed by adding convection or diffusion terms in the governing equation of the present work. In addition, the framework of the PINN can be applied to two or three dimensional problems. Nevertheless, the present study focuses only on the Laplace pressure in onedimensional problems. In the previous studies, it has been confirmed that the PINN is able to predict accurate solutions of several fundamental types of transport equations, for instance, Burgers equation, Navier-Stokes equation and Allen-Cahn equation. These equations involve the advection and diffusion terms which are described by the spatial derivatives up to the second-order, whereas the Laplace pressure term in the liquid film flow is described by the fourth-order derivative. The main purpose of the present study is to confirm whether the PINN can be applied to a such a higher-order PDE, and to extract the computational issues to which the attention should be paid. The leveling process of the thickness undulation can be modeled using the following governing equation F : ∂h ∂t + ∂ ∂x h 3 ∂ 3 h ∂x 3 = 0,(2) where h is the thickness of the liquid film, t is the time, and x is the spatial coordinate. The derivation of Eq. (2) is described, for instance, in Oron et al. [1] or Shiratori et al. [13]. The equations are nondimensionalized using the scale shown in Table 1. The spatial domain is defined as x ∈ [−x L : x L ] and the following boundary conditions are employed: ∂h ∂x = ∂ 3 h ∂x 3 = 0 at x = ±x L ,(3) which are corresponding to symmetry conditions for the thickness h and the curvature ∂ 2 x h, respectively. Division of equations Equation (2) involves the fourth-order derivative, which may incur a large computational cost in the automatic differentiation step in the PINN. To address this, instead of using the original Eq. (2), we divide the equation by introducing intermediate variables. The second-order two PDEs can be obtained by explicitly introducing the Laplace pressure p as G 1 : ∂h ∂t − ∂ ∂x h 3 ∂p ∂x = 0,(4a)G 2 :p + ∂ 2 h ∂x 2 = 0. (4b) Similarly, the original PDE can be divided into first-order four PDEs as by introducing additional variables q 1 , q 2 , and q 3 . Regarding the initial condition for the divided equations, Eq. (1) is applied to h, whereas no conditions are applied to the intermediate variables p, q 1 , q 2 , and q 3 . In the boundary conditions, the curvature gradient condition ∂ 3 x h = 0 is replaced by the lower-order derivative forms: H 1 : ∂h ∂t − 3h 2 q 1 q 3 − h 3 ∂q 3 ∂x = 0,(5a)H 2 :q 1 − ∂h ∂x = 0,(5b)H 3 :q 2 − ∂q 1 ∂x = 0,(5c)H 4 :q 3 − ∂q 2 ∂x = 0,(5d)q 1 q 2 q 3 x t h Input Output (c)∂h ∂x = ∂p ∂x = 0 at x = ±x L , (for 2nd-order problem) (6) ∂h ∂x = q 3 = 0 at x = ±x L , (for 1st-order problem)(7) 3. Physics-informed neural networks Architectures Throughout this study, we have been using simple deep feed-forward neural network architectures, as shown in Fig. 2. All the hidden layers are fully connected dense layers and all the activation functions are hyperbolic tangents tanh. For the fourth-order governing equation Eq. (2), a single-output network ( Fig. 2(a)) is used, whereas a two-output network ( Fig. 2(b)) and a four-output network (Fig. 2(c)) are used for the second-order PDEs Eq. (4) and first-order PDEs Eq. (5), respectively. The number of hidden layers is defined as N and the number of neurons in each hidden layer is denoted as N h . Thus, the total number of the training parameters is (N in + 1)N h + (N − 1)(N h + 1)N h + (N h + 1)N out , where N in and N out are the number of inputs and outputs, respectively. Loss function The loss function for training is defined as follows: J = J f + J b + J o + J p ,(8a)J f = 1 N f N f i=1 N k k=1 G k t i f , x i f 2 ,(8b)J b = 1 N b N b i=1 B t i b , x i b − b i 1 2 + ∂ x h t i b , x i b − b i 2 2 ,(8c)J o = 1 N o N o i=1 h t i o , x i o − h i o 2 ,(8d)J p = 1 N f N f i=1 f p h t i f , x i f ,(8e) where J f is the mean squared error (MSE) of the governing equations G k (t, x). Although the governing equations must be mathematically satisfied, residuals remain when the equation is evaluated numerically by a neural network using automatic differentiation. The components J b and J o are the MSEs of the boundary and initial conditions, respectively. {t i f , x i f } N f i=1 denote the collocation points for the governing equation, whereas {t i b , x i b , b i 1 , b i 2 } N b i=1 and {t i o , x i o , h i o } N o i=1 are the boundary and initial conditions of the training data set, respectively. The values for the boundary and initial conditions are described in the next subsection. The concrete forms of G k and B are dependent on the problem, as shown in Table 2. Here, the number of PDEs is denoted by N k . The evaluation of J b and J o requires supervisor data, whereas the term J f does not need supervisor. J p is the loss based on a penalty function, defined as f p (h) = max 0, exp (−h) − 1 ,(9) which is introduced so that the network does not predict a negative value of h. In the governing equations Eqs. (2), (4) and (5) the fourth-order terms represent the diffusion of the Laplace pressure with the term h 3 being the diffusion coefficient. Thus, if the PINN unexpectedly predicts a negative value of h, the equation behaves as anti-diffusion, rendering computation unstable. The penalty function Eq. (9) is introduced in the loss function to avoid this situation. Training data The collocation points {t i f , x i f } N f i=1 for the governing equation are generated according to the following procedures. First, one-dimensional arrays are generated for both time {t s } N t s=1 and space {x s } N x s=1 . The array x s is generated with equidistant space as x s = x L −1 + 2 s − 1 N s − 1 , s = 1, · · · , N s ,(10) whereas the array for the time points is evenly spaced on a logarithmic scale as a s = log Γ t min t max N s − s N s − 1 t s = t max Γ a s            s = 1, · · · , N s ,(11)G k (t, x) N k B(t, x) 4th-order one PDE F(t, x) 1 ∂ 3 x h 2nd order two PDEs G k (t, x) 2 ∂ x p 1st-order four PDEs H k (t, x) 4 q 3 where Γ is the base of the logarithm, t min is the smallest time interval, and t max is the upper bound of the time range under consideration. For comparison, uniform time spacing t s = t max s − 1 N s − 1 , s = 1, · · · , N s ,(12) is also investigated. Using these one-dimensional arrays {t s } N t s=1 and {x s } N x s=1 , two-dimensional N t × N x grid points are generated by combining two arrays in round-robin style. From these parent grid points, the final collocation points of N f are randomly sampled. The size of the parent arrays is N t × N x , which is sufficiently larger than the final collocation points N f . Example collocation points are shown in Fig. 3. The training dataset for the boundary condition {t i b , x i b , h i b } N b i=1 is generated as follows: a i = log Γ t min t max N b − i N b − 1 t i b = t max Γ a i x i b = ±x L b i 1 = b i 2 = 0                          i = 1, . . . , N b .(13) The same condition is applied at both boundaries x = ±x L ; Thus, the total size of the data set for the boundary conditions is 2N b . For the initial condition, the training dataset {t i o , x i o , h i o } N o i=1 is generated on the equally-spaced spatial grids as: t i o = 0 x i o = x L −1 + 2 i − 1 N o − 1 h i i = h ini (x i o )                  i = 1, . . . , N o ,(14) where h ini (x) is the function defined in Eq. (1). Implementation For optimization we selected the L-BFGS-B method, which is a quasi-Newton, full-batch gradient-based optimization algorithm [30]. The code for the PINN is implemented in Python using the tensorflow-1.8 machine learning library [31], and executed with assistance of GPUs of the NVIDIA Tesla P100 and the NVIDIA A100. Finite difference method To confirm whether the PINN can correctly predict the solution, the same problem is also calculated using the finite difference method (FDM). The governing equation Eq. (2) is discretized by finite differences on an equidistant grid of N x points with an interval ∆x. The positivity-preserving scheme [32] is applied to h 3 of the Laplace pressure term in Eq. (2), because the standard finite difference schemes do not necessarily preserve positivity of h, and the occurrence of non-positive solutions introduces artificial instability [33]. Due to the fourth spatial derivative of h in Eq. (2), a time step restriction for the numerical stability requires ∆t < O ∆x 4 , which is a severe condition in practice. To cope with this restriction, the fully implicit time integration of the Newton-Kantrovich method is applied with the Crank-Nicholson method, according to Diez and Kondic [15]. The resulting linear systems are solved using BiCGStab iterations [34] with an auto-accelerated incomplete LU factorization (ILU) preconditioner [35]. The code for the FDM is implemented in C++ with OpenMP parallelization and executed using an Intel Xeon E5-2695v4 CPU. The accuracy of the present FDM solver has been validated through comparison with the benchmark problems provided by Dies and Kondic [15]. Although the result of the FDM is not an exact analytical solution, the prediction by the PINN is validated by comparing it with the FDM's result, which is calculated with sufficiently high resolution for the spatial grid (∆x = 5 × 10 −3 ) and time step (∆t = 4 × 10 −5 ). Results and discussions Representative case First, we demonstrate a representative case for the condition shown in Tables 3 and 4. The blue line in Fig. 4 shows the history of the loss function during training. The loss function J is reduced below 1 × 10 −3 after approximately 1 × 10 6 iterations. Because the loss function J is evaluated using only the selected sampling points (t i , x i ), it is not guaranteed that the PINN can predict an accurate solution for any given point, even if the value of the loss function is sufficiently small. Thus, the PINN solutions were compared in detail with results obtained using the FDM. Figure 5 summarizes the result of the representative case. The color contours in Fig. 5(a,b) show the spatio-temporal solutions h(t, x), where the color contours are scaled as log 10 (h). For comparison, the results calculated by the FDM are shown in Fig. 5(a), and the results predicted by the PINN are in Fig. 5(b). The lower three sub-figures Fig. 5(c,d,e) are selected snapshots of thickness variations at different time instants t = 0, 0.5 and 1.0. The training of the PINN is sufficiently converged, and the trained PINN can predict solutions sufficiently close to those of the FDM, even for the points (t, x) that are not provided as training data. In order to discuss the accuracy quantitatively, the difference between the results of the PINN and FDM is evaluated by the following root mean squared relative error: E FDM (t) = 1 N x N x i h PINN (t, x i ) − h FDM (t, x i ) h FDM (t, x i ) 2 ,(15) where h PINN and h FDM are the solutions predicted by the PINN and FDM, respectively. Figure 6 shows the instantaneous E FDM as a function of time t. The time-averaged value of the E FDM was 5.80 × 10 −3 , as indicated in Fig. 6, which can be considered sufficiently small. The maximum value of E FDM occurs at t = 0, even if the initial condition is applied. In classical numerical methods, the discretized governing equation is numerically integrated in time, starting from the initial condition. In the PINN, the initial condition is given as part of the loss function to be minimized; thus, it is not guaranteed that the initial condition is always satisfied. In the present problem, in which the initial thickness undulation is planarized by Laplace pressure, the temporal thickness variation is fastest at t = 0, and this is the reason why the E FDM reaches its maximum at t = 0. Table 5 shows the computational time required for the time integration of the FDM and training iterations in the PINNs. Because of the difference in the computational environments for the FDM and PINN, it is difficult to compare the computational time quantitatively. The code for the FDM is parallelized with OpenMP and executed in a shared memory CPU environment, whereas the code for the PINN is executed with the assistance of a GPU. Nevertheless, we note that training the PINN is computationally expensive. Once training of the PINN converges, the solution h(t, x) can be predicted immediately without any time integration. However, computational time for training is regarded as a major issue from a practical viewpoint. In the field of machine learning, many computationally efficient methods for training have been developed, and such methods can be easily applied to the framework of the PINN. In this study, we consider a PINN-specific method to reduce the computational requirements for training. Effect of order of derivatives in PDEs Automatic differentiation (AD) can be regarded as a bottleneck in the computational time for the PINN. In this subsection, we investigate the computational cost for AD, especially the dependence on the order of the derivatives in PDE. To this end, the computational time for AD is evaluated by the code whose main part is illustrated in the following Python code snippet: Figure 7 shows the computational time required for AD as a function of the order of derivatives. The dashed line shows the fitted function, which clearly indicates an exponential dependence of computational time on the order of the derivative. The time required for the 4th-order derivative is approximately 10 times longer than that for the 2nd-order. As far as the authors are aware, all the AD implementations available to date calculate the higher-order derivatives by recursion of the single-order AD. Therefore, the computational time for AD is exponentially dependent on the order of the derivatives. Regarding the PINN for the present liquid film flow problem, AD for the 4th-order spatial derivative seems to dominate the computational time. In this light, we consider reducing the order of the derivatives by dividing the 4th-order PDE into the coupled 2nd-order or 1st-order PDEs by introducing the intermediate variables, as shown in Eqs. (4) and (5). The neural network used in this investigation has multiple outputs, as shown in Fig. 2(b,c). The training data set is the same as that used with the 4th-order PDE. The history of the loss function J during training is indicated by the green line in Fig. 4. The trajectory and converged value of J for the 2nd-order and 1st-order PDEs are almost the same as those for the 4th-order PDE. With respect to the difference between the PINN and the FDM, the evaluated values of E FDM were almost the same as that for the 4th-order PDE. The entire computational time required for training is summarized in Table 5. The computational time taken for the training of the 2nd-order PINN is approximately 35 % of that for the 4th-order, whereas for the 1st-order PINN the computational time is approximately 26 %. By division of the PDE, the maximum order of the derivative decreases, whereas the number of terms to be evaluated increases. From the present investigation, it was confirmed that the decrease of the order of derivatives in the AD outweighs the increase in the number of equations, resulting in a net decrease in computational time required. This competition is strongly dependent on the efficiency of the algorithm for the higher-order AD, which may be improved in future. Precision of floating-point numbers In this section, we show how the precision of floating-point operations affects the training performance of the PINN. For the 1-dimensional Burgers equation, which is a 2nd-order PDE, Raissi et al. showed that a PINN can be trained using single-precision floating-point operations (FP32) with an activation function of tanh, and can achieve sufficient accuracy compared with the analytical solution [29]. For the present study, PINN training failed when FP32 precision was selected. Figure 8 indicates the history of the loss function J during training for the cases with different floating-point precision (FP32/FP64) and different orders of derivatives (2nd-/4th-order). The loss functions J for all cases show similar behavior, allowing for differences in the initial values because of the randomly initialized weights. Computation for the two FP64 cases continued until the number of iterations reached the selected maximum value of 2 × 10 6 . The FP32 cases, on the other hand, were terminated after approximately 2 × 10 4 iterations; nevertheless, the loss function J can potentially be reduced. It should be noted that the FP32 computation failed even when the governing equation was divided into two 2nd-order PDEs, which are denoted as the '2-Eqs. 32bit' case in Fig. 8. In the following, we consider the reason why the FP32 computation fails. In the present work, the L-BFGS-B method in the SciPy implementation is employed for optimization, and this algorithm stops its iteration when either or both of the following conditions are satisfied: J k − J k+1 max J k , J k+1 , 1 ftol, (16a) max i {|g i |} gtol,(16b) where J is the loss function, k is an index for iterations, and g i is the i-th component of the projected gradient of J. ftol and gtol are the tolerance values, selected as ftol = 2.2 × 10 −16 and gtol = 1 × 10 −15 , respectively. In the present FP32 computations, the iteration is stopped by the condition Eq. (16a); nevertheless, the projected gradient g is still much larger than the tolerance gtol. In general, the gradient-based optimization method first finds a descent direction p k along which the loss function J will be reduced, and then computes a step size α k that adequately reduces J(x k + α k p k ) relative to J(x). The step size α k is computed using line search methods, which have several variations. In an inexact line search, which is implemented in SciPy, the step size α k is determined such that the sufficient decrease condition and the curvature condition are satisfied [36]. In the present PINN computations with FP32, the step size α k becomes as small as O(10 −10 ) at termination, thus the update of the loss function J becomes vanishingly small and the stopping condition Eq. (16a) is satisfied. Such a small value of α k might occur when the loss function J has nearly stationary points and the parameter x gets trapped therein. The FP32 computation is likely to lose precision in the evaluation of J due to the cancellation of significant digits. It is well known that the activation function tanh is likely to suffer from the so-called vanishing gradient problem [37]. The vanishing gradient problem occurs because the tanh function has gradients in a narrow range, and backpropagation computes gradients using the chain rule. Thus, the gradient decreases exponentially during backpropagation, and the weights in the early layers receive vanishingly small updates. The PINN for the present work may suffer from the vanishing gradient problem, because the number of chain rule operations increases exponentially as the order of the derivative increases, as shown in Fig. 7. The use of rectifier functions such as ReLU is expected to reduce the effect of the vanishing gradient problem, because these functions saturate in only one direction. To investigate the effect of the activation function, we trained the PINN for the 4th-order PDE of Eq. (2) with FP32 precision, by changing the activation function tanh to the swish function, which is defined as However, training with FP32 using this activation function failed for the same reason as for tanh. Although there are many other rectifier functions, their effects have not been investigated in the present study. Even if some activation function enables successful training with FP32 precision, there still exists problem of selecting an appropriate activation function. a(z) = z 1 + exp(−z) .(17) As described above, the FP32 computation failed even when the governing equation is divided into two 2nd-order PDEs. This suggests that the failure of FP32 computation is not caused by higher-order automatic differentiation, which requires many multiplications in the chain rule. Based on the investigations described so far, the failure of PINN training with FP32 precision seems to be caused by the loss of significant digits during the line search when the loss function has nearly stationary points. The failure of training with FP32 precision can potentially be avoided by the selection of an appropriate optimization method or the adjustment of the numerical parameters for optimization. However, from a comprehensive standpoint, FP32 computation has a risk of failure for the present problem compared with FP64. Density distribution of collocation points It was found that the convergence and accuracy of the PINN are strongly affected by the sampling density of the collocation points. We considered two types of scaling of points in time, as described in Sec. 3.3. One of them is a linear scale sampling, where the time points are evenly spaced as in Eq. (12). The other is a log scale sampling, where the time points are evenly spaced on a logarithmic scale as in Eq. (11). The examples of the collocation points are shown in Fig. 3. The convergence and the accuracy of the PINN are investigated for both sampling types. Figure 9 indicates the history Loss function Figure 11. History of the loss function J during training for different numbers of hidden layers N and numbers of neurons in each hidden layer N h . of the loss function J during the training. The blue line corresponds to log scale sampling, which is the same as the green line in Fig. 4. The orange line in Fig. 9 is the result of linear scale sampling. It looks at first glance as though the linear scale provides faster convergence and a smaller final value of J compared to the log scale. However, it should be noted that the solutions predicted by the trained PINN are not guaranteed to be accurate for any given point (t, x) even if the loss function J converged to a sufficiently small value. Because the loss function J is evaluated using only the selected sampling points of (t i , x i ), the sampling of the collocation points significantly affects the absolute value of J. Thus, the solutions are compared with the result of the FDM. Figure 10 shows the selected snapshots of the thickness distributions predicted by the trained PINN for the case of linear scale sampling. Although the loss function J converges and the final value of J for the case of the linear scale is smaller than that for the log scale, the predicted solutions differ unacceptably from the result of the FDM. Appropriate sampling of collocation points surely depends on the problem to be solved. It seems that the collocation points should be dense where the solution has large variations in spatial and/or temporal directions. For the present problem, log-scale sampling is effective because the rapid thickness variation occurs at earlier time ranges. Iterations ℓ =2, ℎ =10 ℓ =4, ℎ =10 ℓ =6, ℎ =10 ℓ =8, ℎ =10 ℓ =2, ℎ =20 ℓ =4, ℎ =20 ℓ =6, ℎ =20 ℓ =8, ℎ =20 ℓ =2, ℎ =40 ℓ =4, ℎ =40 ℓ =6, ℎ =40 ℓ =8, ℎ =40 Training efficiency might be further improved by more optimized sampling of the collocation points. One possible approach is a self-adaptive approach, which has been proposed by McClenny & Braga-Neto [38]. In their approach, the trainable weights are introduced as a soft multiplicative mask for the mean squared error for each collocation point, and these weights are optimized concurrently with the network weights. Effect of network size In the results described so far, the network architecture is kept fixed unless otherwise noted, as shown in Fig. 2(a). In the present study, the network is composed of fully-connected hidden layers, thus the network size is simply determined by the number of the hidden layers N and the number of neurons N h in each layer. With a network size of N = 8 and N h = 20, it has been shown that the PINN can be trained with sufficient accuracy. The PINN can be regarded as a kind of fitting of the nonlinear mapping between inputs (time and space) and outputs (solution) by the linear combinations and simple nonlinear activation functions. Therefore, how complicatedly the mapping can be reproduced must depend on the number of degrees of freedom of the neural network. In this section, the effect of the network size is investigated by varying N and N h . Fig. 11 shows the history of the loss function J during training. The final values of J after training is terminated are summarized in Table 6. The trajectory and converged values of J are strongly dependent on the network sizes N and N h . The definition of the loss function J itself is not dependent on the network size, and the number of sampling points is kept fixed for all cases as N f = 6000, N b = 1000 and N o = 400. Thus, the difference in the value of J can be regarded as an effect of the network size. As can be easily predicted, for larger values of N and N h , the loss functions J converged to smaller values. As described in Sec. 5.1, the PINN with network size (N , N h ) = (8,20) can be trained successfully. The root mean square relative error between the PINN prediction and the result of the FDM was 5.80 × 10 −3 on average. With this network size, the final value of J was 6.36 × 10 −4 , which can be considered as sufficient convergence. Among all variations of N and N h in the present work, the final values of the loss function surpassed J 10 −3 for five cases of (N , N h ) = (6, 20), (8,20), (4,40), (6,40), and (8,40). To estimate the necessary values for N and N h such that sufficient convergence of J can be achieved, the calculated data are fitted by the following function log 10 [J fit (N , N h )] = a 0 + a 1 exp − N b 1 + a 2 exp − N h b 2 ,(18) where the coefficients are determined by the least square method as a 0 = −3.81, a 1 = 4.39, b 1 = 2.09, a 2 = 2.43, and b 2 = 13.7, with the residual sum of squares of R 2 = 8.88 × 10 −2 . In Fig. 12, the final values of the loss function J are plotted three-dimensionally by the red stars. In addition, the fitted function Eq. (18) is represented by the blue surface. From the fitting function Eq. (18), we can estimate the necessary values for N and N h so that sufficient convergence of J can be achieved. This necessary condition is plotted in Fig. 12 as a black dashed line, by considering sufficient convergence to be J 10 −3 . At the bounds of the range investigated in the present study, this condition corresponds to N = 3.9 for N h = 40 and N h = 16.7 for N = 8. When the degrees of freedom in the network exceed the above condition, the PINN can reproduce the nonlinear mapping between x, t, and h sufficiently with the present governing equation. Robustness of trained model When the physics-informed neural network is applied to the practical problems, for instance, the data assimilation, the input data may include noise. For such a situation, the solutions predicted by the trained PINN should not largely deviate from those for the clean (noise-free) inputs. For the trained PINN of the present work, the effect of the noise was investigated as follows. First, the evaluation points x i , t i are generated with equally-spaced grids of (N x , N t ) = (1000, 400). The noisy inputs are generated by adding the normally-distributed noises: δx, δt ∼ N (0, σ n ) ,(19) where δx and δt are the noise for x and t, respectively. σ n is the standard deviation of the normal distribution. The solutions of the PINN are calculated for both of the noisy and clean inputs, then the difference of them ∆h i and the root mean squared error E n are defined as ∆h i = h PINN (x i + δx i , t i + δt i ) − h PINN (x i , t i ) ,(20)E n =         1 N e N e i=1 (∆h i ) 2         1 2 ,(21) where N e = N x × N t is the total number of the evaluation points. The values of E n are calculated by changing the noise corruption levels. Figure 13 indicates the root mean squared error E n as a function of the standard deviation σ n for the applied noise. It can be seen that the E n is approximately the same order of magnitude with the σ n . This means that the thickness for the noise-corrupted inputs can be obtained with the error of the the same order of magnitude of the noise included. It can be regarded that the PINN of the present work was not overfitted for the training dataset, and it can provide the accurate solutions even for the inputs that was not encountered during the training. The robustness of the trained PINN was also investigated from another viewpoint: how the PINN predicts the solution when the time input is outside of the training data regime. In general function fitting methods, the function is fitted for the values that span a specified dataset only. For the outside of the dataset, output of the fitted function are not guaranteed, and the output values might be largely different from those expected from the original dataset. Although the PINN is one of the function fittings, in contrast to the conventional methods, the loss function to be minimized is based not on the function value itself but on the governing equation. The weights of the PINN are optimized so that not only the field variable, but also the time and spatial derivatives satisfy the governing equation. Thus the solutions of the PINN may satisfy the governing equation in certain degree, even for the time outside of the dataset. In this light, the prediction of the PINN were evaluated for the wide time range. Figure 14 shows the instantaneous root mean squared relative error E FDM between the solutions of the PINN and the FDM, which is defined as Eq. (15). The collocation time points t i for the training are sampled from the time horizon 0 t 2, which is indicated by the blue line in Fig. 14. The plot for this time range is the same as that indicated in Fig. 6. The red line in Fig. 14 indicates the error E FDM for the time range 2 < t, which is outside of the training dataset. Although the value of E FDM increases with time t, it does not diverge and takes at most E FDM = 1.71 × 10 −1 in the evaluated time range t 10. Figure 15 shows the selected snapshots of the instantaneous spatial thickness distribution h, where the solutions obtained by the FDM are also plotted. It can be seen that the solution of the PINN at t = 3.0 is close to that obtained by the FDM (Fig. 15(a)). For the later time range (Fig. 15(b,c)), the solutions of the PINN clearly differ from those of the FDM. Though in the qualitative sense, some similarities can be recognized between solutions of the PINN and the FDM: the extrema and their locations, gradient at the boundaries. The spatial symmetry in h is not retained for the later time range. As shown in Fig. 15(d), the solutions of the PINN are also evaluated for the very large time value. For the sinusoidal deformation of the wavelength λ m , the time constant for the decay by the Laplace pressure can be derived as τ = 3µ σh 3 0 λ 4 m 16π 4 ,(22) from the analysis of the linearized governing equation. Using the time scale t * = 3µλ 4 /σh 3 0 of the current problem, as described in Table 1, the nondimensional form of Eq. (22) can be written asτ =λ 4 m /16π 4 , whereλ m stands for the nondimensional wavelength. In this problem, the longest wavelength can be estimated asλ m ≈ 2x L = 20, which is corresponding to the time constantτ ≈ 100. Therefore, the time instances are selected as t = 10 2 , 10 3 and 10 4 . For these time instances, the thickness must be flattened by the Laplace pressure. Accordingly, this situation can be confirmed in Fig. 15(d), where the solutions of the PINN are completely flat distribution. It is noted that the three plots for t = 10 2 , 10 3 , 10 4 are overlapped in the figure. By integrating the initial condition Eq. (1), the exact value of the averaged thickness can be analytically calculated as h ave = 1. The flat thickness at t = 10 4 is h = 1.028 ( Fig. 15(d)), which is 2.8 % higher than the exact solution. Concluding remarks In the present study, a physics-informed neural network was applied to a partial differential equation of liquid film flows. The PDE considered is the time evolution of the thickness distribution h(x, t) owing to Laplace pressure that involves the 4th-order spatial derivative and 4th-order nonlinear term. Even for such a PDE, it was confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some aspects are needed to improve training convergence and accuracy of the solutions. Calculation of the automatic differentiation (AD) dominates the computational time required for training, and becomes exponentially longer as the order of derivatives increases. By splitting the original 4th-order single PDE into lower-order coupled PDEs, the computational time for a single training iteration was greatly reduced. The precision of the floatingpoint numbers is a critical issue for the present PDE. When the calculation is executed with FP32 (single precision), training terminated due to the loss of significant digits, despite the loss functions not being reduced sufficiently. The density distribution of the collocation points for the training data also significantly affected training convergence. For the problem considered in the present study, convergence was improved by allowing the sampling density to be higher in earlier time ranges, where the rapid diffusion of the thickness occurs. To clarify the degrees of freedom in the neural network required to reproduce the nonlinear mapping between input and output, the accuracy of the PINN prediction was investigated by varying the number of hidden layers N and the number of neurons in each layer N h . From the fitting function Eq. (18), the necessary values for N and N h are evaluated. From a comprehensive point of view, the original form of the PINN requires a lot of attention to ensure successful training. This is especially true in the case of the higher-order PDEs dealt with the present study. A lot of preliminary investigation is needed to find appropriate configuration for the network size and selection of the collocation points. In addition, the long computational time required for training may also be practical bottleneck. Further research is needed to overcome these problems. Regarding computational time, fast algorithms and an efficient implementations for automatic differentiation, especially for higher orders of derivative, are needed. Regarding the selection of the collocation points, it is expected that self-adaptive approaches will improve the training performance. Another possible approach is an evolutional deep neural network (EDNN), which has been recently proposed by Du & Zaki [39]. In EDNN, the neural network is trained only for the initial condition, and the network weights are updated based on the time evolutional equation derived from the governing equation. Use of this improvement is expected to contribute to practical application of the PINN. Figure 1 . 1System to be considered. A liquid film of average thickness h 0 is initially bumped with maximum height h 1 , and decays due to Laplace pressure. Figure 2 . 2Network structures of the PINNs used in the present study. (a) single-output network for the 4th-order equation, (b) twooutput network for the 2nd-order equations, and (c) four-output network for the 1st-order equations. Figure 3 . 3Distributions of collocation points (t f , x f ) for the evaluation of governing equations (a) logarithmically sampled time points, and (b) uniformly sampled time points. The data sets (t o , x o ) and (t b , x b ) are also plotted. Figure 4 . 4History of the loss function during training. The blue line indicates the result for the representative case shown in Tables 3 and 4. The green and red lines depict the cases where the governing equation is divided into two 2nd-order PDEs and four 1st-order PDEs, respectively. Figure 5 . 5Comparison between results of the FDM and PINN. Upper: the spatio-temporal variation of the thickness h(t, x) predicted by (a) Finite Differences and (b) PINN. Lower: the selected snapshots of the instantaneous spatial thickness distribution at (c) t = 0, (d) t = 0.5, and (e) t = 1.0. Figure 6 . 6Instantaneous root mean squared relative error E FDM between the results of the PINN and FDM. Figure 7 . 7Computational time for automatic differentiation as a function of the order of derivatives. The number of hidden layers is N = 30, the number of neurons in each layer is N h = 200, and the size of the data set is N f = 20 000. The dashed line depicts the fitted exponential function 9.12 × 10 −2 exp(1.35x). This code is executed using tensorflow-1.8 by supplying the data set {t i , x i } N f i=1 of N f = 20 000 with a network size of (N , N h ) = (30, 200). Figure 8 . 8History of the loss function J during training for different floating-point precision and different orders of derivatives. Figure 9 .Figure 10 . 910History of the loss function J during training for different sampling density of the collocation points. The blue line indicates the result of log-scale time sampling, whereas the orange line represents linear-scale time sampling. Selected snapshots of the instantaneous spatial thickness distribution at (a) t = 0.5 and (b) t = 1.0, for the case in which collocation points are sampled on the linear scale. 66 × 10 −2 3.31 × 10 −2 1.10 × 10 −2 4 1.36 × 10 −2 2.54 × 10 −3 7.18 × 10 −4 6 4.83 × 10 −3 9.34 × 10 −4 3.22 × 10 −4 8 2.35 × 10 −3 6.36 × 10 −4 3.65 × 10 −4 Figure 12 . 12Final values of the loss function J when the optimization iteration is terminated, as a function of the number of neurons in each hidden layer N h and the number of hidden layers N . The red stars are calculated cases, and the blue surface indicates a fitted function log 10 [J fit (N , N h )] = a 0 +a 1 exp(−N /b 1 )+a 2 exp(−N h /b 2 ) with coefficients a 0 = −3.81, a 1 = 4.39, b 1 = 2.09, a 2 = 2.43, and b 2 = 13.7. The final sum of squares of residuals is R 2 = 8.88 × 10 −2 . The black dashed line drawn on the base plane indicates the values of N and N h where the loss function reaches J = 10 −3 . Figure 13 . 13Root mean squared error E n defined by Eq. (21) as a function of the standard deviation of noise added to the inputs. The dashed line indicates the y = x. Figure 14 .Figure 15 . 1415Instantaneous root mean squared relative error E FDM between the results of the PINN and FDM. The collocation time points t i for the training are sampled from the time horizon 0 t 2, which is indicated by the blue line. Selected snapshots of the instantaneous spatial thickness distribution for the time instances outside of the time horizon of collocation points for the training. (a) t = 3.0, (b) t = 5.0, (c) t = 10.0, and (d) t = 10 2 , 10 3 , 10 4 . In the sub-figures except (d), the solutions obtained by the finite difference method are also plotted for the comparison. Table 1 . 1Scales of the dimensionless variables.Lateral length Thickness Time Pressure λ h 0 3µλ 4 /(σh 3 0 ) h 0 σ/λ 2 Table 2 . 2Correspondence of equations depending on the problem.Governing PDEs Number of PDEs Curvature boundary condition Problem Table 3 . 3Methods employed.Table 4. Parameters and representative values used.Setting Base value Activation function tanh Precision of floating-point numbers FP64 Type of collocation time points log-scale Method of optimization L-BFGS-B Parameter Symbol Value Time range t max 2 Spatial extent x L 10 Maximum height h 1 /h 0 3 Data size for bulk equation N f 6000 Data size for boundary conditions N b 1000 Data size for initial conditions N o 500 Number of hidden layers N 8 Number of neurons in each layer N h 20 Table 5. Execution environments and elapsed calculation time required for the FDM and PINNs.def auto_diff (x , order ): h = neural_net (x , weights , biases ) for i in range ( order ): h = tf . gradients (h , x )[ ] return h PINN Method FDM 4th-order 2nd-order 1st-order CPU/GPU Xeon E5-2603v4 NVIDIA A100 Performance 163.2 GFLOPS 19.5 TFLOPS Time steps/epochs 5 × 10 4 1 × 10 6 1 × 10 6 1 × 10 6 Computation time 25 049 s 140 277 s 48 587 s 36 034 s (6.96 h) (38.97 h) (13.50 h) (10.01 h) Time-averaged E FDM - 5.80 × 10 −3 4.97 × 10 −3 4.99 × 10 −3 Table 6 . 6Final values of the loss function J when the optimization iteration is terminated. AcknowledgmentsThis work was supported by JSPS KAKENHI (Grant Number JP19K04175). The calculations shown in the present work were executed on the SGI Rackable C2112-4GP3/C1102-GP8 (Reedbush-U/H) and the FUJITSU Supercomputer PRIMEHPC FX1000 and FUJITSU Server PRIMERGY GX2570 (Wisteira/BDEC-01) in the Information Technology Center, The University of Tokyo. 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{'abstract': 'A physics-informed neural network (PINN), which has been recently proposed by Raissi et al. [J. Comp. Phys. 378, pp. 686-707 (2019)], is applied to the partial differential equation (PDE) of liquid film flows. The PDE considered is the time evolution of the thickness distribution h(x, t) owing to the Laplace pressure, which involves 4th-order spatial derivative and 4th-order nonlinear term. Even for such a PDE, it is confirmed that the PINN can predict the solutions with sufficient accuracy. Nevertheless, some improvements are needed in training convergence and accuracy of the solutions. The precision of floating-point numbers is a critical issue for the present PDE. When the calculation is executed with a single precision floating-point number, the optimization is terminated due to the loss of significant digits. Calculation of the automatic differentiation (AD) dominates the computational time required for training and becomes exponentially longer with increasing order of derivatives. By splitting the original 4th-order single PDE into lower-order coupled PDEs, the computational time for each training iteration is greatly reduced. The sampling density of training data also significantly affects training convergence. For the problem considered in this study, improved convergence was obtained by allowing the sampling density of training data to be greater in earlier time ranges, where the rapid flattening of the thickness occurs.', 'arxivid': '2109.04104', 'author': ['Yo Nakamura \nGraduate School of Integrative Science and Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Suguru Shiratori \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Ryota Takagi \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Michihiro Sutoh \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Iori Sugihara \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Hideaki Nagano \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n', 'Kenjiro Shimano \nDepartment of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan\n'], 'authoraffiliation': ['Graduate School of Integrative Science and Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan', 'Department of Mechanical Systems Engineering\nTokyo City University\nSetagaya-ku158-8557TokyoJapan'], 'corpusid': 237452206, 'doi': '10.1002/fld.5093', 'github_urls': [], 'n_tokens_mistral': 20862, 'n_tokens_neox': 17595, 'n_words': 10870, 'pdfsha': '860c23fd34d306c4409e41d5917ce3fd1f457945', 'pdfurls': ['https://arxiv.org/pdf/2109.04104v2.pdf'], 'title': ['Physics-informed neural network applied to surface-tension-driven liquid film flows', 'Physics-informed neural network applied to surface-tension-driven liquid film flows'], 'venue': []}

arxiv

Automated Data Denoising for Recommendation Yingqiang Ge yingqiang.ge@rutgers.edu Rutgers University Texas A & M University Inc., Inc., Inc Mostafa Rahmani Rutgers University Texas A & M University Inc., Inc., Inc Athirai Irissappane Rutgers University Texas A & M University Inc., Inc., Inc Jose Sepulveda Rutgers University Texas A & M University Inc., Inc., Inc James Caverlee caverlee@gmail.com Rutgers University Texas A & M University Inc., Inc., Inc Fei Wang feiww@amazon.com Rutgers University Texas A & M University Inc., Inc., Inc Automated Data Denoising for Recommendation 10.1145/nnnnnnn.nnnnnnnCCS CONCEPTS • Computing methodologies → Machine learningReinforce- ment learning• Information systems → Recommender sys- temsInformation retrieval In real-world scenarios, most platforms collect both large-scale, naturally noisy implicit feedback and small-scale yet highly relevant explicit feedback. Due to the issue of data sparsity, implicit feedback is often the default choice for training recommender systems (RS), however, such data could be very noisy due to the randomness and diversity of user behaviors. For instance, a large portion of clicks may not reflect true user preferences and many purchases may result in negative reviews or returns. Fortunately, by utilizing the strengths of both types of feedback to compensate for the weaknesses of the other, we can mitigate the above issue at almost no cost. In this work, we propose an Automated Data Denoising framework, AutoDenoise, for recommendation, which uses a small number of explicit data as validation set to guide the recommender training. Inspired by the generalized definition of curriculum learning (CL), AutoDenoise learns to automatically and dynamically assign the most appropriate (discrete or continuous) weights to each implicit data sample along the training process under the guidance of the validation performance. Specifically, we use a delicately designed controller network to generate the weights, combine the weights with the loss of each input data to train the recommender system, and optimize the controller with reinforcement learning to maximize the expected accuracy of the trained RS on the noise-free validation set. Thorough experiments indicate that AutoDenoise is able to boost the performance of the state-of-the-art recommendation algorithms on several public benchmark datasets. INTRODUCTION Recommender systems (RS) are an essential part of modern life, which are widely deployed in almost every corner of our daily routines and facilitate the human decision-making process by providing relevant suggestions. Various techniques have been adapted to enhance the capabilities of deep recommendation systems with the common goal of predicting more accurate user preferences, including but not limited to meticulously selecting user/item interaction features [19,27], or proposing novel loss functions [25,49]. However, very few works doubt the reliability of the data source. General recommender systems focus on modeling the compatibility between users and items, based on historical user-item interaction (e.g. clicks, purchases, likes). Existing literature usually separate user-item interaction into two categories [7,12,14,31]: explicit feedback and implicit feedback. On one hand, explicit feedback, such as ratings and reviews, are collected when users actively and explicitly tell the system about their preferences for an item, which are usually small-scale yet highly relevant. On the other hand, implicit feedback, such as user clicks or purchases, are passively recorded when users interact with the website interface, so they are easily collectable and large-scale but can only implicitly reflect the part of the users' preferences, as they are easily affected by the first impression of users and other factors, such as click baits [42], position bias [16], etc. In order to mitigate the data sparsity issue during model training, most deep recommenders utilize implicit data as input, however, prior work [14,31,44] points out that it is more challenging to utilize implicit data since there is an inevitable gap between implicit feedback and the actual user satisfaction due to the prevailing presence of noisy interactions (a.k.a, false-positive interactions), where the users may dislike certain interacted item. For example, in e-commerce platforms, large portions of clicks do not align well with user preferences, and many purchases end up with negative reviews or being returned. Moreover, existing studies [41,44] have further demonstrated the detrimental effect of such noisy data samples through both offline evaluation and online tests. However, even though it is of critical significance to account for the inevitable noisy nature of implicit feedback for recommender training, little work has been done yet. Existing efforts dedicated to denoising user-item interactions can be separated into two categories: 1) leveraging additional feedback (e.g., dwell time, gaze pattern, favorite, skip) either to predict then remove the noisy data or to incorporate these feedback into the training process [28,44,46]; 2) utilizing domain expert knowledge to eliminate the effects of false-positive interactions, such as setting threshold for losses [41]. These represent two extreme solutions: the former always use explicit data, which needs additional feedback and extensive manual label work, making it infeasible in most cases; the latter does not use any explicit data since it is based on strong assumptions, while such assumptions are made by domain experts for specific datasets and recommenders. Moreover, both of them neglect the fact that the platforms usually have both large-scale but noisy implicit feedback, and small-scale yet highly relevant explicit feedback. Explicit data suffers from the scale issue while implicit data may not, and implicit data is noisy but explicit data may not. Thus, leveraging each other's strengths to make up for the other's weaknesses provides a natural, feasible and fully automatic solution for the implicit data denoising problem. To this end, we propose an Automated Data Denoising framework, named AutoDenoise, for recommenders, which exploits a small volume of noise-free data (i.e., those true-positive user-item interactions) to guide the recommenders to recommend more satisfying (or true-positive) items to users. Inspired by generalized curriculum learning (CL) [3,43], AutoDenoise learns to automatically and dynamically select the most suitable examples, or assign the most appropriate weights to each sample along the training process under the guidance of the performance on the noise-free data, where the former is named as Hard AutoDenoise (AutoDenoise-H) and the latter is called Soft AutoDenoise (AutoDenoise-S). Moreover, unlike traditional CL, our framework allows examples to appear many times (i.e., receive weights greater than one) so as to make a clearer difference between different samples, which helps to generalize data weighting, filtering, and fine-tuning schemes. Technically, following neural architecture search (NAS) in [35,53], we use a reinforcement learning approach involving a learned agent, named controller, whose task is to learn to assign weights to each implicit data sample. Under the guidance of noise-free data samples, the domain knowledge of true-positive interactions is automatically learned by the controller network. Specifically, we use a meticulously designed controller network to generate (hard or soft) weights, combine the weights with the loss of each input data into a weighted loss to train the recommender system, and optimize the controller with REINFORCE [45] to maximize the expected accuracy of the trained RS on the clean and noise-free data. In our implementation, to make more use of these noise-free data, we also turn it into a validation set, so its recommendation accuracy can be further used for both hyper-parameter tuning and controller's reward signal. The contributions of this work can be summarized as follows: • We propose an end-to-end framework, AutoDenoise, which is able to automatically and dynamically select the most appropriate data instances (or assign the most appropriate weights to each sample) for RS training to help reduce the noisy signals and improve the recommendation performance; • We utilize the expected accuracy on the small-scale noise-free validation set to guide the training process, leveraging the strengths of implicit data and explicit data to compensate for each other's weaknesses, which is a natural and elegant solution; • Extensive experiments on three benchmarks using both classical and deep recommendation models validate the effectiveness and generalizability of AutoDenoise. RELATED WORK In this section, we will briefly introduce some background knowledge about noise in recommendation, curriculum learning in recommendation and automated machine learning in recommendation. Noise in Recommendation Generally, there are two types of noises in recommender systems [34]: malicious noise (shilling noise, e.g., injection attacks) and nonmalicious noises (natural noise, e.g., human errors). The former is usually the result of malicious users deliberately manipulating the predicted scores of certain items, while the latter is related to random user behaviors during the selection of items. In collaborative filtering recommendation systems, there have already been many approaches focusing on detecting malicious noise, while, given that natural noise is often hidden in the user's behavior, there are only a few studies related to this problem. For instance, Li et al. [24] utilized a novel real-time quadratic optimization algorithm for identifying and removing noisy non-malicious users (NNMUS), which deals with natural noise through detecting noise but not malicious users if the user's rating for closely related items has the same score. Moreover, some existing work collected the various users' feedback (e.g., dwell time [20], gaze patterns [48], and skip [6]) and the item characteristics [31] to predict the user's satisfaction, while other approaches [28,44,46] directly incorporated additional feedback into training. For example, Wen et al. [44] used three types of items: "click-complete", "click-skip", and "non-click", to train the recommendation model, where the last two types of items are both treated as negative samples but with different weights. All these methods need additional feedback and extensive manual label work, e.g., users have to tell their satisfaction of each interaction. Furthermore, the quantification of item quality largely relies on the manual feature design and the labeling of domain experts [31]. The unaffordable labor cost impedes the practical application of these methods, especially in the scenarios where item pools change over time, such as news recommendation, movie recommendation, etc. On the contrary, our proposed framework, AutoDenoise, takes advantage of the fact that the platforms usually have both largescale noisy implicit feedback and small-scale yet noise-free explicit feedback, which makes it possible to leverage each other's strengths to make up for the other's weakness in the data denoising problem, leading to a feasible and elegant solution. Curriculum Learning in Recommendation The idea of curriculum learning (CL) was popularized by Bengio et al. [3], who viewed it as a way to improve convergence by presenting heuristically identified easy examples first [3]. The core idea of CL is that easier instances should be involved first in the model learning, and then more complex ones are gradually considered. This tactic is empirically evaluated to be beneficial for the learning process to alleviate the bad local minimum, leading to the superior performance in some tasks [38]. To make a recommender model understand user preferences more precisely, various studies have suggested curriculum learning during model training [14,33,47]. For example, in movie recommendation, a self-paced learning (SPL) based reweighting [23] strategy was proposed by Zhang et al. [47] to select the samples for reweighting under a predefined weighting scheme. As another example, Liu et al. [30] designed a bounded SPL learning paradigm with a parameter to control how many instances will be finally induced in the model learning, which tries to learn the model mainly on clean data and exclude noisy instances. Moreover, among works on handling noisy data in recommendation, the closest to ours is Wang et al. [41], which proposed two strategies to deal with noisy data: 1) truncating the loss values of hard interactions to 0 with a dynamic threshold; 2) assigning hard interactions with smaller weights based on a variant of Focal Loss [26]. Nevertheless, this setting has two major drawbacks: first, it still uses fixed weighting functions designed with domain expert knowledge, which is time consuming and inflexible; second, it exploits a strong assumption that deep models will learn the easy and clean patterns in the early stage, which has been proven to be not always true in many other cases [40,52]. Our proposed framework is neither based on the "easier first" assumption nor the "harder first" assumption [43], instead it learns to allocate the most appropriate training data in a trial and error fashion, i.e., reinforcement learning. We also replace the fixed predefined weighting function with a learnable deep neural network, which enhances its ability of generalization. AutoML in Recommendation The proposed framework is closely related to the field of Automated Machine Learning (AutoML) as it focuses on automating data processing tasks such as data selection and reweighting. In the light of this, we provide a brief overview of how AutoML is applied to the field of recommendation. AutoML techniques have been widely introduced in recommendation systems. Specifically, existing works mainly focused on three research directions: 1) the design of the embedding layer, 2) the selection of feature interaction patterns, and 3) the generation of loss function. For the embedding layer design, several works [17,29,50] proposed novel methods to automatically select the most appropriate embedding size for different feature fields. For example, Liu et al. [29] proposed to dynamically search embedding sizes for users and items based on their popularity by introducing a novel embedding size adjustment policy network (ESAPN). Similarly, Ginart et al. [8] proposed to use mixed dimension (MD) embedding layer, which consists of variable embedding sizes for each feature, and Cheng et al. [5] proposed to perform embedding dimension selection with a soft selection layer, making the dimension selection more flexible. For feature selection, Luo et al. proposed AutoCross in [32], which enables explicit high order feature interaction search on a tree-structured search space by implementing greedy beam search. In [39], the authors proposed to interpret feature interactions from a source recommender model and then encode these interactions in a target recommender model, where both source and target models are black-box models. For loss function generation, there have been only a few pioneering works, such as [25,49]. For instance, Li et al. [25] proposed an automatic loss function generation framework, AutoLossGen, which is able to generate loss functions directly constructed from basic mathematical operators without prior knowledge on the loss structure. PROBLEM FORMULATION In this section, we first provide an overview of the general recommendation problem, and then propose to integrate curriculum learning into the recommendation task to mitigate the impact of false-positive training samples. Recommendation Formulation Suppose we have a user set with users denoted as U, an item set V with items and their interaction set D = {( , , , )| ∈ U, ∈ V, ∈ {0, 1}, ∈ R + }, represents whether the user interacted with item , and is the timestamp. All users share an embedding matrix U ∈ R × , and items share an embedding matrix V ∈ R × , where is the size of embedding vector. Accordingly, each user's or item's latent vector e ∈ R or e ∈ R is the corresponding row in the embedding matrix: e = U ; e = V . The embedding matrices U and V for users and items are exactly what to learn in the training process, with supervised ground-truth of user-item interactions. We define a model , which is a function parameterized with , to predict the user-item ranking scoreˆ, for user and item , aŝ , = (e , e | , Θ)(1) where Θ = {U, V, } contains all learnable model parameters, and represents all other auxiliary information. Depending on the application, could be rating scores, clicks, text, images, etc., and is optional in the recommendation model (we will omit in the rest of our paper for simplicity). The goal of a recommendation system can be stated as follows: learning a recommendation model from D so that it can capture user preferences and make high-quality recommendations. L (D|Θ) = 1 |D| ∑︁ ∈ D ℓ (e , e | Θ),(2) where ℓ (·) can be any suitable loss functions, e.g., Binary Cross Entropy loss (BCE), Bayesian Personalized Ranking loss (BPR) [37], Softmax loss, etc. After training is complete, the top-items generated by the ranking score function except the interacted items are recommended to the user . Curriculum Learning for Recommendation Usually, Eq. 2 can be directly optimized by stochastic gradient descent (SGD) method. However, the noisy training samples will be treated equally as good samples for the update of the parameters during the process of model training, and the noisy training instances will greatly harm the effectiveness of the model. To overcome the above issue, curriculum learning (CL) mechanism is introduced. In this work, we mainly focus on automatic curriculum learning since traditional predefined curriculum learning methods heavily rely on human prior knowledge [43]. Concretely, we aim to learn a Symbol Description U The set of users in a recommender system V The set of items in a recommender system D The set of user-item interactions The number of users The number of items A user ID in a recommender system An item ID in a recommender system A user embedding matrix A item embedding matrix e User 's embedding e Item 's embedding The dimension of user/item embedding Ground-truth value of the pair ( , ) Predicted value of the pair ( , ) The length of the recommendation list Θ (·) The recommendation model parameterized with Θ Φ (·) The controller network parameterized with Φ The weight assigned to pair ( , ) w The weight vector for all user-item interactions R Reward on validation set reweighting of each input sample and minimize a weighted loss, L ′ (D |Θ, w) = 1 |D| ∑︁ ( , ) ∈ D · ℓ (e , e | Θ),(3) In classical CL, w = [ ] | D | ∈ [0, 1] | D | represents the weight vector, and each element measures the easiness of each useritem pair ( , ) ∈ D to determine whether an instance is selected or not. Considering the large number of , we replace w with a controller network in Eq. 3, which is a deep neural network parameterized with Φ. The new objective function is shown as follows, L ′ (D |Θ, Φ) = 1 |D | ∑︁ ( , ) ∈ D (e , e | Φ)ℓ (e , e | Θ),(4) Finally, our new optimization problem is, min Θ,Φ L ′ (D|Θ, Φ) + 1 ||Θ|| 2 + 2 ||Φ|| 2(5) where 1 ∈ [0, 1] and 2 ∈ [0, 1] are hyper-parameters used to control the weights between the three terms, and ||Θ|| 2 and ||Φ|| 2 represent L2 penalty. Most of the important symbols used in the paper can be referred in Table 1. Moreover, without losing generality, we only formulate the problem based on collaborative filtering methods and focus on reweighting the training loss. However, the proposed framework can be applied to any differentiable recommendation models. For example, the user embedding e can also be replaced with other hidden representations, such as the representation of user behavior sequences in sequential recommendation. We also demonstrate the generalizability of our framework by applying it to various recommendation algorithms in the experimental section. AUTODENOISE With the above definitions, the objective of our AutoDenoise framework is to automatically and dynamically assign the most appropriate weights (discrete or continuous values) to each implicit data sample through learning Θ * and Φ * in Eq. 5. We will introduce in details on how to leverage reinforcement learning (RL) to optimize the controller network Φ (·) and the recommendation model Θ (·). Overall Procedure In AutoDenoise, the parameters to be optimized are from two networks, namely Θ and Φ . For RS model, we perform stochastic gradient descent (SGD) to update Θ on the implicit training data, while for the controller, inspired by the success of neural architecture search (NAS) in [35,53], we apply the reinforcement learning algorithm to update Φ, and the accuracy on the explicit validation dataset is used as the reward signal for policy gradient. This forms a bi-level optimization problem [2], where controller parameters Φ and RS parameters Θ are considered as the upper-and lower-level variables. The optimization problem is formulated as: max Φ E w∼ ( D |Φ,Θ * ) [R ] s.t. Θ * = arg min Θ L ′ (D|Θ, Φ * )(6) where R represents the reward (i.e., accuracy) on validation set. To solve this problem, one can update Θ and Φ in an alternative manner. Specifically, Θ and Φ are alternately updated on training and validation data by minimizing the training loss L ′ and maximizing the expected validation performance E w∼ ( D |Φ,Θ * ) [R ], respectively. As shown in Fig. 1, at each training epoch, the controller will dynamically select examples for training according to the recommender's feedback, and the trained recommender will result in an accuracy on the validation set. Using this accuracy as the reward signal, we can compute the policy gradient to update the controller. Moreover, the weight assignment is taken as the action in the RL schemes, and the feedback is taken as the state and reward. Training with REINFORCE In our setup, the environment is the recommender system and its training data, as illustrated in Fig. 1. The weight vector w that the controller predicts can be viewed as an action for the training of the recommender. We can use the accuracy on the validation set as the reward signal R and use reinforcement learning to train the controller. More concretely, to find the optimal architecture, we require our controller to maximize its expected reward, represented by (Φ): (Φ) = E w∼ ( D |Φ) [R ](7) Since the reward signal R is non-differentiable, we need to use a policy gradient method to iteratively update Φ. In this work, we use the REINFORCE in [45] as follows, ∇ Φ (Φ) = (w|Φ) [∇ Φ log (w|Φ) · R ](8) An empirical approximation of the above quantity is: ∇ Φ (Φ) ≈ 1 |D| ∑︁ ( , ) ∈ D ∇ Φ log ( |Φ) · R(9) The above update is an unbiased estimate for our gradient, but has a very high variance. In order to reduce the variance of this estimate, following [35,53], we employ a baseline function , which is the moving average of the previous reward signals: 1 |D | ∑︁ ( , ) ∈ D ∇ Φ log ( |Φ) · (R − )(10) The Controller Network In this subsection, we first introduce the concrete concepts for state , action and reward , then provide the details of model structure of the hard and soft controller network. State Representation. In order to effectively and efficiently represent state, the state representation should include both arrived training data and the status of current recommendation model [9]. Here, we adopt three commonly-used categories of features [9,22,43]: 1) Data features, containing information for data instance, such as user and item embeddings or feature vectors; 2) Recommender model features, including the signals reflecting how well current neural network is trained. We collect one simple feature, which is the number of epochs; 3) Features that represent the combination of both data and learner model. By using these features, we aim to represent how important the arrived training data is for the current leaner. We mainly use the loss of a certain user-item pair ℓ , which also appears frequently in curriculum learning [23]. Reward Function. We use the Area under the ROC Curve (AUC) to evaluate the performance, which is a commonly used reward in many AutoML works [25,51] (one can also select other alternative metrics, such as hit rate, NDCG, and so on), and obtain the reward on validation set since they are prediction-sensitive, i.e., the metric values will be different with a very small fluctuation on predictions [1,4], so that non-trivial reward can be calculated for better update on the controller. ) for short. In order to generate discrete actions in the hard action space, we use the multinoulli distribution (also called a categorical distribution), which is, ℎ ∼ M (softmax(ℎ ))(12) where M represents the multinoulli distribution, which samples integers from {0, . . . , − 1} where is the number of actions given their probabilities, and softmax(ℎ ) outputs the probabilities of each action. In the experiments, we define three kinds of actions to represent {0,1,2}, which represent "delete sample", "keep sample" and "augment sample", respectively. For continuous actions in the soft action space, we use the Gaussian distribution. Concretely, we pass ℎ through two separate output layers, one for mean and another for variance (shown in Eq. 13 and Eq. 14), and get and 2 , respectively. = +1 ℎ + +1(13)2 = softplus( ′ +1 ℎ + ′ +1 )(14) Then, we use the mean and the variance to sample a continuous weight based on Gaussian distribution, as ∼ N ( , 2 )(15) Considering the current weight can be either positive or negative, which is not stable for the model training, we finally pass through a softplus layer to make sure it is always positive. = softplus( )(16) Training Procedure We also present the detailed training procedure of the proposed framework in Algorithm 1. In each epoch, there are two phases -recommender updating phase (line 6-10) and controller updating phase (line [11][12][13][14]. The former phase is similar to the normal updating except for a weighted loss. In the latter phase, we receive the reward based on the recommendation performance on the validation set, and then combine other values stored in the recommender updating phase, such as the log probabilities of the actions, to update the controller network. Fix Φ, update Θ through backpropagation. Fix Θ and pass the validation set through ; 12 Get the AUC value on the validation set as reward; 13 Update Φ based on Eq. (10); 14 Update . end EXPERIMENTS In this section, we first introduce the datasets, the base recommenders, the comparable baselines, then discuss and analyse the experimental results. Dataset Description To evaluate the models under different data scales, data sparsity and application scenarios, we perform experiments on three real-world public datasets. • Movielens dataset [10]: One of the most frequently used benchmark dataset for personalized recommendation. We use Movielens-100K-latest 1 which includes about one hundred thousand user transactions (user id, item id, rating, timestamp, etc.) and was generated on September 26, 2018. Moreover, we choose a larger one-Movielens-1M 2 including over one million user-item interactions. • Amazon dataset [11]: This dataset contains user reviews on products in Amazon e-commerce system. 3 It has 29 sub-datasets corresponding to 29 product categories. We adopt Electronics dataset to evaluate our method. The original data is huge and highly sparse, especially Amazon dataset, which makes it challenging to evaluate. Therefore, similar to previous work [12,18], we filter out users and items with fewer than ten interactions. Then, for each dataset, we sort the records of each user based on the timestamp, and split the records into training, validation, and testing sets chronologically by 4:1:1. Moreover, to mimic the real-world scenarios as close as possible (large-scale implicit feedback and only a small volume of explicit feedback), we make some modifications to the traditional training, validation, testing process. First, for training set, we keep all observed interactions in the training set and keep them implicit (i.e., without using any explicit rating information). Second, for validation-and testing sets, we only keep users' true-positive interactions since we need to use a small number of noise-free validation set to guide the model training, and evaluate the recommendation performance on a holdout clean testing set with only true-positive interactions kept, i.e., the evaluation focuses on recommending more satisfying items to users. Following [41], we use the groundtruth rating scores to filter out false-positive data samples in the original validation-and testing sets. More specific, an interaction is identified as false-positive or true-positive according to the explicit feedback, and we define an interaction as false-positive if its rating score ( [1,5]) < 3. Some basic statistics of the filtered experimental datasets are shown in Table 2, including number of users, number of items, number of interactions and density. Experimental Setup Base Recommenders. We compare our proposed method with the following baselines, including both traditional and reinforcement learning based recommendation models. • MF: Collaborative Filtering based on matrix factorization (MF) [21] is a representative method for rating prediction task. In our implementation, we turn the rating prediction task into ranking prediction and use Softmax loss. • GRU4Rec [13]: It applies Gated Recurrent Units (GRU) to model user click sequence for session-based recommendation. It models user interactions with items, such as clicks, by analyzing sequences of interactions. In our implementation, we represent the items using embedding vectors rather than one-hot vectors. • SASRec [18]: It is a deep learning model for session-based recommendation. It uses self-attention mechanism to weight the importance of different items in a user's interaction history, allowing it to better capture the user's preferences. Reprocessing Schemes. To evaluate the effectiveness of the reweighting scheme, we incorporate the following four reweighting schemes with all base recommenders. • Default: The default scheme involves no sample reweighting. • Heuristic: The heuristic scheme involves positive sample upweighting based on a pre-defined heuristic function. Specifically, following [15,36], we evaluate the commonly-used proportion method that weights each example by the inverse frequency, where frequency is the number of times an item appears in the training data. • Adaptive Denoising Training with Truncated Loss (ADT-TL): Wang et al. [41] proposed ADT strategies for recommender systems, which dynamically prunes the large-loss interactions Table 3: Summary of the performance on three benchmark datasets. We evaluate , , 1 and , in percentage (%) values (% symbol is omitted in the table for clarity), whiles is the length of recommendation list. When AutoDenoiseaugmented methods are the best, i.e., AutoDenoise-H or AutoDenoise-S, its improvements against the best baseline are significant at p < 0.01. Bold scores are used for the largest values. Methods Precision GRU4Rec and SASRec, since they are sequence models, we use a sliding window over the items with maximum 10 items. We tune the hyper-parameters using the validation set and terminate training when the performance on the validation set does not change within 6 epochs. (%) ↑ Recall (%) ↑ F1 (%) ↑ NDCG (%) ↑ K=10 K=20 K=50 K=10 K=20 K=50 K=10 K=20 K=50 K=10 K=20 Moreover, for AutoDenoise-augmented recommender systems, e.g., MF-AutoDenoise-H, MF-AutoDenoise-S, GRU4Rec-AutoDenoise-H, GRU4Rec-AutoDenoise-S, SASRec-AutoDenoise-H and SASRec-AutoDenoise-S, we design the controller network by using a Multi-Layer Perceptron (MLP) with two layers and ReLU function as the embedding layer, and the output layer of AutoDenoise-H follows Eq. 12, and that of AutoDenoise-S follows Eq. 15. We optimize the controller using Adam optimizer. Besides, we consider the learning rate from {1e-2, 5e-3, 1e-3, 5e-4, 1e-4, 5e-5}, and the L2 penalty is chosen from {1e-3, 1e-4, 1e-5, 1e-6}. Evaluation Metrics. We select several most commonly used top-ranking metrics to evaluate each model's recommendation performance, including Precision, Recall, F1 Score, and NDCG, with different lengths of recommendation lists, i.e., @10, @20, @50. Experimental Results The major experimental results on Movielens-100K-lateset, Movielens-1M and Electronics datasets are shown in Table 3. We analyze and discuss the results in terms of the following perspectives. 5.3.1 Shallow models vs. Deep models. Among all the base recommenders, we can see that both deep models (GRU4Rec-Default, SASRec-Default) are better than the shallow model (MF-Default) in most cases. Specifically, when averaging across all metrics on all three datasets, GRU4Rec-Default gets 45.86% improvement than MF-Default, and SASRec-Default even achieves 136.77% improvement. The greatest improvement is achieved by SASRec-Default on Electronics dataset on NDCG@10, which is 816.0%. These observations verify the effectiveness of deep recommendation models and their abilities to learn more accurate user preferences. The reason behind is that MF uses a user's historical interaction to learn the static preference, while sequential recommendation models leverage the fact that the next behavior of a user not only depends on the static long-term preference, but also relies on the current intent [13,18]. Another explanation is related to data density. As we can see that for deep sequential recommenders, the sparser the dataset is, the greater the improvement is. and AutoDenoise's discrete and continuous versions in Table 3, we can easily find that AutoDenoise-S outperforms AutoDenoise-H, and ADT-RL outperforms ADT-TL in most cases. For example, ADT-RL gets 11.37% improvement than ADT-RL on Movielens-100Klatest, 1.98% on Movielens-1M. Similarly, AutoDenoise-S achieves 12.22% improvement than AutoDenoise-H on Movielens-100K-latest, 8.49% on Movielens-1M, and 7.26% on Electronics when averaging across all metrics, which shows the strong generalizability of continuous action space. Such action space is able to provide more differences of interaction importance so as to better denoise the data and improve the recommendation performance. Additionally, when averaging across all metrics on three datasets using different base rankers, AutoDenoise-S achieves 4.47% improvement than AutoDenoise-H on MF, 7.80% on GRU4Rec, and 16.18% on SASRec, which also shows a stronger generalizability to fit different base rankers. In-depth Analysis To gain deeper understandings of the inner mechanism of the proposed AutoDenoise framework, we further analyze the learned weights after model convergence and examine the selection of action space for hard AutoDenoise. Study of Learned Weights. In order to study the cause of the improvements observed in AutoDenoise, we collect the weights assigned to each training sample after model convergence, and analyse them by comparing with the ground-truth labels. Specifically, we first label each training sample to either true-positive or false-positive based on its rating score. Then, we calculate the averaged weights for both true-positive sample group and falsepositive sample group, and plot the results from all three datasets in Fig. 2. For the sake of convenience, all the results are from SASRec-AutoDenoise-H and SASRec-AutoDenoise-S on Movielens-100Klatest, Movielens-1M and Electronics datasets. Similar patterns can also be found using other recommendation methods. In Fig. 2, we can easily see that the average weights learned for true-positive samples are much greater than these for false-positive ones, which indicates that the proposed method is able to mitigate the noise in the training data and achieve better recommendation performance. Specifically, the average value of learned weights of true-positive training samples is 1.114 when averaging across all datasets, and that of false-positive ones is 0.132. 5.4.2 Study of Action Space. In order to explain some of our original design choices, we study the relationship between the size of the action space for hard AutoDenoise and its performance. For convenience, we present results from MF-AutoDenoise-H on Movielens-100K-latest, Movielens-1M and Electronics by setting different numbers of actions for the controller network. Similar patterns can also be found using other base rankers. As shown in Fig. 3, the recommendation performance (NDCG@20 on y-axis) increases as we increase the discrete action space, reaching a peak at 3 on Movielens-100K and Electronics, or at 5 on Movielens-1M, then decreases. On the x-axis, 1 means only keeping the sample, 2 is either deleting or keeping the sample, 3 represents {delete the sample, keep the sample, duplicate the sample two times} and so on. This observation explains why we choose to expand the action space for data augmenting in AutoDenoise-H or set weights greater than one in AutoDenoise-S, rather than directly following the traditional design of curriculum learning. Even if two true-positive items are both preferred by a certain user, there is still a preference difference between them, similar for the false-positive ones. Using a larger action space or even a continuous space provides more freedom of choices, thus, resulting in better performance. CONCLUSION In this work, we aim to deal with the noisy nature of implicit feedback for model training in recommendation. Specifically, we select a small number of explicit feedback as validation set to guide the recommender training process, and propose an Automated Data Denoising framework (AutoDenoise) based on reinforcement learning to automatically and adaptively learn the most appropriate weights for each implicit data sample under the guidance of the noise-free validation set. We conduct thorough experiments, which indicate that the proposed framework is capable of boosting the performance of the state-of-the-art recommendation algorithms on several benchmark datasets. Figure 1 : 1Illustration of the proposed method. 4.3. 2 2Action. The controller has two types of actions-hard and soft. First, hard actions are denoted via = | D | =1 ∈ {0, 1, 2} | D | , representing to delete ( = 0) or keep ( = 1) or augment ( = 2) the -th instance in D. Second, soft actions are denoted as = | D | =1 ∈ R + | D | , representing the training weight of the -th instance in D as a real-value positive number. We will introduce two different network architectures to realize each of them in Section 4.3.4. 4.3. 4 4Network Structure. Given the state representation of an input user-item pair ( , ), the controller will pass it through a multi-layer perceptron and output a hidden vector ℎ , which is as shown below,ℎ = (. . . ( 1 1 ( (e ,e , ℓ , )) + 1 ) . . .) + (11) where , (·), represent weights, activation functions and bias values; is the current training epoch number,(·) is the concatenation function and ℓ = ℓ ( (e , e | Θ), Figure 2 : 2The average values of learned weights for ground-truth true-positive and false-positive training samples in all three datasets. The legend indicates that TP (blue bar) stands for true-positive and FP (green bar) stands for false-positive data samples. The x-axis includes AutoDenoise-H and AutoDenoise-S with SASRec as the base ranker. The y-axis represents the average values of the learned weights.along the training process. Specifically, there are two paradigms formulating the training loss, and ART-TL using Truncated Loss, which truncates the loss values of hard interactions to 0 with a dynamic threshold function.• Adaptive Denoising Training with Reweighted Loss (ADT-RL): This method is another ADT strategy proposed by Wang et al. [41]. Specifically, ADT-RL using Reweighted Loss, which assigns hard interactions with smaller weights based on a variant of Focal Loss [26]. • Hard AutoDenoise (AutoDenoise-H): AutoDenoise-H is our proposed AutoDenoise scheme with discrete action space. • Soft AutoDenoise (AutoDenoise-S): AutoDenoise-S is our proposed AutoDenoise scheme with continuous action space. 5.2.3 Implementation Details. We implement MF-Default, MF-Heuristic, MF-ADT-TL, MF-ADT-RL, GRU4Rec-Default, GRU4Rec-Heuristic, GRU4Rec-ADT-TL, GRU4Rec-ADT-RL, SASRec-Default, SASRec-Heuristic, SASRec-ADT-TL, SASRec-ADT-RL using Pytorch with Adam optimizer. For all of them, we consider latent dimensions from {32, 64, 128}, learning rate from {1e-2, 5e-3, 1e-3, 5e-4, 1e-4, 5e-5}, L2 penalty is chosen from {1e-3, 1e-4, 1e-5, 1e-6}, batch size is select from {256, 512, 1024, 2048}, and use Softmax loss. For 5.3.2 Default scheme vs. Reweighting schemes. Among all the reweighting schemes, we can see that reweighting methods are able (a) Results in Movielens-100K-latest (b) Results in Movielens-1M (c) Results in Electronics Figure 3 : 3Relationship between number of actions and recommendation performance for MF- Table 1 : 1Summary of the notations in this work. Algorithm 1 : 1Parameters Training for AutoDenoise1 Input: User-item interaction history D, batch size B, learning rate for recommender and learning rate for controller. 2 Output: parameters Θ * and Φ * 3 Randomly initialize Θ and Φ; 4 b = 0; 5 while not converged do 6 for = 1 ... |D|/ do 7 Sample a minibatch with size B from D; 8 Calculate the weighted training loss based on Eq. (4); 9 Table 2 : 2Basic statistics of the experimental datasets. We use #users represents the number of users, #items the number of items and #act. represents the total number of interactions.Dataset #users #items #act. density Movielens-100K-latest 610 2,270 81,109 5.857% Movielens-1M 6,023 3,044 956,851 5.218% Electronics 47,726 30,115 1,143,343 0.079% AutoDenoise-H on three datasets. -axis is the number of actions chosen for hard AutoDenoise method; -axis represents NDCG@20 on test set after convergence.to improve the default recommendation performance in many cases. For instance, when averaging across all metrics and all base rankers, Heuristic gets 3.24% on Movielens-100K-latest and 0.74% improvement on Electronics, ADT-TL gets 0.26% on Movielens-1M and 7.78% on Electronics, and ADT-RL, the strongest baseline, gets 1.32% on Movielens-100K-latest, 2.12% on Movielens-1M and 3.44% on Electronics. Moreover, our proposed AutoDenoise-augmented methods are always better than the default methods. Specifically, when averaging across all metrics on all three datasets, AutoDenoise-H gets overall 7.29% improvement than Default, with 7.32% on Movielens-100K-latest, 5.20% on Movielens-1M and 9.35% on Electronics; and AutoDenoise-S gets overall 17.44% improvement, with 20.75% on Movielens-100K-latest, 13.83% on Movielens-1M and 17.73% on Electronics.5.3.3Hard policy vs. Soft policy. 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Byoungju Yang, Sangkeun Lee, Sungchan Park, Sang-Goo Lee, Proceedings of the 21st International Conference on World Wide Web. the 21st International Conference on World Wide WebByoungju Yang, Sangkeun Lee, Sungchan Park, and Sang-goo Lee. 2012. Exploit- ing various implicit feedback for collaborative filtering. In Proceedings of the 21st International Conference on World Wide Web. 639-640. Discrete ranking-based matrix factorization with self-paced learning. Yan Zhang, Haoyu Wang, Defu Lian, Ivor W Tsang, Hongzhi Yin, Guowu Yang, Proceedings of the 24th ACM SIGKDD. the 24th ACM SIGKDDYan Zhang, Haoyu Wang, Defu Lian, Ivor W Tsang, Hongzhi Yin, and Guowu Yang. 2018. Discrete ranking-based matrix factorization with self-paced learning. In Proceedings of the 24th ACM SIGKDD. 2758-2767. Gaze prediction for recommender systems. Qian Zhao, Shuo Chang, Maxwell Harper, Joseph A Konstan, Proceedings of the 10th ACM Conference on Recommender Systems. the 10th ACM Conference on Recommender SystemsQian Zhao, Shuo Chang, F Maxwell Harper, and Joseph A Konstan. 2016. Gaze prediction for recommender systems. In Proceedings of the 10th ACM Conference on Recommender Systems. 131-138. Autoloss: Automated loss function search in recommendations. Xiangyu Zhao, Haochen Liu, Wenqi Fan, Hui Liu, Jiliang Tang, Chong Wang, Proceedings of the 27th ACM SIGKDD. the 27th ACM SIGKDDXiangyu Zhao, Haochen Liu, Wenqi Fan, Hui Liu, Jiliang Tang, and Chong Wang. 2021. Autoloss: Automated loss function search in recommendations. In Proceed- ings of the 27th ACM SIGKDD. 3959-3967. Memory-efficient embedding for recommendations. Xiangyu Zhao, Haochen Liu, Hui Liu, Jiliang Tang, Weiwei Guo, Jun Shi, Sida Wang, Huiji Gao, Bo Long, arXiv:2006.14827arXiv preprintXiangyu Zhao, Haochen Liu, Hui Liu, Jiliang Tang, Weiwei Guo, Jun Shi, Sida Wang, Huiji Gao, and Bo Long. 2020. Memory-efficient embedding for recom- mendations. arXiv preprint arXiv:2006.14827 (2020). Ruiqi Zheng, Liang Qu, Bin Cui, Yuhui Shi, Hongzhi Yin, arXiv:2203.13922AutoML for Deep Recommender Systems: A Survey. arXiv preprintRuiqi Zheng, Liang Qu, Bin Cui, Yuhui Shi, and Hongzhi Yin. 2022. AutoML for Deep Recommender Systems: A Survey. arXiv preprint arXiv:2203.13922 (2022). Minimax curriculum learning: Machine teaching with desirable difficulties and scheduled diversity. Tianyi Zhou, Jeff Bilmes, International Conference on Learning Representations. Tianyi Zhou and Jeff Bilmes. 2018. Minimax curriculum learning: Machine teach- ing with desirable difficulties and scheduled diversity. In International Conference on Learning Representations. Neural Architecture Search with Reinforcement Learning. Barret Zoph, V Quoc, Le, 5th ICLR. Barret Zoph and Quoc V. Le. 2017. Neural Architecture Search with Reinforcement Learning. In 5th ICLR. https://openreview.net/forum?id=r1Ue8Hcxg

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{'abstract': 'In real-world scenarios, most platforms collect both large-scale, naturally noisy implicit feedback and small-scale yet highly relevant explicit feedback. Due to the issue of data sparsity, implicit feedback is often the default choice for training recommender systems (RS), however, such data could be very noisy due to the randomness and diversity of user behaviors. For instance, a large portion of clicks may not reflect true user preferences and many purchases may result in negative reviews or returns. Fortunately, by utilizing the strengths of both types of feedback to compensate for the weaknesses of the other, we can mitigate the above issue at almost no cost. In this work, we propose an Automated Data Denoising framework, AutoDenoise, for recommendation, which uses a small number of explicit data as validation set to guide the recommender training. Inspired by the generalized definition of curriculum learning (CL), AutoDenoise learns to automatically and dynamically assign the most appropriate (discrete or continuous) weights to each implicit data sample along the training process under the guidance of the validation performance. Specifically, we use a delicately designed controller network to generate the weights, combine the weights with the loss of each input data to train the recommender system, and optimize the controller with reinforcement learning to maximize the expected accuracy of the trained RS on the noise-free validation set. Thorough experiments indicate that AutoDenoise is able to boost the performance of the state-of-the-art recommendation algorithms on several public benchmark datasets.', 'arxivid': '2305.07070', 'author': ['Yingqiang Ge yingqiang.ge@rutgers.edu \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n', 'Mostafa Rahmani \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n', 'Athirai Irissappane \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n', 'Jose Sepulveda \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n', 'James Caverlee caverlee@gmail.com \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n', 'Fei Wang feiww@amazon.com \nRutgers University\nTexas A & M University\nInc., Inc., Inc\n'], 'authoraffiliation': ['Rutgers University\nTexas A & M University\nInc., Inc., Inc', 'Rutgers University\nTexas A & M University\nInc., Inc., Inc', 'Rutgers University\nTexas A & M University\nInc., Inc., Inc', 'Rutgers University\nTexas A & M University\nInc., Inc., Inc', 'Rutgers University\nTexas A & M University\nInc., Inc., Inc', 'Rutgers University\nTexas A & M University\nInc., Inc., Inc'], 'corpusid': 258676165, 'doi': '10.48550/arxiv.2305.07070', 'github_urls': [], 'n_tokens_mistral': 18041, 'n_tokens_neox': 15666, 'n_words': 9200, 'pdfsha': '1c69b0adcce48eb2a99c30dcce8bf228761377d3', 'pdfurls': ['https://export.arxiv.org/pdf/2305.07070v2.pdf'], 'title': ['Automated Data Denoising for Recommendation', 'Automated Data Denoising for Recommendation'], 'venue': []}

arxiv

LORENTZ-COVARIANT ANALYSIS OF A QUANTUM SOLITON arXiv:hep-th/9509122v1 22 Sep 1995 June 1995 Andrei Dubikovsky dubikovs@sunny.bog.msu.su Department of Physics Moscow State University 119899MoscowRussia LORENTZ-COVARIANT ANALYSIS OF A QUANTUM SOLITON arXiv:hep-th/9509122v1 22 Sep 1995 June 1995Talk given at the XVIII International Workshop on High Energy Physics and Field Theory, Protvino, Russia, A set of integral relations for rotational and translational zero modes in the vicinity of the classical soliton solution are derived from the particle-like properties of the latter. The validity of these all relations is considered for a number of soliton models in 2+1-and 3+1-dimensions. * Theories with non-trivial classical solutions, such as the skyrmion models of baryons [1], are object of intensive investigations. Quantization of hedgehog-type configurations including translational and rotational degrees of freedom [2] leads to the quantum Hamiltonian which contains a full bilinear form in conjugated momenta with nontrivial couplings between different collective variables [3][4][5]. However, for a particle-like classical solution one should expect additive diagonal contributions of kinetic and centrifugal terms to the Hamiltonian, at least to the lowest orders in the appropriate weak coupling expansion. Actually, it is a part of the general problem of decoupling upon quantization of various soliton degrees of freedom, which takes place for any type of field models with classical solutions. In this report we'll present a consistent general analysis of this problem, based on the particle-like properties of the classical solution combined with Lorentz covariance and virial relations. In the present report, we also give an analysis for the soliton with spin, quantized by means of translational and rotational collective coordinates, into corresponding representation of the Poincaré group. Let us consider a field theory in d + 1 space-time dimensions described by the Lagrangian density L(ϕ), which possesses a classical particle-like solution ϕ c (x). It is generally accepted, that if in the rest frame ϕ c is static with finite and localized energy density, then in quantum version of the theory such configuration describes an extended particle. Now we'll show, that there exists a set of nontrivial integral relations, fulfilled by ϕ c (x), which provide the validity of these assumptions. For a given static solution ϕ c ( x) the moving one is constructed via Lorentz boost, what results in the replacement x i → Λ −1 i ν x ν(1) in the arguments of ϕ c , where Λ µ ν is the corresponding Lorentz matrix. The momentum of the moving solution is P µ = T µ0 (ϕ c ( x, x 0 )) d x,(2) where T µν (ϕ c ) is the energy-momentum tensor. Transforming the r.h.s. in (2) to the rest frame, one gets P µ = Λ µ µ ′ Λ 0 ν ′ T µ ′ ν ′ (ϕ c ( ξ)) J d ξ,(3) where J = Λ 0 0 −1 is the Jacobian of transition from d x to the rest frame spatial variable. On the other hand, the l.h.s. of (2) should be the momentum of a particle with the mass M, that is P µ = Λ µ 0 M.(4) From eqs. (3) and (4) for ν = 0 we get P 0 = M, P i = 0, just that we should expect for a static solution. However for µ = i, ν = j we obtain ∂L(ϕ c ) ∂∂ j ϕ c ( ξ) ∂ i ϕ c ( ξ) d ξ = Mδ ij .(5) So we get the first set of conditions (5), which holds for a particle-like classical configuration ϕ c ( ξ) in the rest frame. Now let us consider the orbital part of the 4-rotation tensor (without the spin term) L µν = x ν T µ0 (ϕ c ( x, x 0 )) − x µ T ν0 (ϕ c ( x, x 0 )) d x.(6) Analogous calculations leads to the following relations (for definiteness, we take d = 3) ε lij ξ i ∂ j ϕ c ( ξ) ∂L(ϕ c ) ∂∂ k ϕ c ( ξ) d ξ = 0,(7) since L 0i vanish in the rest frame by assumption. This is the second set of relations on ϕ c ( ξ), following from the Lorentz covariance and particle-likeness of the classical solution. So each particle-like solution should be subject of conditions (5) and (7). It should be noted, that the relation (4) for µ = 0 reproduces nothing else but the relativistic mass-energy relation. For the moving ϕ 4 -kink solution this relation has been explicitly verified in [5], and for the moving skyrmion -in [6]) by direct calculations. However, the eqs. (5) are more general and, moreover, the eqs. (7) also take place. Note also, that these relations, being consistent with the field equations and conservation laws, do not be the direct consequences of the latters, and should be considered separately. As a direct result of these relations we get the orthogonality of the zero-frequency eigenfunctions in the neighborhood of the classical particle-like solution [7]. Let us discuss the theory of a nonlinear scalar field in 3 spatial dimensions, described by the Lagrangian density L = 1 2 (∂ µ ϕ) 2 − U(ϕ),(8) which possesses a static soliton solution ϕ c (x) = u( x).(9) In the general case the non-spherical configuration u( x) yields 6 zero-frequency modes -three translational ones ψ i ( x) = ∂ i u( x) and three rotational f i ( x) = ε ijk x j ∂ k u( x) . Then from eqs. (5) and (7) one immediately obtains d ξ ψ i ( ξ) ψ j ( ξ) = Mδ ij ,(10)d ξ f i ( ξ) ψ j ( ξ) = 0.(11) Further, by spatial rotations one can always achieve that d ξ f i ( ξ) f j ( ξ) = Ω ij = Ω i δ ij ,(12) where Ω i are the moments of inertia of the classical configuration. Obviously, the relations (10) and (11) remain unchanged. So the particle-likeness of the classical solution results in the diagonality of the zero-frequency scalar product matrix. This diagonality plays an essential role in the procedure of quantization in the vicinity of a classical soliton solution by means of collective coordinates [3,5]. Following the conventional procedure [4], let us consider the field ϕ( x) in the Schrödinger picture in the vicinity of the solution u( x). The substitution, introducing translational and rotational collective coordinates, reads [7] ϕ ( x) = u R −1 ( c)( x − q) + Φ R −1 ( c)( x − q) ,(13) where Φ is the meson field, R( c) is the rotation matrix, q and c are the translational and rotational collective coordinates correspondingly. In order to keep the number of degrees of freedom we impose on the field Φ( y) 6 subsidiary conditions, which in the theory of a weak coupling are usually taken as linear combinations d y N (α) ( y) Φ( y) = 0, α = 1, . . . , 6.(14) The set {N (α) ( y)} should ensure the condition of orthogonality of the meson field Φ( y) to zero-frequency modes and is chosen as [8] N (α) ( y) = {ψ i ( y)/M, f i ( y)/Ω i }. It is this relation, that ensures the additive form of the collective coordinate part of the Hamiltonian within the weak coupling expansion in powers of the meson fields. Considering the condition (14) as relation, defining q and c as functionals of ϕ( x) and calculating the conjugate momentum π( x) = −i δ δϕ( x) as a composite derivative, we can obtain finally for the Hamiltonian the following lowest-order expression H = M + K 2 2M + 1 2 i I i 2 Ω i .(15) In eq. (15) K and I are the momentum and the spin of the field, corresponding to the rotating frame (for details see ref. [8]). It is indeed such form of the Hamiltonian, that provides to interpret the resulting ground state as non-relativistic particle with the mass M and moments of inertia Ω i . So the correct form of the Hamiltonian with additive kinetic and centrifugal terms, that means the absence of correlations between translational and rotational degrees of freedom, is ensured by the diagonality of zero-frequency scalar product matrix (10)- (12). In turn, this is a direct consequence of relations (5) and (7). Note also, that this result will be actually valid for any field model in the neighborhood of the suitable soliton solution. These general considerations can be easily illustrated by concrete models. Firstly, we consider the theory of a scalar field in 1+1-dimensions, described by the Lagrangian density (8). In this case we have only one relation (5) dx (ϕ ′ (x)) 2 = M,(16) where the mass M is given by M = dx 1 2 (ϕ ′ (x)) 2 + dx U(ϕ(x)).(17) Performing the dilatation ϕ(x) → ϕ(λx) and demanding for the solution at λ = 1, i. e. dM (λ) dλ λ=1 = 0, we find the well-known Hobart-Derrick virial relation 1 2 dx (ϕ ′ (x)) 2 = dx U(ϕ(x)),(18) owing to which the "particle-likeness condition" (16) is fulfilled automatically. In 2+1-dimensions, the solitons in CP N -models are interesting examples with such particle-like properties. As it is well-known, for N = 1 the CP N -model is reduced to O(3)-model, described by L = 1 2 ∂ µ ϕ a ∂ µ ϕ a , ϕ a ϕ a = 1.(19) The standard one-particle solution of the model is given by ϕ 1 = φ(r) cos nϑ, ϕ 2 = φ(r) sin nϑ, ϕ 3 = (1 − φ 2 ) 1/2 ,(20) where r, ϑ are polar coordinates and φ(r) = 4r n r 2n +4 , and describes the "babyskyrmion" configuration with the topological charge Q = n and the mass M = 4πQ. Inserting the expression (20) into conditions (5) and (7) we obtain, that the conditions of particle-likeness for the solution (20) are satisfied. As a more nontrivial example, we consider the SU(2)-Skyrme model in 3+1dimensions [1], including the break-symmetry pion mass term L = − f 2 π 4 tr L 2 µ + 1 32g 2 tr [L µ L ν ] 2 + m 2 π 4 tr (U + U + − 2),(21) where, as usually, L µ = U −1 ∂ µ U is the left chiral current and U = σ + iτ a π a is the quaternion field. Supposing the conventional "hedgehog" Ansatz σ = cos φ(r), π a = r a r sin φ(r) we find that the first particle-likeness condition for the skyrmion is fulfilled due to virial relation. Further, inserting the substitution (22) into eqs. (7) we find in the same way, that the second set of relations for the skyrmion is provided by the symmetry properties. So we'll get upon quantization, that the full bilinear form considered in [2], automatically simplifies up to a diagonal construction similar to eq. (15), and therefore the quantized skyrmion describes an extended non-relativistic particle. Finally, we consider the 't Hooft-Polyakov monopole for the SU(2)-Yang-Mills-Higgs theory, described by the Lagrangian L = − 1 4 (F a µν ) 2 + 1 2 (D µ φ a ) 2 − V (φ)(23) with the monopole solution φ a = 1 g r a r 2 H(r), A a i = 1 g ε aij r j r 2 (1 − K(r)), A a 0 = 0.(24) We can find that the particle-likeness condition (5) is fulfilled due to virial relation, and the condition (7) is fulfilled due to symmetry properties of the expression (24), just as in the case of skyrmion. So we have proved the validity of the "particle-likeness" conditions (5) and (7) for the most important soliton solutions. Note, that there is a close connection between the condition (5) and virial relations for the static configuration by homogeneous dilatations. The relations (7) are usually fulfilled on account of symmetry properties of classical solutions. Now we give an analysis for the soliton with spin, into corresponding representation of the Poincaré group. As a first step, we consider a nonlinear scalar field in 3 spatial dimensions, described by the Lagrangian density (8) which possesses a static soliton solution (9). According to the virial theorem, such solutions are unstable in more then one spatial dimension, but for our purposes it is not so important compared to simplicity of presentation. The angular momenta J i and Lorentz boosts K i are given by J i = d x ε ijk x j T k0 = d x ε ijk x j ∂ k ϕ π.(25)K i = d x x 0 T i0 − x i T 00 = x 0 P i − d x x i H,(26) In these expressions P i = d x T i0 are the spatial momenta, H = T 00 = πφ − L(ϕ) is the Hamiltonian density and π = ∂L/∂φ is the canonical field momentum. Now we put the quasiclassical soliton field u(R −1 ( x − q)) and its canonical momentum in the leadind qusiclassical approximation (for details see ref. [11]) into corresponding Noether expressions for Lorentz generators (25), (26) and demand for their coincidence with the corresponding one-particle representation of the Poincaré group with the same mass M and spin S. It means, that the Lorentz generators J µν should take the form [9] J i = ε ijk q j P k + S i , K i = q 0 P i − q i P 0 − ε ijk P j S k P 0 + M .(27) Firstly, it is a trivial task to verify, that inserting into eq. (25) the soliton operators, one gets identically the eq. (27), provided by the orthogonality conditions (10)- (12). Applying the same procedure to the Lorentz boost operators (26), we find that the final result is the following set of subsidiary conditions imposed on u( x) d ξ ξ i ψ j ( ξ) ψ k ( ξ) = 0,(29)d ξ ξ i f j ( ξ) f k ( ξ) = 0,(30)d ξ ξ i ψ j ( ξ) f k ( ξ) = 1 2 ε ijl Ω lk .(31) These relations can be understood as a criterion of "particle-likeness" for the classical soliton field, describing a spinning particle. It should be noted, that whereas the orthogonality conditions (10)- (12) are valid for any static classical solution due to the general properties of lorentz-covariance and so are automatically consistent with equations of motion, the relations (29)-(31) are more strong and restrictive. Namely, the eqs. (11) and (12) are the direct consequences from eqs. (29) and (31) correspondingly. Moreover, there might exist a static solution u( x), that describes a two-soliton configuration and so cannot be consistent with the one-particle representation of the Poincaré group. In this case the relations (29)-(31) obviously do not hold. On account of these general considerations we show now that the typical hedgehog configurations of nonlinear σ-models describe spinning particles independently of the profile of their chiral angles. In two spatial dimensions, we consider the O(3) σ-model, described by the Lagrangian density (19), with one-particle solution (20). This theory is hoped to reveal the fractional spin and statistics after adding the Hopf term. In the case of 2 spatial dimensions we have two translational ψ a i = ∂ i ϕ a , (i = 1, 2) and only one rotational f a = ε ij ξ i ∂ j ϕ a zero modes for each isospin component ϕ a . By straightforward substitution it is easy to verify, that the configuration (20) leads to fulfilment of conditions (29)-(31). Thus, the baby-skyrmion solution (20) corresponds to the spinning particle for any choice of the chiral angle. In 3+1-dimensional space-time, we consider the SU(2)-Skyrme model described by the Lagrangian L = − 1 4 tr L 2 µ + 1 32 tr [L µ L ν ] 2 .(32) In terms of three independent fields φ a the expression (32) can be rewritten as [10] L = 1 2φ a M ab ( φ)φ b − V ( φ)(33) (the definition of M ab ( φ) and V ( φ) see in [10,11]). From the Lagrangian (33) we find the Hamiltonian density H = 1 2 π a M −1 ab ( φ) π b + V ( φ),(34) where the canonical field momentum is π a = M ab ( φ)φ b . The treatment of the Skyrme model differs from the theories considered below in that point, that it contains terms of the 4th order in derivatives. As a result, all the scalar products of the model, in particular, the orthogonality conditions (10)- (12) in the vicinity of the static classical solution φ a c ( ξ) acquire a nontrivial integration measure. It is easy to verify, that for the Lagrangian (33) the weight function in the integration measure is M ab (φ c ( ξ)) . Now we verify the conditions of particle-likeness for the standard hedgehog configuration φ a = r a r φ(r). After some algebra we obtain, that the conditions of particle-likeness for this configuration are satisfied in 3 spatial dimensions as well, and once more it holds independently of the profile of the function φ(r). Thus, the soliton (35) of the SU(2)-Skyrme model might be embedded into the irreductible representation of the Poincaré group for the particle with spin without any restrictions on the shape of the chiral angle. To conclude let us mention, that the present analysis can be easily extended to other soliton models including vector fields, etc. On the other hand, the relations (10)- (12) and (29)-(31), being independent of equations of motion, can play an essential role of additional constraints in approximate calculations as well. For example, they can be explored as a test for various sample functions, used in describing the shape of the skyrmion. Concerning the Skyrme model, our analysis is consistent with the well-known result [12], that the spin of SU(2)-skyrmion can be arbitrary. I am grateful to Prof. K. A. Sveshnikov in collaboration with whom the presented results were obtained. . I Zahed, G E Brown, Phys. Rep. 1421I. Zahed and G. E. Brown, Phys. Rep. 142 (1986) 1; . G Holzwarth, B Schwesinger, Rep. Prog. Phys. 49825G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825. . H Verschelde, H Verbeke, Nucl. Phys. 495523H. Verschelde and H. Verbeke, Nucl. Phys. A495 (1989) 523; . H Yamagishi, I Zahed, Phys. Rev. 43891H. Yamagishi and I. Zahed, Phys. Rev. D43 (1991) 891. . J.-L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.-L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038. . O A Khrustalev, A V Razumov, A Yu, Taranov, Nucl. Phys. 17244O. A. Khrustalev, A. V. Razumov, and A. Yu. Taranov, Nucl. Phys. B172 (1980) 44. R Rajaraman, Solitons and Instantons. North-Holland, AmsterdamR. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). . D P Cebula, A Klein, N R Walet, J. Phys. G: Nucl. Part. Phys. 18499D. P. Cebula, A. Klein and N. R. Walet, J. Phys. G: Nucl. Part. Phys. 18 (1992) 499. K A Sveshnikov, Doctoral Dissertation. MoscowMoscow State UniversityK. A. Sveshnikov, Doctoral Dissertation (Moscow State University, Moscow, 1990). . A Dubikovsky, K Sveshnikov, Phys. Lett. 317581A. Dubikovsky and K. Sveshnikov, Phys. Lett. B317 (1993) 581. J Schwinger, Particles, Sources and Fields. Reading, MassachusettsAddison-Wesley Publishing CompanyJ. Schwinger, Particles, Sources and Fields (Addison-Wesley Publishing Com- pany Reading, Massachusetts, 1970). . D P Cebula, A Klein, N R Walet, J. Phys. G: Nucl. Part. Phys. 18499D. P. Cebula, A. Klein and N. R. Walet, J. Phys. G: Nucl. Part. Phys. 18 (1992) 499. . A Dubikovsky, K Sveshnikov, Phys. Lett. 32180A. Dubikovsky and K. Sveshnikov, Phys. Lett. B321 (1994) 80. . G Adkins, C Nappi, E Witten, Nucl. Phys. 228552G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552; . G Adkins, C Nappi, ibid. B233109G. Adkins and C. Nappi, ibid. B233 (1984) 109.

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{'abstract': 'A set of integral relations for rotational and translational zero modes in the vicinity of the classical soliton solution are derived from the particle-like properties of the latter. The validity of these all relations is considered for a number of soliton models in 2+1-and 3+1-dimensions. *', 'arxivid': 'hep-th/9509122', 'author': ['Andrei Dubikovsky dubikovs@sunny.bog.msu.su \nDepartment of Physics\nMoscow State University\n119899MoscowRussia\n'], 'authoraffiliation': ['Department of Physics\nMoscow State University\n119899MoscowRussia'], 'corpusid': 16143891, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5996, 'n_tokens_neox': 5173, 'n_words': 3190, 'pdfsha': '7932da99ba578d3502e0e6713297a7e22c80df24', 'pdfurls': ['https://arxiv.org/pdf/hep-th/9509122v1.pdf'], 'title': ['LORENTZ-COVARIANT ANALYSIS OF A QUANTUM SOLITON', 'LORENTZ-COVARIANT ANALYSIS OF A QUANTUM SOLITON'], 'venue': []}

arxiv

Møller's Energy in the Kantowski-Sachs Space-time 27 Mar 2008 M Abdel-Megied Mathematics Department Faculty of Science Minia University 61915El-MiniaEGYPT Ragab M Gad Mathematics Department Faculty of Science Minia University 61915El-MiniaEGYPT Møller's Energy in the Kantowski-Sachs Space-time 27 Mar 2008 We present a counter example to paper [1] and show that the result obtained is correct for a class of metric but not general. We calculate the total energy of the Kantowski-Sachs space-time by using the energy-momentum definitions of Møller in the theory of general relativity and the tetrad theory of gravity. Introduction Since the birth of the theory of general relativity and this theory has been accepted as a superb theory of space-time and gravitation, as many physical aspects of nature have been experimentally verified in this theory. However, this theory is still incomplete theory, namely, it lacks definition of energy and momentum. In this theory many physicist have introduced different types of energy-momentum complexes [2], each of them being a pseudotensor, to solve this problem. The non-tensorial property of these complexes is inherent in the way they have been defined and so much so it is quite difficult to conceive of a proper definition of energy and momentum of a given system. The recent attempt to solve this problem is to replace the theory of general relativity by another theory, concentrated on the gauge theories for the translation group, the so called teleparallel equivalent of general relativity. We were hoping that the theory of teleparallel gravity would solve this problem. Unfortunately, the localization of energy and momentum in this theory is still an open, unresolved and disputed problem as in the theory of general relativity. Møller modified the theory of general relativity by constructing a gravitational theory based on Weitzenböck space-time. This modification was to overcome the problem of the energy-momentum complex that appears in Riemannian space. In a series of paper [3]- [5], he was able to obtain a general expression for a satisfactory energy-momentum complex in the absolute parallelism space. In this theory the field variable are 16 tetrad components h µ a , from which the Riemannian metric arises as g µν = η ab h a µ h b ν . (1.1) The basic purpose of this paper is to obtain the total energy of the Kantowski-Sachs space-time by using the energy-momentum definitions of Møller in the theory of general relativity and the tetrad theory of gravity. The standard representation of Kantowski and Sachs space-times are given by [7] ds 2 = dt 2 − A 2 (t)dr 2 − B 2 (t)(dθ 2 + sin 2 θdφ 2 ), (1.2) where the functions A(t) and B(t) are function in t and determined from the field equations. For more detailed descriptions of the geometry and physics of this space-time see for example [7], [8] and [9]. On the fourth component of Einstein's complex Prasanna have shown that space-times with purely time dependent metric potentials have their components of total energy and momentum for any finite volume (T 4 i + t 4 i ) identically zero. He had used the Einstein complex for the general Riemannian metric dS 2 − g ij (x 0 )dx i dx j ,(2.1) and concluded the following: For space-times with metric potentials g ij being functions of time variable alone and independent of space variable the components (T 4 i + t 4 i ) vanish identically as a consequence of conservation law. Unfortunately the conclusion above is not the solution to the problem considered, in the sense that it does not give the same result for all metrics have form (2.1), using Einstein complex. If (2.1) is given in spherical coordinates, then Prasanna's conclusion is correct by using Møller's complex but not correct for all metrics by using Einstein's complex. Because Møller's complex could be utlized to any coordinate system, but Einstein's complex give meaningful result if it is evaluated in Cartesian coordinates. In the present paper we have found that the total energy for the Kantwaski-Sachs space-time is identically zero by using Møller's complex, but not zero by using Einstein's complex. In a recent paper [10], Gad and Fouad have found the energy and momentum distribution of Kantowaski-Scahs space-time, using Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy momentum complexes. In this section we restrict our attention to the Einstein's complex which is defined by [11] θ k i = T k i + t k i = u [jk] i ,k , (2.2) with u [jk] i ,k = 1 κ g in √ −g [ − g(g kn g lm − g ln g km )] ,m . (2. 3) The energy and momentum in the Einstein's prescription are given by P i = θ 0 i dx 1 dx 2 dx 3 . (2.4) The Einstein energy-momentum complex satisfies the locall conservation law ∂θ k i ∂x k = 0. (2.5) The energy density for the space-time under consideration, in the Cartesian coordinates, obtained in [10] is (2.6) and the total energy is θ 0 0 = 1 8πAr 4 (A 2 r 2 − B 2 ),E Ein = P 0 = 1 2Ar (A 2 r 2 + B 2 ), Following the approach in [10], we obtain the following components of θ k 0 θ 1 0 = − x 8πA 2 r 4 [ȦA 2 r 2 − B(2AḂ − BȦ)], θ 2 0 = − y 8πA 2 r 4 [ȦA 2 r 2 − B(2AḂ − BȦ)], θ 3 0 = − z 8πA 2 r 4 [ȦA 2 r 2 − B(2AḂ − BȦ)]. (2.7) The components (2.6) and (2.7) satisfy the conservation law (2.5). Hence from equations (2.6) and (2.7), we have θ i 0 = T i 0 + t i 0 = 0 consequently θ 0 0 = T 0 0 + t 0 0 is not identically zero. Energy in the theory of General Relativity In the general theory of relativity, the energy-momentum complex of Møller in a four dimensional background is given as [3] ℑ k i = 1 8π χ kl i,l , (3.1) where the antisymmetric superpotential χ kl i is χ kl i = −χ lk i = √ −g( ∂g in ∂x m − ∂g im ∂x n )g km g nl ,(3. 2) ℑ 0 0 is the energy density and ℑ 0 α are the momentum density components. Also, the energy-momentum complex ℑ k i satisfies the local conservation laws: ∂ℑ k i ∂x k = 0 (3.3) The energy and momentum components are given by (3.5) P i = ℑ 0 i dx 1 dx 2 dx 3 = 1 8π ∂χ 0l i ∂x l dx 1 dx 2 dx 3 . Using these components in equation (3.1), we get the energy and momentum densities as following ℑ 0 0 = 0. (3.6) ℑ 0 1 = ℑ 0 3 = 0, ℑ 0 2 = −A(t) cos θ. (3.7) From equation (3.4) and (3.5) and applying the Gauss theorem, we obtain the total energy and momentum components in the following form 9) 4 Energy in the Tetrad Theory of gravity P 0 = E = 0, (3.8) P α = 0,(3. The super-potential of Møller in the tetrad theory of gravity is given by (see [4]- [6]) U bc a = √ −g 2κ P dbc ef h [Φ f g eh g ad − λg ad γ ef h − (1 − 2λ)g ad γ hf e ],(4.1) where P dbc ef h = δ d e g bc f h + δ d f g bc he − δ d h g bc ef , with g bc f h being a tensor defined by g bc f h = δ b f δ c h − δ b h δ c f , γ abc is the con-torsion tensor given by γ abc = h iµ h i ν;ρ (4.2) and Φ a is the basic vector defined by Φ a = γ b ab . The energy in this theory is expressed by the following surface integral E = lim r→∞ r=const. U 0α 0 n α dS, (4.3) where n α is the unit three vector normal to the surface element dS. The tetrad components of the space-time (1.2), using (1.1), are as following h a µ = [1, A(t), B(t), B(t) sin θ], h µ a = [1, A −1 (t), B −1 (t), B −1 (t) sin θ ]. (4.4) Using these components in (4.2), we get the non-vanishing components of γ µνβ as following Consequently, The only non-vanishing components of basic vector field are γ 011 = −γ 101 = −A(t)Ȧ(t), γ 022 = −γ 202 = −B(t)Ḃ(t), γ 033 = −γ 303 = −B(t)Ḃ(t) sin 2 θ, γ 233 = −γ 323 = −B 2 (t) sin θ cos θ.Φ 0 = −2{Ȧ (t) A(t) +Ḃ (t) B(t) }, Φ 2 = cot θ B 2 (t) . Summary and Discussion In this paper we have shown that the fourth component of Einstein' complex for the Kantowski-Sachs space-time is not identically zero. This give a counter example to the result obtained by Prasanna [1]. We calculated the total energy of Kantowski-Sachs space-time Using Møller's tetrad theory of gravity. We found that the total energy is zero in this space-time. This result do not agree with the previous results obtained in the both theories of general relativity [10] and teleparallel gravity [12], using Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum complexes. In both theories the energy and momentum densities for this space-time are finite and reasonable. We notice that these results are not in conflict with that given by Møller's values for the energy and momentum densities if r tends to infinity. . A R Prasanna, Progress of Theoretical Phsics. 451330A. R. Prasanna, Progress of Theoretical Phsics, 45, 1330 (1971). . R C Tolman, Relativity, Thermodynamics and Cosmology. 227Oxford University PressR. C. Tolman, "Relativity, Thermodynamics and Cosmology, (Oxford University Press, Oxford), p. 227 (1934); L D Landau, E M Lifshitz, The Classical Theory of Fields. Reading, MAAddison-Wesley Press317L. D. Landau and E. M. Lif- shitz, "The Classical Theory of Fields", (Addison-Wesley Press, Read- ing, MA) p. 317 (1951); . A Papapetrou, Proc. R. Ir. Acad. 5211A. Papapetrou, Proc. R. Ir. Acad. A52, 11 (1948); . P G Bergmann, R Thompson, Phys. Rev. 89400P. G. Bergmann and R. Thompson, Phys. Rev. 89, 400 (1953); Gravitation and Cosmology: Principles and Applications of General Theory of Relativity. S Weinberg, Wiley165New YorkS. Weinberg, "Gravitation and Cosmology: Principles and Applications of General Theory of Relativity" ( Wiley, New York) 165 (1972). . C Møller, Ann. Phys. (NY). 4347C. Møller, Ann. Phys. (NY) 4, 347 (1958). . C Møller, Mat. Fys. Medd. K. Vidensk. Selsk. 3110C. Møller, Mat. Fys. Medd. K. Vidensk. Selsk. 31, 10 (1961). . C Møller, Mat. Fys. Medd. K. Vidensk. Selsk. 3913C. Møller, Mat. Fys. Medd. K. Vidensk. Selsk. 39, 13 (1978). . F I Mikhail, M I Wanas, A Hindawi, E I Lashin, Int. J. Theor. Phys. 321627F. I. Mikhail, M. I. Wanas, A. Hindawi and E. I. Lashin, Int. J. Theor. Phys., 32, 1627 (1993). . R Kantowski, R K Sachs, J. Math. Phys. 7443R. Kantowski and R. K. Sachs, J. Math. Phys. 7, 443 (1966). . A S Kompaneets, A S Chernov, Sov. Phys. JETP. 201303A. S. Kompaneets and Chernov A. S., Sov. Phys. JETP 20, 1303 (1965). . C B Collins, J. Math. Phys. 182116C. B. Collins, J. Math. Phys. 18, 2116 (1977). . R M Gad, A Fouad, Astrophys. Space Sci. 310135R. M. Gad and A. Fouad, Astrophys. Space Sci., 310, 135 (2007). . A Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.). 778A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 778 (1915). . R M Gad, Int. J. Theor. Phys. 463263R. M. Gad, Int. J. Theor. Phys. 46, 3263 (2007).

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{'abstract': 'We present a counter example to paper [1] and show that the result obtained is correct for a class of metric but not general. We calculate the total energy of the Kantowski-Sachs space-time by using the energy-momentum definitions of Møller in the theory of general relativity and the tetrad theory of gravity.', 'arxivid': '0803.3953', 'author': ['M Abdel-Megied \nMathematics Department\nFaculty of Science\nMinia University\n61915El-MiniaEGYPT\n', 'Ragab M Gad \nMathematics Department\nFaculty of Science\nMinia University\n61915El-MiniaEGYPT\n'], 'authoraffiliation': ['Mathematics Department\nFaculty of Science\nMinia University\n61915El-MiniaEGYPT', 'Mathematics Department\nFaculty of Science\nMinia University\n61915El-MiniaEGYPT'], 'corpusid': 18581683, 'doi': '10.1155/2010/379473', 'github_urls': [], 'n_tokens_mistral': 3818, 'n_tokens_neox': 3269, 'n_words': 1897, 'pdfsha': '9e73d023f84d0e22d19c8657ddf3bf36e1c6f701', 'pdfurls': ['https://arxiv.org/pdf/0803.3953v1.pdf'], 'title': ["Møller's Energy in the Kantowski-Sachs Space-time", "Møller's Energy in the Kantowski-Sachs Space-time"], 'venue': []}

arxiv

An exact representation isotropic and anisotropic noncommutative phase spaces, and their relations 19 Mar 2014 H Kakuhata M Nakamura Faculty of Science University of Toyama 3190, 930-8555Gofuku, ToyamaJapan Faculty of Engineering University of Toyama 3190, 930-8555Gofuku, ToyamaJapan Introduction An exact representation isotropic and anisotropic noncommutative phase spaces, and their relations 19 Mar 2014UT-HET 088 Noncommutative phase space of an arbitrary dimension is considered. The both of operators coordinates and momenta in noncommutative phase space may be noncommutative. In this paper, we introduce momentum-momentum noncommutativity in addition to coordinate-coordinate noncommutativity. We find an exact form for the linear coordinate transformation which relates a noncommutative phase space to the corresponding ordinary one. As an example, the Hamiltonian of a three-dimensional harmonic oscillator is examined. Introduction Noncommutativity has become a vital field of research owing to its development in string theories, quantum field theories and quantum mechanics [1]. The open string end points are noncommutative in the presence of the background NS-NS B-field which indicates that the coordinates of D-branes are noncommutative [2]. There has been a lot of works based on perturbative and non-perturbative field theories in noncommutative space [3]. An extensive research has also been done on noncommutative quantum mechanical systems [4,5,6,7]. Quantum mechanics in the noncommutative configuration space has attracted much attention. The noncommutative configuration space is characterized by the commutation relations between N-dimensional coordinates q i and momenta p i as [q i , q j ] = iθ ij , [q i , p j ] = iδ ij , [p i , p j ] = 0,(1) where, θ is an N ×N antisymmetric matrix. The noncommutative quantities q i and p i are known to be expressed by the use of the commutative phase space variables Q i and P i as follows: [9,10,11,12]        q i = Q i − N j=1 1 2 θ ij P j , p i = P i , i = 1, . . . , N(2) with [Q i , Q j ] = 0, [Q i , P j ] = iδ ij , [P i , P j ] = 0.(3) Therefore, the Hamiltonian in the noncommutative phase space is rewritten as H(q, p) = H(Q i − N j=1 1 2 θ ij P j , P i ).(4) The noncommutative classical Lagrangian can be written in terms of the commutative coordinates and momenta as PQ − H(Q − P/2θ, P ). And by using the inverse transformation of (2), we can obtain the noncommutative classical Lagrangian with the noncommutative variables [12]. Then the noncommutative commutation relations are given as the Dirac brackets. In the analysis of the noncommutative phase space, the momentum variables are usually regarded as commutative ones. In our opinion, however, it is not natural to introduce only coordinate-coordinate noncommutativity. Because in the Hamiltonian treatment the momenta p are variables independent of and on an equal footing with the coordinates q [8]. In this paper, we introduce not only coordinate-coordinate noncommutativity, but also momentum-momentum noncommutativity. Then, we have a following commutation relations: [q i , q j ] = iθ ij , [q i , p j ] = iδ ij , [p i , p j ] = iη ij ,(5) where θ and η are N × N antisymmmetric matrices. An exact representation of the noncommutative phase space variables (5) by the commutative ones (2) is presented. Similar transformations have been proposed by several authors [13,14,15,16]. For general antisymmetric matrices θ and η, however, those proposed forms make [q i , p j ] non-diagonal which may destroy the canonical cummutation relations. In §2, first, we study transformation from the commutative phase space coordinates to the noncommutative ones, for θ = η. And we show that the isotropic case (θ = η) can be related to anisoropic case (θ = η), by using an regular matrix. As an example, the energy levels of the three-dimensional harmonic oscillator are examined in §3. The last section is devoted to conclusion. Transformation First, we consider the isotropic noncommutative phase space in which θ = η = α 2, . . . , N). (6) Here, α c is a constant, and ǫ ij is the Levi-Civita symbol. The N × N antisymmetric matrix α is written as α ≡ α c ǫ ij . To obtain the representations of the noncommutative q i and p i in term of the corresponding commutative coordinate Q i and momentum P i , we assume [q i , q j ] = iα c ǫ ij , [q i , p j ] = iδ ij , [p i , p j ] = α c ǫ ij , (i, j = 1,             q i = N j=1 (a ij Q j + b ij P j ), p i = N j=1 (c ij Q j + d ij P j ), i = 1, . . . , N,(7) or, equivalently in the matrix notation, q = aQ + bP, p = cQ + dP.(8) Substituting (8) into (5) and using (3), we obtain conditions ab T − ba T = α, ad T − bc T = 1, cd T − dc T = α,(9) where 1 is the identity matrix, the superscript T denotes the matrix transposition. In order to find the forms of the coefficient matrices a, b, c and d, let us impose the following ansatz: a = A(α 2 ), b = B(α 2 )α, c = C(α 2 )α, d = D(α 2 ).(10) Here A and B (C and D) are analytic functions of α 2 , whose forms are to be fixed. For arbitrary functions f (α 2 ), we have f (α 2 )α = αf (α).(11) These relations can easily be seen by noting that a function regular at α c = 0 is a power series of the variable. 1 By using the relations (11) and noting [f (α 2 )α] T = −f (α 2 )α, the conditions (9) reduce to −[A(α 2 )B(α 2 ) + B(α 2 )A(α 2 )]α = α, A(α 2 )D(α 2 ) + B(α 2 )C(α 2 )α 2 = 1, α[C(α 2 )D(α 2 ) + D(α 2 )C(α 2 )] = α.(12) Hence, we have A(α 2 )B(α 2 ) = − 1 2 ,(13)A(α 2 )D(α 2 ) + B(α 2 )C(α 2 )α 2 = 1,(14)C(α 2 )D(α 2 ) = 1 2 .(15) Consequently, we obtain A(α 2 )B(α 2 ) = − 1 2 ,(16)A(α 2 )D(α 2 ) = 1 ± √ 1 + α 2 2 ,(17)C(α 2 )D(α 2 ) = 1 2 ,(18)B(α 2 )C(α 2 )α 2 = 1 ∓ √ 1 + α 2 2 .(19) Note that (1 + √ 1 ± α 2 )/2 is the analytic function of the matrix α 2 . When α 2 = −1, we can easily find that this transformation (8) has no inverse transformation. Thus, we find that the transformation (8) should be of the following form:              q = A(α 2 )Q − 1 2 (A(α 2 )) −1 αP, p = D(α 2 )P + 1 2 (D(α 2 )) −1 αQ, A(α 2 )D(α 2 ) = 1 + √ 1 + α 2 2 .(20) The inverse transformation can be easily obtained as 2        Q = 1 √ 1 + α 2 D(α 2 )q + 1 2 (A(α 2 )) −1 αp , P = 1 √ 1 + α 2 A(α 2 )p − 1 2 (D(α 2 )) −1 αq .(21) In this case, we can consider the commutative limit α c → 0. In addition, we consider the following transformation q ≡ sq,p ≡ (s −1 ) T p.(22) Here s is an arbitrary regular matrix which does not satisfy s † s = 1. The variablesq andp satisfy the commutation relations, [q i ,q j ] = iθ ij , [q i ,p j ] = iδ ij , [p i ,p j ] = iη ij .(23) Here θ ≡ sαs T and η ≡ (s −1 ) T αs −1 are antisymmetric matrices. Thus (q,p) also has the noncommutative commutation relations. Then the variables Q and P transform as follows: Q ≡ sQ,P ≡ (s −1 ) T P.(24) 2 √ 1 + α 2 is also analytic function of matrix α. And the transformation (20) can be rewritten in terms ofq,p,Q,P , θ and η as follows:             q = A(θη)Q − 1 2 (A(θη)) −1 θP , p = D(θη)P + 1 2 (D(θη)) −1 ηQ, A(θη)D(θη) = 1 + √ 1 + θη 2 .(25) This expression tell us, we can choose any noncommutative parameters η and θ, by using the transformation (22). In other words, an anisotropic noncommutative phase space can be representated as a rescaling of an arbitrary isotropic noncommutative phase space. The transformation (20) and (25) are linear, and we can also easily find matrix form of the transformation which corresponds to (20) and (25) by using the regular matrix s. The antisymmetric matrix α can be block-diagonalized as α BD , by using an orthogonal matrix R. When R is taken as s, A(α 2 BD ) and D(α 2 BD ) should be diagonal matrix. By R, the coordinates and momenta are converted into q BD and p BD . Thus, in the noncommutative phase space (q BD , p BD ), transformation (25) can be written an exact matrix form. For the noncommutative phase space (q BD , p BD ), we take the regular matrix s as a product of a diagonal matrix and the orthogonal matrix R T , then one can find an exact matrix form of (25). For example, we take the matrix s as diag( β 1 √ αc , β 2 √ αc , . . . , β N √ αc ). Here α c and β i are constants. Then the variablesq andp satisfy the commutation relations (23). And the noncommutative parameters are defined as θ ij ≡ ǫ ij β i β j and η ij ≡ ǫ ij α 2 c β i β j (repeated indices are not summation convention), (20) can be rewritten as follows,             q = A(α 2 )Q − 1 2 (A(α 2 )) −1 θP , p = D(α 2 )P + 1 2 (D(α 2 )) −1 ηQ, A(α 2 )D(α 2 ) = 1 + √ 1 + α 2 2 .(26) Thus, this phase space is anisotropically noncommutative. In the phase space (q,p), we can consider α c → 0 limit. Then the commutation relations (23) turns to [q i ,q j ] = iθ ij , [q i ,p j ] = iδ ij , [p i ,p j ] = 0.(27) Furthermore, when we choose diag −1 ( β 1 √ αc , β 2 √ αc , . . . , β N √ αc ) as the regular matrix s, then θ ij and η ij are defined as ǫ ij α 2 c β i β j and ǫ ij β i β j . And we can consider α c → 0 limit in the phase space (q,p), then the commutation relations (23) can be rewritten as follows: [q i ,q j ] = 0, [q i ,p j ] = iδ ij , [p i ,p j ] = iη ij .(28) Note that, in above two cases, the commutative limit should be taken β i β j → 0 after the limit α c → 0. Through these resutlts, while keeping the commutation relation between coordinate and momentum, increasing of coordinate and decreasing of momentum (or conversely decreasing coordinate and increasing mommentum ) is changing the noncommutative parameter α. In other words, an arbitrary noncommutative phase space can be tranformed into the other noncommutative phase space by rescaling the noncommutative variables keeping the commutation relation between the coordinates and the momenta. Thus, we can regard an isotropic noncommutative quantum mechanics which has the commutation relation (6) as a basis. Three-demensional Harmonic oscillator In this section, we consider the three-dimensional harmonic oscillator in the noncommutative phase space. Under the condition A(α 2 ) = D(α 2 ), the transformation (20) is rewritten as              q = x + (α 2 )Q − 1 2 (x + (α 2 )) −1 αP, p = x + (α 2 )P + 1 2 (x + (α 2 )) −1 αQ, x + (α 2 ) = 1 + √ 1 + α 2 .(29) In the following, we adopt this expression instead of (20), and we apply it to the three-dimensional harmonic oscillator in the noncommutative phase space. The Hamiltonian for the three-dimensional harmonic oscillator is given by H = 1 2M 3 i=1 p i p i + M 2 ω 2 q i q i .(30) First, we consider the isotropic noncommutative phase space which has the commutation relation [q i , q j ] = iα c ǫ ij , [q i , p j ] = iδ ij , [p i , p j ] = iα c ǫ ij (i = 1, 2, 3).(31) The antisymmetric matrix α can be block-diagonalized by using the orthogonal matrix R. After block-diagonalization of α, [q a ,q b ] = iᾱǫ ab , [q a ,p b ] = iδ ab , [p a ,p b ] = iᾱǫ ab , [q a ,q 3 ] = 0, [q 3 ,p 3 ] = i, [p a ,p 3 ] = 0, (a, b = 1, 2).(32) And we take the regular matrix s as s ≡ diag Mωβ α , Mωβ α , Mωβ α R,(33) hereᾱ and β are constants. This case corresponds to that θ and η are simultaneously diagonalizable. Then the Hamiltonian (30) is rewritten as H = 1 2M 3 i p ipi + M 2 ω 2q iqi (i = 1, 2, 3), = Mω 2 χ A † + A + + A † − A − + 1 2 Θ(A † + A + − A † − A − ) + 1 + Mω 2 (2A † 3 A 3 + 1),(34) where χ ≡ (β 2 M 2 ω 2 + x 4 + ) (ᾱ 4 + x 4 + β 2 M 2 ω 2 ) 2βMx 2 + ω , Θ ≡ x 2 + (ᾱ 2 + β 2 M 2 ω 2 ) (β 2 M 2 ω 2 + x 4 + ) (ᾱ 4 + β 2 M 2 x 4 + ω 2 ) ,(35) and the creation and annihilation operators A † ρ , A ρ are defined as A † ± ≡ 1 √ 2 (a † 1 ± ia † 2 ), A † 3 ≡ a † 3 , A ± ≡ 1 √ 2 (a 1 ∓ ia 2 ), A 3 ≡ a 3 .(36) Here a † i and a i are also creation and annihilation operators respectively. They can be written in terms ofQ andP as a † i ≡ 1 √ 2 1 ξ iQ i − iξ iPi , a i ≡ 1 √ 2 1 ξ iQ i + iξ iPi ,(37) where ξ i have following forms: ξ 1 = ξ 2 ≡ βMω 4 β 2 M 2 ω 2 + x 4 + α 4 + β 2 M 2 x 4 + ω 2 , ξ 3 ≡ √ Mω.(38) A ρ and a i satisfy the commutator algebra [a i , a j ] = 0, [a i , a † j ] = δ ij , [a † i , a † j ] = 0, (i, j = 1, 2, 3) (39) [A ρ , A σ ] = 0, [A ρ , A † σ ] = δ ρσ , [A † ρ , A † σ ] = 0. (ρ, σ = ±, 3)(40) In the sequel, we obatain the Schrödinger equation in the noncommutative phase space in terms of the commutative variables H|n ± , n 3 = Mω 2 χ N + + N − + 1 2 Θ(N + − N − ) + 1 + (2N 3 + 1) |n ± , n 3 ,(41) where N ± ≡ A † ± A ± and N 3 ≡ A † 3 A 3 . Since [N ρ , N σ ] = 0, the eigenvector |n ± , n 3 is given by |n ± , n 3 = 1 N + ! 1 N − ! 1 √ N 3 ! A † + n + A † − n − A † 3 n 3 |0, 0, 0 .(42) Futhermore, it is useful to recognize that the system under consideration possesses, its commutative couterpart as well as, the SU(2) symmetry whose generators (J 1 , J 2 , J 3 ) and the Casimir operator (J 2 ) are J 1 = 1 2 (a † 1 a 2 + a 1 a † 2 ),(43)J 2 = i 2 (a † 1 a 2 − a 1 a † 2 ),(44)J 3 = 1 2 (a † 1 a 1 − a † 2 a 2 ),(45)J 2 = 3 k J k J k = N 2 N 2 + 1 ,(46) where N = a † 1 a 1 + a † 2 a 2 . We can easily find N = N + + N − and J 2 = 1 2 (N + − N − ). It is straightforward to verify that (43)-(46) implies [J k , J l ] = iǫ klm J m and [J 2 , J k ] = 0, as it must be. Since [N, J k ] = 0, it follows from (41) that the energy eigenvalue problem leads H|j, m, n 3 = Mω 2 [χ (N + 2ΘJ 2 + 1) + (2N 3 + 1)] |j, m, n 3(47) where |j, m is the common eigenvector of J 2 and J 2 , and |j, m, n 3 = |n ± , n 3 . They are labeled by the eigenvalues of J 2 and J 2 , named j and m, respectively. As is well known, j = 0, 1/2, 1, 3/2, ..., − j ≤ m ≤ j,(48) while n = 2j. Θ is not equal to zero for any real number θ and η. Thus the angular monmentum term J 2 is always appear in the noncommutative harmonic oscillator, and it can have the half integer eigenvalue. It is easy to find that this term J 2 is also appear in a pair of harmonic oscillators A † + A + and A † − A − . Certainly, this angular momentum term also appears when we treate the noncommutative harmonic oscillator which has only coordinatecoordinate noncommutativity in the Moyal product method [17,18]. Naturaly, this angular momentum term which is appear in the pair of harmonic oscillator is canceled in the commutative limit [4,5,6,7]. Next, we comment on that θ = sαs T and η = (s −1 ) T αs −1 aren't simultaneously block-diagonalizable. Then the regular matrix s can be given s ≡ diag Mωβ α (1 + ε 1 ), Mωβ α (1 + ε 2 ), Mωβ α c (1 + ε 3 ) R,(49) whereᾱ, β and ε i are constants. ε i are shifts of the noncommutative parameters from the antisymmetric matrices θ and η are simultaneously diagonalizable case. The noncommutative parameter is considered very small, so ε i is also very small. The Hamiltonian is H = H + H ′ ,(50) where H is (41), and H ′ is defined as H ′ = O 1 (ε i )A † + A + + O 2 (ε i )A † − A − + O 3 (ε i )A † 3 A + + O 4 (ε i )A † + A 3 + O 5 (ε i )A † 3 A − + O 6 (ε i )A † − A 3 + O 7 (ε i )A + A − + O 8 (ε i )A † + A † − + O 9 (ε i )A − A + + O 10 (ε i )A † − A † + + O 11 (ε i )A 3 A + + O 12 (ε i )A † 3 A † + + O 13 (ε i )A 3 A − + O 14 (ε i )A † 3 A † − .(51) In this case, the second term H ′ may be treated as a perturbation. We give that the three dimensional harmonic oscillator in the noncommutative phase space which the coordinate and momentum satisfy the commutation relation (5) may calculate as a pertabative method for the arbitrary noncommutative parameter θ and η. Conclusion We have obtained the exact linear transformation in the phase space which relates an arbitrary noncommutative phase space to a commutative one. Using a regular matrix, an arbitrary isotropic noncommutative parameter is converted into any anisotropic noncommutative parameter (or conversely an arbitrary anisotropic noncommutative parameter is converted into the isotrocpic noncommutative parameter). And we can obtain an exact matrix form of transfromation by using suitable regular matirx. Thus, we can consider (20) as a basis of the transformation. It is a special case that either coordinates or momenta are noncommutative and the other are commutative. The energy of the three-dimensional harmonic oscillator described by the noncommutative coordinate and momentum variables was calculated as an example. When the noncommutative parameter θ and η are simultaneously diagonalizable, the three dimensional harmonic oscillator is described by the one-dimensional harmonic oscillator and the two-dimensional noncommutative harmonic oscillator. Except for the commutative limit, a pair of harmonic oscillators with the angular momentum terms always appear. The angular momentum terms with oppsite sign each other, which can have a half integer eigenvalue, and those are canceled as a result of the commutative limit. In the case of that θ and η aren't simultaneously diagonalizable, interaction term of the two-dimensional noncommutative harmonic oscillator and the commutative oscillator appears in the Hamiltonian as a pertabation. 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{'abstract': 'Noncommutative phase space of an arbitrary dimension is considered. The both of operators coordinates and momenta in noncommutative phase space may be noncommutative. In this paper, we introduce momentum-momentum noncommutativity in addition to coordinate-coordinate noncommutativity. We find an exact form for the linear coordinate transformation which relates a noncommutative phase space to the corresponding ordinary one. As an example, the Hamiltonian of a three-dimensional harmonic oscillator is examined.', 'arxivid': '1403.2171', 'author': ['H Kakuhata ', 'M Nakamura \nFaculty of Science\nUniversity of Toyama\n3190, 930-8555Gofuku, ToyamaJapan\n', '\nFaculty of Engineering\nUniversity of Toyama\n3190, 930-8555Gofuku, ToyamaJapan\n', '\nIntroduction\n\n'], 'authoraffiliation': ['Faculty of Science\nUniversity of Toyama\n3190, 930-8555Gofuku, ToyamaJapan', 'Faculty of Engineering\nUniversity of Toyama\n3190, 930-8555Gofuku, ToyamaJapan', 'Introduction\n'], 'corpusid': 119134330, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9316, 'n_tokens_neox': 7702, 'n_words': 4255, 'pdfsha': '29374e1cb666d193586599e8b9e444412ebe48e4', 'pdfurls': ['https://arxiv.org/pdf/1403.2171v2.pdf'], 'title': ['An exact representation isotropic and anisotropic noncommutative phase spaces, and their relations', 'An exact representation isotropic and anisotropic noncommutative phase spaces, and their relations'], 'venue': []}

arxiv

RELATIONS BETWEEN CUSP FORMS SHARING HECKE EIGENVALUES Aug 2022 Dipendra Prasad Ravi Raghunathan RELATIONS BETWEEN CUSP FORMS SHARING HECKE EIGENVALUES Aug 2022 In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on GL n (A F ) is contained in the set of Hecke eigenvalues of a cusp form on GL m (A F ) for n ≤ m. This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work. INTRODUCTION Let F be a number field. Each automorphic representation π of GL d (A F ) gives rise to Hecke eigenvalues (also called the Satake parameter), a d-tuple of (unordered) nonzero complex numbers H(π v ) = (a 1v , · · · , a dv ), at each place v of F where π is unramified, and thus at almost all places of F . Let π 1 and π 2 be two irreducible automorphic representations of GL n (A F ) which are written as isobaric sums: π 1 = π 11 ⊞ π 12 ⊞ · · · ⊞ π 1ℓ , π 2 = π 21 ⊞ π 22 ⊞ · · · ⊞ π 2ℓ ′ , where π 1j and π 2k are irreducible cuspidal automorphic representations of GL d j (A F ) and GL d k (A F ) respectively. Then by the strong multiplicity one theorem due to Jacquet and Shalika cf. [JS1], [JS2], if π 1 and π 2 have the same Hecke eigenvalues H(π 1,v ) = H(π 2,v ) at almost all places v of F where π 1 , π 2 are unramified, then ℓ = ℓ ′ , and up to a permutation of indices, π 1j = π 2j . In this paper, we will consider a variant of the strong multiplicity one theorem, identified in the following definition. Definition: Given automorphic representations π 1 on GL(n, A F ) and π 2 on GL(m, A F ), we say that π 1 is immersed in π 2 , written π 1 π 2 , if the Hecke eigenvalues of π 1 (counted with multiplicity) are contained in the Hecke eigenvalues of π 2 (counted with multiplicity) for almost all primes of the number field F . On the other hand, we say that π 1 is embedded in π 2 , written as π 1 ⊂ π 2 if there is an automorphic representation π 3 such that, π 2 = π 1 ⊞ π 3 . The following is the motivating question for this paper. Question: (1) Can it happen for distinct cuspidal representations that π 1 π 2 ? (2) If yes, can we classify all such pairs of cuspidal representations π 1 π 2 ? One would have liked to assert that for cuspidal representations π 1 π 2 never happens if n < m, but that is not true. For example, let π be a cuspidal non-CM automorphic representation of PGL 2 (A F ). At any unramified place v of F , if (a v , a −1 v ) are the Hecke eigenvalues of π v , then for the automorphic representation Sym 2 (π) of PGL 3 (A F ), the Hecke eigenvalues at the place v of F , are (a 2 v , 1, a −2 v ). Thus the Hecke eigenvalues of the trivial representation of GL 1 (A F ) are contained in the set of Hecke eigenvalues of the cuspidal automorphic representation Sym 2 (π) of PGL 3 (A F ) at each unramified place of π. This paper is written in the hope that although H(π 1,v ) can be contained in H(π 2,v ) of π 2 , at almost all the unramified places of π 1 and π 2 , without π 1 being the same as π 2 , this happens rarely, and only for pairs of representations (π 1 , π 2 ) which are related in some well-defined way. We begin by proving the following proposition. Proposition 1. Let π 1 (resp. π 2 ) be an irreducible cuspidal automorphic representation of GL n (A F ) (resp. GL n+1 (A F )). Then H(π 1,v ) of π 1 cannot be contained in H(π 2,v ) of π 2 at almost all places of F where π 1 , π 2 are unramified. Proof. The proof is a simple consequence of the strong multiplicity one theorem of Jacquet-Shalika recalled at the beginning of this paper. Let ω 1 (resp. ω 2 ) be the central character of π 1 (resp. π 2 ); these are Grössencharacters of GL 1 (A F ). It is easy to see that, if H(π 1,v ) is contained in H(π 2,v ) at almost all places v of F , then, π 1 ⊞ (ω 2 /ω 1 ) = π 2 , which is not allowed by the strong multiplicity one theorem. Here is another similarly 'negative' result, this time proved with considerably more effort. Proposition 2. Let π be an irreducible cuspidal automorphic representation of GL 4 (A F ). Then H(π v ) cannot contain 1 at almost all places v of F where π is unramified. Proof. We will prove the proposition by contradiction, so assume that H(π v ) contains 1 at almost all places of F where π is unramified. Observe that to say that H(π v ) contains 1 is equivalent to saying that det(1 − H(π v )) = 0, which translates into the following identity (assuming that H(π v ) operates on a 4 dimensional vector space V ): (One way to think of this identity is in the Grothendieck group of representations of an abstract group G which comes equipped with a 4-dimensional representation V of G such that the action of any g ∈ G on V has a nonzero fixed vector.) Thus we get the identity: 1 + Λ 4 (V ) + Λ 2 (V ) = V + Λ 3 (V ). Let the central character of π be ω : A × F /F × → C × . Since we know by the work of Kim, cf. [Kim], that Λ 2 (π) is automorphic, by the strong multiplicity one theorem, we get an identity of the isobaric sum of automorphic representations: 1 ⊞ ω ⊞ Λ 2 (π) = π ⊞ ω · π ∨ . Observe that the right hand side of this equality is a sum of two cuspidal representations on GL 4 (A F ), whereas there are two one dimensional characters of A × F /F × on the left hand side. This is not allowed by the strong multiplicity one theorem, completing the proof of the proposition. The following question lies at the basis of this work. Question 1. Let π 1 (resp. π 2 ) be an irreducible cuspidal automorphic representation of GL n (A F ) (resp. GL n+2 (A F )). Suppose that H(π 1,v ) is contained in H(π 2,v ) at almost all places of F where π 1 and π 2 are unramified. Then is there an automorphic representation π of GL 2 (A F ) with central character ω : A × F /F × → C × , and a character χ : A × F /F × → C × , such that, π 1 = χ · ω ⊗ Sym n−1 (π), π 2 = χ ⊗ Sym n+1 (π), i.e., up to twist by a character, is (π 1 , π 2 ) = (ω ⊗ Sym n−1 (π), Sym n+1 (π))? We will provide an affirmative answer to this question for n = 1, 2, 3 in this paper. On the other hand, in section 5, we will provide counter-examples to this question using the strong Artin conjecture for all pairs of integers (q − 1, q + 1) where q ≥ 5 is a prime power. The work [Ca] proves strong Artin conjecture for certain cases for q = 5, which allows us to construct an unconditional counter example for the pair (GL 4 , GL 6 ) over Q in section 5. The question studied in this paper can be studied from a purely group theoretic point of view, and is discussed in section 5 from this perspective. We are unaware of this group theoretic point of view to have been put to use earlier; it seems of interest for connected reductive groups too. In section 6, we prove that the group theoretic question has an affirmative answer for groups which are not virtually abelian, i.e., do not contain a subgroup of finite index which is abelian. Thus in the automorphic context, when one of the representations π 1 , or π 2 is Steinberg at a finite place, or has regular infinitesimal character at one of the archimedean places of F , our question should have an affirmative answer, but for the moment, we do not know how to deal with it. Remark 1. Here is a geometric analogue of the questions being discussed in this paper. 1 Let A and B be abelian varieties over a number field F with A simple. For v any finite place of F where both A and B have good reduction, let A v , B v denote their reductions mod v (thus A v , B v are abelian varieties over finite fields). Assume that there are isogenies from A v to B v (not surjective as we are not assuming dim(A) = dim(B)) for almost all places v of F where 1 This question has now been settled in a recent work of Khare and Larsen, cf. [KL]. A and B have good reduction. Then the question is if there is an isogeny from A to B? If dim(A) = dim(B), this is a consequence of the famous theorem of Faltings. Remark 2. The paper was inspired by the notion of relevance introduced in [GGP], and to understand whether two global A-parameters which are locally relevant at all places must be globally relevant. This is not the case, and exactly for the reason discussed in this paper: that the Hecke eigenvalues of the cuspidal representation π 1 may be contained in the set of Hecke eigenvalues of the cuspidal representation π 2 at almost all places of the number field without π 1 being the same as π 2 . Remark 3. Most of the paper deals with cusp forms π 1 on GL n (A F ), and π 2 on GL m (A F ) for the restricted pairs (n, m) with m = n + 2, as the first non-obvious case beyond m = n and m = n + 1, such that the Hecke eigenvalues of π 1 are contained in the Hecke eigenvalues of π 2 at almost all places of F . However, one might begin at the other extreme (n, m) = (1, m), and try to classify cuspforms π 2 on GL m (A F ) such that the Hecke eigenvalues of π 2 at almost all places of F contain the eigenvalue 1. By Proposition 3 below, there is a nice answer for (n, m) = (1, 3), whereas by Proposition 2, there are none in the case (n, m) = (1, 4). It is easy to see that the cuspidal representations π 2 of GL 6 (A F ) which arise as the basechange of a cuspidal selfdual representation of PGL 3 (A E ) for E/F quadratic, have Hecke eigenvalue 1 at almost all places of F . Using Theorem 2 of the paper [Yam] and using a similar identity as in the proof of Proposition 2 which this time would be: 1 + Λ 2 + Λ 4 (V ) + Λ 6 (V ) = V + Λ 3 (V ) + Λ 5 (V ), a cuspidal representations π 2 of GL 6 (A F ) having Hecke eigenvalue 1 at almost all places of F arises as the basechange of a cuspidal representation of GL 3 (A E ) for E/F quadratic (which is most likely selfdual, and on PGL 3 (A F ), but this we have not proved). We have not investigated the situation for general pairs (1, m). We end the introduction by remarking that the last two sections of the paper, sections 5 and 6, are written for finite groups and Lie groups respectively, and are quite independent of the earlier sections. Section 5 eventually has implications for automorphic representations through the known cases of (strong) Artin's conjecture. Since automorphic representations on GL n (A F ), F a function field, are characterised by their Galois representations by the work of L. Lafforgue, these sections also construct both counter-examples to Question 1 above, and to assert that the only counter-examples come from potentially abelian automorphic representations. Acknowledgements: The authors thank F. Calegari and D. Ramakrishnan for helpful correspondences, and the referee for careful reading of our paper and several useful comments. PRELIMINARIES For automorphic representations π 1 , π 2 on GL a (A F ), GL b (A F ), we denote by π 1 ⊞ π 2 the isobaric sum of π 1 , π 2 , which is an automorphic form on GL a+b (A F ). If H(π 1,v ) and H(π 2,v ) are the Hecke eigenvalues of π 1 and π 2 , then the Hecke eigenvalues of π 1 ⊞π 2 is the union (with multiplicities) of H(π 1,v ), H(π 2,v ). We will also use the notation A ⊞ B where A (resp. B) is any collection of a (resp. b) nonzero complex numbers defined for almost all finite places v of F . In this generality, we will use partial L function L(s, A), where the Euler product is taken outside a finite set S of places, S containing the places at infinity. In the same spirit, for A, B as in the last para, we will define A ⊠ B to be a collection of a · b nonzero complex numbers for almost all finite places v of F , and the associated partial Rankin product L function L(s, A ⊠ B), again where the Euler product is taken outside a finite set S of places, S containing the places at infinity. Lemma 1. Suppose C, D are automorphic representations on GL c (A F ), GL d (A F ), and χ a Grössencharacter on F . Suppose A is any collection of c + d − 1 nonzero complex numbers defined for almost all finite places v of F such that A ⊞ χ = C ⊞ D. Suppose that L(χ −1 A, s) is known to have meromorphic continuation to the entire complex plane with no zero at s = 1. Then there is an automorphic representation π 3 of GL c+d−1 (A F ) whose Hecke eigenvalues are equal to A at almost all finite places v of F . Proof. Expand χ −1 (C ⊞ D) as an isobaric sum of cusp forms, and note that for any cusp form π on GL m (A F ), L(π, s), the partial L-function without regard to omitted set of places, has a pole at s = 1 if and only if m = 1, and π = 1. Therefore by what is given for L(χ −1 A, s), the isobaric sum decomposition of χ −1 (C ⊞ D) in terms of cusp forms must contain the trivial representation of GL 1 (A F ), omitting which from χ −1 (C ⊞ D) defines χ −1 A as an automorphic representation. Lemma 2. Suppose π 1 is a cuspidal automorphic representation on GL n (A F ) and π 2 is a cuspidal automorphic representation on GL n+2 (A F ) such that at almost all unramified places of π 1 and π 2 , H(π 1,v ) ⊆ H(π 2,v ). Let ω 1 (resp. ω 2 ) be the central character of π 1 (resp. π 2 ) which is a Grössencharacter on GL 1 (A F ). Suppose that Λ 2 (π 2 ), Sym 2 (π 1 ) are known to be automorphic. Then, (1) The Rankin product π 1 ⊠ π 2 is automorphic. (2) We have the isobaric decomposition of automorphic representations: π 2 ⊠ π 1 ⊞ ω 2 /ω 1 = Λ 2 (π 2 ) ⊞ Sym 2 (π 1 ). Proof. We first prove the identity expressed in (2), i.e., that the two sides have the same Hecke eigenvalues at almost all the primes of F . This task is made more transparent by looking at vector spaces V, W, A with V = W + A with A two dimensional, and noting the identity: V ⊗ W + Λ 2 (A) = W ⊗ W + A ⊗ W + Λ 2 (A) = Λ 2 (W ) + A ⊗ W + Λ 2 (A) + Sym 2 (W ) = Λ 2 (V ) + Sym 2 (W ). Now, Lemma 1 proves the automorphy of π 2 ⊠ π 1 since its L-function is known to be entire and non-vanishing on the line Re(s) = 1 by the Rankin-Selberg theory, see Theorem 5.2 of Shahidi [Sha]. Lemma 3. Suppose π 1 , π 2 , π 3 are cuspidal automorphic representations on GL n i (A F ) (for i = 1, 2, 3). Suppose that the Rankin products π 1 ⊠ π 2 and π 2 ⊠ π 3 are known to be automorphic. Then in the isobaric sum decomposition of π 1 ⊠ π 2 , π ∨ 3 ⊂ π 1 ⊠ π 2 if and only if in the isobaric sum decomposition of π 2 ⊠ π 3 , π ∨ 1 ⊂ π 2 ⊠ π 3 . Proof. Since the Rankin products π 1 ⊠ π 2 is given to be automorphic, by Jacquet-Shalika, the L-function, L(s, π 1 ⊠ π 2 ⊠ π 3 ), has a pole at s = 1 if and only if π ∨ 3 ⊂ π 1 ⊠ π 2 . Same triple-product L-function dictates π ∨ 1 ⊂ π 2 ⊠ π 3 . Besides the strong multiplicity one theorem of Jacquet-Shalika, we will use the symmetric square lift of Gelbart-Jacquet cf. [GJ] which we state in the form we will use. The work of Gelbart-Jacquet was to establish the symmetric square lift from GL 2 to GL 3 ; for characterising the image of the symmetric square lift, we refer to [Ra1]. Theorem 1. Let π 2 be a cuspidal automorphic representation of PGL 3 (A F ) with π 2 ∼ = π ∨ 2 . Then π 2 arises as the adjoint lift of an automorphic representation π of GL 2 (A F ), i.e., π 2 ∼ = Ad (π) = ω −1 ⊗ Sym 2 (π), where ω is the central character of π, a Grössencharacter on GL 1 (A F ). Corollary 1. Let π 2 be a cuspidal automorphic representation of GL 3 (A F ) with π 2 ∼ = χ ⊗ π ∨ 2 for χ a Grössencharacter on GL 1 (A F ). Then π 2 can be written as π 2 = λ ⊗ Sym 2 (π), where π is a cuspidal automorphic representation of GL 2 (A F ), and λ a Grössencharacter on GL 1 (A F ). Proof. Let ω 2 be the central character of π 2 . Comparing the central characters for the given isomorphism: (1) π 2 ∼ = χ ⊗ π ∨ 2 , it follows that, ω 2 2 = χ 3 . Therefore for µ = χ/ω 2 , the isomorphism in (1) can be rewritten as: (2) π 2 ∼ = µ −2 ⊗ π ∨ 2 , or, (3) (µ ⊗ π 2 ) ∼ = (µ ⊗ π 2 ) ∨ . Therefore, the representation µ⊗π 2 of GL 3 (A F ) is selfdual. Comparing the central characters on the two sides of equation (3), we find that the central character ω of the representation µ ⊗ π 2 of GL 3 (A F ) is quadratic. Twisting the representation µ ⊗ π 2 of GL 3 (A F ) by ω, we find that the representation (ωµ) ⊗ π 2 of GL 3 (A F ) is both selfdual and of trivial central character, so Theorem 1 applies, proving that π 2 is a symmetric square up to a twist. THE RESULTS We introduce the following notation keeping Question 1 in mind. Suppose π 1 is a cuspidal automorphic representation on GL n (A F ) and π 2 is a cuspidal automorphic representation on GL m (A F ) such that at almost all unramified places of π 1 and π 2 , H(π 1,v ) ⊆ H(π 2,v ), we write π 1 π 2 . We observe that we may twist the pair (π 1 , π 2 ) appearing in Question 1 by a Grössencharacter. Accordingly, in the following proposition that provides an affirmative answer to Question 1 for n = 1 we may assume that π 1 = 1. Proposition 3. Let 1 denote the trivial representation of GL 1 (A F ) and suppose that π 2 is a cuspidal automorphic representation on GL 3 (A F ) such that 1 π 2 . Then π 2 is a self-dual representation of PGL 3 (A F ), and arises as ω −1 · Sym 2 (π) (the adjoint lift) of a cuspidal automorphic form π on GL 2 (A F ) with central character ω : A × F /F × → C × . Proof. The proof will be a simple consequence of the strong multiplicity one theorem of Jacquet-Shalika recalled in the beginning of this paper and Theorem 1 due to Gelbart-Jacquet. Let ω 2 be the central character of π 2 which is a Grössencharacter on GL 1 (A F ). By Lemma 2 in this case for (π 1 , π 2 ) = (1, π 2 ), it follows that Λ 2 (π 2 ) ⊞ 1 = π 2 ⊞ ω 2 . Therefore, by the strong multiplicity one theorem, we deduce that: (1) ω 2 = 1, (2) Λ 2 (π 2 ) = π 2 . By (1) and (2), we find that π 2 ∼ = π ∨ 2 . Therefore, by Theorem 1 due to Gelbart-Jacquet, π 2 arises as the adjoint lift from a cuspidal automorphic form π on GL 2 (A F ), i.e., π 2 = ω −1 · Sym 2 (π), proving the proposition. The following proposition provides an affirmative answer to Question 1 for n = 2. Proposition 4. Suppose that π 1 is a cuspidal automorphic representation on GL 2 (A F ) with central character ω 1 : A × F /F × → C × , and that π 2 is a cuspidal automorphic representation on GL 4 (A F ), and that π 1 π 2 . Then, (1) π 1 cannot be CM (a CM representation is one defined using a Grössencharacter on a quadratic extension E of F ). (2) π 2 = ω −1 1 ⊗ Sym 3 (π 1 ). Proof. Let ω 2 be the central character of π 2 which is a Grössencharacter on GL 1 (A F ). By Lemma 2, (1) π 2 ⊠ π 1 ⊞ ω 2 /ω 1 = Λ 2 (π 2 ) ⊞ Sym 2 (π 1 ), where all the terms appearing above are automorphic: Λ 2 (π 2 ) by Kim [Kim], Sym 2 (π 1 ) by Gelbart-Jacquet [GJ], and π 2 ⊠ π 1 by Lemma 2. We first assume that π 1 is CM. Observe that since π 1 is a cusp form on GL 2 (A F ) and π 2 a cusp form on GL 4 (A F ), π 2 ⊠π 1 cannot contain any Grössencharacter. Therefore, by the isobaric decomposition (1), exactly one of the two terms Λ 2 (π 2 ) or Sym 2 (π 1 ) may contain a Grössencharacter. Since we have assumed that π 1 is CM, Sym 2 (π 1 ) contains a Grössencharacter, and therefore, Λ 2 (π 2 ) cannot contain a Grössencharacter if π 1 is CM. Since π 1 is CM, we can write, Sym 2 (π 1 ) = π 3 ⊞ χ 3 , where χ 3 is a Grössencharacter, and π 3 must be cuspidal (because the left hand side of (1) has only one Grössencharacter in its isobaric decomposition). By (1), χ 3 = ω 2 /ω 1 , and we can simplify (1) to π 2 ⊠ π 1 = Λ 2 (π 2 ) ⊞ π 3 . From Lemma 3, it follows that, π 2 = π 1 ⊠ π ∨ 3 . Therefore, Λ 2 (π 2 ) = [Sym 2 (π 1 ) ⊠ Λ 2 (π ∨ 3 )] ⊞ [Λ 2 (π 1 ) ⊠ Sym 2 (π ∨ 3 )]. Since Sym 2 (π 1 ) = π 3 ⊞ χ 3 , therefore as Λ 2 (π ∨ 3 ) is a Grössencharacter, we find that Λ 2 (π 2 ) contains a Grössencharacter which is not allowed, proving that π 1 cannot be a CM form. Now we turn to the case when π 1 is not CM in which case it is known by Gelbart-Jacquet that Sym 2 (π 1 ) is a cuspidal automorphic representation of GL 3 (A F ). Therefore by the strong multiplicity one theorem applied to (1), we make the following conclusions: (1) the character ω 2 /ω 1 : A × F /F × → C × must belong to the isobaric sum decomposition of Λ 2 (π 2 ), in particular, π 2 ∼ = π ∨ 2 ⊗ (ω 2 /ω 1 ), i.e., π 2 has parameter in the symplectic similitude group, and considering the similitude factor, we find: (2) (ω 2 /ω 1 ) 2 = ω 2 , i.e., ω 2 = ω 2 1 . (2) Sym 2 (π 1 ) must be contained in the isobaric sum decomposition of π 2 ⊠ π 1 . Since by [KS1], [KS2], π 2 ⊠ π 1 and Sym 2 (π 1 ) ⊠ π 1 are known to be automorphic, we can apply Lemma 3, and conclude that: (3) π ∨ 2 = π 2 ⊗ (ω 1 /ω 2 ) ⊂ π 1 ⊠ Sym 2 (π ∨ 1 ) = π 1 ⊠ ω −2 1 ⊠ Sym 2 (π 1 ) . It is easy to see that, π 1 ⊠ Sym 2 (π 1 ) = (ω 1 ⊗ π 1 ) ⊞ Sym 3 (π 1 ), therefore we can write (3) as: (4) π 2 ⊗ (ω 1 /ω 2 ) ⊂ (ω −1 1 ⊗ π 1 ) ⊞ ω −2 1 ⊠ Sym 3 (π 1 ) . Since π 2 is a cuspidal automorphic representation of GL 4 (A F ), applying the strong multiplicity one theorem to (4), the only option we have (after using (2)) is that: π 2 = ω −1 1 ⊗ Sym 3 (π 1 ) , proving the proposition. The following proposition provides an affirmative answer to Question 1 for n = 3. The proof of this proposition will use the unproved cases of functorialty for Λ 2 (π 2 ) where π 2 is a cuspform on GL 5 (A F ), as well as Sym 6 (π) for π a cusp form on GL 2 (A F ). It may be mentioned that although automorphy of Sym 2 (π 1 ) for π 1 a cuspform on GL 3 (A F ) is not known, in our context below, it will be applied to π 1 which is selfdual up to a twist, and hence is a symmetric square of a cuspform on GL 2 (A F ) up to a twist by Corollary 1 which allows one to conclude automorphy of Sym 2 (π 1 ) using known cases of functoriality for Sym 4 (π). Proposition 5. Suppose that π 1 is a cuspidal automorphic representation on GL 3 (A F ) and that π 2 is a cuspidal automorphic representation on GL 5 (A F ) such that π 1 π 2 . Then, there exists a cuspidal automorphic representation π of GL 2 (A F ) of central character ω such that up to simultaneous twisting of the pair (π 1 , π 2 ) by a Grössencharacter, we have: π 1 = Sym 2 (π), π 2 = ω −1 ⊗ Sym 4 (π). Proof. Let ω 1 (resp. ω 2 ) be the central character of π 1 (resp. π 2 ) which is a Grössencharacter on GL 1 (A F ). By Lemma 2, for any Grössencharacter χ on F : (1) L(s, π 2 × π 1 × χ)L(s, ω 2 /ω 1 × χ) = L(s, [Λ 2 (π 2 ) ⊕ Sym 2 (π 1 )] × χ), Since π 2 is a cuspidal automorphic representation of GL 5 (A F ), it is known by Jacquet-Shalika, cf. [JS], that L(s, Λ 2 (π 2 )) cannot have a pole at s = 1. Therefore for χ = ω 1 /ω 2 , since the left hand side of the product of L-functions in (1) has a simple pole at s = 1, right hand side of (1) too must have a simple pole, contributed therefore by L(s, Sym 2 (π 1 ) ⊗ ω 1 /ω 2 ). In particular, π ∨ 1 ∼ = π 1 ⊠ ω 1 /ω 2 . By Corollary 1, such representations of GL 3 (A F ) arise as a twist of a symmetric square: π 1 ∼ = λ ⊗ Sym 2 (π), for a cuspidal automorphic representation π of GL 2 (A F ) of central character ω, and λ a Grössencharacter on A × F . Twisting the pair (π 1 , π 2 ) by λ −1 , we assume that π 1 ∼ = Sym 2 (π). Since π 1 = Sym 2 (π), and it is easy to see that, Sym 2 (π 1 ) = Sym 2 (Sym 2 (π)) = ω 2 + Sym 4 (π), and therefore by Kim, cf. [Kim], since Sym 4 (π) is known to be automorphic, so is Sym 2 (π 1 ). Since we are assuming that Λ 2 (π 2 ) is known to be automorphic, Lemma 2 applies, allowing us to conclude that π 2 ⊠ π 1 is automorphic and we have the isobaric decomposition: (2) π 2 ⊠ π 1 ⊞ ω 2 /ω 1 = Λ 2 (π 2 ) ⊞ Sym 2 (π 1 ). Therefore, by the strong multiplicity one theorem applied to (2), we conclude Sym 2 (π 1 ) must be contained in the isobaric sum decomposition of π 2 ⊠ π 1 as a direct summand. Applying Lemma 3 to (2), we conclude that: (3) π 2 ⊂ π ∨ 1 ⊠ Sym 2 (π 1 ), again as a direct summand, since as we will see now, π ∨ 1 ⊠ Sym 2 (π 1 ) is automorphic by our assumption that Sym 6 (π) is automorphic. Since π 1 = Sym 2 (π), and Sym 2 (Sym 2 (π)) = ω 2 + Sym 4 (π), we find that: π 1 ⊠ Sym 2 (π 1 ) = Sym 2 (π) ⊠ (ω 2 ⊞ Sym 4 (π)), = ω 2 Sym 2 (π) ⊞ Sym 2 (π) ⊠ Sym 4 (π), = ω 2 Sym 2 (π) ⊞ ω 2 Sym 2 (π) ⊞ ωSym 4 (π) ⊞ Sym 6 (π), if particular, if Sym 6 (π) is automorphic, so is π ∨ 1 ⊠ Sym 2 (π 1 ). Since π 2 ⊂ π ∨ 1 ⊠ Sym 2 (π 1 ) = ω −2 π 1 ⊠ Sym 2 (π 1 ), we find that: (4) π 2 ⊂ Sym 2 (π) ⊞ Sym 2 (π) ⊞ ω −1 Sym 4 (π) ⊞ ω −2 Sym 6 (π). Now π 2 is a cuspidal representation on GL 5 (A F ), and by Proposition 6 proved in the next section, isobaric decomposition of Sym 6 (π) cannot have a cuspidal representation of GL 5 (A F ). Therefore applying the strong multiplicity one theorem to (4), we find that the only option we have is that Sym 4 (π) is cuspidal, and π 2 = ω −1 Sym 4 (π), proving the proposition. Remark 4. The identity proved in Lemma 2: (1) π 2 ⊗ π 1 + ω 2 /ω 1 = Λ 2 (π 2 ) + Sym 2 (π 1 ), holds, as the proof shows, among any two representations (π 1 , V 1 ) and (π 2 , V 2 ) of an abstract group G when dim(π 2 ) − dim(π 1 ) = 2 and when for any g ∈ G, the set of eigenvalues of π 1 (g) acting on V 1 , counted with multiplicity, is contained in the set of eigenvalues of π 2 (g) acting on V 2 , counted with multiplicity. All the proofs in this section of Propositions 3, 4, 5 giving an affirmative answer to Question 1 for the pair (GL n , GL n+2 ) for n = 1, 2, 3 use this identity (1) crucially, answering Question 1 for any group G, and then we had to carefully transport that proof (for arbitrary group G) to the world of automorphic forms using the strong multiplicity one theorem about isobaric decomposition of automorphic forms, and instances of functoriality. However, when in section 6, we answer Question 1 in the affirmative for general groups which are not virtually abelian, we do not rely on the identity (1). Remark 5. (Local-global compatibility of automorphic representations) The identities among Hecke eigenvalues used in this paper, such as in Lemma 2: (1) π 2 ⊗ π 1 + ω 2 /ω 1 = Λ 2 (π 2 ) + Sym 2 (π 1 ), motivates us to ask a general question: if polynomial identities among Hecke eigenvalues of cuspforms on GL n i (A F ) at almost all unramified places of F remain valid for the associated representations of the Weil-Deligne (or, Weil) group at each finite and infinite place v of F . We do not know if principle of functoriality alone would suffice to prove this. We take a moment to make the question precise. For this, recall that the representation ring of polynomial representations of GL n i (C) is the polynomial ring Z[ω 1 n i , ω 2 n i , · · · , ω n i n i ] where ω a n i represents the irreducible representation of GL n i (C) on Λ a (C n i ). Given cuspidal automorphic representations Π 1 , Π 2 , · · · , Π d of GL n i (A F ), i = 1, 2, · · · , d, by a polynomial identity among Π 1 , Π 2 , · · · , Π d , we will mean an element P of the polynomial ring Z[ω m i n i ] where 1 ≤ i ≤ d, and 1 ≤ m i ≤ n i such that the evaluation of the polynomial P on the Hecke eigenvalues of Π i is identically zero at almost all the finite places v of F where each of the Π i is unramified. We write a polynomial identity among the automorphic representations Π i as P (Π 1 , Π 2 , · · · , Π d ) = 0 although we need to keep in mind that the "variables" in the polynomial P is Π i together with its exterior powers Λ m (Π i ), 1 ≤ m ≤ n i . The question is: if P (Π 1 , Π 2 , · · · , Π d ) = 0 in the sense we just defined, then P (Π 1v , Π 2v , · · · , Π dv ) = 0 as a representation W ′ Fv (the Weil-Deligne, or Weil, group) at each finite or infinite place v of F where, now, Π iv is, by abuse of notation, a representation of W ′ Fv associated by the local Langlands correspondence to the local component of Π i at the place v of F . ISOBARIC TYPES OF Sym 6 (π) For an automorphic representation π 1 of GL n (A F ) with isobaric decomposition π 1 = π 11 ⊞ π 12 ⊞ · · · ⊞ π 1ℓ , where π 1j are irreducible cuspidal automorphic representations of GL d j (A F ), we call the set of un-ordered integers {d j } which forms a partition of n to be the isobaric type of π 1 . Proposition 6. Let π be a cuspidal non-CM automorphic representation of GL 2 (A F ) with central character ω : A × F /F × → C × . Assume that Sym i (π) are automorphic for i ≤ 6, and that Sym 6 (π) is not cuspidal. Then the isobaric type of Sym 6 (π) is (3, 3, 1) if π is tetrahedral, (4, 2, 1) if π is octahedral, and of type (4, 3) otherwise. Proof. We will split the proof according to the different cases for π. Tetrahedral case: In this case, one knows that Sym 3 (π) is reducible, and by Theorem 2.2.2 of , (1) Sym 3 (π) = χ 1 π ⊞ χ 2 π, for certain Grössencharacters χ 1 , χ 2 of F . Since, (2) Sym 2 Sym 3 (π) = Sym 6 (π) ⊞ ω 2 Sym 2 (π), equation (1) gives: (3) Sym 6 (π) ⊞ ω 2 Sym 2 (π) = χ 2 1 Sym 2 (π) ⊞ χ 2 2 Sym 2 (π) ⊞ χ 1 χ 2 π ⊠ π. Since we are assuming that π is non-CM, Sym 2 (π) is a cuspidal automorphic representation of GL 3 (A F ), and thus the only option for the isobaric type of Sym 6 (π) is (3,3,1). Octahedral case: In this case, one knows that Sym 2 (π), Sym 3 (π) are irreducible, but Sym 4 (π) is reducible, and by Theorem 3.3.7(3) of Kim-Shahidi [KS3], (4) Sym 4 (π) = χ 1 π ⊞ χ 2 Sym 2 (π). Since, (5) Λ 2 Sym 4 (π) = ωSym 6 (π) ⊞ ω 2 Sym 2 (π), using (4) and (5) we have, (6) ωSym 6 (π) ⊞ ω 2 Sym 2 (π) = χ 2 1 Λ 2 (π) ⊞ χ 2 2 Λ 2 Sym 2 (π) ⊞ χ 1 χ 2 π ⊠ Sym 2 (π). On the other hand, π ⊠ Sym 2 (π) = Sym 3 (π) ⊞ ωπ. Using (6) and (7) we find that the isobaric type of Sym 6 (π) is (4,2,1). Rest of the cases when Sym 6 (π) is not cuspidal: (Although one expects this case to consist exactly of icosahedral representations, this seems not known. Such automorphic representations of GL 2 (A F ) are called quasi-icosahedral in [Ra]. For our proof below, this lack of knowledge does not matter.) By Theorem A ′ of Ramakrishnan [Ra], if we are not in the above two cases, and Sym 6 (π) is not cuspidal, then Sym i (π) are cuspidal for all i ≤ 5, and (8) Sym 5 (π) = χπ ⊠ Sym 2 (π ′ ) where χ is a Grössencharacter on F , and π ′ is a cuspidal automorphic representation on GL 2 (A F ) such that Sym 2 (π) and Sym 2 (π ′ ) are not twist equivalent. We use the identity: (9) π ⊠ Sym 5 (π) = Sym 6 (π) ⊞ ωSym 4 (π), therefore using (8) we have, (10) χπ ⊠ π ⊠ Sym 2 (π ′ ) = Sym 6 (π) ⊞ ωSym 4 (π), or, (11) χ[ω ⊞ Sym 2 (π)] ⊠ Sym 2 (π ′ ) = Sym 6 (π) ⊞ ωSym 4 (π), which under the assumption that Sym 6 (π) is automorphic, is an identity of isobaric automorphic representations. Since Sym 2 (π ′ ) is a cuspidal representation on GL 3 (A F ), there is a GL 3 -cuspform in the isobaric decomposition on the left hand side of the identity (11). Further, we are forced to have a GL 5 -cuspform on the left hand side of the identity (11) to account for Sym 4 (π) on the right (which is given to be cuspidal), so the possible isobaric types on the left hand side of the identity (11) are (3,5) + a partition of 4. Thus we deduce that the isobaric types for Sym 6 (π), an automorphic representation on GL 7 (A F ), is 3 + a partition of 4. However, Sym 6 (π) cannot have, in its isobaric decomposition, a representation of GL 1 or a representation of GL 2 as follows from the isobaric decomposition (9) above. For example, if Sym 6 (π) had the shape π 2 + π 4 with π 2 an automorphic representation of GL 2 and π 4 on GL 4 , then clearly the Rankin product of the right hand side of (9) with π ∨ 2 will have a pole (at s = 1) whereas the left hand side of the identity (11) which will be π ⊠ Sym 5 (π) ⊠ π ∨ 2 does not have a pole since in our case, Sym 5 (π) is a cuspidal representation. Thus the only option for the isobaric decomposition of Sym 6 (π) is (4, 3) GROUP THEORETIC ANALOGUE In this paper we have answered the Question 1 for (GL n , GL n+2 ) for n = 1, 2, 3 in the positive. Thus the first case one may say is left unsettled is (GL 4 , GL 6 ). In this section, in Example 1 we construct instances where our Question 1 has a positive answer (assuming strong Artin conjecture) for (GL 4 , GL 6 ), a case not treated by our work, and then in the final remark of the section, we construct instances where our Question 1 has a negative answer, using Calegari's work in [Ca], for (GL 4 , GL 6 ). We begin with some generalities on finite groups, focusing eventually on GL 2 (F q ). For representations V 1 and V 2 of a group G, define a relationship V 1 V 2 (to be read as V 1 immersed in V 2 ) if for each element g ∈ G, the set of eigenvalues of the action of g on V 1 (counted with multiplicities) is contained in the set of eigenvalues of g acting on V 2 (counted with multiplicities). Thus if V 1 ⊂ V 2 as representations of G, then V 1 V 2 . If V 1 V 2 and dim(V 1 ) = dim(V 2 ), then of course, V 1 ∼ = V 2 as G-modules, whereas just as in Proposition 1, if dim(V 1 ) + 1 = dim(V 2 ), then also V 1 V 2 implies V 1 ⊂ V 2 as G-modules. However, it is not true in general that if V 1 V 2 , then V 1 ⊂ V 2 as G-modules as we will now see. Proposition 7. Let G = GL 2 (F q ). Let C be an irreducible cuspidal representation of GL 2 (F q ) of dimension (q − 1), and P an irreducible principal series representation of GL 2 (F q ) of dimension (q + 1). Assume that the central character of C and P are the same, which is ω : Z = F × q → C × . Then, C P. Proof. One knows that: (1) The restriction of C to the diagonal torus T = F × q × F × q in GL 2 (F q ) is the set of all characters with multiplicity 1 of T whose restriction to the center Z is ω. These characters of T are also contained in the restriction of P to T (there are two characters {χ 1 × χ 2 , χ 2 × χ 1 } of T appearing in the restriction of P to T with multiplicity 2 which are the characters {χ 1 × χ 2 , χ 2 × χ 1 } of T used to define the principal series P ). (2) The restriction of C to the anisotropic torus S = F × q 2 in GL 2 (F q ) is the set of all characters with multiplicity 1 of S whose restriction to the center Z is ω except that the two characters {χ,χ} of S used to define the cuspidal representation C do not appear. On the other hand, the restriction of P to S is the set of all characters of S with multiplicity 1 whose restriction to the center Z is ω. (3) The restriction of C to the upper triangular unipotent group U = F q in GL 2 (F q ) is the regular representation of U except that the trivial representation of U does not appear in C. On the other hand, the restriction of P to U is the regular representation of U, except that the trivial representation appears twice. It follows that C P , with dim P − dim C = 2. Now we note the following proposition whose obvious proof will be omitted. Using this proposition, and the example of C P , we get counter examples to Question 1 at the beginning of the paper. (The group GL 2 (F q ), q = 2, 3, 4 has no two dimensional irreducible (projective) representation, thus we cannot realize C and P as symmetric powers of a two dimensional representation of a central cover of GL 2 (F q ).) Proposition 8. If G is a finite group realized as a Galois group of number fields G = Gal(E/F ), thus any two representations of G, V 1 of dimension n and V 2 of dimension m, gives rise to Artin L-functions, which assuming the strong Artin conjecture give rise to cuspidal automorphic representations π 1 of GL n (A F ) and π 2 of GL m (A F ). If V 1 V 2 , then the Hecke eigenvalues of the automorphic representation π 1 are contained in the Hecke eigenvalues of the automorphic representation π 2 . Example 1. Here is a nice example to illustrate the use of finite groups for Question 1. The details of the example are taken from Lemma 5.1 and 5.3 of Kim [Kim2] whose notation we will follow. The group SL 2 (F 5 ) has two 2-dimensional irreducible representations σ, σ τ (favorite of representation theorists, the odd Weil representation!). These have character values in Q( √ 5), and are Galois conjugate. We have Sym 3 (σ) ∼ = Sym 3 (σ τ ), an irreducible 4 dimensional representation of SL 2 (F 5 ) extending to a cuspidal representation C of GL 2 (F 5 ) of non-trivial central character. Further, we have, Sym 5 (σ) ∼ = Sym 5 (σ τ ) ∼ = Sym 2 (σ) ⊗ σ τ ∼ = Sym 2 (σ τ ) ⊗ σ, giving the unique irreducible representation of SL 2 (F 5 ) of dimension 6 extending to a principal series representation P of GL 2 (F 5 ) of non-trivial central character, same as that of C. In particular, for the automorphic representations π 1 of GL 4 (A F ) associated to Sym 3 (σ) and for the automorphic representations π 2 of GL 6 (A F ) associated to Sym 5 (σ), for which since Sym 3 (σ) Sym 5 (σ), Proposition 8 applies, constructing an instance where Question 1 has an affirmative answer (using the group G = SL 2 (F 5 ).) Remark 6. By the work of Calegari, cf. [Ca], irreducible representations of Gal(Q/Q) of dimension 4 and 6 factoring through a Galois extension E of Q with Gal(E/Q) = S 5 ∼ = PGL 2 (F 5 ) which arise by taking a cuspidal representation of PGL 2 (F 5 ) of dimensions 4 = q − 1, and a principal series of dimension 6 = q + 1, give rise to cuspidal automorphic representations Π and Π ′ of GL 4 (A Q ) and GL 6 (A Q ) respectively. Considerations of this section will then give a counter-example to the Question 1 for the pair of cuspidal automorphic representations π 1 = Π and π 2 = Π ′ of GL 4 (A Q ) and GL 6 (A Q ), since by Lemma 4 below, the automorphic representation Π of GL 4 (A Q ) does not arise as Sym 3 (π) of an automorphic representation π of GL 2 (A Q ). We leave checking that results of this section are in conformity with the earlier results in the paper for q = 2, 3, 4 to the reader as a curious exercise! The proof of the following lemma is due to F. Calegari. Lemma 4. Let Π be an automorphic representation on GL 4 (A F ) which is an Artin representation coming from the standard 4-dimensional irreducible representation of S 5 (realized as the Galois group of an extension E/F ). Then Π cannot be written as Sym 3 (π) of an automorphic representation π of GL 2 (A F ). Proof. Choose a place v of F such that the Frobenius conjugacy class for the extension E/F in S 5 is the conjugacy class of a transposition in S 5 , and therefore, the Hecke eigenvalues of Π at the place v is: (1, 1, 1, −1). Suppose Π v = Sym 3 (π v ), with Hecke eigenvalues of π v being (α v , β v ). Therefore we have the equality of un-ordered quadruples (1, 1, 1, −1) and (α 3 v , α 2 v β v , α v β 2 v , β 3 v ) . Assume without loss of generality that α 3 v = 1, and one of α 2 v β v , α v β 2 v is also 1. Thus either α v = β v or α v = −β v . Neither is an option if the un-ordered quadruples (1, 1, 1, −1) and (α 3 v , α 2 v β v , α v β 2 v , β 3 v ) are the same, and α 3 v = 1. The following question was posed by F. Calegari. Question 2. (F. Calegari) If the Hecke eigenvalues of an automorphic representation Π of GL n+1 (A F ) is of the form (α n v , α n−1 v β v , · · · , α v β n−1 v , β n v ) at almost all places v of F , then is Π = Sym n (π) for an automorphic form π of GL 2 (A F )? VIRTUALLY NON-ABELIAN GROUPS In this section we prove that the group theoretic analogue of the Question 1 has an affirmative answer as long as the group is not 'virtually abelian', i.e., does not contain a finite index subgroup which is abelian. In the following proposition, we call a connected reductive group Q of type A 1 if its derived subgroup is PGL 2 (C) or SL 2 (C). Proposition 9. Let G be a connected reductive algebraic group over C. Let π 1 and π 2 be two finite dimensional representations of G with π 1 π 2 such that dim(π 2 ) − dim(π 1 ) = 2. Then either π 2 = π 1 + λ + µ, where λ, µ are one dimensional representations of G, or G has a reductive quotient Q of type A 1 and π ′ 1 , π ′ 2 irreducible representations of Q of dimensions d and d + 2 respectively (d = 0 allowed) such that, π 1 = π + π ′ 1 , π 2 = π + π ′ 2 , for a finite dimensional representation π of G. Proof. Let T be a maximal torus in G, and W its Weyl group. Clearly, π 1 π 2 if and only if the weights of π 1 for the torus T are contained in the weights of π 2 (with multiplicity). Since weights are W -invariant, if π 1 π 2 with dim(π 2 ) − dim(π 1 ) = 2, we see that there is a set of two (not necessarily distinct) weights of T (that of π 2 − π 1 ) which is W -invariant. By Lemma 5 below, this means that either these are weights of T invariant under W , hence arise from characters λ, µ : G → C × , or the group G has a quotient Q (obtained by dividing G by all normal simple groups except one which is PGL 2 (C) or SL 2 (C)), with a quotient S of T as a maximal torus in Q. Further, by the same Lemma 5 below, these two characters of T are pulled back of characters of S via the map T → S. In the first case, i.e., when these are two characters λ, µ : G → C × , the two representations of G, π 2 and π 1 + λ + µ, have the same characters on T , therefore must be isomorphic. In the second case, we appeal to the elementary fact that for a reductive group Q of type A 1 , with maximal torus S, any set of two distinct characters of S of the form {χ, χ w } where w is the unique nontrivial element of the Weyl group of Q, χ + χ w is the difference of two irreducible representations π ′ 1 , π ′ 2 of Q of dimensions d and d + 2 respectively (d = 0 allowed), therefore, we have π 2 − π 1 = π ′ 2 − π ′ 1 , as T -modules, and therefore π 2 + π ′ 1 = π ′ 2 + π 1 , as G-modules, and the conclusion of the proposition follows. Lemma 5. Let G be a connected reductive group with T a maximal torus and W its Weyl group. Then if χ is a character of T whose W -orbit has ≤ 2 elements, then either χ is the restriction of a one dimensional representation of G to T , or G has a quotient Q of type A 1 with S a maximal torus of Q which is a quotient of T , and χ is a character of T factoring through S. Proof. It suffices to prove the lemma for semisimple groups where it easily reduces to a simple group. The lemma for a simple group reduces to the assertion that if G is a simple group which is not PGL 2 (C) or SL 2 (C), and if χ is a character of T whose W -orbit has ≤ 2 elements, then character must be trivial. For this, observe the well-known fact, cf. [Hum], Lemma B of section 10.3, that the stabilizer of χ (in W ), an element in the character group X ⋆ (T ), which we assume without loss of generality to belong to fundamental Weyl chamber, is generated by the simple reflections fixing χ, hence in particular, it is the Weyl group W X of the associated Levi subgroup. Now |W/W X | ≥ 3 can be easily proved for groups G of rank ≥ 2, by an easy reduction to rank 2 where it is clear. The following corollary follows by an application of Clifford theory (applied to the normal subgroup G 0 of G) combined with Proposition 9 applied to G 0 , we omit its proof. Corollary 2. Let G be an algebraic group over C, with G 0 , the connected component of identity, a non-abelian reductive group. Assume G has irreducible finite dimensional representations π 1 and π 2 , with π 1 π 2 (when restricted to G 0 ) such that dim(π 2 ) − dim(π 1 ) = 2, and the action of G on π 1 + π 2 is faithful. Then, both π 1 , π 2 remain irreducible when restricted to G 0 , and their restriction to G 0 arises from a quotient Q (not necessarily semi-simple) of G of type A 1 . Remark 7. By proposition 9, there are no relations π 1 π 2 among irreducible representations of a connected simple algebraic group with dim(π 2 )−dim(π 1 ) = 2, other than the obvious ones for G = SL 2 (C), and G = PGL 2 (C). Without the constraint on dim(π 2 ) − dim(π 1 ) = 2, there are naturally many more representations, such as the pair of representations Λ k (C n ), Sym k (C n ) of GL n (C). It seems interesting to classify all possible pairs of irreducible representations (π 1 , π 2 ) with π 1 π 2 for connected simple algebraic groups with dim π 2 − dim π 1 , a fixed integer. 2020 Mathematics Subject Classification. Primary 11F70; Secondary 22E55. DP thanks Science and Engineering research board of the Department of Science and Technology, India for its support through the JC Bose National Fellowship of the Govt. of India, project number JBR/2020/000006. His work was also supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the agreement 14.W03.31.0030 dated 15.02.2018. − V + Λ 2 (V ) − Λ 3 (V ) + Λ 4 (V ) = 0. The Artin conjecture for some S 5 -extensions. F Calegari, Math. Ann. 3561F. Calegari, The Artin conjecture for some S 5 -extensions. Math. Ann. 356 (2013), no. 1, 191-207. Branching laws for classical groups: The non-tempered case. W T Gan, B Gross, D Prasad, Compositio Math. 15611W.T. Gan, B. Gross and D.Prasad Branching laws for classical groups: The non-tempered case. Compo- sitio Math, vol 156, No. 11 (2020) 2298-2367. A relation between automorphic representations of GL(2) and GL(3). S Gelbart, H Jacquet, Ann. Sci.École Norm. Sup. 4S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3). Ann. Sci.École Norm. Sup. (4) 11 (1978), no. 4, 471-542. Introduction to Lie algebras and representation theory. J E Humphreys, Graduate Texts in Mathematics. 9Springer-VerlagJ. E. Humphreys, Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972. Exterior square L-functions. H Jacquet, J Shalika, Automorphic forms, Shimura varieties, and Lfunctions. Ann Arbor, MI; Boston, MAAcademic PressIIH. Jacquet and J. Shalika, Exterior square L-functions. Automorphic forms, Shimura varieties, and L- functions, Vol. II (Ann Arbor, MI, 1988), 143-226, Perspect. Math., 11, Academic Press, Boston, MA, 1990. On Euler products and the classification of automorphic representations. I. Amer. H Jacquet, J Shalika, J. Math. 1033H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations. I. Amer. J. Math. 103 (1981), no. 3, 499-558. On Euler products and the classification of automorphic forms II. H Jacquet, J Shalika, Amer. J. Math. 103H. Jacquet and J. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math. 103 (1981), 777-815. Abelian varieties with isogenous reductions. C Khare, M Larsen, C. R. Math. Acad. Sci. Paris. 3589C. Khare and M. Larsen Abelian varieties with isogenous reductions, C. R. Math. Acad. Sci. Paris 358 (2020), no. 9-10, 1085-1089. Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2 . With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. H Kim, J. Amer. Math. Soc. 161H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2 . With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type. H Kim, Invent. Math. 1563H. Kim, An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type. Invent. Math. 156 (2004), no. 3, 495-502. Symmetric cube L-functions for GL 2 are entire. H Kim, F Shahidi, Ann. of Math. 150H. Kim and F. Shahidi, Symmetric cube L-functions for GL 2 are entire, Ann. of Math. 150 (1999), 645- 662. Functorial products for GL 2 × GL 3 and the symmetric cube for GL 2 . With an appendix by. H Kim, F Shahidi, Ann. of Math. Colin J. Bushnell and Guy Henniart2H. Kim and F. Shahidi, Functorial products for GL 2 × GL 3 and the symmetric cube for GL 2 . With an appendix by Colin J. Bushnell and Guy Henniart. Ann. of Math. (2) 155 (2002), no. 3, 837-893. Cuspidality of symmetric powers with applications. H Kim, F Shahidi, Duke Math. J. 1121H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications. Duke Math. J. 112 (2002), no. 1, 177¡V197. Remarks on the symmetric powers of cusp forms on GL(2). D Ramakrishnan, Contemp. Math. 488Amer. Math. SocAutomorphic forms and L-functions I. Global aspectsD. Ramakrishnan, Remarks on the symmetric powers of cusp forms on GL(2). Automorphic forms and L-functions I. Global aspects, 237-256, Contemp. Math., 488, Amer. Math. Soc., Providence, RI, 2009. An exercise concerning the selfdual cusp forms on GL(3). D Ramakrishnan, Indian J. Pure Appl. Math. 455D. Ramakrishnan, An exercise concerning the selfdual cusp forms on GL(3). Indian J. Pure Appl. Math. 45 (2014), no. 5, 777-785. On certain L-functions. F Shahidi, Amer. J. Math. 1032F. Shahidi, On certain L-functions. Amer. J. Math. 103 (1981), no. 2, 297-355. On poles of the exterior cube L-functions for GL 6. S Yamana, INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. 279400076POWAIMath. Zeit.S. Yamana, On poles of the exterior cube L-functions for GL 6 . Math. Zeit. (2015) 279, 267-270. INDIAN INSTITUTE OF TECHNOLOGY BOMBAY, POWAI, MUMBAI-400076 RUSSIA Email address: prasad.dipendra@gmail.com Email address: ravir@math.iitb.ac. Dp: St Petersburg State, University, St Petersburg, DP: ST PETERSBURG STATE UNIVERSITY, ST PETERSBURG, RUSSIA Email address: prasad.dipendra@gmail.com Email address: ravir@math.iitb.ac.in

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{'abstract': 'In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on GL n (A F ) is contained in the set of Hecke eigenvalues of a cusp form on GL m (A F ) for n ≤ m. This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.', 'arxivid': '2007.14639', 'author': ['Dipendra Prasad ', 'Ravi Raghunathan '], 'authoraffiliation': [], 'corpusid': 220845829, 'doi': '10.1090/ert/626', 'github_urls': [], 'n_tokens_mistral': 17686, 'n_tokens_neox': 14794, 'n_words': 9925, 'pdfsha': '5db73a50085353cdf2a90a54626dc8e7c228f96c', 'pdfurls': ['https://export.arxiv.org/pdf/2007.14639v2.pdf'], 'title': ['RELATIONS BETWEEN CUSP FORMS SHARING HECKE EIGENVALUES', 'RELATIONS BETWEEN CUSP FORMS SHARING HECKE EIGENVALUES'], 'venue': []}

arxiv

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO POROUS MEDIUM EQUATION AND PSEUDO-PARABOLIC EQUATION, I. STRATIFIED GROUPS 2 Jun 2021 Michael Ruzhansky Bolys Sabitbek Berikbol Torebek GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO POROUS MEDIUM EQUATION AND PSEUDO-PARABOLIC EQUATION, I. STRATIFIED GROUPS 2 Jun 2021 In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in [38] for stratified groups. Introduction The main purpose of this paper is to study the global existence and blow-up of the positive solutions to the initial-boundary problem of the nonlinear porous medium equation      u t − L p (u m ) = f (u), x ∈ D, t > 0, u(x, t) = 0, x ∈ ∂D, t > 0, u(x, 0) = u 0 (x) ≥ 0, x ∈ D, (1.1) and the nonlinear pseudo-parabolic equation      u t − ∇ H · (|∇ H u| p−2 ∇ H u t ) − L p u = f (u), x ∈ D, t > 0, u(x, t) = 0, x ∈ ∂D, t > 0, u(x, 0) = u 0 (x) ≥ 0, x ∈ D, (1.2) where m ≥ 1 and p ≥ 2, f is locally Lipschitz continuous on R, f (0) = 0, and such that f (u) > 0 for u > 0. Furthermore, we suppose that u 0 is a non-negative and non-trivial function in C 1 (D) with u 0 (x) = 0 on the boundary ∂D for p = 2 and in L ∞ (D) ∩S 1,p (D) for p > 2, respectively. Thus, the functional classS 1,p (D) can be defined as the completion of C 1 0 (D) in the norm generated by J p , see e.g. [7]. A Lie group G = (R n , •) is called a stratified (Lie) group if it satisfies the following conditions: (a) For some integer numbers N 1 + N 2 + ... + N r = n, the decomposition R n = R N 1 × . . . × R Nr is valid, and for any λ > 0 the dilation δ λ (x) := (λx ′ , λ 2 x (2) , . . . , λ r x (r) ) is an automorphism of G. Here x ′ ≡ x (1) ∈ R N 1 and x (k) ∈ R N k for k = 2, . . . , r. (b) Let N 1 be as in (a) and let X 1 , . . . , X N 1 be the left-invariant vector fields on G such that X k (0) = ∂ ∂x k | 0 for k = 1, . . . , N 1 . Then the Hörmander rank condition must be satisfied, that is, rank(Lie{X 1 , . . . , X N 1 }) = n, for every x ∈ R n . Then, we say that the triple G = (R n , •, δ λ ) is a stratified (Lie) group. Recall that the standard Lebesgue measure dx on R n is the Haar measure for G (see e.g. [14], [39]). The left-invariant vector field X j has an explicit form: X k = ∂ ∂x ′ k + r l=2 N l m=1 a (l) k,m (x ′ , ..., x (l−1) ) ∂ ∂x (l) m ,(1.4) see e.g. [39]. The following notations are used throughout this paper: ∇ H := (X 1 , . . . , X N 1 ) for the horizontal gradient, and L p f := ∇ H · (|∇ H f | p−2 ∇ H f ), 1 < p < ∞, (1.5) for the p-sub-Laplacian. When p = 2, that is, the second order differential operator L = N 1 k=1 X 2 k , (1.6) is called the sub-Laplacian on G. The sub-Laplacian L is a left-invariant homogeneous hypoelliptic differential operator and it is known that L is elliptic if and only if the step of G is equal to 1. One of the important examples of the nonlinear parabolic equations is the porous medium equation, which describes widely processes involving fluid flow, heat transfer or diffusion, and its other applications in different fields such as mathematical biology, lubrication, boundary layer theory, and etc. Existence and nonexistence of solutions to problem (1.1) for the reaction term u m in the case m = 1 and m > 1 have been actively investigated by many authors, for example, [3,4,9,11,12,15,16,20,21,22,28,30,41,42,43], Grillo, Muratori and Punzo considered fractional porous medium equation [17,18], and it was also considered in the setting of Cartan-Hadamard manifolds [19]. By using the concavity method, Schaefer [44] established a condition on the initial data of a Dirichlet type initial-boundary value problem for the porous medium equation with a power function reaction term when blow-up of the solution in finite time occurs and a global existence of the solution holds. We refer for more details to Vazquez's book [45] which provides a systematic presentation of the mathematical theory of the porous medium equation. The energy for the isotropic material can be modeled by a pseudo-parabolic equation [10]. Some wave processes [6], filtration of the two-phase flow in porous media with the dynamic capillary pressure [5] are also modeled by pseudo-parabolic equations. The global existence and finite-time blow-up for the solutions to pseudoparabolic equations in bounded and unbounded domains have been studied by many researchers, for example, see [26,27,33,34,37,47,48,49] and the references therein. In [46], Veron and Pohozaev have obtained blow-up results for the following semilinear diffusion equation on the Heisenberg groups ∂u(x, t) ∂t − Lu(x, t) = |u(x, t)| p , (x, t) ∈ H × (0, +∞). Also, blow-up of the solutions to the semi-linear diffusion and pseudo-parabolic equations on the Heisenberg groups was derived in [1,2,13,24,25]. In addition, in [40] the authors found the Fujita exponent on general unimodular Lie groups. In some of our considerations a crucial role is played by • The condition αF (u) ≤ u m f (u) + βu pm + αγ, u > 0, (1.7) where F (u) = pm m + 1 u 0 s m−1 f (s)ds, m ≥ 1, introduced by Chung-Choi [8] for a parabolic equation. We will deal with several variants of such condition. • The Poincaré inequality established by the first author and Suragan in [38] for stratified groups: Lemma 1.2. Let D ⊂ G be an admissible domain with N 1 being the dimension of the first stratum. Let 1 < p < ∞ with p = N 1 . For every function u ∈ C ∞ 0 (D\{x ′ = 0}) we have D |∇ H u| p dx ≥ |N 1 − p| p (pR) p D |u| p dx, (1.8) where R = sup x∈D |x ′ |. Note that it is possible to interpret the constant |N 1 −p| p (pR) p as a measure of the size of the domain D. Then β in (1.7) is dependent on the size of the domain D. Our paper is organised so that we discuss the existence and nonexistence of positive solutions to the nonlinear porous medium equation in Section 2 and the nonlinear pseudo-parabolic equation in Section 3. Nonlinear porous medium equation In this section, we prove the global solutions and blow-up phenomena of the initialboundary value problem (1.1). Blow-up solutions of the nonlinear porous medium equation. We start with the blow-up properly. Theorem 2.1. Let G be a stratified group with N 1 being the dimension of the first stratum. Let D ⊂ G be an admissible domain. Let 2 ≤ p < ∞ with p = N 1 . Assume that function f satisfies αF (u) ≤ u m f (u) + βu pm + αγ, u > 0, (2.1) where F (u) = pm m + 1 u 0 s m−1 f (s)ds, m ≥ 1, for some γ > 0, 0 < β ≤ |N 1 − p| p (pR) p (α − m − 1) m + 1 and α > m + 1, where R = sup x∈D |x ′ | and x = (x ′ , x ′′ ) with x ′ being in the first stratum. Let u 0 ∈ L ∞ (D) ∩S 1,p (D) satisfy the inequality J 0 := − 1 m + 1 D |∇ H u m 0 (x)| p dx + D (F (u 0 (x)) − γ)dx > 0. (2.2) Then any positive solution u of (1.1) blows up in finite time T * , i.e., there exists 0 < T * ≤ M σ D u m+1 0 (x)dx , (2.3) such that lim t→T * t 0 D u m+1 (x, τ )dxdτ = +∞, (2.4) where M > 0 and σ = √ pmα m+1 − 1 > 0. In fact, in (2.3), we can take M = (1 + σ)(1 + 1/σ)( D u m+1 0 (x)dx) 2 α(m + 1)J 0 . Remark 2.2. Note that condition on nonlinearity (2.1) includes the following cases: 1. Philippin and Proytcheva [35] used the condition (2 + ǫ)F (u) ≤ uf (u), u > 0, (2.5) where ǫ > 0. It is a special case of an abstract condition by Levine and Payne [31]. 2. Bandle and Brunner [4] relaxed this condition as follows (2 + ǫ)F (u) ≤ uf (u) + γ, u > 0, (2.6) where ǫ > 0 and γ > 0. These cases were established on the bounded domains of the Euclidean space, and it is a new result on the stratified groups. Proof of Theorem 2.1. Assume that u(x, t) is a positive solution of (1.1). We use the concavity method for showing the blow-up phenomena. We introduce the functional J(t) := − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx,(2.7) and by (2.2) we have J(0) = − 1 m + 1 D |∇ H u m 0 (x)| p dx + D (F (u 0 (x)) − γ)dx > 0. (2.8) Moreover, J(t) can be written in the following form J(t) = J(0) + t 0 dJ(τ ) dτ dτ, (2.9) where t 0 dJ(τ ) dτ dτ = − 1 m + 1 t 0 D d dτ |∇ H u m (x, τ )| p dxdτ + t 0 D d dτ (F (u(x, τ )) − γ)dxdτ = − p m + 1 t 0 D |∇ H u m (x, τ )| p−2 ∇ H u m · ∇ H (u m (x, τ )) τ dxdτ + t 0 D F u (u(x, τ ))u τ (x, τ )dxdτ = p m + 1 t 0 D [L p (u m ) + f (u)](u m (x, τ )) τ dxdτ = pm m + 1 t 0 D u m−1 (x, τ )u 2 τ (x, τ )dxdτ. Define E(t) = t 0 D u m+1 (x, τ )dxdτ + M, t ≥ 0, with M > 0 to be chosen later. Then the first derivative with respect t of E(t) gives E ′ (t) = D u m+1 (x, t)dx = (m + 1) D t 0 u m (x, τ )u τ (x, τ )dτ dx + D u m+1 0 (x)dx. By applying (2.1), Lemma 1. 2 and 0 < β ≤ |N 1 −p| p (pR) p (α−m−1) m+1 , we estimate the second derivative of E(t) as follows E ′′ (t) = (m + 1) D u m (x, t)u t (x, t)dx = −(m + 1) D |∇ H u m (x, t)| p dx + (m + 1) D u m (x, t)f (u(x, t))dx ≥ −(m + 1) D |∇ H u m (x, t)| p dx + (m + 1) D [αF (u(x, t)) − βu pm (x, t) − αγ] dx = α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx + (α − m − 1) D |∇ H u m (x, t)| p dx − β(m + 1) D u pm (x, t)dx ≥ α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx + |N 1 − p| p (pR) p (α − m − 1) − β(m + 1) D u pm (x, t)dx ≥ α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx = α(m + 1)J(t) = α(m + 1)J(0) + pαm t 0 D u m−1 (x, τ )u 2 τ (x, τ )dxdτ. By employing the Hölder and Cauchy-Schwarz inequalities, we obtain the estimate for [E ′ (t)] 2 as follows [E ′ (t)] 2 ≤ (1 + δ) D t 0 (u m+1 (x, τ )) τ dτ dx 2 + 1 + 1 δ D u m+1 0 (x)dx 2 = (m + 1) 2 (1 + δ) D t 0 u m (x, τ )u τ (x, τ )dxdτ 2 + 1 + 1 δ D u m+1 0 (x)dx 2 = (m + 1) 2 (1 + δ) D t 0 u (m+1)/2+(m−1)/2 (x, τ )u τ (x, τ )dxdτ 2 + 1 + 1 δ D u m+1 0 (x)dx 2 ≤ (m + 1) 2 (1 + δ) D t 0 u m+1 dτ 1/2 t 0 u m−1 u 2 τ (x, τ )dτ 1/2 dx 2 + 1 + 1 δ D u m+1 0 (x)dx 2 ≤ (m + 1) 2 (1 + δ) t 0 D u m+1 dxdτ t 0 D u m−1 u 2 τ (x, τ )dxdτ + 1 + 1 δ D u m+1 0 (x)dx 2 , for arbitrary δ > 0. So we have [E ′ (t)] 2 ≤ (m+1) 2 (1+δ) t 0 D u m+1 dxdτ t 0 D u m−1 u 2 τ dxdτ + 1 + 1 δ D u m+1 0 dx 2 . (2.10) The previous estimates together with σ = δ = √ pmα m+1 − 1 > 0 where positivity comes from α > m + 1, imply E ′′ (t)E(t) − (1 + σ)[E ′ (t)] 2 ≥ αM(m + 1) − 1 m + 1 D |∇ H u m 0 | p dx + D (F (u 0 ) − γ)dx + pmα t 0 D u m+1 (x, τ )dxdτ t 0 D u 2 τ (x, τ )u m−1 (x, τ )dxdτ − (m + 1) 2 (1 + σ)(1 + δ) t 0 D u m+1 dxdτ t 0 D u m−1 u 2 τ (x, τ )dxdτ − (1 + σ) 1 + 1 δ D u m+1 0 (x)dx 2 ≥ αM(m + 1)J(0) − (1 + σ) 1 + 1 δ D u m+1 0 (x)dx 2 . By assumption J(0) > 0, thus if we select M = (1 + σ) 1 + 1 δ D u m+1 0 (x)dx 2 α(m + 1)J(0) , that gives E ′′ (t)E(t) − (1 + σ)(E ′ (t)) 2 ≥ 0. (2.11) We can see that the above expression for t ≥ 0 implies d dt E ′ (t) E σ+1 (t) ≥ 0 ⇒ E ′ (t) ≥ E ′ (0) E σ+1 (0) E 1+σ (t), E(0) = M. Then for σ = √ pmα m+1 − 1 > 0, we arrive at − 1 σ E −σ (t) − E −σ (0) ≥ E ′ (0) E σ+1 (0) t, and some rearrangements with E(0) = M give E(t) ≥ 1 M σ − σ D u m+1 0 (x)dx M σ+1 t − 1 σ . Then the blow-up time T * satisfies 0 < T * ≤ M σ D u m+1 0 dx . That completes the proof. 2 ≤ p < ∞ with p = N 1 . Assume that αF (u) ≥ u m f (u) + βu pm + αγ, u > 0, (2.12) where F (u) = pm m + 1 u 0 s m−1 f (s)ds, m ≥ 1, for some γ ≥ 0, α ≤ 0 and β ≥ |N 1 − p| p (pR) p (α − m − 1) m + 1 , where R = sup x∈D |x ′ | and x = (x ′ , x ′′ ) with x ′ being in the first stratum. Assume also that u 0 ∈ L ∞ (D) ∩S 1,p (D) satisfies inequality J 0 := D (F (u 0 (x)) − γ)dx − 1 m + 1 D |∇ H u m 0 (x)| p dx > 0. (2.13) If u is a positive local solution of problem (1.1), then it is global and satisfies the following estimate: D u m+1 (x, t)dx ≤ D u m+1 0 (x)dx. Proof of Theorem 2.3. Recall from the proof of Theorem 2.1, the functional J(t) := − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx = J 0 + pm m + 1 t 0 D u m−1 (x, τ )u 2 τ (x, τ )dxdτ. Let us define E(t) = D u m+1 (x, t)dx.E ′ (t) = (m + 1) D u m (x, t)u t (x, t)dx = (m + 1) D u m (x, t)∇ H · (|∇ H u m (x, t)| p−2 ∇ H u m (x, t)) + D u m (x, t)f (u(x, t))dx = (m + 1) − D |∇ H u m (x, t)| p dx + D u m (x, t)f (u(x, t))dx ≤ (m + 1) − D |∇ H u m (x, t)| p dx + D [αF (u(x, t)) − βu pm (x, t) − αγ] dx = α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx − (m + 1 − α) D |∇ H u m (x, t)| p dx − β(m + 1) D u pm (x, t)dx ≤ α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| p dx + D (F (u(x, t)) − γ)dx − |N 1 − p| p (pR) p (m + 1 − α) + β(m + 1) D u pm (x, t)dx ≤ α(m + 1) − 1 m + 1 D |∇ H u m (x, t)| 2 dx + D (F (u(x, t)) − γ)dx = α(m + 1)J(t). We can rewrite E ′ (t) by using (2.9) and α ≤ 0 as follows E ′ (t) ≤ α(m + 1)J(0) + pαm t 0 D u m−1 (x, τ )u 2 τ (x, τ )dxdτ ≤ 0. (2.14) That gives E(t) ≤ E(0). This completes the proof of Theorem 2.3. Nonlinear pseudo-parabolic equation In this section, we prove the global solutions and blow-up phenomena of the initialboundary value problem (1.2). Assume that αF (u) ≤ uf (u) + βu p + αγ, u > 0, (3.1) where F (u) = u 0 f (s)ds, for some α > p and 0 < β ≤ |N 1 − p| p (pR) p (α − p) p , (3.2) γ > 0 and R = sup x∈D |x ′ |. Assume also that u 0 ∈ L ∞ (D) ∩S 1,p (D) satisfies F 0 := − 1 p D |∇ H u 0 (x)| p dx + D (F (u 0 (x)) − γ)dx > 0. (3.3) Then any positive solution u of (1.2) blows up in finite time T * , i.e., there exists 0 < T * ≤ M σ D u 2 0 + 2 p |∇ H u 0 | p dx , (3.4) such that lim t→T * t 0 D [u 2 + 2 p |∇ H u| p ]dxdτ = +∞, (3.5) where σ = α 2 − 1 > 0 and M = (1 + σ) 1 + 1 σ D u 2 0 + 2 p |∇ H u 0 | p dx 2 2αF 0 . Proof of Theorem 3.1. The proof is based on a concavity method. The main idea is to show that [E −σ p (t)] ′′ ≤ 0 which means that E −σ p (t) is a concave function, for E p (t) defined below. Let us introduce some notations: F (t) := − 1 p D |∇ H u(x, t)| p dx + D (F (u(x, t)) − γ)dx, and F (0) := − 1 p D |∇ H u 0 (x)| p dx + D (F (u 0 (x)) − γ)dx, with F (u) = u 0 f (s)ds. We know that F (t) = F (0) + t 0 dF (τ ) dτ dτ, (3.6) where t 0 dF (τ ) dτ dτ = − 1 p t 0 D d dτ |∇ H u| p dxdτ + t 0 D d dτ (F (u) − γ)dxdτ = − t 0 D |∇ H u| p−2 ∇u · ∇ H u τ dxdτ + t 0 D F u (u)u τ dxdτ = t 0 D [L p u + f (u)]u τ dxdτ = t 0 D u 2 τ − u τ ∇ H · (|∇ H u| p−2 ∇ H u τ )dxdτ = t 0 D u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 dxdτ. Let us define E p (t) := t 0 D [u 2 + 2 p |∇ H u| p ]dxdτ + M, t ≥ 0, with a positive constant M > 0 to be chosen later. Then E ′ p (t) = D [u 2 + 2 p |∇ H u| p ]dx = t 0 d dτ D [u 2 + 2 p |∇ H u| p ]dxdτ + D u 2 0 + 2 p |∇ H u 0 | p dx. (3.7) Now we estimate E ′′ p (t) by using assumption (3.1) and integration by parts, that gives E ′′ p (t) = 2 D uu t dx + 2 p D (|∇ H u| p ) t dx = 2 D [uL p u + u∇ H · (|∇ H u| p−2 ∇ H u t ) + uf (u)]dx + 2 p D (|∇ H u| p ) t dx = −2 D [|∇ H u| p + |∇ H u| p−2 ∇ H u · ∇ H u t ]dx + 2 D uf (u)dx + 2 p D (|∇ H u| p ) t dx ≥ −2 D |∇ H u| p dx + 2 D [αF (u) − βu p − αγ] dx = 2α − 1 p D |∇ H u| p dx + D (F (u) − γ)dx + 2(α − p) p D |∇ H u| p dx − 2β D u p dx. Next we apply Lemma 1.2, which gives ≥ 2α − 1 p D |∇ H u| p dx + D (F (u) − γ)dx + 2 |N 1 − p| p (pR) p (α − p) p − β D u p dx ≥ 2α − 1 p D |∇ H u| p dx + D (F (u) − γ)dx = 2αF (t), with F (t) as in (3.6), then E ′′ p (t) can be rewritten in the following form E ′′ p (t) ≥ 2αF (0) + 2α t 0 D [u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 ]dxdτ. (3.8) Also we have for arbitrary δ > 0, in view of (3.7), [E ′ p (t)] 2 ≤ (1 + δ) t 0 d dτ D [u 2 + 2 p |∇ H u| p ]dxdτ 2 + 1 + 1 δ D [u 2 0 + 2 p |∇ H u 0 | p ]dx 2 . Then by taking σ = δ = α 2 − 1 > 0, we arrive at E ′′ p (t)E p (t) − (1 + σ)[E ′ p (t)] 2 ≥ 2αMF (0) + 2α t 0 D [u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 ]dxdτ t 0 D [u 2 + 2 p |∇ H u| p dx]dτ − (1 + σ)(1 + δ) t 0 d dτ D [u 2 + 2 p |∇ H u| p ]dxdτ 2 − (1 + σ) 1 + 1 δ D [u 2 0 + 2 p |∇ H u 0 | p ]dx 2 = 2αMF (0) − (1 + σ) 1 + 1 δ D [u 2 0 + 2 p |∇ H u 0 | p ]dx 2 + 2α t 0 D [u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 ]dxdτ t 0 D [u 2 + 2 p |∇ H u| p dx]dτ − t 0 D [uu τ + |∇ H u| p−2 ∇ H u · ∇ H u τ ]dxdτ 2 ≥ 2αMF (0) − (1 + σ) 1 + 1 δ D u 2 0 + 2 p |∇ H u 0 | p dx 2 . Note that in the last line we have used the following inequality t 0 D [u 2 + |∇ H u| p ]dxdτ t 0 D [u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 ]dxdτ − t 0 D [uu τ + |∇ H u| p−2 ∇ H u · ∇ H u τ ]dxdτ 2 ≥ D t 0 u 2 dτ dx 1 2 D t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ dx 1 2 − D t 0 |∇ H u| p dτ dx 1 2 D t 0 u 2 τ dτ dx 1 2 2 ≥ 0, where making use of the Hölder inequality and Cauchy-Schawrz inequality we have t 0 D [uu τ + |∇ H u| p−2 ∇ H u · ∇ H u τ ]dxdτ 2 ≤ D t 0 u 2 dτ 1 2 t 0 u 2 τ dτ 1 2 dx + D t 0 |∇ H u| p dτ 1 2 t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ 1 2 dx 2 = D t 0 u 2 dτ 1 2 t 0 u 2 τ dτ 1 2 dx 2 + D t 0 |∇ H u| p dτ 1 2 t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ 1 2 dx 2 +2 D t 0 u 2 dτ 1 2 t 0 u 2 τ dτ 1 2 dx D t 0 |∇ H u| p dτ 1 2 t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ 1 2 dx ≤ D t 0 u 2 dτ dx D t 0 u 2 τ dτ dx + D t 0 |∇ H u| p dτ dx D t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ dx +2 D t 0 u 2 dτ dx D t 0 u 2 τ dτ dx D t 0 |∇ H u| p dτ dx D t 0 |∇ H u| p−2 |∇ H u τ | 2 dτ dx 1 2 . By assumption F (0) > 0, thus we can select M = (1 + σ) 1 + 1 δ D u 2 0 + 2 p |∇ H u 0 | p dx 2 2αF (0) , that gives E ′′ p (t)E p (t) − (1 + σ)[E ′ p (t)] 2 ≥ 0. (3.9) We can see that the above expression for t ≥ 0 implies d dt E ′ p (t) E σ+1 p (t) ≥ 0 ⇒ E ′ p (t) ≥ E ′ p (0) E σ+1 p (0) E 1+σ p (t), E p (0) = M. Then for σ = α 2 − 1 > 0, we arrive at E p (t) ≥ 1 M σ − σ D [u 2 0 + 2 p |∇ H u 0 | p ]dx M σ+1 t − 1 σ . Then the blow-up time T * satisfies 0 < T * ≤ M σ D [u 2 0 + 2 p |∇ H u 0 | p ]dx . This completes the proof. 3.2. Global solution for the pseudo-parabolic equation. We now show that positive solutions, when they exist for some nonlinearities, can be controlled. Now we estimate E ′ (t) by using assumption (3.10), that gives E ′ (t) = 2 D uu t dx + 2 p D (|∇ H u| p ) t dx = 2 D [uL p u + u∇ H · (|∇ H u| p−2 ∇ H u t ) + uf (u)]dx + 2 p D (|∇ H u| p ) t dx = −2 D [|∇ H u| p + |∇ H u| p−2 ∇ H u · ∇ H u t ]dx + 2 D uf (u)dx + 2 p D (|∇ H u| p ) t dx ≤ 2α − 1 p D |∇ H u| p dx + D (F (u) − γ)dx − 2(p − α) p D |∇ H u| p dx − 2β D u p dx ≤ 2α − 1 p D |∇ H u| p dx + D (F (u) − γ)dx − (p − α)[E p (t) − D u 2 dx]dx − 2β D u 2 dx, = 2αF (t) − (p − α)E(t) + [p − α − 2β] D u 2 dx, with F (t) := − 1 p D |∇ H u(x, t)| p dx + D (F (u(x, t)) − γ)dx = F 0 + t 0 D u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 dxdτ. Since β ≥ p−α 2 we arrive at E ′ (t) + (p − α)E(t) ≤ 2α F 0 + t 0 D u 2 τ + |∇ H u| p−2 |∇ H u τ | 2 dxdτ ≤ 0. This implies, E(t) ≤ exp(−(p − α)t)E(0), finishing the proof. Definition 1. 1 . 1Let G be a stratified group. 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P.: Blow-Up in Quasilinear Parabolic Equations. in: De Gruyter Expositions in Mathematics, vol. 19, Walter de Gruyter Co., Berlin, 1995. Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in R n. P Souplet, J. Funct. Anal. 272Souplet P.: Morrey spaces and classification of global solutions for a supercritical semi- linear heat equation in R n . J. Funct. Anal., 272, 2005-2037 (2017) Blow-up phenomena in some porous medium problems. P W Schaefer, Dyn. Sys. and Appl. 18Schaefer P. W.: Blow-up phenomena in some porous medium problems. Dyn. Sys. and Appl., 18, 103-110 (2009) J L Vazquez, The Porous Medium Equation: Mathematical Theory. Oxford University PressVazquez J. L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, 2006. Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group. L Véron, S I Pohozaev, Manuscripta Math. 102Véron L., Pohozaev S. I.: Nonexistence results of solutions of semilinear differential in- equalities on the Heisenberg group. Manuscripta Math., 102, 85-99 (2000) Global existence and finite time blow-up for a class of semilinear pseudoparabolic equations. R Z Xu, J Su, J. Funct. Anal. 264Xu, R.Z., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo- parabolic equations. J. Funct. Anal., 264:12, 2732-2763 (2013) Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. R Z Xu, X C Wang, Y B Yang, Appl. Math. Lett. 83Xu, R. Z., Wang, X. C., Yang, Y. B.: Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy. Appl. Math. Lett., 83, 176-181 (2018) Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. X C Wang, R Z Xu, Adv. Nonlinear Anal. 101Wang, X. C., Xu, R. Z.: Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal., 10:1, 261-288 (2021) Berikbol Torebek, Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University Belgium and Institute of Mathematics and Mathematical Modeling Almaty. Kazakhstan E-mail address berikbol.torebek@ugent.beBerikbol Torebek: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University Belgium and Institute of Mathematics and Mathematical Modeling Almaty, Kazakhstan E-mail address berikbol.torebek@ugent.be

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{'abstract': 'In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in [38] for stratified groups.', 'arxivid': '2106.00406', 'author': ['Michael Ruzhansky ', 'Bolys Sabitbek ', 'Berikbol Torebek '], 'authoraffiliation': [], 'corpusid': 235303675, 'doi': '10.1007/s00229-022-01390-2', 'github_urls': [], 'n_tokens_mistral': 15322, 'n_tokens_neox': 12836, 'n_words': 7131, 'pdfsha': 'a3fb02535a7d87423ff252e3ffdef81d7fed4f55', 'pdfurls': ['https://arxiv.org/pdf/2106.00406v2.pdf'], 'title': ['GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO POROUS MEDIUM EQUATION AND PSEUDO-PARABOLIC EQUATION, I. STRATIFIED GROUPS', 'GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS TO POROUS MEDIUM EQUATION AND PSEUDO-PARABOLIC EQUATION, I. STRATIFIED GROUPS'], 'venue': []}

arxiv

Implementation of Cluster expansion for hot QCD matter Niels-Uwe Friedrich Bastian University of Wrocław pl Maksa Borna 950-204WrocławPoland Pasi Huovinen Incubator of Scientific Excellence-Centre for Simulations of Superdense Fluids University of Wrocław pl Maksa Borna 950-204WrocławPoland Elizaveta Nazarova Joint Institute for Nuclear Research 141980DubnaRussia Implementation of Cluster expansion for hot QCD matter We present a cluster expansion EoS model for strongly-interacting matter based on the generalized Beth-Uhlenbeck formalism to describe hadrons as bound clusters of quarks. This formalism can describe both confined and deconfined phases. Our emphasis is on the region of vanishing baryon densities, where numerical solutions available from Lattice QCD predict a smooth crossover transition from hadron to quark matter. Medium effects are taken into account as self energies, which are motivated from both perturbative QCD calculations and phenomenological models. Parameters are tuned to Lattice QCD data and result in a good agreement of the thermodynamics. Introduction The equation of state (EoS) of strongly-interacting matter is a subject of active investigation within nuclear and high-energy physics communities. In the region of phase diagram, which corresponds to high temperatures (T ) and vanishing baryo-chemical potential (µ), the Lattice Quantum Chromodynamics (LQCD) can be used, predicting a crossover transition between hadronic matter and a phase of deconfined quarks (quark-gluon plasma (QGP)) [5]. Still, due to the known sign problem [7], Lattice QCD is not applicable in the region of finite densities (potentials). On the other hand, pQCD can describe matter at asymptotically high densities, predicting a phase of deconfined quark matter. However, the region of phase diagram "in between" remains for the most part unknown. This in turn means, that the question about the location of transition between hadron and quark matter, as well as the order of this transition, remains open. In this regard there are two possibilities: either somewhere at finite µ there exists at least one critical endpoint (CEP), denoting the change from crossover to first-order phase transition, or, alternatively, the crossover phase transition is present through the entire phase diagram, and the CEP does not exist. It should be noted, that several non-pertubative approaches to the description of quantum chromodynamics (QCD) phase diagram in the regimes of finite T and µ are being developed over the last decades. For example, the Dyson-Schwinger equation approach [17] has achieved remarkable progress towards a unified EoS of quark-hadron matter [8]. The phase diagram with a CEP could be obtained using the Polyakov-Quark-Meson model [20,21], and recently this approach has included the formulation of hadronization (describing hadrons as bound states of quarks) [1]. However, both of these approaches lack baryons in their description. Finally, another systematic non-pertubative approach based on the application of the functional renormalization group methods is being developed [16]. The concept of the model used in this work is outlined in [3]. In this approach the thermodynamics of a dense system of hadronic states (including their dissociation into the QGP) is described on the basis of the underlying quark dynamics (taking into account that hadrons are bound states of quarks): The generalized Beth-Uhlenbeck EoS that uses phase shifts in order to describe correlations and their modifications in a hot and dense environment is used for consistent description of bound and scattering states. To overcome the limitations of the standard Beth-Uhlenbeck approach, fully dressed quasi-particle propagators are used. Moreover, the self energies for the quasi particle properties are being modelled by a relativistic density functional (RDF) approach, which is capable to treat even such intricate effects as confinement and chiral symmetry breaking (χSB) (as well as their medium dependence). In [4] the model was applied to high net-baryon densities and low temperatures, showing its feasibility to describe a crossover or a first-order phase transition ending in a CEP at higher temperatures. Therefore, the model may be applicable to the entire phase diagram of QCD. We test the validity of this approach by implementing it to high temperatures and low densities probed in ultrarelativistic heavy-ion collisions (HIC), and comparing to the results of Lattice QCD calculations. The paper is structured as follows. Section 2 provides a detailed description of the general model of cluster expansion of strongly-interacting matter, followed by the specifics of the implementation to the regime of high temperatures and vanishing densities in section 3. Section 4 gives the details about the parametrisation realized for the high temperature region and the transition area. Finally, section 5 covers discussion of the results and further development of the model. Formalism The cluster expansion model for strongly interacting matter is described in Refs. [4,3,19]. The idea behind it is to describe matter, that consists of colored and colorless particles, utilizing the generalized Beth-Uhlenbeck approach for the description of a cluster expansion of strongly-correlated quark matter, where the clusters represent hadrons with spectral properties (bound states of quarks). The model then will describe thermodynamics of strongly-interacting matter, where partons are taken as quasi particles and hadrons as composites of these partons. To avoid confusion we will follow a convention, in which the index j can assume elementary partons (quarks and gluons), the index i compound hadrons (mesons and baryons) and the index l are all particles in the system (partons and hadrons). In a cluster expansion, the particle density of elementary partons j = {u, d, s, g} can be expressed as sum n j = l A l,j n l (T, µ)(1) over all species l in the system (here: partons and hadrons), while for all fermionic contributions (quarks and baryons) their anti-particles are included implicitly. The matrix A l,j is formed by the number of constituents j in the cluster l. In the generalized Beth-Uhlenbeck approach, the parton densities can be written as quasi particles n l={u,d,s,g} (T, µ) = n id (T, g l , M l ,μ l ) ,(2) with the ideal Fermi (+) or Bose (−) gas expressions for particle density n id (T, g l , M l ,μ l ) = g j d 3 p (2π) 3 1 e ( √ p 2 +M 2 j +V j −µ j )/T ± 1 ,(3) the degeneracy factor g l , the effective mass M l and the effective chemical potential µ l . The density of multi-particle states can be expressed by the generalized Beth-Uhlenbeck formula n i (T, µ) = g i d 3 pdE (2π) 4 f i (E)2 sin 2 δ i (E) dδ i (E) dE ,(4) where the phase shift δ i (E) is a medium-dependent quantity, which includes all properties of the particle species i, corresponding to its spectrum of bound and scattering states. The species i has degeneracy factor g i and obeys the Fermi (+) or Bose (−) distribution f i (E) = (exp [(E − µ i )/T ]±1) −1 , respectively. After substituting the integration over energy dE by an integration over effective mass dM , using the quasi-particle dispersion relation E i = p 2 + (M i ) 2 + V i , where M i = m i + S i is the effective mass, S i (V i ) are scalar (vector) self energies and M thr i is the threshold mass, defined as the sum of constituent masses of a compound particle (hadron). Following this approach we get for the (bound) hadrons the expression n i = g i d 3 p (2π) 3 1 e ( √ p 2 +M 2 i +V i −µ i )/T ± 1 − 1 e ( √ p 2 +(M thr i ) 2 +V i −µ i )/T ± 1 Θ(M thr i − M i ) = (n id (T, g i , M i ,μ i ) − n id (T, g i , M thr i ,μ i ))Θ(M thr i − M i )(6) which contains one usual "free" contribution, subtracted by another contribution, based on the mass of the multi-quark continuum threshold M thr i . Analogously the expressions for scalar density and entropy density can be derived with their ideal gas expressions: n S id (T, g j , M j ,μ j ) = g j d 3 p (2π) 3 M i p 2 + M 2 j 1 e ( √ p 2 +M 2 j −μ j )/T + 1 ,(7)s id (T, g j , M j ,μ j ) = g j d 3 p (2π) 3 4 3 p 2 + M 2 i T p 2 + M 2 j 1 e ( √ p 2 +M 2 j −μ j )/T + 1 .(8) In order to implement our model to the region of high temperature and vanishing densities (and compare with Lattice QCD), we are including up, down, strange quarks and gluons as partons and considering all hadrons in the hadron resonance gas (HRG) as clusters. Now, the total scalar (S) and vector density (V) of the parton j can be expressed as n (S,V) j (T, {µ k }) = i A ij n (S,V) i (T, µ i ) + n (S,V) j,unbound (T, µ j ) .(9) Here A ij is the amount of partons j ∈ {u, d, s, g} contained in the hadron i ∈ HRG. The effective quark mass M j = m j + S j is based on the bare mass m j and its scalar self energy S j and the earlier introduced threshold mass M thr i = j A ij M j of a hadron depends on the effective masses of its quark content. In a similar way we can obtain the expression for the total entropy density of the system s(T, {µ k }) = i A ij s i (T, µ i ) + s j,unbound (T, µ j )(10) and then the pressure as p(T, {µ k }) = T 0 dT s(T , {µ k }).(11) If one assumes that the self energies are derived as RDF in analogy to [10], the pressure would take the form p(T, {µ k }) = i∈HRG p H i (T, µ i ) + j={u,d,s} p Q j (T, µ j ) + p g (T, µ g ) + Θ .(12) Unfortunately, a consistent cluster mean field formulation for the RDF approach is still an open problem, and therefore the corresponding expressions for the self energies S j and the term Θ are not known. Therefore, in our work we utilise the approach of [22,23]. Within this approach every self energy is expressed by the sum of a primary energy shift and a rearrangement contribution S j = ∆m j + m R j .(13) While the primary energy shift ∆m j can be chosen almost arbitrarily, the rearrangement contributions m R j ensure of thermodynamic consistency and depend on the primary shifts of all particle species: m R i = j n s j ∂∆m j ∂n s i(14) Now we have all the basic tools at hand. However, a primary shift for every particle type in the system needs to be assigned. Functional form of self energies In the region of vanishing particle density it is sufficient to define scalar self energies since there are no contributions to the vector self energy. However, in future those will need to be introduced in order to derive and discuss higher baryon susceptibilities. In order to create a model, where particles' self energies are consistently sensitive to each other, we introduce a generating density, which counts the presence of color charges based on the partial scalar densities (irrespective of its particular color): n s = i∈HRG A i n s,i + j={u,d,s} n s,j + 2n s,g .(15) Here baryons contain three color charges (A i = 3), while mesons and gluons contain two color charges (A i = 2). We concentrate first on the high temperature limit, where hadronic contributions can be neglected. Here we can utilize results from pQCD in order to deduce the functional behaviour of the partonic masses at high temperatures. As the next step, we discuss the inclusion of confinement with the help of linear string potential. Last, but not least, we introduce necessary corrections for hadrons. Asymptotic limit At high temperatures we can assume, that all hadrons have disappeared and only quarks and gluons remain in the system, while asymptotically reaching the values or pQCD. In [11] an expression for the pressure, which can be considered applicable at sufficiently high temperatures, was derived. It can be shown (see appendix Appendix A), that the leading terms of this expression can be reformulated in quasi-particle terms as: p/T 4 ∼ d A π 2 45 + i=u,d,s p qu i + An S i + B(n S i ) 2 .(16) Here the first term is the Stefan-Boltzmann limit of gluons and A, B are non-trivial parameters, which are left open here, as we are only utilizing the functional form of the result. The same behaviour can be reproduced in a RDF approach (for example described in [2]) using the generating functional U ∼ n s + n 2 s ,(17) leading to a following expression for scalar self energy of the particles: S ∼ const + n s .(18) Therefore, the highest order for scalar self energy is linear in scalar density. It should be noted, that this diverging self energy does not contradict the Stefan-Boltzmann limit, because the thermal contributions are rising faster at T → ∞. String potential and confinement At moderate temperatures non-perturbative contributions appear (e.g. confinement), which are up to now not fully understood and can not be analytically derived from QCD. In this work we are choosing a phenomenological approach, inspired by the linear string potential, which is discussed, for example, in [12,10,2]. Accordingly, we introduce the following contributions to the scalar self energy of quarks and gluons: S ∼ Cn 1/3 s + Dn −1/3 s ,(19) where C is the coefficient for the one-gluon exchange and D is the effective coefficient of linear string tension, which can be density-dependent in order to take into account color saturation [18]. The second term is diverging at low scalar densities (and hence low temperatures), leading to a statistical confinement of quarks and gluons due to their diverging masses. Hadronic corrections At high temperatures quarks and gluons should dominate due to their small masses. In particular, the Heaviside function in eq. (6) will invoke a Mott transition once the threshold mass drops below the mass of the bound state. On the other hand, the earlier introduced linear term for the quark masses will negate this effect, because quark masses will rise again over hadronic masses and an unphysical rehadronisation would occur. In order to deal with this caveat, a linear contribution for hadrons S ∼ n s needs to be introduced, but to not change the behaviour at low temperatures or contradict the knowledge about hadronic in-medium effects in astrophysics, it should be folded with a form factor, resulting in: S ∼ n s 1 exp[(B 0 − n s )/B] − 1 .(20) This contribution would allow the hadrons to grow mass at sufficiently high temperatures, in order to suppress the unphysical rehadronisation, but does not affect lower temperatures. It also does not interfere at all with applications at zero temperature and high density, because the scalar density in this region is comparably low. Resulting particle properties and self energies Now we are ready to write full expressions for particle properties, utilizing the assumptions, described in the previous subsections. The resulting primary shifts for eq. (13) are: for hadrons. The corresponding rearrangement contributions for each particle species are: m R j={u,d,s} = m R ,(23)m R g = 2m R ,(24)m R i∈HRG = A i m R(25) with the common term m R = j={u,d,s,g} σ 4,j + C j n −2/3 s + D j n −4/3 s e −βn 2 s + D j n −1/3 s e −βn 2 s (−2βn s ) n s,j + A exp[(B 0 − n s )/B] − 1 + An s /B exp[(n s − B 0 )/B] − 1 n s,h .(26) Here n s,h denotes the sum of scalar densities of all hadrons. The arising coefficients σ 2,j , σ 4,j , C j , D j , β, A, B and B 0 are open parameters, subject to a fit and will be discussed in the following section. It will be shown, that the number of parameters can be drastically reduced, because many species dependencies are not necessary. Parametrisation We aim to create a model, whose thermodynamics are comparable to the results of Lattice QCD in [6]. In order to optimize the fit procedure and high-temperature adjustments for the model, we need an analytic parametrisation, that provides us data in the whole temperature range. First, we considered to use the fit of [15], because it also provides parametrisations for higher susceptibilities, which will be important in our future work. Unfortunately, it has unphysical behaviour at high temperatures, and therefore we are using in this work the fit from [9]. It proved to be effective to separate the region of high temperatures in which the system can be assumed to have only elementary partons and use it to fit only parameters which affect partons at high temperatures. Afterwards those parameters are being fixed and the remaining ones are obtained by fitting the transition region, including all possible species. High temperature parametrisation Besides the absence of bound hadrons, the high temperature region has the advantage, that the confinement contributions do not play a role. Instead of fitting our model here directly to the Lattice QCD thermodynamics, we use the effective masses of another quasiparticle approach for QGP [14]. This model describes the system of gluons (g), quarks (u, d and s) and their antiparticles with the effective quasi-particle masses dependent on the dynamically generated self-energies Π i as: M 2 i = m 2 i + Π i ,(27) where Π i (T ) = a i m i G(T ) 2 6 T 2 + G(T ) 2 6 T 2 ,(28) Note, that the perturbative couplings have been replaced by an effective coupling G(T ), which in the high-temperature regime resembles the perturbative coupling for thermal momenta. This model can be modified to describe pure Yang-Mills thermodynamics by setting the number of flavours equal to the spin-degeneracy factors of quarks and therefore introduces additional constraints for our model parameters. In the setup above the only unknown quantity is G(T ), which can be obtained by inverting eq. (29) with the help of aforementioned data for entropy density. The resulting effective masses can now be fitted using our analytical approach for the scalar self energy. We are concentrating on high temperatures (above 500 MeV) and neglect the confinement contribution in eq. (21) as low-temperature term, to obtain the high-temperature behaviour of the primary mass shift: ∆m j={u,d,s,g} = σ 2,j + σ 4,j n s + C j n 1/3 s (30) with the rearrangement term m R = j σ 4,j + C j n −2/3 s n s,j ,(31) which goes into eq. (13) as before. Figure 1 shows the results of this fit as the dependency of masses on the scalar density, and the extracted parameters are: One can see that our functional in eq. (30) reproduces excellently the masses of a model, which has been derived in a completely different way. It is worth noting, that the parameters σ 4 and C are species-independent. Parametrisation of the transition area Now that we have obtained the parameters σ 2,j , σ 4 and C from the high-temperature fit, the remaining parameters are only the confinement parameters D, D g and β and the hadronic corrections A, B and B 0 . Them we obtain by fitting to the Lattice QCD thermodynamic data directly. The fit has been performed using the entropy density, while ensuring to have reasonable particle fractions. It turns out, that the best result could be achieved with vanishing color saturation β = 0. The best fit values of the remaining parameters are Discussion In fig. 2 the thermodynamic quantities of the model, compared to the data of Lattice QCD, can be seen. The fit was mainly done with the entropy density, while the other properties and the particle fraction has also been kept in mind. Figure 3 shows the effective mass M π of the pion, as lightest hadron, and the sum of two light quarks, representing the threshold mass of the pion M thr π . Further it shows the effective mass of the gluon M g for comparison. The masses of partons diverge at low 250 MeV the linear contributions become dominant, which will become the leading term at asymptotic temperatures. The structure at T ≈ 140 MeV, which is a local maximum for gluons and pions and a shoulder for light quarks, arises from the mixed terms in the rearrangement contributions, since there are no clear dominant terms around that temperature. Up to this region the effective mass of pions is constant. The particle fraction can be seen in fig. 4. It is calculated as the ratio of the color scalar density of that species to the over-all color scalar density. This means for the gluon fraction x g = n s,g /n s , the quark fraction x q = j={u,d,s} n s,j /n s , the meson fraction x m = i∈mesons n s,i /n s , and the baryon fraction x b = i∈baryons n s,i /n s and the over-all sum results by definition in 1 = x g + x q + x m + x b . Up to a temperature T ≈ 80 MeV mesons dominate the system. Contributions of baryons stay over the entire temperature scale very small, due to their significantly higher masses, compared to mesons and the lack of chemical potential, which makes them dominant at high densities. The first partons to populate the system are gluons, due to their initially smaller masses. The correlation between effective mass and particle fraction can be clearly seen, as well as the non-trivial mixed term, which arises, when all species have similar fractions. On a qualitative level, the model reproduces the thermodynamics very well. Quantitatively, there are noticeable deviations, particularly in the transition area at T ≈ 150 MeV. Also the inclination point, which would later define the phase diagram, is slighly different from the data, but here one needs to take into account, that the results from Lattice QCD are also subject to uncertainties. The microscopic quantities of effective masses and particle fractions are model predictions, which can only be compared to predictions of other phenomenological models. It is questionable, whether an onset of deconfined partons below T = 100 MeV is physical, but this behaviour can be explained by the simplicity of the model assumptions at this stage. In particular, replacing the hadron phase shifts by experimentally observed scattering phase shifts [13] will suppress partonic contributions at lower temperatures. Furthermore, it will be interesting to see whether the existence of partons at surprisingly low temperatures and hadron-like states in high temperatures will affect the susceptibilities and higher-order fluctuations and force the use of more refined phase shifts. Inclusion of the vector self energies, and calculating these fluctuations is the next step in the development of this model. As was discussed in [4], cluster expansion model can be used to create an EoS with a critical endpoint when nucleons are the only hadronic degrees of freedom. We have now generalised the model to include strangeness and the whole zoo of hadronic resonances, and shown that with even very crude choices of phase shifts it qualitatively reproduces the Lattice QCD results. Thus cluster expansion model is a viable framework for constructing an Equation of State which covers the entire QCD phase diagram from low density-high temperature limit to high density-low temperature region, with great freedom to choose the location of the critical endpoint. Constructing such a "coast-to-coast" EoS is in our plans for near future. In [11] higher order corrections to the asymptotic limit of QCD are discussed. Here we want to show that the leading terms can be expressed in a quasi-particle picture and therefore used to derive the functional form of self-energy shifts in our work. The physical pressure of hot QCD is written in the form p QCD T 4 ≈ α MS E1 +ĝ 2 3 α MS E2 , (A.1) withĝ 2 3 = g 2 , as we are only interested in the first order. The prime at the coupling factor g was added to not confuse it with the previously used in this work degeneracy factor g. The α MS are evaluated for N f = 3 flavours resulting in: α MS E1 = d A π 2 45 + 4C A N f i=1 F 1 m 2 i T 2 , µ i T (A.2) α MS E2 = − d A C A 144 − d A N f i=1 1 6 F 2 m 2 i T 2 , µ i T 1 + 6F 2 m 2 i T 2 , µ i T + m 2 i 4π 2 T 2 3 lnμ m i + 2 F 2 m 2 i T 2 , µ i T − 2m 2 i T 2 F 4 m 2 i T 2 , µ i T (A.3) Extracting the leading orders at asymptotically high temperatures gives us a constant term, which represents the Stefan-Boltzmann result for gluons followed by the expression: p QCD T 4 T →∞ ∼ 4C A N f i=1 F 1 m 2 i T 2 , µ i T − g 2 d A N f i=1 1 6 F 2 m 2 i T 2 , µ i T 1 + 6F 2 m 2 i T 2 , µ i T . (A.4) The thermal functions F 1 and F 2 are defined as: which can be recognized as the ideal gas pressure p id of particles with mass m, the degeneracy factor g, temperature T and chemical potential µ. Considering, that the mass m is not constant, but rather has a logarithmic temperature dependence, this term can be treated as a quasi particle expression. Utilizing the same substitution, the expression for F 2 takes the form which results in the scalar density n id s of the same ideal gas. Now we can write the leading terms of (A.4) as F 2 ( m 2 T 2 , µ T ) = 1 12π 2 T 2p QCD T →∞ == N f i=1 p id (T, g i , m i , µ i ) + A N f i=1 n id s (T, g i , m i , µ i ) + B N f i=1 n id s (T, g i , m i , µ i ) 2 . (A.8) All non-vanishing coefficients are absorbed in the coefficients A and B, whose exact value is not relevant for the schematic discussion here. It should be noted, that the temperature dependence of the coupling constant g and the mass m compensate in B, but slightly remain in A, which causes non-leading effects. we choose to follow a simple ansatz for the phase shifts (they can only attain values of nπ): δ i (M ) = πΘ(M − M i )Θ(M thr i − M ) ∆m j={u,d,s,g} = σ 2,j + σ 4,j n s + C j n HRG = An s 1 exp[(B 0 − n s )/B] − 1 . taking the bare masses of particles as m g = 0 MeV, m i={u,d} = 5 MeV, m s = 95 MeV, while the coefficients are a g a i={u,d,s} = 2. Now the entropy density of the system can be computed by summing up the expression of eq (8) s = j=u,d,s,g s id (T, g j , M j , 0) . σ 2,l = 226.7 MeV, σ 2,s = 182.2 MeV, σ 2,g = 342.2 MeV, σ 4 = 0.1414 MeV fm 3 , C = 35.02 MeV fm. Figure 1 : 1Mass fit for high temperatures (T ≥ 500M eV ), at which hadrons can be neglected. Vertical dotted line is threshold from which the curve is fitted. Dash-dotted line shows temperature. A = 7. 0 0MeV fm 3 , B = 2.8 fm −3 , B 0 = 7.6 fm −3 , D = 200.0 MeV fm, D g = 100.0 MeV fm. Figure 2 : 2Thermodynamic observables of the model (dashed), compared to Lattice QCD data (shaded bands). The presented observables are the trace anomaly ε − 3p, pressure p, energy density ε and entropy density s. The dotted line on top is the Stefan-Boltzmann limit of p, ε, and s.temperatures due to the confinement contribution (n −1/3 s ). At moderate temperatures of T ≈ [80 − 250] MeV the parton masses decrease and they can populate the system accordingly. At higher temperatures of T Figure 3 : 3Effective masses of pions, two light quarks and gluons. The mott transition, shown as crossing point between pions and two light quarks is outside the picture at T ∼ 600 MeV. Both the quark and the gluon masses diverge at low temperatures, but the gluon line is cut due to numerical reasons. Figure 4 : 4Fractions of particles in the system, evaluating their respective color scalar density. Acknowledgement This work was supported by Polish National Science Center (NCN) under the grant No. 2019/32/C/ST2/00556 (N.-U.F.B.) and by the program Excellence Initiative-Research University of the University of Wrocław of the Ministry of Education and Science (P.H.). Appendix A. High temperature QCD in a quasiparticle picture Bound State Properties from the Functional Renormalization Group. Reinhard Alkofer, 10.1103/PhysRevD.99.054029Phys. Rev. D. 99554029Reinhard Alkofer et al. "Bound State Properties from the Functional Renormalization Group". In: Phys. Rev. D 99.5 (2019), p. 054029. doi: 10.1103/PhysRevD.99.054029. Phenomenological Quark-Hadron Equations of State with First-Order Phase Transitions for Astrophysical Applications. F Niels-Uwe, Bastian, 10.1103/PhysRevD.103.023001arXiv:2009.10846Physical Review D. 103223001Niels-Uwe F. Bastian. "Phenomenological Quark-Hadron Equations of State with First- Order Phase Transitions for Astrophysical Applications". 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In: PoS LAT2009.arXiv:1005.0539 (2009), p. 010. doi: 10.22323/1.091.0010. QCD at Finite Temperature and Chemical Potential from Dyson-Schwinger Equations. Christian S Fischer, 10.1016/j.ppnp.2019.01.002arXiv:1810.12938doi: 10. 1016/j.ppnp.2019.01.002Prog.Part.Nucl.Phys. 105Christian S. Fischer. "QCD at Finite Temperature and Chemical Potential from Dyson-Schwinger Equations". In: Prog.Part.Nucl.Phys. 105.arXiv:1810.12938 (2019), pp. 1-60. doi: 10. 1016/j.ppnp.2019.01.002. QCD Equation of State and Hadron Resonance Gas. Pasi Huovinen, Pter Petreczky, 10.1016/j.nuclphysa.2010.02.015Nucl. Phys. A. 837Pasi Huovinen and Pter Petreczky. "QCD Equation of State and Hadron Resonance Gas". In: Nucl. Phys. A 837 (2010), pp. 26-53. doi: 10.1016/j.nuclphysa.2010.02. 015. Quark-Nuclear Hybrid Star Equation of State with Excluded Volume Effects. 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{'abstract': 'We present a cluster expansion EoS model for strongly-interacting matter based on the generalized Beth-Uhlenbeck formalism to describe hadrons as bound clusters of quarks. This formalism can describe both confined and deconfined phases. Our emphasis is on the region of vanishing baryon densities, where numerical solutions available from Lattice QCD predict a smooth crossover transition from hadron to quark matter. Medium effects are taken into account as self energies, which are motivated from both perturbative QCD calculations and phenomenological models. Parameters are tuned to Lattice QCD data and result in a good agreement of the thermodynamics.', 'arxivid': '2302.10117', 'author': ['Niels-Uwe Friedrich Bastian \nUniversity of Wrocław\npl Maksa Borna 950-204WrocławPoland\n', 'Pasi Huovinen \nIncubator of Scientific Excellence-Centre for Simulations of Superdense Fluids\nUniversity of Wrocław\npl Maksa Borna 950-204WrocławPoland\n', 'Elizaveta Nazarova \nJoint Institute for Nuclear Research\n141980DubnaRussia\n'], 'authoraffiliation': ['University of Wrocław\npl Maksa Borna 950-204WrocławPoland', 'Incubator of Scientific Excellence-Centre for Simulations of Superdense Fluids\nUniversity of Wrocław\npl Maksa Borna 950-204WrocławPoland', 'Joint Institute for Nuclear Research\n141980DubnaRussia'], 'corpusid': 257038816, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11697, 'n_tokens_neox': 9666, 'n_words': 5787, 'pdfsha': '16bc9fb3d822f129ed2f3a54b6ca7d2b0523da92', 'pdfurls': ['https://export.arxiv.org/pdf/2302.10117v1.pdf'], 'title': ['Implementation of Cluster expansion for hot QCD matter', 'Implementation of Cluster expansion for hot QCD matter'], 'venue': []}

arxiv

Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields 1 Feb 2019 Yukiyoshi Nakkajima Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields 1 Feb 2019 In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields. Introduction In this article we discuss a new topic-rational points of the underlying schemes of log schemes in the sense of Fontaine-Illusie-Kato over the log point of a finite fieldfor interesting log schemes. First let us recall results on rational points of (proper smooth) schemes over a finite field. The following is famous Ax' and Katz' theorem: Theorem 1.1 ( [A], [Katz]). Let F q be the finite field with q = p e -elements, where p is a prime number. Let n and r be positive integers. Let D i (1 ≤ i ≤ r) be a hypersurface of P n Fq of degree d i . If r i=1 d i ≤ n, then #( r i=1 D i )(F q k ) ≡ 1 mod q k . In [Es] Esnault has proved the following theorem generalizing this theorem in the case where r i=1 D i is smooth over F q and geometrically connected: Theorem 1.2 ( [Es,Corollary 1.3]). Let X be a geometrically connected projective smooth scheme over F q . If X/F q is a Fano variety (i. e., the inverse of the canonical sheaf ω −1 X/Fq of X/F q is ample), then #X(F q k ) ≡ 1 mod q k (k ∈ Z ≥1 ). In [Ki] Kim has proved the following theorem and he has reproved Esnault's theorem as a corollary of his theorem by using the Lefschetz trace formula for the crystalline cohomology of X/F q : Theorem 1.3 ( [Ki,Theorem 1]). Let κ be a perfect field of characteristic p > 0. Set W := W(κ) and K 0 := Frac(W). Let X be a projective smooth scheme over κ. If X/κ is a Fano variety, then H i (X, W(O X )) ⊗ W K 0 = 0 for i > 0. In [GNT] Gongyo, Nakamura and Tanaka have proved the following theorem generalizing (1.2) for the 3-dimensional case by using methods of MMP(=minimal model program) in characteristic p ≥ 7: Theorem 1.4 ( [GNT,Theorem (1.2), (1.3)]). Let κ be as in (1.3). Assume that p ≥ 7. Let X be a geometrically connected proper variety over κ. Let ∆ be an effective Q-Cartier divisor on X. Assume that (X, ∆) is klt(=Kawamata log terminal) pair over κ and that −(K X + ∆) is a Q-Cartier ample divisor on X, where K X is the canonical divisor on X. Then the following hold: (1) H i (X, W(O X )) ⊗ W K 0 = 0 for i > 0. (2) Assume that κ = F q . Then #X(F q k ) ≡ 1 mod q k (k ∈ Z ≥1 ). See [NT] for the case where −(K X + ∆) is nef and big and (X, ∆) is log canonical. In this article we give other generalizations of the Theorems (1.2) and (1.3) under the assumption of certain finiteness: we give the definition of a log Fano variety and we prove a log and stronger version (1.5) below of Kim's theorem under the assumption as a really immediate good application of a recent result: Nakkajima-Yobuko's Kodaira vanishing theorem for a quasi-F -split projective log smooth scheme of vertical type ( [NY]). In this vanishing theorem, we use theory of log structures due to Fontaine- , [Kato2]) essentially. (See §3 for the precise statement of this vanishing theorem.) As a corollary of (1.5), we obtain the congruence of the cardinality of rational points of a log Fano variety over the log point of F q ((1.6) below). To state our result (1.5), we first recall the notion of the quasi-Frobenius splitting height due to Yobuko, which plays an important role for log Fano varieties in this article. Let Y be a scheme of characteristic p > 0. Let F Y : Y −→ Y be the Frobenius endomorphism of Y . Set F := W n (F * Y ) : W n (O Y ) −→ F Y * (W n (O Y ) ). This is a morphism of W n (O Y )-modules. In [Y] Yobuko has introduced the notion of the quasi-Frobenius splitting height h F (Y ) for Y . (In [loc. cit.] he has denoted it by ht S (Y ).) It is the minimum of positive integers n's such that there exists a morphism ρ : F Y * (W n (O Y )) −→ O Y of W n (O Y )-modules such that ρ • F : W n (O Y ) −→ O Y is the natural projection. (If there does not exist such n, then we set h F (Y ) = ∞.) This is a highly nontrivial generalization of the notion of the Frobenius splitting by Mehta and Ramanathan in [MR] because they have said that, for a scheme Z of characteristic p > 0, Z is a Frobenius splitting(=F -split) scheme if F : O Z −→ F Z * (O Z ) has a section of O Z -modules. Because the terminology "quasi Frobenius splitting height" is too long, we call this Yobuko height. Let κ be a perfect field of characteristic p > 0. Let s be a log scheme whose underlying scheme is Spec(κ) and whose log structure is associated to a morphism N ∋ 1 −→ a ∈ κ for some a ∈ κ. That is, s is the log point of κ or (Spec(κ), κ * ). Let X/s be a proper (not necessarily projective) log smooth scheme of pure dimension d of vertical type with log structure (M X , α : M X −→ O X ). Here "vertical type" means that α(I X/s )O X = O X , where I X/s is Tsuji's ideal sheaf of the log structure M X of X defined in [Ts] and denoted by I f in [loc. cit.], where f : X −→ s is the structural morphism. (In §3 below we recall the definition of I X/s .) For example, the product of (locally) simple normal crossing log schemes over s defined in [Nakk1], [NY] and [Nakk6] is of vertical type. Let • X be the underlying scheme of X. Let Ω i X/s be the sheaf of log differential forms of degree i on • X, which has been denoted by ω i X/s in [Kato1]. Set ω X/s := Ω d X/s . We say that X/s is a log Fano scheme if ω −1 X/s is ample. Moreover, if • X is geometrically connected, then we say that X/s a log Fano variety. In this article we prove the following: Theorem 1.5. Let X/s be a log Fano scheme. Assume that h F ( • X) < ∞. Then H i (X, W n (O X )) = 0 for i > 0 and for n > 0. Consequently H i (X, W(O X )) = 0 for i > 0. As mentioned above, we obtain this theorem immediately by using Nakkajima-Yobuko's Kodaira vanishing theorem for a quasi-F -split projective log smooth scheme of vertical type ( [NY]). As a corollary of this theorem, we obtain the following: Corollary 1.6. Let X/s be a log Fano variety. Assume that κ = F q and that h F ( • X) < ∞. Then # • X(F q k ) ≡ 1 mod q k (k ∈ Z ≥1 ). (1.6.1) In particular • X(F q ) = ∅. This is a generalization of Esnault's theorem (1.2) under the assumption of the finiteness of the Yobuko height. To derive (1.6) from (1.5), we use (A):Étess-Le Stum's Lefschetz trace formula for rigid cohomology (with compact support) ( [EL]) and (B) Berthelot-Bloch-Esnault's calculation of the slope < 1-part of the rigid cohomology (with compact support) via Witt sheaves ( [BBE]) as in [BBE], [GNT] and [NT]. However our proofs of (1.5) and (1.6) are very different from Esnault's, Kim's and Gongyo-Nakamura-Tanaka's proofs of (1.2), (1.3) and (1.4) in their articles because we do not use the rational connectedness of a Fano variety which has been used in them. We guess that the assumption of the finiteness of the Yobuko height is not a strong one for log Fano schemes. However this assumption is not always satisfied for smooth Fano schemes because the Kodaira vanishing holds if the Yobuko height is finite and because the Kodaira vanishing does not hold for certain Fano varieties ( [LR], [HL], [To]); the Yobuko heights of them are infinity. Hence to calculate the Yobuko heights of (log) Fano schemes is a very interesting problem. The conclusion of (1.6) holds for a proper scheme Y /F q such that H i (Y, O Y ) = 0 (i > 0). H. Tanaka has kindly told me that it is not known whether there exists an example of a smooth Fano variety over κ for which this vanishing of the cohomologies does not hold. (In [J] Joshi has already pointed out this; Shepherd-Barron has already proved that this vanishing holds for a smooth Fano variety of dimension 3 ( [SB,(1.5 )])). On the other hand, it is not clear at all that there is a precise rule as above about congruences of the cardinalities of the rational points of varieties except Fano varieties. One may think that there is no rule for them. In this article we show that this is not the case for log Calabi-Yau varieties over s of any dimension when • s = Spec(F q ); we are more interested in the cardinalities of the rational points of log Calabi-Yau varieties than those of log Fano varieties. First let us recall the following suggestive observation, which seems well-known ( [B]). Let E be an elliptic curve over F p . It is well-known that E is nonordinary if and only if #E(F p ) = p + 1 (1.6.2) if p ≥ 5. By the purity of the weight for E/F p : |#E(F p ) − (p + 1)| ≤ 2 √ p, (1.6.3) this equality is equivalent to a congruence #E(F p ) ≡ 1 mod p (1.6.4) since √ p > 2. In this article we generalize the congruence (1.6.4) for higher dimensional (log) varieties as follows. (We also generalize (1.6.2) for for any nonordinary elliptic curve over F q when p ≥ 5.) Let X/s be a proper (not necessarily projective) simple normal crossing log scheme of pure dimension d. Recall that, in [NY], we have said that X/s is a log Calabi-Yau scheme of pure dimension d if H i (X, O X ) = 0 (0 < i < d) and ω X/s ≃ O X . Moreover, if • X is geometrically connected, then we say that X/s is a log Calabi-Yau variety of pure dimension d. (This is a generalization of a log K3 surface defined in [Nakk1] .) Note that H d (X, O X ) = H d (X, ω X/s ) ≃ κ. The last isomorphism is obtained by log Serre duality of Tsuji ([Ts,(2.21)]). More generally, we consider a proper scheme Y of pure dimension d satisfying only the following four conditions: (a) H 0 (Y, O Y ) = κ, (b) H i (Y, W(O Y )) K0 = 0 for 0 < i < d − 1, (c) H d−1 (Y, O Y ) = 0 if d ≥ 2, (d) H d (Y, O Y ) ≃ κ. Let Φ Y /κ be the Artin-Mazur formal group of Y /κ in degree d, that is, Φ Y /κ is the following functor: Φ Y /κ (A) := Φ d Y /κ (A) := Ker(H d et (Y ⊗ κ A, G m ) −→ H d et (Y, G m )) ∈ (Ab) for artinian local κ-algebras A's with residue fields κ. Then Φ Y /κ is pro-represented by a commutative formal Lie group over κ ( [AM]). Denote the height of Φ Y /κ by h(Y /κ). We prove the following: Theorem 1.7. Let Y /κ be as above. Assume that κ = F q . Set h := h(Y /F q ). Then the following hold: (1) Assume that h = ∞. Then (1.7.1) #Y (F q k ) ≡ 1 mod q k (k ∈ Z ≥1 ). In particular, Y (F q ) = ∅. (2) Assume that 2 ≤ h < ∞. Let ⌈ ⌉ be the ceiling function: ⌈x⌉ := min{n ∈ Z | x ≤ n}. Then (1.7.2) #Y (F q k ) ≡ 1 mod p ⌈ek(1−h −1 )⌉ (k ∈ Z ≥1 ). In particular, Y (F q ) = ∅ (recall that e = log p q). (3) Assume that h = 1. Then (1.7.3) #Y (F q k ) ≡ 1 mod p (k ∈ Z ≥1 ). (In particular Y (F q k ) can be empty.) To give the statement (1.7) is a highly nontrivial work. However the proof of (1.7) is not difficult. (It does not matter whether the proof is not difficult.) As far as we know, (1.7) even in the 2-dimensional trivial logarithmic and smooth case, i. e., the case of K3 surfaces over finite fields, is a new result. Even in the case d = 1, Y need not be assumed to be an elliptic curve over F q . The heights of Artin-Mazur formal groups describe the different phenomena about the congruences of rational points for schemes satisfying four conditions (a), (b), (c) and (d). By using (1.7), we raise an important problem how the certain supersingular prime ideals are distributed for a smooth Calabi-Yau variety of dimension less than or equal to 2 over a number field. (I think that there is no relation with Sato-Tate conjecture in non-CM cases.) To obtain (1.7), we use the theorems (A) and (B) explained after (1.6) again and the determination of the slopes of the Dieudonné module D(Φ Y /κ ) of Φ Y /κ . The contents of this article are as follows. In §2 we recallÉtess-Le Stum's Lefschetz trace formula for rigid cohomology, Berthelot-Bloch-Esnault's theorem and the congruence of the cardinality of rational points of a separated scheme of finite type over a finite field. In §3 we prove (1.5) and (1.6). In §4 we prove (1.7). We also raise the important problem about the distribution of supersingular primes already mentioned. In §5 we give the formulas of two kinds of zeta functions of a few projective SNCL(=simple normal crossing log) schemes over the log point of a finite field. One kind of them gives us examples of the conclusions of the congruences in (1.6) and (1.7). In §6 we give a remark on Van der Geer and Katsura's characterization of the height h(Y /κ) ( [vGK1]). Acknowledgment. I have begun this work after listening to Y. Nakamaura's very clear talk in which the main theorem in [NT] has been explained in the conference "Higher dimensional algebraic geometry" of Y. Kawamata in March 2018 at Tokyo University. The talk of Y. Gongyo in January 2017 at Tokyo Denki university for the explanation of the main theorem in [GNT] has given a very good influence to this article. Without their talks, I have not begun this work. I would like to express sincere gratitude to them. I would also like to express sincere thanks to H. Tanaka and S. Ejiri for their kindness for informing me of the articles [LR], [HL], [To] and giving me an important remark. Notations. (1) For an element a of a commutative ring A with unit element and for an A-modules M , M/a denotes M/aM . (2) For a finite field F q , s Fq denotes the log point whose underlying scheme is Spec(F q ). Preliminaries In this section we recallÉtess-Le Stum's Lefschetz trace formula for rigid cohomology with compact support ( [EL]) and Berthelot-Bloch-Esnault's calculation of the slope < 1-part of the rigid cohomology with compact support via Witt sheaves with compact support ( [BBE]). Let K 0 (F q ) be the fraction field of the Witt ring W(F q ) of F q . Let Y be a separated scheme of finite type over F q of dimension d. Let F q : Y −→ Y be the q-th power Frobenius endomorphism of Y . The following isÉtess-Le Stum's Lefschetz trace formula proved in [EL, Théorème II]: #Y (F q ) = 2d i=0 (−1) i Tr(F * q |H i rig,c (Y /K 0 (F q ))). (2.0.1) Let {α ij } j be an eigenvalue of F * q on H i rig,c (Y /K 0 (F q )). Then #Y (F q ) = 2d i=0 (−1) i ( j α ij ). (2.0.2) By [CLe,(3.1.2)] (see also [Nakk4,(17.2)]), max{0, i − d} ≤ ord q (α ij ) ≤ min{i, d}. (2.0.3) Henceforth we consider the equalities (2.0.1) and (2.0.2) as the equalities in the integer ring W(F q ) of an algebraic closure of K 0 (F q ). Let κ be a perfect field of characteristic p > 0. Let Y /κ be a separated scheme of finite type. Let K 0 be the fraction field of the Witt ring W of κ. Let H i rig,c (Y /K 0 ) [0,1) be the slope < 1-part of the rigid cohomology H i rig,c (Y /K 0 ) with compact support with respect to the absolute Frobenius endomorphism of Y . Let H i c (Y, W(O Y,K0 )) be the cohomology of the Witt sheaf with compact support of Y /K 0 defined by Berthelot, Bloch and Esnault in [BBE]: H i c (Y, W(O Y,K0H i c (Y, W(O Y,K0 )) −→ H i (Z, W(O Z,K0 )) −→ H i (Z, W(O Z /I) K0 ) (2.0.4) −→ H i c (Y, W(O Y,K0 )) −→ · · · . By replacing Z by the closure of Y in Z, we see that H i c (Y, W(O Y,K0 )) = 0 (2.0.5) if i > d. Then they have proved that there exists the following contravariantly functorial isomorphism H i rig,c (Y /K 0 ) [0,1) ∼ −→ H i c (Y, W(O Y,K0 )) (2.0.6) ([BBE, Theorem (1.1)]). Now let us come back to the case κ = F q . Since H i rig,c (Y /K 0 (F q )) = d−1 j=0 H i rig,c (Y /K 0 (F q )) [j,j+1) ⊕ H i rig,c (Y /K 0 (F q )) [d] , #Y (F q ) = 2d i=0 (−1) i d−1 j=0 Tr(F * q |H i rig,c (Y /K 0 (F q )) [j,j+1) ) + 2d i=d (−1) i Tr(F * q |H i rig,c (Y /K 0 (F q )) [d] ). (2.0.7) Hence we have the following congruence by (2.0.5) and (2.0.6): #Y (F q ) ≡ d i=0 (−1) i Tr(F * q |H i c (Y, W(O Y,K0 ))) mod q (2.0.8) in W(F q ). Remark 2.1. In [BBE,(1.4)], the following zeta function Z W (Y /F q , t) := d i=0 det(1 − tF * q |H i c (Y, W(O Y )) K0 ) (−1) i+1 which is equal to the zeta function Z <1 (Y /F q , t) := ordq(αij )<1 (1 − α ij t) (−1) i+1 has been considered. In this article we do not need this zeta function. We do not need Ax's theorem in [A] (see [BBE,Proposition 6.3]) either. 3 Proofs of (1.5) and (1.6) It is well-known that the analogue of Kodaira's vanishing theorem for projective smooth schemes over a field of characteristic 0 ( [Ko]) do not hold in characteristic p > 0 in general ( [R]). However, in [NY], we have proved the Kodaira vanishing theorem in characteristic p > 0 under the assumption of the finiteness of the Yobuko height. To state this theorem precisely, we recall the definition of the vertical type for a relative log scheme. For a commutative monoid P with unit element, an ideal is, by definition, a subset I of P such that P I ⊂ I. An ideal p of P is called a prime ideal if P \ p is a submonoid of P ([Kato2, (5.1)]). For a prime ideal p of P , the height ht(p) is the maximal length of sequence's p p 1 · · · p r of prime ideals of P . Let h : Q −→ P be a morphism of monoids. A prime ideal p of P is said to be horizontal with respect to h if h(Q) ⊂ P \p ([Ts, (2.4)]). Let Y −→ Z be a morphism of fs(=fine and saturated) log schemes. Let h : Q −→ P be a local chart of g such that P and Q are saturated. Set I := {a ∈ P | a ∈ p for any horizontal prime ideal of P of height 1 with respect to h}. [Ts,(2.6)] Tsuji has proved that I Y /Z is independent of the choice of the local chart h. Let Let I Y /Z be the ideal sheaf of M Y generated by Im(I −→ M Y ). InI Y /Z O Y be the ideal sheaf of O Y generated by the image of I Y /Z . Definition 3.1. We say that Y /Z is of vertical type if I Y /Z O Y = O Y . In [NY,(1.9)] we have proved the following theorem: Theorem 3.2 (Log Kodaira Vanishing theorem). Let Y −→ s be a projective log smooth morphism of Cartier type of fs log schemes. Assume that • Y is of pure dimension d. Let L be an ample invertible sheaf on • Y . Assume that h F ( • Y ) < ∞. Then H i (Y, I Y /s ω Y /s ⊗ OY L) = 0 for i > 0. In particular, if Y /s is of vertical type, then H i (Y, ω Y /s ⊗ OY L) = 0 for i > 0. Now let us prove (1.5) and (1.6) quickly. Let the notations be as in (1.6). Since ω −1 X/s is ample, H i (X, O X ) = 0 for i > 0 by (3.2). Hence, by the following exact sequence 0 −→ W n−1 (O X ) V −→ W n (O X ) −→ O X −→ 0, (3.2.1) H i (X, W n (O X )) = 0 for i > 0 and n > 0. Hence H i (X, W(O X )) = (lim ← − n H i (X, W n (O X ))) = 0. (3.2.2) Thus we have proved (1.5). Next let us prove (1.6). It suffices to prove (1.6) for the case k = 1 by considering the base change X ⊗ Fq F q k . Because H 0 (X, W(O X )) = W(F q ) and F * q = id on H 0 (X, W(O X )) , we obtain the following by (2.0.8): # • X(F q ) ≡ 1 mod q (3.2.3) in W(F q ). This shows (1.6). Remark 3.3. (1) If X is a Fano variety over Q, then the reduction X mod p of a flat model X over Z of X for p ≫ 0 is a Fano variety and F -split ([BM, Exercise 1.6. E5]). In particular, h F (X mod p) < ∞ for p ≫ 0. (2) As pointed out in [BM,p. 58], a Fano variety X is not necessarily F -split. The Kodaira vanishing theorem does not hold for certain Fano varieties ( [LR], [HL], [To]). By (3.2) we see that the Yobuko heights of them are infinity. (3) Let X/s be an SNCL Fano scheme of pure dimension d. Then any irreducible component of • X i of • X is Fano. Indeed, since ω −1 X/s is ample, ω −1 X/s ⊗ OX O Xi = ω −1 • Xi/κ (− j log D j ) is also ample. Here {D j } j is the set of the double varieties in • X i . Hence −K • X i − j D j is ample. Consequently −K • X i is ample. Remark 3.4. Let X/F q be a separated scheme of finite type. Assume that X is geometrically connected. By the argument in this section, it is obvious that, if H i (X, O X ) = 0 (∀i > 0), then the congruence (1.6.1) holds for X/F q . In partic- ular, if d = 2, if X/F q is smooth, if H 1 (X, O X ) = 0 and if H 0 (X, Ω 2 X/s ) = 0, then the congruence (1.6.1) holds for X/F q . Such an example can be given by a proper smooth Godeaux surface. Other examples are given by proper smooth unirational threefolds because H i (X, O X ) = 0 (∀i > 0) by [Ny,Introduction,(2.5)]. Let X/s be an SNCL(=simple normal crossing log) classical Enriques surface X/s for p = 2, i.e., (Ω 2 X/s ) ⊗2 is trivial and the correspondingétale covering X ′ to Ω 2 X/s is an SNCL K3 surface (In [Nakk1, (7.1)] we have proved that H i (X, O X ) = 0 for i > 0.). Hence the congruence (1.6.1) also holds for X/s Fq . See (5.4) below for the zeta function of this example. By the formulas for the zeta function ((5.4.1), (5.4.2)), we can easily verify that # • X(F q ) indeed satisfies the congruence (1.6.1). More generally, if H i (X, W(O X )) K0 = 0 (i > 0), then the congruence (1.6.1) holds for X/F q by the proof of (1.5). By the main theorem of [BBE], one obtains such examples which are special fibers of regular proper flat schemes over discrete valuation rings of mixed characteristics whose generic fibers are geometrically connected and of Hodge type ≥ 1 in positive degrees. See also [Er] for a generalization of the main theorem in [BBE]. Example 3.5. Let n and N be positive integers. Set X 1 := P N W(κ) . Blow up X 1 along an κ-rational hyperplane of P N κ and let X 2 be the resulting scheme. Let • X 1 and • X n be the irreducible components of the special fiber X 2 . Blow up X 2 again along • X 1 ∩ • X n and let X 3 be the resulting scheme. Let • X 1 , • X n and • X n−1 be the irreducible components of the special fiber X 3 . Blow up X 3 again along • X 1 ∩ • X n−1 . Continuing this process (n − 1)-times, we have a projective semistable family X n over Spec(W(κ)). Let • X i (1 ≤ i ≤ n) be the the irreducible components of the special fiber X n . Let X be the log special fiber of X n . Then X is a projective SNCL scheme over s. Let • X (i) (i = 0, 1) be the disjoint union of (i + 1)-fold intersections of the irre- ducible components of • X. Then • X (0) = P N κ (P N −1 κ × κ P 1 κ ) · · · (P N −1 κ × κ P 1 κ ) n−1 times and • X (1) = P N −1 κ · · · P N −1 κ n−1 times . Using the following spectral sequence E ij 1 = H j (X (i) , O X (i) ) =⇒ H i+j (X, O X ) (3.5.1) and noting that the dual graph of • X is a segment, we see that H i (X, O X ) = 0 (i > 0). If s = s Fq , then it is easy to check that • X(F q ) = q N +1 − 1 q − 1 + (n − 1) q N − 1 q − 1 q 2 − 1 q − 1 − (n − 1) q N − 1 q − 1 = q N +1 − 1 q − 1 + (n − 1)q q N − 1 q − 1 . In particular, # • X(F q ) ≡ 1 mod q. The restriction of ω X/s to [BM,(1.1.5)]) and because we have the following exact sequence • X i is isomorphic to O • Xi (−N ) for i = 0, N and O • Xi (−(N − 1)) for 0 < i < N . Hence ω −1 X/s is ample if N ≥ 2. Since each • X i is F -split (the F -splitting is given by the "p −1 -th power" of the canonical coordinate of • X i (see0 −→ O X −→ N i=1 O Xi −→ N −1 i=1 O Xi∩Xi+1 , • X is F -split. 4 Proof of (1.7) Let the notations be as in (1.7). In this section we prove (1.7). We may assume that k = 1. Since H d−1 (Y, O Y ) = 0, we see that H d−1 (Y, W(O Y )) = lim ← − n H i (Y, W n (O Y )) = 0 (4.0.1) by the same proof as that of (1.5). Set Y := Y ⊗ Fq F q and e := log p q. By [AM,II (4.3)] the Dieudonné module M : H, V (28.3.10)]). Let F : M −→ M be the operator "F " on the Dieudonné module M . By abuse of notation, we denote the induced morphism M K0 −→ M K0 by F . By (2.0.8) we have the following congruence = D(Φ Y /κ ) of Φ Y /κ is equal to H d (Y, W(O Y )). Let h be the height of Φ Y /κ . Hence Φ Y /κ is a commutative formal Lie group over κ of dimension 1 and the Dieudonné module M is a free W-module of rank h if h < ∞ ([#Y (F q ) ≡ 1 + Tr(F e |M K0 ) mod q (4.0.2) in W(F q ). Set m := ord p (Tr(F e |M K0 ) ). If m ≤ e = ord p (q), then we obtain the following congruence by (4.0.2): #Y (F q ) ≡ 1 mod p ⌈m⌉ (4.0.3) in Z. First we give the proof of (1.7) (1). Proof of (1.7) (1). Assume that h = ∞. Then D(Φ Y /Fq ) is W(F q )-torsion. By [AM, II (4.3)], H d (Y , W(O Y )) K0 = D(Φ Y /Fq ) K0 = 0. By [I1, I (1.9.2)], W(O Y ) = W(O Y ) ⊗ W(Fq) W(F q ). Since • Y is separated, we obtain the following equality H d (Y, W(O Y )) = H d (Y, W(O Y )) ⊗ W(Fq) W(F q ) by usingČech cohomologies. Hence H d (Y, W(O Y )) K0(Fq) = 0. (To obtain this vanishing, one may use the fact that the Dieudonné module commutes with base change (cf. the description of D(Φ Y /Fq ) in [Mu2,p. 309].)) By (2.0.8) this means the congruence (1.7.1). Now assume that h < ∞. Next we give the proof (1.7) (2). Proof of (1.7) (2). Let us recall the following well-known observation ([Li, Exercise 6.13]): Proposition 4.1. Let G be a commutative formal Lie group of dimension 1 over a perfect field κ of characteristic p > 0. Assume that the height h of G is finite. Then the slopes of the Dieudonné module of D(G) is 1 − h −1 . Proof. Let D(κ) be the Cartier-Dieudonné algebra over κ. We may assume that κ is algebraically closed. In this case, the height is the only invariant which determines the isomorphism class of a 1-dimensional commutative formal group law over κ ( [H, (19.4 .1)]). Hence D(G) ≃ D(κ)/D(κ)(F − V h−1 ) ([vGK1, p. 266]). Express F (1, V, · · · , V h−1 ) = (1, V, · · · , V h−1 )A, where A ∈ M h (W) (as if F were W-linear). Then det(tI − A) = t h − p h−1 . Hence the slopes of D(G) is ord p ((p h−1 ) h −1 ) = 1 − h −1 . By (4.1) and (4.0.3), we obtain the following congruence #Y (F q ) ≡ 1 mod p ⌈e(1−h −1 )⌉ (4.1.1) in Z. Lastly we give the proof of (1.7) (3) in the following. Proof of (1.7) (3). Let κ be a perfect field of characteristic p > 0. Let Y be a proper scheme over κ of pure dimension d ≥ 1. (We do not assume that Y is smooth over κ.) Assume that H d (Y, O Y ) ≃ κ and that H d−1 (Y, O Y ) = 0 if d ≥ 2. Then the following morphism H d (Y, W(O Y ))/p −→ H d (Y, O Y ) is an isomorphism. Indeed, this is surjective and dim κ (H d (Y, W(O Y ))/p) = dim κ (M/p) = 1 = dim κ H d (Y, O Y ). Since h = 1, F on H d (Y, W(O Y ))⊗ W(κ) W(κ) is an isomorphism, Hence F : H d (Y, O Y ) −→ H d (Y, O Y ) is an isomorphism. Hence #Y (F q ) ≡ 1 + α mod q for a unit α ∈ W(F q ) * . Now (1.7) (3) follows. Remark 4.2. (1) If H i (Y, O Y ) = 0 for 0 < i < d − 1 ( this is stronger than (c) in the Introduction), then (1.7) (3) also follows from Fulton's trace formula ( [Fu]): [B,Proposition 5.6]). #Y (F q ) mod p ≡ d i=0 (−1) i Tr(F * q |H i (Y, O Y )) ∈ F q (cf. (2) Let X/s be a log Calabi-Yau scheme. In [NY,(10.1)] we have proved a fundamental equality h F (X/κ) = h(X/κ). Hence X is quasi-F -split (resp. F -split) if and only if h(X/κ) < ∞ (resp. h(X/κ) = 1). Though the following corollary immediately follows from [BBE,(1.6)], we state it for the convenience of our remembrance. Corollary 4.3. Let Y be as in (1.7). Let f : Z 1 −→ Z 2 be a morphism of proper schemes over F q . Assume that Z 1 or Z 2 is isomorphic to Y over F q . Assume that Φ Zi/Fq (i = 1, 2) is representable. If the pull-back f * : H i (Z 2 , O Z2 ) −→ H i (Z 1 , O Z1 ) is an isomorphism, then the natural morphism Φ Z2/Fq −→ Φ Z1/Fq is an isomorphism. In particular, h(Z 1 /F q ) = h(Z 2 /F q ) and (1.7) for #Z i (F q ) holds. Proof. By the assumption, we have an isomorphism f * : H i (Z 2 , W(O Z2 )) ∼ −→ H i (Z 1 , W(O Z1 )). Hence the natural morphism D(Φ Z2/Fq ) −→ D(Φ Z1/Fq ) is an isomorphism. By Cartier theory, the natural morphism Φ Z2/Fq −→ Φ Z1/Fq is an isomorphism. This implies that h(Z 2 /F q ) = h(Z 2 /F q ) and (1.7) for #Z i (F q ) holds. The following corollary immediately follows from the proof of [BBE,(6.12)]. (1) Let E/F p be an elliptic curve. It is very well-known that E/F p is supersingular if and only if #E(F p ) = p + 1 if p ≥ 5 ([Si1, V Exercises 5.9]). As observed in [B,Example 5.11], this also follows from the purity of the weight for an elliptic curve over F p : |#E(F p ) − (p + 1)| ≤ 2 √ p and Fulton's trace formula. In fact, we can say more in (4.8) below. (2) Let d ≥ 3 be a positive integer such that d ≡ 0 mod p. Consider a smooth Calabi-Yau variety X /W(F q ) in P d−1 W(Fq) defined by the following equation: a 0 T d 0 + · · · + a d−1 T d d−1 = 0 (a 0 , . . . , a d−1 ∈ W(F q ) * ). Set a := a 0 · · · a d−1 ∈ W(F q ). Let X/F q be the reduction mod p of X /W(F q ). By [St,Theorem 1] (see also [loc. cit., Example 4.13]), the logarithm l(t) of Φ X /W(Fq) is given by the following formula: l(t) = ∞ m=0 a m (md)! (m!) d t md+1 md + 1 . (a) If p ≡ 1 mod d, then pl(t) = pt + p p−1 i=2 c i t i + (unit)t p + (higher terms than t p ) for some c i ∈ W(F q ) in W(F q )[[t] ]. Hence l −1 (pl(t)) mod p ≡ t p + · · · and the height of Φ X/Fq is equal to 1. (b) If p ≡ 1 mod d, then pl(t) ∈ pW(F q )[[t]] . Hence the height of Φ X/Fq is equal to ∞. (a) and (b) above are much easier and much more direct proofs of [vGK2, Theorem 5.1]. (3) Especially consider the case N = 3 in (2) and let X/F p be a closed subscheme of P N Fp defined by the following equation: T 4 0 + T 4 1 + T 4 2 + T 4 3 = 0. (a) If p = 3, then #X(F 3 ) ≡ 1 mod 3 by (1.7) (1). In fact, it is easy to see that #X(F 3 ) = 4 = 1 + 3 2 − 3 × 2. (This X and X in (c) are Tate's examples in [Ta1] of a supersingular K3-surface (in the sense of T. Shioda) over F 3 and F 7 ), respectively.) (b) If p = 5, then #X(F 5 ) ≡ 1 mod 5 by (1.7) (3). In fact, it is easy to see that #X(F 5 ) = 0. More generally, for a power q of a prime number p, let X q be a closed subscheme of P q−1 Fq defined by the following equation: a 0 T q−1 0 + · · · + a q−2 T q−1 q−2 = 0 (a 0 , . . . , a q−2 ∈ F * q , (a 0 , . . . , a q−2 ) = (0, . . . , 0)), where a 0 , . . . , a q−2 satisfying the following condition: for any nonempty set I of {0, . . . , q − 2}, j∈I a j = 0 in F q . Then #X q (F q ) = 0. (c) If p = 7, then #X(F 7 ) ≡ 1 mod 7 by (1.7) (1). In fact, one can check that #X(F 7 ) = 64 = 1 + 7 2 + 7 × 2. In general, if Φ(X/F q ) is supersingular, then #X(F q ) = 1+q 2 +qα for some |α| ≤ 22 by the purity of the weight and by b 2 (X) = 22. Here b 2 (X/F q ) is the second Betti number of X/F q . (We do not know an example of the big |a|.) (4) See [YY,(4.8)] for explicit examples of X/F q 's such that h(Φ X/Fq ) = 2. See also [vGK2,§6]. Example 4.6. (1) Let n be a positive integer. Let X be an n-gon over F q . Then, by [Nakk5, (6.7) (1)], X is F -split. In particular, h F (X) = h(X/F q ) = 1. Then, by (1.7) (3), #X(F q ) ≡ 1 mod p. In fact, it is easy to see that #X(F q ) = n(q + 1) − n = q. Compare this example with the example in (3.5). (2) Let κ be a perfect field of characteristic p > 0. Let X be an SNCL(=simple normal crossing log) K3-surface over κ, that is, an SNCL Calabi-Yau variety of dimension 2 ( [Nakk1]). In [Nakk5, (6.7) (2)] we have proved the following: (1) Let X/F q and X * /F q be a strong mirror Calabi-Yau pair in the sense of Wan ([Wan2]), whose strict definition has not been given. Then he conjectures that #X(F q ) ≡ #X * (F q ) mod q ([Wan2, (1.3)]). Hence the following question seems natural: does the equality h(Φ X/Fq ) = h(Φ X * /Fq ) hold? If his conjecture is true, only one of h(Φ X/Fq ) and h(Φ X * /Fq ) cannot be 1 by (1.7). This is compatible with Wan's generically ordinary conjecture in [loc. cit., (8.3)]. (2) If X satisfies the conditions (a), (c) and (d) in the Introduction and if X is a special fiber of a regular proper flat scheme over a discrete valuation ring of mixed characteristics whose generic fibers are geometrically connected and of Hodge type ≥ 1 in degrees in [1, d − 2], then we see that X satisfies the condition (b) by [BBE]. We conclude this section by generalizing (4.5) (1) by using (1.7) and raise an important question: Proposition 4.8. Let C be a proper smooth curve over F q such that H 0 (C, O C ) ≃ F q ≃ H 1 (C, O C ). Recall that e = log p q. Then the following hold: (1) Assume that e is odd and p ≥ 5. Then h C/Fq = 2 if and only if #C(F q ) = 1+q. (2) Assume that e is odd and p = 3 or 2. Then h C/Fq = 2 if and only if #C(F q ) = 1 + q or 1 + q ± p e+1 2 . (3) Assume that e is even. Then h C/Fq = 2 if and only if #C(F q ) = 1 + q + αp e 2 , where α ∈ N and |α| ≤ 2. Proof. By the purity of weight, we have the following inequality: for m ∈ N. By (4.8.1) we have the following inequality: Since p ≥ 5, m = p e−1 2 . Hence #C(F q ) = 1 + q. Conversely, assume that #C(F q ) = 1 + q. Then C can be an elliptic curve over F q . Hence h C/Fq = 1 or 2 ([Si1, IV (7.5)]). By (1.7.3) and (1.7.2), h C/Fq = 2. |#C(F q ) − (1 + q)| ≤ 2 √ q. (2): Assume that h C/Fq = 2. Then, by (4.8.2), m = p e−1 2 or m = ±1 + p e−1 2 . Hence #C(F q ) = 1 + q or #C(F q ) = 1 + (±1 + p e−1 2 )p e+1 2 = 1 + q ± p e+1 2 . The proof of the converse implication is the same as that in (1) The proof of the converse implication is the same as that in (1). Remark 4.9. Assume that e is even. By Honda-Tate's theorem for elliptic curves over finite fields (4.10) below, the case |α| = 1 occurs only when p ≡ 1 mod 3; the case α = 0 occurs only when p ≡ 1 mod 4. Theorem 4.10 (Honda-Tate's theorem for elliptic curves ( [Wat2,(4.1)], [P, (4.8)])). For an elliptic curve E/F q , set t E := 1+q−#E(F q ). Consider the following well-defined injective map: {isogeny classes of elliptic curves E/F q } ∋ E −→ t E ∈ {t ∈ Z | |t| ≤ 2 √ q}. (This map is indeed injective by Tate's theorem ([Ta2,Main Theorem].) The image of HT consists of the following values: (1) t is coprime to p. (2) e is even and t = ±2 √ q. (3) e is even and p ≡ 1 mod 3 and t = ± √ q. (4) e is odd and p = 2 or 3 and t = ±p e+1 2 . (5) e is odd, or e is even and p ≡ 1 mod 4 and t = 0. The case (1) arises from ordinary elliptic curves over F q . The case (2) arises from supersingular elliptic curves over F q having all their endomorphisms defined over F q ; the rest cases arises from supersingular elliptic curves over F q not having all their endomorphisms defined over F q . Problem 4.11. Let K be an algebraic number field and O K the integer ring of K. Let x be a positive real number. (1) Consider the following set P(x) := {p ∈ Spec(O K ) | N K/Q (p) ≤ x and log p (#(O K /p)) is even}, where p = ch(O K /p). Assume that p ≥ 5. Let E/K be an elliptic curve. Let α be an integer such that |α| ≤ 2. Consider the following set P ′ (x; E/K, α) := {p ∈ P(x) | E has a good reduction E 0 at p and #E 0 (F q ) = 1 + q + α √ q}. Set P(x; E/K, α) :=      P ′ (x; E/K, α) (|α| = 2), {p ∈ P(x; E/K, α) | p ≡ 1 mod 3} (|α| = 1), {p ∈ P(x; E/K, α) | p ≡ 1 mod 4} (α = 0). Then, what is the function x −→ #P(x; E/K, α) #P(x) when x → ∞? (I do not know whether lim x→∞ P(x; E/K, α) = ∞ for each α such that |α| ≤ 2 for any non-CM elliptic curve over K (see [Si2,p. 185 Exercise 2.33 (a), (b)] for a CM elliptic curve over Q( √ −1): in this example, lim x→∞ P(x; E/K, 2) = ∞, but P(x; E/K, α) = 0 for α = 2 and for any x). If [K : Q] is odd or if K has a real embedding, then lim x→∞ |α|≤2 P(x; E/K, α) = ∞ by Elkies' theorems ([El1,Theorem 2], [El2,Theorem]).) When p = 2 or 3, we can give a similar problem to the problem above by using (4.8) (2). (2) Consider the following set P(x) := {p ∈ Spec(O K ) | N K/Q (p) ≤ x}. Let S/K be a K3 surface. Let α be an integer such that |α| ≤ 22. Consider the following set P ′ (x; S/K, α) := {p ∈ P(x) | S has a good reduction S 0 at p and #S 0 (F q ) = 1 + q 2 + αq}. Then, what is the function Let Y be a separated scheme of finite type over F q . Set x −→ #P ′ (x; S/K, α) #P(x)0 − −−− → U − −−− → V − −−− → W − −−− → 0 α   β   γ   0 − −−− → U − −−− → V − −−− → W − −−− → 0. Let t be a variable. Note that det(1 − tβ|V ) = det(1 − tα|U )det(1 − tγ|W ). If V = {0}, we set det(1 − t0|V ) = 1 (1 ∈ F ). We[(E p (Y /F q ), F * q )] := ∞ i=0 (−1) i [(H i rig,c (Y /K 0 (F q )), F * q )] ∈ K(K 0 (F q )), (5.0.2) where E p means the Euler-characteristic. Let Z(Y /F q , t) := exp ∞ n=0 #Y (F q n ) n t n (5.0.3) be the zeta function of Y /F q . We can reformulate (2.0.1) as the following formula: Z(Y /F q , t) = Z([(E p (Y /F q ), F * q )]) −1 . (5.0.4) Proposition 5.1. Let Y be a proper SNC (not necessarily log) scheme over F q . Let Y (i) (i ∈ Z ≥0 ) be the disjoint union of the (i + 1)-fold intersections of the irreducible components of Y . Then Z(Y /F q , t) = i,j≥0 det(1 − tF * q |H j rig (Y (i) /K 0 (F q ))) (−1) i+j+1 . (5.1.1) Proof. Let Y • be theČech diagram of an affine open covering of Y by finitely many affine open subschemes U j 's of Y . Set U (i) j := Y (i) j ∩ U j , Y (i) 0 := j U (i) j and Y (i) n := cosk Y (i) 0 (Y (i) 0 ) n (n ∈ N). Let Y • ⊂ −→ P • be a closed immersion into a formally smooth formal scheme over Spf(W(F q )). Then we have a closed immersion Y (i) 0 ⊂ −→ (i) P 0 , where (i) P 0 is a finite sum of P 0 which depends on i. Let δ j : Y (i+1) 0 −→ Y (i) 0 (0 ≤ j ≤ i) be the standard face morphism. Then we have a natural morphism ∆ j : (i+1) P 0 −→ (i) P 0 fitting into the following commutative diagram Y (i+1) 0 δj − −−− → Y (i) 0     (i+1) P 0 ∆j − −−− → (i) P 0 and satisfying the standard relations. Set P (i) • := cosk W(Fq) 0 ( (i) P 0 ). Let sp : ]Y (i) • [ P (i) • −→ Y (i) • be the specialization map. Then, as in [C, (2.3)], the following sequence 0 −→ sp * (Ω • ]Y•[P • ) −→ sp * (Ω • ]Y (0) • [ P (0) • ) −→ sp * (Ω • ]Y (1) • [ P (1) • ) −→ · · · is exact. Hence we have the following spectral sequence E ij 1 = H j rig (Y (i) /K 0 (F q )) =⇒ H i+j rig (Y /K 0 (F q )). (5.1.2) By (5.0.4) and this spectral sequence, we obtain the following formula: [(E p (Y /F q ), F * q )] = i,j≥0 (−1) i+j [H j rig (Y (i) /K 0 (F q )), F * q )] ∈ K(K 0 (F q )). (5.1.3) This formula implies (5.1.1). Corollary 5.2. Let X/F q be a non-smooth combinatorial K3 surface ( [Ku], [FS], [Nakk1]). (We do not assume that X has a log structure of simple normal crossing type.) Let m be the summation of the times of the processes of blowing downs making all irreducible components relatively minimal. Let M 1 (resp. M 2 ) be the cardinality of the irreducible components of • X whose relatively minimal models are P 2 Fq (resp. Hirzeburch surfaces =relatively minimal rational ruled surfaces). Let M be the cardinality of the irreducible components of • X. Then the following hold: (1) If X is of Type II with double elliptic curve E/F q , then (5.2.1) Z( • X/F q , t) = det(1 − qtF * q |H 1 rig (E/K 0 (F q ))) M−2 (1 − t)det(1 − tF * q |H 1 rig (E/K 0 (F q )))(1 − qt) M1+2M2+M−3+m (1 − q 2 t) M . (2) Assume that X is of Type III. Let d be the cardinality of the double curves of • X. Then (5.2.2) Z( • X/F q , t) = 1 (1 − t) 2 (1 − qt) M1+2M2+m−d (1 − q 2 t) M . Proof. First we give a remark on the rigid cohomology of a smooth projective rational surface S over F q . Set S F q n := S ⊗ Fq F q n and S Fq : = S ⊗ Fq F q . Let S min be a relatively minimal model of S F q . If S min ≃ P 2 Fq , we see that the motive H(S F q ) is as follows by [DMi,(6.12)]: H(S F q ) ≃ H(P 2 Fq ) ⊕ H(D)(−1), where D is the disjoint sum of 0-dimensional points. Since H 2 (P 2 Fq ) is isomorphic to a Tate-twist, if a natural number n is big enough, then (F * q ) n on H 2 rig (S F q n /K 0 (F q n )) is diag(q n , . . . , q n ). Hence the eigenvalues of (F * q ) n are q n (n ≫ 0) and thus the eigenvalues of F * q are q. If S min is isomorphic to a relatively minimal ruled surface over a smooth curve C over F q , the motive H(S F q ) is as follows by [DMi,(6.10), (6.12)]: H(S Fq ) ≃ H(C) ⊕ H(C)(−1) ⊕ H(D)(−1). Hence we see that F * q on H 2 rig (S/K 0 (F q )) is diag(q, . . . , q) as above. (1): It is easy to check that H i rig ( • X (0) /K 0 (F q )) =                K 0 (F q ) M (i = 0) H 1 rig (E/K 0 (F q )) ⊕M−2 (i = 1) K 0 (F q )(−1) M1+2M2+2(M−2)+m (i = 2) H 1 rig (E/K 0 (F q ))(−1) ⊕M−2 (i = 3) K 0 (F q )(−2) M (i = 4) and H i rig ( • X (1) /K 0 (F q )) =      K 0 (F q ) M−1 (i = 0) H 1 rig (E/K 0 (F q )) ⊕M−1 (i = 1) K 0 (F q )(−1) M−1 (i = 2). Now (5.2.1) follows from (5.1.1). (2): Let T be the cardinality of the triple points of • X. It is easy to check that H i rig ( • X (0) /K 0 (F q )) =                K 0 (F q ) M (i = 0) 0 (i = 1) K 0 (F q )(−1) M1+2M2+m (i = 2) 0 (i = 3) K 0 (F q )(−2) M (i = 4), H i rig ( • X (1) /K 0 (F q )) =      K 0 (F q ) d (i = 0) 0 (i = 1) K 0 (F q )(−1) d (i = 2) and H 0 rig ( • X (2) /K 0 (F q )) = K 0 (F q ) T . Because the dual graph of • X is a circle, M − d + T = χ(S 1 ) = 2. Now (5.2.2) follows from (5.1.1) Remark 5.3. If p = 2, we can prove that T is even (cf. [FS]). However we do not use this fact in this article. Corollary 5.4. Assume that p = 2. Let X/F q be a non-smooth combinatorial classical Enriques surface ( [Ku], [Nakk1]). Let M 1 , M 2 , M , m and d be as in (5.2). Then the following hold: (1) If X is of Type II with double elliptic curve E/F q , then (5.4.1) Z( • X/F q , t) = det(1 − qtF * q |H 1 rig (E/K 0 (F q ))) M−1 (1 − t)(1 − qt) M1+2M2+M−1+m (1 − q 2 t) M . (2) Assume that X is of Type III. Let d and T be the cardinalities of the double curves of • X and the triple points of • X, respectively. Then (5.4.2) Z( • X/F q , t) = 1 (1 − t)(1 − qt) M1+2M2+m−d (1 − q 2 t) M . Proof. (1): It is easy to check that H i rig ( • X (0) /K 0 (F q )) =                K 0 (F q ) M (i = 0) H 1 rig (E/K 0 (F q )) ⊕M−1 (i = 1) K 0 (F q )(−1) M1+2M2+2(M−1)+m (i = 2) H 1 rig (E/K 0 (F q ))(−1) ⊕M−1 (i = 3) K 0 (F q )(−2) M (i = 4) and H i rig ( • X (1) /K 0 (F q )) =      K 0 (F q ) M−1 (i = 0) H 1 rig (E/K 0 (F q )) ⊕M−1 (i = 1) K 0 (F q )(−1) M−1 (i = 2). (2): It is easy to check that H i rig ( • X (0) /K 0 (F q )) =                K 0 (F q ) M (i = 0) 0 (i = 1) K 0 (F q )(−1) M1+2M2+m (i = 2) 0 (i = 3) K 0 (F q )(−2) M (i = 4), H i rig ( • X (1) /K 0 (F q )) =      K 0 (F q ) d (i = 0), 0 (i = 1) K 0 (F q )(−1) d (i = 2) and H 0 rig ( • X (2) /K 0 (F q )) = K 0 (F q ) T . Because the dual graph of • X is P 2 (R), M − d + T = χ(P 2 (R)) = 1. Lastly we consider another type of local zeta functions. Let V be a complete discrete valuation ring of mixed characteristics with finite residue field F q and let K be the fraction field of V. Let Y be a proper smooth scheme over K of dimension d and let I be the inertia group of the absolute Galois group Gal(K/K). Then the zeta function of Y is defined as follows: Z(Y, t) := Π 2d i=0 det(1 − tσ|H i et (Y ⊗ K K, Q l ) I ) (−1) h+1 , where σ ∈ Gal(K/K) is a lift of the geometric Frobenius of Gal(F q /F q ) and l is a prime which is prime to q. If Y is the generic fiber of a proper semistable family Y over V with special fiber Y , then the following formula holds by [FuK] ( [I2]): Z(Y, t) = Π 2d h=0 det(1 − tσ|H h ket (Y , Q l ) I ) (−1) i+1 . Let X be a proper strict semistable family of surfaces over V with log special fiber X over s Fq . Then [Mo,(6.3.3)] tells us that Z(X K , t) can be described by the log crystalline cohomologies by the coincidence of the monodromy filtration and the weight filtration ( [Nakk3,(8.3)], [Mo,(6.2.4)]; however see [Nakk2,(11.15)] and [Nakk3,(7.1)].): Z(X K , t) = Π 4 i=0 det(1 − tF * q |(H i crys (X/W(s Fq )) K0(Fq) ) N =0 ) (−1) i+1 , where W(s Fq ) is the canonical lift of s Fq over W(F q ), H i crys (X/W(s Fq )) is the i-th log crystalline cohomology of X/W(F q ) and N : H i crys (X/W(s Fq )) K0(Fq) −→ H i crys (X/W(s Fq )) K0(Fq) (−1) is the p-adic monodromy operator. More generally, for a proper SNCL scheme Y /s Fq of pure dimension d, set (5.4.4) Let us recall the following result due to the author ( [Nakk7,(8.3)], (cf. [Mat,(2.2)], [CLa,(6.4)])): Z(H i (Y /K 0 (s Fq )), t) := det(1 − tF * q |(H i crys (X/W(s Fq )) K0(Fq) ) N =0 ) (−1) i+1 (5.4.3) and Z(Y /s Fq , t) := 2d i=0 Z(H i (Y /K 0 (s Fq )), t) (−1) i+1 . Theorem 5.5 ( [Nakk7,(8.3)]). Let κ be a perfect field of characteristic p > 0. Let X/s be an SNCL K3 surface. Let H i log (X) (i ∈ N) be the i-th log crystalline cohomology or the i-th Kummerétale cohomology of X/s. Then the following hold: (1) The ⋆-adic ( ⋆ = p, l ) monodromy filtration and the weight one on H i log (X) coincide. (2) The following hold: Proof. For the completeness of this article, we give the proof of (5.5). We give the proof of this theorem in the p-adic case because the proof in the l-adic case is the same as that in the p-adic case. Recall the following weight spectral sequence ( [Mo,3.23], [Nakk2,(2.0.1)]): E −k,i+k 1 = j≥max{−k,0} H i−2j−k rig ( • X (2j+k) /K 0 )(−j − k) =⇒ H i crys (X/W(s)) K0 . (5.5.1) (See [Nakk2] for the mistakes in [Mo].) Here we have used Berthelot's comparison isomorphism H i crys (Y /W) K0 = H i rig (Y /K 0 ) (i ∈ N) for a proper smooth scheme Y over κ. By [Nakk2,(3.6)] this spectral sequence degenerates at E 2 . (The l-adic analogue of this spectral sequence also degenerates at E 2 by Nakayama's theorem ( [Nak,(2.1)]).) (1): We may assume that κ is algebraically closed. If X/s is of Type I, there is nothing to prove. If X is of Type III, the double curves and the irreducible components are rational, and hence E 0,1 1 = E 1,1 1 = E 0,3 1 = E −1,3 1 = 0. By [Nakk1,(3.5) 3)], H 1 log-crys (X/W) = 0 and hence we have E −1,2 2 = 0. (Note that we also have the similar vanishing for the first Kummerétale cohomology of X by the vanishing above and the existence of the Q-structure of E −1,2 2 (cf. the proof of [Nakk3,(8.3)]). By taking the duality in [Nakk2,(10.5)], E 1,2 2 = 0. By [Mo,6.2.1] the p-adic monodromy operator N : H 2 crys (X/W(s)) −→ H 2 crys (X/W(s))(−1) induces an isomorphism N 2 : [Nakk1,(3.5) 3)] again, H 1 log-crys (X/W) = H 3 log-crys (X/W) = 0. Hence E ij 2 = 0 for i + j = 1, 3. Because N : H 2 crys (X/W(s)) −→ H 2 crys (X/W(s))(−1) induces an isomorphism E −1,3 2 ≃ −→ E 1,1 2 (−1) by [Mo,6.2.2], we have proved (1). E −2,4 2 ≃ −→ E 2,0 2 (−2) = K 0 . If X is of Type II, E −2,4 1 = E 2,0 1 = 0. By (2): (2) follows from (1) and the non-vanishings of E 1,1 2 in the Type II case and E 2,0 2 in the Type III case, respectively. Remark 5.6. The author has found the theorem (5.5) in December 1996 by using the p-adic weight spectral sequence (5.5.1). The key point of the proof is to notice to use the p-adic weight spectral sequence of X/s instead of the Clemens-Schmid exact sequence used in Kulikov's article [Ku]. (In fact, the complex analogue (5.7) below of (5.5) holds; this is a generalization of Kulikov's theorem in [loc. cit.] and the proof of (5.5) is simpler than that in [loc. cit.]. To my surprise, mathematicians who are working over C have not used the weight spectral sequence (5.7.1).) The author has finished writing the preprint [Nakk7] by 2000 at the latest (cf. [Nak,Remark 2.4 (3)]). However, after that, he has noticed that there are too many non-minor mistakes in theory of log de Rham-Witt complexes in Hyodo-Kato's article [HK] and Mokrane's article [Mo] as pointed out in [Nakk2]. Because he has used Hyodo-Kato's and Mokrane's theory in [Nakk7] heavily, he has to use their results in correct ways. However he has used his too much time for correcting their results in [Nakk2], he has no will to publish [Nakk7] now (because [Nakk7] is quite long and because he has to use more time for adding comments about Hyodo-Kato's and Mokrane's articles in [Nakk7]). For example, ν is in [Mo] is not a morphism of complexes, the left N in the diagram in [Mat,(2.2)] is incorrect. In [Mat,(2.2)] Matsumoto has proved (5.5) for semistable algebraic spaces of K3surfaces after looking at the proof in [Nakk7]. (See "Proof of p-adic case" in the proof of [Mat,Proposition 2.2].) Theorem 5.7 (cf. [Ku]). Let s be the log point of C. Let X/s be an analytic SNCL K3 surface. Let X ∞ be the base change of the Kato-Nakayama space X log of X ( [KN]) with respect to the morphism R ∋ x −→ exp(2π √ −1) ∈ S 1 . Let N : H i (X ∞ , Q) −→ H i (X ∞ , Q)(−1) (i ∈ N) be the monodromy operator constructed in [FN]. Then the following hold: (1) The weight filtration on H i (X ∞ , Q) constructed in [FN] coincide with the monodromy filtration on H i (X ∞ , Q) (2) The following hold: (a) X is of Type I if and only if N = 0 on H 2 (X ∞ , Q). (b) X is of Type II if and only if N = 0 and N 2 = 0 on H 2 (X ∞ , Q). (c) X is of Type III if and only if N 2 = 0 on H 2 (X ∞ , Q). Proof. By [Nakk3, (2.1.10)] we have the following weight spectral sequence: (5.7.1) E −k,h+k 1,∞ = j≥max{−k,0} H h−2j−k ( • X (2j+k+1) , Q)(−j − k) =⇒ H h (X ∞ , Q). By [Fr,(5.9)], if X is a combinatorial Type II or Type III K3 surface over C, then H 0 (X, Ω 1 X/C ) = 0. (Of course, if X is of Type I, then H 0 (X, Ω 1 X/C ) = 0 by Hodge symmetry.) Hence H 1 (X ∞ , C) = H 1 dR (X/C) = H 0 (X, Ω 1 X/s ) ⊕ H 1 (X, O X ) = 0. Here we have used the isomorphism between Steenbrink complexes A Q ⊗ Q C and A C of X and the isomorphism between A C and Ω • X/s ( [FN]). By the duality of the E 2 -terms of (5.7.1) ( [Nakk3,(5.15) (2)]) and the degeneration at E 2 of (5.7.1) (by Hodge theory), we obtain the vanishing of H 3 (X ∞ , C). The rest of the proof is the same as that of (5.5). Theorem 5.8 ([Nakk7, (15.1)]). Let X/s Fq be a projective SNCL K3 surface. Then the following hold: (1) Z(H i (X/K 0 (s Fq )), t) =      1 − t (i = 0) 1 (i = 1, 3) 1 − q 2 t (i = 4). (2) If X is of Type II with double elliptic curve E, then Z(H 2 (X/K 0 (s Fq )), t) = det(1 − tF * q |H 1 rig (E/K 0 ))(1 − qt) 18 . Consequently Z(X/s Fq , t) = 1 (1 − t)det(1 − tF * q |H 1 rig (E/K 0 ))(1 − qt) 18 (1 − q 2 t) . (3) If X is of Type III, then Z(H 2 (X/K 0 (s Fq )), t) = (1 − t)(1 − qt) 19 . Consequently Z(X/s Fq , t) = 1 (1 − t) 2 (1 − qt) 19 (1 − q 2 t) . Proof. By [Nakk1,(3.5)], H i crys (X/W(s Fq )) = 0 (i = 1, 3). Thus Z(H i (X/K 0 (s Fq )), t) = 1 (i = 1, 3). By [Nakk1,(6.9)], X is the log special fiber of a projective semistable family X over Spec(W (F q )). By [Nakk1,(6.10)], the generic fiber of X is a K3 surface. Hence, by Hyodo-Kato's isomorphism ( [HK,(5.1)]) (however see [Nakk2,§7] for incompleteness of the proof of Hyodo-Kato isomorphism), dim K0(Fq) H 2 crys (X/W(s Fq )) K0(Fq) = 22. (1): In this case, by (5.5), N = 0, N : E −1,2 2 −→ E 1,1 2 is an isomorphism, N 2 = 0 on H 2 crys (X/W (s Fq )) K0(Fq) and E −2,4 2 = E 2,0 2 = 0. Hence we have the following exact sequence by (5.5): 0 −→ E 1,1 2 −→ Ker(N ) −→ E 0,2 2 −→ 0. Because E 1,1 2 ≃ H 1 rig (E/K 0 (F q )), det(1 − tF * q |E 1,1 2 ) = det(1 − tF * q |H 1 rig (E/K 0 (F q )). On the other hand, E 0,2 2 is a subquotient of H 0 rig (X (2) /K 0 (F q ))(−1) ⊕ H 2 rig (X (1) /K 0 (F q )). Hence F * q on E 0,2 2 is diag(q, . . . , q) as shown in the proof of (5.2). Since Ker(N ) is 20-dimensional, we obtain (2). (2): In this case, by (5.5), N 2 = 0, N 3 = 0 and E −1,3 2 = E 1,1 2 = 0. Because N 2 : E −2,4 2 −→E 2,0 2 (−2) is an isomorphism, N : E 0,2 2 −→E 2,0 2 (−1) is surjective and hence the kernel of N is 20-dimensional. Obviously F * q = id on E 2,0 2 . As in (1), F * q on E 0,2 2 is diag(q, . . . , q). Hence we obtain (2). Theorem 5.9 ( [Nakk7,(15.2)]). Let X/s Fq be a projective non-smooth SNCL classical Enriques surface. Then Z(H i (X/K 0 (s Fq )), t) =          1 − t (i = 0), 1 (i = 1, 3), (1 − qt) 10 (i = 2) 1 − q 2 t (i = 4). Consequently Z(X/s Fq , t) = 1 (1 − t)(1 − qt) 10 (1 − q 2 t) . Proof. By [Nakk1, (7.1)], H i crys (X/W(s Fq )) = 0 (i = 1, 3) and hence Z(H i (X/K 0 (s Fq )), t) = 1 (i = 1, 3). By [Nakk1,(7.1)] and the argument in [Nakk1,(6.8), (6.11)], X is the log special fiber of a projective semistable family X over W(F q ) and the generic fiber of X is a classical Enriques surface. Hence dim K0(Fq) H 2 crys (X/W(s Fq )) K0(Fq) = 10. The rest of the proof is the same as that of (5.8) by noting that 0 = E −1,3 2 = E 11 2 = E 20 2 = E −2,4 2 , where E •• 2 's are E 2 -terms of the spectral sequence (5.5.1). A remark on Katsura and Van der Geer's result In this section we generalize the argument in the proof of (1.7) (3). First we recall the following theorem in [NY]. This is a generalization of Katsura and Van der Geer's theorem ([vGK1, (5.1), (5.2), (16.4)]). Theorem 6.1 ( [NY,(2.3)]). Let κ be a perfect field of characteristic p > 0. Let Y be a proper scheme over κ. (We do not assume that Y is smooth over κ.) Let q be a nonnegative integer. Assume that H q (Y, O Y ) ≃ κ, that H q+1 (Y, O Y ) = 0 and that Φ q Y /κ is pro-representable. Assume also that the Bockstein operator β : H q−1 (Y, O Y ) −→ H q (Y, W n−1 (O Y )) arising from the following exact sequence 0 −→ W n−1 (O Y ) V −→ W n (O Y )−→O Y −→ 0 is zero for any n ∈ Z ≥2 . Let V : Proposition 6.2. Let the notations be as in (6.1). Let D(κ) be the Cartier-Dieudonné algebra over κ. Then the following hold: (1) length W H q (Y, W n (O Y )) = n (n ∈ Z ≥1 ). (2) Set h := h(Φ q Y /κ ). Assume that h < ∞. Let us consider the following natural surjective morphism H q (Y, W(O Y )) −→ H q (Y, W h (O Y )). Then this morphism induces the following isomorphism H q (Y, W(O Y ))/p ∼ −→ H q (Y, W h (O Y )) (6.2.1) of D(κ)/p-modules. Proof. (1): By the assumptions we have the following exact sequence 0 −→ H q (Y, W n−1 (O Y )) V −→ H q (Y, W n (O Y )) −→ H q (Y, O Y ) −→ 0. (1) immediately follows from this. (2): First assume that h = 1. Then H q (Y, W(O Y )) ≃ W. In this case, (2) is obvious. Next assume that 1 < h < ∞. Set M n := H q (X, W n (O X )) (n ∈ Z ≥1 ) and M := H q (X, W(O X )). Consider the following exact sequence 0 −→ W m−n (O Y ) V n −→ W m (O Y ) −→ W n (O Y ) −→ 0 for m > n. By the assumption and (3.2.1) we see that H q+1 (Y, W m (O X )) = 0 for any m. Hence the natural morphism M m −→ M n is surjective and consequently the natural morphism M −→ M n is surjective. In particular, the natural morphism M −→ M h is surjective. Let η be an element of M h . We claim that pη = 0. We have to distinguish the operator F : M n −→ M n and the operator F : M n −→ M n−1 . The latter "F is equal to R n F , where R n : M n −→ M n−1 is the projection. We denote R n F by F n to distinguish two F 's. Since the following diagram (6.2.2) M h R h − −−− → M h−1 F     F M h R h − −−− → M h−1 is commutative, we have the following: pη = V F h (η) = V R h F (η) = V F R h (η). Since F = 0 on M h−1 by (6.1), the last term is equal to zero. Hence pη = 0. Consequently the natural morphism H (1), the morphism (6.2.1) is an isomorphism. Remark 6.3. Let the notations be as in (1.7) (2). By using only (6.2) for the case q = d, we can prove that (6.3.1) q (Y, W(O Y )) −→ H q (Y, W h (O Y )) factors through the projection H q (Y, W(O Y )) −→ H q (Y, W(O Y ))/p. Since the morphism (6.2.1) is surjective and dim κ H q (Y, W(O Y ))/p = h = dim κ H q (Y, W h (O Y )) by#Y (F q k ) ≡ 1 mod p [ ke+1 2 ] (k ∈ Z ≥1 ), where [ ] is the Gauss symbol. However the congruence (6.3.1) is not sharper than (1.7.2); only in the case h = 2, (6.3.1) is equivalent to (1.7.2). )) := H i (Y, W(I K0 )), where W(I K0 ) := Ker(W(O Z ) K0 −→ W(O Z /I) K0 ) and I is a coherent ideal sheaf of O Z for an open immersion Y ⊂ −→ Z into a proper scheme over κ such that V (I) = Z \ Y . They have proved that H i c (Y, W(O Y,K0 )) is independent of the choice of the closed immersion. By the definition of H i c (Y, W(O Y,K0 )), we have the following exact sequence Corollary 4. 4 . 4Let Y be as in (1.7). Let G be a finite group acting on Y /F q such that each orbit of G is contained in an affine open subscheme of Y . If #G is prime to p and the induced action on H d (Y, O Y ) is trivial, then h((Y /G)/F q ) = h(Y /F q ) and (1.7) for #(Y /G)(F q ) holds. Example 4. 5 . 5We give examples of trivial logarithmic cases. of Type II ([Nakk1, §3]), then X is F -split if and only if the isomorphic double elliptic curve is ordinary. In this case, h(X/κ) = 1. If this is not the case, of Type III ([loc. cit.]), then X is F -split and h(X/κ) = 1. See (5.2) below for the zeta function of these examples. By the formulas for the zeta function ((5.2.1) and (5.2.2)), we can easily verify that # • X(F q ) indeed satisfies the congruences (1.7.3) and (1.7.2). when x → ∞? (I do not know even whether lim x→∞ |α|≤22 P ′ (x; S/K, α) = ∞.) 5 Two kinds of zeta functions of degenerate SNCL schemes over the log point of F q In this section we give a few examples of two kinds of local zeta functions of a separated scheme Y of finite type over F q : one of them is defined by rational points of Y ; the other is defined by the Kummerétale cohomology of Y when Y is the underlying scheme of a proper log smooth scheme over the log point s Fq . First we introduce a Grothendieck group which is convenient in this section. Let F be a field. Consider a Grothendieck group K(F ) with the following generators and relations: the generators of K(F ) are [(V, β)]'s, where V is a finitedimensional vector space over F and β is an endomorphism of V over F . The relations are as follows: [(V, β)] = [(U, α)]+[(W, γ)] for a commutative diagram with exact rows have a natural map(5.0.1) det(1 − t • |•) : K(F ) −→ F (t) * ∩ (1 + tF [[t]]) *of abelian groups. 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{'abstract': 'In this article we give the definitions of log Fano varieties and log Calabi-Yau varieties in the framework of theory of log schemes of Fontain-Illusie-Kato and give congruences of the cardinalities of rational points of them over the log points of finite fields.', 'arxivid': '1902.00189', 'author': ['Yukiyoshi Nakkajima '], 'authoraffiliation': [], 'corpusid': 119618800, 'doi': '10.18642/jantaa_7100122080', 'github_urls': [], 'n_tokens_mistral': 29742, 'n_tokens_neox': 26599, 'n_words': 14624, 'pdfsha': 'f1eacb5d5e6f3ac9faa5ca64804f736f0fef3cae', 'pdfurls': ['https://arxiv.org/pdf/1902.00189v1.pdf'], 'title': ['Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields', 'Congruences of the cardinalities of rational points of log Fano varieties and log Calabi-Yau varieties over the log points of finite fields'], 'venue': []}

arxiv

On the Discrepancy of pp,pp Total Cross Sections at √ s = 1.8TeV between E710, E811 and CDF arXiv:hep-ph/0512135v1 11 Dec 2005 Keiji Igi Muneyuki Ishida Department of Physics School of Science and Engineering Meisei University 191-8506HinoTokyoJapan Theoretical Physics Laboratory RIKEN 351-0198WakoSaitamaJapan On the Discrepancy of pp,pp Total Cross Sections at √ s = 1.8TeV between E710, E811 and CDF arXiv:hep-ph/0512135v1 11 Dec 20051 Based on the previous approach, we have investigated a possibility to resolve the discrepancy between the E710, E811 and CDF at √ s = 1.8TeV, using the experimental data of the pp,pp total cross sections σ (+) tot and ρ (+) ratio up to the SPS experiments ( √ s = 0.9TeV) as inputs. We predict σp p tot and ρp p at the Tevatron energy ( √ s = 1.8TeV) as σp p tot = 75.9±1.0mb, ρp p = 0.136 ± 0.005.It turns out that only the data of E710 is consistent with the prediction in the one standard deviation. So we can conclude that E710 is preferable but we can exclude neither CDF nor E811 results. §1. Introduction Recently, 1) we have searched for the simultaneous best fit of the average ofpp, pp total cross sections( σ (+) tot ), and the ratio of the real to imaginary part of the forward scattering amplitude( ρ (+) ) for 70GeV < P lab < P large as inputs in terms of highenergy parameters c 0 , c 1 , c 2 and β P ′ constrained by the FESR with N (≃ 10GeV). Block and Halzen 2), 3) also reached to the similar conclusions independently based on duality in a different approach. We first chose P large = 2100GeV corresponding to the ISR region( √ s ≃ 60GeV ). Secondly we chose P large = 2×10 6 GeV corresponding to the Tevatron energy( √ s ≃ 2TeV ). We then predicted σ (+) tot and ρ (+) at the LHC and the high-energy cosmic-ray energy regions. It turned out that the prediction of σ (+) tot agrees with pp experimental data at the cosmic-ray regions 4)-6) within errors in the first case( ISR ). It has to be noted that the energy range of predicted σ (+) tot , ρ (+) is several orders of magnitude larger than the energy regions of σ (+) tot , ρ (+) input. If we use data up to Tevatron( the second case ), the situation has been much improved although there are some systematic uncertainty coming from discrepancy of the data between E710, 7) E811 8) and CDF 9) at √ s = 1.8TeV. 1) Finally we concluded that the precise measurements of σ pp tot in the coming LHC experiments will resolve this discrepancy at √ s = 1.8TeV. The purpose of this paper is to investigate a possibility to resolve this discrepancy using the experimental data of σ typeset using PTPT E X.cls Ver.0.9 §2. The general approach As in the previous paper, 1) let us first consider the crossing-even forward scattering amplitude defined by F (+) (ν) = fp p (ν) + f pp (ν) 2 with Im F (+) (ν) = k σ (+) tot (ν) 4π . (2 . 1) We also assume Im F (+) (ν) = Im R(ν) + Im F P ′ (ν) = ν M 2 c 0 + c 1 log ν M + c 2 log 2 ν M + β P ′ M ν M α P ′ (2 . 2) at high energies (ν > N ). It is to be noted that c 0 , c 1 , c 2 and β P ′ are dimensionless. We have defined the functions R(ν) and F P ′ (ν) by replacing µ by M in Eq. (3) of ref. 10). Here, M is the proton( anti-proton) mass and ν, k are the incident proton(anti-proton) energy, momentum in the laboratory system, respectively. Since the amplitude is crossing-even, we have R(ν) = iν 2M 2 2c 0 + c 2 π 2 + c 1 log e −iπ ν M + log ν M +c 2 log 2 e −iπ ν M + log 2 ν M , (2 . 3) F P ′ (ν) = − β P ′ M (e −iπ ν/M ) α P ′ + (ν/M ) α P ′ sinπα P ′ ,(2 . 4) and subsequently obtain Re R(ν) = πν 2M 2 c 1 + 2c 2 log ν M , (2 . 5) Re F P ′ (ν) = − β P ′ M ν M 0.5 ,(2 . 6) substituting α P ′ = 1 2 in Eq. (2 . 4). FESR: The FESR corresponding to n = 1 11), 12) is: M 0 νIm F (+) (ν)dν + 1 4π N 0 k 2 σ (+) tot (k)dk = N 0 νIm R(ν)dν + N 0 νIm F P ′ (ν)dν . (2 . 7) We call Eq. (2 . 7) as the FESR which we use in our analysis. The ρ (+) ratio: The ρ (+) ratio, the ratio of the real to imaginary part of F (+) (ν) was obtained from Eqs. (2 . 2), (2 . 5) and (2 . 6) as ρ (+) (ν) = Re F (+) (ν) Im F (+) (ν) = Re R(ν) + Re F P ′ (ν) Im R(ν) + Im F P ′ (ν) On the Discrepancy of pp,pp Total Cross Sections at √ s = 1.8TeV 3 = πν 2M 2 c 1 + 2c 2 log ν M − β P ′ M ν M 0.5 kσ (+) tot (ν) 4π . (2 . 8) Although the numerator of Eq. (2 . 8) becomes large for large values of ν, a real constant has to be introduced in principle since the dispersion relation for Re F (+) (ν) requires a single subtraction constant F (+) (0). 2), 13) So, we also add F (+) (0) in the numerator as ρ (+) (ν) = πν 2M 2 c 1 + 2c 2 log ν M − β P ′ M ν M 0.5 + F (+) (0) kσ (+) tot (ν) 4π . (2 . 9) As will be discussed in the Appendix, the introduction of this constant slightly modifies the value of ρ (+) (ν) although it will not affect the value of σ (+) tot . So, we use the Eq. (2 . 9) as the value of ρ (+) (ν) in this analysis. The FESR, Eq. (2 . 7), has some problem. i.e., there are the so-called unphysical regions coming from boson poles below thepp threshold. So, the contributions from unphysical regions of the first term of the right-hand side of Eq. (2 . 7) have to be calculated. These contributions can be estimated to be an order of 0.1% compared with the second term. 1) Thus, it can easily be neglected. Therefore, the FESR, the formula of σ 2)) and the ρ (+) ratio (Eq. (2 . 9)) are our starting points. Armed with the FESR, we express high-energy parameters c 0 , c 1 , c 2 , β P ′ in terms of the integral of total cross sections up to N . Using this FESR as a constraint for β P ′ = β P ′ (c 0 , c 1 , c 2 ), there are four independent parameters including F (+) (0). We then search for the simultaneous best fit to the data points of σ (+) tot (k) and ρ (+) (k) for 70GeV≤ k ≤ P large corresponding to the SPS energy (P large ≃ 0.43 × 10 6 GeV ( √ s = 0.9TeV)), to determine the values of c 0 , c 1 , c 2 and F (+) (0) giving the least χ 2 . We thus predict the σ tot and ρ (+) in the Tevatron energy region ( √ s = 1.8TeV). §3. Predictions for σ (+) tot and ρ (+) at √ s = 1.8TeV Using the data up to √ s = 0.9TeV ( SPS ), we predict σ tot (k) = 3403 ± 20GeV * ) for N = 10GeV in Eq. (2 . 7). The result of the fit is shown in Fig. 1. The values of parameters and resulting χ 2 are given in Tables I and II, respectively. Table I. The values of parameters in the best fit to the data up to SPS energy ( √ s = 0.9TeV) in the analysis 1(fit to the data in 70GeV < k < P large = 4.3 × 10 5 GeV ). The error estimations are done as follows: The c2 is fixed with a value deviated a little from the best-fit value, and then the χ 2 -fit is done by three parameters c0, c1 and F (+) (0), where β P ′ is represented by the other parameters through FESR(Eq. (3 . 1)). When the resulting χ 2 is larger than the least χ 2 of the four-parameter fit by one, the corresponding value of c2 gives one standard deviation. The higher and lower dot-dashed lines in Fig. 1 represent this deviation of c2. The errors of the other parameters are estimated through similar procedures. c2 c1 c0 β P ′ F (+) (0) Analysis 1 0.0466 ± 0.0047 −0.161 ∓ 0.078 6.27 ± 0.33 7.45 ∓ 0.51 12.65 ± 5.66 Table II. The values of χ 2 for the fit to data in 70GeV < k < P large = 4.3 × 10 5 GeV(Analysis 1): NF and Nσ(Nρ) are the degree of freedom and the number of σ (+) tot (ρ (+) ) data points in the fitted energy region. χ 2 /NF χ 2 σ /Nσ χ 2 ρ /Nρ Analysis 1 8.1/20 5.7/17 2.4/8 In terms of the best-fit values of parameters in Table I Table I.) The equation ( where we note that the difference between σp p tot and σ (+) tot is negligible at the relevant energy. It is worthwhile to notice that only the data of E710 7) is consistent with the prediction, Eq. Table I.) The corresponding values of parameters are (c2, c1, c0, β P ′ , F (+) (0)) = (0.0466 ± 0.0047, −0.161 ∓ 0.077, 6.27 ± 0.31, 7.45 ∓ 0.48, 12.65 ± 0.69). So we can conclude that E710 is preferable but we can exclude neither CDF nor E811 results. The predictions at LHC energy ( √ s = 14TeV) in terms of the best fit values of high-energy parameters in Table I are σ pp tot = 107.2 ± 2.8 mb, ρ pp = 0.128 ± 0.005 ,(3 . 4) where the errors correspond to one standard deviation of c 2 . We should note that Eq. (3 . 4) is consistent with the recent prediction by Block and Halzen, 3) σ pp tot = 107.3 ± 1.2 mb, ρ pp = 0.132 ± 0.001. An interesting observation: We can make the following interesting observation. We fitted the data for σ (+) tot and ρ (+) above 70GeV, as is shown by the arrow in the Fig. 1(a), Fig. 1(d) to predict higher-energy data. It is interesting to observe that the prediction of σ (+) tot are also in good agreement with experiments, even below 70GeV. The reason is as follows: The requirement of FESR, Eq. (2 . 7) is nearly equal to require that the theoretical value of σ (+) tot is nearly equal to the experimental value at the upper limit of the integral N = 10GeV since higher side of the integral is enhanced because of k 2 in the integral. Because of this observation, we can apply the same formula to fit the data in the lower energy region than in the analysis 1. Analysis 2: Data in 10GeV < k < P large = 4.3 × 10 5 GeV(4.54GeV < √ s < 0.9TeV ) are fitted through the same formula in the analysis 1. Additionally 15(2) data points are included in σ (+) tot (Re F (+) ). The result of the fit is shown in Fig. 2. The values of parameters and resulting χ 2 are given in Tables III and IV, respectively. Table III. The values of parameters in the best fit to the data up to the SPS energy ( √ s = 0.9TeV) in the analysis 2(fit to the data in 10GeV < k < P large = 4.3 × 10 5 GeV ). We obtain smaller error of F (+) (0) than in analysis 1 (Table I), since, as is seen in Eq. (2 . 9), F (+) (0) has sizable effects only in the low energy region. For errors, see the caption in Table I. Table IV. The values of χ 2 for the fit to data in 10GeV < k < P large = 4.3 × 10 5 GeV(Analysis 2). For NF and Nσ(Nρ), see the caption in Table. II. In §3, we have investigated a possibility to resolve the discrepancy between E710, E811 and CDF, using the experimental data of σ (+) tot and ρ (+) up to the SPS experiments ( √ s = 0.9TeV). χ 2 /NF χ 2 σ /Nσ χ 2 ρ /Nρ We came to the conclusion that only the data of E710 is consistent with the prediction, Eq. (3 . 2) in the one standard deviation although we can exclude neither CDF nor E811 results in the two standard deviations. In our previous paper, ref. 1) we concluded that the precise measurements of σ pp tot in the coming LHC measurements will resolve this discrepancy at √ s = 1.8TeV. It would still be worthwhile , however, to fix this problem in the CDF and D0 experiments, since these values play an important role to search for σ (+) tot and ρ (+) in the higher energy regions. Appendix A Reanalysis of our predictions at the LHC ( √ s=14TeV) with F (+) (0) parameter In our previous work, 1) we exploited the experimental data σ tot is almost the same as the previous one. The obtained values of parameters and the resulting χ 2 are given in Table V and Table VI, respectively. The fit to ρ (+) in the lower energy region is improved in comparison with the previous result, as can be seen in Fig. 3. Correspondingly much smaller χ 2 ρ is obtained in Table VI, which is compared with the previous values, χ 2 ρ =8.4(6.9) for fit 2(3). Table I.) The corresponding values of parameters are (c2, c1, c0, β P ′ , F (+) (0)) = (0.0479 ± 0.0037, −0.186 ∓ 0.056, 6.38 ± 0.21, 7.26 ∓ 0.31, 10.19 ± 0.31). Predicted values of σ (+) tot and ρ (+) at LHC energy( √ s=14TeV) and at cosmic-ray energy (P lab =5 × 10 20 eV) are given in Table VII. and ρ (+) at the LHC energy √ s = Ecm = 14TeV(P lab =1.04×10 8 GeV), and at a very high energy P lab = 5 · 10 20 eV ( √ s=Ecm=967TeV.) in the cosmic-ray region. The errors correspond to one standard deviation of c2. σ (+) tot ( √ s=14TeV) ρ (+) ( √ s=14TeV) σ (+) tot (P lab =5 · 10 20 eV) ρ (+) (P lab =5 · 10 20 eV) fit 2 104.2 ± 2.3mb 0.123 ± 0.004 191 ± 8mb 0.100 ± 0.003 fit 3 109.3 ± 2.4mb 0.130 ± 0.004 206 ± 8mb 0.105 ± 0.003 The predictions combining the two results in Table VII are σp p tot = 106.8 ± 5.1 syst ± 2.4 stat mb, ρp p = 0.127 ± 0.007 syst ± 0.004 stat σ pp tot = 198 ± 16 syst ± 8 stat mb, ρ pp = 0.103 ± 0.004 syst ± 0.003 stat (A . 1) at the LHC energy( √ s = E cm = 14TeV) and the cosmic-ray energy (P lab = 5 × Table I.) The corresponding values of parameters are given in Table V. ρ (+) up to the SPS experiments ( √ s = 0.9TeV). ρ (+) at the Tevatron energy ( √ s = 1.8TeV ). Analysis 1: As was explained in the general approach ( §2), both σ (+) tot and Re F (+) data in 70GeV < k < P large = 4.3 × 10 5 GeV( √ s = 0.9TeV ) are fitted simultaneously through the formula of σ (+) tot ( Eqs. (2 . 1) and (2 . 2)) and the ρ (+) ratio ( Eq. (2 . 9)) with the FESR ( Eq. (2 . 7)) as a constraint. The σ (+) tot (k) data points are obtained by averaging σp p tot and σ pp tot data points 14) when they are listed at the same value of k. For the details of data treatment of σ (+) tot and Re F (+) , see ref. 1). The FESR gives us 8.87 = c 0 + 2.04c 1 + 4.26c 2 + 0.367β P ′ 75.9 ± 1.0 mb, ρ (+) = 0.136 ± 0.005 , (3 . 2) where the errors correspond to the one standard deviation of c 2 , since the c 2 log 2 (ν/M )term in Eq. (2 . 2) is most relevant for predicting σ (+) tot in high energy region. (See the caption in (3 . 2) in the one standard deviation ( 72.8 + 3.1 = 75.9 ). If one tolerates two standard deviations, both CDF 9) (80.03 − 2.24 × 2 = 75.55) andE8118) (71.71 + 2.02 × 2 = 75.75) are consistent with the predictions Eq. (3 . 2). * ) This value is obtained by numerically integrating the experimental k 2 σ (+) tot = k 2 (σp p tot +σ pp tot )/2. See, ref. 1) for details.On the Discrepancy of pp,pp Total Cross Fig. 1 . 1Predictions for σ (+) and ρ (+) in terms of the Analysis 1. The fit is done for the data up to SPS energy, in the region 70GeV≤ k ≤ 4.3 × 10 5 GeV(11.5GeV ≤ √ s ≤ 0.9TeV) which is shown by the arrow. Total cross section σ(+) tot in (a) all energy region, versus log10P lab /GeV, (b) low energy region (up to ISR energy), versus P lab /GeV and (c) high energy (Tevatron-collider, LHC and cosmic-ray energy) region, versus center of mass energy Ecm in TeV unit. (d) gives the ρ (+) (= Re F (+) /Im F (+) ) in high energy region, versus Ecm in terms of TeV. The thin dot-dashed lines represent the one standard deviation of c2.(See the caption in 0479 ± 0.0037 −0.186 ∓ 0.057 6.38 ± 0.22 7.26 ∓ 0.33 10.19 ± 1.72 at LHC energy ( √ s = 14TeV) in terms of the best fit values of high-energy parameters inTable IIIare σ pp tot = 107.8 ± 2.4 mb, ρ pp = 0.129 ± 0.004 , errors correspond to the one standard deviation of c 2 . Essentially the same prediction are obtained as Eq. (3 . 4) of the analysis 1, although the errors are slightly smaller. Our result is stable independently of the choices of the fitting energy range. §4. Concluding remarks ρ (+) above P lab =70GeV up to Tevatron energy ( √ s = 1.8TeV) to predict σ (+) tot and ρ (+) in the LHC region, based on Eq. (2 . 8) of ρ (+) , not by Eq. (2 . 9). Although the effect of the parameter F (+) (0) in the new formula (Eq. (2 . 9)) is not large in the high energy region, we show the results of the analyses based on Eq. (2 . 9) here for completeness. Corresponding to ref. 1) two independent analyses are done: one includes the E710/E811 data at √ s=1.8TeV denoted as fit 2 in ref. 1), and the other includes the CDF datum of σ (+) tot at the same energy denoted as fit 3 in ref. 1). The results of the simultaneous fit to σ (+) tot and ρ (+) are compared with the previous results 1) in Fig. 3. The fit to ρ (+) is slightly improved in the lower energy region, while the result of σ (+) Fig. 2 . 2Predictions for σ (+) and ρ (+) in terms of the Analysis 2. The fit is done for the data up to SPS energy, in the region 10GeV≤ k ≤ 4.3×10 5 GeV(4.54GeV ≤ √ s ≤ 0.9TeV) which is shown by the arrow. For each figure, see the caption in Fig. 1. The thin dot-dashed lines represent the one standard deviation of c2.(See the caption in Fig. 3 . 3Predictions for σ (+) and ρ (+) compared with the previous results: The new results using F (+) (0) parameter are shown by right figures, (b) and (d), respectively, which are compared with the left figures, (a) and (c), of the previous analyses. 1) Predictions in terms of the fit 2(3) are shown by green(blue) lines, and the thin dot-dashed lines represent the one standard deviation of c2. (See the caption in Table V . VThe best-fit values of parameters in the fit 2 (fit up to Tevatron-collider energy includingE710/811 data) and fit 3 (including CDF datum). The errors here correspond to the one standard deviation of c2. (See the caption in Table I .) c2 c1 c0 β P ′ F (+) (0) fit 2 0.0424 ± 0.0041 −0.099 ∓ 0.069 6.04 ± 0.28 7.61 ∓ 1.55 12.48 ± 0.73 fit 3 0.0496 ± 0.0043 −0.205 ∓ 0.072 6.44 ± 0.29 7.20 ∓ 0.81 12.78 ± 0.72 Table VI . VIThe values of χ 2 for the fit 2 and fit 3. NF and Nσ(Nρ) are the degree of freedom and the number of σ(+) tot (ρ (+) ) data points in the fitted energy region. χ 2 /NF χ 2 σ /Nσ χ 2 ρ /Nρ fit 2 11.6/22 7.9/18 3.7/9 fit 3 10.9/22 8.7/18 2.1/9 √ s = 1.8TeV 9 Table VII . VIIThe predictions of σ(+) tot 10 20 eV), respectively. The above results are almost the same as the previous ones, Eq. (13) of ref. 1). Here we obtain fairly large systematic uncertainty again coming from the data treatment at the Tevatron-energy. . K Igi, M Ishida, Phys. Lett. B. 622286K. Igi and M. Ishida, Phys. Lett. B 622 (2005), 286 . M M Block, F Halzen, 036006: Erratum 039902Phys. Rev. D. 72M. M. Block and F. Halzen, Phys. Rev. D 72 (2005), 036006: Erratum 039902. . M M Block, F Halzen, hep-ph/0510238M. M. Block and F. Halzen, hep-ph/0510238. . M Honda, Akeno Collab.Phys. Rev. Lett. 70525M. Honda et al. (Akeno Collab.), Phys. Rev. Lett. 70 (1993), 525. Fly's Eye Collab. R M Baltrusaitis, Phys. Rev. Lett. 521380R. M.Baltrusaitis et al. 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{'abstract': 'Based on the previous approach, we have investigated a possibility to resolve the discrepancy between the E710, E811 and CDF at √ s = 1.8TeV, using the experimental data of the pp,pp total cross sections σ (+) tot and ρ (+) ratio up to the SPS experiments ( √ s = 0.9TeV) as inputs. We predict σp p tot and ρp p at the Tevatron energy ( √ s = 1.8TeV) as σp p tot = 75.9±1.0mb, ρp p = 0.136 ± 0.005.It turns out that only the data of E710 is consistent with the prediction in the one standard deviation. So we can conclude that E710 is preferable but we can exclude neither CDF nor E811 results.', 'arxivid': 'hep-ph/0512135', 'author': ['Keiji Igi ', 'Muneyuki Ishida \nDepartment of Physics\nSchool of Science and Engineering\nMeisei University\n191-8506HinoTokyoJapan\n', '\nTheoretical Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan\n'], 'authoraffiliation': ['Department of Physics\nSchool of Science and Engineering\nMeisei University\n191-8506HinoTokyoJapan', 'Theoretical Physics Laboratory\nRIKEN\n351-0198WakoSaitamaJapan'], 'corpusid': 119470666, 'doi': '10.1143/ptp.115.601', 'github_urls': [], 'n_tokens_mistral': 7628, 'n_tokens_neox': 6219, 'n_words': 3666, 'pdfsha': '0b9dcdb6d514d4a2d6d9af05cd375451bd7fa3bd', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ph/0512135v1.pdf'], 'title': ['On the Discrepancy of pp,pp Total Cross Sections at √ s = 1.8TeV between E710, E811 and CDF', 'On the Discrepancy of pp,pp Total Cross Sections at √ s = 1.8TeV between E710, E811 and CDF'], 'venue': []}

arxiv

Remarks on the Hardy type inequalities with remainder terms in the framework of equalities 11 Nov 2016 Shuji Machihara Tohru Ozawa Hidemitsu Wadade Remarks on the Hardy type inequalities with remainder terms in the framework of equalities 11 Nov 2016Dedicated to Professor Nakao Hayashi on the occasion of his sixtieth birthday We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.2010 Mathematics Subject Classification. 26D10, 46E35. §1. Introduction and the main results Let Ω be a domain in R n with n ≥ 3 and assume 0 ∈ Ω. The classical Hardy inequality states that the inequality n − 2 2 2 Ω |f | 2 |x| 2 dx ≤ Ω |∇f | 2 dx (1. 1) holds for all f ∈ H 1 0 (Ω), where the constant n−2 2 2 is best-possible. It is also well-known that the inequality (1.1) admits no nontrivial extremizers, and this fact implies a possibility for (1.1) to be improved by adding some remainder terms. In fact, the authors in [9] proved that the following improved Hardy inequality n − 2 2 2 Ω |f | 2 |x| 2 dx + Λ Ω |f | 2 dx ≤ Ω |∇f | 2 dx (1.2) holds for all f ∈ H 1 0 (Ω) provided that Ω is bounded, where the constant Λ in (1.2) is given by Λ = Λ(n, Ω) = z 2 0 ω 2 n n |Ω| − 2 n , and ω n and |Ω| denote the Lebesgue measures of the unit ball and Ω on R n , respectively, and the absolute constant z 0 denotes the first zero of the Bessel function J 0 (z). The constant Λ is optimal if Ω is a ball, but still the inequality (1.2) admits no nontrivial extremizers. More generally, the authors in [9] obtained the inequality for f ∈ H 1 0 (Ω), where 1 < p < 2n n−2 andΛ is a positive constant independent of u. Similar improvements have been done for the Hardy inequality not only in the L 2 -setting but in L p -setting with some remainder terms, see for instance [3,7,8,21,35]. Hardy type inequalities are known as useful mathematical tools in various fields such as real analysis, functional analysis, probability and partial differential equations. In fact, Hardy type inequalities and their improvements are applied in many contexts. For instance, Hardy type inequalities were utilized in investigating the stability of solutions of semi-linear elliptic and parabolic equations in [9,11]. As for the existence and asymptotic behavior of solutions of the heat equation involving singular potentials, see [10,35]. Among others we refer to [1,5,16,18,27,31] for the concrete applications of Hardy type inequalities. We also refer to [13,33] for a comprehensive understanding of Hardy type inequalities. Based on the historical remarks on the Hardy type inequalities, our purpose in this paper is to establish the classical Hardy inequalities in the frame work of equalities which immediately imply the Hardy inequalities by dropping the remainder terms. At the same time, those equalities characterize the form of the vanishing remainder terms. Our method on the basis of equalities presumably provides a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers. In what follows, we always assume Ω = R n and the standard L 2 (R n ) norm is denoted by · 2 . Then the Hardy type inequalities in L 2 -setting that we discuss in this paper are the following: f |x| 2 ≤ 2 n − 2 x |x| · ∇f 2 , n ≥ 3, (1.3) sup R>0 f − f R |x| n 2 log R |x| 2 ≤ 2 1 |x| n 2 −1 x |x| · ∇f 2 , n ≥ 2, (1.4) ∞ 0 x −p−1 x 0 f (y)dy 2 dx ≤ 2 p 2 ∞ 0 x −p+1 |f (x)| 2 dx, (1.5) ∞ 0 x p−1 ∞ x f (y)dy 2 dx ≤ 2 p 2 ∞ 0 x p+1 |f (x)| 2 dx, (1.6) where f R (x) = f R x |x| and p > 0. The inequalities (1.3), (1.5), and (1.6) are standard (see [19] for instance), while (1.4) is rather new (see [28,30]). In addition, as we noticed in [30], the logarithmic Hardy inequality (1.4) has a scaling property. We state our main theorems. We denote by ∂ r the radial derivative defined by ∂ r = x |x| · ∇ = n j=1 xj |x| ∂ j . The space D 1,2 (R n ) denotes the completion of C ∞ 0 (R n ) under the Dirichlet norm ∇ · 2 . Also the notation S n−1 denotes the unit sphere in R n endowed with the Lebesgue measure σ. Our first theorem now reads: Theorem 1. Let n ≥ 3. Then the equalities n − 2 2 2 f |x| 2 2 = ∂ r f 2 2 − ∂ r f + n − 2 2|x| f 2 2 (1.7) = ∂ r f 2 2 − |x| − n−2 2 ∂ r (|x| n−2 2 f ) 2 2 (1.8) hold for all f ∈ D 1,2 (R n ). Moreover, the second term in the right hand side of (1.7) or (1.8) vanishes if and only if f takes the form (1.9) f (x) = |x| − n−2 2 ϕ x |x| for some function ϕ : S n−1 → C, which makes the left hand side of (1.7) infinite unless S n−1 |ϕ(ω)| 2 dσ(ω) = 0: (1.10) |f | 2 |x| 2 = ϕ x |x| 2 |x| n / ∈ L 1 (R n ). We remark that as in (1.10), functions of the form (1.9) imply the nonexistence of nontrivial extremizers for (1.3). The corresponding integral diverges at both origin and infinity. A similar result to Theorem 1 can be found in [6,15]. However, the essential ideas for the proofs are different. Indeed, the proof in [6] is done by direct calculations with respect to the quotient with the optimizer of a Hardy type inequality. On the other hand, we shall prove Theorem 1 by applying an orthogonality argument in general Hilbert space settings. More precisely, an equality (1.11) n − 2 2 2 f |x| 2 2 = ∇f 2 2 − ∇f + n − 2 2 x |x| 2 f 2 2 has been observed in [6,15]. We should remark that (1.7) and (1.11) are the same for radially symmetric functions and are not the same for nonradial functions. In fact, the Dirichlet integral is decomposed into radial and spherical components as ∇f 2 2 = ∂ r f 2 2 + n j=1 ∂ j − x j |x| ∂ r f 2 2 . Next, we state the logarithmic Hardy type equalities in the critical weighted Sobolev spaces. Theorem 2. Let n ≥ 2. Then the equalities 1 4 f − f R |x| n 2 log R |x| 2 2 = 1 |x| n 2 −1 ∂ r f 2 2 − 1 |x| n 2 −1 ∂ r f + f − f R 2|x| log R |x| 2 2 (1.12) = 1 |x| n 2 −1 ∂ r f 2 2 − log R |x| 1 2 |x| n 2 −1 ∂ r   f − f R log R |x| 1 2   2 2 (1.13) hold for all R > 0 and all f ∈ L 1 loc (R n ) with 1 |x| n 2 −1 ∇f ∈ L 2 (R n ), where f R is defined by f R (x) = f R x |x| . Moreover, the second term in the right hand side of (1.12) or (1.13) vanishes if and only if f − f R takes the form (1.14) f (x) − f R (x) = log R |x| 1 2 ϕ x |x| for some function ϕ : S n−1 → C, which makes the left hand side of (1.12) infinite unless S n−1 |ϕ(ω)| 2 dσ(ω) = 0: (1.15) |f − f R | 2 |x| n log R |x| 2 = ϕ x |x| 2 |x| n log R |x| / ∈ L 1 (R n ). As in (1.15), functions of the form (1.14) imply the nonexistence of nontrivial extremizers for (1.4). The corresponding integral diverges at both origin and infinity and, in addition, on the sphere of radius R > 0. The final theorem in this paper is one-dimensional Hardy type equalities stated as follows. Theorem 3. Let n = 1 and p > 0. Then: (1) The equalities p 2 2 ∞ 0 x −p+1 1 x x 0 f (y)dy 2 dx (1.16) = ∞ 0 x −p+1 |f (x)| 2 dx − ∞ 0 x −p+1 f (x) − p 2x x 0 f (y)dy 2 dx = ∞ 0 x −p+1 |f (x)| 2 dx − ∞ 0 x d dx x − p 2 x 0 f (y)dy 2 dx hold for all f ∈ L 1 loc (0, ∞) with |x| −p+1 |f | 2 ∈ L 1 (0, ∞). Moreover, the second term in the right hand side of (1. 16 ) vanishes if and only if x 0 f (y)dy = cx p 2 for some c ∈ C, which makes the left hand side of (1.16) infinite unless c = 0: x −p+1 1 x x 0 f (y)dy 2 = |c| 2 x −1 / ∈ L 1 (0, ∞). (2) The equalities p 2 2 ∞ 0 x p+1 1 x ∞ x f (y)dy 2 dx (1.17) = ∞ 0 x p+1 |f (x)| 2 dx − ∞ 0 x p+1 f (x) − p 2x ∞ x f (y)dy 2 dx = ∞ 0 x p+1 |f (x)| 2 dx − ∞ 0 x d dx x p 2 ∞ x f (y)dy 2 dx hold for all f ∈ L 1 loc (0, ∞) with |x| p+1 |f | 2 ∈ L 1 (0, ∞). Moreover, the second term in the right hand side of (1.17) vanishes if and only if ∞ x f (y)dy = cx − p 2 for some c ∈ C, which makes the left hand side of (1.17) infinite unless c = 0: x p+1 1 x ∞ x f (y)dy 2 = |c| 2 x −1 / ∈ L 1 (0, ∞). In the recent paper [26], the authors actually extended the equalities in Theorem 2 with n = 2 in L n (R n )-settings. However, we shall re-prove Theorem 2 in this paper in order to compare the sub-critical case, the critical case and the one-dimensional case corresponding to Theorem 1, Theorem 2 and Theorem 3, respectively. The proofs of all theorems in this paper is essentially based on orthogonality in general Hilbert space settings. Therefore, there would be a possibility to establish other equalities on the functional spaces equipped with the Hilbert structure by applying our method. Concerning related references to Hardy type inequalities, we also refer to [9], [12], [14], [21] and [32]. Indeed, when a function f is radially symmetric on the unit ball at the origin, the corresponding inequality to (1.7) or (1.8) was obtained by [9, p.454]. In [21], authors established the Hardy inequalities with remainder terms on bounded domains by utilizing the argument used in [9]. The papers [12], [14] and [32] proposed simple proofs of classical Hardy, Rellich and Caffarelli-Kohn-Nirenberg inequalities. We study the Hardy type inequalities in a different perspective from those papers above in the sense that we derive the inequalities from equalities. We prove Theorems 1 -3 in subsequent sections. For simplicity, we prove the theorems for f ∈ C ∞ 0 (R n ; C). The proofs are completed by density (see [28,30]). The main idea of the proofs is given by the following "orthogonality lemma." Lemma 1.1. Let X be a scalar product space with scalar product (·|·). Let c > 0. Then the following statements are equivalent. (1) The equality (1.18) u 2 = −2cRe(u|v) holds for all u, v ∈ X. (2) The equality (1.19) u 2 = 4c 2 v 2 − u + 2cv 2 holds for all u, v ∈ X. Proof. The lemma follows by another equivalent equality: (1.20) Re(u|u + 2cv) = 0. Q.E.D. The standard proof of Hardy inequality is based on the inequality (1.21) u 2 ≤ 2c u v by the Cauchy-Schwarz inequality applied to (1.18). In this paper, we regard (1.18) as the orthogonality relation (1.20) of u and u + 2cv and then we regard 2cv as a difference: 2cv = (u + 2cv) − u. The equality (1.19) gives an explicit condition for the case of equality in (1.21). There are many papers on the Hardy type inequalities and related subjects. We refer the readers to [2]- [34] and references therein. §2. Proof of Theorem 1 We introduce polar coodinates (r, ω) = (|x|, x |x| ) ∈ (0, ∞) × S n−1 and we write the integral on the left hand side of (1.7) as R n |f (x)| 2 |x| 2 dx = ∞ 0 r n−3 S n−1 |f (rω)| 2 dσ(ω)dr = − 2 n − 2 Re ∞ 0 r n−2 S n−1 f (rω)ω · ∇f (rω)dσ(ω)dr = − 2 n − 2 Re R n f (x) |x| ∂ r f (x)dx, where we have carried out integration by parts in the radial variable. By Lemma 1.1 with c = 1 n−2 , u = f |x| , and v = ∂ r f , in the Hilbert space L 2 (R n ), we obtain (1.7) as (1.19). By a direct calculation, (1. Let R > 0. In the same way as in the proof of Theorem 1, we calculate |x|<R |f (x) − f R (x)| 2 |x| n log R |x| 2 dx = R 0 1 r log R r 2 S n−1 |f (rω) − f (Rω)| 2 dσ(ω)dr = −2Re R 0 1 log R r S n−1 (f (rω) − f (Rω))ω · ∇f (rω)dσ(ω)dr = −2Re |x|<R f (x) − f R (x) |x| n 2 log R |x| ∂ r f (x) |x| n 2 −1 dx, where we have carried out integration by parts in r and the contribution on the boundary vanishes since log R r = R r 1 dt t ≥ R r −1 R r = R−r R ≥ 0 and |f (rω) − f (Rω)| 2 ≤ ∇f 2 ∞ (R − r) 2 .f − f R |x| n 2 log R |x| 2 L 2 (B(0;R)) = ∂ r f |x| n 2 −1 2 L 2 (B(0;R)) − 1 |x| n 2 −1 ∂ r f + f − f R 2|x| log R |x| 2 L 2 (B(0;R)) . (3.2) Similarly, we obtain |x|>R |f (x) − f R (x)| 2 |x| n log R |x| 2 dx = ∞ R 1 r log R r 2 S n−1 |f (rω) − f (Rω)| 2 dσ(ω)dr = −2Re ∞ R 1 log R r S n−1 (f (rω) − f (Rω))ω · ∇f (rω)dσ(ω)dr = −2Re |x|>R f (x) − f R (x) |x| n 2 log R |x| ∂ r f (x) |x| n 2 −1 dx, where the contribution on the boundary vanishes due to (3.1) and log R r = log r R = r R 1 dt t ≥ r R − 1 r R = r − R r = |R − r| r . In the same way as in the derivation of (3.2), we obtain 1 4 f − f R |x| n 2 log R |x| 2 L 2 (R n \B(0;R)) = ∂ r f |x| n 2 −1 2 L 2 (R n \B(0;R)) − 1 |x| n 2 −1 ∂ r f + f − f R 2|x| log R |x| 2 L 2 (R n \B(0;R)) . (3.3) Then (1.12) follows by adding both sides of (3.2) and (3.3). By a direct calculation, (1.13) follows from (1.12), where we notice that ∂ r f R = 0. The rest of the statements follows in the same way as in the proof of Theorem 1. §4. Proof of Theorem 3 The integral on the left hand side of (1.16) is rewritten by integration by parts as ∞ 0 x −p+1 1 x x 0 f (y)dy 2 dx = ∞ 0 x −p−1 x 0 f (y)dy 2 dx = 2 p Re ∞ 0 x −p f (x) x 0 f (y)dydx = 2 p Re ∞ 0 x −p+1 1 x x 0 f (y)dy f (x)dx. By Lemma 1.1 with c = 1 p , u = 1 x x 0 f (y)dy, and v = −f , in the Hilbert space L 2 (0, ∞; x −p+1 dx), we obtain (1.16) as (1.19). Similarly, the integral on the left hand side of (1.17) is rewritten as ∞ 0 x p+1 1 x ∞ x f (y)dy 2 dx = ∞ 0 x p−1 ∞ x f (y)dy 2 dx = 2 p Re ∞ 0 x p f (x) ∞ x f (y)dydx = 2 p Re ∞ 0 x p+1 1 x ∞ x f (y)dy f (x)dx. By Lemma 1.1 with c = 1 p , u = 1 x ∞ x f (y)dy, and v = −f , in the Hilbert space L 2 (0, ∞; x p+1 dx), we obtain (1.17) as (1.19). The rest of the statements follows by a direct calculation. 8) follows from (1.7). The remainder terms vanish if and only if ∂ r (|x| is a function on the sphere. This completes the proof of Theorem 1. §3. 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Mitidieri, A simple approach to Hardy inequalities, Mathematical notes, 67 (2000), 563-572. Hardy-type inequalities. B Opic, A Kufner, Longman Scientific & Technical. B. Opic and A. Kufner, "Hardy-type inequalities," Longman Scientific & Technical, (1990). A simple proof of Hardy's inequality in a limiting case. F Takahashi, Arch. Math. (Basel). 104F. Takahashi, A simple proof of Hardy's inequality in a limiting case, Arch. Math. (Basel)., 104 (2015), 77-82. The Hardy inequality and the asymptotic behavior of the heat equation with and inverse-square potential. J L Vazquez, E Zuazua, J. Funct. Anal. 173J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with and inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. Saitama 338-8570, Japan E-mail address: machihar@mail. S Machihara, Department of Mathematics, Faculty of Science, Saitama UniversityS. Machihara : Department of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan E-mail address: machihar@mail.saitama-u.ac.jp . T Ozawa, txozawa@waseda.jpJapan E-mail addressDepartment of Applied Physics, Waseda UniversityT. Ozawa : Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan E-mail address: txozawa@waseda.jp . H Wadade, Kanazawa; JapanFaculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa UniversityH. Wadade : Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kanazawa, 920-1192, Japan

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{'abstract': 'We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers.2010 Mathematics Subject Classification. 26D10, 46E35.', 'arxivid': '1611.03580', 'author': ['Shuji Machihara ', 'Tohru Ozawa ', 'Hidemitsu Wadade '], 'authoraffiliation': [], 'corpusid': 119669536, 'doi': '10.2969/aspm/08110247', 'github_urls': [], 'n_tokens_mistral': 8904, 'n_tokens_neox': 7497, 'n_words': 4291, 'pdfsha': '8f5230aeee50157eea7f8849fc77559470d4e236', 'pdfurls': ['https://arxiv.org/pdf/1611.03580v1.pdf'], 'title': ['Remarks on the Hardy type inequalities with remainder terms in the framework of equalities', 'Remarks on the Hardy type inequalities with remainder terms in the framework of equalities'], 'venue': []}

arxiv

K Y Zhang Institute of Nuclear Physics and Chemistry CAEP 621900Mianyang, SichuanChina School of Physics State Key Laboratory of Nuclear Physics and Technology Peking University 100871BeijingChina P Papakonstantinou Rare Isotope Science Project Institute for Basic Science 34000DaejeonKorea M.-H Mun Department of Physics and Origin of Matter and Evolution of Galaxy Institute Soongsil University 06978SeoulKorea Y Kim Center for Exotic Nuclear Studies Institute for Basic Science 34126DaejeonKorea H Yan Institute of Nuclear Physics and Chemistry CAEP 621900Mianyang, SichuanChina Institute of Nuclear Physics and Chemistry Key Laboratory of Neutron Physics CAEP 621900Mianyang, SichuanChina X.-X Sun School of Nuclear Science and Technology University of Chinese Academy of Sciences 100049BeijingChina Institute of Theoretical Physics CAS Key Laboratory of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina arXiv:2305.02636v1 [nucl-th] 4 May 2023 (Dated: May 5, 2023) Collapse of the N = 28 shell closure in the newly discovered 39 Na and the development of deformed halos towards the neutron dripline Halos and changes of nuclear magicities have been extensively investigated in exotic nuclei during past decades. The newly discovered 39 Na with the neutron number N = 28 provides a new platform to explore such novel phenomena near the neutron dripline of the sodium isotopic chain. We study the shell property and the possible halo structure in 39 Na within the deformed relativistic Hartree-Bogoliubov theory in continuum. It is found that the lowering of 2p orbitals in the spherical limit results in the collapse of the N = 28 shell closure in 39 Na, and a well deformed ground state is established. The pairing correlations and the mixing of pf components driven by deformation lead to the occupation of weakly bound or continuum p-wave neutron orbitals. An oblate halo is therefore formed around the prolate core in 39,41 Na, making 39 Na a single nucleus with the coexistence of several exotic structures, including the quenched N = 28 shell closure, Borromean structure, deformed halo, and shape decoupling. The microscopic mechanisms behind the shape decoupling phenomenon and the development of halos towards dripline are revealed. Atomic nuclei cannot be made from arbitrary numbers of protons and neutrons. Their existence ends at the dripline, which marks a boundary of the nuclear territory. Mapping the dripline has always been a major goal of modern nuclear physics, as it is crucial for understanding the nuclear force and exploring the origin of elements [1]. The proton dripline has been experimentally delineated up to neptunium (atomic number Z = 93) [2]. The neutron dripline, however, is only known up to neon (Z = 10) [3], because the production cross sections for the most neutron-rich isotopes are extremely low. Recently, further efforts were dedicated to probe the neutron-rich limits beyond Z = 10 and nine events of sodium-39 ( 39 Na, Z = 11) were observed, while the existence of heavier sodium isotopes was not excluded, thus not determining its neutron dripline [4]. The nucleus 39 Na has a neutron number of N = 28, which is normally a magic number. The disappearance of traditional magic numbers and the appearance of new ones in exotic nuclei near the dripline have attracted a lot of attention [5][6][7][8][9][10][11][12]. It would be therefore interesting to investigate the shell property of 39 Na. Another novel phenomenon discovered in exotic nuclei is the halo [13], which also shares common interests in atomic and molecular physics [14]. Nuclear halo phenomenon is not rare in light nuclei near the dripline. As * Electronic address: sunxiangxiang@ucas.ac.cn shown in Fig. 1, it has been experimentally suggested in every isotopic chain from helium (Z = 2) to phosphorus (Z = 15) either on the neutron-rich or the proton-rich side [13,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], except sodium and silicon. The newly discovered 39 Na and the worldwide development of radioactive ion beam facilities are providing an excellent platform to study more halo nuclei and, in particular, to explore whether halos exist in neutron-rich sodium isotopes. Meanwhile, timely theoretical studies based on advanced nuclear models are desired to guide the forthcoming experiments. In this work, we study the shell property of 39 Na and explore possible halo structures in neutron-rich sodium isotopes by using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) [36][37][38][39][40][41]. Many successful applications of the DRHBc theory have been realized in the past dozen years, such as the study of halo nuclei 17,19 B [42, 43], 15,19,22 C [44, 45], 31 Ne [46], and 42,44 Mg [36,37,47], the investigation of deformation effects on the location of dripline [48], the prediction of stability peninsulas beyond the primary neutron dripline [49][50][51], the revelation of shape coexistence from light to heavy nuclei [52][53][54], and the exploration of rotational excitations of exotic nuclei through the combination with the angular momentum projection [55,56]. The details of the DRHBc theory can be found in Refs. [37,40]. Here the formalism is only briefly presented. In [13,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] as well as proton [30][31][32][33][34] halo nuclei/candidates are indicated by grey, olive/green, and orange/yellow colors, respectively. Here a confirmed halo nucleus means that both enhancement of the cross section and narrow momentum distribution are observed [35]. Otherwise it is treated as a candidate. Bogoliubov (RHB) equation reads [57] h D − λ ∆ −∆ * −h * D + λ U k V k = E k U k V k ,(1) where λ is the Fermi energy, and E k and (U k , V k ) T are the quasiparticle energy and wave function, respectively. h D is the Dirac Hamiltonian, h D (r) = α · p + V (r) + β[M + S(r)],(2) where S(r) and V (r) are the scalar and vector potentials, respectively. ∆ is the pairing potential, ∆(r 1 , r 2 ) = V pp (r 1 , r 2 )κ(r 1 , r 2 ),(3) with a density-dependent force of zero range, V pp (r 1 , r 2 ) = V 0 1 2 (1 − P σ )δ(r 1 − r 2 ) 1 − η ρ(r 1 ) ρ sat γ ,(4) and the pairing tensor κ [58]. In the pairing channel, 1 2 (1 − P σ ) is the projector for the spin-zero component, η = 0 corresponds to a volume pairing, η = 1 & γ = 1 corresponds to a surface pairing, and η = 0.5 & γ = 1 corresponds to a mixed one. A finite-range pairing force, such as the Gogny [59] or separable one [60], is also expected to be implemented in the future. The pairing tensor and various densities and potentials in coordinate space are expanded in terms of the Legendre polynomials, f (r) = λ f λ (r)P λ (cos θ), λ = 0, 2, 4, · · ·(5) The RHB equations (1) are solved using a Dirac Woods-Saxon basis [61,62], which has a wave function with a more appropriate asymptotic behavior compared to the commonly used harmonic oscillator basis and is suitable for the description of weakly bound nuclei. In Eq. (4), η = 1 and γ = 1, i.e., a surface pairing is adopted, which has been one of the usual choices in the study of nuclear halos [36, 37, 42-47, 59, 63-65]. The pairing strength V 0 = −325 MeV fm 3 , the saturation density ρ sat = 0.152 fm −3 , and a pairing window of 100 MeV are taken. This pairing reproduces well the odd-even mass differences for not only calcium isotopes in the same mass region of 39 Na, but also lead isotopes in the heavy mass region, and, thus, is suggested for the DRHBc mass table calculations [40]. For the Dirac Woods-Saxon basis, the energy cutoff E + cut = 300 MeV and the angular momentum cutoff J max = 23/2 are adopted. In Eq. (5), the Legendre expansion truncation is chosen as λ max = 6. The blocking effects in odd-mass or odd-odd Na isotopes are taken into account in the present DRHBc calculations via the equal filling approximation [38,41,66]. The above numerical details are the same as those used in the global DRHBc mass table calculations over the nuclear chart [40,41,67]. Our calculations are carried out with density functionals PC-PK1 [68], PC-F1 [69], NL3* [70], NL-SH [71], and PK1 [72]. In the calculated results, the binding energy of 38 Na is smaller than that of 37 Na, meaning that it is unstable against one-neutron emission. 39 Na is deformed in its ground state with a prolate deformation β 2 ≈ 0.45 and weakly bound with a two-neutron separation energy S 2n 1 MeV. The results are therefore in agreement with the experimental hints that 39 Na has a Borromean structure [3,4], i.e., it is a bound three-body ( 37 Na + n + n) system, even though no pair of its constituents is a bound system. In the PC-PK1 results, 41 Na is less deformed with β 2 = 0.37 and the last bound Na isotope, i.e., the neutron dripline, with S 2n = 0.49 MeV. The NL3* and PK1 results also suggest 41 Na as the neutron dripline, while PC-F1 and NL-SH support 39 Na. The accurate prediction of the dripline location is an ambitious goal in both nonrelativistic and relativistic density functional theories [67,[73][74][75], and it depends on the employed density functional and pairing interaction, which will not be discussed in detail in this work. Future experimental determination of the neutron drip line as well as measurement on, e.g., quadrupole deformations along the dripline, could be helpful in constraining and optimizing the density functionals and pairing interactions. Since PC-PK1 has been successfully used for a global description of nuclear ground-state properties [67,75], in the following we explore the possible exotic structure of 39 Na based on the PC-PK1 results. The large prolate deformation of 39 Na implies the collapse of the N = 28 shell closure. To study the shell structure of 39 Na, the evolution of single-neutron levels around the Fermi energy with the quadrupole deformation obtained from the constrained calculations is shown in Fig. 2. In the spherical limit, the orbitals 2p 3/2 and 1f 7/2 are nearly degenerate and close to the Fermi energy, while 2p 1/2 is just above them within 1 MeV and close to the particle emission threshold. In the traditional shell model [58], there is a considerable energy gap between 1f 7/2 and 2p 3/2 , forming the N = 28 shell closure and making the spherical shape energetically favored. From Fig. 2 it becomes clear that the lowering of 2p orbitals results in the collapse of the N = 28 shell closure in 39 Na. The stable quadrupole deformation of 39 Na could also be explained as a result of the nuclear Jahn-Teller effect [76], induced by the near-degeneracy of the pf orbitals in close proximity to the particle emission threshold. Such near-degeneracy has also been predicted in 39 Na and 40 Mg by relativistic continuum Hartree-Bogoliubov theory with NL-SH density functional [77] and the DRHBc theory with PC-F1 [56], respectively. In the relativistic Hartree-Bogoliubov calculations with DD-PC1 density functional [78], the single-neutron energy difference between 1f 7/2 and 2p 3/2 orbitals in the spherical limit is around 2 MeV for 40 Mg and around 3 MeV for 42 Si. In the relativistic Hartree-Fock calculations with DD-ME2 and PKA1 density functionals [79], such difference extracted from nuclear masses in an effective way is around 2 MeV for 42 Si. Our additional DRHBc calculations using NL3* and PK1 density functionals show that the 1f 7/2 -2p 3/2 energy difference is smaller than 1.0 MeV for 40 Mg and smaller than 2.5 MeV for 42 Si in the spherical limit, and both have deformed ground states with |β 2 | > 0.3, providing a consistent explanation for the suggested N = 28 shell quenching [9,11]. Another N = 28 isotone, 41 Al, may even exhibit triaxial deformation as predicted by the triaxial relativistic Hartree-Bogoliubov theory in continuum [80], and the 1f 7/2 -2p 3/2 energy difference in the spherical limit is around 1 MeV in the PC-PK1 result. Thus, the proximity of 2p 3/2 and 1f 7/2 orbitals in the spherical limit might be common in the covariant density functional theory for N = 28 isotones from 39 Na to 42 Si, which are very neutron-rich. Note that in stable nuclei the pseudospin symmetry would lead to the near-degeneracy of 2p 3/2 and 1f 5/2 orbitals [81]. In fact, the erosion of N = 28 shell closure near the neutron dripline for 10 ≤ Z ≤ 14 is also manifested in terms of the evolution of two-neutron separation energies and twoneutron gaps from the DRHBc mass table for even-even nuclei [67]. In axially deformed cases, each nlj orbital is split into 2j + 1 ones ( 2j+1 2 displayed in Fig. 2 because of Kramers' degeneracy) with quantum numbers Ω π , where π is the parity and Ω the projection of angular momentum on the symmetry axis. The pairing correlations and the pf component mixing driven by deformation lead to the partial occupation of the 1/2 − and 3/2 − orbitals with certain p-wave components. The valance neutrons with sor p-wave nature in weakly bound nuclei can tunnel far out into the classically forbidden region, and a diffuse neutron density distribution or possibly a neutron halo occurs [35,[82][83][84]. The lowered 2p states have been revealed to play a crucial role in the formation of halos close to the island of inversion, e.g., in 29 F [26,85,86] and 31 Ne [28,46]. Given its small two-neutron separation energy, a two-neutron halo might be expected in 39 Na, which was also indicated by the neutron density profiles obtained from the Skyrme-Hartree-Fock-Bogoliubov calculations [87,88]. To examine further the possible halo in 39 Na, the neutron density distributions along and perpendicular to the symmetry axis for neutron-rich odd-even sodium isotopes with N ≥ 20 are shown in Fig. 3. The ground-state de- formation of 31 Na is spherical due to the N = 20 shell closure, and 33,35,37 Na are well deformed with quadrupole deformation β 2 > 0.35. Therefore, the significant increase in density distribution along the symmetry axis from 31 Na to others shown in Fig. 3(a) can be understood from the deformation effects. From 33 Na to 41 Na, the neutron density distribution along the symmetry axis gradually becomes more diffuse with the increasing neutron number. In Fig. 3(b), similar gradual growth is seen in the neutron density distribution perpendicular to the symmetry axis from 31 Na to 37 Na, while remarkable increases at large r ⊥ are found for 39 Na and 41 Na, even though they are prolately deformed. This suggests that an oblate neutron halo that is mainly distributed perpendicular to the symmetry axis is formed around the prolate core in 39,41 Na. To understand the formation of the oblate halo, the root-mean-square (rms) radii and main spherical components of the single-neutron levels around the Fermi energy are shown in Fig. 4 for 39 Na. The levels are labelled by numbers according to the single-neutron energies in Fig. 4(a), and their quantum numbers are respectively 3/2 − , 1/2 − , 5/2 − , 1/2 − , and 3/2 − as seen in Fig. 2. The rms radii of levels 4 and 5 that are embedded in the continuum and the bound level 2 are notably larger than those of levels 1 and 3. This can be understood from the spherical components shown in Fig. 4(b). The considerable components of 2p 1/2 and 2p 3/2 account for the large rms radii of levels 2, 4, and 5. In contrast, the large centrifugal barrier of f -wave components hinders strongly the spatial extension of wave functions for levels 1 and 3. Note that level 2 is relatively deep bound with an energy ǫ ≈ −2 MeV and there are already nearly two valance neutrons occupying the levels above the Fermi energy. Thus level 2 would not contribute to the halo. Next we analyze the shape decoupling phenomenon. For level 4 with Ω π = 1/2 − , both spherical harmonic functions |Y 10 (θ, ϕ)| 2 and |Y 1±1 (θ, ϕ)| 2 from 2p states contribute according to the angular momentum coupling, and the latter dominates. For level 5 with Ω π = 3/2 − , only |Y 1±1 (θ, ϕ)| 2 contributes. The angular distribution of |Y 10 (θ, ϕ)| 2 ∝ cos 2 θ is prolate, while that of |Y 1±1 (θ, ϕ)| 2 ∝ sin 2 θ is oblate. Therefore, an oblate halo is expected for 39 Na. This also demonstrates that the deformation of a halo depends essentially on the quantum numbers of the halo orbitals and their main components. As shown in Fig. 3, a more pronounced neutron halo might develop in 41 Na. This is because the increase of neutron numbers occupying the halo orbitals 1/2 − and 3/2 − (labelled by Nos. 4 and 5 in Fig. 4) after adding two more neutrons; although the considerable 2p components give rise to the formation of the halo in 39 Na, the halo orbitals are only partially occupied as shown in Fig. 2, resulting in a limited number of p-wave neutrons. To see this intuitively, the number of p-wave neutrons in the single-neutron levels split from 2p and 1f 7/2 orbitals are shown in Fig. 5. This number is calculated by g. s. 2v 2 × |C p | 2 for each level, where v 2 is its occupation probability and C p s are the coefficients of its wave function on the p-wave Dirac Woods-Saxon bases. For the ground state of 39 Na, each halo orbital contributes less than 0.3 p-wave neutrons, summing to 0.56, about 30% of the valance neutrons. For 41 Na, the ground state is less deformed compared to 39 Na, which, together with the mean field change induced by adding neutrons, lowers the halo orbitals 1/2 − and 3/2 − . As a result, the two more neutrons in 41 Na make the halo orbitals more occupied and contribute more p-wave neutrons. It turns out that more than 1.2 p-wave neutrons are contributed, leading to a more prominent halo in 41 Na. In summary, the newly discovered 39 Na with the neutron number N = 28 is investigated within the deformed relativistic Hartree-Bogoliubov theory in continuum. Based on several relativistic density functionals, 39 Na is found to be well deformed in its ground state. From the single-neutron levels around the Fermi energy, it is revealed that the lowering of 2p 1/2 and 2p 3/2 orbitals in the spherical limit results in the collapse of the N = 28 shell closure in 39 Na. The pairing correlations and the pf component mixing driven by deformation lead to the partial occupation of the 1/2 − and 3/2 − orbitals with certain p-wave components, giving rise to the formation of a neutron halo. The density profiles of neutron-rich sodium isotopes suggest that an oblate halo is formed around the prolate core in 39,41 Na, adding them as new candidates of deformed halo nuclei with shape decoupling. A microscopic examination of the rms radii, the main components, and the neutron numbers occupying pwave orbitals unravels the mechanisms behind the shape decoupling phenomenon and the development of halos towards the dripline. The calculated p orbital percentage of ∼ 30% for the valance neutrons in 39 Na would be valuable for the spectroscopic factors obtained in future experiments. In addition, the two-neutron halo in 39 Na would belong to the category of "Borromean halos". This particular three-body dynamics would be relevant to its further study through different nuclear reactions, e.g., the electric dipole response of low-lying excitations. The measurements of reaction cross sections, core momentum distributions, and Coulomb dissociations would also be helpful in uncovering the mystery of 39 Na. Finally, the question of which observables are closely related to shape decoupling effects in deformed halos remains unanswered. It is anticipated that future precise measurements on the density distribution and scattering of hadronic probes may reveal signals that shed light on this novel phenomenon. Acknowledgments Fruitful discussions with members of the DRHBc Mass Table Collaboration are FIG. 2 : 2Single-neutron levels around the Fermi energy λn (dashed line) of 39 Na in the canonical basis from constrained calculations. Their quantum numbers nlj in the spherical limit and Ω π on the prolate side are labeled, where π is the parity and Ω the projection of angular momentum on the symmetry axis. The occupation probability v 2 is scaled by colors. The grey vertical line corresponds to the ground state (g. s.) of 39 Na. FIG. 3 : 3Neutron density distributions (a) along (θ = 0 • ) and (b) perpendicular to (θ = 90 • ) the symmetry axis for neutron-rich odd-even sodium isotopes 31,33,··· ,41 Na. In (b), r ⊥ = x 2 + y 2 . FIG. 4 : 4(a) The rms radius versus the energy ǫ for singleneutron levels around the Fermi energy λn in the canonical basis for 39 Na. The thickness of each level is proportional to its occupation probability. (b) The main spherical components of the single-neutron levels labelled by numbers 1-5 in (a). FIG. 5 : 5Same as Fig. 2 but only with 2p and 1f 7/2 orbitals.Here the occupation number of p-wave neutrons is scaled by colors. gratefully appreciated. This work was partly supported by the National Natural Science Foundation of China (Grant Nos. U2230207, U2030209, 11935003, 11875075, 11975031, 12141501, 12070131001, and 12205308), the National Key R&D Program of China (Grant Nos. 2020YFA0406001, 2020YFA0406002, and 2018YFA0404400), the China Postdoctoral Science Foundation (Grant No. 2022M713106), and the Institute for Basic Science (IBS-R031-D1). P. P. was supported by the Rare Isotope Science Project of Institute for Basic Science, funded by the Ministry of Science and ICT (MSICT), and National Research Foundation of Korea (2013M7A1A1075764). M.-H. M. was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean government (Ministry of Science and ICT) (Grant Nos. NRF-2021R1F1A1060066, NRF-2020R1A2C3006177, and NRF-2021R1A6A1A03043957). The results described in this paper were obtained using High-performance computing Platform of Peking University and the High-Performance Computing Cluster of the Institute of Theoretical Physics, Chinese Academy of Sciences. A portion of the computational resources were provided by the National Supercomputing Center including technical support (No. KSC-2022-CRE-0333). the DRHBc theory, the relativistic Hartree-45 Si 47 P 46 P 45 P 44 Si 43 Al Proton halo candidates Proton halo nuclei 41 Al 38 Mg 37 Mg 36 Mg 35 Mg 34 Mg 33 Mg 32 Mg 31 Mg 30 Mg 29 Mg 28 Mg 27 Mg 26 Mg 25 Mg 24 Mg 23 Mg 22 Mg 21 Mg 20 Mg 37 Na 35 Na 34 Na 33 Na 32 Na 31 Na 30 Na 29 Na 28 Na 27 Na 26 Na 25 Na 24 Na 23 Na 22 Na 21 Na 20 Na 34 Ne 32 Ne 31 Ne 30 Ne 29 Ne 28 Ne 27 Ne 26 Ne 25 Ne 24 Ne 23 Ne 22 Ne 21 Ne 20 Ne 19 Ne 18 Ne 17 Ne 31 F 29 F 27 F 26 F 25 F 24 F 23 F 22 F 21 F 20 F 19 F 18 F 17 F 24 O 23 O 22 O 21 O 20 O 19 O 18 O 17 O 16 O 15 O 14 O 13 O 23 N 22 N 21 N 20 N 19 N 18 N 17 N 16 N 15 N 14 N 13 N 12 N 22 C 20 C 19 C 18 C 17 C 16 C 15 C 14 C 13 C 12 C 11 C 10 C 9 C 19 B 17 B 15 B 14 B 13 B 12 B 11 B 10 B 8 B 14 Be 11 Li 9 Li 8 Li 7 Li 6 Li 4 He 3 He 8 He 40 Mg 6 He 7 Be 9 Be 10 Be 11 Be 12 Be 40 Al 39 Al 38 Al 37 Al 36 Al 35 Al 34 Al 33 Al 32 Al 31 Al 30 Al 29 Al 28 Al 27 Al 26 Al 25 Al 24 Al 23 Al 22 Al 42 Al 42 Si 41 Si 40 Si 39 Si 38 Si 37 Si 36 Si 35 Si 34 Si 33 Si 32 Si 31 Si 30 Si 29 Si 28 Si 27 Si 26 Si 25 Si 24 Si 23 Si 43 Si 26 P 27 P 28 P 29 P 30 P 31 P 32 P 33 P 34 P 35 P 36 P 37 P 38 P 39 P 40 P 41 P 42 P 43 P 44 P Stable nuclei 39 Na newly discovered Neutron halo nuclei Neutron halo candidates FIG. 1: Experimentally known nuclear landscape from helium to phosphorus, where stable nuclei and experimentally con- firmed/suggested neutron M Thoennessen, The Discovery of Isotopes. 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J Meng, I Tanihata, S Yamaji, Phys. Lett. B. 4191J. Meng, I. Tanihata, and S. Yamaji, Phys. Lett. B 419, 1 (1998). . Z P Li, J M Yao, D Vretenar, T Nikšić, H Chen, J Meng, Phys. Rev. C. 8454304Z. P. Li, J. M. Yao, D. Vretenar, T. Nikšić, H. Chen, and J. Meng, Phys. Rev. C 84, 054304 (2011). . M Moreno-Torres, M Grasso, H Liang, V De Donno, M Anguiano, N Van Giai, Phys. Rev. C. 8164327M. Moreno-Torres, M. Grasso, H. Liang, V. De Donno, M. Anguiano, and N. Van Giai, Phys. Rev. C 81, 064327 (2010). . K Y Zhang, S Q Zhang, J Meng, arXiv:2212.05703nucl-thK. Y. Zhang, S. Q. Zhang, and J. Meng, arXiv:2212.05703 [nucl-th] . . H Liang, J Meng, S.-G Zhou, Phys. Rep. 5701H. Liang, J. Meng, and S.-G. Zhou, Phys. Rep. 570, 1 (2015). . P G Hansen, A S Jensen, B Jonson, Annu. Rev. Nuc. Part. Sci. 45591P. G. Hansen, A. S. Jensen, and B. Jonson, Annu. Rev. Nuc. Part. Sci. 45, 591 (1995). . J Meng, H Toki, S G Zhou, S Q Zhang, W H Long, L S Geng, Prog. Part. Nucl. Phys. 57470J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, and L. S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006). . J Meng, S G Zhou, J. Phys. G. 4293101J. Meng and S. G. Zhou, J. Phys. G 42, 093101 (2015). . J Casal, J Singh, L Fortunato, W Horiuchi, A Vitturi, Phys. Rev. C. 10264627J. Casal, J. Singh, L. Fortunato, W. Horiuchi, and A. Vitturi, Phys. Rev. C 102, 064627 (2020). . K Fossez, J Rotureau, Phys. Rev. C. 10634312K. Fossez and J. Rotureau, Phys. Rev. C 106, 034312 (2022). . Q Z Chai, J C Pei, N Fei, D W Guan, Phys. Rev. C. 10214312Q. Z. Chai, J. C. Pei, N. Fei, and D. W. Guan, Phys. Rev. C 102, 014312 (2020). . Q Chai, H Chen, M Zha, J Pei, F Xu, 10.3390/sym14020215Symmetry. 14Q. Chai, H. Chen, M. Zha, J. Pei, and F. Xu, Symmetry 14 (2022), 10.3390/sym14020215.

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{'abstract': 'Collapse of the N = 28 shell closure in the newly discovered 39 Na and the development of deformed halos towards the neutron dripline', 'arxivid': '2305.02636', 'author': ['K Y Zhang \nInstitute of Nuclear Physics and Chemistry\nCAEP\n621900Mianyang, SichuanChina\n\nSchool of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina\n', 'P Papakonstantinou \nRare Isotope Science Project\nInstitute for Basic Science\n34000DaejeonKorea\n', 'M.-H Mun \nDepartment of Physics and Origin of Matter and Evolution of Galaxy Institute\nSoongsil University\n06978SeoulKorea\n', 'Y Kim \nCenter for Exotic Nuclear Studies\nInstitute for Basic Science\n34126DaejeonKorea\n', 'H Yan \nInstitute of Nuclear Physics and Chemistry\nCAEP\n621900Mianyang, SichuanChina\n\nInstitute of Nuclear Physics and Chemistry\nKey Laboratory of Neutron Physics\nCAEP\n621900Mianyang, SichuanChina\n', 'X.-X Sun \nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nInstitute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n'], 'authoraffiliation': ['Institute of Nuclear Physics and Chemistry\nCAEP\n621900Mianyang, SichuanChina', 'School of Physics\nState Key Laboratory of Nuclear Physics and Technology\nPeking University\n100871BeijingChina', 'Rare Isotope Science Project\nInstitute for Basic Science\n34000DaejeonKorea', 'Department of Physics and Origin of Matter and Evolution of Galaxy Institute\nSoongsil University\n06978SeoulKorea', 'Center for Exotic Nuclear Studies\nInstitute for Basic Science\n34126DaejeonKorea', 'Institute of Nuclear Physics and Chemistry\nCAEP\n621900Mianyang, SichuanChina', 'Institute of Nuclear Physics and Chemistry\nKey Laboratory of Neutron Physics\nCAEP\n621900Mianyang, SichuanChina', 'School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina', 'Institute of Theoretical Physics\nCAS Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina'], 'corpusid': 258480022, 'doi': '10.1103/physrevc.107.l041303', 'github_urls': [], 'n_tokens_mistral': 25999, 'n_tokens_neox': 21874, 'n_words': 10481, 'pdfsha': '6f6d97d189e125b0d0e1c596a48f15ff4cb7f052', 'pdfurls': ['https://export.arxiv.org/pdf/2305.02636v1.pdf'], 'title': [], 'venue': []}

arxiv

Shock Waves and the Vacuum Structure of Gauge Theories 1 October 1994 Maurizio Martellini Dipartimento di Fisica Dipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria Dipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica Università di Milano Via Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY Augusto Sagnotti Dipartimento di Fisica Dipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria Dipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica Università di Milano Via Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY Mauro Zeni Dipartimento di Fisica Dipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria Dipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica Università di Milano Via Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY Shock Waves and the Vacuum Structure of Gauge Theories 1 October 1994arXiv:hep-th/9411088v2 9 Dec 1994 1 Contribution to "Quark Confinement and the Hadron Spectrum", Como 20-24 June 1994. In Yang-Mills theory massless point sources lead naturally to shock-wave configurations. Their magnetic counterparts endow the vacuum of the four-dimensional compact abelian model with a Coulomb-gas behaviour whose physical implications are briefly discussed. The current semiclassical picture of the vacuum in gauge theories rests, to a large extent, on the known solutions of the Yang-Mills field equations [1]. It is common wisdom that attaining a detailed understanding of this vacuum is a major challenge for Quantum Field Theory, as well as a crucial step in assessing its actual role in High-Energy Physics. In this talk we consider a class of rather simple shock-wave solutions of the field equations with massless point sources. In the compact abelian gauge theory, their euclidean counterparts exhibit rather neatly a phase transition [2], thus providing a simple explicit realization of the standard picture [3]. Let us begin by considering the Yang-Mills field equation D µ F µν = 4πj ν with the massless point source j a ν = q I a δ u ν δ(u) δ 2 (r) ,(1) where u and v are light-cone coordinates (u = x 0 −x 3 √ 2 ; v = x 0 +x 3 √ 2 ) , r is a space-like coordinate vector orthogonal to u and v, and I a labels the color charge of the point particle. The resulting classical solution, A a µ = − 2 q I a δ u µ δ(u) log r ,(2) where r denotes the length of r, may be obtained by inspection, or by a simple extension of the "cut and paste" procedure [4] used to generate a similar type of gravitational wave. The electric and magnetic field strengths associated to eq. (2), E i = √ 2 q I a δ(u) r i r 2 ,(3)B i = − √ 2 q I a δ(u) ε ij r j r 2 (i = 1, 2) , may also be obtained as singular boosts of a static Coulomb field in the limit where the velocity v/c → 1 [5]. The "two-dimensional" shape of the shock wave applies to arbitrary superpositions of comoving massless point-like currents and may be ascribed to the relativistic contraction of the fields in the longitudinal direction. In the quantum theory, the topological character of the field configuration space plays a crucial role in determining the nature of the correct degrees of freedom. In the prototype example for this type of phenomenon, the 2D XY model, the fundamental field is an angular variable θ(r) and the naive elementary excitations, long-wavelength spin waves, must be supplemented with genuinely new ones, vortices. From the mathematical viewpoint, vortices are singular field configurations that result in effective (quantized) violations of the "Bianchi identity" for dθ. These excitations are responsible for the Kosterlitz-Thouless (KT ) phase transition [6] that separates the weak and strong coupling regimes of the model. Vortices are actually relevant in a number of different models with periodic field configuration spaces, most notably in lattice gauge theories, where they may be associated with magnetic monopoles. Again, in continuum formulations they manifest themselves as singular field configurations that violate the relevant "Bianchi identities". A number of reasons call for a periodic formulation of abelian gauge theories, most notably the fact that in unified models abelian gauge fields typically emerge from the spontaneous breaking of non-abelian gauge symmetries [3]. For compact 3D QED Polyakov [7] has shown that the periodicity results in a vacuum filled with a plasma of monopoles and in a mass gap for the gauge fields, as well as in a confinement phenomenon at all scales. For compact 4D QED a quantitative discussion of the role of vortices has been hampered by the string-like nature of the corresponding excitations, monopole current loops. Still, the basic argument is rather simple. It is best exhibited starting from the Villain [8] form of the action S = 1 4g 2 x,αβ (F αβ (x) + 2πn αβ (x)) 2 ,(4) where the gauge potential is an angular variable and the n αβ (x) are matrices of integers associated to the lattice plaquettes that may be related to integer-valued currents upon integration over elementary cubes ✷ µ of the lattice, ∂✷ µ n αβ (x) = j µ .(5) The Berezinsky decomposition n(x) αβ = ∂ α n β − ∂ β n α + ε αβγδ ∂ γ ϕ δ exhibits the longitudinal part of n, as well as the vector field ϕ µ , whose source is the current of eq. (5) and whose (dual) field strength adds to the usual one. Since the total field strength F ′ αβ = F αβ + 2πn αβ = F αβ + ... + 1 2 ε αβγδ F γδ(6) violates the Bianchi identity of the original gauge field, the current in eq. (5) is of magnetic origin. An elegant lattice description of this model, allowing for electric current (Wilson) loops as well, was presented in [12]. The resulting picture involves interacting monopole and charge loops, but it is difficult to turn it into a quantitative analysis, since in this case the effective sine-Gordon description of 3D QED [7] should be replaced by a theory of monopole loops. Although a proper description of this theory is likely to be complicated, the intuition gained from ordinary string theory [9] suggests a possible way of gaining quantitative insight into the problem. This may be associated to "straight" current loops, and conceivably to massless ones near the phase transition, that according to numerical estimates appears to be of second order [2]. In the remainder of this talk we would like to show how the truncation of the monopole strings to these "zero modes" accounts both for the phase transition and for Wilson's area law [11] in a rather neat fashion. To this end, we need the duals of the fields in eq. (3). They may be derived from a "magnetic" analogue of the vector potential of eq. (2) or, alternatively, from the ordinary potential A µ = (0, 0, A i ) , where A i = q (m) θ(u) ε ij r j r 2 (i, j = 1, 2) ,(7) whose Dirac string B 3 = 2πq (m) θ(u) δ 2 (r)(8) would be ineffective in eq. (4) upon suitable quantization of the monopole charge, a single Dirac quantum corresponding to q (m) ± 1. Interestingly, the potential in eq. (7) plays an important role in knot theory [10]. In computing the action for a pair of waves, it is convenient to resort to a covariant notation, that has the further advantage of illuminating the geometry of the space-like planes where the interactions take place. A "straight" current in a generic direction may be written j µ (x) = 1 2 q (m) k µ dτ δ 4 [x α − x α 0 − k α τ ] ,(9) where in the massless case k µ is a null vector. The total action for a pair (ij) of shock waves is then S ij = πq (m)i q (m)j g 2 log(x ij Π x ij ) ,(10) where x ij is the distance between the two lines and Π is a projector onto the space-like plane orthogonal to the two wave vectors. This result essentially holds in the Wick rotated case as well, where the calculation requires a suitable extension of δ(x) to the complex plane, rather interesting in its own right. The logarithmic interaction closely parallels the state of affairs for the XY model and is just enough to compete with the point-like entropy of these configurations, thus displaying a KT -like phase transition, while the divergent self interactions require globally neutral sets of monopole currents. Interestingly, a transition of this type would be predicted by the Migdal-Kadanoff approximation [13], known to become less accurate as the space dimensionality is increased, consistenly with our neglect of higher extended excitations. The naive estimate of the transition temperature from eq. (10) is rather amusing, since it yields 2g 2 c = π/2 for a pair of fundamental Dirac monopoles with a KT -like measure d 4 x, to be compared with the loop-space estimate for the full gas of monopole strings [12][2], 2g 2 c ≈ 1.57! Finally, the area law [11] may be anticipated by comparing the effective KT structure of our "straight-line" vacuum to the Coulomb-gas picture of the three-dimensional model of ref. [7]. One may then arrive at an effective Sine-Gordon dynamics to infer that, above the critical temperature T c , double layers of monopole lines form around the Wilson loop, thus implying the area law and confinement. One may also envisage a similar analysis of black-hole physics [4], given the formal similarity between the logarithm of the Wilson loop and the Bekenstein entropy. More details, including the explicit invariant measure for the moduli, will be presented elsewhere [14]. AcknowledgementsIt is a pleasure to acknowledge stimulating discussions with T. Banks and A. Di Giacomo. We also benefitted from the kind hospitality of the Physics Departments of the University of Milan and of the University of Rome "Tor Vergata" and of the Centre For a review, see A. Actor. Rev. Mod. Phys. 51461For a review, see A. Actor, Rev. Mod. Phys. 51 (1979) 461. J B For A Review, Kogut, Recent Advances in Field Theory and Statistical Mechanics, Proc. Les Houches. J.B. Zuber and R. StoraNorth-Holland; AmsterdamFor a review, see J.B. Kogut, in Recent Advances in Field Theory and Statistical Mechanics, Proc. Les Houches 1982, eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam 1984). G Hooft, Proc. 1975 EPS Conference. A. Zichichi (Compositori1975 EPS ConferenceBolognaG. 't Hooft, Proc. 1975 EPS Conference, ed. A. Zichichi (Compositori, Bologna, 1976); . S Mandelstam, Phys. Rep. 23245S. Mandelstam, Phys. Rep. 23C (1976) 245; A M Polyakov, Gauge Fields and Strings. New YorkHarwoodA.M. Polyakov, Gauge Fields and Strings (Harwood, New York, 1987). . T Dray, G Hooft, Nucl. Phys. 253173T. Dray and G. 't Hooft, Nucl. Phys. B253 (1985) 173. . R Jackiw, D Kabat, M Ortiz, Phys. Lett. 277148R. Jackiw, D. Kabat and M. Ortiz, Phys. Lett. B277 (1992) 148. . J M Kosterlitz, D J Thouless, J. Phys. 61181J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181; . J M Kosterlitz, J. Phys. 71046J.M. Kosterlitz, J. Phys. C7 (1974) 1046. . A M Polyakov, Phys. Lett. 5979A.M. Polyakov, Phys. Lett. B59 (1975) 79. . J. Villain, J. Physique. 36581J. Villain, J. Physique 36 (1975) 581. M B Green, J H Schwarz, E Witten, Superstring Theory. CambridgeCambridge University PressM.B. Green, J.H. Schwarz and E. Witten, Superstring Theory (Cambridge University Press, Cambridge, 1987). . T Khono, Adv. Studies in Pure Math. 16255T. Khono, Adv. Studies in Pure Math. 16 (1988) 255; . E Guadagnini, M Martellini E, M Mintchev, Nucl. Phys. 336581E. Guadagnini, M. Martellini e M. Mintchev, Nucl. Phys. B336 (1990) 581. . K G Wilson, Phys. Rev. 102445K.G. Wilson, Phys. Rev. D10 (1974) 2445. . T Banks, R Myerson, J Kogut, Nucl. Phys. 129493T. Banks, R. Myerson and J. Kogut, Nucl. Phys. B129 (1977) 493; . A Ukawa, P Windey, A H Guth, Phys. Rev. 211013A. Ukawa, P. Windey and A.H. Guth, Phys. Rev. D21 (1980) 1013. . A A , Sov. Phys. JETP. 42413A.A. Migdal, Sov. Phys. JETP 42 (1975) 413; . L P Kadanoff, Rev. Mod. Phys. 49267L.P. Kadanoff, Rev. Mod. Phys. 49 (1977) 267. . M Martellini, A Sagnotti, M Zeni, in preparationM. Martellini, A. Sagnotti and M. Zeni, in preparation.

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{'abstract': 'In Yang-Mills theory massless point sources lead naturally to shock-wave configurations. Their magnetic counterparts endow the vacuum of the four-dimensional compact abelian model with a Coulomb-gas behaviour whose physical implications are briefly discussed.', 'arxivid': 'hep-th/9411088', 'author': ['Maurizio Martellini \nDipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY\n', 'Augusto Sagnotti \nDipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY\n', 'Mauro Zeni \nDipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY\n'], 'authoraffiliation': ['Dipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY', 'Dipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY', 'Dipartimento di Fisica\nDipartimento di Fisica Università di Milano I.N.F.N. -Sezione di Milano Via Celoria\nDipartimento di Fisica Università di Roma "Tor Vergata" I.N.F.N. -Sezione di Roma "Tor Vergata" Via della Ricerca Scientifica\nUniversità di Milano\nVia Celoria 16, 1, 1620133, 00133, 20133Milano ITALY I.N.F.N. -Sezione di Pavia, Roma, MilanoITALY, ITALY'], 'corpusid': 8832302, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4097, 'n_tokens_neox': 3468, 'n_words': 2062, 'pdfsha': 'c53c07aed3178eb22e4c51db0c5f673df20473c5', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9411088v2.pdf'], 'title': ['Shock Waves and the Vacuum Structure of Gauge Theories 1', 'Shock Waves and the Vacuum Structure of Gauge Theories 1'], 'venue': []}

arxiv

Gate-tunable non-volatile photomemory effect in MoS 2 transistors Andreij C Gadelha Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Alisson R Cadore Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Kenji Watanabe Advanced Materials Laboratory National Institute for Materials Science 1-1 Namiki305-0044TsukubaJapan Takashi Taniguchi Advanced Materials Laboratory National Institute for Materials Science 1-1 Namiki305-0044TsukubaJapan Ana M De Paula Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Leandro M Malard Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Rodrigo G Lacerda Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Leonardo C Campos Departamento de Fisica Universidade Federal de Minas Gerais 31270-901Belo HorizonteMGBrazil Gate-tunable non-volatile photomemory effect in MoS 2 transistors 10.1088/2053-1583/ab0af1This is the Accepted Manuscript version (post peer-reviewed) of Andreij C. Gadelha et al 2D Materials, 2019, 6, 025036. https:// Non-volatile memory devices have been limited to flash architectures that are complex devices. Here, we present a unique photomemory effect in MoS 2 transistors. The photomemory is based on a photodoping effect -a controlled way of manipulating the density of free charges in monolayer MoS 2 using a combination of laser exposure and gate voltage application. The photodoping promotes changes on the conductance of MoS 2 leading to photomemory states with high memory on/off ratio. Such memory states are non-volatile with an expectation of retaining up to 50% of the information for tens of years. Furthermore, we show that the photodoping is gate-tunable, enabling control of the recorded memory states. Finally, we propose a model to explain the photodoping, and we provide experimental evidence supporting such a phenomenon.In summary, our work includes the MoS 2 phototransistors in the non-volatile memory devices and expands the possibilities of memory application beyond conventional memory architectures. Introduction Due to the ultra-thin nature and tunable electrostatic properties of two-dimensional (2D) materials, they have strategical importance for digital electronics and memory applications. The required features (figures of merit) for actual memory devices include the miniaturisation capability, the power consumption (high memory on/off ratio), the operation speed, the memory retention time and the cost. Innately, the 2D materials based devices show advantages for miniaturization, however to date there are no reported simple memory devices that cover most of the required features, notably the memory retention time [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Monolayer MoS 2 is a direct band gap 2D semiconductor material [18,19] that shows high photocurrent response [20][21][22][23][24][25], high photoluminescence emission [26,27] and interesting valleytronic properties [28][29][30][31]. Thus, MoS 2 based flash memory devices with high memory on/off ratio and long memory retention time have emerged. However, the implementation of flash devices is challenging because they require engineering with many elements into complex architectures. More recently, floating-gate tunneling devices using simpler architechtures than flash devices have been proposed but they lack ultrahigh time-stability [15][16][17]. Therefore, the development of alternative, high-performance, simpler memory architectures is strategical. Toward this direction, some reports have investigated a thermally assisted memory effect [6] and an optical memory effect [5] in a MoS 2 field effect transistor (FET) that is also a simpler architecture than flash devices. Nonetheless, in the first case, the memory effect has a drawback of not operating at room temperature and both cases [5,6] showed low memory on/off ratio and short memory retention time. Here we show that we can obtain a non-volatile photomemory effect with high on/off ratio in a MoS 2 FET architecture. Such photomemory effect is based on a photodoping process that changes the MoS 2 conductance in a way that promotes two distinguishable binary photomemory states with on/off ratio up to 10 6 . The photogenerated memory states are persistent and predicted to retain up to 50% of its information for decades, that leads to a non-volatile photomemory. Moreover, it is important to mention that the presented photomemory is gate-tunable. The gate voltage is used to both adjust the memory on/off ratio (with the laser off) and to manipulate the recorded photomemory states during the laser exposure. Finally, we explore and discuss a possible physical mechanism of the photodoping that is also supported by our experimental evidence. In summary, we propose a pho-tomemory effect in MoS 2 FETs that expands the possibilities of memory application beyond conventional memory architectures. Results and Discussion The photomemory effect investigated in this work is due to the modulation of the conductance of a monolayer MoS 2 field effect transistor via a simultaneous application of light and electrostatic gate potential. Along this paper, we show evidence that the main mechanisms for the photomemory relies on the manipulation of a charging effect at the gate-insulator interface of the FET (the interface between the insulator and the material of the gate terminal). Although other mechanisms can have some influence on the photomemory, we show that our model explains well our results. Our FET is a Van der Waals heterostructure consisting of a monolayer MoS 2 supported by a high-quality hexagonal Boron Nitride crystal (BN), see Fig. 1(a). In this case, we use a graphite crystal to provide a flat back gate electrode. In Fig. 1(b) we present an atomic force spectroscopy (AFM) phase image of one of the devices measured in this work. While, in the Supplementary Information we depict the characterization for the second device. We start by presenting the process of photocurrent generation in the MoS 2 FET. Fig. 1(c) shows a typical time-resolved photocurrent measurement of our device. Initially we measure the standard current (I SD ) in dark conditions, then we illuminate the device using the laser (λ = 488 nm) for 20 s and V BG = −5 V. We use the same laser with λ = 488 nm for all the optoelectronic measurements that we present in this text. A careful analysis of the current as a function of time reveals that there are two optical processes generating the photocurrent in the MoS 2 channel. First, we observe a rapid increase of I SD due to excitation of electronhole pairs (see vertical black arrow), then a second and slow process that dominates the photocurrent. We observe the same trend when the laser is turned off. There is a rapid collapse of I SD , due to the recombination of electron-hole pairs, then a prolonged decay process that leads to a persistent photocurrent (PPC). The photodoping effect causes the PPC. We will discuss this process later. For now, we will define the photomemory states "ON" and "OFF". In Fig. 1(c) we ascribe the PPC as the "ON" state, while the current before the laser exposure is the "OFF" state, which are binary photomemory states. The photomemory effect is better observed in Fig. 1(d), where we show the transfer curves, I SD vs back gate voltage (V BG ) curves, on a log scale. In the inset, we plot the the same curves, but on a linear scale. In blue, we plot a transfer curve before the laser exposure. By extrapolating the I SD curve we estimate the threshold voltage (V th ) as V 0 th = −2.2 V, see the inset in Fig. 1(d). After this, the MoS 2 device is exposed to the laser beam with V BG = −5 V until the photocurrent saturates. The reason for waiting the photocurrent saturation is to reach the best response of our device. Then, we turn the laser off and repeat the transfer curve measurement. The data from the transfer curve after the laser exposure (red curve in Fig. 1(d)) displays a significant increase of I SD at all applied gate conditions. It is important to note that there is a shift of V th towards V BG out of the range of the experiment. This shift is a signature of a photodoping effect. It means that the density of free charges of MoS 2 has changed after the laser exposure. We estimate by extrapolating the data that the initial V 0 th = −2.2 V shifts to V L th = −9.8 V, see the inset in Fig. 1(d). Also, the expected change in the density of charge of the MoS 2 due to photodoping is ∆n ph = 6 × 10 12 cm −2 . Which is evaluated using the equation: ∆n ph = ε 0 ε ox e d (V L th − V 0 th )(1) where ε ox and d are the dielectric constant of the insulator and its thickness, respectively. Note that such extra doping is obtained simply by the combination of the laser exposure and the applied gate bias. We now describe the methods used here to define the photomemory states, that can also be used to perform the "read" operations. We consider as an "OFF" state the measured I SD before the laser exposure for a given V BG (no information is recorded in the photomemory), see the blue curve in Fig. 1(d). Similarly, the measured I SD after the laser exposure for the same V BG is considered as an "ON" state, see the red curve Fig. 1(d). Another method to determine, or to "read", the photomemory states is by measuring ∆n ph before and after the laser exposure. We reinforce that we perform the "read" operations with the laser off. We will discuss the "record" operations later, which are the procedures that "write" and "erase" the memory states. Because I SD is a function of V BG , by measuring I SD instead of ∆n ph we have the advantage to use the gate voltage (with the laser off) to optimise the gain of the photomemory [1,32]. We elucidate this fact in Fig. 1(e), where we plot the I ON /I OFF ratio (memory on/off ratio) as function of V BG . The I ON /I OFF ratio changes from 10, for positive gate voltages, to values up to 10 6 , for negative gate voltages. Observe that the high modulation of the memory on/off ratio with gate voltage is an attribute of the photomemory effect. It must be noted that to reach the photocurrent saturation we do exposures of 30 min, for example in Fig. 1(d). However, we can also obtain a high memory on/off ratio of 10 4 with a short exposure time (20 s), see Fig. S14. Another crucial figure of merit of a memory device is the memory retention time. To assess that, we measure the MoS 2 photocurrent decay over time, applying V SD = 0.1 V and V BG = 0 V, see the blue dots in Fig. 1(f). We measure the decay after the photocurrent saturation by the laser exposure. After 15 h the photocurrent barely decreases, suggesting that the photomemory state is permanent. So, the photomemory is a non-volatile memory. To estimate the memory loss over ten years, we employ an exponential decay fit, the red line in Fig. 1(f). From the fitting, we predict that the reminiscent memory current for the photomemory device is approximately 50% of the initial photocurrent. Thus, the devices can retain 50% of the memory for ten years. These values are much better than the ones for the MoS 2 flash memory architectures, where the retention percentage is in the range of 15-30% [1][2][3][4]. We can now describe in more details the photomemory device, which is composed mainly of two elements in the FET architecture. One element is the gate-insulator interface, where possibly the charges are trapped inducing the photodoping. The other element is the semiconductor channel, from which we "read" the photomemory states. In this way, we can design better photomemory devices by choosing other gate-insulator interfaces that can provide higher values of photodoping and retention time. Furthermore, the choices of semiconductors with better mobility and subthreshold swing would enable to achieve higher on/off ratio values. The photomemory achieved on a FET architecture has the advantage that we can improve some features by choosing the adequate gate voltages. One example is the high memory on/off ratio already discussed. Another important feature is that we can select distinguishable photomemory states for the same laser exposure due to the photodoping dependence on the gate voltage. In this way, the gate voltage is used both to "read" and to "record" the photomemory states. Here, we define of "record" operation the procedure of doing in our devices laser exposures concomitantly with the gate voltage application. In Fig. 2 we show results that highlight the "recording" of the photomemory states. Fig. 2(a) reveals the changes in the density of free charges due to the photodoping effect by exhibiting multiple transfer curves at different photomemory states. The blue curve represents the "OFF" state before any laser exposure. After the "OFF" state is measured, by evaluating a transfer curve of the device, we "record" a photomemory state by applying V BG = −2 V and by exposing the photomemory device to the laser for 20 s. After this "record" operation, we measure a new transfer curve with the laser off (black curve in Fig. 2(a)) and from the data of Fig. 2(a) we observe that the MoS 2 sheet acquires a new density of charge after the laser exposure. We evaluate the new density of free charges from the equation 1 as a function of the new V th . To visualise how the transfer curve changes at every "record" operation, the process described above is repeated applying gate voltages during the "record" operations up to V BG = −5 V in steps of −1 V, as shown in Fig. 2(a). It is interesting to note that for each "record" operation with different V BG there is a distinct transfer curve and thus a particular photomemory state. Then, we can choose several "ON" states with distinct electrical conductivity. Here we use the same "record" operation described in Fig. 2(a), but we achieve the initially recorded state by applying a V BG = 0 V during the 20 s of laser exposure. We name this recorded state as "0" state. We "record" the other photomemory states in the arrows indicated sequence by changing the gate voltages in a range of −5 V ≤ V BG ≤ 5 V. The negative gate voltages are used during the "record" operations to monotonically increase the density of charge to set a "1" state and the positive gate voltages are used during the "record" operations to reduce the density of charge and to restore the initial "0" state. We name the process of charge injection in the MoS 2 as a "write" operation (red arrow). We perform the "write" operation by exposing the device to the laser with an applied negative gate voltage. We denominate the process of removing the charges as an "erase" operation (blue arrow). The gate-"erase" operation is performed right after writing the "1" state, but doing "record" operations with gate voltages larger than V BG = −5 V. For example, in fig. 2(b) the "erase" operations are executed with several laser exposures applying V BG = −4 V,−3 V ... 5 V. Also, note that the "erase" operation cannot cancel the photodoping completely, so we still have a reminiscent photodoping of ∆n ph = 10 × 10 11 after the "erase" operation, see Fig. 2(b). Fig. 2(b) also shows that the laser "record" operations with different gate voltages generate distinct ∆n ph values, which correspond to distinct photomemory states. This dependence of the photomemory states on the V BG used during the "record" operations shows that we can use the photomemory for multilevel "ON" memory states operation. However, we must point out that it is not the aim of this work to explore multilevel memory operation. We only elucidate that the gate-tunability property of photomemory can allow this type of operation. Fig. 2(c) demonstrate how this is possible employing current readings. In Fig. 2(c), the dashed black line represents the "OFF" state, which is measured by applying V SD = 0.1 V and V BG = −4 V before any laser exposure. We "record" "ON" states applying laser exposures using different V BG at each "record" operation. More precisely, we use V BG from 0 V to −5 V, with increments of −1 V, and do laser exposures for 20 s, "recording" multilevel states which we denote by "0", " 1 5 ", " 2 5 " ... "1". After each "record" operation, we "read" the photomemory state by measuring the current through the device at the same electrostatic condition used when we "read" the "OFF" state. The difference here is that we use pulses of V SD = 0.1 V for 2 s spaced by 2 s to show that the information is stored in the photomemory even when no V SD is applied. Note that the "OFF" state is shown only for reference and the ratio between ON/OFF states previously discussed does not apply for multilevel operations. However, the gain between such multilevel states can be maximised tuning the gate-potential, but in our presented data it is of the order of ten. Although multilevel memory states are interesting, here they are explored only to demonstrate the usefulness of the gate-tunability property of the photomemory. However, for practical memory operations, it is straightforward to explore the reliability in the "write"-"erase" operations between the binary memory states. In this case, we generate the binary "1" and "0" states by applying V BG = −5 V and V BG = 5 V, respectively, during the laser exposures of 20 s. We show the reproducibility and reliability of the "write"-"erase" operations of the binary memory states in Fig. 2(d), that presents a sequence of successful "write"-"erase" cycles. These results demonstrate the device robustness. In Fig. 2(d), it is also represented the error bars in each "record" operation. The error bars show that the "write"-"erase" operations generate distinguishable photomemory states. It is worth mentioning that the variation of the photodoping between the "1" and the "0" states in Fig. 2(b) is ∆n ph ∼ 10 12 cm −2 , which is evaluated by ∆n ph = ε 0 εox e d (V "1" th − V "0" th ), where V "1" th and V "0" th are the threshold voltage of the device in the "1" and in the "0" states, respectively. Recall that to obtain this modulation of the photodoping we do laser exposures of 20 s together with gate voltage applications. The generated photodoping with a 20 s laser exposure is an order of magnitude lower than the maximum photodoping (∆n ph ∼ 10 13 cm −2 ) obtained in this work, see Fig. 1(d), where the photodoping is maximized by waiting for the saturation of the photocurrent after 30 min laser exposure. Such high photodoping modulation give an ultra-high memory on/off ratio of 10 6 in Fig. 1(e). However, we can still obtain a high memory on/off ratio of 10 4 by using laser exposures of 20 s, see Fig. S14. Finally, we discuss the process of photodoping that possibly generates the PPC and the photomemory effect in our MoS 2 FETs. It is important to mention that the PPC is not a consensus topic. The most discussed explanations for the PPC in MoS 2 is either due to the photo-induced charge transfer from adsorbed gases to the MoS 2 channel [33] or due to the Coulomb interaction with defects at the insulator surface [20-22, 32, 34-37]. We believe that the interactions with adsorbed gases are not a valid explanation in our devices as there is no hysteresis in the σ vs V BG curves when we sweep the voltage in opposite directions [38](see Fig. S7). We believe that the interactions with defects at the insulator surface are not the dominant mechanism, as the devices have a low density of defects when compared with the photodoping observed in our work (10 13 cm −2 ). Furthermore, the fact that we measure the photodoping in a clean and flat BN substrate [39] reinforces this statement. Consider that we use a ∼ 30 nm thick BN, which prevents tunneling as a charge-trapping mechanim like occurs in the reference [15][16][17]. It should be also mentioned that to prepare our Van der Waals heteroestructures we use the same wet-transfer method of the reference [39], which leaves some bubbles and wrinkles between the BN and the graphite flakes, see Fig. S1 and Fig. S2. However, in spite of these issues, the BN is clean and atomically flat in the majority of the surface of the devices. Recall that we do not study the influence of these imperfections between the layers in the photodoping effect, but we do not discard that they can play a role. Thus, we shall attribute a different process to the photodoping in our MoS 2 FET, which we propose to be a photogeneration of trapped holes in the gate-insulator interface. We clarify this mechanism by drawing the energy band diagram of the MoS 2 FET. Fig. 3(a) shows a band diagram of the MoS 2 FET with V BG < 0 V applied to the graphite relative to the MoS 2 . Here χ MoS 2 is the MoS 2 electron affinity, χ BN is the BN electron affinity and Φ M is the graphite work function. We also show in the gate-insulator junction the bending of the graphite band, that generates a built-in electric field. For V BG < 0 V, photons with sufficient For V BG > 0 V, the MoS 2 channel is n-doped, so when we turn the laser on, the electrons from MoS 2 are photoexcited to the conduction band of BN, see Fig. 3(c). In this case, the gate-field drives these electrons through the conduction band of BN to the gate-insulator junction, recombining with some of the trapped holes, reducing the photodoping. We do not achieve the photodoping reduction process totally, because the built-in electric field of the gate-insulator junction prevents some of the electrons to recombine. Fig. 3(d) shows that for V BG > 0 V we can also observe a photo-generated leakage current during the laser exposure. energy (E L > Φ M − χ BN ) Note that for V BG > 0 V the photo-generated leakage current is lower than for V BG < 0 V. We can associate this fact to the density of states of MoS 2 , which is smaller than the graphite flake. It is also important to mention that this asymmetry in the photo-generated leakage current imposes a faster "write" operation relative to the "erase" operation, see Fig. S10. The proposed model in Fig. 3 explains the results of Fig. 2(b), which shows that the [40][41][42]. Therefore, the difference between Φ G and χ BN is around 2.2 eV, so only photons with energy larger than 2.2 eV are predicted to promote photoexcitation, see Fig. 3(a). Therefore, we have done measurements with a laser energy of 1.6 eV and measured an almost negligible photodoping of ∆n ph ∼ 10 10 cm −2 (see Fig. S11). In contrast, for the laser energy of 2.5 eV we have observed a high photodoping of ∆n ph ∼ 10 12 cm −2 (see Fig. S13). The small, but not null, photodoping with the 1.6 eV laser may be due to other minor effects that may also occur, as the excitation of defects from the MoS 2 channel [43]. However, mostly the gate-insulator interface contains the physics of the photodoping, therefore studying other materials may enable photomemory improvements. Conclusion In conclusion, we have demonstrated that it is possible to obtain a non-volatile photomemory effect with high on/off ratio in a FET architecture. We showed that high values of doping are achieved via laser exposure that generates the binary photomemory states with high on/off ratio. We have shown that the photomemory described presents long memory retention time and thus the photomemory states are non-volatile. We have also verified that the photomemory states can be controlled and adjusted by the applied gate voltage, that could also be used to improve the memory on/off ratio. FIG. 1 . 1Photodoping and non-volatile photomemory. (a), sketch of the MoS 2 FET. (b), AFM phase image of the device. (c), time resolved photocurrent, laser exposure at 488 nm with fluence of 60 µW/µm 2 . The parameters are V BG = −5 V and V SD = 0.1 V. (d), The I SD vs V BG measurements on a log scale before (blue) and after (red) the laser exposure, V SD = 0.1 V. In the inset, the same measurements but on a linear scale. The red curve is measured after the 488 nm laser exposure with fluence of 700 µW/µm 2 and V BG = −5 V until photocurrent saturation. (e), I ON /I OFF ratio as a function of the gate voltage. (f ), photocurrent decay after the photodoping induced by the 488 nm laser with fluence of 700 µW/µm 2 until photocurrent saturation. The parameters V BG = 0 V and V SD = 0.1 V are used for this measurement. FIG. 2 . 2Gate-tunable photomemory. (a), I SD vs V BG curves before laser exposure, blue curve, and after laser exposures with V BG values defined in the color bar. (b), ∆n ph vs V BG curve. First the point V BG = 0 V is measured and then the arrows indicates the followed applied gate voltages during the laser exposures. For figures (a)-(b) there is a 488 nm laser exposure on each point for 20 s (at laser fluence of 60 µW/µm 2 ). (c), multilevel photomemory, gate values from 0 V to −5 V are used for the "writings" and 20 s of laser exposure at fluence of 700 µW/µm 2 . For the "readings" a gate value of −4 V and bias pulses of 0.1 V are applied. (d), "write"-"erase" operations, for V BG = −5 V and V BG = 5 V, respectively, and 20 s of laser exposure (fluence of 700 µW/µm 2 ). Fig. 2 2(b) exposes the change of the density of free charges acquired for the MoS 2 after every photomemory "record" operation as compared to the intrinsic density of charge of the photomemory (equation 1). FIG. 3 . 3Physical model for the photodoping. (a), energy band diagram for the MoS 2 /BN/graphite junction as a function of position for V BG < 0 V. (b), photogenerated leakage current, V BG = −5 V, λ = 488 nm and fluence of 60 µW/µm 2 . (c), energy band diagram for the MoS 2 /BN/graphite junction as a function of position for V BG > 0 V. (d), photogenerated leakage current, V BG = 5 V, λ = 488 nm and fluence of 60 µW/µm 2 . promote the electrons from the gate-insulator interface to the conduction band of BN. The applied negative gate voltage drives these photoexcited electrons through the conduction band of BN to the MoS 2 channel, but some holes generated during the photoabsorption process remain trapped at the gate-insulator interface by the electric field of the gate-insulator junction. The positively charged layer generates photodoping in the MoS 2 channel, seeFig. 3(a). According to this energy diagram description, we predict that we should observe a photo-generated leakage current under laser exposure between the drain and gate electrodes. This fact is verified in our experiments, as depicted inFig. 3(b)that exhibits a 10 −8 A leakage current during the laser exposure. applied negative gate bias increase the photodoping, whether positive gate bias reduce the photodoping. Moreover, the threshold energy for photodoping generation (E th = Φ M − χ BN ) in Fig. 3 matches our experimental results. Indeed, a crystal of BN possess a band gap (E BN g ) of 5.2-5.9 eV and electron affinity (χ BN ) of 2.0-2.3 eV [1]. Whereas graphite has a work function (Φ G ) of 4.3-4.6 eV Finally, we have proposed a phenomenological model that agrees well with the experimental observations and clarifies a possible nature of the photodoping effect in MoS 2 FETs. Our results widen the possibilities of memory applications using 2D materials. METHODS Device Fabrication. The devices are obtained by transfer [39] of BN crystals (∼ 30 nm thick) to graphite crystals (∼ 20 nm thick). Metal leads were patterned by electron-beam lithography and subsequent deposition of metals (Cr 1 nm/ Au 50 nm). Monolayer MoS 2 flakes were transferred to this structure by dry viscoelastic stamping technique [44]. For more details see Supplementary Information. Optoelectronic Measurements. To provide a source-drain bias the external DC source of a standard lock-in amplifier (SR830) was used. While to provide a gate bias the DC source of the lock-in amplifier or a Keithley 2400 were used. The current of the devices was collected by a pre-amplifier and then measured by a multimeter (Keithley 2000). To generate the photocurrent in the MoS 2 FET a 488 nm laser beam was focused in the devices by a 50× objective lens (∼ 1 µm spotsize). 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Ohno, David Ginley, and Ryan O'Hayre. Dopant-induced electronic structure modification of hopg surfaces: Implications for high activity fuel cell catalysts. The Journal of Physical Chemistry C. 1141Dopant-induced electronic structure modification of hopg surfaces: Implications for high ac- tivity fuel cell catalysts. The Journal of Physical Chemistry C, 114(1):506-515, 2010. Control of work function of graphene by plasma assisted nitrogen doping. Keishi Akada, Gaku Tomo O Terasawa, Seiji Imamura, Koichiro Obata, Saiki, Applied Physics Letters. 10413131602Keishi Akada, Tomo o Terasawa, Gaku Imamura, Seiji Obata, and Koichiro Saiki. Control of work function of graphene by plasma assisted nitrogen doping. Applied Physics Letters, 104(13):131602, 2014. The intrinsic defect structure of exfoliated mos2 single layers revealed by scanning tunneling microscopy. Péter Vancsó, János Gábor Zsolt Magda, Ji-Young Peto, Yong-Sung Noh, Chanyong Kim, László P Hwang, Levente Biró, Tapasztó, Scientific Reports. 629726Péter Vancsó, Gábor Zsolt Magda, János Peto, Ji-Young Noh, Yong-Sung Kim, Chanyong Hwang, László P. Biró, and Levente Tapasztó. The intrinsic defect structure of exfoliated mos2 single layers revealed by scanning tunneling microscopy. Scientific Reports, 6:29726-, July 2016. Acknowledgments This work was supported by CAPES, Fapemig, CNPq, Rede de Nano-Instrumentação and INCT/Nanomateriais de Carbono. The authors are thankful to the Laboratory of Nano Spectroscopy at UFMG for providing an experimental setup for this work, and to Centro Brasileiro de Pesquisas Fisicas. Andres Castellanos-Gomez, Michele Buscema, Rianda Molenaar, Vibhor Singh, Laurens Janssen, S J Herre, Gary A Van Der Zant, Steele, 2D Materials. 1111002Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping. Centro de Componentes Semicondutores (CCSAndres Castellanos-Gomez, Michele Buscema, Rianda Molenaar, Vibhor Singh, Laurens Janssen, Herre S J van der Zant, and Gary A Steele. Deterministic transfer of two-dimensional materials by all-dry viscoelastic stamping. 2D Materials, 1(1):011002, 2014. Acknowledgments This work was supported by CAPES, Fapemig, CNPq, Rede de Nano-Instrumentação and INCT/Nanomateriais de Carbono. The authors are thankful to the Laboratory of Nano Spectroscopy at UFMG for providing an experimental setup for this work, and to Centro Brasileiro de Pesquisas Fisicas (CBPF) and Centro de Componentes Semicondutores (CCS)

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{'abstract': 'Non-volatile memory devices have been limited to flash architectures that are complex devices. Here, we present a unique photomemory effect in MoS 2 transistors. The photomemory is based on a photodoping effect -a controlled way of manipulating the density of free charges in monolayer MoS 2 using a combination of laser exposure and gate voltage application. The photodoping promotes changes on the conductance of MoS 2 leading to photomemory states with high memory on/off ratio. Such memory states are non-volatile with an expectation of retaining up to 50% of the information for tens of years. Furthermore, we show that the photodoping is gate-tunable, enabling control of the recorded memory states. Finally, we propose a model to explain the photodoping, and we provide experimental evidence supporting such a phenomenon.In summary, our work includes the MoS 2 phototransistors in the non-volatile memory devices and expands the possibilities of memory application beyond conventional memory architectures.', 'arxivid': '2006.09986', 'author': ['Andreij C Gadelha \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n', 'Alisson R Cadore \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n', 'Kenji Watanabe \nAdvanced Materials Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan\n', 'Takashi Taniguchi \nAdvanced Materials Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan\n', 'Ana M De Paula \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n', 'Leandro M Malard \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n', 'Rodrigo G Lacerda \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n', 'Leonardo C Campos \nDepartamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil\n'], 'authoraffiliation': ['Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil', 'Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil', 'Advanced Materials Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan', 'Advanced Materials Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan', 'Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil', 'Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil', 'Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil', 'Departamento de Fisica\nUniversidade Federal de Minas Gerais\n31270-901Belo HorizonteMGBrazil'], 'corpusid': 139288011, 'doi': '10.1088/2053-1583/ab0af1', 'github_urls': [], 'n_tokens_mistral': 15760, 'n_tokens_neox': 13260, 'n_words': 7533, 'pdfsha': 'acee48c651b0a8cddce9b5113f15fa88a66afdb9', 'pdfurls': ['https://arxiv.org/pdf/2006.09986v1.pdf'], 'title': ['Gate-tunable non-volatile photomemory effect in MoS 2 transistors', 'Gate-tunable non-volatile photomemory effect in MoS 2 transistors'], 'venue': []}

arxiv

Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization 17 May 2023 Yankun Huang yankun-huang@uiowa.edu Department of Business Analytics University of Iowa 52242Iowa CityIA Qihang Lin qihang-lin@uiowa.edu Department of Business Analytics University of Iowa 52242Iowa CityIA Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization 17 May 2023Constrained optimizationFirst-order methodNon-smooth optimizationNon-convex optimization We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations. optimization arg min y {f 2 (y) + ρ 2 y − x 2 } for any x. There are relatively fewer works on non-convex nonsmooth constrained problems. An alternating direction method of multipliers (ADMM) and an ALM are studied by [81] and [84], respectively, for non-convex non-smooth problems with linear constraints while our study considers nonlinear non-smooth constraints. The methods by[21]and[13]can be extended to a structured non-smooth case where f = f 1 + f 2 with f 1 being smooth non-convex and f 2 = max y {y ⊤ Ax − φ(y)} with a convex φ, and g has a similar structure. The method by[15]can handle a specific non-smooth non-convex constraint, i.e., g(x) = λ x 1 − h(x) where h is a convex and smooth. Compared to these works, our results apply to a more general non-smooth problem without those structure assumptions.When f and g in (1) are weakly convex and non-smooth, the inexact proximal point method has been studied by[14,58, 42] under different constraint qualifications and different notions of stationarity. Their complexity analysis utilizes the relationship between the gradient of the Moreau envelope of (1) and the near stationarity of a solution, which is originally used to analyze complexity of subgradient methods for weakly convex non-smooth unconstrained problems[25,26,28,2,29,67, 89]. Our analysis utilizes a similar framework. The methods[14,58, 42] are double-loop while our algorithm only uses a single loop and achieves the same complexity of O(1/ǫ 4 ) as them under similar assumptions.The SSG algorithm is first proposed by Polyak [65]. It has been well-studied for convex problems [64,8,52, 73, 78, 79, 71, 74, 72,3]and quasi-convex problems [73]. This paper provides the first complexity analysis for the SSG method under weak convexity assumption. Non-smooth non-convex optimization has also been studied without weak convexity assumption by [88, 49, 70, 50,20,76, 77]. These works analyze the complexity of first-order methods for computing an (ǫ, δ)-Goldstein approximate stationary point, which is a more general stationarity notation than what we consider here. However, these works only focus on unconstrained problems. Introduction Continuous optimization with nonlinear constraints arises from many applications of machine learning and statistics. Examples include Neyman-Pearson classification [68] and learning with fairness constraints [83]. In this paper, we consider the following general nonlinear constrained optimization f * ≡ min x∈X f (x) s.t. g(x) ≤ 0,(1) where X ⊂ R d is a closed convex set that allows a computationally easy projection operator, f is weakly-convex, g is either convex or weakly convex, and functions f and g are not necessarily smooth. When g(x) ≡ max i=1,...,m g i (x), (1) is equivalent to an optimization problem with multiple nonlinear constraints g i (x) ≤ 0 for i = 1, . . . , m. A weakly convex function can be non-convex, so computing an optimal solution of (1) is challenging in general even without constraints. For this reason, theoretical analysis for gradient-based algorithms for non-convex problems mostly focuses on an algorithm's (oracle) complexity for finding an ǫ-stationary solution for (1). When a problem is non-smooth, finding an ǫ-stationary solution is generally difficult even if the problem is convex [50]. Hence, in this paper, we consider finding a nearly ǫ-stationary solution for (1), whose definition will be stated later in Definition 3.2. In the past decade, there have been many studies on non-convex constrained optimization. However, most of the existing algorithms and their theoretical complexity analysis are developed by assuming f and g i 's are all smooth or can be written as the sum of a smooth and a simple non-smooth functions. A non-exhaustive list of the works under such an assumption includes [12,35,86,87,41,43,60,39,59,48,75,47,55,69,56,53,13,16,23,10,62,22,57,44,54]. Their results cannot be applied to (1) due to non-smoothness in the problem. Non-smoothness is common in optimization in machine learning, e.g., when a non-smooth loss function is applied, but there are much fewer studies on non-smooth non-convex constrained optimization. Under the weak-convexity assumption, an effective approach for solving a non-smooth non-convex problem with theoretical guarantees is the (inexact) proximal point method, where a quadratic proximal term is added to objective and constraint functions to construct a strongly convex constrained subproblem and then a sequence of solutions can be generated by solving this subproblem inexactly and updating the center of the proximal term. Oracle complexity for this method to find a nearly ǫ-stationary has been established by [14,58,42] under different constraint qualifications. The inexact proximal point method is a double-loop algorithm where the inner loop is another optimization algorithm for solving the aforementioned strongly convex subproblems. The complexity results in [14,58,42] require the inner loop solves each subproblem to a targeted optimality gap. However, the optimality gap is hard to evaluate and thus cannot be used to terminate the inner loop. Although the number of inner iterations needed to achieve the targeted gap can be bounded theoretically, the bound usually involves some constants that are unknown or can only be estimated conservatively. Hence, using the theoretical iteration bound to stop the inner loop usually leads to significantly more inner iterations than needed, making the whole algorithm inefficient. In practices, users often need to tune the number of inner iterations to improve algorithm's efficiency, which is an inconvenience common to most double-loop methods. On the contrary, a single-loop algorithm is usually easier to implement as it does not require tuning the number of inner iterations. Therefore, the main contribution of this paper is showing that a single-loop first-order algorithm can find a nearly ǫ-stationary point of (1) with oracle complexity O(1/ǫ 4 ), which matches state-of-the-art results obtained only by the double-loop methods [14,58,42]. The algorithm we study is the classical switching subgradient (SSG) method proposed by Polyak [65] in 1967, which is intuitive and easy to implement but has only been analyzed before in the convex case. We first show that the SSG method has complexity complexity O(1/ǫ 4 ) when g is convex. Then, we show that the same complexity result holds when g is weakly convex and appropriate constraint qualification holds. We also present some practical examples where all of our assumptions hold, including a fair classification problem subject to the demographic parity constraint [1]. Our technical novelty is inventing a switching stepzize rule to accompany the switching subgradient. In particular, we use a fixed stepsize when the solution is updated by the objective's subgradient while use an adaptive Polyak's stepsize [34,66] when the solution is updated by the constraint's subgradient. This allows us to leverage the local error bound of the constraint function to keep the solution nearly feasible during the algorithm, which prevents the solution from being trapped at an infeasible stationry point. To the best of our knowledge, this paper is the first to establish the complexity of a single-loop first-order method for weakly convex non-smooth nonlinear constrained optimization. In the appendix, we also extend our algorithms and complexity analysis to the stochastic setting. Related work Non-convex constrained optimization has a long history [37,17,30,18,31,6,19] and the interest on this subject has been still growing in the machine learning community because of its new applications such as learning with fairness constraints (see e.g., [83]). In recent literature, the prevalent classes of algorithms for non-convex constrained optimization include the augmented Lagrangian method (ALM), the penalty method [38,85,86,87,40,41,43,46,60,39,59,48,75,47,55,69,56,53,57,44,54], and the sequential quadratic programming method [12,35,7,13,23,11,10,62,9,22]. Besides, an inexact projected gradient method is developed by [16] and a level conditional gradient method is developed by [21]. However, these works all focus on the case where g is smooth and f is either smooth or equals f 1 + f 2 , where f 1 is smooth and non-convex while f 2 is non-smooth and convex and has a simple structure that allows a closed-form solution for the proximal mapping ζ * f + λ * ζ * g ∈ −N X (x * ), λ * g(x * ) = 0, g(x * ) ≤ 0, λ * ≥ 0, where λ * ∈ R is a Lagrangian multiplier, ζ * f ∈ ∂f (x * ) and ζ * g ∈ ∂g(x * ). Typically, an exact stationary point can only be approached by an algorithm at full convergence, which may require infinitely many iterations. Within a finite number of iterations, an algorithm can only generate an ǫ-stationary point [14], which is a point x ∈ X satisfying dist ζ f + λ ζ g , −N X ( x) ≤ ǫ, | λg( x)| ≤ ǫ 2 , g( x) ≤ ǫ 2 , λ ≥ 0, (2) where λ ∈ R is a Lagrangian multiplier, ζ f ∈ ∂f ( x) and ζ g ∈ ∂g( x). However, because f and g are non-smooth, computing an ǫ-stationary point is still challenging even for an unconstrained problem. Nevertheless, under the weak convexity assumption, it is possible to compute a nearly ǫ-stationary point, which we will introduce next. Givenρ ≥ 0,ρ ≥ 0, γ ∈ R and x ∈ X , we define a quadratically regularized problem of (1) as ϕ(x) ≡ min y∈X f (y) +ρ 2 y − x 2 , s.t. g(y) +ρ 2 y − x 2 ≤ γ ,(3) x(x) ≡ arg min y∈X f (y) +ρ 2 y − x 2 , s.t. g(y) +ρ 2 y − x 2 ≤ γ .(4) Following the literature on weakly convex optimization [25,28,24,14,58, 42], we use the value of x(x) − x as a measure of the quality of a solution x because it can be interpreted Algorithm 1 Switching Subgradient (SSG) Method by Polyak [65] 1: Input: x (0) ∈ X , total number of iterations T , tolerance of infeasibility ǫ t ≥ 0, stepsize η t , and a starting index S for generating outputs. 2: Initialization: I = ∅ and J = ∅. 3: for iteration t = 0, 1, · · · , T − 1 do 4: if g(x (t) ) ≤ ǫ t then 5: Set x (t+1) = proj X (x (t) − η t ζ (t) f ) for any ζ (t) f ∈ ∂f (x (t) ) and, if t ≥ S, I = I ∪ {t}. 6: else 7: Set x (t+1) = proj X (x (t) − η t ζ (t) g ) for any ζ (t) g ∈ ∂g(x (t) ) and, if t ≥ S, J = J ∪ {t}. 8: end if 9: end for 10: Output I: x (τ ) with τ randomly sampled from I using Prob(τ = t) = η t / s∈I η s . 11: Output II: x (τ ) with τ randomly sampled from I ∪ J using Prob(τ = t) = η t / s∈I∪J η s . as a stationarity measure. For the purpose of illustration, we assume for now that x(x) is uniquely defined and there exists a Lagrangian multiplier λ ∈ R such that the following KKT conditions of (4) holds. ζ f +ρ( x(x) − x) + λ ζ g +ρ( x(x) − x) ∈ −N X ( x(x)), λ g( x(x)) +ρ 2 x(x) − x 2 − γ = 0, g( x(x)) +ρ 2 x(x) − x 2 ≤ γ, λ ≥ 0,(5) where ζ f ∈ ∂f ( x(x)) and ζ g ∈ ∂g( x(x)). Therefore, as long as x(x) − x ≤ ǫ , we have dist ζ f + λ ζ g , −N X ( x(x)) ≤ (ρ + λρ)ǫ, | λg( x(x))| = λ(ρǫ 2 /2 + γ), g( x(x)) ≤ γ. (6) This means, when γ ≤ O(ǫ 2 ), x(x) is an O(ǫ)-stationary point of the original problem (1) in the sense of (2). Since x is within an ǫ-distance from x(x), we call such an x a nearly ǫ-stationary point of (1). We formalize this definition as follows. Definition 3.2. Suppose x(x) is defined in (4) with 0 ≤ γ ≤ ǫ 2 . A (stochastic) point x ∈ X is a (stochastic) nearly ǫ-stationary point of (1) if E[ x(x) − x ] ≤ ǫ. Of course, we can claim x(x) is an O(ǫ)-stationary point of (1) based on (6) only when λ in (5) exists and does not go to infinity as ǫ approaches zero. Fortunately, we can show in Lemmas 4.2 and 5.2 that this is true under some constraint qualifications, which justifies Definition 3.2. We present the SSG method in Algorithm 1 for finding a nearly ǫ-stationary point of (1). At iteration t, we check if the current solution x (t) is nearly feasible in the sense that g(x (t) ) ≤ ǫ t for a pre-determined tolerance of infeasibility ǫ t . If yes, the algorithm performs a subgradient step along the subgradient of f . Otherwise, the algorithm switches the updating direction to the subgradient of g. The algorithm records the iteration indexes of the nearly feasible solutions in set I and other indexes in set J. The final output is randomly sampled from the iterates in I or I ∪ J with a distribution weighted by the stepsizes η t 's. An index S is set so the algorithm only starts to record I and J when t ≥ S. We study the theoretical oracle complexity of Algorithm 1 for finding a nearly ǫ-stationary point, which is defined as the total number of times for which the algorithm queries the subgradient or function value of f or g. Our results are presented separately when g is convex and when g is weakly convex. Convex constraints In this section, we first consider a relatively easy case where f is weakly convex but g is convex. In particular, we make the following assumptions in addition to Assumption 3.1 in this section. Assumption 4.1. The following statements hold: A. f (x) is ρ-weakly convex on X and g(x) is convex on X . B. (Slater's condition) There exists x feas ∈ relint(X ) and g(x feas ) < 0. C. There exists D ∈ R such that x − x ′ ≤ D for any x and x ′ in X . In this section, we choose parameters in (3) such that ρ > ρ andρ = γ = 0. Under Assumption 4.1, (7) guarantees that (3) is strictly feasible, its objective function is (ρ − ρ)-strongly convex and its constraint function is convex, so x(x) in (4) is unique and λ in (5) exists. We first present an upper bound of λ that is independent of x. The proof is in Section A.1. Given any x ∈ X , let x(x) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and λ be the associated Lagrangian multiplier satisfying (5). We have λ ≤ Λ := (M D +ρD 2 )/(−g(x feas )). For simplicity of notation, we denote x(x (t) ) defined in (4) by x (t) . Let E τ [·] be the expectation taken only over the random index τ when the algorithms stop. We present the convergence properties of Algorithm 1 when ǫ t and η t are static and diminishing. The proof is provided in Section A.4. (8). Let x(x (t) ) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and x (τ ) is generated by Output I. Algorithm 1 guarantees E τ [ x (τ ) − x (τ ) ] ≤ ǫ and E τ [g(x (τ ) )] ≤ ǫ 2 (ρ−ρ) 1+Λ in either of the following cases. Case I: S = 0, ǫ t = ǫ 2 (ρ−ρ) 1+Λ , η t = 2ǫ 2 (ρ−ρ) 5(1+Λ)M 2 and T ≥ 25M 2 D 2 (1+Λ) 2 4ǫ 4 (ρ−ρ) 2 = O(1/ǫ 4 ). Case II: S = T /2, ǫ t = 5M D √ t+1 , η t = D M √ t+1 and T ≥ 50M 2 D 2 (1+Λ) 2 ǫ 4 (ρ−ρ) 2 = O(1/ǫ 4 ). Algorithm 1 is single-loop with O(1) oracle complexity per iteration, so its total complexity is just T = O(1/ǫ 4 ), which matches the start-of-the-art complexity by [14,58,42]. Remark 4.4. Property E τ [g(x (τ ) )] ≤ ǫ 2 (ρ−ρ) 1+Λ in the theorems above is not required by Definition 3.2. By Assumption 3.1A, E τ [g(x (τ ) )] ≤ E τ [g( x (τ ) )] + M E τ [ x (τ ) − x (τ ) ] ≤ M ǫ, which means a nearly ǫ-stationary point must be O(ǫ)-feasible by definition. Property E τ [g(x (τ ) )] ≤ ǫ 2 (ρ−ρ) 1+Λ implies O(ǫ 2 )-feasibility for the output, which is even better. When g is µ-strongly convex with µ > 0, we can show that the complexity of Algorithm 1 is still O(1/ǫ 4 ) but one can simply set ǫ t = 0, which makes η t the only tuning parameter. This makes this single-loop method even more attractive. Due to space limit, we include this result in Section A.5. We also extend our result to the stochastic case in Section A.6 and A.7. Weakly convex constraints Next we consider the case where both f and g are weakly convex but not necessarily convex. Let g + (x) = max{g(x), 0}, L = {x ∈ X | g(x) = 0} and S = {x ∈ X | g(x) ≤ 0}. We make the following assumptions in addition to Assumption 3.1 in this section. Assumption 5.1. The following statements hold: A. f (x) and g(x) are ρ-weakly convex on X . B. There existǭ > 0, θ > 0 andρ > ρ such that, for anyǭ 2 -feasible solution x, there exists y ∈ relint(X ) such that g(y) +ρ 2 y − x 2 ≤ −θ. C. f := inf x∈X f (x) > −∞. Assumption 5.1B is called the uniform Slater's condition by [58]. We will present two real-world examples in Section B.1 that satisfy this assumption, including a fair classification problem under demographic parity constraint, which is one of the applications in our numerical experiments in Section 6.2. In this section, we choose parameters in (3) such that ρ ≥ρ =ρ > ρ and γ = ǫ 2 with ǫ ≤ǭ. Under Assumption 5.1, (9) guarantees that (3) is uniformly strictly feasible for any ǫ 2 -feasible x, and the objective and constraint functions of (3) are both (ρ − ρ)-strongly convex, so x(x) is uniquely defined and λ in (5) exists. In addition, Assumption 5.1 has the following three implications that play important roles in our complexity analysis. First, λ in (5) can be bounded by a constant independent of x and ǫ as long as x is ǫ 2 -feasible with ǫ ≤ǭ. This result is similar to Lemma 1 by Ma et al. [58] except that they require X to be bounded but we do not. The proof is given in Section B.2. Lemma 5.2. Suppose Assumptions 3.1 and 5.1 hold. Given any ǫ 2 -feasible x with any ǫ ≤ǭ, let x(x) defined as in (4) satisfying (9) and λ is the associated Lagrangian multiplier satisfying (5). We have x(x) − x ≤ M/ρ and λ ≤ Λ ′ := 2M/ 2θ(ρ − ρ).(10) Second, the subgradient of the constraint function g(x) + δ X (x) on L is uniformly away from the origin. The proof is provided in Section B.2. Lemma 5.3. Suppose Assumptions 3.1 and 5.1 hold. It holds for any x ∈ L that min ζ g ∈∂g(x),u∈N X (x) ζ g + u ≥ ν := 2θ(ρ − ρ).(11) Lastly, note that S is the optimal set of min x∈X g + (x), which is a ρ-weakly convex nonsmooth optimization problem with an optimal value of zero. Lemma 5.3 implies that g + (x) is sharp near the boundary of S, meaning that g + (x) satisfies a linear error bound in an O(1)-neighborhood of S. A similar result for a convex g + is given in Lemma 1 in [82]. In the lemma below, we extend their result for a weakly convex g + . The proof is in Section B.2 and the second conclusion is directly from [27]. dist(x, S) ≤ ν ρ that (ν/2)dist(x, S) ≤ g + (x).(12) Moreover, ν ≤ 2M and min x∈X g + (x) has no stationary point satisfying 0 < dist(x, S) < ν ρ . Since g is non-convex, Algorithm 1 may not even find a nearly feasible solution if x (t) is trapped at a stationary point of g with g(x) > 0, that is, a sub-optimal stationary point of min x∈X g + (x). Fortunately, the second conclusion of Lemma 5.4 indicates that this situation can be avoided by keeping dist(x (t) , S) = O(ǫ 2 ) during the algorithm. To do so, we start with x (0) ∈ S and use ǫ t = O(ǫ 2 ) in Algorithm 1. Moreover, we apply a switching stepsize rule that sets η t = O(ǫ 2 ) when t ∈ I and η t = g(x (t) )/ ζ (t) g 2 when t ∈ J, the latter of which is known by the Polyak's stepsize [34,66]. This way, when g(x (t) ) ≤ ǫ t , (12) ensures dist(x (t) , S) = O(ǫ 2 ). When g(x (t) ) > ǫ t , (12) ensures dist(x (t) , S) Q-linearly converges to zero [27], which also guarantees dist(x (t) , S) = O(ǫ 2 ). As a result, we have g(x (t) ) ≤ ǫ 2 for any t, so problem (3) with x = x (t) and (ρ,ρ, γ) satisfying (9) will be strictly feasible according to Assumption 5.1B. This finding is given in the proposition below with its proof in Section B.3. Proposition 5.5. Suppose Assumptions 3.1 and 5.1 hold and ǫ ≤ǭ. Also, suppose x (t) is generated by Algorithm 1 using x (0) ∈ S, ǫ t = ν 4 min ǫ 2 /M, ν/(4ρ) and η t = ν 4M 2 min ǫ 2 /M, ν/(4ρ) if ∈ I g(x (t) )/ ζ (t) g 2 if ∈ J. Then dist(x (t) , S) ≤ min ǫ 2 /M, ν/(4ρ) and g(x (t) ) ≤ ǫ 2 for any t ≥ 0. As a consequence, x (t) is feasible to (3) where x = x (t) and (ρ,ρ, γ) satisfies (9). Let x (t) be x(x (t) ) defined in (4) with (ρ,ρ, γ) satisfying (9). The complexity result for the case of weakly convex constraints is as follows. The proof can be found in Section B.3. E τ [ x (τ ) − x (τ ) ] ≤ Cǫ and E τ [g(x (τ ) )] ≤ ǫ 2 , where C := 2 (1+Λ ′ ) max{Λ ′ ,1} ρ−ρ , if we use Output II and set S = 0 and T ≥ 8M 2 f (x (0) ) − f + 3M 2 /(2ρ) ρ(1 + Λ ′ ) max {Λ ′ , 1} νǫ 2 min {ǫ 2 /M, ν/(4ρ)} = O(1/ǫ 4 ). Let ǫ ′ = max{C, 1} · ǫ = Θ(ǫ). The oracle complexity of Algorithm 1 for finding a nearly ǫ ′ -stationary point is O(1/ǫ 4 ). To obtain an nearly ǫ-stationary point, one only needs to replace ǫ in γ, η t , ǫ t and T by ǫ/ max{C, 1}. This will only change the constant factor in the complexity. Numerical experiments We demonstrate the performance of the SSG method on two fairness-aware classification problems, which are instances of (1) with convex and weakly convex g's, respectively. We compare with two different double-loop inexact proximal point (IPP) methods [14,58,42]. The IPP method approximately solves a strongly convex constrained subproblem in each outer iteration, and we use the SSG method in this paper and the ConEx method in [14] as the solvers (inner loop) because they both have the best theoretical complexity for that subproblem. We use IPP-SSG and IPP-ConEx to denote these two implementations of the IPP method. All experiments are conducted using MATLAB 2022b on a computer with the CPU 3.20GHz x Intel Core i7-8700 and 16GB memory. Classification problem with ROC-based fairness Given a feature vector a ∈ R d and a class label b ∈ {1, −1}, the goal in a binary linear classification problem is to learn a model x ∈ R d to predict b based on the score x ⊤ a. Let D = {(a i , b i )} n i=1 be a training set and ℓ(·) be a convex non-increasing loss function. Model x can be learned by solving L * = min x∈W L(x) := 1 n n i=1 ℓ(b i x ⊤ a i ) ,(13) where W = {x ∈ R d | x ≤ r}. Solving (13) may ensure good classification performance of x but not its fairness. Suppose there exist two additional datasets. One contains the feature vectors of a protected group, denoted by D p = {a p i } np i=1 , and the other one contains the feature vectors of an unprotected group, denoted by D u = {a u i } nu i=1 . We want to enhance the fairness of x between these two groups using the ROC-based fairness metric proposed by [80]. Suppose we set a threshold θ and classify data a as positive if x ⊤ a ≥ θ and as negative otherwise. The ROC-based fairness measure and its continuous approximation are defined as max θ∈Θ 1 n p np i=1 I(x ⊤ a p i ≥ θ) − 1 n u nu i=1 I(x ⊤ a u i ≥ θ) ≈ R(x) := max θ∈Θ 1 n p np i=1 σ(x ⊤ a p i − θ) − 1 n u nu i=1 σ(x ⊤ a u i − θ) ,(14) where σ(z) = exp(z)/(1 + exp(z)) is the sigmoid function and Θ is a finite set of thresholds. If the value of this measure is small, model x produces similar predicted positive rates for the protected and unprotected groups on various θ's, indicating the fairness of the model. To obtain a fair x, we balance (13) and (14) by solving min x∈W R(x) s.t. L(x) ≤ L * + κ,(15) where κ is a slackness parameter indicating how much we are willing to increase the classification loss in order to reduce R(x) to obtain a more fair model. Problem (15) is an instance of (1) satisfying Assumptions 3.1 and 4.1 with ρ = β where β is defined in (56) in Section B.1. [4]. The information of these datasets is given in Table 1. We split each dataset into two subsets with a ratio of 2 : 1. The larger set is used as D in the constraint and the smaller set is further split into D p and D u based on a binary group variable listed in Table 1. In our experiments, we choose ℓ(z) = (1 − z) + and first solve (13) using the subgradient method with a large enough number of iterations to obtain L * and a solution x ERM . Then we set κ = 0.001L * and r = 5 x ERM , and let Θ consist of 400 points equally spaced be- tween min i x ⊤ ERM a i −0.5(max i x ⊤ ERM a i −min i x ⊤ ERM a i ) and max i x ⊤ ERM a i +0.5(max i x ⊤ ERM a i − min i x ⊤ ERM a i ) . All methods are initialized at x ERM . We implemented the SSG method with both static and diminishing stepsizes. For the static stepsize, we select ǫ t from {10 −6 , 2 × 10 −6 , 5 × 10 −6 , 10 −5 } and η t from {2 × 10 −4 , 5 × 10 −4 , 10 −3 , 2 × 10 −3 }. For the diminishing stepsize, we set ǫ t = E 1 √ t+1 and η t = E 2 √ t+1 and select E 1 from {5 × 10 −5 , 10 −4 , 2 × 10 −4 , 5 × 10 −4 } and E 2 from {0.02, 0.05, 0.1, 0.2}. We select the best set of parameters that produces the smallest objective value after 5000 iterations. For IPP, we selectρ from max{ρ, 1}×{1, 1.5, 2} for all three datasets, and the proximal point subproblem is approximately solved by SSG and ConEx both with 100 iterations. For IPP-SSG, we apply a static stepsize with ǫ t and η t tuned in the same way as SSG. For IPP-ConEx, following the notation in [14], we set θ t = t t+1 , η t = c 1 (t + 1) and τ t = c 2 t+1 , and select c 1 from {20, 50, 100, 200} and c 2 from {0.002, 0.005, 0.01, 0.02} by the same procedure adopted by SSG. We report the performances of all methods on each dataset in Figure 1. For SSG, the x-axis represents the total number of iterations while, for IPP, it represents the total number of inner iterations across all outer iterations. The y-axis in each row represents the objective value, infeasibility and near stationarity achieved at each iteration, respectively. To measure near stationarity, we solve (3) with x = x (t) and parameters (7) using the SSG method with 2500 iterations and use the last iterate as an approximation of x(x (t) ). We make sure that the change of x(x (t) ) − x (t) is less than 1% if the number of iterations is increased to 5000. Then we plot x(x (t) ) − x (t) as near stationarity in Figure 1. Since computing x(x (t) ) with a high precision for each t is time-consuming, we only report near stationarity at 100 equally spaced iterations. According to Figure 1, the SSG method with a diminishing stepsize has the best performance on all three datasets in the sense that it reduces the objective value and the (approxi-mate) near stationarity measure faster than others while keeping the solutions nearly feasible. However, the SSG method with a static stepsize is not always better than the IPP methods. This is consistent with our theoretical finding that the SSG and IPP methods have the same oracle complexity. Classification problem with demographic parity Following the notation in the previous subsection, we consider a binary classification problem with a constraint enforcing demographic parity [1]. The measure of demographic parity and its continuous approximation are 1 n p np i=1 I(x ⊤ a p i ≥ 0) − 1 n u nu i=1 I(x ⊤ a u i ≥ 0) ≈ R 0 (x) := 1 n p np i=1 σ(x ⊤ a p i ) − 1 n u nu i=1 σ(x ⊤ a u i ) . (16) Fairness measure R 0 (x) is a special case of R(x) with Θ = {0}. If R 0 (x) is small, model x produces similar predicted positive rates for the protected and unprotected groups. To obtain a fair x, we balance (13) and (16) by solving min L(x) + λSCAD(x) s.t. R 0 (x) ≤ κ.(17) Different from (15), the fairness measure is used as the constraint in (17) while the objective function becomes the empirical hinge loss plus the smoothly clipped absolute deviation (SCAD) regularizer [36] for promoting a sparse solution. Here, λ is a regularization parameter and SCAD(x) := d i=1 s(x i ), s(x i ) =    2|x i | 0 ≤ |x i | ≤ 1 −x 2 i + 4|x i | + 1 1 < |x i | ≤ 2 3 2 < |x i |.(18) We prove in Section B.1 that (17) is an instance of (1) satisfying Assumptions 3.1 and 5.1 with ρ = max{2λ, β} where β is defined in (56) We set λ = 0.2 for all datasets and set κ = 0.005, 0.02 and 0.02 for a9a, bank and COMPAS, respectively. For SSG, we select ǫ t from {10 −6 , 2 × 10 −6 , 5 × 10 −6 , 10 −5 } and select η t from {10 −4 , 2 × 10 −4 , 5 × 10 −4 , 7.5 × 10 −4 } for t ∈ I while set η t = g(x (t) )/ ζ (t) g 2 for t ∈ J. We select the best set of parameters that produces the smallest objective value after 50000 iterations. For IPP, we selectρ from max{ρ, 1} × {1, 1.5, 2} and the proximal point subproblem is approximately solved by SSG and ConEx both with 600 iterations. For IPP-SSG, we apply a static stepsize with ǫ t and η t tuned in the same way as SSG. For IPP-ConEx, following the notation in [14], we set θ t = t t+1 , η t = c 1 (t + 1) and τ t = c 2 t+1 and select c 1 from {20, 50, 100, 200} and c 2 from {0.002, 0.005, 0.01, 0.02} by the same procedure adopted by SSG. We report the performances of all methods on each dataset in Figure 2 where the meanings of axises and the procedure of calculating the (approximate) near stationarity are the same as in Figure 1. Since computing x(x (t) ) with a high precision for each t is time-consuming, we only report near stationarity at 600 equally spaced iterations. Figure 2 shows that the SSG method is only a little faster than the other two methods in reducing the objective value and the (approximate) near stationarity while keeping the solutions nearly feasible. This is consistent with our theoretical finding that the three methods have the same oracle complexity. Conclusion We study the oracle complexity of the switching subgradient (SSG) method for finding a nearly ǫ-stationary point of a non-smooth weakly convex constrained optimization problem. We show that the complexity of the SSG method matches the best result in literature that is achieved only by double-loop methods. On the contrary, the SSG method is single-loop and easier to implement with reduced tuning effort. This is the first complexity result for a single-loop first-order method for a weakly-convex non-smooth constrained problem. [87] Jiawei Zhang, Wenqiang Pu, and Zhi-Quan Luo. On the iteration complexity of smoothed proximal alm for nonconvex optimization problem with convex constraints. arXiv preprint arXiv:2207.06304, 2022. [88] Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka, Ali Jadbabaie, and Suvrit Sra. Complexity of finding stationary points of nonsmooth nonconvex functions. arXiv preprint arXiv:2002.04130, 2020. [89] Siqi Zhang and Niao He. On the convergence rate of stochastic mirror descent for nonsmooth nonconvex optimization. arXiv preprint arXiv:1806.04781, 2018. A Complexity analysis for convex constraints In this section, we present the theoretical complexity analysis for the SSG method when f is non-convex but g is convex. Our analysis will first focus on a more general setting where the oracles of the subgradients and function values of f and g are stochastic. Then the complexity for the case of deterministic oracles is derived as a special case. A.1 Proof of Lemma 4.2 Proof of Lemma 4.2. For simplicity of notation, we denote x(x) in (4) by x. By Assumption 4.1B, there exists a strictly feasible solution x feas ∈ relint(X ) with g(x feas ) < 0. As a result, the Lagrangian multiplier λ ≥ 0 corresponding to x is well-defined and satisfies (5), which means λg( x) = 0 and ζ f +ρ( x − x) + λ ζ g + u = 0,(19) where ζ f ∈ ∂f ( x), ζ g ∈ ∂g( x), u ∈ N X ( x) and N X ( x) is the normal cone of X at x. If λ = 0, the conclusion holds trivially. Hence, we focus on the case that λ > 0. Note that, in this case, we must have g( x) = 0. Taking the inner product between (19) and x − x feas gives 1 λ ζ f +ρ( x − x), x − x feas = ζ g + u/ λ, x feas − x ≤ g(x feas ) − g( x) = g(x feas ),(20) where the inequality is because of convexity of g and the fact that u/ λ ∈ N X ( x). By Assumption 3.1 and 4.1C, we have ζ f ≤ M , x − x feas ≤ D and x − x ≤ D, which imply from (20) that λ ≤ ζ f +ρ( x − x), x feas − x −g(x feas ) ≤ M D +ρD 2 −g(x feas ) . A.2 Stochastic switching subgradient method In this section, we introduce a more general setting where the subgradients and function values of f and g can only be accessed through stochastic oracles, and then propose a stochastic switching subgradient method using these oracles. More specifically, in addition to Assumptions 3.1 and 4.1, we make the following assumption. Assumption A.1. For any x ∈ X , stochastic subgradients ξ f and ξ g and a stochastic value ω can be generated independently such that E (1), and E exp (ω − g(x)) 2 /σ 2 ≤ exp(1) for a constant σ for any x ∈ X . [ω] = g(x), E[ξ f ] ∈ ∂f (x) and E[ξ g ] ∈ ∂g(x). Moreover, it holds that E exp ξ f 2 /M 2 ≤ exp(1), E exp ξ g 2 /M 2 ≤ exp Assumption A.1 means the distributions of ξ f 2 , ξ g 2 and (ω − g(x)) 2 have light tails, which is a common assumption for proving a stochastic first-order method converges in a high probability. See (2.50) in [63] for an example. We follow the convention that σ = 0 when w = g(x) deterministically. Algorithm 2 Stochastic Switching Subgradient Method 1: Input: x (0) ∈ X , total number of iterations T , tolerance of infeasibility ǫ t ≥ 0, stepsize η t , mini-batch size B, and a starting index S for generating outputs. 2: Initialization: I = ∅ and J = ∅. 3: for iteration t = 0, 1, · · · , T − 1 do 4: Generate a mini-batch {ω (t) i } B i=1 of stochastic estimators of g(x (t) ). 5: Setω (t) = 1 B B i=1 ω (t) i . 6: ifω (t) ≤ ǫ t then 7: Generate a stochastic subgradient ξ (t) f of f at x (t) . 8: Set Generate a stochastic subgradient ξ (t) g of g at x (t) . x (t+1) = proj X (x (t) − η t ξ (t) 11: Set x (t+1) = proj X (x (t) − η t ξ (t) g ) and, if t ≥ S, I = I ∪ {t}. 12: end if 13: end for 14: Output: x (τ ) with τ randomly sampled from I using Prob(τ = t) = η t / s∈I η s . Under Assumption A.1, we have to use stochastic subgradients ξ f and ξ g in the SSG method. Moreover, since g(x (t) ) cannot be evaluated exactly, we sample a mini-batch {ω (t) i } B i=1 of stochastic estimators of g(x (t) ) with a batch size of B, and then construct an estimator of g(x (t) ), denoted byw (t) , by averaging the samples. Then, we use the conditionw (t) ≤ ǫ t in the SSG method to determine when the stochastic subgradient should be switched between ξ f and ξ g . The SSG method with those modifications is called the stochastic SSG method and presented in Algorithm 2. Note that Algorithm 2 is reduced to Algorithm 1 when, for t ≥ 0, it holds deterministically that ξ (t) f = ζ (t) f , ξ (t) g = ζ (t) g ,ω (t) = g(x (t) ), σ = 0 and B = 1.(21) The oracle complexity of Algorithm 1 is given in Theorem 4.3 while that of Algorithm 2 is given in Theorems A.11 and A.12. A.3 Technical lemmas and propositions In this section, we first introduce additional notations and then present a few technical lemmas and propositions which are needed to prove the convergence properties of Algorithms 1 and 2. Let I(·) be an indicator of an event, which equals one if the event occurs and zero otherwise. For each iterate x (t) in Algorithms 1 and 2, let ζ (t) f := E[ξ (t) f ] ∈ ∂f (x (t) ) and ζ (t) g := E[ξ (t) g ] ∈ ∂g(x (t) ) and let x (t) := x(x (t) ) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and let λ t ≥ 0 be the corresponding Lagrangian multiplier satisfying (5), which exists by Assumption 4.1B. Under Assumption A.1, let E t [·] := E[· ω (0) , ξ (0) f , ξ (0) g ,ω (1) , ξ (1) f , ξ (1) g , . . . ,ω (t−1) , ξ (t−1) f , ξ (t−1) g ], i.e., the conditional expectation conditioning on all stochastic events before iteration t. Let E τ [·] be the expectation taken only over the random index τ when the algorithm is terminated. The proposition below characterizes the relationship between x (t) −x (t) and the parameters T , S, η t and ǫ t . Proposition A.2. Suppose Assumptions 3.1, 4.1 and A.1 hold and g is µ-strongly convex (µ can be zero). Algorithm 2 guarantees T −1 t=S η tρ λ t I(ω (t) ≤ ǫ t ) + η tρ I(ω (t) > ǫ t ) · µ 2 x (t) − x (t) 2 + T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t g(x (t) ) I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ g(x (t) )I(ω (t) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t ξ (t) f 2 I(ω (t) ≤ ǫ t ) + η 2 t ξ (t) g 2 I(ω (t) > ǫ t ) + T −1 t=S η tρ ξ (t) f − ζ (t) f , x (t) − x (t) I(ω (t) ≤ ǫ t ) + η tρ ξ (t) g − ζ (t) g , x (t) − x (t) I(ω (t) > ǫ t ) . Proof. Let ξ (t) = ξ (t) f if t ∈ I and ξ (t) = ξ (t) g if t ∈ J. Similarly, let ζ (t) = ζ (t) f ∈ ∂f (x (t) ) if t ∈ I and ζ (t) = ζ (t) g ∈ ∂g(x (t) ) if t ∈ J. By the updating equation of x (t+1) , we have x (t+1) − x (t) 2 = proj X (x (t) − η t ξ (t) ) − x (t) 2 = proj X (x (t) − η t ξ (t) ) − proj X ( x (t) ) 2 ≤ x (t) − η t ξ (t) − x (t) 2 = x (t) − x (t) 2 − 2η t ξ (t) , x (t) − x (t) + η 2 t ξ (t) 2 . Multiplying the inequality above byρ/2 and adding f ( x (t) ) to both sides, we obtain f ( x (t) ) +ρ 2 x (t+1) − x (t) 2 ≤ f ( x (t) ) +ρ 2 x (t) − x (t) 2 − η tρ ξ (t) , x (t) − x (t) + η 2 tρ 2 ξ (t) 2 = ϕ(x (t) ) − η tρ ζ (t) , x (t) − x (t) + η 2 tρ 2 ξ (t) 2 − η tρ ξ (t) − ζ (t) , x (t) − x (t) ,(22) where the equality is by the definition of ϕ(x) in (3). Since x (t) is a feasible solution to problem (3) with x = x (t+1) , we have ϕ(x (t+1) ) ≤ f ( x (t) ) +ρ 2 x (t+1) − x (t) 2 , which, together with (22), implies η tρ ζ (t) , x (t) − x (t) ≤ ϕ(x (t) ) − ϕ(x (t+1) ) + η 2 tρ 2 ξ (t) 2 − η tρ ξ (t) − ζ (t) , x (t) − x (t) .(23) Next, we will bound ζ (t) , x (t) − x (t) from below when t ∈ I and t ∈ J, separately. Suppose t ∈ I soω (t) ≤ ǫ t , ζ (t) = ζ (t) f and ξ (t) = ξ (t) f . By the ρ-weak convexity of f , we have ζ (t) , x (t) − x (t) ≥ f (x (t) ) − f ( x (t) ) − ρ 2 x (t) − x (t) 2 = f (x (t) ) − f ( x (t) ) −ρ 2 x (t) − x (t) 2 +ρ − ρ 2 x (t) − x (t) 2 .(24) Consider the convex optimization problem (3) with x = x (t) . By Assumption 4.1B, there exists a Lagrangian multiplier λ t ≥ 0 such that λ t g( x (t) ) = 0 (complementary slackness) and x (t) = arg min x∈X f (x) +ρ 2 x − x (t) 2 + λ t g(x) . Since the objective function above is (ρ − ρ + λ t µ)-strongly convex, we have f (x (t) ) + λ t g(x (t) ) = f (x (t) ) +ρ 2 x (t) − x (t) 2 + λ t g(x (t) ) ≥ f ( x (t) ) +ρ 2 x (t) − x (t) 2 + λ t g( x (t) ) +ρ − ρ + λ t µ 2 x (t) − x (t) 2 , which, by the fact that λ t g( x (t) ) = 0, implies f (x (t) ) − f ( x (t) ) −ρ 2 x (t) − x (t) 2 ≥ − λ t g(x (t) ) +ρ − ρ + λ t µ 2 x (t) − x (t) 2 . Applying this inequality and inequality (24) to (23) leads to η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t g(x (t) ) + η tρ λ t µ 2 x (t) − x (t) 2 ≤ ϕ(x (t) ) − ϕ(x (t+1) ) + η 2 tρ 2 ξ (t) f 2 − η tρ ξ (t) f − ζ (t) f , x (t) − x (t) .(25)Suppose t ∈ J soω (t) > ǫ t , ζ (t) = ζ (t) g and ξ (t) = ξ (t) g . By the µ-strong convexity of g and the fact that g( x (t) ) ≤ 0, we have ζ (t) , x (t) − x (t) − µ 2 x (t) − x (t) 2 ≥ g(x (t) ) − g( x (t) ) ≥ g(x (t) ). Applying this inequality to (23) leads to η tρ g(x (t) ) + η tρ µ 2 x (t) − x (t) 2 ≤ ϕ(x (t) ) − ϕ(x (t+1) ) + η 2 tρ 2 ξ (t) g 2 − η tρ ξ (t) g − ζ (t) g , x (t) − x (t) .(26) Recall that I(t ∈ I) = I(ω (t) ≤ ǫ t ) and I(t ∈ J) = I(ω (t) > ǫ t ). Summing up (25) and (26) for t = S, S + 1, . . . , T − 1, we have T −1 t=S η tρ λ t I(ω (t) ≤ ǫ t ) + η tρ I(ω (t) > ǫ t ) µ 2 x (t) − x (t) 2 + T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t g(x (t) ) I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ g(x (t) )I(ω (t) > ǫ t ) ≤ ϕ(x (S) ) − ϕ(x (T ) ) +ρ 2 T −1 t=S η 2 t ξ (t) f 2 I(ω (t) ≤ ǫ t ) + η 2 t ξ (t) g 2 I(ω (t) > ǫ t ) − T −1 t=S η tρ ξ (t) f − ζ (t) f , x (t) − x (t) I(ω (t) ≤ ǫ t ) + η tρ ξ (t) g − ζ (t) g , x (t) − x (t) I(ω (t) > ǫ t ) . The conclusion is then implied by the facts that ϕ( x (T ) ) = f ( x (T ) )+ρ 2 x (T ) −x (T ) 2 ≥ f ( x (T ) ) and ϕ(x (S) ) = f ( x (S) ) +ρ 2 x (S) − x (S) 2 ≤ f ( x (T ) ) +ρ 2 x (T ) − x (S) 2 ≤ f ( x (T ) ) +ρ D 2 2 . Lemma A.3. Suppose Assumption A.1 holds. Given any x ∈ X , let {ω i } B i=1 be a mini-batch of stochastic estimators of g at x andω = 1 B B i=1 ω i . It holds that, for any δ ∈ (0, 1), Prob ω > g(x) + 3 B σ ln(1/δ) ≤ δ and Prob ω < g(x) − 3 B σ ln(1/δ) ≤ δ. Proof. The conclusion is guaranteed by Assumption A. g(x (t) )I(ω (t) ≤ ǫ t ) ≤ ǫ t + 3 B σ ln((T − S)/δ) I(ω (t) ≤ ǫ t )(27) for t = S, S + 1, . . . , T − 1 and, consequently, with probability at least 1 − δ that T −1 t=S η tρ λ t g(x (t) )I(ω (t) ≤ ǫ t ) ≤ T −1 t=S η tρ λ t ǫ t + 3 B σ ln((T − S)/δ) I(ω (t) ≤ ǫ t ). Similarly, Algorithm 2 guarantees with probability at least 1 − δ that T −1 t=S η tρ g(x (t) )I(ω (t) > ǫ t ) ≥ T −1 t=S η tρ ǫ t − 3 B σ ln((T − S)/δ) I(ω (t) > ǫ t ). Proof. For any t, x (t) is determined byω (0) , ξ (0) f , ξ (0) g ,ω (1) , ξ (1) f , ξ (1) g , . . . ,ω (t−1) , ξ (t−1) f and ξ (t−1) g , whileω (t) is generated after x (t) . Hence, according to Lemma A.3, we have with a probability of at least 1−δ/(T −S) that g(x (t) )I(ω (t) ≤ ǫ t ) ≤ ω (t) + 3 B σ ln((T − S)/δ) I(ω (t) ≤ ǫ t ), which implies (27). The first conclusion is then obtained by taking the union bound for the events above for t = S, S + 1, . . . , T − 1. The second conclusion can be proved in a similar way. Lemma A.5. Suppose Assumption A.1 holds. For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ that T −1 t=S η 2 t ξ (t) f 2 I(ω (t) ≤ ǫ t ) + η 2 t ξ (t) g 2 I(ω (t) > ǫ t ) ≤ T −1 t=S η 2 t M 2 + max 12 ln(2/δ), 4 3 ln(2/δ) T −1 t=S η 4 t M 4 . Proof. For any t, x (t) is determined byω (0) , ξ (0) f , ξ (0) g ,ω (1) , ξ (1) f , ξ (1) g , . . . ,ω (t−1) , ξ (t−1) f and ξ (t−1) g . Also,ω (t) , ξ (t) f and ξ (t) g are independent and generated after x (t) . Hence, by As- sumption A.1, we have E t   exp   η 2 t ξ (t) f 2 I(ω (t) ≤ ǫ t ) + η 2 t ξ (t) g 2 I(ω (t) > ǫ t ) η 2 t M 2     = E t   I(ω (t) ≤ ǫ t ) exp   ξ (t) f 2 M 2     + E t I(ω (t) > ǫ t ) exp ξ (t) g 2 M 2 = E t I(ω (t) ≤ ǫ t ) E t   exp   ξ (t) f 2 M 2     + E t I(ω (t) > ǫ t ) E t exp ξ (t) g 2 M 2 ≤ E t I(ω (t) ≤ ǫ t ) exp (1) + E t I(ω (t) > ǫ t ) exp (1) = exp (1) , where the second equality is by the conditional independence betweenω (t) , ξ T −1 t=S η tρ ξ (t) f − ζ (t) f , x (t) − x (t) I(ω (t) ≤ ǫ t ) + η tρ ξ (t) g − ζ (t) g , x (t) − x (t) I(ω (t) > ǫ t ) ≤ 3 ln(1/δ) T −1 t=S 4η 2 tρ 2 M 2 D 2 . Proof. By Assumption A.1 and Jensen's inequality, we have (1) and exp exp ζ (t) f 2 M 2 ≤ E t exp ξ (t) f 2 M 2 ≤ expζ (t) g 2 M 2 ≤ E t exp ξ (t) g 2 M 2 ≤ exp(1), which, by Jensen's inequality again, implies E t exp 2 ξ (t) f 2 + 2 ζ (t) f 2 4M 2 = E t exp ξ (t) f 2 2M 2 exp ζ (t) f 2 2M 2 ≤ E t exp ξ (t) f 2 M 2 exp 1 2 ≤ exp(1)(28) and E t exp 2 ξ (t) g 2 + 2 ζ (t) g 2 4M 2 = E t exp ξ (t) g 2 2M 2 exp ζ (t) g 2 2M 2 ≤ E t exp ξ (t) g 2 M 2 exp 1 2 ≤ exp(1).(29) For any t, x (t) is determined byω (0) , ξ (0) f , ξ (0) g ,ω (1) , ξ (1) f , ξ (1) g , . . . ,ω (t−1) , ξ (t−1) f and ξ (t−1) g . Also,ω (t) , ξ (t) f and ξ (t) g are independent and generated after x (t) . Hence, by Assumption A.1, we have E t η tρ ξ (t) f − ζ (t) f , x (t) − x (t) I(ω (t) ≤ ǫ t ) + η tρ ξ (t) g − ζ (t) g , x (t) − x (t) I(ω (t) > ǫ t ) = 0, and E t   exp    η tρ ξ (t) f − ζ (t) f , x (t) − x (t) I(ω (t) ≤ ǫ t ) + η tρ ξ (t) g − ζ (t) g , x (t) − x (t) I(ω (t) > ǫ t ) 2 4η 2 tρ 2 M 2 D 2       = E t   I(ω (t) ≤ ǫ t ) exp    ξ (t) f − ζ (t) f , x (t) − x (t) 2 4M 2 D 2       + E t   I(ω (t) > ǫ t ) exp    ξ (t) g − ζ (t) g , x (t) − x (t) 2 4M 2 D 2       ≤ E t I(ω (t) ≤ ǫ t ) E t exp ξ (t) f − ζ (t) f 2 x (t) − x (t) 2 4M 2 D 2 + E t I(ω (t) > ǫ t ) E t exp ξ (t) g − ζ (t) g 2 x (t) − x (t) 2 4M 2 D 2 ≤ E t I(ω (t) ≤ ǫ t ) E t exp 2 ξ (t) f 2 + 2 ζ (t) f 2 4M 2 + E t I(ω (t) > ǫ t ) E t exp 2 ξ (t) g 2 + 2 ζ (t) g 2 4M 2 ≤ E t I(ω (t) ≤ ǫ t ) exp (1) + E t I(ω (t) > ǫ t ) exp (1) = exp (1) , where the first inequality is by the Cauchy-Schwarz inequality and the conditional independence betweenω (t) , ξ (t) f and ξ (t) g , the second inequality is by Assumption 3.1B and the last inequality is by (28) and (29). Then the conclusion is guaranteed by Lemma 2 (Case A) in [51] by choosing Ω = 3 ln(1/δ) in their bound. Suppose g is convex but not strongly convex, i.e., µ = 0. Taking the union bound of the four events in Lemmas A.4, A.5 and A.6 with δ replaced by δ/4 and applying the four inequalities (two from Lemma A.4, one from Lemma A.5 and one from Lemma A.6) holding in these events to Proposition A.2, we have the following bounds. Proposition A.7. Suppose Assumptions 3.1, 4.1 and A.1 hold and g is µ-strongly convex (µ can be zero). Algorithm 2 guarantees with probability at least 1 − δ/(4(T − S)) that g(x (t) )I(ω (t) ≤ ǫ t ) ≤ ǫ t + 3 B σ ln(4(T − S)/δ) I(ω (t) ≤ ǫ t )(30) for t = S, S + 1, . . . , T − 1 and, consequently, with probability at least 1 − δ that T −1 t=S η tρ λ t I(ω (t) ≤ ǫ t ) + η tρ I(ω (t) > ǫ t ) · µ 2 x (t) − x (t) 2 + T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t ǫ t I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(ω (t) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t M 2 +ρ 2 max 12 ln(8/δ), 4 3 ln(8/δ) T −1 t=S η 4 t M 4(31)+ 3 ln(4/δ) T −1 t=S 4η 2 tρ 2 M 2 D 2 + T −1 t=S η tρ 3 B σ ln(4(T − S)/δ) λ t I(ω (t) ≤ ǫ t ) + I(ω (t) > ǫ t ) . A.4 Proof of Theorem 4.3 Although the technical results in the previous sections are derived for Algorithm 2 in the stochastic case, they all apply to Algorithm 1 which is a special case of Algorithm 2 when all oracles are deterministic, i.e., when (21) holds. Hence, we can uses these technical results to prove Theorem 4.3 as follows. Proof of Theorem 4.3. Because Algorithm 1 is a deterministic special case of Algorithm 2 when (21) holds and also because we do not assume strong convexity in g (µ = 0), we can simplify the inequality in Proposition A.2 as follows T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t g(x (t) ) I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ g(x (t) )I(g(x (t) ) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t ζ (t) f 2 I(g(x (t) ) ≤ ǫ t ) + η 2 t ζ (t) g 2 I(g(x (t) ) > ǫ t ) . By the facts that g(x (t) )I(g(x (t) ) ≤ ǫ t ) ≤ ǫ t I(g(x (t) ) ≤ ǫ t ) and g(x (t) )I(g(x (t) ) > ǫ t ) ≥ ǫ t I(g(x (t) ) > ǫ t ), the inequality above implies T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t ǫ t I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(g(x (t) ) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t ζ (t) f 2 I(g(x (t) ) ≤ ǫ t ) + η 2 t ζ (t) g 2 I(g(x (t) ) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t M 2 ,(32) where the second inequality is because of Assumption 3.1. We first prove that, if S, T , η t and ǫ t are chosen such that ǫ t (1 + λ t ) ≤ ǫ 2 (ρ − ρ)(33) and T −1 t=S η tρ ǫ t >ρ D 2 2 +ρ 2 T −1 t=S η 2 t M 2 ,(34) we must have g(x (t) ) ≤ ǫ t for at least one t in {S, S + 1, . . . , T − 1} (i.e., I = ∅) and E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 (so E τ [ x (τ ) − x (τ ) ] ≤ ǫ). Suppose (34) holds and g(x (t) ) > ǫ t for t = S, S + 1, . . . , T − 1, i.e., I = ∅. (32) becomes exactly the opposite of (34). This contradiction means g(x (t) ) ≤ ǫ t for at least one t in {S, S + 1, . . . , T − 1}. Suppose (33) and (34) hold but E τ [ x (τ ) − x (τ ) 2 ] > ǫ 2 . Since τ is generated by Output I, we have ǫ 2 < E τ [ x (τ ) − x (τ ) 2 ] = T −1 t=S η t I(g(x (t) ) ≤ ǫ t ) x (t) − x (t) 2 T −1 t=S η t I(g(x (t) ) ≤ ǫ t ) . ( Note that the right-hand side of (35) is well-defined because we just proved I = ∅. (35) and (33) imply T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t ǫ t I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(g(x (t) ) > ǫ t ) > T −1 t=S η tρ (ρ − ρ)ǫ 2 − η tρ λ t ǫ t I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(g(x (t) ) > ǫ t ) ≥ T −1 t=S η tρ ǫ t I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(g(x (t) ) > ǫ t ) = T −1 t=S η tρ ǫ t ,(36) where the second inequality is because of (33). Combining this inequality and (32) leads to the opposite of (34). This contradiction means E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 . Given the result above, we only need to show that the two choices of S, T , η t and ǫ t ensure (33) and (34). In Case I, (33) holds because of Lemma 4.2 and the choice of ǫ t . Let η = η t = 2ǫ 2 (ρ−ρ) 5(1+Λ)M 2 for any t. Using Lemma 4.2 and plugging the values of S, T , η t and ǫ t in (34), we can show that (34) is equivalent to T ηρǫ 2 (ρ − ρ) 1 + Λ >ρ D 2 2 +ρ 2 T η 2 M 2 , which, after dividing both sides by T ηρ, can be equivalently written as ǫ 2 (ρ − ρ) 1 + Λ > D 2 2T η + ηM 2 2 . By the values of η and T , each summand in the right-hand side of the inequality above is no more than ǫ 2 (ρ−ρ) 5(1+Λ) so the right-hand side of the inequality above no more than 2ǫ 2 (ρ−ρ) 5(1+Λ) which is strictly less than the left-hand side. This means (34) holds with this choice of parameters and thus E τ [ x (τ ) − x (τ ) ] ≤ ǫ. By the convexity of g and the choices of η t and ǫ t , we have E τ [g(x (τ ) )] ≤ ǫ 2 (ρ−ρ) 1+Λ . In Case II, by the choices of ǫ t and T , we have, for any t ∈ {S, S + 1, . . . , T − 1}, ǫ t = 5M D √ t + 1 ≤ 5M D √ S + 1 = 5M D T /2 + 1 ≤ ǫ 2 (ρ − ρ) 1 + Λ .(37) This further implies (33) because of Lemma 4.2. Note that η t and ǫ t are decreasing in t. Hence, the left-hand side of (34) satisfies T −1 t=S η tρ ǫ t > T 2 η Tρ ǫ T = 5T 2T + 2ρ D 2 ≥ 5ρD 2 4 .(38) The right-hand side of (34) satisfieŝ ρD 2 2 +ρ 2 T −1 t=S η 2 t M 2 =ρ D 2 2 +ρ 2 D 2 T −1 t=S 1 t + 1 ≤ρD 2 ,(39) where the equality is obtained by plugging in the definition of η t and the inequality is because T −1 t=S 1 t+1 ≤ T S 1 t dt = ln(T /S) = ln(2) ≤ 1. The right-hand side of (38) is strictly greater than the right-hand side (39), which means (34) holds and thus E τ [ x (τ ) − x (τ ) ] ≤ ǫ. By the convexity of g and the choices of η t and ǫ t , we have E τ [g(x (τ ) )] ≤ ǫ S ≤ ǫ 2 (ρ−ρ) 1+λ according to (37). A.5 Complexity analysis for strongly convex constraints Suppose f and g in (1) are deterministic, i.e., (21) holds, and g is µ-strongly convex and µ > 0. The complexity of Algorithm 1 is characterized by the following theorem. Before presenting its proof, we would like to make a few remarks. Remark A.9. According to Theorem A.8, strong convexity in the constraint function g does not reduce the O(1/ǫ 4 ) complexity of the SSG method for finding a nearly ǫ-stationary point. However, strong convexity brings benefit on other aspects. First, we can simply set ǫ t = 0 when g is strongly convex, which makes stepsize η t the only tuning parameter. Second, the theoretical complexity no longer depends on Λ, the upper bound of the dual variables, so it can be strictly better than the one in Theorem 4.3 when Λ is very large. E τ [ x (t) − x (t) 2 ] ≤ 1 min{ρ − ρ, µ/2} DM √ T + DM √ T ≤ ǫ 2 2 + ǫ 2 2 = ǫ 2 , where, in the first inequality, we use the facts that T −1 t=S η t ≥ T 2 η T −1 = √ T D 2M and that T −1 t=S η 2 t = T −1 t=S D 2 M 2 (t+1) ≤ T S D 2 M 2 t dt = D 2 M 2 ln(T /S) = D 2 M 2 ln(2) ≤ D 2 M 2 , and the second inequality is by the choice of T . A.6 Complexity analysis when subgradient oracle is stochastic In this section, we will analyze the oracle complexity of Algorithm 2 under Assumption A.1 with an additional condition that ω (t) = g(x (t) ), σ = 0 and B = 1,(42) which is a relaxation of (21). In this case, the function value of g is can be accessed deterministically while the subgradients of f and g are still stochastic. This holds in the setting of zeroth-order or derivative-free optimization where one can evaluate f and g exactly as black boxes and then use their values to construct stochastic approximate subgradients. We will show that the complexity of Algorithm 2 when (42) holds is the same as Algorithm 1 (equivalent to Algorithm 2 when (21) holds). For simplicity of notation, we denote x(x (t) ) by x (t) . Theorem A.11. Suppose Assumptions 3.1, 4.1 and A.1 hold and Λ is as in (8). Also, suppose (42) holds. Let x(x (t) ) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and x (τ ) is generated by Algorithm 2. Algorithm 2 guarantees E τ [ x (τ ) − x (τ ) ] ≤ ǫ and E τ [g(x (τ ) )] ≤and T ≥ 2E 2 M 2 D 2 (1+Λ) 2 ǫ 4 (ρ−ρ) 2 = O(1/ǫ 4 ). This theorem shows that, the complexity remains O(1/ǫ 4 ) if the subgradient oracles are stochastic but the function value oracles remain deterministic. The only difference is that the result holds in a high probability. This complexity matches the lower-bound complexity for stochastic smooth non-convex unconstrained optimization [5,32], so it is optimal. Proof of Theorem A. 11. By (42) and the fact that µ = 0, we can drop the last term in the right-hand side of (31) and have T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t ǫ t I(g(x (t) ) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(g(x (t) ) > ǫ t ) ≤ρ D 2 2 +ρ 2 T −1 t=S η 2 t M 2 +ρ 2 max 12 ln(8/δ), 4 3 ln(8/δ) T −1 t=S η 4 t M 4(43)+ 3 ln(4/δ) T −1 t=S 4η 2 tρ 2 M 2 D 2 . with a probability of at least 1 − δ. In the rest of the proof, we always assume (43) holds. We first prove that, if S, T , η t and ǫ t are chosen such that (33) holds and T −1 t=S η tρ ǫ t >ρ D 2 2 +ρ 2 T −1 t=S η 2 t M 2 +ρ 2 max 12 ln(8/δ), 4 3 ln(8/δ) T −1 t=S η 4 t M 4(44)+ 3 ln(4/δ) T −1 t=S 4η 2 tρ 2 M 2 D 2 , we must have g(x (t) ) ≤ ǫ t for at least one t in {S, S + 1, . . . , T − 1} (i.e., I = ∅) and E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 (so E τ [ x (τ ) − x (τ ) ] ≤ ǫ). Suppose (44) holds and g(x (t) ) > ǫ t for t = S, S + 1, . . . , T − 1, i.e., I = ∅. (43) becomes exactly the opposite of (44). This contradiction means g(x (t) ) ≤ ǫ t for at least one t in {S, S + 1, . . . , T − 1}. Suppose (33) and (44) hold but E τ [ x (τ ) − x (τ ) 2 ] > ǫ 2 . Since τ is generated by Algorithm 2, we have (35). Note that the right-hand side of (35) is well-defined because we just proved I = ∅. (35) and (33) imply (36). Combining (36) and (44) leads to the opposite of (43). This contradiction means E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 . Given the result above, we only need to show that the choices of S, T , η t and ǫ t ensure (33) and (44). In Case I, (33) holds because of Lemma 4.2 and the choice of ǫ t . Let η = η t = 2ǫ 2 (ρ−ρ) 5(1+Λ)M 2 for any t. Using Lemma 4.2 and plugging the values of S, T , η t and ǫ t in (44), we can show that (44) is equivalent to T ηρǫ 2 (ρ − ρ) 1 + Λ >ρ D 2 2 +ρ 2 T η 2 M 2 +ρ 2 max 12 ln(8/δ), 4 3 ln(8/δ) √ T η 2 M 2 + 2 3 ln(4/δ)ρ √ T ηM D, which, after dividing both sides by T ηρ, can be equivalently written as ǫ 2 (ρ − ρ) 1 + Λ > D 2 2T η + ηM 2 2 + 1 2 max 12 ln(8/δ), 4 3 ln(8/δ) ηM 2 √ T + 2 3 ln(4/δ)M D √ T . By the values of η and T , each summand in the right-hand side of the inequality above is no more than ǫ 2 (ρ−ρ) 5(1+Λ) so the right-hand side of the inequality above no more than 4ǫ 2 (ρ−ρ) 5(1+Λ) which is strictly less than the left-hand side. This means (44) holds with this choice of parameters and thus E τ [ x (τ ) − x (τ ) ] ≤ ǫ. We can prove E τ [g(x (τ ) )] ≤ ǫ 2 (ρ−ρ) 1+λ in the same way as in the proof of Theorem 4.3. In Case II, by the choices of ǫ t and T , we have, for any t ∈ {S, S + 1, . . . , T − 1}, ǫ t = EM D √ t + 1 ≤ EM D √ S + 1 = EM D T /2 + 1 ≤ ǫ 2 (ρ − ρ) 1 + Λ .(45) value of g instead of its stochastic subgradient. Note that, in the fully deterministic case, Algorithm 1 essentially updates x (t) along a hybrid subgradient I(g(x (t) ) ≤ ǫ t )ζ (t) f + I(g(x (t) ) > ǫ t )ζ (t) g .(48) If only the subgradients are stochastic but the function value g(x (t) ) remains deterministic, the hybrid stochastic subgradient I(g(x (t) ) ≤ ǫ t )ξ (t) f + I(g(x (t) ) > ǫ t )ξ (t) g provides an unbiased estimation of (48). In this case, we can still obtain complexity of O(1/ǫ 4 ) (see Theorem A.11) by a proof similar to the deterministic case. However, when g(x (t) ) must be queried through some unbiased estimator w (t) , the naively constructed direction I(w (t) ≤ ǫ t )ξ (t) f + I(w (t) > ǫ t )ξ (t) g is not an unbiased estimator of (48). To tackle this issue, we have to query a mini-batch of w (t) of size B, i.e., {ω (t) i } B i=1 to constructw (t) as an estimation of g(x (t) ) with a high accuracy in a high probability. In this way, we can use I(w (t) ≤ ǫ t )ξ (t) f + I(w (t) > ǫ t )ξ (t) g as a nearly unbiased estimator of (48), which leads to the switching condition in Algorithm 2. As a consequence, the oracle complexity of the function value of g increases from O(1/ǫ 4 ) tõ O(1/ǫ 8 ). 1 although the oracle complexity of stochastic subgradients remains O(1/ǫ 4 ). Theorem A.12. Suppose Assumptions 3.1, 4.1 and A.1 hold and Λ is as in (8). Let x(x (t) ) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and x (τ ) is generated by Algorithm 2. Algo- rithm 2 guarantees E τ [ x (τ ) − x (τ ) ] ≤ ǫ and E τ [g(x (τ ) )] ≤ 2ǫ 2 (ρ−ρ) 1+Λ with probability at least 1 − δ in either of the following cases. and B = 3T σ 2 ln(2T /δ)(1+Λ) 2 2M 2 D 2 . In each iteration of Algorithm 2, we query one stochastic subgradient of f or g but B stochastic function values of g. In both Case I and Case II, we have T = O(1/ǫ 4 ) and B =Õ(1/ǫ 4 ) so the subgradient oracle complexity is still O(1/ǫ 4 ) but the function value oracle complexity becomesÕ(1/ǫ 8 ), which is higher than the O(1/ǫ 6 ) complexity by the double-loop methods in [14,58]. It is our future work to reduce the complexity when g is stochastic, for example, by a single-loop primal-dual method that uses a hybrid subgradient like ξ (t) f + λ t ξ (t) g with the dual variable λ t updated by only one sample of w (t) . Proof of Theorem A.12. By Proposition A.7, with a probability of at least 1 − δ, we simultaneously have (31) with µ = 0 and (30) for t = S, S + 1, . . . , T − 1. In the rest of the proof, we assume (31) with µ = 0 and (30) hold for t = S, S + 1, . . . , T − 1. 1Õ (·) suppresses logarithmic factors in the order of complexity. We first prove that, if S, T , B, η t and ǫ t are chosen such that (33) holds and + 3 ln(4/δ) T −1 t=S 4η 2 tρ 2 M 2 D 2 + T −1 t=S η tρ 3 B σ ln(4(T − S)/δ)(1 + Λ), we must haveω (t) ≤ ǫ t for at least one t in {S, S + 1, . . . , T − 1} (i.e., I = ∅) and E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 (so E τ [ x (τ ) − x (τ ) ] ≤ ǫ). Suppose (49) holds andω (t) > ǫ t for t = S, S + 1, . . . , T − 1, i.e., I = ∅. (31) contradicts with (49) as Λ ≥ λ t . This contradiction meansω (t) ≤ ǫ t for at least one t in {S, S+1, . . . , T −1} so I = ∅. Suppose (33) and (49) hold but E τ [ x (τ ) − x (τ ) 2 ] > ǫ 2 . Since τ is generated by Algorithm 2, we have ǫ 2 < E τ [ x (τ ) − x (τ ) 2 ] = T −1 t=S η t I(ω (t) ≤ ǫ t ) x (t) − x (t) 2 T −1 t=S η t I(ω (t) ≤ ǫ t ) . (50) Note that the right-hand side of (50) is well-defined because we just proved I = ∅. (50) and (33) imply T −1 t=S η tρ (ρ − ρ) x (t) − x (t) 2 − η tρ λ t ǫ t I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(ω (t) > ǫ t ) > T −1 t=S η tρ (ρ − ρ)ǫ 2 − η tρ λ t ǫ t I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(ω (t) > ǫ t ) ≥ T −1 t=S η tρ ǫ t I(ω (t) ≤ ǫ t ) + T −1 t=S η tρ ǫ t I(ω (t) > ǫ t ) ≥ T −1 t=S η tρ ǫ t ,(51) where the second inequality is because of (33). Combining (51) and (49) leads to the opposite of (31). This contradiction means E τ [ x (τ ) − x (τ ) 2 ] ≤ ǫ 2 . Given the result above, we only need to show that the choices of S, T , B, η t and ǫ t ensure (33) and (49). In Case I, (33) holds because of Lemma 4.2 and the choice of ǫ t . Let η = η t = 2ǫ 2 (ρ−ρ) By the values of η, B and T , each of the first four summands on the right-hand side of the inequality above is no more than ǫ 2 (ρ−ρ) 5(1+Λ) while the last summand is no more than ǫ 2 (ρ−ρ) 10(1+Λ) , so the right-hand side of the inequality above no more than 9ǫ 2 (ρ−ρ) 10(1+Λ) which is strictly less than the left-hand side. This means (34) holds with this choice of parameters and thus E τ [ x (τ ) − x (τ ) ] ≤ ǫ. Moreover, by (30), we have E τ [g(x (τ ) )] = T −1 t=0 η t g(x (t) )I(ω (t) ≤ ǫ t ) T −1 t=0 η t I(ω (t) ≤ ǫ t ) ≤ T −1 t=0 η t ǫ t + 3 B σ ln(4T /δ) I(ω (t) ≤ ǫ t ) T −1 t=0 η t I(ω (t) ≤ ǫ t ) ≤ ǫ 2 (ρ − ρ) 1 + Λ + 3 B σ ln(4T /δ) ≤ 2ǫ 2 (ρ − ρ) 1 + Λ , where the last inequality is because of the choice of B. In Case II, by the choice of ǫ t , we have (45) holds. This further implies (33) because of Lemma 4.2. Note that η t and ǫ t are decreasing in t. Hence, we also have (46). The right-hand side of (49) satisfieŝ ρD 2 2 +ρ 2 Remark A.13. When g is µ-strongly convex with µ > 0, under the same assumptions as Theorem A.11, we can establish O(1/ǫ 4 ) oracle complexity for Algorithm 2. Similarly, under the same assumptions as Theorem A.12, we can also establishÕ(1/ǫ 8 ) oracle complexity for Algorithm 2. These two complexity results of Algorithm 2 can be proved in a similar way as Theorem A.8. Since those results do not provide additional insights on the complexity and analysis for the SSG method, we do not include them in the paper. Similar to Theorem A.8, when µ > 0, we only need to set ǫ t = 0 in Algorithm 2 and at last generate τ by Output II in Algorithm 1. This indicates that strong convexity in g also reduces the number of tuning parameters in Algorithm 2 just as it does for Algorithm 1. B Convergence analysis for weakly convex constraints In this section, we present two non-trivial practical examples that satisfy Assumption 5.1 and then analyze the complexity of Algorithm 1 when f and g are both weakly convex. B.1 Examples that satisfy Assumption 5.1 In this section, we present two examples that satisfy Assumption 5.1, especially, Assumption 5.1B. B.1.1 Demographic parity constraint Problem (17) is an instance of (1) with X = R d , f (x) = L(x) + λSCAD(x), g(x) = R 0 (x) − κ, where L, R 0 and SCAD are defined in (13), (16) and (18), respectively. It is easy to verify that problem (17) satisfies Assumption 3.1. In this section, we will prove that problem (17) also satisfies Assumption 5.1. Since L(x) is convex and SCAD(x) is 2-weakly convex, f (x) is 2λ-weakly convex. Let h(x) = 1 n p np i=1 σ(x ⊤ a p i ) − 1 n u nu i=1 σ(x ⊤ a u i ) (53) so g(x) = |h(x)| − κ. Recalling that σ(z) = exp(z)/(1 + exp(z)), we have ∇h(x) = 1 n p np i=1 σ(x ⊤ a p i )(1 − σ(x ⊤ a p i ))a p i − 1 n u nu i=1 σ(x ⊤ a u i )(1 − σ(x ⊤ a u i ))a u i .(54) It is easy to prove that h(x) is Lipschitz continuous with a constant of α = 1 4n p np i=1 a p i + 1 4n u nu i=1 a u i(55) and that ∇h(x) is Lipschitz continuous with a constant of β = 1 4n p np i=1 a p i 2 + 1 4n u nu i=1 a u i 2 ,(56) B.1.2 Smoothly clipped absolute deviation constraint Suppose f (x) in (1) is any training loss that satisfies Assumptions 3.1, 5.1A and 5.1C. We consider a SCAD constraint for promoting sparsity of model x. It is formulated as g(x) = SCAD(x) − κ = d i=1 s(x i ) − κ ≤ 0,(65) where s(x i ) is defined in (18). Jia and Grimmer [42] show that, if κ is not divisible by three, constraint (65) satisfies the strong Mangasarian-Fromovitz constraint qualification (MFCQ), which is used in [42] to bound the Lagrangian multiplier in a way similar to (10). Motivated by the finding in [42], we show that (65) also satisfies the uniform Slator's condition in Assumption 5.1B. Suppose g(x) ≥ −(κ−3k)/2 so SCAD(x) ∈ [(κ+3k)/2, (κ+3(k+1))/2]. By (66), there exists an index i such that dist(x i , Z) ≥ q, which implies x i ∈ [q, 2−q] or x i ∈ [−2+q, −q]. Without loss of generality, we assume i = 1 and x i ∈ [q, 2 − q] since the proof when x 1 ∈ [−2 + q, −q] is the same. By the definition of s in (18), we have s ′ (x 1 ) ∈ [2q, 1]. Let y i = x i for i = 1 and y 1 = x 1 − ηs ′ (x 1 ) with η = min{q/2, 1/(2 +ρ)}. We then have y 1 ≥ x 1 − q/2 ≥ q/2. Since the function s(y 1 ) +ρ 2 (y 1 − x 1 ) 2 is smooth on [q/2, x 1 ] and its gradient with respect to y 1 is (2 +ρ)-Lipschitz continuous, we have g(y) +ρ 2 y − x 2 ≤ g(x) + s ′ (x 1 ) · (y 1 − x 1 ) + 2 +ρ 2 (y 1 − x 1 ) 2 ≤ǭ 2 − η 1 − 2 +ρ 2 η [s ′ (x 1 )] 2 ≤ǭ 2 − 1 2 min q 2 , 1 2 +ρ [s ′ (x 1 )] 2 ≤ǭ 2 − 2 min q 2 , 1 2 +ρ q 2 ≤ − min q 2 , 1 2 +ρ q 2 ≤ −θ,(70) where the second inequality is by the definition of y 1 and the fact g(x) ≤ǭ 2 , the third inequality is because η = min{q/2, 1/(2 +ρ)}, the fourth is because s ′ (x 1 ) ∈ [2q, 1], the fifth is becauseǭ 2 ≤ min{q/2, 1/(2 +ρ)}q 2 , and the last by the definition of θ. Inequalities (69) and (70) Proof of Lemma 5.2. For simplicity of notation, we denote x(x) in (4) by x. Suppose x is ǫ 2 -feasible. According to Assumption 5.1B, there exists y ∈ relint(X ) such that g(y) +ρ 2 y − x 2 − γ ≤ g(y) +ρ 2 y − x 2 − γ ≤ −θ − γ ≤ −θ. This means (3) satisfies the Slater's condition, so there exists λ ≥ 0 that satisfies the KKT condition together with x. In particular, we have λ(g( x) +ρ 2 x − x 2 − γ) = 0 and ζ f +ρ( x − x) + λ( ζ g +ρ( x − x)) + u = 0, where ζ f ∈ ∂f ( x), ζ g ∈ ∂g( x), u ∈ N X ( x) and N X ( x) is the normal cone of X at x. Taking inner product between (72) and x − x gives 0 ≥ − u, x − x = ζ f + λ ζ g , x − x +ρ(1 + λ) x − x 2 ≥ − ζ f + λ ζ g 2 2ρ(1 + λ) −ρ (1 + λ) 2 x − x 2 +ρ(1 + λ) x − x 2 , where the first inequality is because u ∈ N X ( x), the second inequality is by Young's inequality. Reorganizing the terms in this inequality and using the facts that ζ f ≤ M and ζ g ≤ M , we obtainρ (1 + λ) x − x 2 ≤ ζ f + λ ζ g 2 ρ(1 + λ) ≤ (1 + λ)M 2 ρ , which further implies the first inequality in (10). If λ = 0, the conclusion holds trivially. Hence, we focus on the case that λ > 0. Note that, in this case, we must have g( x) +ρ 2 x − x 2 = γ. Since g(z) +ρ 2 z − x 2 + δ X (z) is (ρ − ρ)-strongly convex in z and u/ λ ∈ N X ( x) = ∂δ X ( x), we have g(y) +ρ 2 y − x 2 − γ ≥ g( x) +ρ 2 x − x 2 − γ + ζ g +ρ( x − x) + u/ λ, y − x +ρ − ρ 2 y − x 2 = ζ g +ρ( x − x) + u/ λ, y − x +ρ − ρ 2 y − x 2 . Applying (71) to the inequality above and arranging terms give −θ ≥ ζ g +ρ( x − x) + u/ λ, y − x +ρ − ρ 2 y − x 2 ≥ − ζ g +ρ( x − x) + u/ λ 2 2(ρ − ρ) , which, together with (72) and the first inequality in (10), implies λ = ζ f +ρ( x − x) ζ g +ρ( x − x) + u/ λ ≤ 2M 2θ(ρ − ρ) . Proof of Lemma 5.3. Consider x ∈ L. Since x is ǫ 2 -feasible, by Assumption 5.1B, there exists y ∈ relint(X ) such that g(y) +ρ 2 y − x 2 ≤ −θ. Note that function g(z) +ρ 2 z − x 2 + δ X (z) is (ρ − ρ)-strongly convex with respect to z and its subdifferential with respect to z at location z = x is ∂g(x) + N X (x). We then have g(x) +ρ 2 x − x 2 + ζ g + u, y − x +ρ − ρ 2 y − x 2 ≤ g(y) +ρ 2 y − x 2 ≤ −θ for any ζ g ∈ ∂g(x) and any u ∈ N X (x). Since g(x) = 0 when x ∈ L, applying Young's inequality to the inequality above yields − ζ g + u 2 2(ρ − ρ) ≤ ζ g + u, y − x +ρ − ρ 2 y − x 2 ≤ −θ, which implies the conclusion. Proof of Lemma 5.4. For any x ∈ X satisfying dist(x, S) ≤ ν ρ , we define x † = proj S (x) = arg min y∈X ,g(y)≤0 1 2 y − x 2 .(73) Since the first conclusion of this lemma holds trivially if x ∈ S, we assume x ∈ X \S, which implies g(x † ) = 0 and x † ∈ L. By Assumption 5.1B, there exists y ∈ relint(X ) such that g(y) ≤ g(y) +ρ 2 y − x † 2 ≤ −θ. Hence, the Slater's condition holds for (73), so there exist a scalar λ † ≥ 0, a subgradient ζ g ∈ ∂g(x † ) and a vector u ∈ N X (x † ) such that x † − x + λ † ζ g + u = 0.(74) Since u, x − x † ≤ 0, we assert that λ † > 0 because, otherwise, (74) implies 0 = x † − x + u, x − x † ≤ − x − x † 2 ≤ 0 and thus x = x † , contradicting with the fact that x ∈ X \S. By the ρ-weak convexity of g and (74), we have λ † g(x) − g(x † ) ≥ λ † ζ g , x − x † − ρ 2 x − x † 2 ≥ λ † ζ g + u, x − x † − λ † ρ 2 x − x † 2 = x − x † 2 − λ † ρ 2 x − x † 2 . Since λ † > 0, dividing both sides of the inequalities above leads to g(x) − g(x † ) ≥ x − x † 2 λ † − ρ 2 x − x † 2 = x − x † · ζ g + u/λ † − ρ 2 x − x † 2 ≥ ν x − x † − ρ 2 x − x † 2 ≥ ν 2 x − x † , where the second inequality is by Lemma 5.3 and the last inequality is because x − x † = dist(x, S) ≤ ν ρ . The first conclusion is thus proved by the facts that g(x) = g + (x) for x ∈ X \S and that g(x † ) = 0. The second conclusion is directly from Lemma 3.1 and 3.2 in [27]. B.3 Proof of Proposition 5.5 and Theorem 5.6 To prove Proposition 5.5 and Theorem 5.6, we need the following lemma from Theorem 4.1 in [27] whose proof is provided only for completeness. It shows that, when the Polyak's stepsize is used, the subgradient method can utilize the local error bound condition in Lemma 5.4 to ensure dist(x (t) , S) Q-linearly converges to zero, which makes sure dist(x (t) , S) is small even when t ∈ J and prevents x (t) from being trapped in an infeasible stationary point given the second conclusion of Lemma 5.4. Lemma B.3 ([27] ). Suppose Assumptions 3.1 and 5.1 hold. Also, suppose the sequence {x (t) } t≥0 is generated by applying the projected subgradient method to min x∈X g + (x) using a Polyak's stepsize, namely, x (t+1) = proj X (x (t) − η t ζ (t) g ), η t = g + (x (t) )/ ζ (t) g 2 if ζ (t) g = 0 0 if ζ (t) g = 0 , for t = 0, 1, . . . , (75) where ζ (t) g ∈ ∂g + (x (t) ). If dist(x (0) , S) ≤ ν/(4ρ), we have dist 2 (x (t+1) , S) ≤ 1 − ν 2 8M 2 dist 2 (x (t) , S)(76) and dist(x (t) , S) ≤ dist(x (0) , S), ∀t ≥ 0. Proof. We prove (77) by induction. (77) holds trivially for t = 0. Suppose that (77) holds up to iteration t of (75). We want to prove that it also holds for iteration t + 1. By the induction hypothesis, we have dist(x (t) , S) ≤ ν/(4ρ). Lemma 4. 2 . 2Suppose Assumptions 3.1 and 4.1 hold. Theorem 4. 3 . 3Suppose Assumptions 3.1 and 4.1 hold and Λ is as in Lemma 5 . 4 . 54Suppose Assumptions 3.1 and 5.1 hold. It holds for any x satisfying Theorem 5 . 6 . 56Under the same assumptions as Proposition 5.5, Algorithm 1 guarantees Figure 1 : 1Performances on classification problems with ROC-based fairness. Figure 2 : 2Performances on classification problems with demographic equity constraint. f ) and, if t ≥ S, I = I ∪ {t}. 1 and Lemma 2 ( 2Case A) in [51] by choosing Ω = 3 ln(1/δ) in their bound. Lemma A.4. Suppose Assumption A.1 holds. For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ/(T − S) that the conclusion is guaranteed by Lemma 2 (Case B) in [51] by choosing Ω = max{ 12 ln(2/δ), (4/3) · ln(2/δ)} in their bound. Lemma A.6. Suppose Assumptions 3.1 and A.1 hold. For any δ ∈ (0, 1), Algorithm 2 guarantees with probability at least 1 − δ that Theorem A. 8 . 8Suppose Assumptions 3.1 and 4.1 hold. Let x(x (t) ) be defined as in (4) with (ρ,ρ, γ) satisfying (7) and x (τ ) is generated by Output II. Algorithm 1 guarantees E τ [ x (τ ) − x (τ ) ] ≤ ǫ in either of the following cases. Case I: S = 0, ǫ t = 0, η t = ǫ 2 min{ρ−ρ,min{(ρ−ρ) 2 ,µ 2 /4} = O(1/ǫ 4 ). Case II: S = T /2, ǫ t = 0, η t = D M √ t+1 and T ≥ 4M 2 D 2 ǫ 4 min{(ρ−ρ) 2 ,µ 2 /4} = O(1/ǫ 4 ). probability at least 1 − δ in either of the following cases. Case I: S, ǫ t and η t are chosen as Case I in Theorem 4.3 and T ≥ 25M 2 D 2 (1 + Λ) 2 4ǫ 4 (ρ − ρ) 2 , max 12 ln(8/δ), 16 9 ln 2 (8/δ) , 300 ln(4/δ)M 2 D 2 (1 + Λ) 2 ǫ 4 (ρ − ρ) 2 = O(1/ǫ 4 ). Table 1 : 1Information of the datasets. Groups are males VS females in a9a, users with age within [25, 60] VS outside [25, 60] in bank, and caucasian VS non-caucasian in COMPAS. 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Case II: S and η t are chosen as Case II in Theorem 4.3, ǫ t = EM D√ t+1 where E is any positive constant such that E ≥ 4 + 2π √ 6 max 12 ln(8/δ), 4 3 ln(8/δ) + 8 3 ln(4/δ) Case I: If S, ǫ t , η t and T are chosen as Case I in Theorem A.11 and B = 300σ 2 ln(4T /δ)(1+Λ) 4 . Case II: If S, ǫ t , η t and T are chosen as Case II in Theorem A.11 except that E is any positive constant such thatǫ 4 (ρ−ρ) 2 E ≥ 8 + 2π √ 6 max 12 ln(8/δ), 4 3 ln(8/δ) + 8 3 ln(4/δ) 5 ( 51+Λ)M 2 for any t. Using Lemma 4.2 and plugging the values of S, T , B, η t and ǫ t in (49), we can show that (49) holds if which, after dividing both sides by T ηρ, can be equivalently written asT ηρǫ 2 (ρ − ρ) 1 + Λ >ρ D 2 2 +ρ 2 T η 2 M 2 +ρ 2 max 12 ln(8/δ), 4 3 ln(8/δ) √ T η 2 M 2 + 2 3 ln(4/δ)ρ √ T ηM D + T ηρ 3 B σ ln(4T /δ)(1 + Λ), ǫ 2 (ρ − ρ) 1 + Λ > D 2 2T η + ηM 2 2 + 1 2 max 12 ln(8/δ), 4 3 ln(8/δ) ηM 2 √ T + 2 3 ln(4/δ)M D √ T + 3 B σ ln(4T /δ)(1 + Λ). T −1 t=S η tρ λ t I(g(x (t) ) ≤ 0) + η tρ I(g(x (t) ) > 0) µ 2After dropping some non-negative terms, the left-hand side of the inequality above can be bounded from below asCombining the two inequalities above giveŝ ρ min{ρ − ρ, µ/2}where the last inequality is because of Assumption 3.1. Since τ is generated by Output II, after organizing terms, we haveIn Case I, let η = η t = ǫ 2 min{ρ − ρ, µ/2}/M 2 for any t. By the choices of η t , S and T , (41) impliesThis further implies (33) because of Lemma 4.2. Note that η t and ǫ t are decreasing in t.Hence, the left-hand side of (44) satisfiesThe right-hand side of (44) satisfieŝwhere the equality is obtained by plugging in the definition of η t and the inequality is because(t+1) 2 ≤ π 2 /6. By the condition satisfied by E, the right-hand side of (46) is greater than or equal to the right-hand side (47). This means (34) holds with this choice of parameters and thus1+λ in the same way as in the proof of Theorem 4.3.A.7 Complexity analysis when subgradient and function oracles are both stochasticIn this section, we will analyze the oracle complexity of Algorithm 2 under Assumption A.1 without requiring(21)or (42). In this case, not only the subgradient oracles are stochastic but also the function value g can only be accessed through a stochastic oracle. When the function value of g is stochastic, the complexity analysis becomes fundamentally more challenging than the case of a deterministic g. In fact, the challenge comes only from the stochastic functionwhere the equality is obtained by plugging in the definition of η t and the inequality is because of the definition of B and the facts thatBy the condition of E, the righthand side of (46) is strictly greater than the right-hand side (52). This means (49) holds with this choice of parameters and thus E τ [ x (τ ) − x (τ ) ] ≤ ǫ. Moreover, since(27)holds with δ replaced by δ/4 for t = S, S + 1, . . . , T − 1, we havewhere the last inequality is because of (45) and the choices of B and T . so g is β-weakly convex according toLemma 4.2 in [33]. Hence, (17) satisfies Assumption 5.1A with ρ = max{2λ, β}.are the feature vectors of protected and unprotected groups, respectively. We assume without generality that the first feature of a p i , denoted by [a p i ] 1 , equals one and the first feature of a u i , denoted by [a u i ] 1 , equals negative one. In fact, this feature can be the one used to split data into the protected group or the unprotected group (see the group variables inTable 1for example). We can also simply add this feature to the data if it does not exist originally. With this feature, we can show that (17) satisfies Assumption 5.1B under mild conditions.The following statements hold.1. There exist κ and κ in K such that (κ, κ) contains κ but no numbers in K.It holds thatwhere h is defined in (53) and3. Problem (17) satisfies Assumption 5.1B withfor anyρ > ρ and anyǭ > 0 that satisfȳProof. Since κ / ∈ K and K is discrete, Statement 1 holds trivially. In fact, one can sort the numbers in K and let κ and κ be the two consecutive numbers with κ ∈ (κ, κ).Suppose q = 0. There must exist a sequence {x (t) } t≥0 such that |h(x (t) )| ∈ [(κ + κ)/2, (κ + κ)/2] for any t ≥ 0 and lim t→∞ q(x (t) ) = 0. Since σ(z) ∈ (0, 1) for any z ∈ R, any sequence like σ(z (t) )(1 − σ(z (t) )) will converge to zero only when all limiting points of σ(z (t) ) are in {0, 1}. By this observation and the definition of q in (58), after passing to a subsequence if necessary, we haveThis means lim t→∞ |h(x (t) )| ∈ K, contradicting with Statement 1 and the fact that |h(x (t) )| ∈ [(κ+κ)/2, (κ+κ)/2] ⊂ (κ, κ). This contradiction indicates that q > 0, which proves Statement 2.Consider anyρ > ρ and anyǭ > 0 that satisfy (59). Because [a p i ] 1 = 1 for i = 1, . . . , n p and [a u i ] 1 = −1 for i = 1, . . . , n u , it is easy to show thatwhere ∇h is given in (54). As a result, it holds thatwhere the second inequality is becauseǭWe assume h(x) > 0 and the proof when h(x) < 0 is the same. Let y be generated from x by running a gradient descent step on h(x) with a stepsize of η > 0, namely, y = x − η∇h(x). We choose η = min{1/(β +ρ), κ/(4α 2 )}.Since h(y) +ρ 2 y − x 2 is smooth and its gradient with respect to y is (β +ρ)-Lipschitz continuous, we havewhere the second inequality is by the definition of y and the fact that h(x) = |h(x)| = g(x) + κ ≤ κ+ǭ 2 , the third inequality is because η = min{1/(β +ρ), κ/(4α 2 )}, and the last is by (61) and the fact thatwhich indicates g(y) = |h(y)| − κ = h(y) − κ. Hence, (63) and (59) implyInequalities(62)and(64)1. There exists a non-negative integer k such that κ ∈ (3k, 3(k + 1)).It holds thatConstraint (65) satisfies Assumption 5.1B withfor anyρ > ρ and anyǭ > 0 that satisfȳProof. Since κ is not divisible by three, Statement 1 holds trivially. In addition, by (67), we also have 3k < κ + 3k 2 < κ < κ +ǭ 2 ≤ κ + 3(k + 1) 2 < 3(k + 1).As observed by [42], 0 ∈ ∂s(x i ) only when x i ∈ Z, or equivalently, s(x i ) ∈ {0, 3} for any i. Suppose (66) is not true. There must exist a sequence {x (t) } t≥0 such that SCAD(x (t) ) ∈ [(κ + 3k)/2, (κ + 3(k + 1))/2] for any t ≥ 0 and lim SCAD(x (t) ) is divisible by three, contradicting with (68) and the fact that SCAD(x (t) ) ∈ [(κ + 3k)/2, (κ + 3(k + 1))/2]. This contradiction proves Statement 2.Consider a solution x with g(x) ≤ǭ 2 . Suppose g(x) ≤ −(κ − 3k)/2. We can set y = x and haveSuppose ζ (t) g = 0 so x (t+1) = x (t) . By Lemma 5.4, we must have dist(x (t) , S) = 0 and thus dist(x (t+1) , S) = 0 ≤ dist(x (0) , S) and (76) holds.Suppose ζ (t) g = 0. Let x †(t) = proj S (x (t) ) for t ≥ 0. By the nonexpansiveness of prox X (·), we havewhere the second equality is because g + (x †(t) ) = 0, the third inequality is by the ρ-weak convexity of g + , the fourth is by Lemma 5.4, the fifth is by the hypothesis that dist(x (t) , S) ≤ dist(x (0) , S) ≤ ν/(4ρ), and the last is by Lemma 5.4 and Assumption 3.1A. This inequality further implieswhich proves (76) and also proves (77) by induction.Proof of Proposition 5.5. The choices of η t and η t implyfor t ∈ I. We first prove dist(x (t) , S) ≤ min ǫ 2 /M, ν/(4ρ) by induction on t. Since x (0) ∈ S, this conclusion holds trivially for t = 0. Suppose it holds up to iteration t. We want to prove it also holds for iteration t + 1.≤ η t M, which, by triangle inequality and the induction hypothesis, implies thatSince g(x (t) ) ≤ ǫ t for t ∈ I, we have g(x (t+1) ) ≤ g(x (t) ) + M x (t+1) − x (t) ≤ ǫ t + η t M 2 . By Lemma 5.4, we have dist(x (t+1) , S) ≤ 2ǫ t /ν + 2η t M 2 /ν ≤ min ǫ 2 /M, ν/(4ρ) .Proposition B.4. Under the same assumptions as Proposition 5.5, Algorithm 1 guaranteeswhere Λ ′ is defined in(10).Let ϕ(x) and x be defined in(3)and(4)with (ρ,ρ, γ) satisfying (9). For simplicity of notation, we denote x(x (t) ) by x (t) . Since x (t) is ǫ 2 -feasible by Proposition 5.5 and γ = ǫ 2 , ϕ(x (t) ) and x (t) are well defined for any t ≥ 0.By Assumption 5.1B, problem (3) with x = x (t) is strongly convex and has a strictly feasible solution, so there exists a Lagrangian multiplier λ t ≥ 0 satisfying (5) andBy the updating equation of x (t+1) , we haveMultiplying the inequality above byρ(1 + λ t+1 )/2 and adding f (which, together with (80), impliesNext, we will bound ζ (t) , x (t) − x (t) from below when t ∈ I and t ∈ J, separately.f . By the ρ-weak convexity of f , we haveSince the objective function in(79)which, by the facts that g(x (t) ) ≤ ǫ t , λ t ≥ 0 and λ t g(Applying this inequality and inequality (82) to (81) leads toSuppose t ∈ J so g(x (t) ) > ǫ t and ζ (t) = ζ (t)g . By the ρ-weak convexity of g and the fact that g( x (t) ) +ρ 2 x (t) − x (t) 2 ≤ γ, we haveApplying this inequality to (81) leads toSumming up (83) and (84) for t = S, S + 1, . . . , T − 1, we haveFinally, from Assumption 5.1C and the first inequality in (10), it can be easily shown thatThen, the conclusion is derived from by applying these two inequalities to (85).We are now ready to prove the main theorem for the weakly convex case.Proof of Theorem 5.6. The inequality E τ [g(x (τ ) )] ≤ ǫ 2 is a directly consequence of Proposition 5.5.Applying the upper bound Λ ′ of λ t from Lemma 5.2 to the inequality in Proposition B.4, we obtainFor t ∈ I, we have from Assumption 3.1 and the definition of η t thatFor t ∈ J, we have η t = g + (x (t) )/ ζ (t) g 2 sowhere the second inequality is from Proposition 5.5 that implies g + (x (t) ) = g(x (t) ) ≤ γ . Applying (87) and (88) to (86) and organizing term lead tôwhere the second inequality is by the fact that γ = ǫ 2 and the definition of ǫ t which implies ǫ t ≤ ǫ 2 ν/(2M ) ≤ ǫ 2 . According to the definition of η t for t ∈ I and t ∈ J, we haveApplying (90) as well as the definitions of S, η t , ǫ t and γ to the right hand side of (89) gives which implies the conclusion of this theorem after plugging in the value of T . A reductions approach to fair classification. Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, Hanna Wallach, International Conference on Machine Learning. 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Yi Cheng, Guanghui Lan, H Edwin Romeijn, arXiv:2210.05108arXiv preprintYi Cheng, Guanghui Lan, and H. Edwin Romeijn. Functional constrained optimization for risk aversion and sparsity control. arXiv preprint arXiv:2210.05108, 2022. Worst-case complexity of an sqp method for nonlinear equality constrained stochastic optimization. Frank E Curtis, Michael J O&apos;neill, Daniel P Robinson, arXiv:2112.14799arXiv preprintFrank E. Curtis, Michael J. O'Neill, and Daniel P. Robinson. Worst-case complexity of an sqp method for nonlinear equality constrained stochastic optimization. arXiv preprint arXiv:2112.14799, 2021. Inexact sequential quadratic optimization for minimizing a stochastic objective function subject to deterministic nonlinear equality constraints. Frank E Curtis, Daniel P Robinson, Baoyu Zhou, arXiv:2107.03512arXiv preprintFrank E. Curtis, Daniel P. Robinson, and Baoyu Zhou. Inexact sequential quadratic optimization for minimizing a stochastic objective function subject to deterministic non- linear equality constraints. arXiv preprint arXiv:2107.03512, 2021. Complexity of finding near-stationary points of convex functions stochastically. Damek Davis, Dmitriy Drusvyatskiy, arXiv:1802.08556arXiv preprintDamek Davis and Dmitriy Drusvyatskiy. Complexity of finding near-stationary points of convex functions stochastically. arXiv preprint arXiv:1802.08556, 2018. Stochastic subgradient method converges at the rate o(k −1/4 ) on weakly convex functions. Damek Davis, Dmitriy Drusvyatskiy, arXiv:1802.02988arXiv preprintDamek Davis and Dmitriy Drusvyatskiy. Stochastic subgradient method converges at the rate o(k −1/4 ) on weakly convex functions. arXiv preprint arXiv:1802.02988, 2018. Stochastic model-based minimization of weakly convex functions. Damek Davis, Dmitriy Drusvyatskiy, SIAM Journal on Optimization. 291Damek Davis and Dmitriy Drusvyatskiy. Stochastic model-based minimization of weakly convex functions. SIAM Journal on Optimization, 29(1):207-239, 2019. Subgradient methods for sharp weakly convex functions. Damek Davis, Dmitriy Drusvyatskiy, Kellie J Macphee, Courtney Paquette, Journal of Optimization Theory and Applications. 1793Damek Davis, Dmitriy Drusvyatskiy, Kellie J. MacPhee, and Courtney Paquette. Sub- gradient methods for sharp weakly convex functions. Journal of Optimization Theory and Applications, 179(3):962-982, 2018. Proximally guided stochastic subgradient method for nonsmooth, nonconvex problems. Damek Davis, Benjamin Grimmer, SIAM Journal on Optimization. 293Damek Davis and Benjamin Grimmer. Proximally guided stochastic subgradient method for nonsmooth, nonconvex problems. SIAM Journal on Optimization, 29(3):1908-1930, 2019. Minibatch and momentum model-based methods for stochastic weakly convex optimization. Qi Deng, Wenzhi Gao, Advances in Neural Information Processing Systems. 34Qi Deng and Wenzhi Gao. Minibatch and momentum model-based methods for stochas- tic weakly convex optimization. Advances in Neural Information Processing Systems, 34:23115-23127, 2021. On the exactness of a class of nondifferentiable penalty functions. Di Gianni, Luigi Pillo, Grippo, Journal of optimization theory and applications. 573Gianni Di Pillo and Luigi Grippo. On the exactness of a class of nondifferentiable penalty functions. Journal of optimization theory and applications, 57(3):399-410, 1988. Exact penalty functions in constrained optimization. Di Gianni, Luigi Pillo, Grippo, SIAM Journal on control and optimization. 276Gianni Di Pillo and Luigi Grippo. Exact penalty functions in constrained optimization. SIAM Journal on control and optimization, 27(6):1333-1360, 1989. The complexity of finding stationary points with stochastic gradient descent. Yoel Drori, Ohad Shamir, International Conference on Machine Learning. PMLRYoel Drori and Ohad Shamir. The complexity of finding stationary points with stochastic gradient descent. In International Conference on Machine Learning, pages 2658-2667. PMLR, 2020. Since indexes t ′ + 1, t ′ + 2, ..., t are in J, Algorithm 1 essentially performs the projected subgradient method to min x∈X g + (x) using a Polyak's stepsize during iterations t ′ + 1, t ′ + 2. S) ≤ min ǫ 2 /M, ν/(4ρ) for any t ≥ 0. As a result, g(x (t) )+ρ 2 x (t) −x (t) 2 = g(x (t) ) ≤ M ·dist(x (t) , S) ≤ M. Hence, by Lemma B.3, we have dist(x (t+1) , S) ≤ dist(x (t ′ +1) , S) ≤. ·min ǫ 2 /M, ν/(4ρ) ≤ ǫ 2 = γ, meaning that x (t) is feasible forSuppose t ∈ J. Let t ′ be the largest index in I that is smaller than t. By the same proof as in the previous case, we have dist(x (t ′ +1) , S) ≤ min ǫ 2 /M, ν/(4ρ) . Since indexes t ′ + 1, t ′ + 2, ..., t are in J, Algorithm 1 essentially performs the projected subgradient method to min x∈X g + (x) using a Polyak's stepsize during iterations t ′ + 1, t ′ + 2, ..., and t. Hence, by Lemma B.3, we have dist(x (t+1) , S) ≤ dist(x (t ′ +1) , S) ≤ min ǫ 2 /M, ν/(4ρ) . By induction, we have prove that dist(x (t) , S) ≤ min ǫ 2 /M, ν/(4ρ) for any t ≥ 0. As a result, g(x (t) )+ρ 2 x (t) −x (t) 2 = g(x (t) ) ≤ M ·dist(x (t) , S) ≤ M ·min ǫ 2 /M, ν/(4ρ) ≤ ǫ 2 = γ, meaning that x (t) is feasible for (3). The following proposition provides the main inequality needed for proving Theorem 5.6. where the second inequality is by the definition of ϕ(x) in (3) and the fact that λ t+1 ≥ 0 and g( x (t) the definitions of ϕ(x (t+1) ) and λ t+1. we have the following equation similar to (79) ϕ(x (t+1) ) = f ( x (t+1) ) +ρThe following proposition provides the main inequality needed for proving Theorem 5.6. where the second inequality is by the definition of ϕ(x) in (3) and the fact that λ t+1 ≥ 0 and g( x (t) the definitions of ϕ(x (t+1) ) and λ t+1 , we have the following equation similar to (79) ϕ(x (t+1) ) = f ( x (t+1) ) +ρ

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{'abstract': 'We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations. optimization arg min y {f 2 (y) + ρ 2 y − x 2 } for any x. There are relatively fewer works on non-convex nonsmooth constrained problems. An alternating direction method of multipliers (ADMM) and an ALM are studied by [81] and [84], respectively, for non-convex non-smooth problems with linear constraints while our study considers nonlinear non-smooth constraints. The methods by[21]and[13]can be extended to a structured non-smooth case where f = f 1 + f 2 with f 1 being smooth non-convex and f 2 = max y {y ⊤ Ax − φ(y)} with a convex φ, and g has a similar structure. The method by[15]can handle a specific non-smooth non-convex constraint, i.e., g(x) = λ x 1 − h(x) where h is a convex and smooth. Compared to these works, our results apply to a more general non-smooth problem without those structure assumptions.When f and g in (1) are weakly convex and non-smooth, the inexact proximal point method has been studied by[14,58, 42] under different constraint qualifications and different notions of stationarity. Their complexity analysis utilizes the relationship between the gradient of the Moreau envelope of (1) and the near stationarity of a solution, which is originally used to analyze complexity of subgradient methods for weakly convex non-smooth unconstrained problems[25,26,28,2,29,67, 89]. Our analysis utilizes a similar framework. The methods[14,58, 42] are double-loop while our algorithm only uses a single loop and achieves the same complexity of O(1/ǫ 4 ) as them under similar assumptions.The SSG algorithm is first proposed by Polyak [65]. It has been well-studied for convex problems [64,8,52, 73, 78, 79, 71, 74, 72,3]and quasi-convex problems [73]. This paper provides the first complexity analysis for the SSG method under weak convexity assumption. Non-smooth non-convex optimization has also been studied without weak convexity assumption by [88, 49, 70, 50,20,76, 77]. These works analyze the complexity of first-order methods for computing an (ǫ, δ)-Goldstein approximate stationary point, which is a more general stationarity notation than what we consider here. However, these works only focus on unconstrained problems.', 'arxivid': '2301.13314', 'author': ['Yankun Huang yankun-huang@uiowa.edu \nDepartment of Business Analytics\nUniversity of Iowa\n52242Iowa CityIA\n', 'Qihang Lin qihang-lin@uiowa.edu \nDepartment of Business Analytics\nUniversity of Iowa\n52242Iowa CityIA\n'], 'authoraffiliation': ['Department of Business Analytics\nUniversity of Iowa\n52242Iowa CityIA', 'Department of Business Analytics\nUniversity of Iowa\n52242Iowa CityIA'], 'corpusid': 258762243, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 41633, 'n_tokens_neox': 36124, 'n_words': 20310, 'pdfsha': '22a6b4056bd715830e98e1bcc40a6b5e094c81ec', 'pdfurls': ['https://export.arxiv.org/pdf/2301.13314v2.pdf'], 'title': ['Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization', 'Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization'], 'venue': []}

arxiv

Optimal locally recoverable codes with hierarchy from nested F -adic expansions Austin Dukes Giacomo Micheli Vincenzo Pallozzi Lavorante Optimal locally recoverable codes with hierarchy from nested F -adic expansions 1 In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of [1],[11]with an algebraic number theoretical approach that allows to give a finer tuning of the minimum distance of the intermediate code (allowing larger dimension of the final code), and to remove restrictions on the arithmetic properties of q compared with the size of the locality sets in the hierarchy. In turn, we manage to obtain codes with a wide set of parameters both for the size q of the base field, and for the hierarchy size, while keeping the optimality of the codes we construct. I. INTRODUCTION Various classes of locally recoverable codes have received great attention in recent times due to their applications to cloud and distributed storage systems [2], [3], [4], [5], [7], [8], [9], [12], [14], [15]. In this paper we produce new optimal hierarchical locally recoverable codes (HLRC). HLRCs are suitable solutions that address the problem of recovering lost information in a distributed storage system, and they have been widely studied in [1], [6], [11], [16]. HLRCs allow to recover certain patterns of erasures by gradually looking at more components depending on the number of erasures that occurred. One can then design codes that recover one erasure by looking at at most b other components; λ erasures by looking at a other components; and d − 1 erasures by looking at at most k components, where k is the dimension of the code. This is impactful from a practical perspective, as one can deal with the most likely scenario (one erasure) in the optimal way, with the less likely scenario (λ erasures) in an acceptable G. Micheli is with the Department of Mathematics, University of South Florida, ZIP, Tampa, US (e-mail: gmicheli@usf.edu). Manuscript received ...; revised ... arXiv:2207.10383v1 [cs.IT] 21 Jul 2022 way, and still be able to recover d − 1 erasures by accessing a large number of nodes. Tuning these parameters in an efficient way depends on the reliability of the servers and the required efficiency of the system in terms of node retrieval. One of the features that one would desire from this kind of code is that λ is not too large, as the second most likely scenario is the failure of only a few other nodes more than 1 (and not too many others, that can anyway be recovered using the minimum distance). We address this problem by writing a sharper Singleton bound for this regime of parameters and then constructing codes that achieve the bound. Let us now define the main objects we will be treating in this paper. A. Definitions In the rest of the paper we will consider the occurence of either one, λ, or d − 1 erasures, as these arise most commonly from applications (instead of the more general setting where one allows λ 1 , λ 2 , or d−1 erasures). Let n,k,b be positive integers with k ≤ n. A locally recoverable code (LRC) C having parameters [n,k,b] is an F q -subspace of F n q of dimension k such that if one deletes one component of any v ∈ C, this can be recovered by accessing at most b other components of v. If d is the minimum distance of the code, we will write that C is an [n,k,d,b] LRC. We now give the following definition which will be useful in the rest of the paper. Definition 1.1: Let n be a positive integer, C ⊆ F n q be a linear code, and S be a subset of the set of indices {1,...,n}. We say that C can tolerate x erasures on S if, whenever there are x erasures on components of a codeword with indices belonging to S, the missing components can be recovered by looking at |S| − x other coordinates in S. In this paper we construct new locally recoverable codes with hierarchy of locality sets. Our Definition 1.2 is equivalent to the one of hierarchical codes in [1] but we find it slightly easier to employ ours for practical situations, as we keep direct track of the size of the "hierarchy". Definition 1.2: Let n,k,d,b,a,λ be positive integers with n > k and 2 ≤ λ ≤ b . An [n,k,d,b,a,λ] hierarchical locally recoverable code (HLRC) is an [n,k,d]-linear code such that • (a + λ) | n, • (b + 1) | (a + λ), • the codeword indices are partitioned into ≥ 1 distinct sets A i , each of size a + λ, such that C tolerates λ erasures on A i for every i ∈ {1, . . . , }, and • each A i can be partitioned into B i,j , each of size b + 1, such that C tolerates 1 erasure on each B i,j for every i ∈ {1, . . . , } and every j ∈ {1, . . . , (a + λ)/(b + 1)}. B. Motivation Let us now briefly explain the motivation behind codes with hierarchical locality. Let T be the time needed to replace a failed node. Suppose that a second node fails in the same locality set as the first node during the time T . An [n,k,d,b] locally recoverable code will still need to access k information symbols, as the 1-locality procedure is not guaranteed to work anymore. However, an [n,k,d,b,a,λ] hierarchical locally recoverable code only requires accessing at most a information symbols. Since the failure of only a few nodes, say λ < d − 1, is significantly more likely than the failure of d − 1 nodes in the span of time T , it is convenient to have a code which addresses separately the case in which only λ nodes fail. The codes in [1] address this issue, but they are restricted to certain λ's, as we explain in subsection III-E and moreover in many cases they require restrictions on the arithmetic of q and the size of the hierarchy (see for example the case of power functions in [1, Section IV.A, Example]). C. Our contribution In this paper we provide new constructions of optimal codes with hierarchical locality and an improved bound for HLRC for a special set of parameters. Our construction is based on the ideas in [1] combined with powerful techniques from algebraic number theory, allowing us to remove arithmetic restrictions on the size of the hierarchy compared with q or q − 1. Structure of the paper: • In Section I and its subsections we explain the basic coding theoretical definitions and provide the practical motivations for the study of such codes. • In Section II, for some regime of parameters, we provide a stronger Singleton bound than the one already present in the literature for HLRC [11]. Our bound beats the previous bound for an infinite set of parameters (see for example Remark 1). • In Section III we achieve our new bound with a new construction of HLRC that covers a set of parameters that are not available using previous constructions (see subsection III-E). In subsection III-F we construct one of our codes to show how a generator matrix looks like in practice. • In Section IV we show that our codes are constructible without requiring arithmetic arithmetic restriction of q,q −1, the locality parameters, and the sizes of the sets in the hierarchy. • Using the existential results provided in Section IV, in Section V we provide some practical choice of parameters for codes with large length. II. AN IMPROVED BOUND FOR HIERARCHICAL LOCALLY RECOVERABLE CODES A. The Singleton bound for [n,k,d,b,a,λ] hierarchical LRCs with λ ≤ b. Let M m×n (q) denote the set of all matrices of dimension m×n defined over F q . The following is a well-known proposition, but we include a proof for completeness. Proposition 2.1: Let C be an [n,k,d] q code with generator matrix G ∈ M k×n (q) and let S ∈ M k×t (q) be a submatrix of G. If rk(S) ≤ k − 1 then t ≤ n − d. Proof: Let S = [S 1 , . . . , S t ], where S i is a column of G for i ∈ {1, . . . ,t}. DefineS : F k q → F t q such that x →S(x) = xS = [xS 1 , . . . , xS t ]. Since rk(S) ≤ k − 1 and we can writẽ S(x) = x 1 R 1 + . . . x k R k , where R i are the rows of S, there exists x ∈ F k q such thatS(x ) = 0. Assuming without loss of generality that S consists of the first t columns of G, there exists a codeword c = [S(x ),y t+1 , . . . ,y n ] whose weight equals n − t. Hence d ≤ n − t. To help the reader understand the more complex bound we propose on hierarchical LRCs, we include here a proof of the standard Singleton bound for LRCs, which does not make formal use of an algorithm like in other proofs in the literature (the ideas in the proof are the same). k − 1 + k − 1 b ≤ n − d. Proof: We know that every set of b + 1 columns has rank b by the locality condition. This means that we can choose a set S of k−1 b (b + 1) + k−1 b b columns in such a way that rk(S) ≤ k − 1 (here {x} = x − x ) . Thus, by applying Proposition 2.1, we have the following: k − 1 b + k − 1 b b + k − 1 b = k − 1 + k − 1 b ≤ n − d. Notice that the above is equivalent to the well-known bound d ≤ n − k − k/b + 2. We aim to generalize the bound in Corollary 2.2 when C is an [n,k,d,b,a,λ] HLRC. The key observation is that one can partition the columns of the generator matrix into sets of a + λ columns so that each set has rank strictly less than a, and each set of a + λ columns can be partitioned further into sets of b + 1 columns so that each set has rank at most b. To see this, note that each set S i of a + λ columns (corresponding to A i ) can be divided into β = (a + λ)/(b + 1) sets, say S i,j for j ∈ {1, . . . ,β}, of b + 1 columns (corresponding to B i,j ) with rank at most b for i ∈ {1, . . . , } by the definition of the code. Now, in the first set S i,1 we have λ columns which are in the span of the other a columns in S i . This means that we can choose b + 1 − λ columns from S i,1 and b columns from each of the other S i,j , with j = 1, and be able to recover any λ of the a + λ columns in S i . Therefore, the rank of each S i is at most ρ := (a + λ)/(b + 1) − 1 β−1 b + (b + 1 − λ) ≤ a. Theorem 2.3: Let C be a [n,k,d,b,a,λ] HLRC with λ ≤ b, and let ρ = b(a+λ)/(b+1)−(λ−1) . Then k − 1 ρ (a + λ) + k 1 + k 1 b ≤ n − d,(1) where k − 1 ≡ k 1 (mod ρ) and 0 ≤ k 1 < ρ. Proof: Given k−1 ρ locality sets A i , say i ∈ {1, . . . , k−1 ρ }, denoteb (b + 1) + { k 1 b }b, such that rk(S 1 ) = k 1 (explicitly, S 1 is the union of columns which correspond to B i,j , for j ∈ {1, . . . , k 1 b }) . Hence, rk(S ∪ S 1 ) = k − 1 ρ ρ + k 1 = k − 1, by the definition of k 1 . Applying Proposition 2.1 we have k − 1 ρ (a + λ) + k 1 b (b + 1)+ k 1 b b ≤ n − d. Now, since k 1 b = k 1 b + { k 1 b } we have k − 1 ρ (a + λ) + k 1 + k 1 b ≤ n − d.d = n − k − 1 ρ (a + λ) + k 1 + k 1 b , for k − 1 ≡ k 1 (mod ρ) and 0 ≤ k 1 < ρ. Remark 1: Notice that our bound improves upon the bound in [11] for infinitely many parameters, but ours holds only for λ ≤ b. In fact, for any length n, and for parameters k = 6, a = 4, r 1 = ρ = 3, r 2 = b = 2, δ 1 = λ + 1 = 3 and δ 2 = 2 [11, Theorem 2.1] gives d ≤ n − 8 when instead our bound gives d ≤ n − 9. The moral reasons for this are that we are taking into account a finer arithmetic of the parameters which involves the reduction of the dimension modulo the upper level hierarchical locality, and we are restricting to the case in which the number of nodes that we simultaneously erase is strictly smaller than the size of the smaller locality set . III. OUR CONSTRUCTION OF OPTIMAL HLRCS USING NESTED f -ADIC EXPANSIONS A. Main tool for the construction Lemma 3.1: Let f,h ∈ F q [X] be non-constant polynomials. Suppose there is some t 0 ∈ F q such that f (h(X)) − t 0 splits completely (i.e., factors into (deg f )(deg h) distinct factors) over F q . Then the set of roots of f (h(X))−t 0 , say A 0 , can be partitioned into sets B 1 , . . . , B deg f ⊆ F q which satisfy the following: • h(B i ) = c i ∈ F q for each 1 ≤ i ≤ deg f , • the cardinality of each B i is deg h, and • h(B i ) = h(B j ) whenever i = j. Proof: By the hypothesis we may write f (h(X)) − t 0 = (deg f )(deg h) i=1 (X − x i ) for distinct elements x 1 , . . . , x (deg f )(deg h) ∈ F q . Notice now that if f (h(X)) − t 0 splits completely, then f (X) − t 0 splits completely. If we let α 1 , . . . , α deg f ∈ F q be the (distinct) roots of f (X) − t 0 , then we may also write f (h(X)) − t 0 = deg f i=1 (h(X) − α i ) ∈ F q [X] . Combining these two factorizations and relabeling the x i appropriately yields deg f i=1 deg h j=1 (X − x i,j ) = deg f i=1 (h(X) − α i ), where deg h j=1 (X − x i,j ) = h(X) − α i for each 1 ≤ i ≤ deg f . In particular, it follows that α i ∈ F q for each i. Write B i = {x i,j : 1 ≤ j ≤ deg h}. Then we have h(B i ) = α i for each i, proving the first statement. The second and third statements both follow from the fact that the x i,j 's are pairwise distinct and the B i 's are respectively and disjoint. Definition 3.1: For f,h ∈ F q [X], we say that a set A ⊂ F q is a nest for (f,h) if A is the set of preimages of t 0 ∈ F q such that f (h(X)) − t 0 is totally split. Furthermore, we say that B ⊂ A is a sub-nest if h is constant on B and |B| = deg h. B. The main construction We present a general method of constructing linear codes with the nested locality property. Later we will show that these codes are optimal in the sense of Section III. In line with the notion of (r, )-good polynomials in [10], we now begin defining our nested polynomials. Definition 3.2 ( -nested): Let f,h ∈ F q [X] and let be a positive integer. Then f and h are said to be -nested if f (h(X)) − t 0 splits completely over F q for at least elements t 0 ∈ F q . Remark 2: Note that if f and h are -nested, then from Lemma 3. 1 there exist A 1 , . . . ,A distinct nests for (f,h) such that • for any i ∈ {1, . . . , }, f (h(A i )) = {t i } for some t i ∈ F q , • |A i | = deg f deg h, • A i ∩ A j = ∅ for any i = j, and • each A i can be partitioned into sub-nests B i,j for (f,h). Those properties will be the key of our next construction. m(X) = s i=0 deg f −2 j=0 g i,j (X)h(X) j +g i (X)h(X) deg f −1 f (h(X)) i ,(2) where g i,j ∈ F q [X] ≤deg h−2 andg i ∈ F q [X] ≤deg h−λ−1 . Let n = (deg f deg h) and let k be the dimension of M as an F q -vector space. Define C := {(m(x), x ∈ A) | m ∈ M}.(3) We will prove that C is an optimal [n,k,b,a,λ]-HLRC over F q . C. Locality Since we evaluate at n distinct points of F q , we need q ≥ n. Write n = (a + λ)(s + 1) and recall that b + 1 divides a + λ. Take a ∈ F k q and write Enc C (a) = c = c i,j 1 ,j 2 for 1 ≤ i ≤ s + 1, 1 ≤ j 1 ≤ (a + λ)/(b + 1), 1 ≤ j 2 ≤ b + 1. Note that the index i determines a nest A i , j 1 determines a sub-nest B i,j 1 and j 2 a precise element of the sub-nest considered, which is denoted by c i,j 1 ,j 2 indeed. We begin by showing that the code C described in Construction 3.2 allows one to recover a single missing component of c by accessing at most b other components of c. Fix ≥ 1 and let f,h ∈ F q [X] be the -nested polynomials from which C is obtained. Write A = {A 1 , . . . , A } with A i = deg f j=1 B i,j 1 and B i,j 1 = {x i,j 1 ,j 2 : 1 ≤ j 2 ≤ b + 1} as in Remark 2. Without loss of generality, assume that the missing component is c 1,1,b+1 = m a (x 1,1,b+1 ), where m a ∈ M. Observe immediately that because both of f • h and h are constant on B 1,1 , the restriction m a | B 1,1 can be written as a polynomial of degree max{deg h − 2, deg h − λ − 1} = deg h − 2 = b − 1. Since x 1,1,j 2 ∈ B 1,1 for each j 2 , we have that m a | B 1,1 (x 1,1,j 2 ) = m a (x 1,1,j 2 ) = c 1,1,j 2 . Using Lagrange interpolation on the points (x 1,1,j 2 , c 1,1,j 2 ) for 1 ≤ j 2 ≤ b, we obtain a polynomial ∆ B 1,1 of degree b − 1 which agrees with m a | B 1,1 at b distinct points, so the two polynomials must be equal. Thus we can recover c 1,1,b+1 by evaluating ∆ B 1,1 at the element x 1,1,b+1 . Let us now consider the case of λ erasures (in the practical example we will take λ = 2, as that is the second most likely scenario of failures. Among these λ erasures, the erasures which are isolated in locality sets B i,j can be recovered by using the 1-locality, so the interesting case is when multiple erasures occur in the same B i,j . Let us assume that λ ≥ 2 erasures occur in the same locality set B i,j . In this case, since f • h is constant on A i , the restriction m a | A i is a polynomial of degree deg f deg h − λ − 1 = a + λ − λ − 1 = a − 1 D. Optimality of the code We dedicate this subsection to proving the optimality of our code C. Therefore, we will be computing the values of k and d. Lemma 3.3: Let C be the code in (3). Then k = (s + 1)((deg f − 1)(deg h − 1) + deg h − λ) Proof: Since in particular deg(g i,j h j + degg i h deg f −1 ) ≤ deg(f • h) and deg g i,j , degg i ≤ deg h, by uniqueness of F -adic expansion both for F = f • h and for F = h, we have k = dim Fq M = (s + 1)((deg f − 1)(deg h − 1) + (deg h − λ)), as we wanted to prove. δ =((deg f deg h)s + (deg f − 1) deg h + deg h − λ − 1) =(s + 1) deg h deg f − λ − 1,(4) and this proves the claim. δ = n − d = k − 1 ρ (a + λ) + k 1 + k 1 b . Note that k − 1 ρ = 1 − deg f deg h + deg f + λ − 1 + s + 1 = s, since λ ≤ deg h − 1, and k 1 b = deg f − λ deg h − 1 = deg f − λ deg h − 1 , in fact k 1 = deg f (−(b + 1)s + deg h(s + 1) − 1) − λ = deg f (deg h − 1) − λ.δ − δ = (s + 1) deg h deg f − λ − 1 − s deg f deg h − k 1 − k 1 b = deg f deg h − λ − 1 − (deg f (deg h − 1) − λ) − deg f + λ deg h − 1 = λ deg h − 1 − 1(5) and since λ deg h−1 − 1 = 0 for λ ≤ deg h − 1, the code is optimal. E. Comparison with the optimal hierarchical RS-like code in [1] A construction of optimal hierarchical LRCs for a certain set of parameters is presented in [1, Proposition IV.2]. Let us fix the parameters for which that construction exists, i.e., r 1 = sr 2 (we note that we do not require such a constraint, but that even in this scenario we show that we can construct codes that are not available from [1, Proposition IV.2]) . The set of parameters of the codes in [1, Proposition IV.2], given also in our notation, is as follows: • the length of the codes in both settings is n, • each small locality set (at the bottom level of the hierarchy) has size r 2 + 1, so in our case each has size b + 1, • their ν is our a + λ, • the middle code has distance r 2 + 3 and hence can tolerate r 2 + 2 erasures, so their r 2 + 2 corresponds to our λ, • their r 1 is our ρ, • the code is optimal, with distance d = n − t(r 1 + r 2 + 1 + s) + r 2 + 3, for some t,s, • the two level hierarchy have locality parameters (r 1 ,r 2 + 3) and (r 2 ,2). This shows immediately that our class of codes is different from the codes in [1,Proposition IV.2]. In fact, the optimality of our codes strongly relies on the assumption λ ≤ r 2 , which is not the case in the construction in [1, Proposition IV.2], in which instead λ = r 2 + 2. It follows that our class of codes contains codes which are not covered by this construction, as we can construct optimal hierarchical codes with two level hierarchy having locality parameters (r 1 ,λ) and (r 2 ,2), for any λ ≤ r 2 , such as λ = 2. We emphasize that in [11] it is necessary to set a fixed λ = r 2 + 2 since in this way one can reach optimality using the bound in [11, Theorem 2.1], while, using our improved bound and enhancing the construction in [1], one is allowed more flexibility as we explained. Moreover, we will see in detail how to construct our codes without the arithmetic restrictions appearing in the examples which use monomials or linearized functions (see Section IV). For a better comparison and to simplify the understanding, in the next paragraph, we will still use monomials for the toy example, even if it is not a requirement as we explain in Section IV. F. Toy example Suppose one desires a code over F 19 of dimension 6 which can recover 1, 2, and 8 lost nodes by accessing at most 2, 4, and 6 other nodes, respectively (i.e., the distance of the code equal 9). This is not possible using standard Tamo-Barg construction since, to recover more than 1 node, one would need to access as many nodes as the dimension of the code, that is, 6 nodes. Another option is to consider codes with availability using an orthogonal partition of the multiplicative group of F 19 that includes C 3 (as one wants the locality to be 3). But this does not work in this case either as the only other option is C 9 and C 3 ⊆ C 9 (since F * q is cyclic for any prime power q). Moreover [11, Proposition IV.2] does not hold for λ = 2. Our construction instead provides a code that allows these recovery cababilities and is information theoretically optimal in the sense of the Singleton bound in Subsection II-A. Suppose we choose f = X 2 and h = X 3 (so b = 2 and a = 4). Then a general information polynomial is given by ≤0 . In particular theg i (X) =g i are constants (notice that the internal sum in j in (2) disappears as deg(f ) = 2). Therefore, by evaluating the messages at the preimage of the the 3 totally split places of x 6 = f • h, we get a code of length n = 18, dimension k = 6, b = 2 and a = 4. Notice that this code can recover 1 erasure by looking at b = 2 other nodes. Moreover, if two erasures occur, we have two possibilities: either the erasures occur in the same nest for (f,h), in which case one needs to access (in the worst case scenario) at most 4 other nodes, or the erasures occur in different nests, in which case one can use twice the locality (that is, 2) to recover each node so that one again needs to access at most 4 other nodes. Since we are evaluating polynomials of degree at most 9, the distance of the code is 18 − 9 = 9 and therefore one has a fault tolerance of 8 erasures. Practically, given those 18 nodes, we are looking at the following disposition of hierarchy: Fig. 1: point-sets which corresponds to the following matrix: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 11 8 12 18 2 3 14 5 16 17 4 6 9 10 13 15 1 1 1 18 18 18 8 8 8 11 11 11 7 7 7 12 12 12 1 1 1 1 1 1 7 7 7 7 7 7 11 11 11 11 11 11 1 7 11 8 12 18 14 2 3 16 17 5 6 9 4 15 10 13 1 1 1 18 18 18 18 18 18 1 1 1 1 1 1 18 18 18 m(X) = 1 i=0 g i (X) +g i (X)h(X) f (h(X)) i , for g i ∈ F q [X] ≤1 andg i ∈ F q [X]             1             , where the rows correspond to (the evaluator of) the basis {1,x,x 3 ,x 6 ,x 7 ,x 9 } and the columns to the elements of F * 19 ordered as in Figure 1. This means that to check the locality of each set one just needs to check the rank of the corresponding set of columns in the above matrix. For example, suppose we want to recover the third column, which corresponds to the symbol 11. We can do that using only the first two columns, since the matrix             1 1 1              has rank equal to 2. Similarly, we can recover any two lost symbols using either 3 (if they belong to the same large orbit) or 4 (otherwise, if they belong to distinct large orbits) other symbols. IV. EXISTENTIAL RESULTS VIA CHEBOTAREV DENSITY THEOREM In this section we explain how to apply Chebotarev Density Theorem to count the places t 0 ∈ F q such that f (h) − t 0 is totally split. A lower bound on this quantity determines directly a lower bound on the size of the hierarchy in our construction. This determines completely the range of parameters of our hierarchical codes, and in turn it shows that they always exist for q large enough, without arithmetic restrictions on the localities and the size of the base field. A. Background on Galois theory We begin by recalling a few preliminary definitions. Let K and M be fields. We will write K[X] to denote the polynomial ring in the indeterminate X over K. The field extension K ⊆ M will be written as M/K, and its degree, that is, the dimension of M as a K-vector space, as [M : K]. For q a power of a prime, let F q be the finite field with q elements and let F * q = F q \{0} be its (cyclic) multiplicative subgroup. Let t be transcendental over F q and denote by F q (t) the rational function field in t over F q . We follow closely the notation and terminology in [13] throughout this section, and we provide the essential notions here. A finite-dimensional extension There is a one-to-one correspondence between places of M/K and valuation rings O of M/K, so we will write O P to denote the valuation ring whose maximal ideal is P . We will write P M to denote the set of all places of M/K and P 1 M ⊆ P M to denote the set of rational places of M/K. Let K ⊆ M be an extension of function fields. The automorphism group of M/K, that is, the group of all automorphisms of M which fix K element-wise, is denoted by Aut(M/K). When |Aut(M/K)| = [M : K], we say that the extension M/K is Galois with Galois group Gal(M/K) = Aut(M/K). For places P ∈ P K and Q ∈ P M , we say that Q lies above P (and write Q | P ) if P ⊆ Q. We denote the ramification index and relative degree of the extension of places Q | P by e(Q | P ) and by f (Q | P ), respectively. We say that a polynomial f ∈ F q [X] is separable over F q if f ∈ F q [X p ], where p = char F q , and for such an f , the polynomial f − t is seen to be a separable irreducible polynomial over F q (t). We will write M f to denote the splitting field of f − t over F q (t). Equivalently, M f denotes the Galois closure of the extension F q (x) : F q (t), where x is any root of f (X) − t in the algebraic closure F q (t) of F q (t). The field of constants of M f will be denoted by k f , and we note that it is possible to have k f F q . Let G f be the monodromy group (sometimes called the arithmetic Galois group) of f , that is, the Galois group of the extension M f /F q (t). B. The number of totally split places t 0 of f (h) − t We will appeal to the Chebotarev density theorem as in Proposition 3.1 of [10] since this formulation is the most convenient for our purposes. We provide a full exposition in this section, but we briefly describe in the next paragraph the general procedure and ideas. For the extension M/F q (t), let G = Gal(M/F q (t)) be its arithmetic Galois group and let N be its geometric Galois group. Since we are interested in the number of places P ⊆ F q (t) of degree 1 which are totally split in M , by Proposition 3.4 of [10] we may assume that M ∩F q = F q is the field of constants of the extension M/F q (t) since otherwise = 0. Hence G = N . Lemma 4.1: Let f,h ∈ F q [X] be nonzero polynomials having positive degrees. Define G f = Gal(f (X) − t : F q (t)) and similarly for G h . Then the number of t 0 ∈ F q such that f (h(X)) − t 0 splits completely into distinct (linear) factors over F q is at least 1 |G h | deg f |G f | q + O( √ q), where the implied constant can be chosen explicitly and is independent of q. Proof: Denoting the number of t 0 ∈ F q we are considering by |T 1 split (f •h)|, from Proposition 3.1(ii) of [10] we immediately have |T 1 split (f • h)| ≥ q + 1 − 2g √ q |G| − #Ram 1 (M : F q (t)) 2 .(6) We proceed by proving an upper bound on the size of G, which in turn gives the wanted lower bound for |T 1 split (f • h)| Let T be the rooted tree of height 2 with deg f branches and deg h roots adjacent to each branch. One can easily see that G ⊆ Aut(T ), so because Aut(T ) is isomorphic to the wreath product (G h × · · · × G h deg f ) G f , we have |G| ≤ |G h | deg f |G f |. Combining (6) with the bound on |G|, we obtain ≥ q + 1 − 2g √ q |G h | deg f |G f | − Ram 1 (M : F q (t)) 2 . Note that the bound given in the previous lemma can be written more explicitly as ≥ (q + 1) − 2g √ q |G h | deg f |G f | − (deg f )(deg h)/2.≥ q |G h | deg f |G f | + O( √ q), where the implied constant can be made explicit, and G h (resp G f ) is the Galois group of f − t (resp. h − t), and k is as in Lemma 3.3. Remark 3: Notice that the condition of having trivial constant field extension is automatic once there is a single totally split place, and it is the generic situation if the polynomials are chosen at random. Proof: Since M has trivial constant field F q , Lemma 4.1 guarantees that there exist at least ≥ 1 |G h | deg f |G f | q + O( √ q) totally split places, i.e. elements t 0 of F q such that f (h) − t 0 is totally split. Now Lemma 3.1 guarantees that the evaluation set T consisting of the preimages of the t 0 's forms a nest for the pair (f,h) (see Remark 2). Now construct the code by evaluating the polynomials in (2) Section III-C. This means that we are able to recover either 1 (resp. 2) lost node(s) by looking either at 2 (resp. 5) other nodes. We point out that the Tamo Corollary 2 . 2 : 22Let C be a [n,k,d,b] LRC. Then by S the set of the corresponding columns of the generator matrix G of the code. Thus, |S| = k−1 ρ (a + λ) and by the above discussion, we have rk(S) = ρ k−1 ρ ≤ k − 1. This allows us to add more columns to S until the rank equals k − 1 using a smaller locality set. More precisely, we can always choose a set of the remaining columns of G, say S 1 , of size k 1 Construction 3. 2 ( 2HLRC): Let f,h ∈ F q [X] be -nested, with 3 ≤ deg h = b + 1 and deg f (X) = a+λ b+1 for some integer 2 ≤ λ ≤ b, and let A = ∪ i=1 A i , where {A 1 , .. . ,A } is a set of nests for (f,h). a positive integer s ≥ 1, consider the set M of polynomials of the form . Thus Lagrange interpolation on a set of a points of A i on which no erasure occurred yields a polynomial ∆ A i which agrees with m a on all of A i . Hence the missing components can be obtained by evaluating ∆ A i at each of the corresponding locations in A i . Lemma 3 . 4 : 34Let C be the code in(3). Then d ≥ n − δ, forδ = (s + 1) deg h deg f − λ − 1Proof: A lower bound for the minimum distance is obtained by subtracting δ from n, where δ is the upper bound for the maximum number of zeros of m ∈ M. We compute Theorem 3 . 5 : 35Let C be the code obtained by using Construction 3.2. Then C is an optimal [n,k,b,a,λ] HLRC. Proof: Let ρ = (a + λ)/(b + 1)b − (λ − 1) and k 1 = k − 1 − k−1 ρ ρ. Moreover, we recall that a + λ = deg f deg h and deg h = b + 1. Let d denote the optimal distance, such that K of F q (t) is called a (global) function field over F q . A valuation ring of a function field M/K is a ring O such that K O M and which contains at least one of z or z −1 for every z ∈ M . A place P of M/K is the unique maximal ideal of some valuation ring O of M/K, and the degree of P is defined to be deg P = [O/P : K]. In particular, P is called a rational place of M/K if [O/P : K] = 1. For polynomials f,h ∈ F q [X], consider the composition f (h). By the lower bound in [10,Proposition 3.1] on the number of t 0 ∈ F q such that f (h) − t 0 splits into linear factors over F q ,we have that for large enough q it is guaranteed to have a large number of totally split places ofdegree 1 of F q (x)/F q (t) when f (h) ischosen correctly. Now, we may assume that the field of constants k f (h) of M f (h) is trivial since otherwise there cannot be a totally split place of degree 1. Since we want to be as large as possible, one quickly sees from the lower bound in [10, Proposition 3.1] that minimizing the size of the monodromy group G f (h) of f (h) achieves this goal. Thus our construction always effectively results in an optimal code as long as the size of the alphabet verifies a certain lower bound. at the subset A of preimages of T via f (h), i.e. A = (f • h) −1 (T ), which has size deg(f (h)) . The hierarchy is now given by the nest structure in the sense of Remark 2, and the parameters obtained from Section III. V. PRACTICAL CHOICE OF PARAMETERS TO CONSTRUCT OPTIMAL HLRC The construction we presented in the previous sections allows us to exhibit some interesting examples of HLRCs. To begin with, we consider the case F 64 . Choosing f and h such that deg f = deg h = 3 and = 7, our construction gives rise to a (63,k,d,2,5,2)-HLRC, where the values of k and d depend on the choice of s in Construction 3.2. In fact, the first locality b equals deg h − 1, whereas the second locality (a = 5) can be computed by following the passages of -Barg construction for availability over the field of size 64, under the same first locality assumption (b = 2), forces to have length 21 (with locality sets of size 3 and 7), whereas ours permits to have lenght 63, leading to a much better minimum distance and a larger number of servers allowed. More precisely, the Tamo-Barg construction requires the use of two orthogonal partitions, and this can be achieved by using 21 symbols corresponding to the action of x 3 and of x 7 on F 64 \ {0}. Note further that their construction has a larger second locality: 7, against our better parameter a = 5. VI. ACKNOWLEDGEMENTS This research is supported by the National Science Foundation under Grant No. Lavorante graduated at the University of Perugia in October 2018. He completed his Ph.D. with distinction at the University of Modena and Reggio Emilia in February 2022 under the supervision of Prof. Massimo Giulietti. He is currently a Post-Doc researcher at the University of South Florida. Proposition 4.2: Let f,h ∈ F q [x] be polynomials such that f (h) − t has Galois group G and the splitting field M of f (h) − t has constant field equal to F q . Then there exists an optimal H-LRC, with parameters [deg(f (h)) ,k,d, deg(h) − 1, deg(f (h)) − λ,λ] for any λ < deg(h) and Codes with hierarchical locality from covering maps of curves. S Ballentine, A Barg, S Vlȃduţ, IEEE Transactions on Information Theory. 6510S. Ballentine, A. Barg, and S. Vlȃduţ. Codes with hierarchical locality from covering maps of curves. IEEE Transactions on Information Theory, 65(10):6056-6071, 2019. Locally recoverable codes from algebraic curves and surfaces. A Barg, K Haymaker, E W Howe, G L Matthews, A Várilly-Alvarado, Algebraic Geometry for Coding Theory and Cryptography. SpringerA. Barg, K. Haymaker, E. W. Howe, G. L. Matthews, and A. Várilly-Alvarado. Locally recoverable codes from algebraic curves and surfaces. In Algebraic Geometry for Coding Theory and Cryptography, pages 95-127. Springer, 2017. Locally recoverable codes on algebraic curves. A Barg, I Tamo, S Vlȃduţ, 2015 IEEE International Symposium on. IEEEInformation Theory (ISITA. Barg, I. Tamo, and S. Vlȃduţ. Locally recoverable codes on algebraic curves. In Information Theory (ISIT), 2015 IEEE International Symposium on, pages 1252-1256. IEEE, 2015. Locally recoverable codes from automorphism group of function fields of genus g ≥ 1. D Bartoli, M Montanucci, L Quoos, IEEE Transactions on Information Theory. 6611D. Bartoli, M. Montanucci, and L. Quoos. Locally recoverable codes from automorphism group of function fields of genus g ≥ 1. IEEE Transactions on Information Theory, 66(11):6799-6808, 2020. Optimal selection for good polynomials of degree up to five. Designs, Codes and Cryptography. A Dukes, A Ferraguti, G Micheli, 90A. Dukes, A. Ferraguti, and G. Micheli. Optimal selection for good polynomials of degree up to five. Designs, Codes and Cryptography, 90(6):1427-1436, 2022. Locally repairable codes with availability and hierarchy: matroid theory via examples. R Freij-Hollanti, T Westerbäck, C Hollanti, International Zurich Seminar on Communications-Proceedings. ETH ZurichR. Freij-Hollanti, T. Westerbäck, and C. Hollanti. Locally repairable codes with availability and hierarchy: matroid theory via examples. In International Zurich Seminar on Communications-Proceedings, pages 45-49. ETH Zurich, 2016. Explicit maximally recoverable codes with locality. P Gopalan, C Huang, B Jenkins, S Yekhanin, IEEE Trans. Information Theory. 609P. Gopalan, C. Huang, B. Jenkins, and S. Yekhanin. Explicit maximally recoverable codes with locality. IEEE Trans. Information Theory, 60(9):5245-5256, 2014. Codes with local regeneration and erasure correction. G M Kamath, N Prakash, V Lalitha, P V Kumar, IEEE Transactions on information theory. 608G. M. Kamath, N. Prakash, V. Lalitha, and P. V. Kumar. Codes with local regeneration and erasure correction. IEEE Transactions on information theory, 60(8):4637-4660, 2014. New constructions of optimal locally recoverable codes via good polynomials. J Liu, S Mesnager, L Chen, IEEE Transactions on Information Theory. 642J. Liu, S. Mesnager, and L. Chen. New constructions of optimal locally recoverable codes via good polynomials. IEEE Transactions on Information Theory, 64(2):889-899, 2018. Construction of locally recoverable codes which are optimal. G Micheli, IEEE transactions on information theory. 661G. Micheli. Construction of locally recoverable codes which are optimal. IEEE transactions on information theory, 66(1):167-175, 2020. Codes with hierarchical locality. B Sasidharan, G K Agarwal, P V Kumar, 2015 IEEE International Symposium on Information Theory (ISIT). IEEEB. Sasidharan, G. K. Agarwal, and P. V. Kumar. Codes with hierarchical locality. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 1257-1261. IEEE, 2015. Optimal locally repairable codes via rank-metric codes. N Silberstein, A S Rawat, O O Koyluoglu, S Vishwanath, 2013 IEEE International Symposium on Information Theory. IEEEN. Silberstein, A. S. Rawat, O. O. Koyluoglu, and S. Vishwanath. Optimal locally repairable codes via rank-metric codes. In 2013 IEEE International Symposium on Information Theory, pages 1819-1823. IEEE, 2013. Algebraic function fields and codes. H Stichtenoth, Springer Science & Business Media254H. Stichtenoth. Algebraic function fields and codes, volume 254. Springer Science & Business Media, 2009. A family of optimal locally recoverable codes. I Tamo, A Barg, IEEE Transactions on Information Theory. 608I. Tamo and A. Barg. A family of optimal locally recoverable codes. IEEE Transactions on Information Theory, 60(8):4661- 4676, 2014. Bounds on the parameters of locally recoverable codes. I Tamo, A Barg, A Frolov, IEEE Transactions on information theory. 626I. Tamo, A. Barg, and A. Frolov. Bounds on the parameters of locally recoverable codes. IEEE Transactions on information theory, 62(6):3070-3083, 2016. PLACE PHOTO HERE Austin Dukes is a graduate student at the University of South Florida working in Applied Algebra and Coding Theory. I Tamo, D S Papailiopoulos, A G Dimakis, IEEE Transactions on Information Theory. 6212Optimal locally repairable codes and connections to matroid theoryI. Tamo, D. S. Papailiopoulos, and A. G. Dimakis. Optimal locally repairable codes and connections to matroid theory. IEEE Transactions on Information Theory, 62(12):6661-6671, 2016. PLACE PHOTO HERE Austin Dukes is a graduate student at the University of South Florida working in Applied Algebra and Coding Theory. He completed his Ph.D. with distinction at the Zurich Graduate School in Mathematics in October 2015 under the supervision of Prof. Place Photo Here Giacomo, Micheli, La Sapienza. University of South Florida and Co-Director of the Center for Cryptographic Research at USFJoachim Rosenthal. He is currently a tenure-track assistant professor at thePLACE PHOTO HERE Giacomo Micheli graduated at the University of Rome "La Sapienza" in July 2012. He completed his Ph.D. with distinction at the Zurich Graduate School in Mathematics in October 2015 under the supervision of Prof. Joachim Rosenthal. He is currently a tenure-track assistant professor at the University of South Florida and Co-Director of the Center for Cryptographic Research at USF.

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{'abstract': 'In this paper we construct new optimal hierarchical locally recoverable codes. Our construction is based on a combination of the ideas of [1],[11]with an algebraic number theoretical approach that allows to give a finer tuning of the minimum distance of the intermediate code (allowing larger dimension of the final code), and to remove restrictions on the arithmetic properties of q compared with the size of the locality sets in the hierarchy. In turn, we manage to obtain codes with a wide set of parameters both for the size q of the base field, and for the hierarchy size, while keeping the optimality of the codes we construct.', 'arxivid': '2207.10383', 'author': ['Austin Dukes ', 'Giacomo Micheli ', 'Vincenzo Pallozzi Lavorante '], 'authoraffiliation': [], 'corpusid': 250917090, 'doi': '10.48550/arxiv.2207.10383', 'github_urls': [], 'n_tokens_mistral': 13029, 'n_tokens_neox': 11520, 'n_words': 7707, 'pdfsha': '9ba23c44b94485feece95eb1017f54cd25a7af9b', 'pdfurls': ['https://arxiv.org/pdf/2207.10383v1.pdf'], 'title': ['Optimal locally recoverable codes with hierarchy from nested F -adic expansions', 'Optimal locally recoverable codes with hierarchy from nested F -adic expansions'], 'venue': []}

arxiv

The energetic particle environment of a GJ 436 b-like planet D Rodgers-Lee Dublin Institute for Advanced Studies School of Cosmic Physics 31 Fitzwilliam PlaceDublin D02 XF86Ireland School of Physics Trinity College Dublin University of Dublin Dublin 2D02 PN40College GreenIreland P B Rimmer Cavendish Laboratory University of Cambridge JJ Thomson AveCB3 0HECambridgeUnited Kingdom A A Vidotto Leiden Observatory Leiden University P.O. Box 95132300 RALeidenThe Netherlands A J Louca Leiden Observatory Leiden University P.O. Box 95132300 RALeidenThe Netherlands A M Taylor Deutsches Elektronen-Synchrotron ZeuthenGermany A L Mesquita Leiden Observatory Leiden University P.O. Box 95132300 RALeidenThe Netherlands Y Miguel Leiden Observatory Leiden University P.O. Box 95132300 RALeidenThe Netherlands SRON Netherlands Institute for Space Research Niels Bohrweg 42333 CALeidenthe Netherlands O Venot Université Paris Cité Univ Paris Est Creteil CNRS F-75013ParisLISAFrance Ch Helling Space Research Institute Austrian Academy of Sciences Schmiedlstrasse 6A-8042GrazAustria Institute for Theoretical Physics and Computational Physics Centre for Exoplanet Science Graz University of Technology Petersgasse 168010Graz 10 St Andrews University of St Andrews North Haugh KY16 9SSSt AndrewsUK P Barth Space Research Institute Austrian Academy of Sciences Schmiedlstrasse 6A-8042GrazAustria School of Physics and Astronomy SUPA University of St Andrews North Haugh KY16 9SSSt AndrewsUK School of Earth & Environmental Sciences University of St Andrews Bute BuildingKY16 9TSTerrace, St AndrewsQueen'sUK E Lacy Dublin Institute for Advanced Studies School of Cosmic Physics 31 Fitzwilliam PlaceDublin School of Physics University College Dublin Belfield, Dublin 4Ireland The energetic particle environment of a GJ 436 b-like planet MNRAS 000Accepted xxxx xxxxxx xx. Received xxxx xxxxxx xx; in original form xxxx xxx xxPreprint 14 March 2023 Compiled using MNRAS L A T E X style file v3.0planetary systems -planets and satellites: atmospheres -cosmic rays -methods: numerical -stars: low-mass -stars: winds, outflows A key first step to constrain the impact of energetic particles in exoplanet atmospheres is to detect the chemical signature of ionisation due to stellar energetic particles and Galactic cosmic rays. We focus on GJ 436, a well-studied M dwarf with a warm Neptune-like exoplanet. We demonstrate how the maximum stellar energetic particle momentum can be estimated from the stellar X-ray luminosity. We model energetic particle transport through the atmosphere of a hypothetical exoplanet at orbital distances between a = 0.01 − 0.2 au from GJ 436, including GJ 436 b's orbital distance (0.028 au). For these distances we find that, at top-of-atmosphere, stellar energetic particles ionise molecular hydrogen at a rate of ζ StEP,H2 ∼ 4 × 10 −10 − 2 × 10 −13 s −1 . In comparison, Galactic cosmic rays alone lead to ζ GCR,H2 ∼ 2 × 10 −20 − 10 −18 s −1 . At 10au we find that ionisation due to Galactic cosmic rays equals that of stellar energetic particles: ζ GCR,H2 = ζ StEP,H2 ∼ 7×10 −18 s −1 for the top-of-atmosphere ionisation rate. At GJ 436 b's orbital distance, the maximum ion-pair production rate due to stellar energetic particles occurs at pressure P ∼ 10 −3 bar while Galactic cosmic rays dominate for P > 10 2 bar. These high pressures are similar to what is expected for a post-impact early Earth atmosphere. The results presented here will be used to quantify the chemical signatures of energetic particles in warm Neptune-like atmospheres.Galactic cosmic rays interact with, and their fluxes are suppressed by, the magnetised winds of low-mass stars (e.g. Potgieter (2013) for the Sun and Rodgers-Lee et al. (2021b) for INTRODUCTION Spectroscopic observations with JWST (Gardner et al. 2006;Rigby et al. 2022) and future dedicated exoplanet missions, such as Ariel (Tinetti et al. 2021), will characterise the composition of exoplanet atmospheres. Chemical models have predicted that energetic particles lead to specific chemical signatures in exoplanet atmospheres, such as H3O + for gas giants Barth et al. 2021). Bourgalais et al. (2020) produced synthetic JWST and Ariel transmission spectra of a sub-Neptune and showed that absorption due to H3O + should be observable. The detection of such a E-mail: dlee@cp.dias.ie chemical signature would be a key step to constrain the energetic particle fluxes impacting on planets outside the solar system. Here, we focus on modelling the fluxes of two types of energetic particles: stellar energetic particles from the host star (also known as stellar cosmic rays) and Galactic cosmic rays from the interstellar medium (ISM). In the future, it may be possible to disentangle the effect of stellar energetic particles and Galactic cosmic rays by detecting chemical signatures of energetic particles for exoplanets at different orbital distances (as suggested in Rodgers-Lee et al. 2020, for instance). solar-like stars). Previous studies have focused on calculating the Galactic cosmic ray fluxes in the habitable zone (i.e. the orbital distances where liquid water can exist, Kasting et al. 1993;Pierrehumbert & Gaidos 2011;Abe et al. 2011;Kopparapu et al. 2013;Zsom et al. 2013) and at the orbital distance of known exoplanets for a number of well-studied solar-type (Rodgers-Lee et al. 2021b) and M dwarf stars (Herbst et al. 2020;Mesquita et al. 2021Mesquita et al. , 2022a. Here, our aim is to combine the energetic particle transport through the stellar system with the subsequent transport through the exoplanet atmosphere. We relate the maximum energy of the stellar energetic particles accelerated by stellar flares to stellar magnetic field strength, following Rodgers-Lee et al. (2021a), and show how this can be related to stellar X-ray luminosity. Previous studies have investigated the distribution of stellar energetic particles in the Trappist-1 and AU Mic systems (Fraschetti et al. 2019(Fraschetti et al. , 2022. In the latter case, following a perturbation of the interplanetary medium by a coronal mass ejection (CME). Separately, the propagation, and chemical effect, of energetic particles through a number of Earth-like (e.g. Grießmeier et al. 2015;Tabataba-Vakili et al. 2016;Herbst et al. 2019b;Scheucher et al. 2020) and hot Jupiter/brown dwarf Barth et al. 2021) atmospheres has been studied. Here, we focus on the environment of the warm Neptune exoplanet GJ 436 b. GJ 436 b is scheduled for JWST observations and is also an Ariel target (Edwards & Tinetti 2022). We calculate the ionisation rate, ion-pair production rate and skin-depth equivalent dose rate due to energetic particles at different heights in the exoplanet atmosphere. In addition to studying GJ 436 b, we also investigate the effect of artificially changing the exoplanet's orbital distance on these quantities. The energetic particle ionisation rate is an important input for studying disequilibrium chemistry in exoplanet atmospheres, with chemical kinetic models such as ARGO (Rimmer & Helling 2016. Here, for simplicity, we do not account for the effect of planetary magnetic fields. However, planetary magnetic fields will reduce the energetic particle fluxes penetrating the exoplanet atmosphere (Herbst et al. 2013(Herbst et al. , 2019bGrießmeier et al. 2015Grießmeier et al. , 2016. This study has been separated into two companion papers. This paper (Paper I) focuses on characterising the energetic particle environment of a GJ 436 b-like planet. The paper is structured as follows: Section 2 introduces the models that are used for the energetic particle transport and in Section 3 our results are presented. The discussion and conclusions are given in Sections 4 and 5. Paper II (Rimmer et al. 2023) will focus on the chemical consequences of the energetic particles for this system. ENERGETIC PARTICLE SPECTRA We consider the transport of both Galactic cosmic rays from the ISM and stellar energetic particles, described in Sections 2.1 and 2.2. In this section we also describe our method to construct the stellar energetic particle spectra. Fig. 1 shows a schematic of the system indicating the two sources of energetic particles. The exoplanet atmosphere profiles that we use are described in Section 2.3. Using these, we then model the energetic particle propagation through the exoplanet atmosphere, which is described in Section 2.4. Galactic cosmic ray fluxes The Galactic cosmic ray fluxes throughout the GJ 436 system were previously calculated (Mesquita et al. 2021) using a diffusion-advection cosmic ray transport model (described in Rodgers-Lee et al. 2020). The stellar wind velocity and magnetic field profile are important quantities to model the cosmic ray transport through the stellar system. The Galactic cosmic ray fluxes that we adopt here correspond to 'Case A' from Mesquita et al. (2021), using the stellar wind properties from Mesquita & Vidotto (2020). Fig. 2(a) shows the differential intensity, j (i.e. the number of energetic particles per unit area, steradian, time and kinetic energy), as a function of energetic particle kinetic energy for the Galactic cosmic rays. The black line represents the Galactic cosmic ray fluxes at the orbital distance of GJ 436 b (∼ 0.03 au). The coloured lines represent j at the different orbital distances between a = 0.01 − 0.2 au. For the temperature pressure profiles in Section 2.3, we focus on a range of orbital distances between a = 0.01 − 0.2 au in increments of 0.01 au. We select the Galactic cosmic ray fluxes from the cosmic ray transport model closest to these orbital distances. However, the cosmic ray transport uses a logarithmic grid and due to the model resolution some of the lines overlap. This occurs at larger orbital distances, for a > 0.11 au in all but one case, where the fluxes are already very similar (see Fig. 2(a)). Thus, the overlap will not affect our results significantly. The Galactic cosmic ray fluxes vary significantly for 10 GeV energy cosmic rays depending on the orbital distance considered. The fluxes for 10 GeV energy cosmic rays are also much smaller than the values observed at Earth (see Fig.5 from Mesquita et al. 2021) at all of the orbital distances considered here. Case A corresponds to a stellar wind model which is more magnetically-dominated than the more thermally-dominated wind ('Case B' in Mesquita et al. 2021). Fig. 2 from Mesquita et al. (2021) shows the stellar wind velocity and magnetic field profiles for Case A and B. For Case A, the surface stellar magnetic field adopted was B = 4 G, the stellar wind terminal velocity, v∞, was found to be 1250 km s −1 and the stellar mass loss rate wasṀ = 1.2 × 10 −15 M yr −1 . The astrospheric size (Rast), which determines how far the Galactic cosmic rays must travel through the stellar wind, was calculated using v∞ andṀ to be 33 au (see Eq. 11 in Mesquita et al. 2021). Another important quantity for the transport model is the assumed Galactic cosmic ray spectrum outside of the astrosphere. A model fit to the local interstellar spectrum (LIS) for Galactic cosmic rays (Vos & Potgieter 2015) from Voyager 1 data (Stone et al. 2013) was used as the boundary condition for these simulations. Finally, the assumed turbulence properties for the stellar wind are important by dictating the diffusion of the Galactic cosmic rays from the ISM through the astrosphere. For more details see Mesquita et al. (2021). Stellar energetic particle fluxes Unlike the LIS for Galactic cosmic rays, less is known about stellar energetic particle spectra. It is generally assumed that stars more active than the Sun will produce higher fluxes of stellar energetic particles (e.g. Feigelson et al. 2002). The higher flux is expected due to the increased flare energies Figure 1. Schematic of the GJ 436 system (not to scale) and its surrounding environment embedded in the ISM. The curved blue and green arrows represent the advection and diffusion of Galactic cosmic rays (labelled as 'GCRs') and stellar energetic particles (labelled as 'StEPs'), respectively. The straight blue and green arrows represent their ballistic transport through the exoplanet atmosphere which does not account for the effect of a planetary magnetic field. The annotations indicate where the stellar wind model and the various energetic particle transport models are described or were first presented. It is important to note, as indicated in the figure, that the stellar energetic particle and Galactic cosmic ray fluxes are assumed to be isotropic at the top of the atmosphere. This assumption is discussed further in Section 2.4. and flaring rate of active stars (e.g. from Kepler and TESS observations, Maehara et al. 2015;Günther et al. 2020). It is also very likely for active stars that stellar energetic particles will be accelerated to higher energies than for the Sun. The Sun itself has been inferred to accelerate particles to GeV energies from γ−ray observations (Ajello et al. 2014;Ackermann et al. 2014). Considering the maximum energy that stellar energetic particles are accelerated to by their host star is important because the energy loss rate for energetic particles is energy dependent. The energy loss rate for GeV energy particles is much lower than for ∼MeV energy particles (see Fig. 5.6 in Longair 2011). Thus, GeV energy particles are far more penetrating than ∼MeV energy particles and can, for instance, lead to showers of secondary particles that can reach the surface of a rocky exoplanet (Atri 2017). Following Rodgers-Lee et al. (2021a), there are three quantities that we use to obtain a stellar energetic particle spectrum. These are (i) the power law index for the spectrum, (ii) the total energy available to produce stellar energetic particles and (iii) the maximum stellar energetic particle momentum (pmax) before the spectrum decays exponentially. The power law index is set to be α = 2 (representative of diffusion shock acceleration, Krymskii 1977;Bell 1978;Blandford & Ostriker 1978, and compatible with acceleration due to magnetic reconnection) such that dN/dp ∝ p −α e −p/pmax , where N and p are the number and momentum of the particles, respectively. The total luminosity injected in stellar energetic particles (LCR) that we adopt is 0.1PSW, where PSW =Ṁ v 2 ∞ /2 is the stellar wind kinetic power. This is in line with efficiency estimates from supernova remnants (Vink et al. 2010). We calculate LCR = 6 × 10 25 erg s −1 for GJ 436 withṀ = 1.2 × 10 −15 M yr −1 and v∞ = 1250 km s −1 . Maximum stellar energetic particle momentum In Rodgers-Lee et al. (2021a), the Hillas criterion (Hillas 1984) was used to estimate the maximum stellar energetic particle momentum that the Sun would have accelerated particles to in the past. The Hillas criterion posits that the maximum energy achieved by an accelerator is limited by the accelerator size, i.e. if the accelerated particle's Larmor radius is larger than the accelerator region it will escape. Using the Hillas criterion, the maximum momentum, pmax, of accelerated particles from a star can be expressed as pmax pmax, = β B R β B R(1) where R and R are the solar and stellar radii, respectively. B and B are the surface solar and stellar magnetic field strengths, respectively. B = 1.3 G and pmax, = 0.2 GeV/c. The parameters β and β are the solar and stellar shock velocity in units of the speed of light, respectively. Here, we assume β ∼ β as a first approximation. We consider flare accelerated particles injected at ∼ 1.4R rather than particles accelerated by CMEs (see Rodgers-Lee et al. 2021a, for a discussion of this). Eq. 1 is the same as Eq. 7 from Rodgers-Lee et al. (2021a) for the Sun. The only difference is that R is also included in Eq. 1. The Zeeman Doppler Imaging (ZDI) technique has provided average large-scale stellar magnetic field strengths for hundreds of low-mass stars (e.g. Donati et al. 2006;Morin 2012;Moutou et al. 2017). This technique has traditionally focused on active stars which generally have strong magnetic fields, best suited for the observations. However, transiting exoplanets are typically detected around less active stars because increased stellar activity, in the form of stellar spots, tends to obscure exoplanet signatures. Thus, the overlap between these samples is currently extremely limited and stellar magnetic field strengths for less active stars often must be estimated. For instance, Vidotto et al. (2014) presented the relation between stellar X-ray luminosity (LX) and the large-scale magnetic flux (Φ) derived from a large sample of stars with ZDI maps, as LX ≈ 10 −13.7 Φ 1.80±0.20 erg s −1 where Φ = 4πR 2 |B| (Mx) and |B| is the average largescale magnetic field strength derived using the ZDI technique. Using Eq. 2, we can now re-write Eq. 1 in terms of LX as pmax pmax, = B R 2 B R 2 R R = Φ Φ R R = LX, LX, γ R R .(3) where γ = 0.56 −0.06 +0.125 . Thus, in the absence of a magnetic field strength measurement the maximum stellar energetic particle momentum can be estimated using the stellar X-ray luminosity. Here however, we have used B = 4 G for GJ 436 to be consistent with Mesquita et al. (2021) 1 . This results in pmax = 0.3GeV/c using Eq. 1. In comparison, using Eq. 3, 1 As mentioned in Mesquita et al. (2021), B = 4 G was chosen because the stellar wind model from Mesquita & Vidotto (2020) with this B value predicts an X-ray luminosity that is consistent with what is observed for GJ 436. we find pmax = 0.1 − 0.7GeV/c for LX, = 2.7 × 10 26 − 4.7 × 10 27 erg s −1 (spanning the range of values from solar minimum to maximum, Peres et al. 2000) with γ = 0.56 and LX, = 5.7 × 10 26 erg s −1 for GJ 436 (Mesquita & Vidotto 2020). We construct the stellar energetic particle spectrum close to the surface of the star using the values motivated above for LCR, α and pmax. This spectrum is used in the energetic particle transport equation (Eq. 1 from Rodgers-Lee et al. 2021a) to calculate the resulting differential intensity, j, as a function of orbital distance. Fig. 2(b) shows j as a function of energetic particle kinetic energy for stellar energetic particles at different orbital distances. The black line represents the energetic particle fluxes at the orbital distance of GJ 436 b (∼ 0.03 au). The coloured lines represent j at different orbital distances between a = 0.01 − 0.2 au in increments of 0.01 au. The stellar energetic particle fluxes vary significantly for 10 GeV energy cosmic rays depending on the orbital distance considered. The stellar energetic particle fluxes are much higher than the Galactic cosmic ray fluxes at MeV energies (see Fig. 2(a)). For GeV energies the Galactic cosmic rays begin to dominate due to the exponential cut-off assumed for the stellar energetic particle spectrum; this is similar to what Barth et al. (2021) found in their Fig. 4. For a more active/younger star than GJ 436 it would be reasonable to assume a higher cut-off energy, as motivated by Eq. 1 which would result in stellar energetic particles dominating up to higher energies (as discussed in Rodgers-Lee et al. 2021a, in the context of a young Sun-like star). Other methods for estimating stellar energetic particle spectra use relationships between solar far-UV (Youngblood et al. 2017), X-ray flare energy (Herbst et al. 2019a) or starspot size (Herbst et al. 2021) and > 10 MeV proton flux. This is similar to our scaling of the overall spectra with stellar wind kinetic energy. The stellar energetic particle spectrum is normalised using these proton fluxes once a spectral shape has been selected. The shape of the spectrum is often based on a solar energetic particle event (e.g. from Mewaldt et al. 2005), as adopted by Rab et al. (2017) and Barth et al. (2021) for instance, or the strongest ground level enhancement event measured on Earth (e.g. Herbst et al. 2019b;Scheucher et al. 2020). This is the main difference between our model and those based on solar energetic particle events. The spectral shape obtained from our model depends on stellar magnetic field strength which varies from star to star. As mentioned above, we consider flare accelerated particles rather than those from CMEs as stellar CME properties remain elusive (e.g. Veronig et al. 2021, for a recent detection of coronal dimming of X-ray emission due to a CME and references therein). However, Hu et al. (2022) recently modelled the expected stellar energetic particle fluxes from a CME associated with a young solar-like star and found higher maximum particle energies for higher CME speeds. On the other hand, previous stellar wind simulations including CMEs (Alvarado-Gómez et al. 2018) have suggested that strong solar-like CMEs may be suppressed by a largescale dipole stellar magnetic field of 75 G which would affect the corresponding stellar energetic particle fluxes. Fraschetti et al. (2022) showed for the AU Mic system how the stellar energetic particle flux distribution reaching the planetary orbits is strongly altered by the passage of a CME. To complicate matters further, solar CME-CME interactions account for more than 25% of the major geomagnetic storms observed (Zhang et al. 2007). Using models, Koehn et al. (2022) suggest that successive CMEs can lead to extreme conditions at Earth. Thus, it would be expected that stellar CME-CME interactions will also affect the associated stellar energetic particle fluxes. Exoplanet atmosphere model The exoplanet atmosphere density (n(z)) in units of cm −3 , is important for the propagation of the energetic particles through the exoplanet atmosphere. The 1D temperature pressure profiles for a Neptune-like planet in the GJ 436 system are found using the radiative transfer code HELIOS (Malik et al. 2017(Malik et al. , 2019. We assume an adiabatic atmosphere for pressures, P > 1 bar 2 . The density is then calculated using the ideal gas law for the entire atmosphere. The profiles, shown in Fig. 3, are calculated for a Neptune-like planet at different orbital distances, a, ranging from 0.01-0.2 au in steps of 0.01 au which includes GJ 436 b's observed orbital distance (i.e. 0.028 au). The required inputs for HELIOS are the planetary parameters (i.e. mass, radius and orbital distance), the opacities and chemical abundances of included species and the stellar spectrum. For the opacities we make use of the line lists from the DACE database (Grimm & Heng 2015;Grimm et al. 2021) and we only take the most relevant species into account a=0.01au a=0.2au GJ436b Figure 3. Temperature profiles of a hydrogen-dominated warm Neptune atmosphere for orbital distances between 0.01 au (rightmost blue line) and 0.2 au (leftmost green line) in the GJ 436 system. The black line represents the temperature profile at the orbital distance of GJ 436 b. The profiles were determined using HE-LIOS when pressure, P < 1 bar, and assuming an adiabat when P > 1 bar, indicated by the grey dotted line. (i.e. CH4, CN, CO, CO2, H2O, H2S, NH3, NaH, PH3, SiO, TiO, and VO) considering solar elemental abundance (Asplund et al. 2009) in chemical equilibrium (as described in Louca et al. 2022, Fig. 3). The stellar XUV spectrum is taken from the MUSCLES survey (France et al. 2016;Youngblood et al. 2016;Loyd et al. 2016;Youngblood et al. 2017). For higher wavelengths PHOENIX models were used and stitched to the XUV spectrum. The stellar and planetary parameters for the GJ 436 system are given in Table 1. For the HELIOS models where the planet had an orbital distance (period) <0.069 au (<10 days), the model assumed a tidally locked planet, with no heat-redistribution. Planets with orbital distances 0.069 au were not assumed to be tidally locked anymore and global heat-redistribution was included. The final temperature pressure profiles are averaged over the irradiated hemisphere. Energetic particle transport in exoplanet atmospheres The energetic particle propagation through the exoplanet atmosphere is performed using a Monte Carlo model (from Rimmer et al. 2012;Rimmer & Helling 2013). Here, we briefly outline the method which is described in detail in Rimmer et al. (2012); Rimmer & Helling (2013). The model is based on the continuous slowing down approximation where the energetic particles travel through a column density, n(z)dz, of the exoplanet atmosphere. Here, we take 5 × 10 5 particles to ensure that the fluctuations per energy bin are negligible. In order to calculate the energetic particle energy losses for each height element, dz, in the exoplanet atmosphere, each energetic particle is assigned two values: an individual kinetic energy, E, such that the energetic particle energies are normalised to match the top-of-atmosphere fluxes shown in Fig. 2 and a random number, N , of uniform distribution between 0 and 1. The random number is compared with the energetic particle 'optical depth', σ ion p,X n(z)dz, for the cell dz where σ ion p,X (E) is the ionisation cross section. The subscript 'X' refers to the ionisation cross sections for H2 and He from Rudd et al. (1985) and Padovani et al. (2009). If N < σ ion p,X n(z)dz, the energetic particle collides and loses energy. This requires that σmax,Xn(z)dz < 1, where σmax,X is the maximum cross section value for H2 and He. Following Rimmer et al. (2012), based on Cravens & Dalgarno (1978), the average energy loss per collision, W , is given by W = 7.92E 0.082 + 4.76, where W and E have units of eV. For this model, we assume that the atmosphere is composed of 80% H2 and 20% He. We have adopted the same expression for W for H2 and He. This step is repeated for each height element and new random numbers are assigned. The transport for stellar energetic particles and Galactic cosmic rays is treated separately. This is because if the fluxes are combined, the energetic particle flux is dominated by the stellar energetic particles and the high-energy Galactic cosmic rays are poorly sampled by the Monte Carlo method. In our model we assume that the energetic particle fluxes are isotropically distributed at the top of the atmosphere, as shown in Fig. 1. We also do not account for the effect of a planetary magnetic field. First, a planetary magnetic field would act to deflect low-energy energetic particles towards the magnetic poles. Thus, the energetic particle fluxes would no longer be isotropically distributed at the top of the atmosphere and only particles with Larmor radii comparable to or larger than the planetary radius would penetrate the magnetosphere. Second, while diffusion tends to isotropise energetic particle fluxes, such as for Galactic cosmic rays, the stellar energetic particles are more influenced by advection processes at such small orbital distances (see Fig. 4 from Mesquita et al. 2021). Thus, for close-in exoplanets the stellar energetic particle fluxes may be higher on the side of the exoplanet facing the star which we do not account for in our model. The quantities, presented in Section 3, relevant for chemical modelling of exoplanet atmospheres are the ionisation rate, ζ (s −1 ), and the ion-pair production rate, Q = nζ (cm −3 s −1 ). Following Padovani et al. (2009), the ionisation rate of molecular hydrogen, ζH 2 , by protons is calculated from the energetic particle differential intensities, j(E), as: ζH 2 = 4π Emax I(H 2 ) j(E)[1 + φ(E)]σ ion p,H 2 (E) dE(4) where I(H2) = 15.603 eV is the ionisation potential of H2 and Emax is the maximum energy of the energetic particle spectrum. The parameter, φ(E), is a correction factor that accounts for additional ionisation of H2 by secondary elec- a=0.01au GJ436b a = 0.2au (b) Figure 4. The H 2 ionisation rates are shown as a function of pressure in an exoplanet atmosphere at different orbital distances in the GJ 436 system for (a) Galactic cosmic rays and (b) stellar energetic particles. The grey dotted line denotes P = 1 bar. It is important to note that the x-axis ranges are different between the two plots which is indicated by the grey dashed lines between the panels. trons given by: φ(E) = 1 σ ion p,H 2 (E) Ee,max I(H 2 ) 1 σ ion p,H 2 (E) dσ ion p,H 2 (E) dEe Pe(E,Ee) σ ion e (Ee) dEe,(5) where Ee (eV) is the secondary electron energy, σ ion e (Ee) (cm 2 ) is the ionisation cross section of H2 by electrons and Pe(E, Ee) is the probability density that a secondary electron with energy Ee is produced as a result of an initial ionising proton of energy E. This probability density is represented in terms of the total ionisation cross-section (σ ion p,H 2 (E), cm 2 ), and differential cross-section (dσ ion p,H 2 (E)/dEe, cm 2 MeV −1 ). From Cravens et al. (1975), dσ ion p,H 2 (E) dEe = A(E) 1 + Ee E 0 2.1 ,(6) where E0 = 8.3 eV. Given that, by definition: ∞ 0 Pe(E, Ee) dEe = 1, A(E) can be expressed as A(E) = Cσ ion p,H 2 (E) where the value of C is determined from numerical integration. In addition to being important for chemical modelling, energetic particles will impact life on other planets leading to increased mutation rates (for instance see discussion in Scalo et al. 2007;Dartnell 2011). The equivalent dose rate, D (Sv s −1 ), which is the energy absorbed per unit time and mass, provides a measure of how damaging energetic particle fluxes are for life-forms. We estimate the skin-depth equivalent dose rate from primary protons as: D ∼ 2πW Emax E min E j(E) R(E) dE(7) where W = 2 is the radiation weighting factor for protons and R(E) (g cm −2 ) is the proton range in water 3 . The quantity E j(E) in Eq. 7 represents the particle number flux in an energy bin, dE. We have assumed that the energetic particles only impact from above. In Section 3.3, we presentḊ in units of mSv day −1 . RESULTS Here we present our results for the energetic particle propagation in the atmosphere of a GJ 436 b-like planet at different orbital distances. Using the energetic particle fluxes (shown in Fig. 2) and the atmospheric profiles (Fig. 3), we calculate the ionisation rate (Section 3.1), the ion-pair production rate (Section 3.2) and the skin-depth equivalent dose rate (Section 3.3). Fig. 4 shows ζH 2 from Galactic cosmic rays ( Fig. 4(a)) and stellar energetic particles ( Fig. 4(b)) as a function of pressure for different orbital distances (a = 0.01 − 0.2 au) with different coloured lines. The black line represents ζH 2 at the orbital distance of GJ 436 b. The grey dotted line denotes P = 1 bar. At the top of the atmosphere, ζH 2 from Galactic cosmic rays varies by approximately two orders of magnitude ranging from ∼ 1.7 × 10 −20 − 10 −18 s −1 between a = 0.01 − 0.2 au, shown in Fig. 4(a). In comparison, the LIS for Galactic cosmic ray protons results in a higher value of ζH 2 ∼ 1.6 × 10 −17 s −1 (estimated here as 2ζH calculated in Cummings et al. 2016, from Voyager data). For the orbital distances considered in Fig. 4(a), the top-of-atmosphere ζH 2 from Galactic cosmic rays is largest at the largest orbital distance of 0.2 au. This is because the Galactic cosmic ray fluxes are highest at the LIS values in the ISM which are then suppressed as they travel inwards through the stellar wind. The Galactic cosmic rays diffuse into the stellar system from the ISM. At the same time, the Galactic cosmic ray fluxes are suppressed by advection processes due to the expanding magnetised stellar wind. In Fig. 4(a), ζH 2 starts to decrease rapidly between 3 https://physics.nist.gov/PhysRefData/Star/Text/PSTAR.html. P ∼ 10 −1 − 10 1 bar, depending on the orbital distance considered. This illustrates that the Galactic cosmic rays travel essentially unimpeded through the exoplanet atmosphere in regions where the density is relatively low (P 10 −1 bar). For stellar energetic particles ( Fig. 4(b)), ζH 2 begins to decrease rapidly at P ∼ 10 −3 bar. The top-of-atmosphere ζH 2 values vary by approximately three orders of magnitude ranging from ∼ 2 × 10 −13 − 4 × 10 −10 s −1 . In contrast to ζH 2 from Galactic cosmic rays, the highest top-of-atmosphere ζH 2 values for the stellar energetic particles are found at the smallest orbital distances. This is because the stellar energetic particles originate from a ∼point source close to the stellar surface, causing the fluxes to decrease with increasing orbital distance ( Fig. 2(b)) as they are advected by and diffuse through the stellar wind in our model. As mentioned above, the reverse is true for Galactic cosmic rays which have the lowest fluxes closest to the star (Fig. 2(a)) as they originate from the ISM. The stellar energetic particle values for ζH 2 decrease much more rapidly with increasing pressure than the Galactic cosmic ray values (compare the black lines from Figs. 4(a) and 4(b), for instance). This is because the stellar energetic particles are predominantly at ∼MeV energies which have a higher energy loss rate than GeV energy cosmic rays. Thus, in the upper atmosphere stellar energetic particles dominate while in deeper parts (P 100 bar) Galactic cosmic rays are more important. Transmission spectroscopy with JWST and Ariel will likely probe regions between P ∼ 10 −4 − 10 −1 bar for H2-rich atmospheres (Welbanks & Madhusudhan 2019). Ionisation rates Thus, for the atmospheres and orbital distances (a = 0.01 − 0.2 au) that we consider here for the GJ 436 system, transmission spectroscopy with JWST and Ariel would probe regions where stellar energetic particles are more likely to impact the chemistry than Galactic cosmic rays (also see Section 3.2 where the ion-pair production rates are discussed). While ionisation and dissociation by energetic particles, including photons, are likely the main sources of chemical disequilibrium in the upper atmosphere, vertical mixing and chemical quenching are likely the dominant sources of disequilibrium chemistry deeper in the atmosphere (Tsai et al. 2018;Ohno & Fortney 2022). This trend will be dependent on the specific nature of the temperature profile. Cooler atmospheres, especially those of rocky planets, may have other sources of disequilibrium (e.g. degassing, deposition, condensation, escape, Noack et al. 2014;Tosi et al. 2017;Ortenzi et al. 2020;Zahnle et al. 2020). Very hot atmospheres may never experience chemical quenching ). The total ζH 2 from energetic particles as a function of pressure can be calculated by summing the individual ζH 2 values shown in Fig. 4 for Galactic cosmic rays and stellar energetic particles 4 . These ionisation rates can be used to study the energetic particle-induced chemistry in an exoplanet atmosphere, similar to Barth et al. (2021). Barth et al. (2021) studied the hot Jupiter, HD 189733 b, orbiting its K dwarf host star at 0.031 au, similar to GJ 436 b's orbital distance of 0.028 au. As discussed in Section 2.2, the fact that the adopted stellar energetic particle spectra are different, in combination with the different temperature-pressure profiles, may result in different chemical tracers for energetic particles. The main difference in the stellar energetic particle spectra is that our spectra has more ∼GeV energy particles at 0.03 au, which penetrate further into an exoplanet atmosphere. As shown in Fig. 4, the top-of-atmosphere values ζH 2 values from stellar energetic particles are all larger than those due to Galactic cosmic rays for a = 0.01 − 0.2 au. However, at 10 au we find that the value of ζH 2 = 7 × 10 −18 s −1 due to stellar energetic particles is the same as that obtained from Galactic cosmic rays. The Galactic cosmic ray fluxes, and the resulting ionisation rates, are highest at large orbital distances where they have travelled through relatively little of the stellar wind. Galactic cosmic rays from the ISM lose energy as they diffuse through the stellar wind. At the same time, stellar energetic particle fluxes decrease with distance from the star. Thus, for an exoplanet at an orbital distance > 10 au in the GJ 436 system it is more likely that any observable chemical signatures of energetic particles from the upper atmosphere will be due to Galactic cosmic rays rather than stellar energetic particles. Additionally, as mentioned above, the (higher energy) Galactic cosmic rays lose their energy less rapidly with increasing atmospheric pressure/density in comparison to the (lower energy) stellar energetic particles. This means, for an exoplanet at an orbital distance > 10 au in the GJ 436 system, that Galactic cosmic rays will likely be the dominant source of ionisation due to energetic particles at all heights in the atmosphere. This should be true for many systems, as discussed in Rodgers-Lee et al. (2020), though the exact orbital distance where this occurs will be systemdependent due to different stellar wind conditions. For instance, Fig.3 of Mesquita et al. (2021) shows the effect of assuming two different stellar wind models on the Galactic cosmic ray spectra at various orbital distances for the GJ 436 system. The difference between these stellar wind models is that for one model the wind is more magnetically-dominated (Case A adopted here), whereas the other is more thermallydominated (Case B). Thus, the value of ζH 2 from Galactic cosmic rays at 10 au would decrease somewhat if we had adopted the thermally-dominated stellar wind, largely due to the larger astrosphere resulting from this stellar wind model. For other systems which are dominated more by advective processes (see Rodgers-Lee et al. 2021b, for instance), the difference in Galactic cosmic ray ζH 2 values at large distances assuming different astrosphere sizes would be more significant. Ion-pair production rates The ion-pair production rate, Q = nζ, convolves the number density of the exoplanet atmosphere with the ionisation rate. It gives an indication of where in the exoplanet atmosphere the energetic particles will be important for chemistry by creating the most ions. Thus, high Q values correspond to regions where the energetic particles should be most important. Fig. 5(a) and 5(b) show the ion-pair production rate, Q, as a function of pressure from Galactic cosmic rays and stellar energetic particles, respectively for different orbital distances. The maximum Galactic cosmic ray Q value occurs at P ∼ 10 − 10 2 bar (rightmost green line in Fig. 5(a)). For stellar energetic particles instead, the maximum Q value occurs much higher in the atmosphere between P ∼ 10 −3 −10 −2 bar, with a much greater value (rightmost blue line). Here, we have ignored the diffusion of ions. For comparison, Fig. 6 shows the ion-pair production rate for Galactic cosmic rays (Q GCR , green dashed line) and stellar energetic particles (Q StEP , blue dot-dashed line) at the orbital distance of GJ 436 b as a function of pressure. The thin solid black line represents the combination of both. The blue shaded region illustrates where Q StEP > Q GCR and the green shaded region shows where Q GCR > Q StEP which occurs at ∼1 bar for GJ 436 b. The fact that Galactic cosmic rays are more important than stellar energetic particles at 10 − 10 3 bar is interesting because it has been suggested that a ∼ 10 2 bar atmosphere is expected for a post-impact early Earth atmosphere (Zahnle et al. 2020;Itcovitz et al. 2022). This suggests that Galactic cosmic rays could have been the dominant ionisation source deep in the atmosphere at this time. Herbst et al. (2019b) and Scheucher et al. (2020) calculate the ion-pair production rate from Galactic cosmic rays and solar/stellar energetic particles for an Earth-like atmosphere and for Prox Cen b, respectively. Fig. 6 of Herbst et al. (2019b) shows broadly the same behaviour as our results: Q GCR peaks at a lower value than Q StEP , but occurs at a higher pressure/lower altitude. The main difference is that for our model atmosphere for GJ 436 b, Galactic cosmic rays dominate for a broad range of high pressures. The fact that our maximum Q StEP value peaks at ∼ 10 −3 bar reflects that our assumed stellar energetic particle spectrum decays exponentially at a relatively low cosmic ray kinetic energy. For a more active star, we would expect the maximum Q StEP value to occur at a higher pressure. In terms of the range of values for Q StEP and Q GCR , our peak Q GCR value is lower than that found in Herbst et al. (2019b) which is reasonable since our top-of-atmosphere Galactic cosmic ray spectrum ( Fig. 2(a)) is reduced in comparison to values observed at Earth. Our peak Q StEP value lies between the maximum values found in Herbst et al. (2019b) and Scheucher et al. (2020) which is again consistent since our stellar energetic particle fluxes at ∼ 10 − 100 MeV energies lie between those assumed for the Earth-like atmosphere and for Prox Cen b. Our stellar energetic particle fluxes are higher for GJ 436 b mainly because it orbits much closer to its star than the Earth orbits the Sun, while they are lower than for Prox Cen b. At the same time, it is complex to compare values since Q depends not only on the incident energetic particle fluxes but also on the density profile of the planetary atmosphere. Additionally, Herbst et al. (2019b) include the effect of an Earth-like planetary magnetic field which introduces a low-energy particle cut-off, i.e. energetic particles below a certain energy (510 MeV in their case for latitudes of around 60 • ) cannot penetrate the planet's magnetosphere and do not contribute to Q. In this context, this indicates that our results represent the largest possible values for Q and that exoplanetary magnetic fields would generally result in a decrease of the values obtained for Q. It is likely that the Q StEP values would be more affected than Q GCR . This is because the top-of-atmosphere stellar energetic particle spectra are dominated by low-energy particles, whereas the peaks of the Galactic cosmic ray spectra occur closer to GeV energies (see Fig. 2). Skin-depth equivalent dose rates While warm Neptune exoplanets are not expected to be habitable we can calculate the skin-depth equivalent dose rate for these planets. In the future, the same procedure can be applied to stars like GJ 436 that could have terrestrial planets in their habitable zones. Fig. 7 shows the skin-depth equivalent dose rate from energetic particles as a function of pressure for Galactic cosmic rays (green dashed line) and stellar energetic particles (blue dot dashed line) at a = 0.2 au, which is the inner edge of the habitable zone for the GJ 436 system. The thin black line represents the combination of both. The grey star and magenta pentagon represent the average equivalent dose rate at Earth's surface due to Galactic cosmic rays (9 × 10 −4 mSv day −1 , including however secondary electrons, muons and γ−rays) and background radiation sources (7 × 10 −3 mSv day −1 ). Here we have used the temperature pressure profile at 0.2 au, described in Section 2.3, assuming the same planetary mass and radius as before. For Galactic cosmic rays, the equivalent dose rate is approximately constant for P < 1 bar. Again in comparison to Herbst et al. (2019b), our maximum equivalent dose rates are lower (higher) for Galactic cosmic rays (stellar energetic particles). It is important to note that Herbst et al. (2019b) consider a terrestrial planet atmosphere whereas we are still considering a hydrogen dominated atmosphere which affects the energetic particle transport in the exoplanet atmosphere. In turn, this will impact the dose rates that are calculated. For these equivalent dose rates, and for chemical modelling that include our energetic particle ionisation rates, it is particularly important to consider how constant these rates would be in time. We discuss this further in Section 4.1. While indirect detections of exoplanetary magnetic fields remain challenging, Fig. 9 of Grießmeier et al. (2016) shows that the biological dose rate decreased significantly with increasing magnetic moments for two different atmospheric column depths. This indicates that our results represent an upper limit in the case that the exoplanet possesses a magnetic field. DISCUSSION 4.1 How continuous are stellar energetic particle fluxes? Stellar energetic particle fluxes may vary on timescales of days, depending on stellar activity. Galactic cosmic ray fluxes are more likely to be approximately constant on these 10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 Equivalent dose rate [mSvday 1 ] timescales. Instead, they vary with the solar cycle (e.g. Fig.3 from Vos & Potgieter 2015), for instance, and will also vary on stellar evolutionary timescales (Rodgers-Lee et al. 2020). Similar to Rodgers-Lee et al. (2021a), we plot the flare rate as a function of the maximum momentum of the stellar energetic particles in Fig. 8. This is based on the relation between flare frequency and flare energy (dN/dE flare ∝ E b flare , where the power law index b varies from star to star). Flare energies are related to the stellar energetic particle maximum momenta via pmax ∼ E 1/2 flare , normalised for ∼GeV energetic particle energies at E flare ∼ 10 33 erg and assuming that E flare ∼ B 2 (see Rodgers-Lee et al. 2021a, for more details). The underlying relation between flare rate and energy is plotted in Fig. A1. The relation between flare rate and energy, and therefore the relation between flare rate and maximum stellar energetic particle momentum is unknown for GJ 436. Thus, in Fig. 8, we plot the flare rate versus maximum stellar energetic particle momentum for the Sun and a number of M dwarfs. In Fig. 8, the solid blue line is a fit to the solar flare frequency versus flare energy for a broad range of flare energies using solar observations and Kepler observations of superflares from solar-type stars (Maehara et al. 2015). The solid black line is 10 times the solar flare frequency fit (i.e. 10 × E dN/dE flare , where dN/dE flare ∝ E −1.8 flare ) from Maehara et al. (2015), which is what would be expected for a solar-type star with a rotation period between 5 − 10 days. The grey shaded region is ±0.5 dex of this value. The solid green line reflects a fit of the flare rate versus energy for an M dwarf with a rotation period of ∼27 days from Günther et al. (2020) using TESS data. This M dwarf is selected because it is relatively similar to GJ 436 in terms of stellar effective temperature (3850 K) and has a relatively long rotation period. The grey dashed lines are a number of flare rate fits for M dwarfs with similar spectral types to GJ 436 but with unknown rotation periods (also from Günther et al. 2020). Using 10 times the solar flare rate, Fig. 8 shows that the value of pmax=0.3 GeV/c (kinetic energy of 50 MeV) assumed for GJ 436 corresponds to a flare rate of 0.04 day −1 , i.e. a flare capable of accelerating particles to this momentum occurs approximately every 25 days with E flare = 3 × 10 31 erg (see Fig. A1). Considering the grey shaded region for the flare rate corresponds to a range of a flare every 7 or 72 days. While Günther et al. (2020) suggest that the flare rates given by the dashed grey lines are likely to be associated with stars with Prot < 10 days (in comparison to GJ 436 with Prot ∼ 44 days), it is interesting to note that the flare rate given by the solid green line is for an M dwarf with Prot ∼ 27 days. This rotation period is similar to the Sun's rotation period but with a flare rate 2 orders of magnitude larger than the Sun's. If GJ 436 had such an elevated flare rate, flares capable of accelerating particles to 50 MeV energies would occur approximately once per day. Fig. 8 (and more broadly Fig. 11 from Günther et al. 2020) also shows that the power law fit for the flare rate varies from star to star. Thus, determining stellar flare rates is important to understand how continuous stellar energetic particle fluxes of a certain energy are. Comparing flare rates with chemical recombination rates in the future would shed light on whether the chemistry in exoplanet atmospheres could reset between flares with associated stellar energetic particles. Connection to habitability Understanding exoplanet habitability, and detecting the signature of life as we know it from an exoplanet, is a key goal of modern astronomy. Energetic particles are one of the many factors thought to affect exoplanet habitability (e.g. Meadows et al. 2018). For instance, energetic particles can drive the formation of prebiotic molecules (Airapetian et al. 2016;Barth et al. 2021) that are important for the origin of life. On the other hand, high energetic particle fluxes may be detrimental to existing life by contributing to ozone depletion (Segura et al. 2010) which blocks harmful UV radiation reaching the surface of an Earth-like planet. Studying exoplanets that are not expected to be habitable (such as GJ 436 b), but that will be observed in greater numbers, can provide important information about the conditions present in other stellar systems. While hydrogen dominated gas giants are not generally thought to be conducive to life, Seager et al. (2021) have suggested how life might survive in the cloud decks of these planets, and Seager et al. (2020) have found unique biosignature gas compositions for microbial life as we know it in H2rich environments. Prebiotically relevant molecules, such as the amino acid glycine, may also form more easily on charged cloud particles (Stark et al. 2014). Globus & Blandford (2020) suggested muons (which are on-average spin-polarised) that reach Earth's surface could be responsible for the chirality of the DNA helix by interacting with biological molecules. In this context, it will be of interest in the future to calculate the muon fluxes associated with our (proton) energetic particle fluxes for a terrestrial planet. While genetic mutation rates are known to increase with increasing radiation dose (Muller 1927), it remains unclear what the ultimate effect of high radiation doses due to energetic particles would be on the development of life on another planet. Lifeforms that use radiation energy for metabolism (Matusiak 2016) may dominate instead. In our model, we have assumed that the top-of-atmosphere stellar energetic particle fluxes are isotropic. In reality, stellar energetic particles will travel along spiral flux tubes which would result in anisotropic top-of-atmosphere fluxes. This type of transport can be modelled with the focused transport equation, for instance (Roelof 1969). For tidally locked planets, if the stellar energetic particle flux is not isotropic, one side of the atmosphere, facing toward the star, will likely receive more stellar energetic particles than the side facing away from the star. This angular dependence could affect the energetic particle environment and could introduce heterogeneous chemistry in the upper atmosphere, depending on chemical timescales and efficiency of atmospheric circulation. This could be explored by 3D models, and is outside the scope of this paper. An anisotropic energetic particle flux could also have implications for habitability. Again, exoplanetary magnetic fields will complicate the picture further. For instance, Herbst et al. (2019b) considered the impact of an Earth-like magnetic field and indicated that < 510 MeV energy particles were not able to penetrate an Earth-like magnetosphere at latitudes around 60 • . The atmospheres of the planets considered here are roughly analogous to the most extreme transient post-impact atmospheres predicted for the early Earth (Zahnle et al. 2020;Itcovitz et al. 2022). We can consider the energetic particle flux at different pressures as analogous particle radiation environments for a post-impact early Earth. If Earth postimpact had a 100 bar H2 atmosphere, as proposed by Zahnle et al. (2020), then Galactic cosmic rays determine the surface irradiation environment, and the irradiation dose is much lower than if Earth post-impact had a relatively more tenuous atmosphere, at < 10 bar. Then the irradiation is much more severe and due to stellar energetic particles. This is relevant for habitability around small cool stars, such as GJ 436, where the 0.2 au case is at the inner edge of the liquid water habitable zone of the system. This is also relevant for prebiotic chemistry driven by energetic particle-driven chemistry in aqueous solutions, either on the surface or in aerosols (Kobayashi et al. 2001;Lingam et al. 2018). CONCLUSIONS In this paper, we consider the effect of the stellar magnetic environment and planetary atmosphere on the energy dependence and intensity of energetic particles as applied to the well-studied M dwarf system, GJ 436. GJ 436 b is a warm Neptune planet orbiting at 0.028 au from its host star (scheduled for JWST observations and an Ariel target). We have coupled a stellar wind model with an energetic particle transport model for the large-scale stellar system. We have then propagated the top-of-atmosphere energetic particle fluxes through the atmosphere of a warm Neptune-like exoplanet at various orbital distances from GJ 436, using temperature pressure profiles from the radiative transport code HELIOS. We consider two sources of energetic particles: Galactic cosmic rays and flare-accelerated stellar energetic particles. We show how the maximum momentum that the stellar energetic particles are accelerated to can be related to the stellar X-ray luminosity. Thus, our input stellar energetic particle spectrum reflects in some sense the activity of the star. To our knowledge, no other model relates the cut-off energy of the stellar energetic particle spectrum to a stellar property. The high energy cut-off is very important because the highest energy particles (i.e. GeV energies) are those that can reach the surface of an Earth-like planet, for instance. Our calculations have assumed that the stellar energetic particle fluxes are constant in time up to our calculated maximum stellar energetic particle momentum. These would be associated with a ∼ 10 31 erg flare which would occur once every ∼25 days assuming 10 times the solar flare rate for GJ 436. We calculate top-of-atmosphere energetic particle fluxes as a function of star-planet separation. We also model the transport of energetic particles through a hydrogen-dominated gas giant atmosphere at orbital distances of a = 0.01 − 0.2 au. The peak top-of-atmosphere ionisation rate for Galactic cosmic rays is ∼ 10 6 times less than the peak ionisation rate for stellar energetic particles at orbital distances between a = 0.01 − 0.2 au. However, the stellar energetic particles do not penetrate as deep into the atmospheres. This is because for GJ 436, the stellar energetic particle spectrum that we adopt has a lower cutoff energy than for the Galactic cosmic rays. At the orbital distance of GJ 436 b, we find that the ion-pair production rate, Q (cm −3 s −1 ), peaks at ∼ 100 bar for Galactic cosmic rays and 10 −3 bar for stellar energetic particles. We also estimate the skin-depth equivalent dose rate from the primary energetic particles as a function of height in the hydrogen-dominated atmospheres. Given the close-in orbital distance of the inner edge of GJ 436's habitable zone (0.2 au), the top-of-atmosphere equivalent dose rates that we calculate are substantial (10 4 mSv day −1 ). However, an important future step would be to study the effect of a planetary magnetic field on our results. The stellar energetic particle fluxes decrease with distance from the star and we assume that they are isotropic at the top of the exoplanet atmosphere. At small orbital distances, the fluxes drop close to 1/r 2 as advection processes dominate. At the same time, the magnetised, expanding stellar wind suppresses Galactic cosmic ray fluxes, and so the greater the distance from the star, the more intense the Galactic cosmic ray spectrum. A future step would be to study the effect of anistropic stellar energetic particle fluxes as they travel along magnetic flux tubes. We find that, at a distance of 10 au, the top-of-atmosphere molecular hydrogen ionisation rates from stellar energetic particles and Galactic cosmic rays are equal. Top-ofatmosphere ionisation by energetic particles is dominated by Galactic cosmic rays for a > 10 au. Ionisation by energetic particles at the top of the atmosphere is dominated by stellar energetic particles for a < 10 au. We anticipate that most of the ion chemistry will happen at P < 1µbar, and so the ion chemistry will be determined by stellar energetic particles and UV photons for planets within 10 au based on the topof-atmosphere ionisation rates that we found. Clouds might also form in GJ 436 b's atmosphere because of the low atmospheric temperatures. Their formation and its effects on the atmospheric structure and the chemistry are, however, neglected in this paper. The chemical consequences of these trends and specific predictions for ion-neutral chemistry will be discussed in more detail in Paper II (Rimmer et al. 2023). Günther et al. (2020). The linestyles and the grey shaded region are the same as for Fig. 8. Figure 2 . 2The (a) Galactic cosmic ray and (b) stellar energetic particle differential intensities, j, are plotted as a function of the energetic particle kinetic energy at different orbital distances. The colour of the lines represents the orbital distance. For instance, for the left hand plot, the lowest blue line represents the fluxes at a = 0.01 au and the highest green line at a = 0.2 au. Additionally, the solid black line represents the fluxes at the orbital distance of GJ 436 b. The Galactic cosmic ray fluxes are fromMesquita et al. (2021). Figure 5 . 5The ion-pair production rate, Q, as a function of pressure in an exoplanet atmosphere for different orbital distances in the GJ 436 system for (a) Galactic cosmic rays and (b) stellar energetic particles. The grey dotted line denotes P = 1 bar. The linestyles are the same as forFig. 4. Figure 6 . 6The ion-pair production rate, Q, as a function of pressure at the orbital distance of GJ 436b for Galactic cosmic rays (labelled as 'GCRs', green dashed line) and stellar energetic particles (labelled as 'StEPs', blue dot-dashed line). The solid black line represents the total Q value. The grey dotted line denotes P = 1 bar. Figure 7 . 7The skin-depth equivalent dose rate (Ḋ) is plotted as a function of pressure for a warm Neptune exoplanet atmosphere at an orbital distance of 0.2 au, the inner edge of the habitable zone for the GJ 436 system. The green dashed line representsḊ for Galactic cosmic rays (labelled as 'GCRs') and the blue dotdashed line representsḊ for stellar energetic particles (labelled as 'StEPs').The solid black line represents the total skin-depth equivalent dose rate. The grey star and the magenta pentagon represent the equivalent dose rate at Earth's surface due to Galactic cosmic rays and background radiation sources, respectively. dN/dE flare E 1 Figure 8 . 18Flare rates are plotted as a function of the maximum momentum of the stellar energetic particles accelerated. The solid blue line represents a fit to the solar flare frequency (fromMaehara et al. 2015). The solid black line is 10 times the solar flare rate and the grey shaded region is ±0.5 dex of this value. The solid green line is a flare rate fromGünther et al. (2020) for an M dwarf with a rotation period of ∼ 27 days. The grey dashed lines are flare rates for M dwarfs fromGünther et al. (2020) without a measured rotation period with a spectral type similar to GJ 436. Figure A1 . A1Observationally inferred flare rates (E flare dN/dE flare ) are plotted as a function of flare energy for the Sun, 10 times the solar flare rate and a number of M dwarf flare rates from Table 1 . 1Stellar and planetary parameters for the GJ 436 system.Identifier M R T eff a Prot Ref. (K) (au) (days) GJ 436 0.452M 0.437R 3479 - - 1 GJ 436 b 0.08M J 0.372R J 879 0.028 2.64 2 (1) Knutson et al. (2011); (2) Butler et al. (2004). The effective temperature of GJ 436 b, as shown in this table, is derived using radiative transfer calculations with HELIOS. 250 500 750 1000 1250 1500 1750 2000 T [K] 10 7 10 5 10 3 10 1 10 1 10 3 P [bar] MNRAS 000, 1-13 (xxxx) We have not included any internal temperature in the HELIOS models. Thus, taking into account that the radiative gradient is proportional to the planet luminosity, no adiabat will occur for our HELIOS models. Setting an internal temperature in HELIOS could not be done for the temperature pressure profiles considered due to non-convergence effects. We instead include an adiabat for P > 1 bar. This is a reasonable approximation given the uncertainties in internal temperatures and opacities for exoplanet atmospheres. Alternatively, the top-of-atmosphere energetic particle fluxes could have been summed. However, as described in Section 2.4, we treat the stellar energetic particle and Galactic cosmic ray transport separately.MNRAS 000, 1-13 (xxxx) ACKNOWLEDGEMENTSDRL would like to acknowledge that this publication has emanated from research conducted with the financial support of Science Foundation Ireland under Grant number 21/PATH-S/9339. DRL, ALM and AAV acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 817540, ASTROFLOW). DRL wishes to acknowledge the Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. Research reported in this publication was supported by the Royal Irish Academy. ChH acknowledges funding from the European Union H2020-MSCA-ITN-2019 under Grant Agreement no. 860470 (CHAMELEON). OV acknowledges funding from the ANR project 'EXACT' (ANR-21-CE49-0008-01), from the Centre National d'Études Spatiales (CNES), and from the CNRS/INSU Programme National de Planétologie (PNP). PB acknowledges a St Leonard's Interdisciplinary Doctoral Scholarship from the University of St Andrews. We would like to thank the referee for helpful comments which improved the manuscript.DATA AVAILABILITYThe output data underlying this article will be available via zenodo.org upon publication.APPENDIX A: FLARE RATESSection 4.1 discusses how continuous stellar energetic particle fluxes can be considered for GJ 436.Fig. 8relates the flare rate to a maximum stellar energetic particle momentum. This is based on the underlying relation between flare rate and flare energy which is plotted inFig. A1. The solar flare rate is a fit to solar and stellar super-flare data fromMaehara et al. (2015)with a flare frequency relation of dN/dE flare ∝ E −1.8 flare . The grey shaded region surrounding the solid black line, representing 10 times the solar flare rate, is ±0.5 dex. While the flare rate for GJ 436 is unknown, given the maximum stellar energetic particle momentum that we assume, we indicated the frequency of flares accelerating to this momentum in Section 4.1. . 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{'abstract': "A key first step to constrain the impact of energetic particles in exoplanet atmospheres is to detect the chemical signature of ionisation due to stellar energetic particles and Galactic cosmic rays. We focus on GJ 436, a well-studied M dwarf with a warm Neptune-like exoplanet. We demonstrate how the maximum stellar energetic particle momentum can be estimated from the stellar X-ray luminosity. We model energetic particle transport through the atmosphere of a hypothetical exoplanet at orbital distances between a = 0.01 − 0.2 au from GJ 436, including GJ 436 b's orbital distance (0.028 au). For these distances we find that, at top-of-atmosphere, stellar energetic particles ionise molecular hydrogen at a rate of ζ StEP,H2 ∼ 4 × 10 −10 − 2 × 10 −13 s −1 . In comparison, Galactic cosmic rays alone lead to ζ GCR,H2 ∼ 2 × 10 −20 − 10 −18 s −1 . At 10au we find that ionisation due to Galactic cosmic rays equals that of stellar energetic particles: ζ GCR,H2 = ζ StEP,H2 ∼ 7×10 −18 s −1 for the top-of-atmosphere ionisation rate. At GJ 436 b's orbital distance, the maximum ion-pair production rate due to stellar energetic particles occurs at pressure P ∼ 10 −3 bar while Galactic cosmic rays dominate for P > 10 2 bar. These high pressures are similar to what is expected for a post-impact early Earth atmosphere. The results presented here will be used to quantify the chemical signatures of energetic particles in warm Neptune-like atmospheres.Galactic cosmic rays interact with, and their fluxes are suppressed by, the magnetised winds of low-mass stars (e.g. Potgieter (2013) for the Sun and Rodgers-Lee et al. (2021b) for", 'arxivid': '2303.07058', 'author': ['D Rodgers-Lee \nDublin Institute for Advanced Studies\nSchool of Cosmic Physics\n31 Fitzwilliam PlaceDublin\n\nD02 XF86Ireland\n\nSchool of Physics\nTrinity College Dublin\nUniversity of Dublin\nDublin 2D02 PN40College GreenIreland\n', 'P B Rimmer \nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AveCB3 0HECambridgeUnited Kingdom\n', 'A A Vidotto \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n', 'A J Louca \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n', 'A M Taylor \nDeutsches Elektronen-Synchrotron\nZeuthenGermany\n', 'A L Mesquita \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n', 'Y Miguel \nLeiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands\n\nSRON\nNetherlands Institute for Space Research\nNiels Bohrweg 42333 CALeidenthe Netherlands\n', 'O Venot \nUniversité Paris Cité\nUniv Paris Est Creteil\nCNRS\nF-75013ParisLISAFrance\n', 'Ch Helling \nSpace Research Institute\nAustrian Academy of Sciences\nSchmiedlstrasse 6A-8042GrazAustria\n\nInstitute for Theoretical Physics and Computational Physics\nCentre for Exoplanet Science\nGraz University of Technology\nPetersgasse 168010Graz 10 St Andrews\n\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt AndrewsUK\n', "P Barth \nSpace Research Institute\nAustrian Academy of Sciences\nSchmiedlstrasse 6A-8042GrazAustria\n\nSchool of Physics and Astronomy\nSUPA\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt AndrewsUK\n\nSchool of Earth & Environmental Sciences\nUniversity of St Andrews\nBute BuildingKY16 9TSTerrace, St AndrewsQueen'sUK\n", 'E Lacy \nDublin Institute for Advanced Studies\nSchool of Cosmic Physics\n31 Fitzwilliam PlaceDublin\n\nSchool of Physics\nUniversity College Dublin\nBelfield, Dublin 4Ireland\n'], 'authoraffiliation': ['Dublin Institute for Advanced Studies\nSchool of Cosmic Physics\n31 Fitzwilliam PlaceDublin', 'D02 XF86Ireland', 'School of Physics\nTrinity College Dublin\nUniversity of Dublin\nDublin 2D02 PN40College GreenIreland', 'Cavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AveCB3 0HECambridgeUnited Kingdom', 'Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands', 'Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands', 'Deutsches Elektronen-Synchrotron\nZeuthenGermany', 'Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands', 'Leiden Observatory\nLeiden University\nP.O. Box 95132300 RALeidenThe Netherlands', 'SRON\nNetherlands Institute for Space Research\nNiels Bohrweg 42333 CALeidenthe Netherlands', 'Université Paris Cité\nUniv Paris Est Creteil\nCNRS\nF-75013ParisLISAFrance', 'Space Research Institute\nAustrian Academy of Sciences\nSchmiedlstrasse 6A-8042GrazAustria', 'Institute for Theoretical Physics and Computational Physics\nCentre for Exoplanet Science\nGraz University of Technology\nPetersgasse 168010Graz 10 St Andrews', 'University of St Andrews\nNorth Haugh\nKY16 9SSSt AndrewsUK', 'Space Research Institute\nAustrian Academy of Sciences\nSchmiedlstrasse 6A-8042GrazAustria', 'School of Physics and Astronomy\nSUPA\nUniversity of St Andrews\nNorth Haugh\nKY16 9SSSt AndrewsUK', "School of Earth & Environmental Sciences\nUniversity of St Andrews\nBute BuildingKY16 9TSTerrace, St AndrewsQueen'sUK", 'Dublin Institute for Advanced Studies\nSchool of Cosmic Physics\n31 Fitzwilliam PlaceDublin', 'School of Physics\nUniversity College Dublin\nBelfield, Dublin 4Ireland'], 'corpusid': 257496351, 'doi': '10.1093/mnras/stad900', 'github_urls': [], 'n_tokens_mistral': 26749, 'n_tokens_neox': 21131, 'n_words': 12420, 'pdfsha': '9f60ff2f226f461ad87690fa9005324f1cf24606', 'pdfurls': ['https://export.arxiv.org/pdf/2303.07058v1.pdf'], 'title': ['The energetic particle environment of a GJ 436 b-like planet', 'The energetic particle environment of a GJ 436 b-like planet'], 'venue': ['MNRAS']}

arxiv

RELATIVELY GEOMETRIC ACTIONS OF KÄHLER GROUPS ON CAT(0) CUBE COMPLEXES 10 Apr 2023 Corey Bregman Daniel Groves Kejia Zhu RELATIVELY GEOMETRIC ACTIONS OF KÄHLER GROUPS ON CAT(0) CUBE COMPLEXES 10 Apr 2023arXiv:2210.12850v3 [math.GR] We prove that for n ≥ 2, a non-uniform lattice in PU(n, 1) does not admit a relatively geometric action on a CAT(0) cube complex, in the sense of[10]. As a consequence, if Γ is a nonuniform lattice in a non-compact semisimple Lie group G without compact factors that admits a relatively geometric action on a CAT(0) cube complex, then G is commensurable with SO(n, 1). We also prove that if a Kähler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group.2020 Mathematics Subject Classification. 20F65, 22E40 (primary), 32J27, 32J05, 57N65 (secondary). Introduction A finitely generated group is called cubulated if it acts properly cocompactly on a CAT(0) cube complex. Agol [1], building on the work of Wise [29] and many others, proved that cubulated hyperbolic groups enjoy many important properties, and used this to solve several open conjectures in 3-manifold topology, in particular the Virtual Haken and Virtual Fibering Conjectures. Wise [29, §17] proved the Virtual Fibering Conjecture in the non-compact, finite-volume setting, using the relatively hyperbolic structure of the fundamental group. Einstein-Groves define the notion of a relatively geometric action of a group pair (Γ, P) on a CAT(0) cube complex [10]. For such an action, elements of P act elliptically. This allows the possibility that even though the elements of P might not act properly on any CAT(0) cube complex, there still may be a relatively geometric action. Relatively geometric actions are a natural generalization of proper actions and share many of the same features as in the proper case, especially when Γ is hyperbolic relative to P . Uniform lattices in SO(3, 1) always act geometrically thus relatively geometrically on CAT(0) cube complexes [5]. Bergeron-Haglund-Wise [4] prove that in higher dimensions, lattices in SO(n, 1) which are arithmetic of simplest type are cubulated. It also follows from this and Wise's quasiconvex hierarchy theorem [29] that many "hybrid" hyperbolic n-manifolds have cubulated fundamental groups. In the relatively geometric setting, using the work of Cooper-Futer [7], Einstein-Groves proved that non-uniform lattices in SO(3, 1) also admit relatively geometric actions, relative to their cusp subgroups [10]. In fact, they prove that if (G, P) is hyperbolic relative to free abelian subgroups and the Bowditch boundary ∂(G, P) is homeomorphic to S 2 , then G is isomorphic to a non-uniform lattice in SO (3,1) if and only if (G, P) admits a relatively geometric action on a CAT(0) cube complex. This result is a relative version of the work of Markovic [25] and Haïssinsky [22] in the convex-cocompact setting, giving an equivalent formulation of the Cannon conjecture in terms of actions on hyperbolic CAT(0) cube complexes. It is not known in general whether the above results extend to all lattices in SO(n, 1) for n ≥ 3. In contrast, work of Delzant-Gromov implies that uniform lattices in PU(n, 1) are not cubulated [8]. Recall that a group Γ is Kähler if Γ ≅ π 1 (X) for some compact Kähler manifold X. If Γ ≤ PU(n, 1) is a torsion-free, uniform lattice, then Γ acts freely, properly discontinuously cocompactly on complex hyperbolic n-space H n C . The quotient M = Γ H n C is a closed, negatively curved Kähler manifold, and in particular Γ is a hyperbolic, Kähler group. In this context, Delzant-Gromov showed that any infinite Kähler group that is hyperbolic and cubulated is commensurable to a surface group of genus g ≥ 2 [8]. Thus Γ is not cubulated for n ≥ 2. Since every uniform lattice in PU(n, 1) is virtually torsion-free, it follows that uniform lattices in PU(n, 1) are not cubulated if n ≥ 2. On the other hand, uniform lattices in PU(1, 1) = SO(2, 1), are finite extensions of hyperbolic surface groups, hence are hyperbolic and cubulated. Similarly, non-uniform lattices in PU(1, 1) are the orbifold fundamental groups of surfaces with finitely many cusps, hence virtually free. Such lattices admit both proper cocompact and relatively geometric actions on CAT(0) cube complexes. Since the cusp subgroups of a non-uniform lattice in PU(n, 1) (n ≥ 2) are virtually nilpotent but not virtually abelian, it follows from a result of Haglund [21] that such a lattice does not admit a proper action on a CAT(0) cube complex (see Proposition 4.3 below). However, the parabolic subgroups do not yield such an obstruction to the existence of a relativeley geometric action. Thus, this leaves open the question of whether non-uniform lattices in PU(n, 1) admit relatively geometric actions on CAT(0) cube complexes for n ≥ 2. Our first result answers this question in the negative. Theorem 1.1. Let Γ ≤ PU(n, 1) be a non-uniform lattice with n ≥ 2, and let P be the collection of cusp subgroups of Γ. Then (Γ, P) does not admit a relatively geometric action on a CAT(0) cube complex. Corollary 1.2. Let Γ be a lattice in a non-compact semisimple Lie group G without compact factors. If either (1) Γ is uniform and cubulated hyperbolic, or (2) Γ is non-uniform, hyperbolic relative to its cusp subgroups P, and (Γ, P) admits a relatively geometric action on a CAT(0) cube complex, then G is commensurable to SO(n, 1) for some n ≥ 1. Proof. A uniform lattice (resp. non-uniform lattice) Γ in a semisimple Lie group G is hyperbolic (resp. hyperbolic relative to its cusp subgroups P) if and only if G has rank 1, by a result of Behrstock-Druţu-Mosher [3]. Any rank 1 noncompact semisimple Lie group is commensurable with one of SO(n, 1), PU(n, 1), Sp(n, 1) for n ≥ 2, or the isometry group of the octonionic hyperbolic plane H 2 O . The latter and Sp(n, 1) have Property (T), while SO(n, 1) and PU(n, 1) do not. Hence if Γ is commensurable with a lattice in Sp(n, 1) or Isom(H 2 O ), then Γ has (T). By a result of Niblo-Reeves [26], any action of a group with Property (T) on a CAT(0) cube complex has a global fixed point, so lattices in Sp(n, 1) and Isom(H 2 O ) admit neither geometric nor relatively geometric actions on CAT(0) cube complexes. Hence if Γ is as in the statement of the result, it must be commensurable to a lattice in either PU(n, 1) or SO(n, 1). For n ≥ 2, the uniform case of Γ ≤ PU(n, 1) is eliminated by work Delzant-Gromov [8]. The corollary now follows from Theorem 1.1. We say that a relatively hyperbolic group pair (Γ, P) is properly relatively hyperbolic if P ≠ {Γ}. The following result considers general relatively geometric actions of Kähler relatively hyperbolic groups on CAT(0) cube complexes (when the peripheral subgroups are residually finite). Theorem 1.3. Let (Γ, P) be a properly relatively hyperbolic pair such that each element of P is residually finite. If Γ is Kähler and acts relatively geometrically on a CAT(0) cube complex, then Γ is virtually a hyperbolic surface group. We will deduce Theorem 1.1 from Theorem 1.3 in Section 4. In fact, non-uniform lattices in PU(n, 1) are Kähler for n ≥ 3 [28], hence Theorem 1.1 follows immediately from Theorem 1.3 in this range. However, our proof of Theorem 1.1 will work for all n ≥ 2, and will not use this fact. In [9], Delzant-Py considered actions of Kähler groups on locally finite, finite-dimensional CAT(0) cube complexes that are more general than geometric ones (see Theorem A for precise hypotheses), and showed that every such action virtually factors through a surface group. We remark that the cube complexes appearing in relatively geometric actions will in general not be locally finite. We conclude the introduction with a sample application of Theorem 1.3. Corollary 1.4. Suppose that A and B are infinite residually finite groups which are not virtually free. Any C ′ 1 6 -small cancellation quotient of A * B is not Kähler. Proof. Let Γ be such a small cancellation quotient of A * B. According to [13], Γ is residually finite and admits a relatively geometric action on a CAT(0) cube complex. If Γ were Kähler, it would be a virtual surface group, by Theorem 1.3. However, A embeds in Γ as an infinite-index subgroup, and the only infinite index subgroups of virtual surface groups are virtually free. Outline: In Section 2, we review the definition of a relatively geometric action of a group pair on a CAT(0) cube complex and the notion of group-theoretic Dehn fillings, then collect some known results about these. In Section 3 we prove Theorem 1.3. In Section 4, after reviewing the Borel-Serre and toroidal compactifications of non-uniform quotients of complex hyperbolic space, we prove Theorem 1.1. Acknowledgments: The first author was supported by NSF grant DMS-2052801. The second author was supported by NSF grants DMS-1904913 and DMS-2203343. The third author would like to thank his advisor, Daniel Groves, for introducing him to the subject and answering his questions. He would like to thank his co-advisor, Anatoly Libgober, for his constant support and warm encouragement. He would also like thank Hao Liang and Xuzhi (Carl) Tang for helpful discussions. Actions on CAT(0) Cube Complexes In this section, we review the notion of a relatively geometric action of a group pair (Γ, P) on a CAT(0) cube complex, defined by Einstein and Groves in [10]. We then introduce Dehn fillings of group pairs and recall some useful results from [11]. Definition 2.1. Let Γ be a group and P a collection of subgroups of Γ. An action of Γ on a CAT(0) cube complex X is relatively geometric with respect to P if (1) Γ X is compact; (2) Each element of P acts elliptically on X; (3) Each cell stabilizer in X is either finite or else conjugate to a finite-index subgroup of an element of P. Recall that if (Γ, P) is a relatively hyperbolic group pair and Γ 0 ≤ Γ has finite-index then (Γ 0 , P 0 ) is also a relatively hyperbolic group pair, where P 0 is the set of representatives of the Γ 0 -conjugacy classes of (1) {P g ∩ Γ 0 g ∈ Γ, P ∈ P} Since [Γ∶ Γ 0 ] is finite, P 0 is still a finite collection of subgroups. It follows that if Γ admits a relatively geometric action on a CAT(0) cube complex X, then (Γ 0 , P 0 ) also admits a relatively geometric action on X by restriction. Indeed, (2) and (3) in Definition 2.1 follow immediately and (1) follows from that fact that under the natural map Γ 0 X → Γ X , each cell of Γ X has at most [Γ∶ Γ 0 ] < ∞ pre-images. Hence if Γ X is compact, so is Γ 0 X . We have just proven Lemma 2.2. Let Γ 0 ≤ Γ be a finite-index subgroup. If (Γ, P) has a relatively geometric action on a CAT(0) cube complex X, then the restriction of this action to (Γ 0 , P 0 ) is also relatively geometric, where P 0 is defined as in Equation 1. Dehn fillings. Dehn fillings first appeared in the context of 3-manifold topology and were subsequently generalized to the group-theoretic setting by Osin [27] and Groves-Manning [18]. We now recall the notion of a Dehn filling of a group pair (G, P): Definition 2.4 (Peripherally finite). If each normal subgroup N i has finite-index in P i , the filling is said to be peripherally finite. Definition 2.5 (Sufficiently long). We say that a property X holds for all sufficiently long Dehn fillings of (G, P) if there is a finite subset B ⊂ G ∖ {1} so that whenever N i ∩ B = ∅ for all i, the corresponding Dehn filling G(N 1 , ..., N n ) has property X . The proof of the next theorem relies on the notion of a Q-filling of a collection of subgroups Q of G. Recall from [17] that given a subgroup Q < G, the quotient G(N 1 , . . . , N m ) is a Q-filling if for all g ∈ G, and P i ∈ P, Q ∩ P g i = ∞ implies N g i ⊆ Q. If Q = {Q 1 , . . . , Q l } is a family of subgroups, then G(N 1 , . . . , N m ) is a Q-filling if it is a Q-filling for every Q ∈ Q. Let Q be a collection of finite-index subgroups of elements of P so that any infinite cell stabilizer contains a conjugate of an element of Q. The following is proved in [11]. Theorem 2.6 (Proposition 4.1 and Corollary 4.2 of [11]). Let (Γ, P) be a relatively hyperbolic pair such that the elements of P are residually finite. If (Γ, P) admits a relatively geometric action on a CAT(0) cube complex X then (1) For sufficiently long Q-fillings Γ → Γ = Γ K, the quotient X = K X is a CAT(0) cube complex; and (2) Any sufficiently long, peripherally finite Q-filling of Γ is hyperbolic and virtually special. The following result is implicit in [11]. For completeness, we provide a proof. Lemma 2.7. In the context of Theorem 2.6.(1), the action of Γ on X is relatively geometric. Proof. Since Γ X = Γ X the action is cocompact. Let P be the induced peripheral structure on Γ (the image of P). The fact that elements of P act elliptically on X follows from the fact that elements of P act elliptically on X. Because each cell-stabilizer of Γ ↷ X is either finite or conjugate to a finite-index of subgroup of some P i ∈ P, this implies that the cell-stabilizers of Γ ↷ X are conjugate to finite-index subgroups of P i (K ∩ P i ) (the elements of P). Thus the action of ∆ on Y is relatively geometric. Relatively geometric actions: the Kähler case In this section, we apply Theorem 2.6 to prove Theorem 1.3. The main idea is to use Dehn filling to produce a minimal action of a finite-index subgroup of Γ on a tree with finite kernel. A deep result of Gromov-Schoen implies that any Kähler group admitting a minimal acting on tree with finite kernel must be virtually a hyperbolic surface group [16]. Proof of Theorem 1.3. Suppose that (Γ, P) acts relatively geometrically on a CAT(0) cube complex. Since the elements of P are residually finite, there exists a sufficiently long, peripherally finite Q-filling Γ → Γ = Γ K which satisfies the hypotheses of Theorem 2.6.((2)), so Γ is hyperbolic and X = K X is a CAT(0) cube complex. Let Γ 0 ≤ Γ be a finite-index subgroup such that Γ 0 is torsion-free and Γ 0 X is special, which exists by [11,Theorem 1.4]. Cutting along an embedded essential two-sided hyperplane H in Γ 0 X yields a splitting of Γ 0 according to the complex of groups version of van Kampen's Theorem [6, III.C.3.11.(5), III.C.3.12, p.552]. 1 The edge group of such a splitting is a hyperplane stabilizer for the Γ 0 -action on X, which is relatively quasi-convex by [12,Corollary 4.11], and infinite-index since the hyperplane is essential. The action of Γ 0 on the Bass-Serre tree T associated to this splitting has finite kernel, since any normal subgroup contained in an infinite-index relatively quasi-convex subgroup is finite. Let F denote the kernel of the action of Γ 0 on T . By [16], the induced action of Γ 0 on T factors through a surjective homomorphism ϕ∶ Γ 0 → ∆ 0 , where ∆ 0 ≤ PSL 2 (R) is a cocompact lattice. The kernel of ϕ is contained in F , hence finite, so Γ 0 is commensurable up to finite kernels with ∆ 0 , which is itself virtually a hyperbolic surface group. Since any group commensurable up to finite kernels with a hyperbolic surface group is virtually a hyperbolic surface group, this means that Γ 0 , and hence Γ, is virtually a hyperbolic surface group, as desired. Relatively geometric actions: Lattices in PU(n, 1) Let Γ be a non-uniform lattice in PU(n, 1). Then Γ acts properly discontinuously on complex hyperbolic space H n C and the quotient, which we henceforth denote by M = Γ H n C , is a non-compact orbifold of finite volume with finitely many cusps. Each cusp corresponds to a conjugacy class of subgroups stabilizing a parabolic fixed point in ∂ ∞ H n C . Farb [14] proved that Γ is hyperbolic relative to the collection of these cusp subgroups, which we denote by P. In this section, we prove Theorem 1.1, namely that (Γ, P) does not admit a relatively geometric action on a CAT(0) cube complex. Throughout the course of the proof, we pass freely to finite-index subgroups by invoking Lemma 2.2. In order to streamline the exposition, we do not refer to Lemma 2.2 each time. First we reduce the Theorem 1.1 to the case where Γ is torsion-free. Proof. We have a short exact sequence 1 → Z (n + 1)Z → SU(n, 1) → PU(n, 1) → 1. Restricting to Γ, we get a short exact sequence 1 → Z (n + 1)Z → Λ → Γ → 1, where Λ is the pre-image of Γ in SU(n, 1). Since Γ is finitely generated and Z (n + 1)Z is finite, Λ is finitely generated. As SU(n, 1) is linear, Selberg's lemma implies that Λ has a finite-index torsion-free subgroup, say, Λ 0 . Thus Λ 0 ∩ Z (n + 1)Z = 1 and hence it is mapped isomorphically to finite-index subgroup Γ 0 ≤ Γ. Following Lemma 4.1, for the remainder of this section we assume that Γ ≤ PU(n, 1) is torsionfree. 4.1. The structure of cusps. We now briefly review the geometric structure of cusps in M . For more details see [15]. Recall that up to scaling each horosphere in H n C is isometric to H 2n−1 (R), the (2n − 1)-dimensional real Heisenberg group, equipped with a left-invariant metric. The Heisenberg group is a central extension (2) 1 → R → H 2n−1 (R) → R 2n−2 → 1 with extension 2-cocycle equal to the standard symplectic form ω = 2 n−1 i=1 dx i ∧ dy i , where (x 1 , y 1 , . . . , x n−1 , y n−1 ) are coordinates on R 2n−2 . The Lie algebra h 2n−1 is 2-step nilpotent with basis {X 1 , Y 1 , . . . , X n , Y n , Z} where [X i , Y i ] = Z and all other brackets vanish. Thus Z generates the center of h 2n−1 representing the kernel R in Equation (2), while the remaining coordinates project to the generators of R 2n−2 . Choosing the identity matrix I 2n−1 as the inner product on h 2n−1 , we see that the isometry group of H 2n−1 (R) is isomorphic to H 2n−1 (R) ⋊ U (n − 1), where the H 2n−1 (R) factor is the action of H 2n−1 (R) on itself by left translation, and the unitary group U (n − 1) is the stabilizer of the identity. Indeed, any isometry which fixes 1 ∈ H 2n−1 (R) must also be a Lie algebra isomorphism; it therefore preserves the center ⟨Z⟩ and induces an isometry of R 2n−2 ≅ ⟨X 1 , Y 1 , ⋯, X n−1 , Y n−1 ⟩ preserving ω. We conclude that such an isometry lies in U (n − 1) = O 2n−2 (R) ∩ Sp 2n−2 (R). Definition 4.2. Let π∶ H 2n−1 (R) ⋊ U (n − 1) → U (n − 1) be the projection. For any g ∈ H 2n−1 (R) ⋊ U (n − 1), we call π(g) the rotational part of g. Since the center of H 2n−1 (R) is invariant under any isometry we have a short exact sequence (3) 1 → R = Z(H 2n−1 (R)) → H 2n−1 (R) ⋊ U (n − 1) → R 2n−2 ⋊ U (n − 1) → 1 Since Γ is torsion-free, each cusp subgroup P ≤ Γ is isomorphic to a discrete, torsion-free, cocompact subgroup of Isom(H 2n−1 (R)). In particular, P 0 = P ∩ H 2n−1 (R) is a discrete cocompact subgroup and P ∩ Z(H 2n−1 (R)) ≅ Z. By Equation 3, P fits into a short exact sequence (4) 1 → Z = P ∩ Z(H 2n−1 (R)) → P → Λ → 1 where Λ is a discrete cocompact subgroup of R 2n−2 ⋊ U (n − 1). It follows that Λ has a finite-index subgroup Λ 0 isomorphic to Z 2n−2 , which is the image of P 0 . On the level of quotient spaces, the sequence in Equation (4) has the following translation. The quotient space O = Λ R 2n−2 is a Euclidean orbifold finitely covered by the (2n − 2)-dimensional torus T = Λ 0 C n−1 , and Σ = P H 2n−1 (R) is the total space of an S 1 -bundle over O, i.e., there is a fiber sequence (5) S 1 ↪ Σ → O Since O need not be smooth, this is not generally a locally trivial fibration. However, as P is torsion-free, Σ is smooth. Passing to the torus cover, we obtain an actual fiber bundle S 1 ↪Σ → T The finite group F = P P 0 acts onΣ preserving the fibration, hence defines a finite group of isometries of T . Thus the stabilizer of a point in T acts freely on the S 1 fiber. Since the action of F onΣ is free, it follows that point stabilizers in T must be cyclic of finite order, and act by rotations on the fiber. Since , where at least one k i is coprime to m. See Figure 1 for a schematic. Since F acts by rotation on each fiber, Σ is the boundary of a disk bundle over O, which we denote by E O . Recall that the center of H 2n−1 (R) is quadratically distorted. It follows that the center of P is quadratically distorted as well. By [21,Theorem 1.5], there is no proper action of P on a CAT(0) cube complex. Therefore, we have: [24,2] for more details. When the parabolic elements in Γ have trivial rotational part, then each O i is a (2n − 2)dimensional torus, T (M ) is a smooth Kähler manifold and D is a smooth divisor in T (M ). Moreover, Hummel-Schroeder show that T (M ) admits a nonpositively curved Riemannian metric [24]. In particular, T (M ) is aspherical; if ∆ = π 1 (T (M )) then T (M ) is a K(∆, 1). The following lemma ensures that we can always find a finite cover of M whose toroidal compactification is smooth. Proof. By the main theorem of [23] (p. 2453), there exists a finite subset F ⊂ Γ of parabolic isometries such that if a N ⊴ Γ is a normal subgroup satisfying F ∩ N = ∅, then any parabolic isometry in N has no rotational part. Since Γ is residually finite and F is finite, we can find a finite-index normal subgroup Γ ′ ⊴ Γ such that Γ ′ ∩ F = ∅. Therefore the finite cover M ′ ∶= Γ ′ H n C of M admits a toroidal compactification which is smooth. For the rest of this section, we assume that T (M ) is smooth. Proof of Theorem 1.1. By Lemma 4.1 and Lemma 4.4, we may assume that Γ ≤ SU(n, 1) is torsionfree, and that the toroidal compactification T (M ) is smooth. In particular, Γ and all of its peripheral subgroups are residually finite. Since M 0 ∖ ∂M 0 ≅ M ≅ T (M ) ∖ D, Suppose (Γ, P) admits a relatively geometric action on a CAT(0) cube complex X. Given a finite-index subgroup Γ 0 ≤ Γ, let P 0 be the induced peripheral structure on Γ 0 , and let ∆ 0 be π 1 (T (M 0 )), where M 0 = Γ 0 H n C . Since the kernel of the quotient map Γ 0 → ∆ 0 is normally generated by subgroups in P 0 , we get an induced peripheral structure (∆ 0 , A 0 ), where A 0 is the collection of images of elements of P 0 . Our strategy is to show that there exists a finite-index subgroup Γ 0 ≤ Γ so that the pair (∆ 0 , A 0 ) is relatively hyperbolic and admits a relatively geometric action on a CAT(0) cube complex. Since T (M 0 ) is smooth (since T (M ) is), ∆ is Kähler. Thus, as n ≥ 2, we will get a contradiction by Theorem 1.3. Let P = {P 1 , . . . , P k } be the induced peripheral structure on Γ. Now let Z(P i ) be the center of P i . We apply Theorem 2.6(1) to a sufficiently long Q-filling Z = {Z 1 , . . . , Z k } where Z i ≤ Z(P i ) is a finite-index subgroup. We then obtain a Dehn filling ψ∶ Γ → ∆ = Γ K determined by the Z i so that Y = K X is a CAT(0) cube complex. Let (∆, A) be the induced peripheral structure on ∆. By Theorem 2.6, we know that (∆, A) is relatively hyperbolic. Lemma 2.7 implies that the action of ∆ on Y is relatively geometric. Finally, we claim that there exists a finite-index subgroup ∆ 0 ≤ ∆ that is torsion-free. Since the elements of A are virtually abelian, hence residually finite, we know that ∆ is also residually finite by Corollary 1.7 of [11]. Since Γ is torsion-free, by [19,Theorem 4.1] so long as the filling Γ → ∆ is long enough (which we may assume without loss of generality), any element of finite order in ∆ is conjugate into some element of A. As there are finitely many elements of A, each of which has only finitely many conjugacy classes of finite order elements, we can find a finite-index subgroup ∆ 0 ≤ ∆ which avoids each of these conjugacy classes, hence is torsion-free. The induced peripheral structure (∆ 0 , A 0 ) is relatively hyperbolic and ∆ 0 ↷ Y is relatively geometric by Lemma 2.2. Let Γ 0 = ψ −1 (∆ 0 ) and let P 0 = {P 0,1 , . . . , P 0,r } be the induced peripheral structure on Γ 0 . Then K ≤ Γ 0 , and since ∆ 0 is torsion-free, this implies K ∩ P 0,i = Z(P 0,i ) for each i. As the P is the collection of cusp subgroups of M 0 = Γ 0 H n C , we conclude that ∆ 0 = π 1 (T (M )). Thus, ∆ 0 is Kähler and acts relatively geometrically on Y . By Theorem 1.3, we conclude that ∆ 0 is virtually a hyperbolic surface group, which is impossible because ∆ 0 contains a subgroup isomorphic to Z 2n−2 and n ≥ 2. This contradiction completes the proof. Remark 4.5. In [20, Definition 1.9], Groves-Manning introduce the notion of a weakly relatively geometric action on a CAT(0) cube complex. We can replace "relatively geometric" with "weakly relatively geometric" in Theorem 1.1 using similar arguments. Indeed, after performing the toroidal filling of (Γ, P) to land in the Kähler setting, we can perform a further peripherally finite filling to obtain a hyperbolic quotient, which is virtually special [20,Theorem 4.5]. In this case the cube complex for the quotient is not K X where K is the kernel of the filling homomorphism Γ → Γ. Indeed, the action of Γ on K X in general has cell stabilizers that are virtually free. Nevertheless, Theorem D of [17] implies that in this case Γ is still cubulated hyperbolic. The arguments from the remainder of the proof of Theorem 1.1 still apply. Definition 2. 3 ( 3Dehn Filling). Given a group pair (G, P), where P = {P 1 , ..., P m } and a choice of normal subgroups of peripheral groups N = {N i ⊴ P i }, the Dehn filling of (G, P) with respect to N is the pair (G, P) where G = G K and K = ⟪∪N i ⟫ is the normal closure in G of the group generated by {∪ i N i } andP is the set of images of P under this quotient. The N i are called the filling kernels. When we want to specify the filling kernels we write G(N 1 , . . . , N m ) for the quotient G. Lemma 4.1. Γ has a torsion-free subgroup of finite index.1 One can also see this tree directly by considering the dual tree to the collection of hyperplanes of X which project to H. See [17, Remark 1.1] for more details. F ≤ U (n − 1), any abelian subgroup is diagonalizable. Thus, locally each point in N has a neighborhood of the form (S 1 × D n−1 ) (Z mZ) where D ⊂ C is the open unit disk, and Z mZ acts on S 1 by rotation by 2π m and on the polydisk D n−1 D n− 1 Figure 1 . 11Local picture of the fibration in Equation 5 near a singular point of O. A nonsingular fiber, shown in blue, winds m = 2 times around the singular fiber, shown in red. Proposition 4 . 3 . 43Let Γ ≤ PU(n, 1) be a non-uniform lattice, and suppose Γ acts on a CAT(0) cube complex X. The action of each cusp subgroup of Γ is not proper. In particular, Γ is not cubulated.4.2. The toroidal compactification of M . Another natural compactification of M fills in the cusps with the Euclidean orbifolds described in Section 4.1. Let O i be the Euclidean orbifold quotient of Σ i , with corresponding disk bundle E i . Thus, we can identify E i ∖ O i with the cusp C i , then compactify M by adding ⊔ i O i at infinity. The result is a Kähler orbifold T (M ) with boundary divisor D = ⊔ i O i . The pair (T (M ), D) is called the toroidal compactification of M . See Lemma 4 . 4 . 44Let Γ ≤ PU(n, 1) be torsion-free and let M = Γ H n C be the quotient. There exists a finite cover M ′ → M such that the toroidal compactification of M ′ is smooth. there is a natural map of pairs f ∶ (M 0 , ∂M 0 ) → (T (M ), D) that is a diffeomorphism on the interior of M 0 and sends ∂M = ⊔Σ i → D = ⊔ i O i via the fibering in Equation 5. 4.3. Proof of Theorem 1.1. We now have all the ingredients necessary to prove Theorem 1.1. The virtual Haken conjecture. Ian Agol, Doc. Math. 18With an appendix by AgolIan Agol. The virtual Haken conjecture. Doc. Math., 18:1045-1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning. Avner Ash, David Mumford, Michael Rapoport, Yung-Sheng Tai, Smooth compactifications of locally symmetric varieties. Cambridge Mathematical Library. CambridgeCambridge University Presssecond edition, 2010. With the collaboration of Peter ScholzeAvner Ash, David Mumford, Michael Rapoport, and Yung-Sheng Tai. Smooth compactifications of locally sym- metric varieties. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2010. With the collaboration of Peter Scholze. Thick metric spaces, relative hyperbolicity, and quasiisometric rigidity. Jason Behrstock, Cornelia Druţu, Lee Mosher, Mathematische Annalen. 3443Jason Behrstock, Cornelia Druţu, and Lee Mosher. Thick metric spaces, relative hyperbolicity, and quasi- isometric rigidity. Mathematische Annalen, 344(3):543-595, 2009. Hyperplane sections in arithmetic hyperbolic manifolds. Nicolas Bergeron, Frédéric Haglund, Daniel T Wise, J. Lond. Math. Soc. 832Nicolas Bergeron, Frédéric Haglund, and Daniel T. Wise. Hyperplane sections in arithmetic hyperbolic manifolds. J. Lond. Math. Soc. (2), 83(2):431-448, 2011. A boundary criterion for cubulation. Nicolas Bergeron, Daniel T Wise, American Journal of Mathematics. 1343Nicolas Bergeron and Daniel T Wise. A boundary criterion for cubulation. American Journal of Mathematics, 134(3):843-859, 2012. Metric Spaces of Non-Positive Curvature. Martin Bridson, André Haefliger, SpringerMartin Bridson and André Haefliger. Metric Spaces of Non-Positive Curvature. Springer, 1999. Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds. Daryl Cooper, David Futer, Geom. Topol. 231Daryl Cooper and David Futer. Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds. Geom. Topol., 23(1):241-298, 2019. Cuts in Kähler groups. Thomas Delzant, Misha Gromov, Infinite groups: geometric, combinatorial and dynamical aspects. SpringerThomas Delzant and Misha Gromov. Cuts in Kähler groups. In Infinite groups: geometric, combinatorial and dynamical aspects, pages 31-55. Springer, 2005. . Thomas Delzant, Pierre Py, Cubulable Kähler groups. Geom. Topol. 234Thomas Delzant and Pierre Py. Cubulable Kähler groups. Geom. Topol., 23(4):2125-2164, 2019. Relative cubulations and groups with a 2-sphere boundary. Eduard Einstein, Daniel Groves, Compositio Mathematica. 1564Eduard Einstein and Daniel Groves. Relative cubulations and groups with a 2-sphere boundary. Compositio Mathematica, 156(4):862-867, 2020. Relatively geometric actions on CAT(0) cube complexes. Eduard Einstein, Daniel Groves, Journal of the London Mathematical Society. 1051Eduard Einstein and Daniel Groves. Relatively geometric actions on CAT(0) cube complexes. Journal of the London Mathematical Society, 105(1):691-708, 2022. Separation and relative quasi-convexity criteria for relatively geometric actions. Eduard Einstein, Daniel Groves, Thomas Ng, arXiv:2110.14682arXiv preprintEduard Einstein, Daniel Groves, and Thomas Ng. Separation and relative quasi-convexity criteria for relatively geometric actions. arXiv preprint arXiv:2110.14682, 2021. Relative cubulation of small cancellation free products. Eduard Einstein, Thomas Ng, Eduard Einstein and Thomas Ng. Relative cubulation of small cancellation free products, 2021. Relatively hyperbolic groups. B Farb, Geom. Funct. Anal. 85B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal., 8(5):810-840, 1998. Complex hyperbolic geometry. William M Goldman, Oxford Mathematical Monographs. The Clarendon Press. New YorkOxford Science PublicationsWilliam M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Michael Gromov, Richard Schoen, Publications Mathématiques de l'IHÉS. 76Michael Gromov and Richard Schoen. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publications Mathématiques de l'IHÉS, 76:165-246, 1992. Hyperbolic groups acting improperly. Daniel Groves, Jason Fox Manning, arXiv:1808.02325Geom. Topol. To appear, preprint atDaniel Groves and Jason Fox Manning. Hyperbolic groups acting improperly. Geom. Topol. To appear, preprint at arXiv:1808.02325. Dehn filling in relatively hyperbolic groups. Daniel Groves, Jason Fox Manning, Israel J. Math. 168Daniel Groves and Jason Fox Manning. Dehn filling in relatively hyperbolic groups. Israel J. Math., 168:317-429, 2008. Dehn fillings and elementary splittings. Daniel Groves, Jason Fox Manning, Trans. Amer. Math. Soc. 3705Daniel Groves and Jason Fox Manning. Dehn fillings and elementary splittings. Trans. Amer. Math. Soc., 370(5):3017-3051, 2018. Specializing cubulated relatively hyperbolic groups. Daniel Groves, Jason Fox Manning, J. Topol. 152Daniel Groves and Jason Fox Manning. Specializing cubulated relatively hyperbolic groups. J. Topol., 15(2):398- 442, 2022. Isometries of CAT(0) cube complexes are semi-simple. Frédéric Haglund, Annales mathématiques du Québec. Frédéric Haglund. Isometries of CAT(0) cube complexes are semi-simple. Annales mathématiques du Québec, pages 1-13, 2021. Hyperbolic groups with planar boundaries. Peter Haïssinsky, Invent. Math. 2011Peter Haïssinsky. Hyperbolic groups with planar boundaries. Invent. Math., 201(1):239-307, 2015. Rank one lattices whose parabolic isometries have no rotational part. Christoph Hummel, Proceedings of the. theAmerican Mathematical SocietyChristoph Hummel. Rank one lattices whose parabolic isometries have no rotational part. Proceedings of the American Mathematical Society, pages 2453-2458, 1998. Cusp closing in rank one symmetric spaces. Christoph Hummel, Viktor Schroeder, Invent. Math. 1232Christoph Hummel and Viktor Schroeder. Cusp closing in rank one symmetric spaces. Invent. Math., 123(2):283- 307, 1996. Criterion for Cannon's conjecture. Vladimir Markovic, Geom. Funct. Anal. 233Vladimir Markovic. Criterion for Cannon's conjecture. Geom. Funct. Anal., 23(3):1035-1061, 2013. Groups acting on CAT(0) cube complexes. Graham Niblo, Lawrence Reeves, Geom. Topol. 1Graham Niblo and Lawrence Reeves. Groups acting on CAT(0) cube complexes. Geom. Topol., 1:approx. 7 pp., 1997. Peripheral fillings of relatively hyperbolic groups. Denis V Osin, Invent. Math. 1672Denis V Osin. Peripheral fillings of relatively hyperbolic groups. Invent. Math., 167(2):295-326, 2007. Examples of fundamental groups of compact Kähler manifolds. Domingo Toledo, Bull. London Math. Soc. 224Domingo Toledo. Examples of fundamental groups of compact Kähler manifolds. Bull. London Math. Soc., 22(4):339-343, 1990. The structure of groups with a quasiconvex hierarchy. Daniel T Wise, Annals of Mathematics Studies. 209Princeton University PressDaniel T. Wise. The structure of groups with a quasiconvex hierarchy, volume 209 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, [2021] ©2021.

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{'abstract': 'We prove that for n ≥ 2, a non-uniform lattice in PU(n, 1) does not admit a relatively geometric action on a CAT(0) cube complex, in the sense of[10]. As a consequence, if Γ is a nonuniform lattice in a non-compact semisimple Lie group G without compact factors that admits a relatively geometric action on a CAT(0) cube complex, then G is commensurable with SO(n, 1). We also prove that if a Kähler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a CAT(0) cube complex, then it is virtually a surface group.2020 Mathematics Subject Classification. 20F65, 22E40 (primary), 32J27, 32J05, 57N65 (secondary).', 'arxivid': '2210.12850', 'author': ['Corey Bregman ', 'Daniel Groves ', 'Kejia Zhu '], 'authoraffiliation': [], 'corpusid': 258049350, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11411, 'n_tokens_neox': 9655, 'n_words': 5997, 'pdfsha': '9e29829794b4b39cbd308140eca90cdd1790f224', 'pdfurls': ['https://export.arxiv.org/pdf/2210.12850v3.pdf'], 'title': ['RELATIVELY GEOMETRIC ACTIONS OF KÄHLER GROUPS ON CAT(0) CUBE COMPLEXES', 'RELATIVELY GEOMETRIC ACTIONS OF KÄHLER GROUPS ON CAT(0) CUBE COMPLEXES'], 'venue': []}

arxiv

Semi-analytical solutions of shallow water waves with idealised bottom topographies January 31. 2023. 00 Chang Liu Department of Applied Mathematics and Statistics Johns Hopkins University Baltimore 21218MDUSA Department of Physics University of California 94720Berkeley BerkeleyCAUSA Antwan D Clark Department of Applied Mathematics and Statistics Johns Hopkins University Baltimore 21218MDUSA Semi-analytical solutions of shallow water waves with idealised bottom topographies Month 0000January 31. 2023. 00released July 2022)Adomian decomposition methodshallow water equationsbottom topographies Analysing two-dimensional shallow water equations with idealised bottom topographies have many applications in the atmospheric and oceanic sciences; however, restrictive flow pattern assumptions have been made to achieve explicit solutions. This work employs the Adomian decomposition method (ADM) to develop semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters while capturing the nonlinear phenomenon. Furthermore, we exploit these techniques as reverse engineering mechanisms to develop key connections between some prevalent ansatz formulations in the open literature as well as derive new families of exact solutions describing geostrophic inertial oscillations and anticyclonic vortices with finite escape times. Our semi-analytical evaluations show the promise of this approach in terms of providing robust approximations against several oceanic variations and bottom topographies while also preserving the direct correlation between the physical parameters such as the Froude number, the bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours. Our numerical validations provide additional confirmations of this approach while also illustrating that ADM can also be used to provide insight and deduce novel solutions that have not been explored, which can be used to characterize various types of geophysical flows. Introduction Analysing two-dimensional shallow water equations has been extensively studied in geophysical fluid dynamics to understand a myriad of atmospheric and oceanic phenomena. Some examples include understanding the effects of long-term oceanic waves (Pedlosky 2013, Vallis 2017, analyzing the behaviour of oceanic warm-core rings (Cushman-Roisin 1987), investigating flows in channels and shorelines (Shapiro 1996, Sampson et al. 2005, studying steady-state flows (Iacono 2005, Sun 2016, and grasping the temporal instability of barotropic zonal flows (Clark and Herron 2013). These theoretical analyses also serve as a good basis for numerical simulations and validations. For example, the creators of the Shallow Water Analytic Solutions for Hydraulic and Environmental Studies (SWASHES) software library (Delestre et al. 2013) incorporated a significant number of theoretical solutions of the shallow water equations in the open literature, which has been cited by over 200 research papers currently. Furthermore, several of the solutions in this library are obtained from Thacker (1981) in which have been widely used to demonstrate the validity and accuracy of several numerical schemes including finite volume schemes (Gallardo et al. 2007, Bollermann et al. 2011, Nikolos and Delis 2009) and discontinuous Galerkin methods (Ern et al. 2008, Kesserwani and Liang 2012, Li et al. 2017, Wintermeyer et al. 2018. Some significant advancements include the original works of Ball and Thacker who demonstrated that nonlinear oscillations can be modelled as either low-order polynomials or normal modes (Ball 1963, 1964, 1965, Thacker 1977, 1981. Researchers also developed elliptical vortex solutions to understand the temporal effects of oceanic warm-core rings including stationary clockwise rotations (rodons), pulsating circular eddies (pulsons), and a subclass of these phenomena called pulsrodons (Cushman-Roisin 1987, Cushman-Roisin et al. 1985, Rogers 1989b). Extensions to these approaches have been made, where some examples include the work of Sachdev et al. (1996) who extended the approach of Clarkson and Kruskal (1989) and derived new families of solutions in paraboloidal basins that provided additional insights in terms of describing flow behaviour due to deformation modes. Additionally, Matskevich and Chubarov (2019) extended the results of Ball and Thacker to include the effects of Coriolis forces and bottom friction. Bristeau et al. (2021) also extended the results of Thacker and introduced two respective solutions describing velocity distributed along the vertical axis and velocity accounting for variable density. Group analysis was also explored. Some pioneering works in this area include that of Currò (1989) and Rogers (1989a) who also advanced the works of Thacker and Ball and related several forms of the depth function as well as developed invariance theorems. Levi et al. (1989) developed symmetry reductions for flows with elliptic and circular bottom topographies. Bila et al. (2006) derived Lie point symmetries and conservation laws. Chesnokov (2009) discovered 9-dimensional Lie algebra point symmetries and developed transformations between rotating and non-rotating cases, which were later used to describe spatial oscillations in spinning paraboloids (Chesnokov 2011). Some recent advancements include Meleshko (2020) and Bihlo et al. (2020) who performed group classification and analysis for zero and constant Coriolis parameters. Meanwhile, Meleshko and Samatova (2020) performed similar analysis and considered the beta-plane approximation of the Coriolis parameter and an irregular bottom topography. However, deriving theoretical solutions to the two-dimensional shallow water equations poses the following main challenges. First, these efforts involve making specific assumptions regarding the flow conditions which only satisfy specific cases. Some solutions also contain combinations of special functions and integral expressions (Shapiro 1996, Rogers 1989b, which in turn makes it difficult to determine the correlation between the physical quantities of these models. Finding invariant solutions via group analysis has the additional advantage of deriving conservation laws to these equations. However, this approach depends on the construction of Lie-groups which depend on the problem formulation as well as specific assumptions such as the Coriolis parameter and bottom topography. Therefore, there is a need to find solutions that are not only flexible, in terms of relaxing certain limiting assumptions, but also provide a direct correlation of the physical parameters. This work applies Adomian decomposition method (ADM) (Adomian 1990) to the shallow water equations to provide the following main contributions. First, we present the ADM formulation of the rotating shallow water equations where we also present key connections between the ansatz formulations in the work of Thacker (1981), Shapiro (1996), Matskevich and Chubarov (2019). Next, we derive and present some new families of exact solutions, for flat bottom topographies, that describe inertial oscillations in geostrophic flows and anticyclonic vortices with finite escape times. The rest of this paper is organised in the following manner. Section 2 presents the ADM formulation and initial theoretical formulation of the problem, where we present the connection to fundamental assumptions on the formulation of the solutions. Section 3 presents derivations of new families of solutions and their properties. Section 4 provides numerical experimentation and results. Section 5 provides some concluding remarks, where we also list some future research directions. The non-dimensional form of the governing equations is defined as ∂u ∂t = − u ∂u ∂x − v ∂u ∂y − 1 F 2 ∂h ∂x +f v,(1a)∂v ∂t = − u ∂v ∂x − v ∂v ∂y − 1 F 2 ∂h ∂y −f u,(1b)∂h ∂t = − ∂ ∂x [u(h + D)] − ∂ ∂y [v(h + D)].(1c) This is illustrated in figure 1, where u and v are the flow velocity components, h is the free surface height,f = f L 0 /U 0 is the dimensionless Coriolis parameter (associated with the Coriolis force), and F = U 0 / √ gH 0 is the Froude number. Here, the spatial variables x, y, l, and L are normalised by the horizontal length scale L 0 ; h is normalised by a vertical length scale H 0 ; the horizontal velocities, u and v, are normalised by the characteristic velocity U 0 ; and time t is normalised by L 0 /U 0 . Hence, the dimensionless form of the idealised bottom topography is defined as D(x, y) = D 0 1 − x 2 L 2 − y 2 l 2 ,(2) where D 0 is also normalised by a vertical length scale H 0 . It is noteworthy to mention that other bottom topographies can be determined from (2) such as flat bottom (D 0 = 0), circular paraboloid (l = L), and channel (l → ∞ or L → ∞) terrains. Additionally, D(x, y) can also be used to incorporate linear terms in its description via change of variables (Shapiro 1996, Thacker 1981. The total fluid depth D + h, shown in figure 1, follows the formulations of Thacker (1981) and Shapiro (1996) where D + h = 0 represents a moving shoreline and D + h < 0 represents dry regions. When the moving shoreline is closed, the water mass within the shoreline is conserved (Thacker 1981, Shapiro 1996. When the moving shoreline is open such as in tsunami modelling, then water within a bounded domain will have mass exchange with an infinite mass reservoir. It is also important to mention that our explorations in this section consider flow velocities that are linearly varying spatially while the free surface height either varies linearly or in a quadratic fashion. The initial conditions are given by u(x, y, 0) =u 0 (x, y),(3a) v(x, y, 0) =v 0 (x, y), h(x, y, 0) =h 0 (x, y). Next, u, v, and h are decomposed as follows   u(x, y, t) v(x, y, t) h(x, y, t)   = ∞ n=0   u n (x, y, t) v n (x, y, t) h n (x, y, t)   ,(4) where the initial components are defined by equation (3). Thus, the recurrence relationships to equation (1) (for n ≥ 0) are given by u n+1 (x, y, t) = L −1 t −A n u, ∂u ∂x − A n v, ∂u ∂y − 1 F 2 ∂h n ∂x +f v n , (5a) v n+1 (x, y, t) = L −1 t −A n u, ∂v ∂x − A n v, ∂v ∂y − 1 F 2 ∂h n ∂y −f u n , (5b) h n+1 (x, y, t) = L −1 t − ∂ ∂x [A n (u, h)] − ∂ ∂y [A n (v, h)] − ∂ ∂x [u n D] − ∂ ∂y [v n D] ,(5c) where L t = ∂(·) ∂t , and L −1 t = t 0 (·) dτ, and the Adomian polynomial representing the quadratic nonlinearity is defined as (Adomian 1990(Adomian , 2013 A n (u, h) = n j=0 u j h n−j . It is important to note that equation (6) can be used to approximate the quadratic nonlinear terms, such as uh, as follows uh = ∞ p u p ∞ q h q = ∞ n A n (u, h) and thus the semi-analytical solution to (1) is expressed via the partial sums   u(x, y, t) v(x, y, t) h(x, y, t)   =   S N (u) S N (v) S N (h)   = N n=0   u n (x, y, t) v n (x, y, t) h n (x, y, t)   .(7) Next, the following results connect the properties of the initial conditions to the behaviours of the true solutions via their partial sums. Lemma 2.1: Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal parabolic topography (2). If the initial conditions u 0 ( x, y), v 0 (x, y), h 0 (x, y) are defined such that ∂ 2 u 0 (x, y) ∂x 2 = ∂ 2 v 0 (x, y) ∂x 2 = ∂ 3 h 0 (x, y) ∂x 3 = 0,(8a)∂ 2 u 0 (x, y) ∂y 2 = ∂ 2 v 0 (x, y) ∂y 2 = ∂ 3 h 0 (x, y) ∂y 3 = 0,(8b)∂ 2 u 0 (x, y) ∂xy = ∂ 2 v 0 (x, y) ∂x∂y = ∂ 3 h 0 (x, y) ∂x 2 ∂y = ∂ 3 h 0 (x, y) ∂x∂y 2 = 0. (8c) Then the higher order components u n (x, y, t), v n (x, y, t), h n (x, y, t) also satisfy the same property, where ∂ 2 u n (x, y, t) ∂x 2 = ∂ 2 v n (x, y, t) ∂x 2 = ∂ 3 h n (x, y, t) ∂x 3 = 0,(9a)∂ 2 u n (x, y, t) ∂y 2 = ∂ 2 v n (x, y, t) ∂y 2 = ∂ 3 h n (x, y, t) ∂y 3 = 0,(9b)∂ 2 u n (x, y, t) ∂x∂y = ∂ 2 v n (x, y, t) ∂x∂y = ∂ 3 h n (x, y, t) ∂x 2 ∂y = ∂ 3 h n (x, y, t) ∂x∂y 2 = 0 (9c) for n ∈ N + . Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5). Condition (9a) is demonstrated by examining the following relationships ∂ 2 u n+1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 A n u, ∂u ∂x + A n v, ∂u ∂y + 1 F 2 ∂ 3 h n ∂x 3 −f ∂ 2 v n ∂x 2 ,(10a)∂ 2 v n+1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 A n u, ∂v ∂x + A n v, ∂v ∂y + 1 F 2 ∂ 3 h n ∂x 2 ∂y +f ∂ 2 u n ∂x 2 ,(10b)∂ 3 h n+1 ∂x 3 = − L −1 t ∂ 4 ∂x 4 [A n (u, h)] + ∂ 4 ∂x 3 ∂y [A n (v, h)] + ∂ 4 ∂x 4 [u n D] + ∂ 4 ∂x 3 ∂y [v n D] .(10c) Therefore, when n = 0 equations (10a-c) representing the relationship between the initial and first components for u, v, and h become ∂ 2 u 1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 u 0 ∂u 0 ∂x + v 0 ∂u 0 ∂y + 1 F 2 ∂ 3 h 0 ∂x 3 −f ∂ 2 v 0 ∂x 2 ,(11a)∂ 2 v 1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 u 0 ∂v 0 ∂x + v 0 ∂v 0 ∂y + 1 F 2 ∂ 3 h 0 ∂x 2 ∂y +f ∂ 2 u 0 ∂x 2 ,(11b)∂ 3 h 1 ∂x 3 = − L −1 t ∂ 4 ∂x 4 [u 0 h 0 ] + ∂ 4 ∂x 3 y [v 0 h 0 ] + ∂ 4 ∂x 4 [u 0 D] + ∂ 4 ∂x 3 ∂y [v 0 D] .(11c) Employing (8a-c) it can be shown that equations (11a-c) reduce to the following relationship ∂ 2 u 1 (x, y, t) ∂x 2 = ∂ 2 v 1 (x, y, t) ∂x 2 = ∂ 3 h 1 (x, y, t) ∂x 3 = 0. Continuing this argument for n = {1, 2, . . . , n − 1} yields equation (9a). Similar arguments can be made to produce (9b,c), respectively. Theorem 2.2 : Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal parabolic topography (2). If the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) are defined as (8a-c), then the solutions of u, v, and h have the same property where ∂ 2 u(x, y, t) ∂x 2 = ∂ 2 u(x, y, t) ∂y 2 = ∂ 2 u(x, y, t) ∂x∂y = 0, (12a) ∂ 2 v(x, y, t) ∂x 2 = ∂ 2 v(x, y, t) ∂y 2 = ∂ 2 v(x, y, t) ∂x∂y = 0, (12b) ∂ 3 h(x, y, t) ∂x 3 = ∂ 3 h(x, y, t) ∂x 2 ∂y = ∂ 3 h(x, y, t) ∂x∂y 2 = ∂ 3 h(x, y, t) ∂y 3 = 0. (12c) Consequently, these solutions can be expressed as u(x, y, t) =ũ 0 (t) +ũ x (t)x +ũ y (t)y, (13a) v(x, y, t) =ṽ 0 (t) +ṽ x (t)x +ṽ y (t)y, (13b) h(x, y, t) =h 0 (t) +h x (t)x +h y (t)y + 1 2h xx (t)x 2 + 1 2h yy (t)y 2 +h xy (t)xy, (13c) where the coefficientsũ 0 (t),ũ x (t),ũ y (t),ṽ 0 (t),ṽ x (t),ṽ y (t),h 0 (t),h x (t),h y (t),h xx (t),h yy (t), andh xy (t) are time-dependent. Proof : Applying Lemma 2.1 to each component in (4) yields (12a-c). From (12a), we observe that ∂ 2 u(x, y, t) ∂x 2 = 0 yields u(x, y, t) = C 1 (y, t)x + C 2 (y, t), where the integration constants, C 1 (y, t) and C 2 (y, t), are independent of x. Similarly, we have ∂ 2 u(x, y, t) ∂x∂y = 0 yields C 1 (y, t) =ũ x (t), and ∂ 2 u(x, y, t) ∂y 2 = 0 yields C 2 (y, t) =ũ y (t)y +ũ 0 (t), and thus (13a) is achieved. Similar arguments can be made to achieve (13b,c), respectively. We note the significance of Theorem 2.2. In the works of Thacker (1981), Shapiro (1996), and Matskevich and Chubarov (2019) equations (13a-c) were presented as ansatz solutions, where they were also used to produce the reduced system of shallow water equations to derive closed-form solutions. This theorem removes these assumptions and provides more insight to this behaviour by connecting it to the initial conditions (8a-c). Novel exact solutions for flat bottom topographies with constant Coriolis force Next, we use the ADM construction to derive new families of solutions and their properties that describe other geophysical flows such as inertial oscillations and anticyclonic vortices which have a profound effect on oceanic and atmospheric dynamics (Vallis 2017, Kafiabad et al. 2021. Here, we consider flows over flat bottom topologies where D 0 = 0 in (2) with constant Coriolis parameter (f = 0). Inertial oscillations in geostrophic flows For these types of flows, our analysis considers the following initial conditions. • Condition I u 0 (x, y) = v 0 (x, y) = 0, h 0 (x, y) = η x x + η y y,(14) • Condition II u 0 (x, y) = v 0 (x, y) = 0, h 0 (x, y) = η x x,(15) • Condition III u 0 (x, y) = v 0 (x, y) = 0, h 0 (x, y) = η y y,(16) where η x and η y are the respective constant free surface gradients in the x and y directions. We note that the behaviour of the initial conditions (14) - (16) affect the decomposition of the decomposed functions of u, v, and h as presented in the following lemma. Lemma 3.1: Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h such that their relationship is defined by (5) (for n ∈ N). If D = 0 and the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) satisfy the following properties ∂u 0 (x, y) ∂x = ∂v 0 (x, y) ∂x = ∂ 2 h 0 (x, y) ∂x 2 = 0, (17a) ∂u 0 (x, y) ∂y = ∂v 0 (x, y) ∂y = ∂ 2 h 0 (x, y) ∂y 2 = 0,(17b)∂ 2 h 0 (x, y) ∂x∂y =0. (17c) Then the higher order components u n (x, y, t), v n (x, y, t), h n (x, y, t) also satisfy the property that ∂u n (x, y, t) ∂x = ∂v n (x, y, t) ∂x = ∂h n (x, y, t) ∂x = 0, (18a) ∂u n (x, y, t) ∂y = ∂v n (x, y, t) ∂y = ∂h n (x, y, t) ∂y = 0 (18b) for n ∈ N + . Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in (5). Condition (18a) is demonstrated by examining the following relationships ∂u n+1 ∂x = − L −1 t ∂ ∂x A n u, ∂u ∂x + A n v, ∂u ∂y + 1 F 2 ∂ 2 h n ∂x 2 −f ∂v n ∂x ,(19a)∂v n+1 ∂x = − L −1 t ∂ ∂x A n u, ∂v ∂x + A n v, ∂v ∂y + 1 F 2 ∂ 2 h n ∂x∂y +f ∂u n ∂x ,(19b)∂h n+1 ∂x = − L −1 t ∂ 2 ∂x 2 [A n (u, h)] + ∂ 2 ∂x∂y [A n (v, h)] .(19c) Therefore, when n = 0, equations (19a-c) representing the relationship between the initial and first components for u, v, and h become ∂u 1 ∂x = − L −1 t ∂ ∂x A 0 u, ∂u ∂x + A 0 v, ∂u ∂y + 1 F 2 ∂ 2 h 0 ∂x 2 −f ∂v 0 ∂x , (20a) ∂v 1 ∂x = − L −1 t ∂ ∂x A 0 u, ∂v ∂x + A 0 v, ∂v ∂y + 1 F 2 ∂ 2 h 0 ∂x∂y +f ∂u 0 ∂x ,(20b)∂h 1 ∂x = − L −1 t ∂ 2 ∂x 2 [A 0 (u, h)] + ∂ 2 ∂x∂y [A 0 (v, h)] .(20c) Employing (17a-c) it can be shown that equations (20a-c) reduce to the following relationship ∂u 1 (x, y, t) ∂x = ∂v 1 (x, y, t) ∂x = ∂h 1 (x, y, t) ∂x = 0, and continuing this argument for n ∈ N + yields equation (18a). Following similar arguments yields (18b). From this, the behaviour of uniform u, v over space, and planar free surface h with constant spatial gradients over time can be summarised in the following theorem. Theorem 3.2 : Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N). If D = 0 and the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) satisfy the properties defined in (17a-c), then the solutions u, v, and h have the following properties ∂u(x, y, t) ∂x = ∂u(x, y, t) ∂y = 0,(21a) ∂v(x, y, t) ∂x = ∂v(x, y, t) ∂y = 0, ∂h(x, y, t) ∂x = ∂h(x, y, 0) ∂x ,(21b) ∂h(x, y, t) ∂y = ∂h(x, y, 0) ∂y , ∂ 2 h(x, y, t) ∂x 2 = ∂ 2 h(x, y, t) ∂x∂y = ∂ 2 h(x, y, t) ∂y 2 = 0.(21d) Additionally, u, v, and h are reduced to the following forms u(x, y, t) =ũ 0 (t),(22a)v(x, y, t) =ṽ 0 (t),(22b) h(x, y, t) =h 0 (t) +h x x +h y y, where the coefficientsũ 0 (t),ṽ 0 (t), andh 0 (t) are time-dependent, whileh x andh y are constants. Additionally, (22a-c) satisfy the reduced system of equations d dtũ 0 (t) = − 1 F 2h x +fṽ 0 (t),(23a)d dtṽ 0 (t) = − 1 F 2h y −fũ 0 (t),(23b)d dth 0 (t) = −ũ 0 (t)h x −ṽ 0 (t)h y .(23c) Proof : Applying Lemma 3.1 to each component in (4) yields (21). From (21a), we observe that ∂u(x, y, t) ∂x = 0 yields u(x, y, t) = C 1 (y, t), where the integration constants, C 1 (y, t), are independent of x. Similarly, we have ∂u(x, y, t) ∂y = 0 yields C 1 (y, t) =ũ 0 (t) and thus (22a) is achieved. Similar arguments can be made to achieve (22b,c), respectively. Substituting (22a-c) into (1) achieves the reduced system of equations (23), which completes the proof. Hence, we have the following results for inertial oscillations for geostrophic flows. Theorem 3.3 : Given inertial oscillations over flat bottom topographies with constant Coriolis parameterf = 0, where the initial behaviour is defined by (14). The solutions u, v, and h are expressed as u(x, y, t) = − η x f F 2 sin f t − η ȳ f F 2 1 − cos f t ,(24a)v(x, y, t) = η x f F 2 1 − cos f t − η ȳ f F 2 sin(f t), (24b) h(x, y, t) = η 2 x f 2 F 2 1 − cos f t + xη x + η 2 ȳ f 2 F 2 1 − cos f t + η y y,(24c) where η x and η y are the constant free surface gradients in the x and y directions, respectively. Proof : The initial conditions (14) satisfy (17a-c). Therefore, the sequence of decomposed functions {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} satisfy (18a,b) for n ∈ N + which satisfies Lemma 3.1 and consequently Theorem 3.2. Examining the system of reduced equations (23), the initial conditions (14) also produce the following reduced relationships:h x = η x ,h y = η y , andũ(t = 0) =ṽ(t = 0) =h 0 (t = 0) = 0. Solving this reduced system achieves (24a-c) which proves the theorem. Corollary 3.4: Given inertial oscillations over flat bottom topographies with constant Coriolis parameterf = 0. (i) If the initial behaviour is defined by (15), then the solutions u, v, and h are expressed as u(x, y, t) = − η x f F 2 sin f t ,(25a)v(x, y, t) = η x f F 2 1 − cos f t ,(25b)h(x, y, t) = η 2 x f 2 F 2 1 − cos f t + xη x .(25c) (ii) If the initial behaviour is defined by (16), then the solutions u, v, and h are expressed as u(x, y, t) = − η ȳ f F 2 1 − cos f t ,(26a)v(x, y, t) = − η ȳ f F 2 sin f t ,(26b)h(x, y, t) = η 2 ȳ f 2 F 2 1 − cos f t + η y y.(26c) η x and η y are the constant free surface gradients in the x and y directions, respectively. Proof : This is a special case of Theorem 3.3 for η y = 0 and η x = 0, respectively. Theorem 3.3 and Corollary 3.4 show the explicit relationship between these types of flows with respect to the constant Coriolis parameter, the free surface gradients, and the Froude number where the inertial oscillation frequency is defined by the constant Coriolis parameter f . These results also demonstrate that these oscillations are based on the magnitude of the free surface gradients that depend on the initial behaviour and the geostrophic flows, which are consistent with the results of (Vallis 2017). Moreover, Theorem 3.3 describes these types of oscillations as the interaction between the geostrophic flow fluctuations and the free surface gradients, where Corollary 3.4 considers cases when these gradients are negligible in the x and y directions. Anticyclonic vortices with finite escape times For these types of flows our analysis considers the following initial conditions • Condition IV u 0 (x, y) =f y, v 0 (x, y) = 0, h 0 (x, y) = h 0 ,(27)• Condition V u 0 (x, y) =f y, v 0 (x, y) = −f x +f y, h 0 (x, y) = h 0 ,(28) • Condition VI u 0 (x, y) = 0, v 0 (x, y) = −f x, h 0 (x, y) = h 0 ,(29) • Condition VII u 0 (x, y) =f x +f y, v 0 (x, y) = −f x, h 0 (x, y) = h 0 ,(30) where h 0 is the constant free surface height. These describe anticyclonic vortices for the initial vorticity is proportional to the negative constant Coriolis parameter. The behaviour of the initial conditions (27) -(30) affect the decomposition of the decomposed functions of u, v, and h as presented in the following lemmas. Lemma 3.5: Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom topography D = 0. If the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) are defined such that u 0 (x, y) =f y,(31a)∂ 2 v 0 (x, y) ∂x 2 = ∂h 0 (x, y) ∂x = 0,(31b)∂ 2 v 0 (x, y) ∂y 2 = ∂h 0 (x, y) ∂y = 0,(31c)∂ 2 v 0 (x, y) ∂x∂y =0. (31d) Then the higher order components u n (x, y, t), v n (x, y, t), h n (x, y, t), for n ∈ N + satisfy u n (x, y, t) =0, (32a) ∂ 2 v n (x, y, t) ∂x 2 = ∂h n (x, y, t) ∂x = 0, (32b) ∂ 2 v n (x, y, t) ∂y 2 = ∂h n (x, y, t) ∂y = 0,(32c)∂ 2 v n (x, y, t) ∂x∂y =0. (32d) Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5). Condition (32a) is demonstrated by examining u n+1 = −L −1 t A n u, ∂u ∂x + A n v, ∂u ∂y + 1 F 2 ∂h n ∂x −f v n ,(33) In the case of n = 0 and using (31a) -(31d), it reduces to u 1 = −L −1 t A 0 u, ∂u ∂x + A 0 v, ∂u ∂y + 1 F 2 ∂h 0 ∂x −f v 0 = −L −1 t A 0 v, ∂u ∂y −f v 0 = −L −1 t v 0 ∂u 0 ∂y −f v 0 = 0, and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (32a). Condition (32b) is demonstrated by examining the following relationships ∂ 2 v n+1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 A n u, ∂v ∂x + A n v, ∂v ∂y + 1 F 2 ∂ 3 h n ∂x 2 ∂y +f ∂ 2 u n ∂x 2 ,(34a)∂h n+1 ∂x = − L −1 t ∂ 2 ∂x 2 [A n (u, h)] + ∂ 2 ∂x∂y [A n (v, h)] .(34b) Therefore, when n = 0, equations (34a,b) representing the relationship between the initial and first components for v and h become ∂ 2 v 1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 A 0 u, ∂v ∂x + A 0 v, ∂v ∂y + 1 F 2 ∂ 3 h 0 ∂x 2 ∂y +f ∂ 2 u 0 ∂x 2 ,(35a)∂h 1 ∂x = − L −1 t ∂ 2 ∂x 2 [A 0 (u, h)] + ∂ 2 ∂x∂y [A 0 (v, h)] .(35b) Employing (31a-d), it can be shown that equations (35a,b) reduce to the following relationship ∂ 2 v 1 (x, y, t) ∂x 2 = ∂h 1 (x, y, t) ∂x = 0, and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (32b). Following similar arguments yields (32c,d). Lemma 3.6: Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom topography D = 0. If the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) are defined such that v 0 (x, y, t) = −f x,(36a)∂ 2 u 0 (x, y) ∂x 2 = ∂h 0 (x, y) ∂x = 0,(36b)∂ 2 u 0 (x, y) ∂y 2 = ∂h 0 (x, y) ∂y = 0,(36c)∂ 2 u 0 (x, y) ∂x∂y =0. (36d) Then the higher order components u n (x, y, t), v n (x, y, t), h n (x, y, t), for n ∈ N + satisfy the property v n (x, y, t) =0, (37a) ∂ 2 u n (x, y, t) ∂x 2 = ∂h n (x, y, t) ∂x = 0,(37b)∂ 2 u n (x, y, t) ∂y 2 = ∂h n (x, y, t) ∂y = 0, (37c) ∂ 2 u n (x, y, t) ∂x∂y =0. (37d) Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5). Condition (37a) is demonstrated by examining the following relationships v n+1 = −L −1 t A n u, ∂v ∂x + A n v, ∂v ∂y + 1 F 2 ∂h n ∂y +f u n .(38) At n = 0, we have v 1 = −L −1 t A 0 u, ∂v ∂x + A 0 v, ∂v ∂y + 1 F 2 ∂h 0 ∂y +f u 0 = −L −1 t A 0 u, ∂v ∂x +f u 0 = −L −1 t −u 0f +f u 0 = 0. Employing a similar argument for n = {1, 2, . . . , n − 1}, we have (37a). Equation (37b) is demonstrated by examining the following ∂ 2 u n+1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 A n u, ∂u ∂x + A n v, ∂u ∂y + 1 F 2 ∂ 3 h n ∂x 3 −f ∂ 2 v n ∂x 2 , (39a) ∂h n+1 ∂x = − L −1 t ∂ 2 ∂x 2 [A n (u, h)] + ∂ 2 ∂x∂y [A n (v, h)] .(39b) Therefore, when n = 0, equations (39a,b) representing the relationship between the initial and first components for u and h become ∂ 2 u 1 ∂x 2 = − L −1 t ∂ 2 ∂x 2 u 0 ∂u 0 ∂x + v 0 ∂u 0 ∂y + 1 F 2 ∂ 3 h 0 ∂x 3 −f ∂ 2 v 0 ∂x 2 ,(40a)∂h 1 ∂x = − L −1 t ∂ 2 ∂x 2 [A 0 (u, h)] + ∂ 2 ∂x∂y [A 0 (v, h)] .(40b) Employing (36a-d), it can be shown that equations (40a,b) reduce to the following relationship ∂ 2 u 1 (x, y, t) ∂x 2 = ∂h 1 (x, y, t) ∂x = 0, and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (37b). Following similar arguments yields (37c,d). Therefore, the behaviour of u, v, and h can be summarised in the following theorem. ∂ 2 v(x, y, t) ∂x 2 = ∂ 2 v(x, y, t) ∂y 2 = ∂ 2 v(x, y, t) ∂x∂y = 0,(41a) ∂h(x, y, t) ∂x = ∂h(x, y, t) ∂y = 0. Consequently, these solutions can be expressed as u(x, y, t) =f y, (42a) v(x, y, t) =ṽ 0 (t) +ṽ x (t)x +ṽ y (t)y, (42b) h(x, y, t) =h 0 (t),(42c) where the coefficientsṽ 0 (t),ṽ x (t),ṽ y (t), andh 0 (t) are time-dependent that also satisfy the following reduced system of equations d dtṽ 0 (t) = −ṽ 0 (t)ṽ y (t),(43a) d dtṽ x (t) = −ṽ x (t)ṽ y (t), (43b) d dtṽ y (t) = −fṽ x (t) −ṽ y (t) 2 −f 2 ,(43c)d dth 0 (t) = −h 0 (t)ṽ y (t).(43d) Proof : Applying Lemma 3.5 to each component in (4) yields (41a-c). From (41b), we observe that ∂ 2 v(x, y, t) ∂x 2 = 0 yields v(x, y, t) = C 1 (y, t)x + C 2 (y, t), where the integration constants, C 1 (y, t) and C 2 (y, t), are independent of x. Similarly, we have ∂ 2 v(x, y, t) ∂x∂y = 0 yields C 1 (y, t) =ṽ x (t) and ∂ 2 v(x, y, t) ∂y 2 = 0 yields C 2 (y, t) =ṽ y (t)y +ṽ 0 (t). and thus (42b) is achieved. Similar arguments can be made to achieve (42a,c), respectively. The reduced system of equations (43) is obtained via substituting (42a-c) into (1). Theorem 3.8 : Let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom topography D = 0. If the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) are defined as (36a-d), then the solutions of u, v, and h have the same property, where ∂ 2 u(x, y, t) ∂x 2 = ∂ 2 u(x, y, t) ∂y 2 = ∂ 2 u(x, y, t) ∂x∂y = 0, (44a) v(x, y, t) = −f x,(44b) ∂h(x, y, t) ∂x = ∂h(x, y, t) ∂y = 0. Consequently, these solutions can be expressed as u(x, y, t) =ũ 0 (t) +ũ x (t)x +ũ y (t)y, v(x, y, t) = −f x,(45a)h(x, y, t) =h 0 (t),(45b) where the coefficientsũ 0 (t),ũ x (t),ũ y (t), andh 0 (t), are time-dependent. These coefficients satisfy d dtũ 0 (t) = −ũ 0 (t)ũ x (t),(46a) d dtũ x (t) = −ũ x (t) 2 +fũ y (t) −f 2 ,(46b)d dtũ y (t) = −ũ y (t)ũ x (t),(46c)d dth 0 (t) = −h 0 (t)ũ x (t).(46d) Proof : Applying Lemma 3.6 to each component in (4) yields (44a-c). From (44a), we observe that ∂ 2 u(x, y, t) ∂x 2 = 0 yields u(x, y, t) = C 1 (y, t)x + C 2 (y, t), where the integration constants, C 1 (y, t) and C 2 (y, t), are independent of x. Similarly, we have ∂ 2 u(x, y, t) ∂x∂y = 0 yields C 1 (y, t) =ũ x (t) and ∂ 2 u(x, y, t) ∂y 2 = 0 yields C 2 (y, t) =ũ y (t)y +ũ 0 (t). and thus (45a) is achieved. Similar arguments can be made to achieve (45b,c). The reduced equations (46) is obtained by substituting (45a-c) into (1). Therefore, the following results describe closed-form solutions for anticyclonic vortices with finite escape times. Theorem 3.9 : For any flows over flat bottom topographies (D = 0) with a constant Coriolis parameter (f = 0) and initial constant free surface height (h 0 ), the solutions u, v, and h with respect to their corresponding initial conditions are defined as follows. (i) If the initial behaviour is defined by (27) then u(x, y, t) =f y,(47a)v(x, y, t) = −f y tan(f t),(47b) h(x, y, t) =h 0 sec(f t). (47c) (ii) If the initial behaviour is defined by (28) then u(x, y, t) =f y,(48a) v(x, y, t) =f y sec (f t) −f y tan(f t) + x − d dt tan(f t) + d dt sec(f t) , (48b) h(x, y, t) = h 0 f d dt tan(f t) − d dt sec(f t) . (48c) Furthermore, these solutions describe anticyclonic vortices with finite escape times that are based on the initial zonal velocity being represented as u(x, y, 0) = u 0 (x, y) =f y. Proof : Equations (27) and (28) satisfy Theorem 3.7, where these flows can be represented by (43). The initial conditions (27) requirẽ v 0 (t = 0) =ṽ x (t = 0) =ṽ y (t = 0) = 0,(49a)h 0 (t = 0) =h 0 .(49b) Similarly, the initial conditions (28) requirẽ v 0 (t = 0) =0,ṽ x (t = 0) = −f ,ṽ y (t = 0) =f ,(50a)h 0 (t = 0) =h 0 .(50b) Solving (43) with the initial conditions, defined by (49) and (50), achieves (47) and (48), respectively. Theorem 3.10 : For any flows over flat bottom topographies (D = 0) with a constant Coriolis parameter (f = 0) and initial constant free surface height (h 0 = 0), the solutions u, v, and h with respect to their corresponding initial conditions are defined as follows. (i) If the initial behaviour is defined by (29) then u(x, y, t) = −f x tan(f t), (51a) v(x, y, t) = −f x,(51b) h(x, y, t) =h 0 sec(f t). (51c) (ii) If the initial behaviour is defined by (30) then u(x, y, t) =f x sec(f t) −f x tan(f t) + y d dt tan(f t) − d dt sec(f t) , (52a) v(x, y, t) = −f x,(52b)h(x, y, t) = h 0 f d dt tan(f t) − d dt sec(f t) . (52c) Furthermore, these solutions describe anticyclonic vortices with finite escape times that are based on the initial meridional velocity being represented as v(x, y, 0) = v 0 (x, y) = −f x. Proof : Equations (29) and (30) satisfy Theorem 3.8, where these flows can be represented by (46). The initial conditions (29) requirẽ u 0 (t = 0) =ũ x (t = 0) =ũ y (t = 0) = 0,(53a)h 0 (t = 0) =h 0 .(53b) Similarly, the initial conditions (30) requirẽ u 0 (t = 0) =0,ũ x (t = 0) =ũ y (t = 0) =f ,(54a)h 0 (t = 0) =h 0 .(54b) Solving (46) with the initial conditions, defined by (53) and (54), achieves (51) and (52), respectively. Theorems 3.9 and 3.10 show that the flow velocity components directly depend only on the constant Coriolis parameter whereas the free surface height depends on both the constant Coriolis parameter and the initial free surface height. Since these solutions are valid for t ∈ 0, π/ 2f , these results also represent anticyclonic vortices with finite escape times that rotate faster and are more unstable than cyclonic ones which is consistent with previous observations (Tsang andDritschel 2015, McKiver 2020). These solutions also consider the nonlinear balance between the inertial and Coriolis terms in the momentum portion of the shallow water equations, which is important to understand irregularities between cyclonic and anticyclonic vortices which also improves previous results using quasi-geostrophic approximations (Vallis 2019, McKiver 2020, linear stability analysis techniques (Clark and Herron 2013), and numerical approaches (Tsang and Dritschel 2015). Numerical validation and results Numerical validation is provided via examining the convergence and accuracy of the partial sums of u, v, and h (given by S N (u), S N (v), and S N (h)) against the governing equations (1), the exact solutions (u, v, and h), and numerical solutions (û,v, andĥ) via the relative integral squared error defined as E(N ) = Lx −Lx Ly −Ly T 0 e(N ; x, y, t) dt dx dy Lx −Lx Ly −Ly T 0 (u 2 + v 2 + h 2 ) dt dx dy ,(55) where L x = 1, L y = 1, and T = 1. The convergence E c (N ) is measured by evaluating (55) with e(N ; x, y, t) = ∂S N (u) ∂t + S N (u) ∂S N (u) ∂x + S N (v) ∂S N (u) ∂y + 1 F 2 ∂S N (h) ∂x −f S N (v) 2 + ∂S N (v) ∂t + S N (u) ∂S N (v) ∂x + S N (v) ∂S N (v) ∂y + 1 F 2 ∂S N (h) ∂y +f S N (u) 2 + ∂S N (h) ∂t + ∂ ∂x [S N (u)(S N (h) + D)] + ∂ ∂y [S N (v)(S N (h) + D)] 2 .(56) E ex (N ) is the accuracy of the partial sums of u, v, and h against the exact solutions which is measured via evaluating (55) with e(N ; x, y, t) = (S N (u) − u) 2 + (S N (v) − v) 2 + S N (h) − h 2 .(57) E(N ) is the accuracy of the partial sums of u, v, and h against the numerical solutions which is measured via evaluating (55) with e(N ; x, y, t) = (S N (u) −û) 2 + (S N (v) −v) 2 + S N (h) −ĥ 2 .(58) E ex is the accuracy between the numerical and exact solutions, which is measured via evaluating (55) with e(N ; x, y, t) = (u −û) 2 + (v −v) 2 + h −ĥ 2 .(59) In all evaluations, we follow Matskevich and Chubarov (2019) where F = 1 represents the characteristic velocity as U 0 = √ gH 0 . The summaries of all parameters used for our evaluations are listed in Table 1 below. Equation (55) is discretised with spatial grid spacings of ∆x = 0.1 and ∆y = 0.1 and a temporal grid spacing of ∆t = 0.1. Numerical implementations (û,v, andĥ) are done using the large-particle method as outlined by Matskevich and Chubarov (2019). Condition Ff D 0 L l Other Parameters Exact solutions I 1 0.5 0 --η x = 10 −4 Theorem 3.3 II 1 0.5 0 --η y = 10 −4 Corollary 3.4(i) III 1 0.5 0 --η x = η y = 10 −4 Corollary 3.4(ii) IV 1 0.5 0 --h 0 = 10 −4 Theorem 3.9(i) V 1 0.5 0 --h 0 = 10 −4 Theorem 3.9(ii) VI 1 0.5 0 --h 0 = 10 −4 Theorem 3.10(i) VII 1 0.5 0 --h 0 = 10 −4 Theorem 3.10(ii) 4.1. Results Table 2 presents a summary of the convegence and accuracy results, where the partial sums (for N = 2, 4 and 6) was used to assess the level of convergence. We note the convergence trend where the relative error margins stabilise between O 10 −11 and O 10 −6 at N = 6, which indicate that the Adomian approximations of up to six terms in its partial sum yield effective and robust estimates for Conditions I-VII. This is further validated when examining the accuracy of these partial sums with the numerical solutions, where the accuracies range between O 10 −6 and O 10 −4 . We also note the comparisons between the explicit solutions generated for Conditions I-VII and the numerical solutions, where these deviations are also miniscule. Table 2.: Summary of convergence trend (for N = 2, 4 and 6) and accuracy (for N = 6) via integral squared error E(N ) calculations for Conditions I-VII. E c (N = 2) E c (N = 4) E c (N = 6) E ex (N = 6)Ê(N = 6)Ê ex Minimum 3.3 × 10 −3 1.5 × 10 −6 8.2 × 10 −11 1.6 × 10 −12 3.1 × 10 −6 3.1 × 10 −6 Maximum 6.5 × 10 −3 2.5 × 10 −4 7.0 × 10 −6 1.3 × 10 −7 2.1 × 10 −4 2.1 × 10 −4 Figures 2 through 3 present the behaviour of the ADM partial sums of u, v, and h (for N = 6) along with Conditions II and IV (section 3) are used as examples. In each case we note the direct relationship between the initial conditions (figures 2-3 part (a)) and a temporal snapshot of the behaviour of the corresponding partial sums at t = 1 (figures 2-3 part (b)), which illustrates the velocity vector field u = u (x, y, 1) , v (x, y, 1) over the contour representing the free surface height h (x, y, 1). Figure 2(a) shows the initial zero velocity over constant free surface gradient η x = 10 −4 , which corresponds to the initial conditions represented by (15). Figure 2(b) confirms the temporal behaviour where we note the behaviour of u over the contour, which is analogous to the exact solutions described in equation (25). However, in figure 2(b) we also observe the rotating velocity field u over the contour illustrating the behaviour of inertial geostrophic oscillations. These effects are not only driven by the pressure gradient due to variations in the free surface height but also due to the Coriolis force, which are also noticed analytically when constructing the ADM decompositions. These confirmations continue in figure 3, where part (a) illustrates the behaviour of the initial velocity u 0 = u (x, y, 0) , v (x, y, 0) with respect to the initial free surface height h 0 = h (x, y, 0). Figure 3 also shows the correlation between the initial conditions and analytical solutions to Condition IV while also illustrating the effects of anticyclonic vortices with finite escape time as shown in figure 3 (b). Specifically, we note the clockwise orientation of u that is consistent with the behaviour of anticyclonic vortices which are valid for t ∈ 0, π/ 2f . (a) (b) Figure 3.: Velocity vector field u (arrow) and free surface height h (contour) behaviour for Condition IV: (a) initial condition at t = 0, (b) partial sum approximation based on ADM (with N = 6) at t = 1. Parameters used include F = 1,f = 0.5, D 0 = 0, and h 0 = 10 −4 (Colour online). Discussion This work employs Adomian decomposition method (ADM) to the shallow water equations, where we made the following main contributions. First, we used these methods as reverse engineering mechanisms to develop theoretical connections between the ansatz formulations of previous works, such as Thacker (1981), Shapiro (1996) and Matskevich and Chubarov (2019), as well as develop a connection to the corresponding reduced systems of shallow water equations. Furthermore, we developed some novel families of closed-form solutions that respectively describe inertial oscillations and anticyclonic vortices with finite escape times over flat bottom topographies. We perform various numerical experiments against several cases that yielded relative errors between O 10 −6 and O 10 −4 . Our numerical visualizations further demonstrate the validity of our approach, which illustrate the consistency with the dynamic behaviour for several scenarios while also preserving the correlation between the physical parameters. Our study establishes the flexibility of these methods in terms of not only preserving the correlation of parameters with respect to the overall nonlinear physical behaviour but also alleviating the need to make restrictive assumptions like those based on the overall flow behaviour. Moreover, we illustrate that these techniques can be used to analytically deduce other aspects of shallow water phenomenon based on the characteristics of initial flows in which, to the best of our knowledge, this work is the first to explore these concepts. Therefore, some avenues of future work include extending these techniques to understand the implications of external forces such as the effects of bottom friction which are applicable to understanding various coastal effects such as impacts from tsunamis. Another area of research is extending this framework to analyse practical bottom topographies and shocks, which will consider bottom terrains that extend beyond those of parabolic shapes. Figure 1 . 1: Illustration of a thin layer of incompressible flow under the Earth's rotation described by rotating shallow-water equations with idealised bottom topography. Theorem 3. 7 : 7Given a flat bottom topography, let {u n (x, y, t)}, {v n (x, y, t)}, {h n (x, y, t)} be the sequence of decomposed functions of u, v, and h, defined by (5) (for n ∈ N). If the initial conditions u 0 (x, y), v 0 (x, y), h 0 (x, y) are defined as (31a-d), then the solutions of u, v, and h have the same property where u(x, y, t) =f y, Figure 2 . 2: Velocity vector field u (arrow) and free surface height h (contour) behaviour for Condition II: (a) initial condition at t = 0 and (b) partial sum approximation based on ADM (with N = 6) at t = 1. Parameters used include F = 1,f = 0.5, D 0 = 0, and η x = 10 −4 (Colour online). Table 1 . 1: Summary of evaluation parameters, initial conditions, and applicable exact solutions used to validate Conditions I-VII. 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{'abstract': 'Analysing two-dimensional shallow water equations with idealised bottom topographies have many applications in the atmospheric and oceanic sciences; however, restrictive flow pattern assumptions have been made to achieve explicit solutions. This work employs the Adomian decomposition method (ADM) to develop semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters while capturing the nonlinear phenomenon. Furthermore, we exploit these techniques as reverse engineering mechanisms to develop key connections between some prevalent ansatz formulations in the open literature as well as derive new families of exact solutions describing geostrophic inertial oscillations and anticyclonic vortices with finite escape times. Our semi-analytical evaluations show the promise of this approach in terms of providing robust approximations against several oceanic variations and bottom topographies while also preserving the direct correlation between the physical parameters such as the Froude number, the bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours. Our numerical validations provide additional confirmations of this approach while also illustrating that ADM can also be used to provide insight and deduce novel solutions that have not been explored, which can be used to characterize various types of geophysical flows.', 'arxivid': '2301.02957', 'author': ['Chang Liu \nDepartment of Applied Mathematics and Statistics\nJohns Hopkins University Baltimore\n21218MDUSA\n\nDepartment of Physics\nUniversity of California\n94720Berkeley BerkeleyCAUSA\n', 'Antwan D Clark \nDepartment of Applied Mathematics and Statistics\nJohns Hopkins University Baltimore\n21218MDUSA\n'], 'authoraffiliation': ['Department of Applied Mathematics and Statistics\nJohns Hopkins University Baltimore\n21218MDUSA', 'Department of Physics\nUniversity of California\n94720Berkeley BerkeleyCAUSA', 'Department of Applied Mathematics and Statistics\nJohns Hopkins University Baltimore\n21218MDUSA'], 'corpusid': 255546635, 'doi': '10.1080/03091929.2023.2169283', 'github_urls': [], 'n_tokens_mistral': 20670, 'n_tokens_neox': 17951, 'n_words': 10039, 'pdfsha': '07f6e912fb06ea5dfc52fbdc3ae9fdd539d215df', 'pdfurls': ['https://export.arxiv.org/pdf/2301.02957v3.pdf'], 'title': ['Semi-analytical solutions of shallow water waves with idealised bottom topographies', 'Semi-analytical solutions of shallow water waves with idealised bottom topographies'], 'venue': ['Month']}

arxiv

On the q-Laplace transform in the non-extensive statistical physics 23 Jan 2013 (Dated: December 11, 2013) Sang Chung Department of Physics and Research Institute of Natural Science College of Natural Science Gyeongsang National University 660-701JinjuKorea On the q-Laplace transform in the non-extensive statistical physics 23 Jan 2013 (Dated: December 11, 2013) In this paper, q-Laplace transforms related to the non-extensive thermodynamics are investigated by using the algebraic operation of the non-extensive calculus. The deformed simple harmonic problem is discussed by using the q-Laplace transform. * Electronic address: mimip4444@hanmail.net I. INTRODUCTION Boltzman-Gibbs statistical mechanics shows how fast microscopic physics with short-range interaction has as effect on much larger space-time scale. The Boltzman-Gibbs entropy is given by S BG = −k W i=1 p i ln p i = k W i=1 ln 1 p i(1) where k is a Boltzman constant, W is a total number of microscopic possibilities of the system and p i is a probability of a given microstate among W different ones satisfying W i=1 p i = 1. When p 1 = 1 W , we have S BG = k ln W . Boltzman-Gibbs theory is not adequate for various complex, natural, artificial and social system. For instance, this theory does not explain the case that a zero maximal Lyapunov exponent appears. Typically, such situations are governed by power-laws instead of exponential distributions. In order to deal with such systems, the non-extensive statistical mechanics is proposed by C.Tsallis [1,2]. The non-extensive entropy is defined by S q = k(Σ W i p q i − 1)/(1 − q)(2) The non-extensive entropy has attracted much interest among the physicist, chemist and mathematicians who study the thermodynamics of complex system [3]. When the deformation parameter q goes to 1, Tsallis entropy (2) reduces to the ordinary one (1). The non-extensive statistical mechanics has been treated along three lines: 1. Mathematical development [4,5,6] 2. Observation of experimental behavior [7] 3. Theoretical physics ( or chemistry) development [8] The basis of the non-extensive statistical mechanics is q -deformed exponential and logarithmic function which is different from those of Jackson's [9]. The q-deformed exponential and q-logarithm of non-extensive statistical mechanics is defined by [10] ln q t = t 1−q − 1 1 − q , (t > 0) (3) e q (t) = (1 + (1 − q)t) 1 1−q , (x, q ∈ R)(4)where 1 + (1 − q)t > 0. From the definition of q-exponential and q-logarithm, q-sum, q-difference, q-product and q-ratio are defined by [ 5,6] x ⊕ y = x + y + (1 − q)xy x ⊖ y = x − y 1 + (1 − q)y x ⊗ y = [x 1−q + y 1−q − 1] 1 1−q x ⊘ y = [x 1−q − y 1−q + 1] 1 1−q (5) It can be easily checked that the operation ⊕ and ⊗ satisfy commutativity and associativity. For the operator ⊕, the identity additive is 0, while for the operator ⊗ the identity multiplicative is 1. Indeed, there exist an analogy between this algebraic system and the role of hyperbolic space in metric topology [11]. Two distinct mathematical tools appears in the study of physical phenomena in the complex media which is characterized by singularities in a compact space [12]. For the new algebraic operation, q-exponential and q-logarithm have the following properties: ln q (xy) = ln q x ⊕ ln q y e q (x)e q (y) = e q (x ⊕ y) ln q (x ⊗ y) = ln q x + ln q y e q (x) ⊗ e q (y) = e q (x + y) ln q (x/y) = ln q x ⊖ ln q y e q (x)/e q (y) = e q (x ⊖ y) ln q (x ⊘ y) = ln q x − ln q y e q (x) ⊘ e q (y) = e q (x − y)(6) From the associativity of ⊕ and ⊗, we have the following formula : t ⊕ t ⊕ t ⊕ · · · ⊕ t n times = 1 1 − q {[1 + (1 − q)t] n − 1} (7) t ⊗ n = t ⊗ t ⊗ t ⊗ · · · ⊗ t n times = [nt 1−q − (n − 1)] 1 1−q (8) II. Q-LAPLACE TRANSFORM In this section, we find the q-Laplace transform related to the non-extensive thermodynamics. From the relation e q (x)e q (y) = e q (x ⊕ y), the q-analogue of e nx , (n ∈ Z) is given by e q (n ⊙ t) = [e q (t)] n = e q 1 1 − q [(1 + (1 − q)t) n − 1] = (1 + (1 − q)t) n 1−q(10) Then we have e q (0 ⊙ t) = 1 (11) and the inverse of e q (n ⊙ t) is e q ((−n) ⊙ t). For this adoption, q-Laplace kernel is defined by e q ((−s) ⊙ t) = [e q (t)] −s = e q 1 1 − q [(1 + (1 − q)t) −s − 1] = (1 + (1 − q)t) − s 1−q(12) Therefore q-Laplace transform is defined by L s (F (t)) = ∞ 0 [e q (t)] −s F (t)dt, (s > 0)(13) Limiting q → 1, the eq.(13) reduces to an ordinary Laplace transform. Form now on we assume that s is sufficinetly large. Since, for two functions F (t) and G(t), for which the integral exist L s (aF (t) + bG(t)) = aL s (F (t)) + bL s (G(t)),(14) the q-Laplace transform is linear. For F (t) = t N , (N = 0, 1, 2, · · ·) , we have the following result. Theorem 1 For sufficiently large s , when q < 1, the following holds: L s (t N ) = N ! (s; 1 − q) N +1 (15) where (a; Q) n = 1 (n = 0 ) n k=1 (a − kQ) (n ≥ 1) (16) Proof. Let us assume that the eq.(15) holds for t N . Then, L s (t N +1 ) = ∞ 0 [e q (t)] −s t N +1 dt = ∞ 0 (1 + (1 − q)t) − s 1−q t N +1 dt = (1 + (1 − q)t) − s 1−q +1 1 − q − s t N +1 ∞ 0 + N + 1 s − (1 − q) ∞ 0 (1 + (1 − q)t) − s−(1−q) 1−q t N dt = N + 1 s − (1 − q) L s−(1−q) (t N +1 ) = (N + 1)N ! (s − (1 − q))(s − (1 − q); 1 − q) N +1 = (N + 1)! (s; 1 − q) N +2(17) In a similar way, we can obtain the q-Laplace transform for e q (a ⊙ t) which is given by e q (a ⊙ t) = [e q (t)] a Theorem 2 For sufficiently large s , when q < 1, the following holds: L s (e q (a ⊙ t)) = 1 s − a − (1 − q) (18) Proof. It is trivial. In order to obtain q-Laplace transform for the trigonometric function, we need q-analogue of Euler identity. The q-Euler formula is given by e q (ia ⊙ t) = C q (a ⊙ t) + iS q (a ⊙ t),(19) where q-cosine and q-sine functions are defined by C q (a ⊙ t) = cos( a 1 − q ln(1 + (1 − q)t)) S q (a ⊙ t) = sin( a 1 − q ln(1 + (1 − q)t))(20) and we used the following identity. p i = e i ln p = cos ln p + i sin ln p(21) Indeed, q-cosine and q-sine functions can be expressed in terms of q-exponential as follows: C q (a ⊙ t) = 1 2 [e q (ia ⊙ t) + e q ((−ia) ⊙ t) S q (a ⊙ t) = 1 2i [e q (ia ⊙ t) − e q ((−ia) ⊙ t)(22) Then we have the q-Laplace transform for q-sine and q-cosine functions : Theorem 3 For sufficiently large s , when q < 1, the following holds: L s (C q (a ⊙ t)) = s − (1 − q) (s − (1 − q)) 2 + a 2 L s (S q (a ⊙ t)) = a (s − (1 − q)) 2 + a 2(23) Proof. It is trivial. The eq.(23) can be written in terms of the ordinary sine and cosine functions : L s (cos[ a 1 − q ln(1 + (1 − q)t)]) = s − (1 − q) (s − (1 − q)) 2 + a 2 L s (sin[ a 1 − q ln(1 + (1 − q)t)]) = a (s − (1 − q)) 2 + a 2(24) The q-sine function and q-cosine function have the following zeros: S q (1 ⊙ t n ) = S q (t n ) = 0, C q (1 ⊙ u n ) = C q (u n ) = 0,(25) where t n = ln q e nπ , u n = ln q e (n+ 1 2 )π , n ∈ Z From the zeros of the q-sine function and q-cosine function, we have the following Theorem: Theorem 4 For sufficiently large s , when q < 1, the following holds: S q (t) = t ∞ j=1 1 − t ln q e jπ 1 − t ln q e −jπ (27) C q (t) = 1 − t ln q e −π/2 ∞ j=1 1 − t ln q e (j+1/2)π 1 − t ln q e −(j+1/2)π(28) Proof. From the zeros of the q-sine function , we can set S q (t) t = A ∞ j=1 1 − t ln q e jπ 1 − t ln q e −jπ Because lim t→0 Sq(t) t = 1, we have A = 1, which proves the eq.(27). Similarly we can easily prove the eq.(28). Theorem 4 can be also written as follows : Theorem 5 For sufficiently large s , when q < 1, the following holds: S q (t) = t ∞ j=1   1 + (1 − q)t − (1 − q)t 2 sinh( j(1−q)π 2 ) 2   (29) C q (t) = 1 − t ln q e −π/2 ∞ j=1 1 + (1 − q)t − (1 − q)t 2 sinh( 1 2 (1 − q)(j + 1/2)π) 2 (30) Proof. It is trivial from the formula cosh x − 1 = 2 sinh 2 x 2 . III. Q-LAPLACE TRANSFORM AND DIFFERENTIAL EQUATION Now we discuss the q-differential equation. The main purpose of q-Laplace transform is in converting q-differential equation into simpler forms which may be solved more easily. Like the ordinary Laplace transform , we can compute the q-Laplace transformation of derivative by using the definition of the q-Laplace trnsform, which is given by L s (F ′ (t)) = sL s+1−q (F (t)) − F (0)(31) An extension gives L s (F ′′ (t)) = s(s + 1 − q)L s+2(1−q) (F (t)) − sF (0) − F ′ (0)(32) Generally, we have following theorem : Theorem 6 For sufficiently large s , when q < 1, the following holds: L s (F (n) (t)) = [s; 1 − q] n L s+n(1−q) (F (t)) − n−1 i=0 [s; 1 − q] n−1−i F (i) (0),(33) where F (0) (0) = F (0) and [a; Q] n = 1 (n = 0 ) n k=1 (a + kQ) (n ≥ 1) (34) Proof. Let us assume that the eq.(34) holds for n. Then, L s (F (n+1) (t)) = ∞ 0 (1 + (1 − q)t) − s 1−q F (n+1) (t)dt = sL s+1−q (F (n) (t)) − F (n) (0) = s{[s + 1 − q; 1 − q] n L s+(n+1)(1−q) (F (t)) − n−1 i=0 [s + 1 − q; 1 − q] n−1−i F (i) (0)} − F (n) (0) = [s; 1 − q] n+1 L s+(n+1)(1−q) (F (t)) − n i=0 [s; 1 − q] n−i F (i) (0)(35) We have another formula for the q-Laplace transform of derivative as follows: L s (F ′ (t)) = sL s F (t) 1 + (1 − q)t − F (0) (36) An extension gives L s (F ′′ (t)) = s(s + 1 − q)L s F (t) (1 + (1 − q)t) 2 − sF (0) − F ′ (0)(37) Generally, we have the following theorem: Theorem 7 For sufficiently large s , when q < 1, the following holds: L s (F (n) (t)) = [s; 1 − q] n L s F (t) (1 + (1 − q)t) n − n−1 i=0 [s; 1 − q] n−1−i F (i) (0) (38) Proof. It is not hard to prove Theorem 7. Comparing Theorem 6 with Theorem 7, we have the following Lemma: Lemma 8 For sufficiently large s , when q < 1, the following holds: L s F (t) (1 + (1 − q)t) n = L s+n(1−q) (F (t))(39) Proof. It is trivial. With the help of q-Laplace transform of the derivative, we can solve some differential equation. It is worth noting that e q (t) is not invariant under the ordinary derivative, instead it obeys d dt (e q (t)) = 1 1 + (1 − q)t e q (t)(40) Now consider the following differential equation: F ′ (t) = F (t) 1 + (1 − q)t , F (0) = 1(41) It is evident that e q (t) is a solution of the eq.(40). Let us consider the vertical motion of a body in a resisting medium in which there again exists a retarding force proportional to the velocity. Let us consider that the body is projected downward with zero initial velocity v(0) = 0 in a uniform gravitational field. The equation of motion is then given by m d dt v = mg − kv(t)(42) This equation is not solved by using the q-Laplace transform , instead we solve the following equation : m d dt v = mg − k v(t) 1 + (1 − q)t(43) The solution of the eq.(43) is then given by v(t) = g 1 − q + k m (1 + (1 − q)t − [e q (t)] − k m )(44) Similarly, we can modify the harmonic problem whose equation of motion is given by m d dt 2 x(t) = −mw 2 x(t) (1 + (1 − q)t) 2 ,(45) where x(0) = A, d dt x (0) = 0. The solution of the eq.(45) is then given by x(t) = A(1 + (1 − q)t) 2    C 3−q 2 w 2 − ( q − 1 2 ) 2 ⊙ t) + q − 1 w 2 − ( q−1 2 ) 2 S 3−q 2 w 2 − ( q − 1 2 ) 2 ⊙ t   (46) The eqs.(43)and (45) seem to be too artificial due to the factor 1+(1−q)t. Instead, we can introduce the q-derivative [6] instead of the ordinary time derivative as follows: D t F (t) = lim s→t F (t) − F (s) t ⊖ s = [1 + (1 − q)t] dF dt The Leibniz rule for q-derivative is as follows: D t [F (t)G(t)] = D t [F (t)]G(t) + F (t)D t [G(t)](47) Then the eq.(43) is replaced as follows : mD t v = mg − kv(t)(48) The solution of the eq.(48) is then given by v(t) = mg k (1 − [e q (t)] − k m )(49) Similarly, the eq.(45) is replaced as follows: mD 2 t x(t) = −mw 2 x(t)(50) Using the q-Laplace transform, we get the solution of the eq.(50) : x(t) = AC q (w ⊙ t)(51) Obtaining these solutions, we used the following theorem: Theorem 9 For sufficiently large s , when q < 1, the following holds: L s ((1 + (1 − q)t) n F (n) (t)) = [s − (n + 1)(1 − q); 1 − q] n L s (F (t)) − n−1 i=0 [s − (n + 1)(1 − q); 1 − q] n−1−i F (i) (0) (52) Proof. Let us assume that the eq.(52) holds for n. Then, L s ((1 + (1 − q)t) n F (n+1) (t)) = ∞ 0 (1 + (1 − q)t) − s 1−q +n+1 F (n+1) (t)dt = (s − (n + 1)(1 − q))L s ((1 + (1 − q)t) n F (n) (t)) − F (n) (0) = (s − (n + 1)(1 − q)){[s − (n + 1)(1 − q); 1 − q] n L s (F (t)) − n−1 i=0 [s − (n + 1)(1 − q); 1 − q] n−1−i F (i) (0)} − F (n) (0) = [s − (n + 2)(1 − q); 1 − q] n+1 L s (F (t)) − n i=0 [s − (n + 2)(1 − q); 1 − q] n−i F (i) (0)(53) The eq.(50) is also obtained by using the variational method whose Lagrangian is given by L = dt 1 2 (D t x) 2 − U (x)(54) The equation of motion is then given by D t ∂L ∂(D t x) − ∂L ∂x = 0,(55) where the momentum p is defined by p = ∂L ∂(D t x)(56) For the harmonic potential U = 1 2 mw 2 x 2 , we have the eq.(50). This equation is rewritten by (1 + (1 − q)t) 2ẍ + (1 − q)(1 + (1 − q)t)ẋ = −w 2 x(57) Replacing η = 1 1 − q ln(1 + (1 − q)t),(58) the eq.(57) is then as follows : ∂ 2 x ∂η 2 = −w 2 x(η)(59) Solving the eq.(59), we have x(t) = A cos wη = AC q (w ⊙ t)(60) Here, let us investigate the times t n when a body goes back to the initial position. This time is determined by t n = ln q e 2πn w , (t = 0, 1, 2, · · ·) (61) When q < 1 and w > 0, we have the following inequality: t n+1 − t n > t n − t n−1 (62) Thus, the time when a body goes back to the initial position keeps increasing. IV. CONCLUSION In this paper, we used the algebraic operation and differential calculus related to the non-extensive thermodynamics to investigate the q-Laplace transform. We used the q-Laplace transform to solve some differential equation such as harmonic oscillator problem. We think that this work will be applied to some q-differential equation which might appear in the study of the non-extensive statistical mechanics. We hope that these work and their related topics will be clear in the near future. . C Tsallis, J.Stat.Phys. 52479C.Tsallis, J.Stat.Phys. 52 (1988) 479. . E Curado, C Tsallis, J.Phys. A. 2469E.Curado, C.Tsallis, J.Phys. A 24 (1991) L69. . A Cho, Science. 2971268A.Cho, Science 297 (2002) 1268. . S Plastino, Science. 300250S. Plastino, Science 300 (2003) 250. . L Nivanen, A Le Mehaute, Q Wang, Rep.Math.Phys. 52437L.Nivanen, A.Le Mehaute , Q.Wang, Rep.Math.Phys.52 (2003) 437 . E Borges, Physica A. 34095E. Borges, Physica A 340 (2004) 95. . S Abe, A , Science. 300249S.Abe, A.Rajagopal, Science 300 (2003) 249. . V Latora, A Rapisarda, A Robledo, Science. 300250V.Latora, A,Rapisarda, A.Robledo, Science 300 (2003) 250. . F Jackson, Mess.Math. 3857F. Jackson, Mess.Math. 38 , 57 (1909). . C Tsallis, Quimica Nova. 17479C.Tsallis, Quimica Nova 17 (1988) 479. An Introduction to hyperbolic geometry. A Beardon, Oxford University PressNew YorkA.Beardon, An Introduction to hyperbolic geometry, Oxford University Press, New York, (1991). A Mehaute, R Nigmatullin, L Nivanen, Fleches du temps et geometrie fractale. ParisA.Mehaute, R.Nigmatullin, L.Nivanen, Fleches du temps et geometrie fractale. Hermes, Paris, (1998).

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{'abstract': 'In this paper, q-Laplace transforms related to the non-extensive thermodynamics are investigated by using the algebraic operation of the non-extensive calculus. The deformed simple harmonic problem is discussed by using the q-Laplace transform. * Electronic address: mimip4444@hanmail.net', 'arxivid': '1301.5480', 'author': ['Sang Chung \nDepartment of Physics and Research Institute of Natural Science\nCollege of Natural Science\nGyeongsang National University\n660-701JinjuKorea\n'], 'authoraffiliation': ['Department of Physics and Research Institute of Natural Science\nCollege of Natural Science\nGyeongsang National University\n660-701JinjuKorea'], 'corpusid': 119266895, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6385, 'n_tokens_neox': 5717, 'n_words': 3070, 'pdfsha': 'e0d64280d8d8be79448dad003f8e6b11c3332e13', 'pdfurls': ['https://arxiv.org/pdf/1301.5480v1.pdf'], 'title': ['On the q-Laplace transform in the non-extensive statistical physics', 'On the q-Laplace transform in the non-extensive statistical physics'], 'venue': []}

arxiv

Crystal orientation and grain size: do they determine optoelectronic properties of MAPbI 3 perovskite? Loreta A Muscarella Center for Nanophotonics AMOLF Science Park 1041098 XGAmsterdamThe Netherlands Eline M Hutter Center for Nanophotonics AMOLF Science Park 1041098 XGAmsterdamThe Netherlands Sandy Sanchez Laboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL) Station 61015LausanneSwitzerland Christian D Dieleman Center for Nanophotonics AMOLF Science Park 1041098 XGAmsterdamThe Netherlands Tom J Savenije Department of Chemical Engineering Delft University of Technology Van der Maasweg 92629 HZDelftThe Netherland Anders Hagfeldt Laboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL) Station 61015LausanneSwitzerland Michael Saliba Institute of Materials Science Technical University of Darmstadt Alarich-Weiss-Strasse 2D-64287DarmstadtGermany Bruno Ehrler Center for Nanophotonics AMOLF Science Park 1041098 XGAmsterdamThe Netherlands Crystal orientation and grain size: do they determine optoelectronic properties of MAPbI 3 perovskite? 1 2 It is thought that growing large, oriented grains of perovskite can lead to more efficient devices.We study MAPbI3 films fabricated via Flash Infrared Annealing (FIRA) consisting of highly oriented, large grains. Domains observed in the SEM are often misidentified with crystallographic grains, but SEM images don't provide diffraction information. We measure the grain size, crystal structure and grain orientation using Electron Back-Scattered Diffraction (EBSD) and we study how these affect the optoelectronic properties as characterized by local photoluminescence (PL) and time-resolved microwave conductivity measurements (TRMC).We find a spherulitic growth yielding large (tens of µm), highly oriented grains along the(112)and(400)planes in contrast to randomly oriented, smaller (400 nm) grains observed in films fabricated via conventional antisolvent (AS) dripping. We observe a local enhancement and shift of the photoluminescence emission at different regions of the FIRA clusters, but these can be explained with a combination of light-outcoupling and self-absorption. We observe no effect of crystal orientation on the optoelectronic properties. Additionally, despite a substantial difference in grain size between our FIRA sample and a conventional AS sample, we find similar photoluminescence and charge carrier mobilities and lifetime for the two films. These findings show that the optoelectronic quality is not necessarily related to the orientation and size of crystalline domains in perovskite films indicating that fabrication requirements may be more relaxed for perovskites. Introduction Hybrid perovskites have recently gathered significant attention due to the high efficiency of perovskite-based solar cells and other optoelectronic devices 1,2 . One of the most surprising properties of these materials is that the performance is very tolerant to different methods of fabrication 3,4,5 , different compositions 6,7 , and chemical treatments 8,9,10 . This is reflected in high photoluminescence quantum efficiencies (PLQE) 11,12 , which is a measure of the fraction of radiative versus non-radiative decay, and therefore a direct measurement of the optical quality. In solar cells, the PLQE, for example, is directly related to the open-circuit voltage 13 . In practice, PLQE is reduced by the presence of defects 13,14 which are often related to chemical impurities such as interstitials, vacancies, dangling bonds 15,16 or defects on the surface and grain boundaries 17,18,19 . Both bulk and surface defects have been extensively studied in perovskites 20,21,22,23 , and efficient passivation strategies are now routinely employed to achieve high LED and solar cell efficiencies 24,25,26 . In addition, chemical methods (e.g. Lewis bases 21 or chloride-based additives 27,6 ) in the perovskite precursor are often applied to grow larger grains which has been thought to suppress non-radiative recombination pathways by reducing the number of grain boundaries 28,29,30 . Furthermore, these changes in the synthesis route affect the crystal growth and therefore the preferred crystallographic orientations 31,8,32 . However, it is unclear if the changes in grain size and orientation obtained by these treatments cause the improved optoelectronic properties, or if these are mainly related to passivation effects from the additives. Crystallographic orientation and their relation to the photoluminescence and other properties have not been studied in thin films so far because spatial resolution of the crystallographic parameters was lacking. The morphological "grain" observed in SEM images does not necessarily correspond to a crystallographic grain; so additional information is needed in order to relate the grain size with optoelectronic properties. 4 Here we use EBSD to measure size, orientation and rotation of crystallographic grains in polycrystalline MAPbI3 films with high spatial resolution. We study a MAPbI3 thin film where crystallization is induced by FIRA, a low cost and rapid synthesis method 33,34 . ³⁵These films exhibit highly oriented (112) and (400) planes with large grain size (tens of micrometers). We find that the growth is spherulitic, i.e. needle-like arrays, yielding ~100 micrometer sized clusters that consist of radially grown grains. With EBSD mapping we find that the two crystal orientations in the FIRA films are well-separated in pairs in the large clusters of grains. We compare the PL from these clusters and find that PL intensity and spectrum is the same for the two crystal orientations. These results suggest that the crystallographic orientation does not govern the optoelectronic quality of perovskite thin films. Furthermore, we find enhanced emission and a red-shift at the cluster boundaries and at the nucleation sites, which we attribute to favorable light-outcoupling and self-absorption. Finally, we compare the FIRA sample to one where crystallization is induced by the conventional AS dripping method, from the same precursor solution. This method produces sub-micron grains with random orientation. Both samples show comparable charge carrier mobility and lifetime demonstrating that these properties are not necessarily determined by the grain size, at least for grains above a few hundred nanometers. Results and Discussion To study the relation between the perovskite crystal orientation and its optoelectronic properties we first synthesize MAPbI3 on ITO via FIRA wherein the spin coated perovskite film is annealed using a short (1.2 second), highly intense infrared illumination to induce nucleation, as previously reported 33 . Importantly, to decouple the effect of preferential orientation and the presence of additives on to the optoelectronic properties, we fabricate a highly oriented FIRA samples without the aid of additives. For comparison we also fabricate a 5 sample by the AS method where chlorobenzene is rapidly poured the liquid precursor while spin coating. Both the FIRA and AS sample were fabricated from the same precursors under identical conditions, and thus the only difference is the crystallization process. We initially characterize the morphology of both samples using scanning electron microscopy (SEM) as shown in Figure 1a,b and Figure S1. SEM images show a dramatic difference in cluster size from the antisolvent method (100 nm -2 µm) compared to FIRA (~100 µm). Often, these clusters seen in SEM images are assigned to grains. In crystallography, the term "grain" is defined by a coherently diffracting domain of solid-state matter which has the same structure as a single crystal 36 . Therefore, from SEM images alone it is not possible to define the apparent domains as crystallographic grains because diffraction information is not measured. For this reason, we use "clusters" to describe the large perovskite domains shown in SEM images. We analyze the bulk crystal structure of the two systems deposited on ITO using X-Ray Diffraction (XRD). The AS sample shows a tetragonal XRD pattern where peaks from (110), (112), (220), (310) planes arise from the background 37,38 . In contrast, the FIRA sample shows a strong preferential orientation along the (112) and (400) planes (Figure 1c). A cut-off of the primitive tetragonal cell and the planes which show the highest diffraction peaks are shown next to the diffraction patterns. in agreement with the SEM images. We use polarized light microscopy to find how the two orientations observed in the XRD are spatially distributed. Figure 2b shows the presence of paired oriented domains within FIRA films for all the clusters (see also Figure S2) originated by the change in polarization of the incident polarized light caused by the two different refractive indices in the different crystal directions. This is a indication of spherulitic growth (schematically shown in Figure 2c) via non-crystallographic branching 39 typical for many polymeric materials 40 and inorganic salts 41 . Figure 2Interestingly, there also seems to be a common direction to all clusters, suggesting a global effect from temperature or strain gradient. In general, spherulitic growth requires the use of a saturated solution, high viscosity, and slow crystallization. This growth is also catalyzed by the presence of impurities 39 or strain in the material. In this case, MAPbI3 and the ITO (or quartz) show a substantial difference in the thermal expansion coefficient (αMAPbI3 =6.1 × 10 −5 K −1 , αITO = 0.85 × 10 −5 K −1 , αglass = 0.37 × 10 −5 K −1 42 ), which has been shown to be the origin of strain during the cooling process after the thermal annealing 43 . Thus, strain can be considered as a factor inducing spherulitic growth in our system. To study the crystal grains and their orientation with high spatial resolution (10 nm) we use EBSD which is commonly used for investigations of grains in metal alloys 44 , strain 45 , and the nature of grain boundaries 46 . EBSD is a SEM-based technique where the incident electron beam, with a suitable voltage and current, interacts with a crystalline material and electron backscattered patterns, also called Kikuchi patterns, are produced by incoherent wide-angle electron scattering (thermal diffuse scattering) from a specimen. A scheme of the setup is depicted in Figure 3a. For a full description of EBSD measurements see Supplementary Information S1. The main limitation which has restricted its application in the perovskite field is related to the high electron current needed for the phosphor screen to collect a sufficiently large signalto-noise ratio. In case of perovskites containing organic cations, the use of a current of a few nA can already be damaging to the material. Recently, Adhyaksa et al. 47 have pioneered the application of EBSD for MAPbBr3 using a direct electron detector which allows for low accelerating voltage of 5 kV instead of 30 kV, and low sample currents of pA instead of nA in conventional systems. We use the same detection system to collect the Kikuchi patterns from our MAPbI3 films. The obtained Kikuchi patterns allow for the identification of grains, their size and shape and the nature of boundaries between them. By fitting the patterns, we can identify crystal phase, grain orientation and rotation, as described in Supplementary Information S1. Importantly, since with EBSD diffraction information is measured, we can distinguish clusters from grains and define the crystallographic grain size of MAPbI3 fabricated via FIRA and AS. Figure S4 for orientation distribution). As the axes-dependent orientation is different for the three directions, we can now deduce that the 83% of the grains have an area between 0-30 µm 2 (see Figure S5) as shown in the EBSD, which does not coincide with the size clusters shown in SEM (Figure 3g). Furthermore, the orientation in the x-and y-direction is aligned along the growth of the initial needles, and the subsequent space-filling branches grow off these needles. The grain size obtained from EBSD for the AS sample is much smaller compared to the FIRA sample. with more than 90% of the grains smaller than 1 µm 2 (see Figure S5). Orientation maps along xand y-direction for the AS sample are shown in Figure S6. MAPbI3 has an anisotropic, tetragonal crystal structure, properties like trap-state density have been shown to be dependent upon crystal orientation 48,49 . Here, we study the optoelectronic properties of the two well-characterized and spatially separated orientations shown by FIRA sample using spatially resolved PL. We measure the PL intensity using a confocal imaging microscope using 405nm laser as excitation source with a power density of 0.23 W/cm 2 . As the two orientations on the sample are spatially well-separated, we can map any difference in PL emission between them. We measure a large area of the FIRA and AS samples including a whole FIRA cluster (Figure 4a and Figure S7 for a larger area). In the AS sample the PL intensity is relatively homogeneously distributed across the measured region, varying from cluster to cluster, consistent with many other works 50,51 . In contrast, the PL map of the FIRA sample shows an enhancement in intensity of two to six times at the cluster boundaries and at the nucleation site where the spherulitic growth is initiated. Enhancement in PL is often attributed to the presence of less non-radiative recombination, but light outcoupling must be taken in account as well. From AFM measurements, the AS sample shows only minor height variation between the center and the rest of the grain (RMS roughness 79 nm, Figure 4b). On the contrary, the FIRA sample shows significant height variation at the cluster boundaries (CBs), where boundaries from different clusters impinge with each other, and at the nucleation point (Figure 4b and in Figure S8). In these regions the film is around 400 nm to 1000 nm thicker compared to the interior of the cluster leading to a much larger roughness (RMS roughness 131 nm across the whole grain, between 85-65 nm in the interior of the cluster and 145-200 nm at the cluster boundaries). We note that FIRA clusters also show some local height variation in the interior part although less pronounced compared to the CBs. Thus, the rough nature of the boundary can favor light outcoupling, as shown on patterned perovskite surfaces 52 . We corroborate the assignment that the PL efficiency is constant across the FIRA clusters by measuring PL lifetime maps. The lifetime is constant across the cluster and CBs 12 (with the exception of a few local hotspots) and hence there is no difference in radiative versus non-radiative rates at the darker and brighter regions ( Figure S9). The PL enhancement observed can hence fully be explained by better light outcoupling. Next to these differences, there is no trend across the cluster that would correspond to the two different grain orientations. Thus, there is no direct correlation between the crystal orientation and the PL intensity. The map of the PL peak position of the AS sample (Figure 4c) shows identical emission peak position for every grain. The FIRA sample shows a variation in emission wavelengths at different locations. The PL peak position at the CBs and nucleation point is red-shifted compared to the interior of the cluster (Figure 4c). Normalizing the PL spectra extracted from the boundary region of the map, we also observe asymmetric shape of the peak for the FIRA sample ( Figure S10). This shape, in combination with the red shift of the peak has been assigned to self-absorption process when light travels through the perovskite layers before being emitted. 53 . We calculate the spectra expected from the emitted light passing through different thicknesses of MAPbI3. We observe the red-shift at the boundaries and nucleation points corresponds to the light that has been transmitted through the 400nm -800nm excess material as compared to the cluster interior region before being emitted (horizontal scale bar in Figure 4c, see Supplementary Information S2 for details); this is in good agreement with the observed thickness variation. Again, we see no discrepancy for the regions that correspond to the two different, well-defined crystal orientations. Hence, the variation in PL emission intensity and wavelength cannot be correlated to the crystallographic orientation. In Figure 4d we plot PL spectra from five random regions of the two samples. Here we show that the interior region of the FIRA cluster shows a comparable PL intensity with the AS (FIRA Point 2&3). Importantly, this shows that the PL emission is not solely determined by the grain size (at least for grains >400 nm). Figure 5a,b shows the photoconductance ΔG as a function of time after 14 pulsed excitation of AS and FIRA sample, respectively. The product of the yield of free charges φ and their mobility Σμ (sum of electron and hole mobility) is derived from the maximum signal height (ΔGmax) which was divided by the fraction of absorbed photons for the two samples to take in account difference in absorption. We find a mobility of (15±3) cm 2 /(Vs) for the AS and (19±4) cm 2 /(Vs) for the FIRA sample, which is comparable to sample-to-sample variation. The charge carrier lifetime is obtained from the photoconductance decay. The decay of the photoconductance represents the immobilization of free charges due to trapping or recombination. For both systems, we find that the lifetime of charges is in the order of a few hundred nanoseconds. We observe a slight increase of the effective mobility in the FIRA sample compared to the AS sample likely related to the enlarged grain size 54 . This difference is relatively small, despite the difference in grain size between the FIRA (tens of microns) and the AS sample (hundreds of nanometers). This shows that grain size does not play a major role in charge carrier transport properties. We note that the TRMC measurement mostly probes the local conductivity (~50 nm, more details about the probing length are reported in Conclusion We have shown that the crystallographic orientation of MAPbI3 grains does not determine the optical and local electronic properties. We study a MAPbI3 thin film where crystallization is induced by FIRA. We apply EBSD to extract information about the microstructure of the perovskite thin film with high spatial resolution. The large FIRA clusters consist of grains that are tens of micrometer in size. They are highly oriented along the [112] and [100] direction perpendicular to the substrate. In comparison, a conventional sample fabricated via AS shows randomly oriented grains of hundreds of nanometers. We find increased PL intensity and a spectral shift in the FIRA sample compared to the AS sample, which we can be explained by roughness variations favoring light-outcoupling and selfabsorption. Finally, we investigate charge carrier dynamics and find comparable lifetime and a slight increase in effective mobility in the FIRA and AS samples. We hence conclude that neither the grain size (when larger than a few hundreds of nm) nor the grain orientation are the dominant factor determining the optoelectronic properties of perovskite thin films. This finding implies that efforts towards a more efficient perovskite device may need to focus on reducing defects within the bulk and at the interface as well as impurities within the materials rather than growing large, oriented grains. Experimental methods Thin film preparation The fabrication of the two systems (antisolvent dripping and FIRA method) is reported elsewhere 33 . Thin Film Characterization The X-Ray diffraction pattern of perovskite films deposited on ITO was measured using an Xray diffractometer, Bruker D2 Phaser, with Cu Kα 1.5406 Å as X-Ray source, 0.002° (2θ) as step size, 0.150 s as exposure time. A FEI Verios 460 instrument was used to obtain SEM images. Atomic force microscopy (AFM) measurements were performed on a Veeco Dimension 3100 (Bruker) in tapping mode. Optical microscope (Zeiss, AxioCam ICc 5) equipped with a 10x/0.2 objective EC Epiplan, polarizer and analyzer set at different angles was used for polarized optical microscopy image. We combine the optical microscope in reflection mode with two polarizers, one placed in the light path before the specimen, and a second one, called analyzer, between the objective lenses and eyepieces. Steady-state photoluminescence (PL) of samples deposited on quartz was measured with a home-built setup equipped with a 640 nm continuous-wave laser as source of excitation (PicoQuant LDH-D-C-640) at a power output of 1 mW. Two Thorlabs filters, a long pass, ET655LP, and a notch, ZET642NF, were used to remove the excitation laser from the signal. The PL was coupled into a fiber connected to an OceanOptics USB4000 spectrometer. An integration time of 300 ms was used for each measurement. For EBSD measurements, samples were deposited on ITO to avoid charging effects during the experiment. The detector used is a direct electron detector based on the Timepix sensor from Amsterdam Scientific Instruments (ASI). The best parameters for the scans were found to be 15 keV as voltage, 100 pA as current, 100 ms as exposure time and working distances between 12 mm and 10.1 mm. This translates to the application of 10 nAms electron dose per pixel which is around 10 3 times lower compared conventional measurement reducing sample damage. The step size was chosen depending the cluster size shown by the specific sample, 200 nm and 1 μm for the AS and FIRA sample, respectively. EBSD data were collected using EDAX OIM software and a Python script was used for image processing. The resulting Kikuchi patterns were indexed using tetragonal symmetry, I4/mcm, using 1-3° as degree of tolerance. Detailed procedure for fitting the Hough's space is reported in the Supplementary Note. Spatially resolved PL map was measured using a confocal imaging microscope (WITec alpha300 SR). A 405 nm laser diode (Thorlabs S1FC405) was used as excitation source where the PL intensity of the two MAPbI3 films was collected in reflection mode through a NA 0.9 objective using a spectrometer (UHTC 300 VIS, WITec) leading to a spatial resolution of 0.33 μm. The intensity was measured within the 700 to 840 nm emission wavelength range. A 488 nm LP filter was used to remove the excitation laser from the signal. The light collection was done from the same sample side as the excitation. The PL spectra were converted to the energy scale using a Jacobian transformation 55 . Time-Resolved Microwave Conductivity (TRMC) was measured on AS and FIRA sample deposited on quartz. The thin films were placed in a sealed resonance cavity inside a nitrogen-filled glovebox to avoid degradation due to air exposure. The samples were excited at 485 nm using a pulsed excitation (10 Hz) and the photoconductance probed at different excitation density. Neutral density filters were used to vary the intensity of the incident light. The ΔG signal rise is limited by the width of the laser pulse (3.5 ns FWHM) and the response time of the microwave system (18 ns). The slow repetition rate of the laser of 10 Hz ensures full relaxation of all photo-induced charges to the ground state before the next laser pulse hits the sample. The mobility is derived by the maximum signal height as the following equation 56 Conflict of Interest The authors declare no competing interests. Figure 1 . 1Morphology and crystal orientation of AS and FIRA sample deposited on ITO. SEM image of MAPbI3 crystallized with a) AS and b) FIRA, c) XRD pattern of AS and FIRA MAPbI3 with the corresponding unit cell cut along the (110) and (004) planes for the AS sample and along (112) and (400) for the FIRA sample. To follow the crystal growth of the FIRA film, we apply infrared annealing with varying pulse duration (Figure 2a). After 0.2 s of annealing we observe a needle-like crystal morphology. After 0.5 s the crystals have grown in a space-filling manner by branching from the parental needle until each domain impinges with neighboring domains resulting in straight boundaries. Optical microscopy of the final films (Figure 2b) shows large perovskite domains Figure 2 . 2Spherulitic growth mechanism of MAPbI3 results in paired oriented domains. a) SEM images of FIRA film after 0.2s and 0.5s of infrared annealing, b) Polarized microscopy image of the final FIRA film showing paired oriented perovskite domains, c) Schematic spherulitic growth of perovskite films after 0.2, 0.5, 1, and 1.2 s of FIRA annealing 8 Figure 3b , 3bc shows an overlay of the image quality (brightness, IQ) with Inverse PoleFigure (color, IPF) of the AS and FIRA system along the z-axis. The IQ maps the sharpness of the Kikuchi lines obtained from the EBSD measurement which gives qualitative indications about the crystallinity of the material, topographic effects (e.g. roughness), strain of the microstructure9 and grain boundaries (for more details see Supplementary Information S1). The IPF represents the crystal orientation obtained from fitting the Kikuchi patterns to each pixel of the image with respect to a reference axis. Along the z-axis, the AS sample(Figure 3b), shows randomly oriented grains of hundreds of nanometers. On the contrary, in the FIRA sample (Figure 3c), all grains are aligned along [100] and [112] direction along the z-axis (green and purple color) and the two orientations are paired in larger regions, in agreement with XRD and polarized microscopy measurements. The distribution of orientations for the two samples clearly shows the mostly random orientation for the AS sample, and the bipolar distribution of orientations for the FIRA sample (Figure 3d). Consistent with XRD (Figure 1c), we observe significant orientation along both the [100] and [112] direction, but locally the ratio can vary (Figure S3). To investigate the actual grain size of the sample, we study the crystal orientation along the x-and y-direction for the FIRA sample(Figure 3e,f and Figure 3 . 3EBSD maps reveal the crystal orientation and grain size of the two systems. a) EBSD setup. b) Image Quality (IQ, brightness) overlay with Inverse Pole Figure (IPF) map of AS sample showing crystallographic orientation along z-direction. Inset, a magnification of the typical Kikuchi patterns recorded from the sample. c) Image Quality (IQ, brightness) overlay with Inverse Pole Figure (IPF) map of FIRA sample showing crystallographic orientation along z-direction. Inset, a magnification of the typical Kikuchi patterns recorded from the sample. d) Distribution of orientation for the two samples along z-direction. e) Image Quality (IQ, brightness) overlay with Inverse Pole Figure (IPF) map of FIRA sample showing crystallographic orientation along x-direction and f) y-direction. g) SEM image showing the apparent grain size of a FIRA cluster. The cluster measured with EBSD is highlighted with a dashed line. Figure 4 . 4Emission properties of MAPbI3 varying crystal orientation and grain size. a) Spatially resolved PL of AS and FIRA sample b) AFM image of AS and FIRA sample highlighting the cluster boundary region in the FIRA sample. c) Emission wavelength map of the same region as in figure (a) for the AS and FIRA sample. For the FIRA sample, the emission wavelength is converted into the additional thickness the light has been transmitted through, before it is emitted, d) PL spectra extracted from five random regions indicated in figure (a) in the AS sample showing similar PL intensity and no shift in the peak position, and PL spectra extracted from the cluster boundary and the inner cluster region of FIRA sample showing enhancement in PL at the cluster boundaries, and red-shift of the peak due to self-absorption. Next to the optical properties, the electronic properties have been shown to depend on grain size in some cases 54 . To investigate the mobility and recombination dynamics of photoexcited charge carriers in our two systems we use the time-resolved microwave conductivity (TRMC) technique. The FIRA and AS samples were excited with 485 nm excitation wavelength. Supplementary 15 Figure 5 . 155Note 3). Inter-grain transport across larger distances may show larger differences in crystallographically different systems. This finding is consistent with the similar device performances 33 that have been reported for both FIRA and AS showing similar Jsc, Voc, FF and PCE. Practically, FIRA could allow a lower cost, environmentally friendly fabrication route to produce large scale and reproducible perovskite compared to the AS method 33 . Mobility and lifetime varying the grain size. Time-resolved microwave conductivity (TRMC) traces measured at different carrier density for the a) AS and b) FIRA sample deposited on quartz. is the number of photons per unit area per pulse, β a geometric factor related to the microwave cell, e is the elementary charge and FA the fraction of the light absorbed by the sample at the excitation wavelength used. We assume φ to be unitary for the low exciton binding energy of the material. Time-correlated single photon counting (TCSPC) measurements were performed with a homebuilt setup equipped with PicoQuant PDL 828 ''Sepia II'' and a PicoQuant HydraHarp 400 multichannel picosecond event timer and TCSPC module. A 640 nm pulsed laser (PicoQuant LDH-D-C-640) with a repetition rate of 2 MHz was used to excite the sample. A Thorlabs FEL-700 long-pass filter was used to remove the excitation laser.19 Supporting Information SEM at lower magnification of AS sample; polarized optical microscopy images of FIRA with polarizer and analyzer at different angles; EBSD measurement of a FIRA cluster; crystal orientation distribution along x-,y-direction for AS and FIRA sample; IQ and IPF overlay showing orientation along x-and y-direction for the AS sample; grain size distribution for AS and FIRA obtained from EBSD measurement; spatially resolved PL of bigger area of FIRA sample; AFM of different cluster boundary region in the FIRA sample; Lifetime map of FIRA sample at the cluster boundary region; Normalized PL spectra of five random regions in the AS and FIRA sample; EBSD geometry, Hough transformation and pattern indexing; Additional thickness travelled by light as function of emission wavelength in MAPbI3 fabricated by FIRA; AcknowledgementsThe authors thank Erik C. 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{'abstract': "It is thought that growing large, oriented grains of perovskite can lead to more efficient devices.We study MAPbI3 films fabricated via Flash Infrared Annealing (FIRA) consisting of highly oriented, large grains. Domains observed in the SEM are often misidentified with crystallographic grains, but SEM images don't provide diffraction information. We measure the grain size, crystal structure and grain orientation using Electron Back-Scattered Diffraction (EBSD) and we study how these affect the optoelectronic properties as characterized by local photoluminescence (PL) and time-resolved microwave conductivity measurements (TRMC).We find a spherulitic growth yielding large (tens of µm), highly oriented grains along the(112)and(400)planes in contrast to randomly oriented, smaller (400 nm) grains observed in films fabricated via conventional antisolvent (AS) dripping. We observe a local enhancement and shift of the photoluminescence emission at different regions of the FIRA clusters, but these can be explained with a combination of light-outcoupling and self-absorption. We observe no effect of crystal orientation on the optoelectronic properties. Additionally, despite a substantial difference in grain size between our FIRA sample and a conventional AS sample, we find similar photoluminescence and charge carrier mobilities and lifetime for the two films. These findings show that the optoelectronic quality is not necessarily related to the orientation and size of crystalline domains in perovskite films indicating that fabrication requirements may be more relaxed for perovskites.", 'arxivid': '1907.07363', 'author': ['Loreta A Muscarella \nCenter for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands\n', 'Eline M Hutter \nCenter for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands\n', 'Sandy Sanchez \nLaboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL)\nStation 61015LausanneSwitzerland\n', 'Christian D Dieleman \nCenter for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands\n', 'Tom J Savenije \nDepartment of Chemical Engineering\nDelft University of Technology\nVan der Maasweg 92629 HZDelftThe Netherland\n', 'Anders Hagfeldt \nLaboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL)\nStation 61015LausanneSwitzerland\n', 'Michael Saliba \nInstitute of Materials Science\nTechnical University of Darmstadt\nAlarich-Weiss-Strasse 2D-64287DarmstadtGermany\n', 'Bruno Ehrler \nCenter for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands\n'], 'authoraffiliation': ['Center for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands', 'Center for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands', 'Laboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL)\nStation 61015LausanneSwitzerland', 'Center for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands', 'Department of Chemical Engineering\nDelft University of Technology\nVan der Maasweg 92629 HZDelftThe Netherland', 'Laboratory of Photomolecular Science (LSPM) École Polytechnique Fédérale de Lausanne (EPFL)\nStation 61015LausanneSwitzerland', 'Institute of Materials Science\nTechnical University of Darmstadt\nAlarich-Weiss-Strasse 2D-64287DarmstadtGermany', 'Center for Nanophotonics\nAMOLF\nScience Park 1041098 XGAmsterdamThe Netherlands'], 'corpusid': 197431236, 'doi': '10.29363/nanoge.ngfm.2019.157', 'github_urls': [], 'n_tokens_mistral': 19995, 'n_tokens_neox': 16364, 'n_words': 8526, 'pdfsha': 'e852abf3442d9cf4528164e930d8ed8af395fc75', 'pdfurls': ['https://arxiv.org/pdf/1907.07363v1.pdf'], 'title': ['Crystal orientation and grain size: do they determine optoelectronic properties of MAPbI 3 perovskite?', 'Crystal orientation and grain size: do they determine optoelectronic properties of MAPbI 3 perovskite?'], 'venue': []}

arxiv

Finite-size effect of η-deformed AdS 5 × S 5 at strong coupling 31 Jan 2017 Changrim Ahn ahn@ewha.ac.kr Department of Physics Ewha Womans University DaeHyun 11-1120-750SeoulS. Korea Finite-size effect of η-deformed AdS 5 × S 5 at strong coupling 31 Jan 2017 We compute Lüscher corrections for a giant magnon in the η-deformed (AdS 5 × S 5 ) η using the su(2|2) q -invariant S-matrix at strong coupling and compare with the finite-size effect of the corresponding string state, derived previously. We find that these two results match and confirm that the su(2|2) q -invariant S-matrix is describing world-sheet excitations of the η-deformed background. Introduction AdS/CF T duality [1], a correspondence between string theories in AdS background with certain supersymmetric and conformal Yang-Mills theories on the boundary space-time of the AdS space, has been a hot topic for theoretical researches and produced many important quantitative results and applications (for overview see [2]). In these developments, integrability has played a crucial role on both sides of the correspondence. Two-dimensional world-sheet actions for the string theory moving in the background are described by nonlinear sigma models on coset group manifolds which are classically integrable. Aspects of quantum integrable structure of supersymmetric Yang-Mills theories appear in Bethe ansatz equations and related exact integrable machineries which can determine conformal dimensions of the CFTs. Quantum S-matrices of the world-sheet actions provide integrable framework which interpolate from the strong to weak coupling limits. An important direction of research is to find new AdS/CFT pairs which show novel integrability structures. One such string theory, which has been studied recently, is the type IIB superstring theory in the η-deformed targe space (AdS 5 × S 5 ) η for a real parameter η [3]. The classical integrability of nonlinear sigma model is provided by solutions of the classical Yang-Baxter equation [4]. (See [5,6,7] for related issues.) It has been conjectured in [3] that full quantum S-matrix of the deformed sigma model is given by the R-matrix of the q-deformed Hubbard model which has been proposed much earlier in [8]. When q is a complex phase, the dressing phase of the S-matrix and bound-states have been analyzed in [9]. Scattering amplitudes of bosonic exitations for small values of the world-sheet momentum have been computed and shown to agree with the q-deformed S-matrix in the large string tension (strong coupling) limit for real q with explicit relation with η [10]. Based on the exact S-matrix, thermodynamic Bethe ansatz equations for ground states and dressing phase for real q have been studied in [11]. A pertinent issue which should be mentioned is that the deformed sigma model is not a fully consistent string theory at quantum level. It has been found that this η-deformed sigma model does not solve the type IIB supergravity equations of motion [12], but rather a generalization of them [13]. This generalized ones allow only scale invariance but not full Weyl invariance at one-loop [14]. The Weyl invarince can be restored if the deformation is generalized by some modified solutions of the Yang-Baxter equation [15]. This suggests that one should pay attention to treat the η-deformed theory at quantum level. In this letter, we provide another evidence for the q-deformed S-matrix to describe the string theory on the η-deformed geometry. For this purpose, we consider finite-size effects of a giant magnon state, a classical string configuration living on a subspace of the (AdS 5 ×S 5 ) η [16]. These corrections have been computed for the undeformed AdS 5 ×S 5 in [17,18] and for the γ-deformed AdS 5 ×S 5 in [19,20] from both directions of string solutions and world-sheet S-matrices. For the η-deformed case, this effect has been studied from only string theory side in [21], which will be reviewed in sect.2. Exact q-deformed S-matrix and related formula will be presented in sect.3. We present our computation of the Lüscher corrections for a giant magnon based on q-deformed S-matrix in sect.4 along with a conjecture on the deformed dressing phase in sect.5. In sect.6, we conclude with a short summary and comments. 2 Finite-size effect of a giant magnon in (AdS 5 × S 5 ) η In this section, we give a brief review on computing the energy of a giant magnon using Neumann-Rosochatius ansatz following [21]. The giant magnon is defined in the R t × S 3 η subspace of (AdS 5 × S 5 ) η , where backgound metric and B-field are given by g tt = −1, g φ 1 φ 1 = sin 2 θ, g φ 2 φ 2 = cos 2 θ 1 +η 2 sin 2 θ , g θθ = 1 1 +η 2 sin 2 θ , b φ 2 θ = −η sin 2θ 1 +η 2 sin 2 θ . (2.1) Deformation parameterη is related to original parameter η byη = 2η/(1 − η 2 ). One can solve the giant magnon configuration using an ansatz for the dynamics of the target space coordinates t(τ, σ) = κτ, φ i (τ, σ) = ω i τ + F i (ξ), θ(τ, σ) = θ(ξ), ξ = σ − vτ, i = 1, 2, (2.2) where τ and σ are the string world-sheet coordinates and the Virasoro constraints. If we restrict further to S 2 by setting the isometry angle φ 2 to zero, conserved charges E s , J 1 corrsponding to other isometric coordinates t, φ 1 are given by complete elliptic integrals of first and third kinds (W = κ 2 /ω 2 1 ): E s = 2T η (1 − v 2 ) √ W (χ η − χ m )(χ p − χ n ) K(1 − ǫ),(2. 3) J 1 = 2T η (χ η − χ m )(χ p − χ n ) 1 − v 2 W − χ η K(1 − ǫ) + (χ η − χ p ) Π χ p − χ m χ η − χ m , 1 − ǫ , where the parameters are satisfying χ m = χ η χ p χ η − (1 − ǫ)χ p ǫ, ǫ = χ m (χ η − χ p ) χ p (χ η − χ m ) , (1 − ǫ)χ 2 p − 2ǫχ p χ η − χ 2 η χ η − (1 − ǫ)χ p + 3 − (1 + v 2 )W + 1 η 2 = 0, χ p χ η + ǫχ p χ η (χ p + χ η ) χ η − (1 − ǫ)χ p − 2 − (1 + v 2 )W + (3 − (2 + v 2 (2 − W )) W )η 2 η 2 = 0, ǫχ 2 p χ 2 η χ η − (1 − ǫ)χ p − (1 +η 2 )(1 − W )(1 − v 2 W ) η 2 = 0. The momentum of a giant magnon, which is related to the deficit angle by ∆φ 1 = p, satisfies p = 2ṽ η χ p (χ η − χ m ) − v K(1 − ǫ) + (2.4) + W (χ η − 1)(1 − χ p ) (χ η − χ p ) Π − (χ η − 1)(χ p − χ m ) (χ η − χ m )(1 − χ p ) , 1 − ǫ − (1 − χ p ) K(1 − ǫ) . Eqs.(2.3) and (2.4) generate the dispersion relation of a giant magnon at finite J 1 . In the limit of J 1 ≫ g ≫ 1 one can solve the parameter relations in terms of small ǫ-expansions to determine the energy and angular momentum E s − J 1 = 2g 1 +η 2 η arcsinh η sin p 2 − 8g(1 +η 2 ) 3/2 sin 3 p 2 1 +η 2 sin 2 p 2 exp − J 1 g 1 +η 2 sin 2 p 2 (1 +η 2 ) sin 2 p 2 . (2.5) The first term is the energy dispersion relation of a giant magnon in the infinite volume and the second one is the small finite-size (or finite J 1 ) correction. In next sections, we are going to reproduce this result from the su(2|2) q S-matrix. q-deformed S-matrix The quantum-deformed S-matrix can be written as a graded tensor product of su(2|2) qinvarint matrix as follows: S(p 1 , p 2 ) = S su(2) S⊗Ṡ. (3.6) The overall scalar factor S su(2) is given by [ 10] 1 S su(2) (p 1 , p 2 ) = 1 σ 2 (p 1 , p 2 ) x + 1 + ξ x − 1 + ξ x − 2 + ξ x + 2 + ξ x − 1 − x + 2 x + 1 − x − 2 1 − 1 x − 1 x + 2 1 − 1 x + 1 x − 2 ,(3.7) with q-deformed dressing phase σ. The su(2|2) q -invarint S-matrix has 16 × 16 elements, S i ′ j ′ ij , i, j, i ′ , j ′ = 1,a 1 = 1, a 2 = −q + 2 q x − 1 (1 − x − 2 x + 1 )(x + 1 − x + 2 ) x + 1 (1 − x − 2 x − 1 )(x − 1 − x + 2 ) , a 5 = x + 1 − x + 2 √ qU 1 V 1 (x − 1 − x + 2 ) . (3.9) The parameters x ± satisfy a shortening relation 1 q x + + 1 x + − q x − + 1 x − = q − 1 q ξ + 1 ξ ,(3.10) and related to energy E and momentum p by V 2 = q x + x − x − + ξ x + + ξ ≡ q E , U 2 = 1 q x + + ξ x − + ξ ≡ e ip . (3.11) The constant ξ is related to the string tension g and deformation parameter q by ξ = − i 2 g(q − q −1 ) 1 − g 2 4 (q − q −1 ) 2 . (3.12) It is claimed that the quantum group parameter q is related toη by q = e −ν/g with ν =η 1 +η 2 . (3.13) General energy-momentum dispersion relation follows from this E(p) = 2g ν arcsinh ξ i 1 4g 2 cosh 2 ν 2g + sin 2 p 2 . (3.14) At strong coupling limit g ≫ 1, Eqs. x ± (p) = x ± 0 (p) + 1 g x ± 1 (p) + O(g −2 ),(3.16) where x ± 0 (p) = e ±ip/2 1 +η 2 sin 2 p 2 ∓η sin p 2 , (3.17) x ± 1 (p) = x ± 0 (p) + iη η x ± 0 (p) − i 1 +η 2 x − 0 (p) − x + 0 (p) . (3.18) Also the dispersion relation in Eq.(3.14) becomes E 0 (p) = 2g 1 +η 2 η arcsinh η sin p 2 ,(3.19) which is consistent with that of giant magnon string state given in the first term of Eq.(2.5). Lüscher corrections Leading finite-size corrections in the strong coupling limit are the µ-term Lüscher corrections which arise from residues of S-matrix in the contour integrals of the F -term formula. Explicit µ-term Lüscher formula for one su(2) giant magnon state with su(2|2) index (11) is given by [22,18], δE µ = −i 1 − E ′ (p) E ′ (q ⋆ ) e −iq⋆J 1 j,j,j ′ ,j ′ Res q=q S (11)(j ′j′ ) (11)(jj) (p, q ⋆ (q)) , (4.20) whereq is the location of S-matrix the poles. The physical giant magnon has momentum p and energy given by (3.19), while the momentum q ⋆ of the virtual particle satisfies the following on-shell relation q 2 + E 2 (q ⋆ ) = 0. (4.21) We also use a short notationq ⋆ = q ⋆ (q). We start with locating the poles of the S-matrix. The overall scalar factor S su(2) (p, q ⋆ ) in (3.7) have both s-channel pole at x − (q ⋆ ) = x + (p) and t-channel pole at x − (q ⋆ ) = 1/x + (p). We have checked that the t-channel gives exactly same results as the s-channel. We will present a detailed computation for the s-channel here and multiply a factor 2 at the end. Substituting x + (p) for x − (q ⋆ ) in Eq.(3.10), we can compute x + (q ⋆ ) x + (q ⋆ ) = x + 0 (p) + 3 g x ± 0 (p) + iη η x ± 0 (p) − i 1 +η 2 x − 0 (p) − x + 0 (p) + O(g −2 ). (4.22) From Eq.(3.11), we can obtain e iq⋆ = 1 q x + (q ⋆ ) + ξ x − (q ⋆ ) + ξ = 1 + 1 gη x + 0 (p) + x − 0 (p) − 2i 1 +η 2 x − 0 (p) − x + 0 (p) + O(g −2 ) . (4.23) Using Eq.(3.17), we obtainq ⋆ as follows: iq ⋆ = 1 g 1 +η 2 sin 2 p 2 1 +η 2 sin p 2 + O(g −2 ) . The next factor to consider in the Lüscher formula (4.20) is the energy dispersion. Sincẽ q ⋆ ∼ O(g −1 ) , one should use exact relation (3.14) instead of (3.19) before taking the large g limit along with (4.24). A straightforwad computation yields 1 − E ′ (p) E ′ (q ⋆ ) = (1 +η 2 ) sin 2 p 2 1 +η 2 sin 2 p 2 . (4.26) Now we move on to the residue of the S-matrix, which comes from the scalar factor (3.7) Res q=q S su(2) (p, q ⋆ ) = 2e 3ip/2 1 + ie ip/2η 1 +η 2 sin 2 p 2 −η sin p 2 g 1 +η 2 sin p 2 · σ 2 (p,q ⋆ ) · x − ′ (q ⋆ ) . (4.27) The last factor can be computed by a trick used in [18] dx − (q ⋆ ) dq q=q = dx + (p)/dp dq/dp = −ie ip/2 1 + ie ip/2η 1 +η 2 sin 2 p 2 −η sin p where we used (4.21) for dq/dp. Combining these, we get Res q=q S su(2) (p, q ⋆ ) = 2ie ip 1 +η 2 sin 2 p 2 g 1 +η 2 sin 3 p 2 · σ 2 (p,q ⋆ ) . (4.29) The contribution form each matrix element is from Eq.(3.8) 1 + q 2 a 1 + 1 2 a 2 + 2a 5 2 (4.30) and becomes 1 in the leading order from (3.9). q-deformed Dressing phase The dressing phase has been proposed first in terms of q-deformed Gamma function for q a complex phase, [9] σ 2 (p 1 , p 2 ) = exp i χ(x + 1 , x + 2 ) − χ(x + 1 , x − 2 ) − χ(x − 1 , x + 2 ) + χ(x − 1 , x − 2 ) , (5.31) χ(x 1 , x 2 ) = i |z|=1 dz 2πi 1 z − x 1 |w|=1 dw 2πi 1 w − x 2 log Γ q 2 1 + i 2a (u(z) − u(w)) Γ q 2 1 − i 2a (u(z) − u(w)) , (5.32) where a = ν/g for g ≫ 1 and u(z) is defined by X(z, w) ≡ u(z) − u(w) = i 2ν log z + 1 z + ξ + 1 ξ w + 1 w + ξ + 1 ξ . (5.33) An integral representation for Γ q 2 given in [9] can be analytically continued for real q to get strong coupling limit [10] log Γ q 2 [1 + gX] Γ q 2 [1 − gX] ≈ g 2X(log g − 1) + X log(−X 2 ) − i 2π ν ψ −2 1 − iνX π − ψ −2 1 + iνX π , (5.34) where ψ −2 is the poly-gamma function. The integrals over two unit circles in (5.32) may develop a branch cut for ν ≥ 1/2 but can be handled with proper care as pointed out in [11]. For computing σ(p,q ⋆ ) at strong coupling, we combine the χ functions with arguments x ± (p), x ± (q ⋆ ) given in Eqs. (3.16) and (4.22) to get log σ 2 (q ⋆ , p) = |z|=1 dz 2πi |w|=1 dw 2πi   2iνx + 0 x + 0 + x − 0 + ξ + 1 ξ g(z − x + 0 )(z − x − 0 )(w − x + 0 ) 2   × × g X log g 2 e 2 + X log(−X 2 ) − 2πi ν ψ −2 1 − iνX π − ψ −2 1 + iνX π ,(5.35) with a short notation x ± 0 = x ± 0 (p) given in (3.17). Due to complicated branch cuts appearing in the contour integrals, we could not evaluate this integral analytically. However, we have found numerically that the integration depends onη very insensitively within available accuracy. This leads to our conjecture that the dressing phase with given arguments in the strong coupling limit is σ 2 (q ⋆ , p) = −2g 2 e −ip sin 4 p 2 ,(5.36) which is the result for the undeformed case, computed from the AFS phase [23] in [18]. Combining (4.25), (4.26), (4.29) and (5.36) along with a factor −i in (4.20) and 2 for the t-channel contribution, we get the final µ-term Lüscher correction δE µ = − 8g(1 +η 2 ) 1/2 sin 3 p Conclusion Compared with finite-size giant magnon computation (2.5), the strong coupling Lüscher correction match quite well except 1 +η 2 in the overall factor. We think this factor should be modified in the string theory computation. Apart from this minor discrepancy, both coefficient and exponent of the exponential factor show correct dependence on the momentum and deformation parameter. Our check is valid for the su(2) sector with generic value of p and supports that the q-deformed S-matrix should describe the string theory in the η-deformed AdS background. It will be interesting to further elaborate the q-deformed dressing phase to check (5.36) both numerically and analytically. Another interesting but less studied domain is the weak coupling limit of the S-matrix, which could be related to certain q-deformed spin-chain. ( 3 . 312) and(3.13) lead toξ = iη + O(g −1 ). (3.15)From Eqs.(3.10) and(3.11), one can expand the paramters with the string computations shown in (2.5). Several different candidates have been proposed in[9]. We have checked that only this one is consistent with the finite-size correction. AcknowledgementsWe thank S. van Tongeren for sharing useful information on the dressing phase. This work was supported by the National Research Foundation of Korea (NRF) grant (NRF-2016R1D1A1B02007258). The large N limit of superconformal field theories and supergravity. J Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231J. Maldacena, "The large N limit of superconformal field theories and supergravity," Adv. Theor. Math. 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B880, 225 (2014) [arXiv:1312.4560 [hep-th]]. On deformations of AdS n × S n supercosets. B Hoare, R Roiban, A A Tseytlin, arXiv:1403.5517JHEP. 14062hep-thB. Hoare, R. Roiban and A. A. Tseytlin, "On deformations of AdS n × S n supercosets," JHEP 1406, 002 (2014) [arXiv:1403.5517 [hep-th]]. Quantum Deformations of the One-Dimensional Hubbard Model. N Beisert, P Koroteev, J. Phys. 41255204N. Beisert, P. Koroteev, "Quantum Deformations of the One-Dimensional Hubbard Model," J. Phys. A41, 255204 (2008). q-Deformation of the AdS5 x S5 Superstring S-matrix and its Relativistic Limit. B Hoare, T J Hollowood, J L Miramontes, arXiv:1112.4485JHEP. 120315hep-thB. Hoare, T. J. Hollowood and J. L. Miramontes, "q-Deformation of the AdS5 x S5 Su- perstring S-matrix and its Relativistic Limit," JHEP 1203, 015 (2012) [arXiv:1112.4485 [hep-th]]. S-matrix for strings on η-deformed AdS 5 × S 5. G Arutyunov, R Borsato, S Frolov, arXiv:1312.3542JHEP. 14042hep-thG. Arutyunov, R. Borsato, S. Frolov, "S-matrix for strings on η-deformed AdS 5 × S 5 ," JHEP 1404, 002 (2014) [arXiv:1312.3542 [hep-th]]. On the exact spectrum and mirror duality of the (AdS 5 × S 5 ) η superstring. G Arutyunov, M De Leeuw, S Van Tongeren, arXiv:1403.6104Theor. Math. Phys. 182hep-thG. Arutyunov, M. de Leeuw, S. van Tongeren "On the exact spectrum and mirror duality of the (AdS 5 × S 5 ) η superstring," Theor. Math. Phys. 182, 23 (2015) [arXiv:1403.6104 [hep-th]]. Puzzles of η-deformed AdS5 x S5. G Arutyunov, R Borsato, S Frolov, arXiv:1507.04239JHEP. 151249hep-thG. Arutyunov, R. Borsato and S. Frolov, "Puzzles of η-deformed AdS5 x S5," JHEP 1512 (2015) 049 [arXiv:1507.04239 [hep-th]]. Scale invariance of the η-deformed AdS5 x S5 superstring, T-duality and modified type II equations. G Arutyunov, S Frolov, B Hoare, R Roiban, A A Tseytlin, arXiv:1511.05795Nucl. Phys. 903262hep-thG. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A. A. Tseytlin, "Scale invariance of the η-deformed AdS5 x S5 superstring, T-duality and modified type II equations," Nucl. Phys. B903 (2016) 262 [arXiv:1511.05795 [hep-th]]. Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations. L Wulff, A A Tseytlin, arXiv:1605.04884JHEP. 1606174hep-thL. Wulff and A. A. Tseytlin, "Kappa-symmetry of superstring sigma model and gener- alized 10d supergravity equations," JHEP 1606 (2016) 174 [arXiv:1605.04884 [hep-th]]. Target space supergeometry of and -deformed strings. R Borsato, L Wulff, arXiv:1608.03570JHEP. 161045hep-thR. Borsato and L. Wulff, "Target space supergeometry of and -deformed strings," JHEP 1610 (2016) 045 [arXiv:1608.03570 [hep-th]]. Giant Magnons. D Hofman, J Maldacena, arXiv:hep-th/0604135J. Phys. 3913095D. Hofman, J. Maldacena, "Giant Magnons," J. Phys. A39, 13095 (2006) [arXiv:hep-th/0604135]. Finite-size Effects from Giant Magnons. G Arutyunov, S Frolov, M Zamaklar, arXiv:hep-th/0606126Nucl. Phys. 7781G. Arutyunov, S. Frolov, M. Zamaklar, "Finite-size Effects from Giant Magnons," Nucl. Phys. B778, 1 (2007) [arXiv:hep-th/0606126]. Wrapping interactions at strong coupling -the giant magnon. R A Janik, T Lukowski, arXiv:hep-th/0708.2208Phys. Rev. 76126008R.A. Janik and T. Lukowski, "Wrapping interactions at strong coupling -the giant magnon," Phys. Rev. D76, 126008 (2007) [arXiv:hep-th/0708.2208]. . D V Bykov, S Frolov, arXiv:0805.1070JHEP. 080771D.V. Bykov, S. Frolov, JHEP 0807, 071 (2008) [arXiv:0805.1070]. Finite-size effects of β-deformed AdS5/CFT4 at strong coupling. C Ahn, D Bombardelli, M Kim, Phys. Lett. 71467C. Ahn, D. Bombardelli, M. Kim, "Finite-size effects of β-deformed AdS5/CFT4 at strong coupling," Phys. Lett. B71, 467 (2012). Finite-size giant magnons on η-deformed AdS5×S5. C Ahn, P Bozhilov, arXiv:1406.0628Phys. Lett. 737293hep-thC. Ahn, P. Bozhilov, "Finite-size giant magnons on η-deformed AdS5×S5," Phys. Lett. B737, 293 (2014) [arXiv:1406.0628 [hep-th]]. Volume Dependence Of The Energy Spectrum. M Luscher, Massive Quantum Field Theories. 1. Stable Particle States. 104177M. Luscher, "Volume Dependence Of The Energy Spectrum In Massive Quantum Field Theories. 1. Stable Particle States," Com. Math. Phys. 104, 177 (1986). . G Arutyunov, S Frolov, M Staudacher, arXiv:hep-th/0406256JHEP. 041016G. Arutyunov, S. Frolov and M. Staudacher, JHEP 0410, 016 (2004) [arXiv:hep-th/0406256].

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{'abstract': 'We compute Lüscher corrections for a giant magnon in the η-deformed (AdS 5 × S 5 ) η using the su(2|2) q -invariant S-matrix at strong coupling and compare with the finite-size effect of the corresponding string state, derived previously. We find that these two results match and confirm that the su(2|2) q -invariant S-matrix is describing world-sheet excitations of the η-deformed background.', 'arxivid': '1611.09992', 'author': ['Changrim Ahn ahn@ewha.ac.kr \nDepartment of Physics\nEwha Womans University\nDaeHyun 11-1120-750SeoulS. Korea\n'], 'authoraffiliation': ['Department of Physics\nEwha Womans University\nDaeHyun 11-1120-750SeoulS. Korea'], 'corpusid': 119236903, 'doi': '10.1016/j.physletb.2017.01.063', 'github_urls': [], 'n_tokens_mistral': 9324, 'n_tokens_neox': 7715, 'n_words': 3986, 'pdfsha': '029011e6d84acdcb734cf3b31ebc971e3b858571', 'pdfurls': ['https://arxiv.org/pdf/1611.09992v2.pdf'], 'title': ['Finite-size effect of η-deformed AdS 5 × S 5 at strong coupling', 'Finite-size effect of η-deformed AdS 5 × S 5 at strong coupling'], 'venue': []}

arxiv

Radial distributions of spectral absorption indices in spiral disks 2007 Mercedes Mollá mercedes.molla@ciemat.es CIEMAT Avda. Complutense 2228040MadridSpain Radial distributions of spectral absorption indices in spiral disks Proceedings Title IAU Symposium Proceedings IAU Symposium No. xxx Title IAU Symposium IAU Symposium No. xxx2007Vazdekis et al, eds.Stellar populationsGalaxies: spiralGalaxies: dwarfGalaxies: Abundances We present a grid of 440 spectro-photometric models for simulating spiral and irregular galaxies. They have been consistently calculated with evolutionary synthesis models which use as input the information proceeding from chemical evolution models. The model predictions are spectral energy distributions, brightness and color profiles and radial distributions of spectral absorption stellar indices which are in agreement with observations. Evolutionary Synthesis Models for Spiral Galaxies Spectrophotometric data, such as spectral energy distributions (SED), colors and spectral absorption stellar indices, are interpreted by mean of evolutionary synthesis models (ESM) that usually provide their predictions for the so-called Single Stellar Populations (SSP). These data are also available for spiral and irregular galaxies, although absorption spectral indices, mainly their radial distributions along the disks have been only recently obtained due to their arduous detection, for a certain number of spirals in Beauchamp & Hardy(1997), Mollá, Hardy & Beauchamp(1999), Ryder, Fenner & Gibson(2005). Spiral disks show a large complexity since they are composed by a mixing of stellar populations and show spatial variations along the disks and special phenomena as bars and outflows. The use of the SSPs predictions to analyze these data is inappropriate since in that case, the SED, F λ (t), corresponds to the sum of different SSP SEDs, S λ (t); that is, a convolution with the star formation history (SFH), Ψ(t), must be done: F λ (t) = t 0 S λ (τ, Z)Ψ(t ′ )dt ′ (1.1) where τ = t − t ′ . and S λ are the SED of the SSP's. The SFH and the age-metallicity relation (AMR), Z(t), necessary to assign S λ (t, Z(t)) to each time step, are, however, unknown. But the present time state of a disk galaxy is known since emission lines from Hii regions, from which elemental abundances are estimated, are observed. These data, and other gas information, are usually interpreted by mean of chemical evolution models (CEM). Thus, our idea is to fit the present time data of a given spiral or irregular galaxy with a CEM and then to use the evolutionary histories thus produced as input of a ESM. This technique of combining both types of data, those from the gas, and the spectro-photometric ones from stars, to better search for the possible evolution of a given spiral galaxy, has been successfully used to compute absorption spectral indices in Mollá, Hardy & Beauchamp(1999), by demonstrating its validity. We now apply the outlined method to the grid of CEMs from Mollá & Díaz(2005). The SFH and AMR of each radial region are the input of Eq. 1.1 to calculate F λ (r, t) and thus colors, surface brightness and spectral absorption indices profiles. Fig. 1 shows the model results for NGC 4900, a SBb spiral galaxy. Top panel are results of the CEM which fits the present day data. The oxygen abundance shows a radial gradient with a central value similar to that one estimated by Cantin et al.(2006). The Hi predicted density shows a good fit for the disk, except the inner region with a density lower than observed. The disk modeled star formation rate is in agreement with the data (< Ψ >∼ 2.5 ± 0.5 M ⊙ yr −1 ). The model predicts, however, a value one order of magnitude smaller than observed for the center. An infall of gas from the disk, and a consequent burst of star formation, due to the effect of a bar may explain these differences model-data. They are also apparent in spectral indices such as the bidimensional spectroscopy data from Cantin et al.(2006), marked in Fig. 1, show. In the bar, where young stellar populations over a subjacent old stellar population there exist, the spectral indices F e5270 and higher H β are smaller -green dashed lines-than predicted, and actually observed out of the bar. We conclude that our models are useful to interpret adequately complex disk galaxies, in particular the barred ones. Model results and conclusions Figure 1 . 1Radial distributions for NGC 4900: a) Hi density; b) oxygen abundance; c) Hβ and d) F e5270. Full (red) dots in a) are from Warmels(1988). Data in the other panels are from Cantin et al.(2006) except the cyan dots from Ryder, Fenner & Gibson(2005) for a similar galaxy in d). Warmels, R. H. 1988, A&AS, 72, 427 . D Beauchamp, E Hardy, AJ. 1131666Beauchamp, D., & Hardy, E. 1997, AJ, 113, 1666 . S Cantin, C Robert, A Pellerin, M Mollá, New Astronomy Review. 49536Cantin, S., Robert. C., Pellerin, A. & Mollá, M. 2005, New Astronomy Review, 49, 536 . M Mollá, A I Díaz, MNRAS. 358521Mollá , M., & Díaz , A. I. 2005, MNRAS, 358, 521 . M Mollá, E Hardy, D Beauchamp, ApJ. 513695Mollá, M., Hardy, E., & Beauchamp, D. 1999, ApJ, 513, 695 . S D Ryder, Y Fenner, B K Gibson, MNRAS. 3581337Ryder, S. D., Fenner, Y., & Gibson, B. K. 2005, MNRAS, 358, 1337

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{'abstract': 'We present a grid of 440 spectro-photometric models for simulating spiral and irregular galaxies. They have been consistently calculated with evolutionary synthesis models which use as input the information proceeding from chemical evolution models. The model predictions are spectral energy distributions, brightness and color profiles and radial distributions of spectral absorption stellar indices which are in agreement with observations.', 'arxivid': 'astro-ph/0701655', 'author': ['Mercedes Mollá mercedes.molla@ciemat.es \nCIEMAT\nAvda. Complutense 2228040MadridSpain\n'], 'authoraffiliation': ['CIEMAT\nAvda. Complutense 2228040MadridSpain'], 'corpusid': 14170739, 'doi': '10.1017/s174392130700899x', 'github_urls': [], 'n_tokens_mistral': 1602, 'n_tokens_neox': 1341, 'n_words': 847, 'pdfsha': '30b0d578bc309b0295c9477dd714c58f276cd2b1', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0701655v1.pdf'], 'title': ['Radial distributions of spectral absorption indices in spiral disks', 'Radial distributions of spectral absorption indices in spiral disks'], 'venue': ['Proceedings Title IAU Symposium Proceedings IAU Symposium No. xxx']}

arxiv

FARRELL-JONES SPHERES AND INERTIA GROUPS OF COMPLEX PROJECTIVE SPACES Ramesh Kasilingam FARRELL-JONES SPHERES AND INERTIA GROUPS OF COMPLEX PROJECTIVE SPACES We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds M 2n , where n = 7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C.S. Aravinda and F.T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell-Jones sphere. We also discuss the relationship between inertia groups of CP n and Farrell-Jones spheres.2010 Mathematics Subject Classification. 57R60, 57C24, 57R55, 53C56. Introduction Let Θ m be the group of homotopy spheres defined by M. Kervaire and J. Milnor in [15]. Definition 1.1. We call Σ 2n ∈ Θ 2n (n ≥ 4) a Farrell-Jones sphere if CP n #Σ 2n is not concordant to CP n . The following theorem gives an equivalent definition of Farrell-Jones spheres, which we prove in Section 3: Theorem 1.2. Let Σ 2n be an exotic sphere in Θ 2n (n ≥ 4). Σ 2n is a Farrell-Jones sphere if and only if CP n #Σ 2n is not diffeomorphic to CP n . By [10,Lemma 3.17], there exists a Farrell-Jones sphere Σ m ∈ Θ m for all m = 8n+2 (n ≥ 1) and for m = 8. Also we prove the following theorem in Section 3: Theorem 1.3. The non zero element of Θ 2n ∼ = Z 2 (n = 7 or 8) is a Farrell-Jones sphere. The study of Farrell-Jones spheres is motivated by the following result, which is a slight modification of [10,Theorem 3.20]: Theorem 1.4. Let M 2n be any closed complex hyperbolic manifold of complex dimension n. Let Σ 2n ∈ Θ 2n be a Farrell-Jones sphere. Given a positive real number , there exists a finite sheeted cover N 2n of M 2n such that the following is true for any finite sheeted cover N 2n of N 2n . (i) The smooth manifold N 2n is not diffeomorphic to N 2n #Σ 2n . (ii) The connected sum N 2n #Σ 2n supports a negatively curved Riemannian metric whose sectional curvatures all lie in the closed interval [−4 − , −1 + ]. The proof of the above Theorem 1.4 follows from [10, Corollary 3.14 and Proposition 3.19]. By using Theorem 1.3 and Theorem 1.4, we also construct in section 2 examples of closed negatively curved manifolds M 2n , where n = 7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C.S. Aravinda and F.T. Farrell [5]. Another source for Farrell-Jones spheres is the class of so called Hitchin spheres. In [12], Hitchin showed that if Σ is a homotopy sphere with a metric of positive scalar curvature, then α(Σ) = 0, where α : Ω spin * → KO * is the ring homomorphism which associates to a spin bordism class the KO-valued index of the Dirac operator of a representative spin manifold. The following definition can be found in [19,Remark 3.4]: Definition 1.5. An exotic sphere Σ m ∈ Θ m (m ≥ 1) is called a Hitchin sphere if α(Σ m ) = 0. We prove the following theorem in Section 3: Theorem 1.6. Every Hitchin (8n + 2)-sphere (n ≥ 1) is a Farrell-Jones sphere. Recall that the collection of homotopy spheres which admit an orientation preserving diffeomorphism M → M #Σ form the inertia group of M , denoted by I(M ). There is a canonical topological identification ι : M → M #Σ which is the identity outside of the attaching region; the subset of the inertia group consisting of homotopy spheres that admit a diffeomorphism homotopic to ι is called the homotopy inertia group I h (M ). Similarly, the concordance inertia group of M m , I c (M m ) ⊆ Θ m , consists of those homotopy spheres Σ m such that M and M #Σ m are concordant. By Theorem 1.2, we have that Σ 2n is a Farrell-Jones sphere iff Σ 2n / ∈ I(CP n ) iff Σ 2n / ∈ I c (CP n ) iff Σ 2n / ∈ I h (CP n ). In section 4, we discuss the group I(CP 4n+1 ). Exotic Smooth Structures on Complex Hyperbolic Manifolds The negatively curved Riemannian symmetric spaces are of four types: RH n , CH n , HH n and OH 2 , where R, C, H, O denote the real, complex, quaternion and Cayley numbers, i.e., the four division algebras K over the real numbers whose dimensions over R are d = 1, 2, 4 and 8 respectively. A Riemannian manifold M dn is called a real, complex, quarternionic or Cayley hyperbolic manifold provided its universal cover is isometric to RH n , CH n , HH n and OH 2 , respectively. (Note that we need to consider only the cases n ≥ 2 and when K = O, n = 2.) In [5, p. 2], C.S. Aravinda and F.T. Farrell ask the following: Question 2.1. For each division algebra K over the reals and each integer n ≥ 2 (n = 2 when K = O), does there exist a closed negatively curved Riemannian manifold M dn (where d = dim R K) which is homeomorphic but not diffeomorphic to a K-hyperbolic manifold. Remark 2.2. For K = R and n = 2, 3, this is impossible since homeomorphism implies diffeomorphism in these dimensions [17]. Also when K = R, it was shown in [11] that the answer is yes provided n ≥ 6. When K = C, it was shown in [10] that the answer is yes for n = 4m + 1 for any integer m ≥ 1 and for n = 4. When K = H, the answer is yes for n = 2, 4 and 5, see [5]. The answer to this question is yes for K = O by [4] since only one dimension needs to be considered in this case. In this section, we consider the case K = C and show that the answer is yes for n = 7, 8. Since Borel [6] has constructed closed complex hyperbolic manifolds in every complex dimension m ≥ 1 and by Theorem 1.3 and Theorem 1.4, we have the following result : Theorem 2.3. Let n be either 7 or 8. Given any positive number ∈ R, there exists a pair of closed negatively curved Riemannian manifolds M and N having the following properties: (i) M is a complex n-dimensional hyperbolic manifold. (ii) The sectional curvatures of N are all in the interval [−4 − , −1 + ]. (iii) The manifolds M and N are homeomorphic but not diffeomorphic. Farrell-Jones Spheres and Hitchin Sphere In this section, we give proofs of the Theorems 1.2, 1.3 and 1.6. N 1 → N 2 such that the composition f 2 • g is topologically concordant to f 1 , i.e., there exists a homeomorphism F : N 1 × [0, 1] → M × [0, 1] such that F |N 1 ×0 = f 1 and F |N 1 ×1 = f 2 • g. The set of all such concordance classes is denoted by C(M ). We recall some terminology from [15] : Milnor [15] showed that each Θ m is a finite group; in particular, Θ 8 , Θ 14 and Θ 16 are cyclic groups of order 2, Θ 10 and Θ 20 are cyclic groups of order 6 and 24 respectively and Θ 18 is a group of order 16. Start by noting that there is a homeomorphism h : M n #Σ n → M n (n ≥ 5) which is the inclusion map outside of homotopy sphere Σ n and well defined up to topological concordance. We will denote the class of (M n #Σ n , h) in given by [16, p. 25 and 194], f * M becomes [Σ m ] → [M m #Σ m ]. Definition 3.3. If M is homotopy equivalent to CP n , we will call a generator of H 2 (M ; Z) a c-orientation of M . Hereafter g denotes the conjugation map (z 0 , z 1 , z 2 , z 3 , z 4 , ..., z n ) → (z 0 ,z 1 ,z 2 ,z 3 ,z 4 , ...,z n ) (the complex conjugation) induces the diffeomorphism g : CP n → CP n such that g * (c 1 ) = −c 1 where c 1 is the c-orientation of CP n . Proof of Theorem 1.2: Assume that Σ 2n is a Farrell-Jones sphere. Suppose CP n #Σ 2n and CP n are diffeomorphic. If f : CP n #Σ 2n → CP n is a diffeomorphism, then f induces an isomorphism on cohomology f * : H * (CP n , Z) → H * (CP n #Σ 2n , Z) such that f * (c 1 ) = ±c 2 , where c 1 and c 2 are c-orientation of CP n and CP n #Σ 2n respectively. If f * (c 1 ) = c 2 , then f is a c-orientation preserving diffeomorphism. If f * (c 1 ) = −c 2 , then g•f is a c-orientation preserving diffeomorphism, where g : CP n → CP n is the conjugation map. In both cases, we have that CP n #Σ 2n is c-orientation diffeomorphic to CP n . By [18, Corollary 3, p. 97], CP n #Σ 2n is concordant to CP n . This is a contradiction since Σ 2n is a Farrell-Jones sphere. Thus CP n #Σ 2n and CP n are not diffeomorphic. Conversely, suppose CP n #Σ 2n and CP n are not diffeomorphic. Then, by [18, Corollary 3, p. 97], CP n #Σ 2n is not concordant to CP n . This shows that Σ 2n is a Farrell-Jones sphere. This completes the proof of Theorem 1.2. Proof of Theorem 1.3: Let Σ 2n be the generator of Θ 2n (withn = 7 or 8). Suppose Σ 2n is not a Farrell-Jones sphere. Then CP n #Σ 2n is concordant to CP n . By [18, Corollary 3, p. 97], CP n #Σ 2n is c-orientation diffeomorphic to CP n . Let f : CP n #Σ 2n → CP n be a c-orientation diffeomorphism such that f * (c 1 ) = c 2 , where c 1 and c 2 are corientation of CP n and CP n #Σ 2n respectively. Using properties of the cup product, we have that f * (c n 1 ) = c n 2 . If c 1 = c 2 in H 2 (CP n , Z), then f is an orientation preserving diffeomorphism. If c 1 = c 2 in H 2 (CP n , Z), then g • f is an orientation preserving diffeomorphism with the property that (g • f ) * (c 1 ) = f * (g * (c 1 )) = −c 2 = c 1 ,, where g : CP n → CP n is the conjugation map. In both cases, we have that CP n #Σ 2n is an orientation preserving diffeomorphic to CP n . This is a contradiction because, by [13, Theorem 1], CP n #Σ 2n can not be orientation preserving diffeomorphic to CP n . Thus Σ 2n is a Farrell-Jones sphere. This completes the proof of Theorem 1.3. Recall that the α-invariant is the ring homomorphism α : Ω spin * → KO * which associates to a spin bordism class the KO-valued index of the Dirac operator of a representative spin manifold. We also write α for the corresponding invariant on framed bordism : α : Ω f * → Ω spin * → KO * Under the Pontryagin-Thom isomorphism Ω f * ∼ = π s * , the α-invariant has the following interpretation as Adams d-invariant d R : π s r → KO * , which was used already in [12, p. 44 We start by recalling some facts from smoothing theory [7], which were used already in [ (3.1) ψ * : Θ 8n+2 = π 8n+2 (T op/O) → π 8n+2 (F/O) is an isomorphism for n ≥ 1. The homotopy groups of SF are the stable homotopy groups of spheres π s m ; i.e., π m (SF ) = π s m for m ≥ 1. For n ≥ 1, (3.2) φ * : π s 8n+2 → π 8n+2 (F/O) is an isomorphism. Since every homotopy sphere has a unique spin-structure, we obtain the α-invariant on π s 8n+2 ∼ = π 8n+2 (F/O) ∼ = Θ 8n+2 : α : π s 8n+2 φ * → π 8n+2 (F/O) ψ −1 * → Θ 8n+2 → Ω spin 8n+2 → KO 8n+2 , where ψ * and φ * are the isomorphisms as in Equation (3.1) Let Σ 8n+2 ∈ Θ 8n+2 be a Hitchin (8n + 2)-sphere (withn ≥ 1) and further let η ∈ π s 8n+2 = [S 8n+2 , SF ] be such that ψ −1 * (φ * (η)) = Σ 8n+2 . Recall that [X, SF ] can be identified with the 0 th stable cohomotopy group π 0 (X). Let h : S q+8n+2 → S q represent η. Since Σ 8n+2 is a Hitchin sphere and by Lemma 3.4, we have (3.3) [S 2m , T op/O] = Θ 2m f * CP m − −− → [CP m , T op/O] = C(CP m )   ψ *   ψ * [S 2m , F/O] f * CP m − −− → [CP m , F/O]   φ *   φ * [S 2m , SF ] = π s 2m f * CP m − −− → [CP m ,0 = α(Σ 8n+2 ) = d R (h) = h * ∈ Hom( KO q (S q ), KO q (S q+8n+2 )). Also Adams and Walker [2] showed that Σ q f CP 4n+1 : Σ q CP 4n+1 → S q+8n+2 induces a monomorphism on KO q (.). Consequently the composite map h • Σ q f CP 4n+1 : Σ q CP 4n+1 → S q induces a non-zero homomorphism on KO q (.). This shows that f * CP 4n+1 (η) = [h • Σ q f CP 4n+1 ] = 0, where f * CP 4n+1 : [S 8n+2 , SF ] → [CP 4n+1 , SF ]. Since the homomorphism φ * : [CP m , SF ] → [CP m , F/O] is monic, by the commutative diagram (3.3) where m = 4n + 1, we have ψ * (f * CP 4n+1 (Σ 8n+2 )) = φ * (f * CP 4n+1 (η)) = 0. This implies that f * CP 4n+1 (Σ 8n+2 ) = 0 and hence CP 4n+1 #Σ 8n+2 is not concordant to CP 4n+1 . This shows that Σ 8n+2 is a Farrell-Jones sphere and this completes the proof of Theorem 1.6. Remark 3.5. 1. Let us note that the homotopy sphere Σ 8n+2 (n ≥ 1) given by [10,Lemma 3.17] is the image of Adams element µ 8n+2 of order 2 under the composed isomorphism ψ −1 * • φ * , where ψ * and φ * are the isomorphisms as in Equation This shows that Σ 8n+2 is a Hitchin sphere of order 2 in Θ 8n+2 . By Theorem 1.6, Σ 8n+2 is a Farrell-Jones sphere. 2. Since Θ 18 ∼ = Ker(α) ⊕ Z 2 , where the α-invariant α : Θ 18 → Z 2 satisfies Ker(α) = Z 8 (see [9, p. 12]). This shows that there are exotic spheres of order = 2 in Θ 18 which are not in the kernel of α. This implies that there are Hitchin spheres of order = 2 in Θ 18 which are all Farrell-Jones sphere by Theorem 1.6. 3. In [3], Anderson, Brown and Peterson proved that one has α(Σ m ) = 0 iff m = 8k + 1 or 8k + 2 iff Σ m is an exotic sphere not bounding a spin manifold, where α : Θ m → Ω spin m → KO m is the α-invariant. This implies that Σ m is a Hitchin sphere in Θ m iff Σ m is an exotic sphere not bounding a spin manifold. By Theorem 1.6, every exotic sphere not bounding a spin manifold Σ 8n+2 in Θ 8n+2 is a Farrell-Jones sphere. The Inertia Groups of Complex Projective Spaces In this section, we discuss the relationship between inertia groups of CP n and Farrell-Jones spheres. The proof of theorem 1.6 leads one to the following question: Question 4.4. Let f : CP 4n+1 → S 8n+2 be any degree one map (n ≥ 1). Does there exist an element η ∈ Ker(d R ) ⊂ π s 8n+2 = Θ 8n+2 such that the following is true : ( ) If any map h : S q+8n+2 → S q represents η, then h • Σ q f : Σ q CP 4n+1 → S q is not null homotopic. . If the answer to Question 4.4 is yes, then we have I(CP 4n+1 ) = Ker(d R ), i.e., there exists an exotic sphere Σ bounding spin manifold in Θ 8n+2 such that Σ / ∈ I(CP 4n+1 ). This can be seen as follows : Let η ∈ Ker(d R ) and let h : S q+8n+2 → S q represent η such that h • Σ q f : Σ q CP 4n+1 → S q is not null homotopic. This implies that f * CP 4n+1 (h) = [h • Σ q f CP 4n+1 ] = 0, where f * CP 4n+1 : π 0 (S 8n+2 ) → π 0 (CP 4n+1 ). A similar argument given in the proof of Theorem 1.6 using the commutative diagram (3.3) shows that there exists an exotic sphere Σ ∈ Θ 8n+2 such that ψ −1 * •φ * (η) = Σ, d R (η) = α(Σ) = 0 and CP 4n+1 #Σ is not concordant to CP 4n+1 , where ψ * and φ * are the isomorphisms as in Equation (3.1) and (3.2) respectively. This implies that Σ is a Farrell-Jones sphere such that Σ ∈ Ker(d R ). By Theorem 4.2, I(CP 4n+1 ) = Ker(d R ). 2. If all non-zero elements in Ker(d R ) satisfy the condition ( ) in Question 4.4, then, by the above remark (1), Σ / ∈ I(CP 4n+1 ) for all exotic sphere Σ ∈ Ker(d R ) and hence I(CP 4n+1 ) = 0. Theorem 4.6. Let n be a positive integer such that Θ 8n+2 is a cyclic group of order 2. Then I(CP 4n+1 ) = 0. Proof. Let Σ 8n+2 be the generator of Θ 8n+2 ∼ = Z 2 . Let ψ * : Θ 8n+2 → π 8n+2 (F/O) and φ * : π s 8n+2 → π 8n+2 (F/O) be the isomorphisms as in Equation (3.1) and (3.2). By [1,Theorem 1.2], there exists an element µ 8n+2 of order 2 in π s 8n+2 . This shows that φ −1 * • ψ * (Σ 8n+2 ) = µ 8n+2 . By [1, Theorem 1.2] and Lemma 3.4, we have d R (µ 8n+2 ) = α(Σ 8n+2 ) = 1. This implies that Σ 8n+2 is a Hitchin sphere. By Theorem 1.6 and Theorem 1.2, CP 4n+1 #Σ 8n+2 is not diffeomorphic to CP 4n+1 . This implies that I(CP 4n+1 ) = 0. Definition 3. 1 . 1Let M be a topological manifold. Let (N, f ) be a pair consisting of a smooth manifold N together with a homeomorphism f : N → M . Two such pairs (N 1 , f 1 ) and (N 2 , f 2 ) are concordant provided there exists a diffeomorphism g : Definition 3. 2 . 2(a) A homotopy m-sphere Σ m is an oriented smooth closed manifold homotopy equivalent to S m . (b) A homotopy m-sphere Σ m is said to be exotic if it is not diffeomorphic to S m . (c) Two homotopy m-spheres Σ m 1 and Σ m 2 are said to be equivalent if there exists an orientation preserving diffeomorphism f : Σ m 1 → Σ m 2 . The set of equivalence classes of homotopy m-spheres is denoted by Θ m . The equivalence class of Σ m is denoted by [Σ m ]. When m ≥ 5, Θ m forms an abelian group with group operation given by the connected sum # and the zero element represented by the equivalence class of the round sphere S m . M. Kervaire and J. C(M ) by [M n #Σ n ]. (Note that [M n #S n ] is the class of (M n , id M n ).) Let f M : M n → S n be a degree one map. Note that f M is well-defined up to homotopy. Composition with f M defines a homomorphism f * M : [S n , T op/O] → [M n , T op/O], and in terms of the identifications Θ n = [S n , T op/O] and C(M n ) = [M n , T op/O] (3.1) and (3.2) respectively (see [10, Equation (3.17.4)]). By [1, Theorem 1.2] and Lemma 3.4, we have d R (µ 8n+2 ) = α(Σ 8n+2 ) = 1. 4 . 4By [1, Theorem 7.2], Θ 10 ∼ = Ker(d R ) ⊕ Z 2 such that Ker(d R ) = Z 3 . If Σ 10 is a generator of Ker(d R ), then d R (Σ 10 ) = α(Σ 10 ) = 0. This shows that Σ 10 is not a Hitchin sphere. But, by[10, Lemma 3.17], Σ 10 is a Farrell-Jones sphere. Definition 4 . 1 . 41Let M m be a closed smooth, oriented m-dimensional manifold. Let Σ ∈ Θ m and let g : S m−1 → S m−1 be an orientation preserving diffeomorphism corresponding to Σ. Writing M #Σ as (M m \ int(D m )) ∪ g D m , let ι : M → M #Σ denote the PL homeomorphism defined by ι |M \int(D m ) = id and ι |D m = Cg, where Cg : D m → D m is the cone extension of g. The inertia group I(M ) ⊂ Θ m is defined as the set of Σ ∈ Θ m for which there exists an orientation preserving diffeomorphism φ : M → M #Σ. Define the homotopy inertia group I h (M ) to be the set of all Σ ∈ I(M ) such that there exists a diffeomorphism M → M #Σ which is homotopic to ι. Define the concordance inertia group I c (M ) to be the set of all Σ ∈ I h (M ) such that M #Σ is concordant to M . Clearly, I c (M ) ⊆ I h (M ) ⊆ I(M ). Note that for M = CP n , Theorem 1.2 can be restated as :Theorem 4.2. A sphere Σ 2n ∈ Θ 2n is a Farrell-Jones sphere iff Σ 2n / ∈ I(CP n ). Remark 4. 3 . 3Since I c (CP n ) ⊆ I h (CP n ) ⊆ I(CP n ) and by the above Theorem 4.2, we have that I c (CP n ) = I h (CP n ) = I(CP n ). . By [14, Lemma 9.1], I(CP 4n+1 ) ⊆ Ker(d R ) ] and [9, Lemma 2.12]. Lemma 3.4. Under the Pontryagin-Thom isomorphism Ω f * KO 8n+2 may be identified with d R : π s 8n+2 → KO 8n+2 .∼ = π s * , the α-invariant α : Ω f 8n+2 → 10, Lemma 3.17]. There are H-spaces SF , F/O and T op/O and H-space maps φ : SF → F/O, ψ : T op/O → F/O such that Ker(d R ) consists of framed manifolds which bound spin manifolds.Proof of Theorem 1.6: Consider the following commutative of diagram :and (3.2) respectively. Let Ker(d R ) denotes the kernel of the Adams d-invariant d R : π s 8n+2 → Z 2 . By Lemma 3.4, SF ] SFIn this diagram, φ * and ψ * are induced by the H-space maps φ : SF → F/O, ψ : T op/O → F/O respectively and the homomorphism φ * : [CP m , SF ] → [CP m , F/O] is monic for all m ≥ 1 by a result of Brumfiel [8, p. 77]. Recall that the concordance class [CP m #Σ] ∈ [CP m , T op/O] of CP m #Σ is f * CP m ([Σ]) when m > 2, and that [CP m ] = [CP m #S 2m ] is the zero element of this group. J F Adams, On the groups J(X).IV, Topology. 5J.F. Adams, On the groups J(X).IV, Topology 5 (1966), 21-71. On complex Stiefel manifolds. J F Adams, G Walker, Proc. Cambridge Philos. Soc. 61J.F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 61 (1965), 81-103. The structure of the spin cobordism ring. D W Anderson, E H Brown, F P Peterson, Ann. of Math. 2D.W. Anderson, E.H. Brown and F.P. Peterson, The structure of the spin cobordism ring, Ann. of Math. (2)86 (1967), 271-298. Exotic negatively curved structures on Cayley hyperbolic manifolds. C S Aravinda, F T Farrell, J. Differential Geom. 63C.S. Aravinda and F.T. Farrell, Exotic negatively curved structures on Cayley hyperbolic man- ifolds, J. Differential Geom 63 (2003), 41-62. Exotic structures and quaternionic hyperbolic manifolds. C S Aravinda, F T Farrell, Tata Institute of Fundamental Research Studies in Mathematics. 507524Algebraic Groups and ArithmeticC.S. Aravinda and F.T. Farrell, Exotic structures and quaternionic hyperbolic manifolds, in: Al- gebraic Groups and Arithmetic, Tata Institute of Fundamental Research Studies in Mathematics, Mumbai (2004), 507524. Compact Clifford-Klein forms of symmetric spaces. A Borel, Topology. 2A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111-122. Homotopy equivalences of almost smooth manifolds. G Brumfiel, Comment. Math. Helv. 46G. Brumfiel, Homotopy equivalences of almost smooth manifolds, Comment. Math. Helv. 46 (1971), 381-407. Homotopy equivalence of almost smooth manifolds, in: Algebraic Topology. G W Brumfiel, Proc. Symp. Pure Math. 227379American Mathematical SocietyG.W. Brumfiel, Homotopy equivalence of almost smooth manifolds, in: Algebraic Topology, Proc. Symp. Pure Math. 22, American Mathematical Society, Providence (1971), 7379. The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature. D Crowley, T Schick, Geom. Topol. 1717731789D. Crowley and T. Schick, The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature, Geom. Topol. 17 (2013), 17731789. Complex hyperbolic manifolds and exotic smooth structures. F T Farrell, L E Jones, Invent. Math. 117F.T. Farrell and L.E. Jones, Complex hyperbolic manifolds and exotic smooth structures, In- vent. Math. 117 (1994), 57-74. Hyperbolic manifolds with negatively curved exotic triangulations in dimensions larger than five. F T Farrell, L E Jones, P Ontaneda, J. Differential Geom. 48F.T. Farrell, L.E. Jones and P. Ontaneda, Hyperbolic manifolds with negatively curved exotic triangulations in dimensions larger than five, J. Differential Geom. 48 (1998), 319-322. Harmonic spinors. N Hitchin, Adv. Math. 14N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1-55. Inertia groups of low dimensional complex projective spaces and some free differentiable actions on spheres I. K Kawakubo, Proc. Japan Acad. 44K. Kawakubo, Inertia groups of low dimensional complex projective spaces and some free dif- ferentiable actions on spheres I, Proc. Japan Acad. 44 (1968), 873-875. Smooth structures on S p × S q. K Kawakubo, Osaka J. Math. 6K. Kawakubo, Smooth structures on S p × S q , Osaka J. Math. 6 (1969), 165-196. Groups of homotopy spheres: I, Annals of Math. M Kervaire, J Milnor, M. Kervaire and J. Milnor, Groups of homotopy spheres: I, Annals of Math. (2) 77 (1963), 504-537. R C Kirby, L C Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, With Notes by John Milnor and Michael Atiyah. PrincetonPrinceton University Press88R.C. Kirby and L.C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, With Notes by John Milnor and Michael Atiyah, Ann. of Math. Stud. 88, Princeton University Press, Princeton, 1977. Geometric Topology in Dimensions 2 and 3, Grad. Texts in Math. E E Moise, Springer-Verlag47New YorkE.E. Moise, Geometric Topology in Dimensions 2 and 3, Grad. Texts in Math. 47, Springer- Verlag, New York, 1977. Triangulating and smoothing homotopy equivalences and homeomorphisms. D P Sullivan, The Hauptvermutung Book. A Collection of Papers of the Topology of Manifolds. DordrechtKluwer Academic Publishers169103D.P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms, in: The Hauptvermutung Book. A Collection of Papers of the Topology of Manifolds, K-Monogr. Math. 1, Kluwer Academic Publishers, Dordrecht (1996), 69103. Exotic spheres on the Manifold Atlas. Exotic spheres on the Manifold Atlas : http : //www.map.mpim − bonn.mpg.de/Exotic − spheres.

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{'abstract': 'We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds M 2n , where n = 7 or 8, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a question raised by C.S. Aravinda and F.T. Farrell. We show that every exotic sphere not bounding a spin manifold (Hitchin sphere) is a Farrell-Jones sphere. We also discuss the relationship between inertia groups of CP n and Farrell-Jones spheres.2010 Mathematics Subject Classification. 57R60, 57C24, 57R55, 53C56.', 'arxivid': '1510.03029', 'author': ['Ramesh Kasilingam '], 'authoraffiliation': [], 'corpusid': 119615479, 'doi': '10.1515/forum-2013-0072', 'github_urls': [], 'n_tokens_mistral': 8982, 'n_tokens_neox': 7590, 'n_words': 4462, 'pdfsha': '61fe7e7ff77d9f5797cd985fdd1c9d3fc4e1144c', 'pdfurls': ['https://arxiv.org/pdf/1510.03029v1.pdf'], 'title': ['FARRELL-JONES SPHERES AND INERTIA GROUPS OF COMPLEX PROJECTIVE SPACES', 'FARRELL-JONES SPHERES AND INERTIA GROUPS OF COMPLEX PROJECTIVE SPACES'], 'venue': []}

arxiv

Riccati-parameter solutions of nonlinear second-order ODEs 2008 M A Reyes Instituto de Física Universidad de Guanajuato León, GuanajuatoMexico H C Rosu Potosí Institute of Science and Technology Apdo Postal 3-74, 78231Tangamanga, San Luis PotosíMexico Riccati-parameter solutions of nonlinear second-order ODEs J. Phys. A: Math. Theor 4162008Received 5 March 2008, in final form 13 May 20080230Jr0230Hq1130Pb It has been proven by2]that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions are easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a 'growth' parameter from the trivial null solution up to the particular solution found through the factorization procedure. Introduction Despite powerful integrability methods, such as the Lie group-theoretical approach, Painlevé analysis, existence of Lax representations and the associated inverse scattering transforms, the task of obtaining solutions of nonlinear second order partial and ordinary differential equations (ODEs) remains one of the most difficult problems in mathematical physics; in some cases, even finding one particular solution turns out to be a very difficult matter [3,4]. However, in a number of cases, it has been proven that finding one particular solution turns out to be easier than expected. In the case of polynomial non-linearities, Rosu and Cornejo-Pérez [1,2], working with a factorization procedure stemming from a work of Berkovich [5], have found that if the second order nonlinear differential equation can be factorized into two first order differential operators then it is easy to find a first particular solution for the problem. They considered a nonlinear equation of the typë u + g(u)u + F (u) = 0 ,(1) where the dots represent derivatives with respect to the independent variable τ , which is usually the traveling coordinate of a reaction-diffusion equation [1]. The method they proposed was to factorize this equation in the following form [D − φ 2 (u)] [D − φ 1 (u)] u = 0 ,(2) (where D ≡ d dτ ) which implies the following conditions on the functions φ i (u) − φ 1 + φ 2 + dφ 1 du u = g(u)(3)φ 1 φ 2 = F (u) u .(4) If Eq. (1) can be factorized as in Eq. (2), then a first particular solution, say u 1 , can be easily found by solving [D − φ 1 (u)] u = 0 .(5) Of course, obtaining one solution of a nonlinear second order ODE does not guarantee that one may find more general solutions. However, what Rosu and Cornejo-Pérez found was that in many cases (some of which will be described below) the function φ 1 (u) turned out to be a linear function of the dependent variable u. Hence, Eq. (5) turns out to be a Riccati equation for this variable, which is very fortunate, since we already know how to find the general solution for this equation once a particular solution is known. The appearance of the Riccati equation in linear second order differential equations is very common. In particular, it was very successfully exploited by Mielnik to find potentials which are isospectral to the simple harmonic oscillator potential [6], and it is a cornerstone for all SUSY developments [7]. However, it has not been used to solve nonlinear second-order differential equations at least in the way we present here. Thus, if φ 1 is of the form φ 1 (u) = c 1 u + c 2 , Eq. (5) transforms into the Riccati equatioṅ u − c 1 u 2 − c 2 u = 0 (6) and if a particular solution u 1 of this equation is known, then the general solution can be found as [6] u λ,c1 = u 1 + e I1 λ − c 1 I 2 ,(7) where I 1 (τ ) ≡ τ τ0 (2c 1 u 1 (τ ′ ) + c 2 )dτ ′(8) and I 2 (τ ) ≡ τ τ0 e I1(τ ′ ) dτ ′ .(9) Notice that for λ s = c 1 I 2 (τ ) ,(10) a singularity may develop. Eq. (7) provides in turn what we call as a Riccati-parameter solution of the nonlinear equation. We notice that the first parameter, c 1 , is essentially the slope of the factorization function φ 1 , whereas the λ parameter can be chosen in such a way as to prevent this solution from possessing singularities [6], although for nonlinear differential equations this is not an absolutely prohibitive issue. It will be seen in the examples given in the following that the latter parameter acts like a label in this class of solutions placing them between the trivial null solution and the particular solution given by Eq. (5). Examples of physical interest In this section we find the explicit form of the Riccati-parameter solution for three nonlinear equations of physical interest that are polynomial type Liénard equations, i.e., similar to Eq. (1) but with F (u) a polynomial of order two and three in our cases. Modified Emden equation We start with the modified Emden equation u + α uu + β u 3 = 0 ,(11) for which the first rigorous study has been done by Painlevé more than a century ago [8] who got solutions for β = α 2 /9 and β = −α 2 . Recently, Chandrasekar et al [9] provided a detailed discussion of this equation from the point of view of the modified Prelle-Singer procedure that gives the construction of the solution in terms of elementary functions if such a solution exists [10], although Iacono [11] noticed that much simpler connections with the Abel equation could be used to get the solutions. For the remarkable physical applications, see [9]. Employing φ 1 = a 1 √ β u and φ 2 = a −1 1 √ β u, where a 1 = − α± √ α 2 −8β 4 √ β , one particular solution is [2] u 1 = − 1 a 1 √ β(τ − τ 0 ) .(12)u λ = − 1 a 1 √ β(τ − τ 0 ) + 1 λ(τ − τ 0 ) 2 + a 1 √ β(τ − τ 0 ) .(13) In this case it is instructive to notice that when |λ| runs from zero to infinity, the u λ solution goes from the trivial solution u = 0 to the particular solution u = u 1 , as can be deduced from Eq. (13) and graphically seen in figure 1. Convective Fisher equation We pass now to the convective Fisher equation [12] u + 2(ν − µ u)u + 2u(1 − u) = 0 . The second term corresponding to convection is introduced to describe mechanical transport in competition with diffusive transport or cases when external bias fields are present. In the context of population dynamics which is typical for the Fisher equation, it has been used by Walsh et al [13] to simulate the population mobility according to spatial gradients in the food supply. Rosu and Cornejo-Pérez found that for ν = µ/2 + µ −1 , the factorization functions are φ 1 = −µ(1 − u) and φ 2 = −2/µ. Hence, a particular solution for this equation will be [2] u 1 = [1 ± exp (µ(τ − τ 0 ))] −1 .(15) The λ-parameter solution can be readily obtained in the form u λ = u 1 + e −µ(τ −τ0) e −µ(τ −τ0) ± 1 λ e −µ(τ −τ0) ± 1 − 1 .(16) Once again, u 0 = 0 and u ∞ = u 1 . For the graphical representation see figure 2. Generalized Liénard equation Consider now the generalized Liénard equation for cubic nonlinear oscillators where F 3 (u) = Au + Bu 2 + Cu 3 . The previous equations can be seen as particular cases of this one. With ∆ = √ B 2 − 4AC, Rosu and Cornejo-Pérez found that using u + g(u)u + F 3 = 0 ,(17)φ 1 = a 1 B + ∆ 2 + Cu , φ 2 = a −1 1 B − ∆ 2C + u for g(u) = − B + ∆ 2 a 1 + B − ∆ 2C a −1 1 + 2Ca 1 + a −1 1 u (18) the following particular solution could be obtained [2] u 1 (τ ) = B+∆ 2 exp −a 1 B+∆ 2 (τ − τ 0 ) − C = B+∆ 2 exp a 1 B+∆ 2 (τ − τ 0 ) 1 − C exp a 1 B+∆ 2 (τ − τ 0 ) .(19) Now, using Eq. (7) and denoting ξ = τ − τ 0 , we can see that the two-parameter solution in this case is u λ (ξ) = u 1 + B+∆ 2 exp −a 1 B+∆ 2 ξ exp −a 1 B+∆ 2 ξ − C λ B+∆ 2 − 1 exp −a 1 B+∆ 2 ξ − λC B+∆ 2 .(20) Varying λ between 0 and ∞, the Riccati-parameter solutions go from the null solution to u 1 . Plots of u λ for several values of λ are displayed in figure 3. Other cases of physical interest The examples we have provided here are not the only possible cases that can be solved with this method, but they show the typical solutions to be found. Other examples where Riccati-parameter solutions can be obtained in this way are: the generalized Burgers-Huxley equation with δ = 1, γ = 1, the generalized Fisher equation, with n = 2, the Dixon-Tuszynski-Otwinowski type equation, with n = 4, and the Fitzhugh-Nagumo equation [1]. The linear factorization functions φ 1 of all these cases are given in [1]. Last but not least, we would like to comment on the possible physical interpretation of the Riccati parameter λ. We follow the works of Barton et al [14] and Monthus et al [15] to assert that λ is related to the introduction of finite interval boundaries on the abscissa of the problem. This is very well described in Section II A of [15] to which the interested reader is directed. Essentially, the introduction of boundary conditions at certain points on the axis generates a modulation of the particular solution as presented here and the λ parameter can be fixed through the boundary conditions. When the boundary is sent to infinity the original particular solution is recovered. In conclusion, we introduced here an interesting class of parametric solutions of a number of physically relevant nonlinear differential equations. They cover the space between the null or constant solution and the particular solution obtained by a simple factorization method proposed previously. We would like to thank Dr. O. Cornejo-Pérez for a careful reading of the first draft of this work. The second author wishes to thank CONACyT for partial support through project 46980. Fig. 1 : 1u λ (τ ) in the case of the modified Emden equation for λ = −0.2, −0.4, −1, −6, from top to bottom respectively, and a1 √ β = −1. Fig. 2 : 2Convective Fisher u λ (τ ) for λ = −0.2, −0.4, −1, −6, from top to bottom respectively, and µ = 1. Fig. 3 : 3Cubic Liénard u λ (ξ) solutions for λ = 0.7, 0.9, 1.1, 1.3, from top to bottom respectively, in the case A = C = 1, B = 2 (∆ = 0) and a1 = 1. . H C Rosu, O Cornejo-Pérez, arXiv:math-ph/0401040Phys. Rev. E. 7146607Rosu H C and Cornejo-Pérez O 2005 Phys. Rev. E 71 046607, arXiv: math-ph/0401040. . O Cornejo-Pérez, H C Rosu, arXiv:math-ph/0504055Prog. Theor. Phys. 114533Cornejo-Pérez O and Rosu H C 2005 Prog. Theor. Phys. 114 533, arXiv: math-ph/0504055. . X Wang, Phys. Lett. A. 131277Wang X Y 1988 Phys. Lett. A 131 277 . W Hereman, M Takaoka, J. Phys. A. 234805Hereman W and Takaoka M 1990 J. Phys. A 23 4805 . L Berkovich, Sov. Math. Dokl. 45162Berkovich L M 1992 Sov. Math. Dokl. 45 162 . B Mielnik, J. Math. Phys. 253387Mielnik B 1984 J. Math. Phys. 25 3387 . F Cooper, Khare A Sukhatme, U , arXiv:hep-th/9405029Phys. Rep. 251267For a review, see Cooper F, Khare A and Sukhatme U 1995 Phys. Rep. 251 267, arXiv: hep-th/9405029 . P Painlevé, Acta Math. 251Painlevé P 1902 Acta Math. 25 1 . V K Chandrasekar, M Senthilvelan, M Lakshmanan, J. Phys. A. 404717Chandrasekar V K, Senthilvelan M and Lakshmanan M 2007 J. Phys. A 40 4717 . V K Chandrasekar, M Senthilvelan, M Lakshmanan, arXiv:nlin.SI/0408053Proc. Roy. Soc. Lond. A. 4612451Chandrasekar V K, Senthilvelan M and Lakshmanan M 2005 Proc. Roy. Soc. Lond. A 461 2451, arXiv: nlin.SI/0408053. . R Iacono, J. Phys. A. 4168001Iacono R 2008 J. Phys. A 41 068001 . O Schönborn, R Desai, D Stauffer, arXiv:cond-mat/9403025J. Phys. A. 27251Schönborn O, Desai R C and Stauffer D, 1994 J. Phys. A 27 L251, arXiv: cond-mat/9403025 . O Schönborn, S Puri, R C Desai, arXiv:cond-mat/9312035Phys. Rev. E. 493480Schönborn O, Puri S and Desai R C 1994 Phys. Rev. E 49 3480, arXiv: cond-mat/9312035 . C Walsh, T Ray, Jan N , J. Stat. Phys. 81761Walsh C, Ray T S and Jan N 1995 J. Stat. Phys. 81, 761 . G Barton, A Bray, A J Mckane, Am. J. Phys. 58751Barton G, Bray A J and McKane A J 1990 Am. J. Phys. 58, 751 . C Monthus, G Oshanin, Comtet A Burlatsky, S F , Phys. Rev. E. 54231Monthus C, Oshanin G, Comtet A and Burlatsky S F 1996 Phys. Rev. E 54 231

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{'abstract': "It has been proven by2]that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions are easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a 'growth' parameter from the trivial null solution up to the particular solution found through the factorization procedure.", 'arxivid': 'math-ph/0510072', 'author': ['M A Reyes \nInstituto de Física\nUniversidad de Guanajuato\nLeón, GuanajuatoMexico\n', 'H C Rosu \nPotosí Institute of Science and Technology\nApdo Postal\n3-74, 78231Tangamanga, San Luis PotosíMexico\n'], 'authoraffiliation': ['Instituto de Física\nUniversidad de Guanajuato\nLeón, GuanajuatoMexico', 'Potosí Institute of Science and Technology\nApdo Postal\n3-74, 78231Tangamanga, San Luis PotosíMexico'], 'corpusid': 15284133, 'doi': '10.1088/1751-8113/41/28/285206', 'github_urls': [], 'n_tokens_mistral': 4276, 'n_tokens_neox': 3637, 'n_words': 2207, 'pdfsha': 'f645f7fc04c09227940b1d98578f7a4cfd8ae02a', 'pdfurls': ['https://arxiv.org/pdf/math-ph/0510072v3.pdf'], 'title': ['Riccati-parameter solutions of nonlinear second-order ODEs', 'Riccati-parameter solutions of nonlinear second-order ODEs'], 'venue': ['J. Phys. A: Math. Theor']}

arxiv

Existence of solutions for a k-Hessian equation and its connection with self-similar solutions May 2023 Justino Sánchez jsanchez@userena.cl Departamento de Matemáticas Universidad de La Serena Avenida Cisternas 1200, La SerenaChile Existence of solutions for a k-Hessian equation and its connection with self-similar solutions May 2023 Let α, β be real parameters and let a > 0. We study radially symmetric solutions ofwhere S k (D 2 v) denotes the k-Hessian operator of v. For α ≤ β(n−2k) k and β > 0, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters α and β. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function. Introduction We briefly introduce the class of operators studied in this paper. For a twicedifferentiable function u defined on a domain Ω ⊂ R n , the k-Hessian operator (k = 1, ..., n) is defined by the formula S k (D 2 u) = σ k (Λ) = 1≤i 1 <...<i k ≤n λ i 1 ...λ i k , where Λ = Λ(D 2 u) := (λ 1 , ..., λ n ), the λ's are the eigenvalues of D 2 u and σ k is the k-th elementary symmetric function. Equivalently, S k (D 2 u) is the sum of the k-th principal minors of the Hessian matrix. See, e.g., X.-J. Wang [19,20]. These operators form an important class of second-order operators which contains, as the most relevant examples, the Laplace operator S 1 (D 2 u) = ∆u and the Monge-Ampère operator S n (D 2 u) = det D 2 u. They are fully nonlinear when k > 1. In particular, S 2 (D 2 u) = 1 2 (∆u) 2 − |D 2 u| 2 . The study of k-Hessian equations has many applications in geometry, optimization theory and in other related fields. See [20]. There exists a large literature about existence, regularity and qualitative properties of solutions for the k-Hessian equations, starting with the seminal work of L. Caffarelli, L. Nirenberg and J. Spruck [3]. We point out that the k-Hessian operators are k-homogeneous and also invariant under rotations of coordinates. For more details about these operators, we refer to [20]. As a by-product of our study, we construct self-similar solutions of a k-Hessian evolution equation posed on the whole Euclidean space. We establish that the selfsimilar solutions that we construct present similar properties with those self-similar solutions of the evolution equation u t = ∆u m in the slow diffusion range, m > 1, called the porous medium equation and also with those solutions of the p-Laplacian evolution equation u t = div(|∇u| p−2 ∇u) when p > 2. The self-similar solutions are particular solutions that reflect some symmetries of the underlying equations (whenever they exist) and, although they are isolated objects related to simplified models, they play an important role both in the theory and in the applications. We refer the interested readers to the pioneering book of G.I. Barenblatt [1] for a detailed discussion of this subject. We also point out that there exists an extensive literature about evolution equations that generalize the standard heat equation. This literature addresses, among others equations, the p-Laplacian equation, the porous medium equation and the space-fractional porous medium equation. See e.g. [2,4,5,6,8,9,10,12,13,17,18]. Concerning exact solutions of some nonlinear diffusion equations, we note that in [11] new closed-form similarity solutions of N-dimensional radially symmetric equations were given, which are generalizations of the classical Barenblatt solutions. In [9], the authors studied an explicit equivalence between radially symmetric solutions for two basic models of nonlinear degenerate diffusion equations, namely, the porous medium equation and the p-Laplacian equation. The correspondence in [9] between self-similar radial solutions is obtained by a careful and detailed phase plane analysis. In particular, Iagar et al. derive the existence of new self-similar solutions for the evolution p-Laplacian equation. In [2], a complete classification of radial self-similar solutions of the p-Laplace heat equation is given in the case when p > 2. The method of proof consists in performing a careful phase-plane analysis of the system associated to the second-order equation satisfied by the stationary part of the self-similar solution. In [8] several one-parameter families of explicit self-similar solutions were constructed for the porous medium equations with fractional operators, as for example, u t + (−∆) s u m = 0 and u t = ∇ · (u m−1 ∇(−∆) −s u) for some ranges of s and m. To derive the explicit self-similar solutions, certain special functions are used, such as the modified Bessel function and the (Gauss) hypergeometric function. We prove the existence of radially symmetric solutions of S k (D 2 v) + αv + βξ · ∇v = 0, v > 0 in R n , v(0) = a > 0,(1) where α, β are real parameters, · denotes the standard scalar product in R n , ∇v is the gradient of v and S k (D 2 v) is the k-Hessian operator of v. This equation is related to the study of the self-similar solutions of the k-Hessian evolution equation u t = S k (D 2 u) in (0, T ) × R n ,(2) which can be see as a nonlinear counterpart of the heat equation. Note that this equation satisfies a scaling group invariance: if u is a solution, so is (Su)(t, x) =ũ(t, x) = cu(at, bx), provided that ac k−1 = b 2k , where a, b and c are arbitrary positive numbers (these are the scaling parameters). Those special solutions that are themselves invariant under the scaling group, i.e.,ũ = u are called self-similar solutions. If we impose an extra condition on the solutions, we obtain a corresponding condition on the parameters. Hence, the group of scaling is reduced to a one-parameter family. Note that we recover a well-known scaling for the heat equation (k = 1 in (2)), namely u λ (t, x) = u(λ 2 t, λx) for any positive number λ. The configuration of the parameters α and β in (1) permit us to construct various types of self-similar solutions of equation (2). Following the literature, these are classified into three types, according to the following: Let v be the solution of (1), then the function u(t, x) = t −α v(xt −β ) (called of Type I)(3) is a solution of (2) in (0, ∞) × R n if α(k − 1) + 2kβ = 1. Now, for any T > 0 the function u(t, x) = (T − t) α v(x(T − t) β ) (called of Type II)(4) is a solution of (2) in (0, T ) × R n if α(k − 1) + 2kβ = −1. Finally, the function u(t, x) = e −αt v(xe −βt ) (called of Type III)(5) is an eternal (defined for all times) solution of (2) in (−∞, ∞) × R n if α(k − 1) + 2kβ = 0. As we noted above, the self-similar solutions usually describe the asymptotic behavior of the general solutions of a diffusion equation. Thus, they are of great importance in the general theory of similar equations. We hope that the study of the solutions of (1) is a first step for understand the behavior of solutions of (2). On the other hand, if the parameters α and β are related via α(k − 1) + 2kβ = ρ(6) where ρ ∈ {1, −1, 0} then they are called self-similar exponents. The corresponding solutions v of (1) are then called self-similar profiles, or simply profiles. The relation (6) between the two exponents introduced by (3)-(5) arises from the requirement that v satisfies an equation involving only ξ, i.e., equation (1). Note that, independently of the form of the solutions u(t, x) in ansatz (3)-(5), the profile v satisfies the same profile equation (1). In situations where the problem under consideration satisfies a conservation law (e.g. conservation of mass), the law in question supplies a second condition on α and β, thus fixing the parameters. In this case we speak of self-similar solutions of the first kind. When there are no further restrictions on α and β, one of the exponents α or β is free. In such a case, we speak of self-similar solutions of the second kind. See [1]. For a non-negative function u(t, x), we define the function M(t) by M(t) = R n u(t, x)dx.(7) In some physical models, M(t) represents (when u(t, ·) is integrable on R n ) the total mass in R n at time t of the solution u(t, x) of an evolution PDE. From conservation principles, as we noted above, this quantity remains invariant, that is, it does not depend on the time variable. In Section 3, we construct explicit solutions of type I and type II along which M(t) is constant. We say that a classical solution of (2) blows up at time T < ∞ if lim t↑T sup |u(t, ·)| = ∞. We give examples of solutions of type II, which exhibit finite time blow up for any x in a compact domain or on the whole space. In particular, for the classical heat equation, an infinite family of self-similar solutions of type II that blows up in finite time is given in Section 5. Now, for r = |ξ|, the radially symmetric profile v satisfies the ODE: c n,k r 1−n (r n−k (v ′ ) k ) ′ + αv + βrv ′ = 0, v > 0 on (0, ∞),(8) where prime denotes differentiation with respect to r > 0 and v(0) = a, v ′ (0) = 0.(9) The results obtained here depends strongly on the relation between the parameters α, β, k and the dimension n, as well as on whether k is an odd or an even integer. Theorem 1.1. Let a > 0 and let α, β ∈ R. Let k be an odd integer, 1 ≤ k ≤ n. Assume that α and β satisfy the conditions α ≤ β(n − 2k) k and β > 0.(10) Then there exists a unique solution v of (8) and (9) on (0, ∞). Moreover, for α = 0, δ = β/α, the function E(r) = r 2 v(r) 2δ is such that E ′ (r) > 0 for all r > 0. We will prove this result adapting to our setting a technique of S.-Y. Hsu (Lemma 2.1 in [7]), which avoids the use of the phase-plane method. Remark 1. Although the above result only includes the odd-Hessian, we also find explicit solutions for the even-Hessian for certain values of the parameters α and β. This includes the important case when α = nβ, for which we unify these k-Hessian operators. See Section 3. The plan of the paper is as follows. In Section 2 we will establish the existence and prove various properties of the solutions of (8), (9). The proof of Theorem 1.1 appears in this section. In Section 2 we also give explicit self-similar solutions for particular values of the parameters. Section 3 is devoted to constructing explicit solutions in self-similar form for equation (2) when α = nβ. Using the results of Section 3, in Section 4 we obtain an explicit family of self-similar solutions of type II that blow up in finite time. In Section 5 we find, for the heat equation, a whole family of self-similar solutions that blow up in finite time. This family is given in terms of a confluent hypergeometric function of the first kind, which is called the Kummer function. Finally, the local existence of a solution is proved in the Appendix. Existence of solutions In this section we prove the existence of radially symmetric solutions v of equation (1) which satisfy v(0) = a. Before giving the proof of the existence result, we give an example of explicit solutions for particular values of the parameters α and β. Note that if α = β = 0, then the constant function v = a is the unique solution of (8) and (9), and u is a constant solution of (2). Let α = 0 and β = 0. Here equation (8) can be solved explicitly: either v ′ ≡ 0 (hence v = a and u is a constant solution of (2), as in case α = β = 0), or there exist explicit profiles v(r) = a − C + r 2k k−1 in (0, r) if β > 0 and k even, a + C − r 2k k−1 in (0, ∞) if β < 0,(11) where C ± = k−1 2k (k−1)(±β) (n(k−1)+2k)c n,k 1 k−1 and r = a C + k−1 2k . In particular, for these profiles we have two self-similar solutions of (2), one of type I and other of type II. The corresponding exponents β are 1 k and − 1 k , respectively. More precisely, u(t, x) = a − C + t − 1 k−1 |x| 2k k−1 , 0 ≤ |x| < t 1 2k r, k even, 0, |x| ≥ t 1 2k r(12) and u(t, x) = a + C − (T − t) − 1 k−1 |x| 2k k−1 , (t, x) ∈ (0, T ) × R n .(13) Remark 2. Note that the solution in (12) has compact support in space and this support expands, that is, increases from {0} as t increases from 0. In fact, supp u(t, ·) ⊆ B 0, t 1 2k r . On the other hand, the solution in (13) is defined on the whole space R n and blows up everywhere in x as t ↑ T . Lemma 2.1. Let k be an odd integer, 1 ≤ k ≤ n. Let α, β = 0. Assume that α and β satisfy inequality kα β ≤ n − 2k. For any R 0 > 0 and a > 0, let v be the solution of (8) and (9) in (0, R 0 ). Then v(r) + δrv ′ (r) > 0 in [0, R 0 )(15) and v ′ (r) < 0 in (0, R 0 ) if α > 0, v ′ (r) > 0 in (0, R 0 ) if α < 0.(16) Proof. Let h(r) = v(r) + δrv ′ (r). By (10), we have n − 2k ≥ k/δ. Then, by direct computation, h ′ + n−2k k − 1 δ r + β kc n,k r k (v ′ ) 1−k h = n−2k k − 1 δ r v ≥ 0 in (0, R 0 ).(17) Now let f (r) = exp β kc n,k r 0 ρ k (v ′ (ρ)) 1−k dρ .(18) Note that the function f (r) is well-defined since the integrand has a finite limit at ρ = 0. In fact, using (8), (9) and L'Hopital's rule, we have lim r↓0 v ′ (r) r k = lim rց0 −α c n,k r n−1 h(r) nr n−1 = −αa nc n,k ∈ R \ {0}. By (17) (r n−2k (15) follows. By (8), (9) and (15), k − 1 δ f (r)h(r)) ′ ≥ 0 ∀ 0 < r < R 0 ⇒ r n−2k k − 1 δ f (r)h(r) > 0 ∀ 0 < r < R 0 ⇒ h(r) > 0 ∀ 0 < r < R 0 andc n,k r 1−n (r n−k (v ′ ) k ) ′ = −αh(r) < 0 in (0, R 0 ) if α > 0, > 0 in (0, R 0 ) if α < 0 ⇒ r n−k (v ′ ) k < 0 in (0, R 0 ) if α > 0, r n−k (v ′ ) k > 0 in (0, R 0 ) if α < 0 and (16) follows, since k is an odd integer. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. We may assume that α = 0, since the solutions are given explicitly by (12)-(13) when α = 0. We note that the uniqueness of the solutions of (8) and (9) in (0, ∞) follows from the standard ODE theory. Hence we only need to prove the existence of solutions of (8) and (9) in (0, ∞). The local existence of solutions of (8) and (9) in a neighbourhood of the origin follows from classical arguments involving the Banach fixed point theorem (see the Apenndix). Let (0, R 0 ) be the maximal interval of the existence of solution of (8) and (9). Suppose R 0 < ∞. Then there exists a sequence {r i } ∞ i=1 , r i ↑ R 0 as i → ∞ such that either |v ′ (r i )| → ∞ as i → ∞ or v(r i ) ↓ 0 as i → ∞ or v(r i ) → ∞ as i → ∞. By Lemma 2.1, we have E ′ (r) = 2rv 2δ + 2δr 2 v 2δ−1 v ′ = 2rv 2δ−1 (v + δrv ′ ) > 0 ∀ 0 < r < R 0 .(19) We divide the proof into two cases, depending on the sign of α. Case 1: α > 0. By (10), inequality (14) holds. Hence by (19), E(r) = r 2 v 2δ ≥ E(R 0 /2) > 0 ∀ R 0 /2 ≤ r < R 0 ⇒ v(r) ≥ (R −2 0 E(R 0 /2)) 1 2δ ∀ R 0 /2 ≤ r < R 0(20) By Lemma 2.1, we have v ′ < 0 on (0, R 0 ). Hence 0 < v(r) ≤ v(0) = a ∀ 0 ≤ r < R 0 .(21) By (8), (9) and (21), if 0 < r < R 0 , we have c n,k r 1−n (r n−k (v ′ ) k ) ′ = −(αv + βrv ′ ) ⇒ c n,k r n−k (v ′ ) k = − α r 0 ρ n−1 v(ρ)dρ + β r 0 ρ n v ′ (ρ)dρ ⇒ c n,k r n−k (v ′ ) k = −βr n v(r) + (nβ − α) r 0 ρ n−1 v(ρ)dρ (22) ⇒ c n,k (v ′ ) k = −βr k v(r) + nβ − α r n−k r 0 ρ n−1 v(ρ)dρ ⇒ c n,k |v ′ (r)| k ≤ β + |nβ − α| n R k 0 v(0) c n,k |v ′ (r)| ≤ β + |nβ − α| n 1 k R 0 v(0) 1 k .(23) By (20), (21) and (23), we have obtained a contradiction. Hence no such sequence {r i } ∞ i=1 exists. Thus R 0 = ∞ and there exists a unique solution of (8) and (9) in (0, ∞). Case 2: α < 0. By Lemma 2.1, 0 < v ′ (r) ≤ v(r) |δ| r in (0, R 0 ).(24) Choose r 0 ∈ (0, R 0 ) and let C 0 = max 0≤r≤r 0 v ′ (r). Then by (24) 0 < v ′ (r) ≤ C 0 + v(r) |δ| r ≤ Cv(r) ∀ r ∈ (0, R 0 ), where C = C 0 v(0) + (|δ| r 0 ) −1 > 0. Then v(0) ≤ v(r) ≤ v(0) exp(CR 0 ) ∀ 0 ≤ r < R 0(25) and 0 < v ′ (r) ≤ Cv(0) exp(CR 0 ) ∀ 0 ≤ r < R 0 .(26) By (25) and (26), we again obtain a contradiction. Hence no such sequence {r i } ∞ i=1 exists. Thus R 0 = ∞ and there exists a unique solution of (8) and (9) in (0, ∞). By (19) and Cases 1 and 2, the theorem follows. Proof. Let a > 0 and denote by (0, R 0 ) the maximal interval where a solution v of (8) and (9) exists. Suppose R 0 < ∞. Then there exists a sequence {r i } ∞ i=1 , r i ↑ R 0 as i → ∞ such that either |v ′ (r i )| → ∞ as i → ∞ or v(r i ) ↓ 0 as i → ∞ or v(r i ) → ∞ as i → ∞. Multiplying (8) by r n−1 and taking β = 0, we have (c n,k r n−k (v ′ (r)) k ) ′ = |α| r n−1 v(r) > 0 ∀ 0 < r < R 0 .(27) Since k is odd, it follows that v ′ > 0 on (0, R 0 ). Hence 0 < a = v(0) ≤ v(r) ∀ 0 ≤ r < R 0 .(28) Integrating (27), if 0 < r < R 0 , r n−k (v ′ (r)) k = |α| c n,k r 0 s n−1 v(s)ds ⇒ r n−k (v ′ (r)) k ≤ |α| nc n,k r n v(r) ⇒ (v ′ (r)) k ≤ |α| nc n,k r k v(r) ⇒ 0 < v ′ (r) ≤ |α| nc n,k 1 k r(v(r)) 1 k . Then v(0) ≤ v(r) ≤ v(0) k−1 k + |α| nc n,k 1 k R 2 0 2 k k−1 := C ∀ 0 ≤ r < R 0(29) and 0 < v ′ (r) ≤ |α| C nc n,k 1 k R 0 ∀ 0 ≤ r < R 0 .(30) By (28), (29) and (30), we obtain a contradiction. Hence no such sequence {r i } ∞ i=1 exists. Thus R 0 = ∞ and there exists a unique solution of (8) and (9) in (0, ∞). In particular, when k > 1 is an odd integer, we obtain a self-similar solution of type II for equation (2), which blows up in finite time. The solution is given in separated variables of the form u(t, x) = (T − t) − 1 k−1 v(x), where v(x) = v(|x|) is the unique positive solution of the equation Proof. Let a > 0 and denote by (0, R 0 ) the maximal interval where a solution v of (8) and (9) exists. Suppose R 0 < ∞. Then there exists a sequence S k (D 2 v) = 1 k−1 v in R n .{r i } ∞ i=1 , r i ↑ R 0 as i → ∞ such that either |v ′ (r i )| → ∞ as i → ∞ or v(r i ) ↓ 0 as i → ∞ or v(r i ) → ∞ as i → ∞. We only consider the case 0 > nβ > α (since the case 0 > nβ = α is similar). By (22), c n,k r n−k (v ′ (r)) k > 0 for all 0 < r < R 0 . Hence v ′ (r) > 0 for all 0 < r < R 0 . Then by (22), c n,k r n−k (v ′ ) k ≤ −βr n v(r) − (α − nβ) r 0 ρ n−1 v(r)dρ = |α| n r n v(r) The rest of the proof is as in Lemma 2.2, with the obvious changes in case k = 1. Exact solutions for α = nβ = 0 Let k = 1. In this section we consider the case α = nβ. We construct explicit solutions in self-similar form. By (22), we obtain the separable ODE c n,k r n−k (v ′ (r)) k = −βr n v(r). We solve this equation under the conditions v > 0 and v(0) = a > 0. For this, we consider two cases. Case 1: α < 0. From (31), we obtain (v ′ (r)) k = |β| c n,k r k v(r) = |α| nc n,k r k v(r). Now, according to the parity of k, we have two subcases. • If k is odd, then v ′ (r) = |α| nc n,k 1 k rv(r) 1 k > 0. Integrating this equation from 0 to r, we obtain k k − 1 v k−1 k − a k−1 k = |α| nc n,k 1 k r 2 2 v k−1 k = a k−1 k + k − 1 2k |α| nc n,k 1 k r 2 v(r) = a k−1 k + k − 1 2k |α| nc n,k 1 k r 2 k k−1 ∀r > 0. • If k is even, then |v ′ (r)| = |α| nc n,k 1 k rv(r) 1 k . Note that if v ′ (r) > 0, then we arrive at the same solution as in the case k odd. If v ′ (r) < 0, we have v − 1 k v ′ = − |α| nc n,k 1 k r < 0. Integrating this equation from 0 to r (as long as v > 0), we obtain v(r) = a k−1 k − k − 1 2k |α| nc n,k 1 k r 2 k k−1 ∀ 0 ≤ r < r * , where r * is the positive root of the quadratic equation a To do so, define v(r) = 0 for all r ≥ r * . This has the effect of extending the solution v to all r ≥ 0. Case 2: α > 0. Clearly k odd is allowed and thus v ′ (r) < 0 (as long as v > 0). In this case, from (31), we obtain (−v ′ (r)) k = α nc n,k r k v(r) > 0. Solving this equation we obtain the same solution as in Case 1, where k is even and α < 0. We conclude that, when v ′ < 0 and α = nβ, it is possible to unify Cases 1 and 2 for all k. Further, in this case we can write v(r) = C − k − 1 k β c n,k 1 k r 2 2 k k−1 + , r ≥ 0,(32) where C = a k−1 k and (·) + denotes the positive part. Observe that this class of profiles is similar to the classical class of Barenblatt profiles for the porous medium equation (m > 1) and the p-Laplacian evolution equation (p > 2), as we noted in the Introduction. See, e.g., [9]. Note that the quantity M(t) defined in (7) is conserved for self-similar solutions when α = nβ and the profile v is integrable in R n , such as in (32). This is the case for self-similar solutions of type I with β = β = − 1 n(k−1)+2k < 0. However, M(t) is not conserved for eternal solutions of type III for any β = 0 since, in the case of our equation, for those solutions the exponents α and β have opposite sings. In case of type I solutions, we recover the family of self-similar positive solutions with finite "mass" found in [14]. For completeness, we recall its explicit form: U C (t, x) = t −α C − γ |x| t β 2 k k−1 + ,(33) where (·) + denotes the positive part, C > 0 is an arbitrary constant, and α, β and γ have explicit values, namely α = n n(k − 1) + 2k , β = 1 n(k − 1) + 2k , γ = k − 1 2k β c n,k 1 k , c n,k = 1 n n k . Note that this family is well defined for the full range of k-Hessian operators with k = 1. In the language of [2], U C in (33) has an expanding support. Finally, we can interpret the initial condition U C (0, x) as Mδ 0 (x) in the sense that U C (t, x) → Mδ 0 (x) as t ↓ 0, where δ 0 (x) is the Dirac delta function concentrated at 0 and M = R n U C (t, x) dx. In other words, lim t↓0 R n U C (t, x)ϕ(x)dx = Mϕ(0) for all ϕ ∈ C ∞ c (R n ). This statement is a particular case of Lemma 5.4 in [16] applied to our setting. A family of blow up solutions We now give some explicit type II self-similar blow up solutions for equation (2). By (4), they have the form u(t, x) = (T − t) α v(ξ), ξ = x(T − t) β , t ∈ [0, T ), T < ∞, x ∈ R n .(34) If there is only one condition on the scaling exponents α and β, which here is given by (6) with ρ = −1, that is, α(k − 1) + 2kβ = −1, then it is not enough to determine α and β explicitly. In particular, when k = 1, we obtain β = − 1 2 leaving α free. Thus, for a radially symmetric function v, (8) reduces to the equation −αv + 1 2 rv ′ = r 1−n (r n−1 v ′ ) ′ . However, the choice α = − n 2 (= nβ) allows for the integration of the last equation easily, yielding the profile v(r) = a e r 2 4 , where a > 0 is a constant. Inserting all the information in (34), we write a self-similar solution for equation (2) that blows up in finite time T and has infinite mass u(t, x) = a(T − t) − n 2 e |x| 2 4(T −t) . This solution has the simplest form. In the next section, we obtain a whole family of blows up solutions for the heat equation. Now let k = 1 and α = nβ. Using the results obtained in the previous section, we can write in a closed form the profiles and then derive explicit self-similar solutions which blow up in finite time. A surprising fact is that when k is an even integer we have a profile with compact support and when k is an odd integer we have a profile without compact support. Since the profiles are radial, necessarily the support is a ball or the whole space. Summing up, the following are self-similar solutions of type II for equation (2) that blow up in finite time T : u(t, x) = (T − t) α a k k−1 − k − 1 2k |β| c n,k 1 k |x| 2 (T − t) 2|β| k k−1 + (k even), u(t, x) = (T − t) α a k k−1 + k − 1 2k |β| c n,k 1 k |x| 2 (T − t) 2|β| k k−1 (k odd), where a > 0 is an arbitrary constant, α = − n n(k−1)+2k and β = − 1 n(k−1)+2k . A family of blow up solutions for the heat equation In this section we find, for the heat equation, a whole family of self-similar solutions of type II that blows up in finite time. Recall that, from the previous section, the choice k = 1 and ρ = −1 in (6) fixes the value of the exponent β to β = − 1 2 leaving α free. We are thus left with a continuum of admissible scaling exponents α < 0, as is typical for self-similarity of the second kind. A discretely infinite sequence of exponents α m is, however, selected by requiring that the family be simpler to compute. The ODE equation for the profile v is −αv + 1 2 rv ′ = r 1−n (r n−1 v ′ ) ′ . In order to obtain more precise information about the solutions, we rewrite this equation as r 2 v ′′ + r n − 1 − r 2 2 v ′ + αr 2 v = 0.(35) Since our attention is focused on blow up solutions, we set α = −γ < 0, with the restriction γ ≥ n 2 due to Lemma 2.3. To find explicit solutions, we introduce the new variables z = r 2 4 , w(z) = v(r), with which equation (35) takes the form z d 2 w dz 2 + n 2 − z dw dz − γw = 0. Collecting all the information, we give two explicit self-similar solutions of type II for the heat equation that blow-up in finite time T : u α 1 (t, x) = a(T − t) − n+2 2 1 + |x| 2 2n(T − t) e |x| 2 4(T −t) , u α 2 (t, x) = a(T − t) − n+4 2 1 + |x| 2 n(T − t) + |x| 4 4n(n + 2)(T − t) 2 e |x| 2 4(T −t) , where a > 0 is an arbitrary constant. As far as we know, the family in (39) is new. Finally, if we put in (39) α m = − n 2 − m, with m = 0, 1, 2, ..., we obtain an infinite family of explicit self-similar solutions of type II for the heat equation that blow up in finite time T , namely u αm (t, x) = a(T − t) − n+2m 2 ∞ s=0 ( n 2 + m) s ( n 2 ) s |x| 2 4(T −t) s s! , where a > 0 is an arbitrary constant. Note that when m = 0, we recover the solution obtained in the previous section. for all r ∈ [0, ρ]. Choose ρ small enough so that |J (ϕ)(r) − a| ≤ δ, ϕ ∈ B δ (a). Hence J (ϕ) ∈ B δ (a). Next we show that J is a contraction in some interval [0,r], wherer =r(a). Recall that, ifr is small enough, the ball B δ (a) is invariant under J , i.e., J (B δ (a)) ⊂ B δ (a). For suchr and any pair ϕ, ψ ∈ B δ (a), we have r 2 ϕ − ψ ∞ for any r ∈ [0,r]. Choosingr small enough, J is a contraction. The Banach fixed point theorem now implies the existence of a unique fixed point of J in B δ (a), which is a solution of (40) and, consequently, of (8) and (9). As usual, this solution can be extended to a maximal interval [0, R 0 ), 0 < R 0 ≤ ∞. Lemma 2. 2 . 2Let k odd. Let a > 0, α < 0 and β = 0. Then (8) and (9) have a solution in (0, ∞). Lemma 2 . 3 . 23Let k odd. Let a > 0 and 0 > nβ ≥ α. Then (8) and (9) have a solution in (0, ∞). that we are looking for a solution defined on the whole space. So, if we do not impose the condition v > 0, then the solution may take zero or negative values. To avoid this, we can cut off the unwanted part of the solution. a solution of the equation ϕ = J (ϕ) in B δ (a) where ρ and δ are small positive numbers which will be chosen sufficiently close to zero. Obviously J (ϕ) ∈ C([0, ρ]), and from the definition of B δ (a), ϕ(r) ∈ [a − δ, a + δ] for all r ∈ [0, ρ]. Moreover, simple calculations show that, for small δ, F (ϕ) is positive on [0, ρ] for all ϕ ∈ B δ (a). More precisely, we have F (ϕ)(s) ≥ As k , for all s ∈ [0, As k |F (ϕ)(s)| ds for r ∈ [0, ρ].On the other hand, |F (ϕ)(s)| ≤ Cs k , where C = F (ϕ)(s)) − G(F (ψ)(s))| ds(44)where F (ϕ) is given by (42). Now, defineH(s) = min{F (ϕ)(s), F (ψ)(s)}.As a consequence of estimate (43), we haveH(s) ≥ As k for 0 ≤ s ≤ r ≤r, whence |G(F (ϕ)(s)) − G(F (ψ)(s))| ≤ G(H(s)) H(s) (F (ϕ)(s) − F (ψ)(s)) ≤ G(As k ) As k |F (ϕ)(s) − F (ψ)(s)| . |F (ϕ)(s) − F (ψ)(s)| ≤C ϕ − ψ ∞ s k , n(k−1)+2k > 0, and also type II with AcknowledgementsThe author has been supported by ANID Fondecyt Grant Number 1221928, Chile.The above is a particular case of the so-called confluent hypergeometric equationwith a and b constants. In general, the parameters a, b and the variables z, w may take complex values. This is also known as Kummer's equation, which is one of the most important differential equations in physics, chemistry, and engineering. Equation (36) has a regular singularity at z = 0 with index 0. Thus, we have a power series representing w. This power series is called the confluent hypergeometric function of the first kind, called Kummer's function, which is defined bywhere the Pochhammer symbol (a) s is defined byNote that M(a, b; z) is an entire function of z (provided it exists). Among the many properties of this function, we highlight that M(a, b; 0) = 1 (if b is not a non-positive integer) and M(a, a; z) = e z . When a = γ(= −α), b = n 2 and z = r 2 4 in (37), an explicit solution of equationBy the asymptotic behavior of the Kummer function as |z| → ∞ (if Re(z) > 0), these solutions have very large growth as r → ∞. For other properties of this special function, see[15]. Note that, when α = − n 2 , we recover the profile found in the previous section, namely v(r) = e Therefore, from (38), we obtain a family of self-similar solutions of type II for the heat equation that blows up in finite time, namelywhere a > 0 is an arbitrary constant and α ∈ (−∞, − n 2 ].Appendix: Existence of a local solutionIn this appendix we prove the existence of a local solution v of equation(8)which satisfy v(0) = a > 0. The procedure via fixed point arguments is standard, but for the sake of completeness we will give the corresponding proof. For α = 0, we need to consider two cases, according to the sign of α. We only consider α, β > 0, the other cases being similar. Note that, for α > 0, the equation(8)is equivalent to the integral equationwhere G(s) = αs c n,k Similarity, self-similarity, and intermediate asymptotics. G I Barenblatt, New York-LondonConsultants BureauG.I. Barenblatt. Similarity, self-similarity, and intermediate asymptotics. Con- sultants Bureau [Plenum], New York-London, 1979. Self-similar solutions of the p-Laplace heat equation: the case when p > 2. M F Bidaut-Véron, Proc. Roy. Soc. Edinburgh Sect. A. 1391M.F. Bidaut-Véron. Self-similar solutions of the p-Laplace heat equation: the case when p > 2. Proc. Roy. Soc. Edinburgh Sect. A, 139(1):1-43, 2009. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. L Caffarelli, L Nirenberg, J Spruck, Acta Math. 1553-4L. Caffarelli, L. Nirenberg, and J. Spruck. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math., 155(3-4):261-301, 1985. Rate of convergence to a singular steady state of a supercritical parabolic equation. M Fila, M Winkler, J. Evol. Equ. 84M. Fila and M. Winkler. Rate of convergence to a singular steady state of a supercritical parabolic equation. J. Evol. Equ., 8(4):673-692, 2008. Rate of convergence to separable solutions of the fast diffusion equation. M Fila, M Winkler, Israel J. Math. 2131M. Fila and M. Winkler. Rate of convergence to separable solutions of the fast diffusion equation. Israel J. Math., 213(1):1-32, 2016. A stability technique for evolution partial differential equations. V A Galaktionov, J L Vázquez, Nonlinear Differential Equations and their Applications. Boston, MABirkhäuser Boston, Inc56A dynamical systems approachV.A. Galaktionov and J.L. Vázquez. A stability technique for evolution partial differential equations, volume 56 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA, 2004. A dynamical systems approach. Singular limit and exact decay rate of a nonlinear elliptic equation. S Y Hsu, Nonlinear Anal. 757S.Y. Hsu. Singular limit and exact decay rate of a nonlinear elliptic equation. Nonlinear Anal., 75(7):3443-3455, 2012. Explicit Barenblatt profiles for fractional porous medium equations. Y Huang, Bull. Lond. Math. Soc. 464Y. Huang. Explicit Barenblatt profiles for fractional porous medium equations. Bull. Lond. Math. Soc., 46(4):857-869, 2014. Radial equivalence for the two basic nonlinear degenerate diffusion equations. R Iagar, A Sánchez, J L Vázquez, J. Math. Pures Appl. 899R. Iagar, A. Sánchez, and J.L. Vázquez. Radial equivalence for the two basic nonlinear degenerate diffusion equations. J. Math. Pures Appl. (9), 89(1):1-24, 2008. Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. S Kamin, J L Vázquez, Rev. Mat. Iberoamericana. 42S. Kamin and J.L. Vázquez. Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana, 4(2):339-354, 1988. Exact similarity solutions to some nonlinear diffusion equations. J R King, J. Phys. A. 2316J.R. King. Exact similarity solutions to some nonlinear diffusion equations. J. Phys. A, 23(16):3681-3697, 1990. Classification of type I and type II behaviors for a supercritical nonlinear heat equation. H Matano, F Merle, J. Funct. Anal. 2564H. Matano and F. Merle. Classification of type I and type II behaviors for a supercritical nonlinear heat equation. J. Funct. Anal., 256(4):992-1064, 2009. Superlinear parabolic problems. Birkhäuser Advanced Texts: Basler Lehrbücher. P Quittner, Ph, Souplet, Birkhäuser Advanced Texts: Basel TextbooksP. Quittner and Ph. Souplet. Superlinear parabolic problems. Birkhäuser Ad- vanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Text- books]. Blow-up, global existence and steady states. Birkhäuser VerlagBaselBirkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states. Asymptotic behavior of solutions of a k-Hessian evolution equation. J Sánchez, J. Differential Equations. 2684J. Sánchez. Asymptotic behavior of solutions of a k-Hessian evolution equation. J. Differential Equations, 268(4):1840-1853, 2020. Confluent hypergeometric functions. L J Slater, Cambridge University PressNew YorkL.J. Slater. Confluent hypergeometric functions. Cambridge University Press, New York, 1960. Asymptotic behaviour for the porous medium equation posed in the whole space. J L Vázquez, 3Dedicated to Philippe BénilanJ.L. Vázquez. Asymptotic behaviour for the porous medium equation posed in the whole space. volume 3, pages 67-118. 2003. Dedicated to Philippe Bénilan. The porous medium equation. Mathematical theory. J L Vázquez, Oxford University PressOxfordOxford Mathematical MonographsJ.L. Vázquez. The porous medium equation. Mathematical theory. Oxford Math- ematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. On the Cauchy problem for reaction-diffusion equations. X Wang, Trans. Amer. Math. Soc. 3372X. Wang. On the Cauchy problem for reaction-diffusion equations. Trans. Amer. Math. Soc., 337(2):549-590, 1993. A class of fully nonlinear elliptic equations and related functionals. X.-J Wang, Indiana Univ. Math. J. 431X.-J. Wang. A class of fully nonlinear elliptic equations and related functionals. Indiana Univ. Math. J., 43(1):25-54, 1994. The k-Hessian equation. X.-J Wang, Geometric analysis and PDEs. DordrechtSpringer1977X.-J. Wang. The k-Hessian equation. In Geometric analysis and PDEs, volume 1977 of Lecture Notes in Math., pages 177-252. Springer, Dordrecht, 2009.

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{'abstract': 'Let α, β be real parameters and let a > 0. We study radially symmetric solutions ofwhere S k (D 2 v) denotes the k-Hessian operator of v. For α ≤ β(n−2k) k and β > 0, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters α and β. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function.', 'arxivid': '2305.19364', 'author': ['Justino Sánchez jsanchez@userena.cl \nDepartamento de Matemáticas\nUniversidad de La Serena Avenida Cisternas 1200, La SerenaChile\n'], 'authoraffiliation': ['Departamento de Matemáticas\nUniversidad de La Serena Avenida Cisternas 1200, La SerenaChile'], 'corpusid': 258988579, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12632, 'n_tokens_neox': 11187, 'n_words': 6720, 'pdfsha': '1c1e1281cd8189eb43d0023fe5fcc51bc26118fa', 'pdfurls': ['https://export.arxiv.org/pdf/2305.19364v1.pdf'], 'title': ['Existence of solutions for a k-Hessian equation and its connection with self-similar solutions', 'Existence of solutions for a k-Hessian equation and its connection with self-similar solutions'], 'venue': []}

arxiv

Mimetic Spectral Element Method for Anisotropic Diffusion Marc Gerritsma Artur Palha Varun Jain Yi Zhang Mimetic Spectral Element Method for Anisotropic Diffusion This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids. Introduction Anisotropic and inhomogeneous diffusion appears in many applications such as heat transfer [15], flow through porous media [87], turbulent fluid flow [116], image processing [98] or plasma physics [112]. In 2D, steady, anisotropic diffusion is governed by the following elliptic partial differential equation Here, p is the flow potential, f the source term, with p =p along Γ p and (K∇p, n n n) = u n along Γ u . Here, for all x x x, K(x x x) is a symmetric, positive definite tensor. In the presence of strong anisotropy, i.e. large ratio between the smallest and largest eigenvalues of the diffusion tensor, the construction of robust and efficient discretizations becomes particularly challenging. Under these conditions, the convergence rates of the discretization error can be considerably reduced; this effect is commonly referred in the literature as locking effect, see for example [4,5,12,84]. For sufficiently refined discretizations, the deterioration of the convergence rates eventually disappears. Unfortunately, this may occur only when the grid cell size is prohibitively small. Another important aspect is mesh flexibility. In many applications of diffusion equations, particularly in porous media flow, typical grids are highly irregular. In many of these situations the results obtained are strongly dependent on the grid type, see [11] for a discussion of the use and properties of different grids in reservoir modelling. Overview of standard discretizations In order to overcome these limitations and improve the efficiency and robustness of the discretization of the anisotropic diffusion equations, several approaches have been proposed. The discretization of the anisotropic diffusion equations in complex media in many situations is still a trade-off between, e.g. [89]: • Accuracy in the representation of the medium (complex grids). • Accuracy in the discretization of the equations. The need for such a choice is rooted in the use of numerical schemes based on two-point flux approximations (TPFA), see for example, [3,89,120]. These methods produce good approximations on orthogonal grids when the diffusion tensor K is diagonal, but are known to introduce significant discretization errors in the presence of a non-diagonal diffusion tensor. This introduces severe limitations into the possible grid choices. Under these conditions, the geometric flexibility introduced by perpendicular bisector (PEBI) grids, [11,67,90], is considerably limited, for example. It has been known that the discretization error is related to the misalignment between the grid and the principal directions of the diffusion tensor K. In fact, Aavatsmark showed in [3] that for TPFA this misalignment leads to the discretization of the wrong diffusion tensor. These ideas initially led to the construction of grids aligned with the principal axis of the diffusion tensor, so called K-orthogonal grids, see for example [65,67]. This approach significantly improves the performance of the numerical method but substantially limits the geometric flexibility. More recently, multipoint flux-approximation (MPFA) schemes have been introduced specifically to address these limitations, see e.g. the initial works by Aavatsmark [4,5] or a more recent presentation [2], and by Edwards and Rogers [57]. This method is based on a cell-centred finite volume formulation and introduces a dual grid in order to generate shared sub-cells and sub-interfaces. This in turn produces a discretization of the flux between two cells that involves a linear combination of several adjacent cells. This method is robust and locally conservative but does not guarantee a resulting symmetric discrete diffusion operator. More recently, this work has been connected to the mixed finite element method, [56]. Alternative approaches based on the finite element formulation have also been proposed by several authors. We briefly mention the work on the control-volume finite element discretization by Forsyth [60] and Durlofsky [54], on nodal Galerkin finite elements by Young [122], and on mixed finite elements by Durlofsky [53]. Overview of mimetic discretizations Over the years, the development of numerical schemes that preserve some of the structures of the differential models they approximate has been identified as an important ingredient of numerical analysis. One of the contributions of the formalism of mimetic methods is to identify differential geometry as the proper language in which to encode these structures/symmetries. Another novel aspect of mimetic discretizations is the identification and separation of physical field laws into two sets: (i) topological relations (metric-free), and (ii) constitutive relations (metric dependent). Topological relations are intimately related to conservation laws and can (and should) be exactly represented on the computational grid. Constitutive relations include all material properties and therefore are approximate relations. For this reason, all numerical discretization error should be included in these equations. A general introduction and overview of spatial and temporal mimetic/geometric methods can be found in [39,42,66,100]. The relation between differential geometry and algebraic topology in physical theories was first established by Tonti [117]. Around the same time Dodziuk [52] set up a finite difference framework for harmonic functions based on Hodge theory. Both Tonti and Dodziuk introduce differential forms and cochain spaces as the building blocks for their theory. The relation between differential forms and cochains is established by the Whitney map (k-cochains → k-forms) and the de Rham map (k-forms → k-cochains). The interpolation of cochains to differential forms on a triangular grid was already established by Whitney, [119]. These generalized interpolatory forms are now known as Whitney forms. Hyman and Scovel [75] set up the discrete framework in terms of cochains, which are the natural building blocks of finite volume methods. Later, Bochev and Hyman [18] extended this work and derived discrete operators such as the discrete wedge product, the discrete codifferential, and the discrete inner products. Robidoux,Hyman,Steinberg and Shashkov,[74,[76][77][78]107,108,111,113,114] used symmetry considerations to construct discretizations on rough grids, within the finite difference/volume setting . In a more recent paper by Robidoux and Steinberg [110] a finite difference discrete vector calculus is presented. In that work, the differential operators grad, curl and div are exactly represented at the discrete level and the numerical approximations are all contained in the constitutive relations, which are already polluted by modeling and experimental error. For mimetic finite differences, see also the work of Brezzi et al. [32,33] and Beirão da Veiga et al. [47]. The application of mimetic ideas to unstructured triangular staggered grids has been extensively studied by Perot, [99,[101][102][103]123], specially in [100] where the rationale of preserving symmetries in numerical algorithms is well described. The most geometric approach is presented in the work by Desbrun et al. [49,58,86,97] and the thesis by Hirani [72]. The Japanese papers by Bossavit, [26][27][28][29][30], serve as an excellent introduction and motivation for the use of differential forms in the description of physics and the use in numerical modeling. The field of application is electromagnetism, but these papers are sufficiently general to extend to other physical theories. In a series of papers by Arnold, Falk and Winther, [8][9][10], a finite element exterior calculus framework is developed. Higher order methods are described by Rapetti [104,105] and Hiptmair [71]. Possible extensions to spectral methods were described by Robidoux, [109]. A different approach for constructing arbitrary order mimetic finite elements has been proposed by the authors [31,64,92,96], with applications to advection problems [95], Stokes' flow [81], MHD equilibrium [94], Navier-Stokes [93], and within a Least-Squares finite element formulation [16,62,63,91]. Extensions of these ideas to polyhedral meshes have been proposed by Ern, Bonelle and co-authors in [22][23][24][25], by di Pietro and co-authors in [50,51], by Brezzi and co-authors in [34], and by Beirão da Veiga and co-authors in [44][45][46]48]. These approaches provide more geometrical flexibility while maintaining fundamental structure preserving properties. Mimetic isogeometric discretizations have been introduced by Buffa et al. [40], Evans and Hughes [59], and Hiemstra et al. [70]. Another approach to develop a discretization of the physical field laws is based on a discrete variational principle for the discrete Lagrangian action. This approach has been used in the past to construct variational integrators for Lagrangian systems, e.g. [79,85]. Kraus and Maj [80] have used the method of formal Lagrangians to derive generalized Lagrangians for non-Lagrangian systems of equations. This allows to apply variational techniques to construct structure preserving discretizations on a much wider range of systems. Recently, Bauer and Gay-Balmaz presented variational integrators for elastic and pseudo-incompressible flows [14]. Due to the inherent challenges in discretizing the diffusion equations with anisotropic diffusion tensor K, several authors have explored different mimetic discretizations of these equations. Focussing on generalized diffusion equations we highlight [13, 69, 74, 76-78, 102, 107, 108, 111, 113, 114] for a finitedifference/finite-volume setting, [23,[36][37][38] for polyhedral discretizations, and [19,21,96,106,121] for a finite element/mixed finite element setting. For applications to Darcy flow equations and reservoir modelling see for example [1,6,7,55,73,83,89]. Outline of chapter In Section 2 the topological structure of anisotropic diffusion problems is discussed. In Section 3 spectral basis functions are introduced which are compatible with the topological structure introduced in Section 2. In Section 4 transformation to curvilinear elements is discussed. Results of the proposed method are presented in Section 5. Anisotropic diffusion / Darcy problem Let Ω ⊂ R d be a contractible domain with Lipschitz continuous boundary ∂ Ω = Γ p ∪ Γ u , Γ p ∩ Γ u = ∅. The steady anisotropic diffusion problem is given by − ∇ · (K∇p) = f ,(2) with p =p along Γ p and (K∇p, n n n) =ū n along Γ u . Here, for all x x x, K(x x x) is a symmetric, positive definite tensor, i.e. there exist constants α,C > 0 such that αξ ξ ξ T ξ ξ ξ ≤ ξ ξ ξ T K(x x x)ξ ξ ξ ≤ Cξ ξ ξ T ξ ξ ξ . If Γ p = ∅, then (2) has a unique solution. If Γ p = ∅ then (2) only possesses solutions if ∂ Ωū n dS = Ω f dΩ , in which case the solution, p, is determined up to a constant. An equivalent first order system is obtained by introducing u u u = −K∇p in which case (2) can be written as u u u + K∇p = 0 in Ω ∇ · u u u = f in Ω with (u u u, n n n) =ū n along Γ u p =p along Γ p .(3) An alternative first-order formulation is given by      v v v − ∇p = 0 in Ω u u u + Kv v v = 0 in Ω ∇ · u u u = f in Ω with (u u u, n n n) =ū n along Γ u p =p along Γ p .(4) Formulation (3) is generally referred to as the Darcy problem, while the relation u u u = −K∇p is called Darcy's law, [87]. The Darcy problem plays an important role in reservoir engineering. In this case u u u is the flow velocity in a porous medium and p denotes the pressure. While the formulations (2), (3) and (4) are equivalent, (2) only has 1 unknown, p, (3) has (d + 1) unknowns, p and the d components of u u u, and (4) has (2d + 1) unknowns. Formulation (4) is of special interest, because it decomposes the anisotropic diffusion problem into two topological conservation laws and one constitutive law 1 . By making a suitable choice where and how to represent the unknowns on a grid, the topological relations, v v v − ∇p = 0 and ∇ · u u u = f reduce to extremely simple algebraic relations which depend only on the topology of the mesh and are independent of the mesh size, independent of the shape of the mesh, and independent of the order of the numerical scheme. We will refer to such discretizations as exact discrete representations. Gradient relation Consider two points A, B ∈ Ω and a curve C which connects these two points, then v v v − ∇p = 0 =⇒v v v C := C v v v · dl = B A v v v · dl = B A ∇p · dl = p(B) − p(A) , where dl is a small increment along the curve C . Suppose that we take another curveC which connects the two points A and B then we also havev v vC : = C v v v · dl = p(B) − p(A) ,(6) The integral along C is equal to the integral alongC . We will refer tov v v as an integral value, since it denotes an integral and not a point-wise evaluation of v v v. The advantages of integral values are: 1 An even more extended system is, see for instance [16]            v v v − ∇p = 0 in Ω u u u + Kv v v = 0 in Ω ∇ · u u u − ψ = 0 in Ω ψ = f in Ω with (u u u, n n n) =ū n along Γ u p =p along Γ p .(5) This seems an unnecessarily complicated system. If we eliminate ψ from (5) we obtain (4). The usefulness of this system lies in the fact that by introducing ψ, the conservation ∇ · u u u = f becomes independent of the data of the PDE, in this case the right hand side function. A similar situation occurs when K = I, the identity tensor, then the equation u u u+Kv v v = 0 in (4) seems redundant, but we have good reason to keep this seemingly redundant equation as we will show in this paper. 1. The velocity-gradient relation is exact. It is not obtained by truncated Taylorseries expansions or does not depend on the choice of basis functions/interpolations. 2. Does not depend on mesh parameters. The mesh size h does not appear in (6). Whether the curve which connects two points is straight or curved is irrelevant in this relation, therefore this relation is directly applicable on curved domains. 3. Integral quantities are additive. Consider the points and lines segments as shown in Figure 1. In this figure the arrow along the curves indicates the direction in which v v v is integrated 2 . Application of (6) shows, for instance, thatv v v 14 = P 6 − P 2 . The additivity property implies that P 7 − P 2 =v v v 2 +v v v 15 = P 3 − P 2 + P 7 − P 3 =v v v 14 +v v v 5 = P 6 − P 2 + P 7 − P 6 , and even more paths can be constructed that connect P 2 and P 7 . The independence of the path depends critically on the assumption that the space is contractible, i.e. there are no holes in the domain (Poincaré's Lemma). A special case is the curve from a point to itself, say P 2 → P 2 in Figure 1. This integral is zero and if the integral is independent of the path this implies that, for instance, 0 =v v v 2 +v v v 15 −v v v 5 −v v v 14 = v v v · dl = ∇ × v v v · dS S S = w 2 ,(7) where we once again use the additivity property. We see that the circulation vanishes if v v v is a potential flow, which in turn implies that the circulation of the velocity field over the boundary of any surface vanishes. Or, using Stokes' theorem, the integrated vorticity w vanishes. Here the vorticity w is represented as the integral over a surface. We can collect all the integrated velocity fields and pressures in Figure 1 in the following form                                         v v v 1 v v v 2 v v v 3 v v v 4 v v v 5 v v v 6 v v v 7 v v v 8 v v v 9 v v v 10 v v v 11 v v v 12 v v v 13 v v v 14 v v v 15 v v v 16 v v v 17 v v v 18 v v v 19 v v v 20 v v v 21 v v v 22 v v v 23 v v v 24                                          =                                                                                                            P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P 16                           . If we store allv v v i in a vector v and all P j in a vector P and denote the matrix by E 1,0 , we have v = E 1,0 P . If we now also collect all the integrated vorticities, w i , we can relate them to the integrated velocities in the following way             w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9             =                                                                v v v 1 v v v 2 v v v 3 v v v 4 v v v 5 v v v 6 v v v 7 v v v 8 v v v 9 v v v 10 v v v 11 v v v 12 v v v 13 v v v 14 v v v 15 v v v 16 v v v 17 v v v 18 v v v 19 v v v 20 v v v 21 v v v 22 v v v 23 v v v 24                                          . If we store all vorticity integrals, w i in the vector w, then we can write this as w = E 2,1 v .(8) The matrices E 1,0 and E 2,1 are called incidence matrices. We have E 2,1 · E 1,0 ≡ 0. This identity holds for this particular case, but is generally true; it holds when we would have used triangles or polyhedra instead of quadrilaterals and it holds in any space dimension d. If E 1,0 represents the gradient operation and E 2,1 represents the curl operation, then E 2,1 · E 1,0 ≡ 0 is the discrete analogue of the vector identity ∇ × ∇ ≡ 0, [24-27, 49, 88, 110]. If boundary conditions for p are prescribed along ∂ Ω , then these degrees of freedom can be removed from the grid in Figure 1. If p is known along the boundary then the integral of v v v is also known along the boundary, so the degrees of freedom for v can also be removed. Relabeling the remaining unknowns gives the geometric degrees of freedom as shown in Figure 2.                  v v v 1 v v v 2 v v v 3 v v v 4 v v v 5 v v v 6 v v v 7 v v v 8 v v v 9 v v v 10 v v v 11 v v v 12                   =                   1 0 0 0 −1 1 0 0 0 −1 0 0 0 0 1 0 0 0 −1 1 0 0 0 −1 1 0 0 0 0 1 0 0 −1 0 1 0 0 −1 0 1 0 0 −1 0 0 0 0 −1                      P 1 P 2 P 3 P 4    .(9) Divergence relation Consider a bounded, contractible volume V ⊂ Ω then we have ∇ · u u u = f =⇒ ∂ V u u u · n n n dS = V f dV . If the boundary ∂ V can be partitioned into n sub-boundaries, ∂ V = i Γ i and i Γ i = 0, we have ± n ∑ i=1ū u u i = ± n ∑ i=1 Γ i u u u · n n n dS = V f dV =: f V ,                                                    ū 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 15 u 16 u 17 u 18 u 19 u 20 u 21 u 22 u 23 u 24                                          =             f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 f 9             . Collecting all fluxes and source terms in vectors u and f, respectively, we can write this equation asẼ 2,1 u = f .(10) The matrixẼ 2,1 is the incidence matrix which represents the divergence operator, not to be confused with E 2,1 in (8) which represents the curl operator. If, in the 2D case, the flow field is divergence-free, i.e. f = 0, we know that a stream function ψ ψ ψ exists which is connected to u u u by u x = ∂ ψ ψ ψ ∂ y , u y = − ∂ ψ ψ ψ ∂ x . If we represent the stream function in the nodes of the grid shown in Figure 3, then we have the exact topological equation                                         ū 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 15 u 16 u 17 u 18 u 19 u 20 u 21 u 22 u 23 u 24                                          =                                         1 −1                                                                    ψ 1 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 ψ 10 ψ 11 ψ 12 ψ 13 ψ 14 ψ 15 ψ 16                           . We can write this in terms of incidence matrices as 3 u =Ẽ 1,0 ψ .(11) If the flux u u u is prescribed along the Γ u the associated edges (2D) or surfaces (3D) can be eliminated from the system E d,d−1 u = f and transferred to the right hand side. For the discretization of (4) the first and last equation in that system can be represented on the mesh by 3 Note that if we performed the same steps in 3D, then the divergence relation (10) would bẽ E 3,2 u = f , and the 2D stream function becomes the 3D stream vector field and we would have u =Ẽ 2,1 ψ . So clearly the incidence matricesẼ depend on the dimension of the space d in which the problem is posed. Note that this is not the case for the incidence matrices E. Alternatively, we could refer to the dimension-dependent incidence matrices as E d,d−1 = Ẽ 2,1 if d = 2 E 3,2 if d = 3 and E d−1,d−2 = Ẽ 1,0 if d = 2 E 2,1 if d = 3 , in which case it is immediately clear that these matrices depend on the d. From now on we will use the incidence matrices with the d, because then the results are valid for any space dimension d. v − E 1,0 p = 0 E d,d−1 u = f . Prescription of boundary conditions p along Γ u and u u u along Γ u can be done strongly. The degrees of freedom can be eliminated and transferred to the right hand side. The equation between p and v v v is exact on any grid and the discrete divergence relation between u u u and f is exact on any grid. Note the (v v v, p)-grid is not necessarily the (u u u, f )-grid, so in principle we can use different grids for both equations. Unfortunately, neither of the two problems, v v v = ∇p and ∇·u u u = f has a unique solution on their respective grids. It is the final equation in (4), u u u = −Kv v v, that couples the solution on the two grids and renders a unique solution. It is also in this equation that the numerical approximation is made; the more accurate we approximate this algebraic equation, the more accurate the solution to the first order system (4) will be. For many numerical methods 4 well-posedness requires that the number of discrete degrees of freedomv v v i is equal to the discrete number of degrees of freedom u u u j , or more geometrically, that the number of k-dimensional geometric objects on one grid is equal to the number of (d − k)-dimensional geometric objects on the other grid. Here k = 0 refers to points in the grid, k = 1 to edges in the grid, k = 2 the faces in the grid, and k = 3 the volumes in the grid. The requirement #k = #(d − k) cannot be accomplished on a single grid, so this requires two different grids which are constructed in such a way that #k = #(d − k) holds, [24,25,49,82,88,110]. A dual grid complex is shown in Figure 4. The integral quantities (v v v, p) can be represented on the gray grid. If p is prescribed along the entire boundary, then those degrees of freedom are eliminated (including the gray edges along the boundary for which the integral value v v v is then known also), see for instance Figure 2. In that case flux u u u along the boundary cannot be prescribed. In Figure 4, the number of points in the gray grid, 9, equals the number of surfaces in the black grid, the number of edges in the grey grid is equal to the number of edges on the black grid, 24, and the number of surfaces on the gray grid equals the number of points in the black grid, 16, therefore, we have #k = #(d − k) for d = 2. Alternatively, we could have represented (u u u, f ) on the gray grid with u u u and the stream function ψ prescribed and (v v v, p) on the black grid.In this case Γ u = ∂ Ω and Γ p = ∅. Dual grids If dual grids, such as described above, are employed then we have two properties: 1. There exists a square, invertible matrix H d−1,1 K such that u = H d−1,1 K v. 2. The incidence matrices on the primal and dual grid satisfy 5 E d−k,d−k−1 = E k,k−1 T . If we use dual grids and these properties hold, we can write (4) as        v − E 1,0 p = 0 u − H d−1,1 K v = 0 E 1,0 T u = f ,(12) where the vectors p, v, u and f contain the integral quantities in the mesh as discussed in the previous sections. In the diagram below, we place the various integral values in appropriate 'spaces' p ∈ H 0 E 1,0 / / H d,0 v ∈ H 1 E 2,1 / / H d−1,1 K ξ ∈ H 2 f ∈H d H 0,d O u ∈H d−1 E 1,0 T o o H 1,d−1 K −1 O ψ ∈H d−2 E 2,1 T o o 5 This relation is true if the orientations on primal and dual grid agree. This is not always the case and then the relation reads E d−k,d−k−1 = −E k,k−1 T . A well known example is the duality between grad and div. Here H k denotes the space of values assigned to k-dimensional objects in the H-grid for k = 0, 1, 2. IfH denotes the dual grid, thenH l is the space of values assigned to l-dimensional objects in theH-grid. For dual grids the number of points in the H-grid is equal to the number of ddimensional volumes in the dual gridH. Let H d,0 and H 0,d be square, invertible matrices which map between H 0 andH d as shown in the diagram above. If we eliminate v and u from (12) we have E 1,0 T H d−1,1 K E 1,0 p = f .(13) This discretization corresponds to (2). We will refer to this formulation as the direct formulation. If p ∈H d we can set up the diffusion problem as −H 1,d−1 K −1 u + E d,d−1 T H 0,d p = 0 H 0,d E d,d−1 u = f .(14) This formulation, where we solve for p and u u u simultaneously, resembles (3), and will be called the mixed formulation, [35]. Mimetic spectral element method The incidence matrices introduced in the previous section are generic and only depend on the grid topology. The matrices H which switch between the primal and the dual grid representation explicitly depend on the numerical method that is used. In this section we will introduce spectral element functions which interpolate the integral values in a grid. With these functions we can construct the H-matrices, which turn out to be (weighted) finite element mass matrices. The derivation in this section will be on an orthogonal grid. The extension to curvilinear grids will be discussed in the next section. One dimensional spectral basis functions h i (ξ j ) =    1 if i = j 0 if i = j i, j = 0, . . . N .(15) The explicit form of the Lagrange polynomials in terms of the Legendre polynomials is given by h i (ξ ) = (1 − ξ 2 )L N (ξ ) N(N + 1)L N (ξ i )(ξ i − ξ ) . Let f (ξ ) be a function defined for ξ ∈ [−1, 1] by f (ξ ) = N ∑ i=0 a i h i (ξ ) .(17) Using property (15) we see that f (ξ j ) = a j , so the expansion coefficients in (17) coincide with the value of f in the GLL nodes. We will refer to this expansion as a nodal expansion, because the expansion coefficients, a i in (17) e i (ξ ) = − i−1 ∑ k=0 dh k (ξ ) dξ .(18) The functions e i (ξ ) are polynomials of degree (N − 1). These polynomials satisfy, [61,82,96] ξ j ξ j−1 e i (ξ ) =    1 if i = j 0 if i = j i, j = 1, . . . N .(19) Let a function g(ξ ) be expanded in these functions g(ξ ) = N ∑ i=1 b i e i (ξ ) ,(20) then using (19) ξ j ξ j−1 g(ξ ) = b j . So the expansion coefficients b i in (20) coincide with the integral of g over the edge [ξ i−1 , ξ i ]. We will call these basis functions edge functions and refer to the expansion (20) as an edge expansion, see for instance [16,82,96] for examples of nodal and edge expansions. Let f (ξ ) be expanded in terms Lagrange polynomials as in (17), then the derivative 6 of f is given by, [61,82,96] f (ξ ) = N ∑ i=0 a i h i (ξ ) = N ∑ i=1 (a i − a i−1 )e i (ξ ) .(21) If we collect all the expansion coefficients in a column vector and all the basis functions in a row vector we have f (ξ ) = [h 0 h 1 . . . h N ]        a 0 . . . a N        ,(22) then the derivative is given by 7 (21) f (ξ ) = [e 1 . . . e N ]         −1 1 0 . . . 0 . . . . . . 0 −1 1 0 . . . . . . 0 . . . 0 −1 1                a 0 . . . a N        = [e 1 . . . e N ]E 1,0        a 0 . . . a N        . (23) So taking the derivative essentially consists of two step: Apply the matrix E 1,0 to the expansion coefficients and expand in a new basis. Two dimensional expansions Expanding p (Direct formulation) In finite element methods the direct finite element formulation for the anisotropic diffusion problem is given by: For (K∇p, n n n) = 0 along Γ u and f ∈ H −1 (Ω ), find p ∈ H 1 0,Γ p (Ω ) such that 6 Note that the set of polynomials {h i }, i = 0, . . . , N is linearly dependent and therefore does not form a basis, while the set {e i }, i = 1, . . . , N is linearly independent and therefore forms a basis for the derivatives of the nodal expansion (17). 7 The matrix E 1,0 is the incidence matrix as was discussed in Sections 2.1 and 2.2. It takes the nodal expansion coefficients and maps them to the edge expansion coefficients. The incidence matrix is the topological part of the derivative. It is independent of the order of the method (the polynomial degree N) and the size or the shape of the mesh. The incidence matrix only depends on the topology and orientation of the grid, see [18,81,82]. (∇p, K∇p) = (p, f ) , ∀p ∈ H 1 0,Γ p (Ω ) .(24) where H 1 0,Γ p = {p ∈ H 1 (Ω )|p = 0 on Γ p }. Consider [−1, 1] 2 ⊂ R 2 and let p(ξ , η) be expanded as p(ξ , η) = N ∑ i=0 N ∑ j=0 p i, j h i (ξ )h j (η) .(25) From (15) it follows that p i, j = p(ξ i , η j ). If we take the gradient of p using (21) we have ∇p = ∑ N i=1 ∑ N j=0 (p i, j − p i−1, j )e i (ξ )h j (η) ∑ N i=0 ∑ N j=1 (p i, j − p i, j−1 )h i (ξ )e j (η) (26) = e 1 (ξ )h 0 (η) . . . e N (ξ )h N (η) 0 . . . 0 0 . . . 0 h 0 (ξ )e 1 (η) . . . h N (ξ )e N (η) E 1,0    p 0,0 . . . p N,N    = e 1 (ξ )h 0 (η) . . . e N (ξ )h N (η) 0 . . . 0 0 . . . 0 h 0 (ξ )e 1 (η) . . . h N (ξ )e N (η) E 1,0 p . (27) If we insert this in (24), we have E 1,0 T M (1) K E 1,0 p = f ,(28) where M (1) K = Ω           e 1 (ξ )h 0 (η) 0 . . . . . . e N (ξ )h N (η) 0 0 h 0 (ξ )e 1 (η) . . . . . . 0 h N (ξ )e N (η)           K e 1 (ξ )h 0 (η) . . . e N (ξ )h N (η) 0 . . . 0 0 . . . 0 h 0 (ξ )e 1 (η) . . . h N (ξ )e N (η) dΩ ,(29) and p is the vector which contains the expansion coefficients of p(ξ , η) in (25). The vector f in (28) is given by f = Ω    h 0 (ξ )h 0 (η) . . . h N (ξ )h N (η)    f (ξ , η) dΩ . If we compare (28) with (13), we see that the H d−1,1 K -matrix from (13) is represented in the finite element formulation by the weighted mass matrix M (1) K given by (29), see also [18,115]. Expanding u u u and p (Mixed formulation) The mixed formulation for the anisotropic steady diffusion problem is given by: For p = 0 along Γ p and for f ∈ L 2 (Ω ), find u ∈ H 0,Γ n (div; Ω ) such that −(ũ u u, K −1 u u u) + (∇ ·ũ u u, p) = 0 ∀ũ u u ∈ H 0,Γ u (div; Ω ) (p, ∇ · u u u) = f ∀p ∈ L 2 (Ω ) .(30) where, H 0,Γ u (div; Ω ) = {u ∈ H(div; Ω )|u · n = 0 along Γ u }. In contrast to the pressure expansion in Section 3.2.1 in the direct formulation, (25), in the mixed formulation the pressure is expanded in terms of edge functions p(ξ , η) = N ∑ i=1 N ∑ j=1 p i, j e i (ξ )e j (η) .(31) The velocity u u u is expanded as u u u = u v =   ∑ N i=0 ∑ N j=1 u i, j h i (ξ )e j (η) ∑ N i=1 ∑ N j=0 v i, j e i (ξ )h j (η)   (32) = h 0 (ξ )e 1 (η) . . . h N (ξ )e N (η) 0 . . . 0 0 . . . 0 e 1 (ξ )h 0 (η) . . . e N (ξ )h N (η)           u 0,1 . . . u N,N v 1,0 . . . v N,N           . Application of the divergence operator to (32) and using (21) we obtain ∇ · u u u = N ∑ i=1 N ∑ j=1 (u i, j − u i−1, j + v i, j − v i, j−1 )e i (ξ )e j (η) (33) = e 1 (ξ )e 1 (η) . . . e N (ξ )e N (η) E d,d−1           u 0,1 . . . u N,N v 1,0 . . . v N,N           = e 1 (ξ )e 1 (η) . . . e N (ξ )e N (η) E d,d−1 u . Note that E d,d−1 is the incidence matrix which also appeared in (10) and footnote 3. If we insert the expansion (32) in (ũ u u, K −1 u u u) we obtain (ũ u u, K −1 u u u) =ũ T M (d−1) K −1 u ,(34) with M (d−1) K −1 = (35) Ω           h 0 (ξ )e 1 (η) 0 . . . . . . h N (ξ )e N (η) 0 0 e 1 (ξ )h 0 (η) . . . . . . 0 e N (ξ )h N (η)           K −1 h 0 (ξ )e 1 (η) . . . h N (ξ )e N (η) 0 . . . 0 0 . . . 0 e 1 (ξ )h 0 (η) . . . e N (ξ )h N (η) dΩ .(36) Note that pressure is expanded in the same basis as the divergence of the velocity field, (31) and (33), therefore we can write (p, ∇ · u u u) =p T M (d) E d,d−1 u ,(37) with M (d) = Ω    e 1 (ξ )e 1 (η) . . . e N (ξ )e N (η)    e 1 (ξ )e 1 (η) . . . e N (ξ )e N (η) dΩ . With (34) and (37) we can write (30) as M (d−1) K −1 u + E d,d−1 T M (d) p = 0 M (d) E d,d−1 u = f ,(38)with f = Ω    e 1 (ξ )e 1 (η) . . . e N (ξ )e N (η)    f (ξ , η) dΩ . Comparison of (38) with (14) shows that the topological incidence matrices also appear in the finite element formulation and that the (weighted) mass matrices M (d−1) K −1 and M (d) once again play the role of the H-matrices which connect solutions on dual grids. In this section only the discretization on a single spectral element is discussed. Transformation of the domain [−1, 1] 2 to more general domains will be discussed in Section 4. The use of multiple elements follows the general assembly procedure from finite element methods. Results of this approach are presented in Section 5. Transformation rules The basis functions used in the discretization of the different physical field quantities have only been introduced for the reference domainΩ = [−1, 1] 2 . For these basis functions to be applicable in a different domain Ω , it is fundamental to discuss how they transform under a mapping Φ : (ξ , η) ∈Ω → (x, y) ∈ Ω ⊂ R 2 . Within a finite element formulation this is particularly useful because the basis functions in the reference domainΩ can then be transformed to each of the elements Ω e , given a mapping Φ e :Ω → Ω e . Consider a smooth bijective map Φ : (ξ , η) ∈Ω → (x, y) ∈ Ω such that x = Φ x (ξ , η) and y = Φ y (ξ , η) , and the associated rank two Jacobian tensor J J :=      ∂ Φ x ∂ ξ ∂ Φ x ∂ η ∂ Φ y ∂ ξ ∂ Φ y ∂ η      . The transformation of a scalar function ϕ discretized by nodal values is given bỹ ϕ(ξ , η) = (ϕ • Φ)(ξ , η) and ϕ(x, y) = (φ • Φ −1 )(x, y),(39) and of a scalar function ρ discretized by surface integrals is given bỹ ρ(ξ , η) = det J (ρ • Φ)(ξ , η) and ρ(x, y) = 1 det J (ρ • Φ −1 )(x, y).(40) The transformation of vector fields v v v discretized by line integrals is v v v(ξ , η) = J T (v v v • Φ)(ξ , η) and v v v(x, y) = (J T ) −1 (ṽ v v • Φ −1 )(x, y),(41) and of vector fields u u u discretized by flux integrals is u u u(ξ , η) = det J J −1 (u u u • Φ)(ξ , η) and u u u(x, y) = 1 det J J(ũ u u • Φ −1 )(x, y).(42) These transformations affect only the mass matrices and not the incidence matrices. This is fundamental to ensure the topological nature of the incidence matrices. Numerical results In this section three test cases are presented to illustrate the accuracy of the discretization scheme developed in this work. The first test case, 5.1, is an analytical solution taken from [68] to assess the convergence rates of the method. The second test case, 5.2, is the flow through a system of sand and shale blocks with highly heterogeneous permeability in the domain, see for more details [54]. The third test case, 5.3, is a highly anisotropic and heterogeneous permeability tensor in the domain, see for more details, [53]. Manufactured solution We first test the method using the exact solution p exact (x, y) = sin(πx) sin(πy),(43) with the permeability tensor given by K = 1 (x 2 + y 2 + α) 10 −3 x 2 + y 2 + α 10 −3 − 1 xy 10 −3 − 1 xy x 2 + 10 −3 y 2 + α .(44) The mixed formulation (3) in the form of (38) is then solved in the domain (x, y) ∈ Ω = [0, 1] 2 with the source term f = −∇ · (K∇p exact ) and the Dirichlet boundary condition p| ∂ Ω = 0. A benchmark of this test case for α = 0 using multiple numerical schemes can be found in [68]. When α = 0, K is multi-valued at the origin which makes this test case a challenging one. To see this, we can first convert the Cartesian coordinates (x, y) to polar coordinates (r, θ ) by x = r cos θ , y = r sin θ . Then we have K| α=0 = 10 −3 cos 2 θ + sin 2 θ 10 −3 − 1 cos θ sin θ 10 −3 − 1 cos θ sin θ cos 2 θ + 10 −3 sin 2 θ . It can be seen that we get different K| α=0 when we approach the origin along different angles, θ . It must be noted that inverse of K does not exist at the origin. The inverse of the tensor term appears in 3.2.2. We use Gauss integration and thus the inverse term is not evaluated at the origin. The meshes we use here are obtained by deforming the GLL meshes in the reference domain (ξ , η) ∈ Ω ref = [−1, 1] 2 with the mapping, Φ, given as      x = 1 2 + 1 2 (ξ + c sin(πξ ) sin(πη)) y = 1 2 + 1 2 (η + c sin(πξ ) sin(πη)) ,(46) where c is the deformation coefficient. The two meshes, for c = 0.0 and c = 0.3, are shown in Figure 5. The method is tested for α ∈ {0, 0.01} and c ∈ {0, 0.3}. In Figure 6, the results for ||∇ · u u u h − f h || L 2 are presented. They show that the relation ∇ · u u u h = f h is conserved to machine precision even on a highly deformed and coarse mesh i.e. of 2 × 2 elements with N = 2 and c = 0.3. When α = 0.01, K is no longer multi-valued at the origin. In this case the source term f is smooth over the domain, see Figure 7 (bottom). For this smooth case, the method displays optimal convergence rates on both the orthogonal mesh and the deformed mesh, i.e. see Figure 8 (bottom) and Figure 9 (bottom). When α = 0, both the h-convergence rate and p-convergence rates are suboptimal, see Figure 8 (top) and Figure 9 (top). This is because K is multi-valued and therefore f becomes singular at the origin when α = 0, see Figure 7 (top left). L 2 −error N = 2, ||ph − pexact|| L2 N = 2, ||uh − uexact|| L2 N = 2, ||fh − fexact|| L2 N = 4, ||ph − pexact|| L2 N = 4, ||uh − uexact|| L2 N = 4, ||fh − fexact|| L2 The Sand-Shale system This example is taken from [54,76,78]. The domain is a 2D unit square, Ω = [0, 1] 2 , with 80 shale blocks, Ω s , placed in the domain such that the total area fraction of shale blocks is A shale = 20%, as shown in Figure 10. We solve the mixed formulation (38) with f = 0 in this domain. The flux across the top and the bottom boundaries is u u u · n n n = 0. The flow is pressure driven with the pressure at the left boundary, p = 1, and the pressure at the right boundary, p = 0. The permeability in the domain is defined as K = kI, where k is given by: k = 10 −6 in Ω s 1 in Ω \ Ω s . For this test case an orthogonal uniform grid of 20 × 20 elements is used. The polynomial degree is varied to achieve convergence. Streamlines through the domain for 20 × 20 elements and polynomial degree N = 15 are shown in Figure 11. It can be seen that the streamlines do not pass through, but pass around the shale blocks of low permeability. The ||∇ · u u u h || L 2 over the entire domain as a function of polynomial degree is shown in Figure 12. We observe that ∇ · u u u h = 0 is satisfied up to machine precision. The net flux entering the domain (the same as the net flux leaving the domain) is given in Table 1 for varying polynomial degree. A reference value for this solution is given in [54] as 0.5205, and in [78] as 0.519269. In this work the maximum resolution corresponds to 20 × 20 elements and a polynomial degree N = 19, for which the net flux entering the domain is obtained as 0.52010. In Figure 13 we compare the net flux entering the sand-shale domain, calculated using the mixed and the direct formulation of equations, as a function of polynomial degree for different values of k in the shale blocks. The data for these figures is given in Table 2. Note that the direct formulation converges from above towards the correct inflow flux, whereas the mixed formulation converges from below. The Impermeable-Streak system The next example is from [53,76,78]. The physical domain is a 2D unit square, Ω = [0, 1] 2 . The domain is divided into three different regions, Ω 1 , Ω 2 , and Ω 3 , as shown in Figure 14 (left). For calculations, each region is further divided into K × K elements. Therefore, the total number of elements in the domain is given by K × K × 3. In Figure 14 (right) we show the domain with each region divided into 2 × 2 elements. The mixed formulation (38) is solved, with f = 0 and mixed boundary conditions, such that at the top and the bottom boundaries the net flux u u u · n n n = 0, and at the left and the right boundaries, p = 1 and p = 0, respectively. Permeability in Ω 1 and Ω 3 is given by K = I. Ω 2 has a low permeability and defined such that the component parallel to the local streak orientation is k = 10 −1 , and the component perpendicular to the local streak orientation is k ⊥ = 10 −3 . The analytical expression for the permeability in terms of Cartesian coordinates is given in [76] as, The flow field in the domain is shown in Figure 15. The magnitude of velocity in Ω 2 is small due to low values of the permeability tensor in this region. The velocity vectors bend in the direction of the permeability streak Ω 2 . The L 2 -norm of ∇·u u u over the entire domain as a function of polynomial degree, N, is shown in Figure 16. We can see that the flow field is divergence free up to machine precision because f = 0. The net flux through the system for varying number of elements and polynomial degree is given in Table 3. In this work the finest resolution corresponds to 12×12× 3 elements and N = 15. For this case the net influx at the left boundary is 0.75668. K xx = k (y + 0.4) 2 + k ⊥ (x − 0.1) 2 (x − 0.1) 2 + (y + 0.4) 2 , K xy = −(k − k ⊥ )(x − 0.1)(y + 0.4) (x − 0.1) 2 + (y + 0.4) 2 , K yy = k (x − 0.1) 2 + k ⊥ (y + 0.4) 2 (x − 0.1) 2 + (y + 0.4) 2 .L 2 − error ||∇ · uh|| L 2 K × K = 2 × 2 K × K = 4 × 4 K × K = 6 × 6 The net influx and outflux from the region Ω 1 , Ω 2 and Ω 3 is given in Tables 4, 5, and 6, respectively. The net influx for Ω 1 is larger than the net outflux. And the net outflux for Ω 2 and Ω 3 is larger than the net influx. Table 6: Net flux through the left boundary of the region Ω 3 for K × K elements, K = 4, 6, 8, 10, 12 and N = 1, ..., 15. Future Work In the above sections, mixed and direct formulations of mimetic spectral element method are discussed. The next step is to explore this framework in the direction of hybrid formulations [20,35,43]. Additionally, the focus will be on developing multiscale methods [118], using these formulations, for reservoir modelling applications. Fig. 1 : 1Relation between pressure in points, integrated velocity along line segments and vorticity in surfaces. Fig. 2 : 2Relation between pressure in points and integrated velocity along line segments in case Γ p = ∂ Ω . Fig. 3 : 3where we have the convention that the fluxes,ū u u i , are positive when the flow leaves the volume and negative when the flow enters the volume. For a 2D case the integral flux degrees of freedom,ū u u i are depicted inFigure 3. The arrow in this figure indicates the positive default direction of the fluxes. The integrated values of source function f are shown in the 2D volumes inFigure 3as f i . The topological relation between the fluxes and the integrated source values f i , for the situation shown inFigure 3, is given by Stream function, fluxes and the divergence degrees of freedom. Fig. 4 : 4The primal grid (thin gray) where (v v v, p) are represented and the dual grid (thick black) where (u u u, f ) are represented. Note that Γ p = ∂ Ω and consequently Γ u = ∅. the interval [−1, 1] ⊂ R and the Legendre polynomials,L N (ξ ), of degree N, ξ ∈ [−1, 1]. The (N + 1) roots, ξ i , of the polynomial (1 − ξ 2 )L N (ξ ) satisfy −1 ≤ ξ i ≤ 1. Here L N (ξ )is the derivative of the Legendre polynomial. The roots ξ i , i = 0, . . . , N, are called the Gauss-Lobatto-Legendre (GLL) points,[41]. Let h i (ξ ) be the Lagrange polynomial through the GLL points such that are the value of f (ξ ) in the nodes ξ i . The basis functions h i (ξ ) are polynomials of degree N.From the nodal basis functions, define the polynomials e i (ξ ) by Fig. 5 : 5Example meshes with 3×3 elements of polynomial degree N = 6. Left: c = 0 (orthogonal mesh). Right: c = 0.3 (highly deformed mesh). Fig. 6 : 6The L 2 -norm of (∇ · u u u h − f h ). Left: K × K elements, K = 4, ..., 250, and N = 2, 4. Right: 2 × 2, 6 × 6 elements, and N = 2, ..., 30. Top: α = 0. Bottom: α = 0.01. Fig. 7 : 7Left: the source term f . Right: the log 10 distribution of the projection error of f h for 3 × 3 elements, N = 10 and c = 0.3. Top: α = 0. Bottom: α = 0.01. Fig. 8 : 8The p-convergence for 2 × 2, 6 × 6 elements and N = 2, ..., 30. Left: c = 0. Right: c = 0.3. Top: α = 0. Bottom: α = 0.01. NN = 2, ||ph − pexact|| L2 N = 2, ||uh − uexact|| L2 N = 2, ||fh − fexact|| L2 N = 4, ||ph − pexact|| L2 N = 4, ||uh − uexact|| L2 N = 4, ||fh − fexact|| = 2, ||ph − pexact|| L2 N = 2, ||uh − uexact|| L2 N = 2, ||fh − fexact|| L2 N = 4, ||ph − pexact|| L2 N = 4, ||uh − uexact|| L2 N = 4, ||fh − fexact|| Fig. 9 : 9The h-convergence of the L 2 -error for K × K elements, K = 4, ..., 250 and N = 2, 4. Left: c = 0. Right: c = 0.3. Top: α = 0. Bottom: α = 0.01. Fig. 10 : 10The discretized domain for the sand-shale test case. Black blocks are shale blocks with k = 10 −6 . White blocks are sand blocks with k = 1. Fig. 11 :Fig. 12 : 1112Streamlines through the domain of sand-shale test case. The L 2 -norm of ∇ · u u u h for 20 × 20 elements for a polynomial approximation of N = 1, ..., 19. of net flux through the left boundary of the sand-shale domain using mixed formulation and direct formulation for 20 × 20 elements, N = 1, ..., 10, k = 10 −1 (top-left), 10 −2 (top-right), 10 −3 (bottom-left) and 10 −4 (bottom-right). Fig. 13 : 13Convergence of the net flux through the left boundary of the sand-shale domain using the mixed formulation and the direct formulation for 20×20 elements, N = 1, ..., 10. Top left: k = 10 −1 . Top right: k = 10 −2 . Bottom left: k = 10 −3 . Bottom right: k = 10 −4 . Fig. 14 : 14Three regions of the domain for the impermeable streak test case. The regions are separated by the dashed lines. The solid lines indicate the element boundaries. Left: 1 × 1 element in each region. Right: 2 × 2 elements in each region. Fig. 15 : 15Velocity vectors through the domain of permeability streak test case for 12 × 12 elements, N = 15. Fig. 16 : 16The L 2 -norm of ∇ · u u u h for K × K elements, K = 2, 4, 6, N = 1, ..., 15. flux through the left boundary of the permeability streak test case domain for K × K elements, K = 4, 6, 8, 10, 12 and N = 1, ..., 15. Out flux In flux Out flux In flux Out flux In flux Out flux In flux flux through the left boundary of the region Ω 1 for K × K elements, K = 4, 6, 8, 10, 12 and N = 1, ..., flux through the left boundary of the region Ω 2 for K × K elements, K = 4, 6, 8, 10, 12 and N = 1, ..., flux Out flux In flux Out flux In flux Out flux In flux Out flux In flux Net flux through the left boundary of the sand-shale domain for k = 10 −6 , 20 × 20 elements, N = 1, ..., 19.N Net flux No. of unknowns 1 0.49041 1240 2 0.51247 4880 3 0.51744 10920 4 0.51863 19360 5 0.51931 30200 6 0.51957 43440 7 0.51977 59080 8 0.51985 77120 9 0.51993 97560 10 0.51997 120400 11 0.52001 145640 12 0.52003 173280 13 0.52005 203320 14 0.52007 235760 15 0.52008 270600 16 0.52009 307840 17 0.52009 347480 18 0.52010 389520 19 0.52010 433960 Table 1: The points in the grid shown inFigure 1are also 'oriented', in the sense that when we 'move into a point following the integration direction' we assign a positive value and when we 'leave a point' we assign a negative value. 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{'abstract': 'This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.', 'arxivid': '1802.04597', 'author': ['Marc Gerritsma ', 'Artur Palha ', 'Varun Jain ', 'Yi Zhang '], 'authoraffiliation': [], 'corpusid': 119176238, 'doi': '10.1007/978-3-319-94676-4_3', 'github_urls': [], 'n_tokens_mistral': 32854, 'n_tokens_neox': 27421, 'n_words': 15510, 'pdfsha': 'c62fad4e7e7bc7ef4bf93a53bb9ab8682277dbe1', 'pdfurls': ['https://arxiv.org/pdf/1802.04597v1.pdf'], 'title': ['Mimetic Spectral Element Method for Anisotropic Diffusion', 'Mimetic Spectral Element Method for Anisotropic Diffusion'], 'venue': []}

arxiv

Epistemic AI platform accelerates innovation by connecting biomedical knowledge Da Chen Emily Koo Heather Bowling Kenneth Ashworth David J Heeger Stefano Pacifico Epistemic Ai Epistemic AI platform accelerates innovation by connecting biomedical knowledge Epistemic AI accelerates biomedical discovery by finding hidden connections in the network of biomedical knowledge. The Epistemic AI web-based software platform embodies the concept of knowledge mapping, an interactive process that relies on a knowledge graph in combination with natural language processing (NLP), information retrieval, relevance feedback, and network analysis. Knowledge mapping reduces information overload, prevents costly mistakes, and minimizes missed opportunities in the research process. The platform combines state-of-the-art methods for information extraction with machine learning, artificial intelligence and network analysis. Starting from a single biological entity, such as a gene or disease, users may: a) construct a map of connections to that entity, b) map an entire domain of interest, and c) gain insight into large biological networks of knowledge. Knowledge maps provide clarity and organization, simplifying the day-to-day research processes. Introduction Biomedical knowledge is the key to accelerate discoveries and identify hypotheses leading to novel diagnostic tools, therapies, and vaccines. Biomedical researchers struggle with information overload while attempting to grapple with the vast and rapidly expanding base of biomedical knowledge including documents (e.g., papers, patents, clinical trials) and databases (e.g., databases of genes, proteins, pathways, drugs, diseases, medical terms). This is a major pain point for biomedical researchers and, with no appropriate solution available, they are forced to use search tools (e.g., PubMed and Google Scholar) and cumbersomely navigate manually-curated databases. These tools are suitable for finding documents matching keywords (e.g., a single gene or published journal paper), but not for acquiring a collection of knowledge to explore and learn about a topic area or subdomain (e.g., the long term impact of COVID-19 on cognitive function), or for interpreting the results of high-throughput biology experiments such as gene sequencing, protein expression, or compound screening. We developed the Epistemic AI platform to remedy this problem. The platform is an innovative, AI-powered and interactive platform for researchers to connect with and discover knowledge more quickly and efficiently than ever before, saving time and cost, while also providing completeness and accuracy to make informed decisions, especially at critical stage-gates. This platform is changing the way that biomedical investigators work and think, thereby fueling the next generation of breakthrough discoveries and innovations to improve human health. The core innovation in the Epistemic AI platforms is what we call knowledge mapping. Knowledge mapping uses a knowledge graph in combination with biomedical natural language processing (bioNLP) (Devlin et al, 2018;Przbyła et al, 2018;Afentenos et al, 2005;Elhadad et al, 2005;Yetisgen-Yildiz and Pratt, 2005;Goldstein, 2007;Roberts et al, 2007;Kipper-Schuler et al, 2008;Fiszman et al, 2009;Savova et al, 2010;Luther et al, 2011;see Bretonnel Cohen and Demner-Fushman, 2014 for review), relevance feedback (Agichtein, Brill and Dumais, 2006;States et al, 2009;Yu et al, 2009;Alatrash et al, 2012;Ji et al, 2016;Rocchio et al, 1971), and network analysis (Zhong et al, 2006;Mostafavi et al, 2008;Suthram et al, 2010;Cokol et al, 2011;Green et al, 2011;Ciofani et al, 2012;Marbach et al, 2012;Guimera And Sales-Pardo, 2013;Kurts et al, 2015;Shi et al, 2016;Suresh et al, 2016;Wong et al, 2016;Wang et al, 2017;Gligorijević et al, 2018;Castro et al, 2019;Chasman et al, 2019;Cramer et al, 2019;Miraldi, 2019;Siahpirani et al, 2019), for an interactive knowledge mapping platform. Methods Technical challenges We faced four critical technical challenges in developing the platform: 1) Biomedical knowledge is siloed, locked in text documents or spread across multiple biomedical databases that each provide partially overlapping and complementary information. We addressed this problem by offering a seamless integration of knowledge extracted from text documents and a multitude of structured biomedical databases with a hybrid architecture combining deep learning and statistical machine learning (ML) with knowledge-intensive AI methods (Gabor et al, 2018;Luan et al, 2018). 2) Despite the advances in natural language processing (NLP) and bioNLP (see Introduction for references), a fully automatic solution with acceptable performance for knowledge extraction, representation, and reasoning is not close to being realized (Weischedel and Boschee, 2018), e.g., the failure of IBM Watson Health (Strickland, 2019). Current methods perform best when trained on large, manually-labeled datasets but labeling is timeconsuming, especially in highly specialized tasks such as those in BioNLP. Precision and recall of BioNLP is inadequate (Doğan et al, Wang et al, 2019), and automated reasoning suffers from inadequately modeling the complexity of both the data and the problem domain (Rocktäschel and Riedel, 2017) -the "blocks world" problem of symbolic AI (Slaney and Thiébaux, 2001). To circumvent these problems, rather than fully automating knowledge discovery, our approach relies on the premise that machine amplification of skilled experts can unlock enormous value. Consequently, we utilize an interactive platform; precision and recall are increased as the investigator adds landmarks to their knowledge map, and the human domain expert does the hard reasoning. Rather than attempt to replace the expertise of biomedical researchers with a fully automated system, the Epistemic AI platform augments the user's expertise with an interactive process for knowledge mapping and knowledge discovery. This lightens the demands of the AI/ML technology, so that it is fully capable of supporting the platform, with a human-in-the-loop. 3) Information retrieval and ranking is context-and task-dependent -two users submitting the same query may have different goals, and consequently are seeking different information. Our knowledge mapping platform deals with this naturally via user interaction. An investigator begins by entering terms in a text box. Our information retrieval algorithms find relevant biomedical entities, documents and relationships. The investigator may then select any of the results and add them as landmarks in their knowledge map, providing context to re-rank the search results. This process is repeated iteratively with the option to replenish (re-rank) the results with respect to proximity to the landmarks in the map at each stage. Precision and recall (measures of relevancy) increase through the interaction as the investigator adds more landmarks to the map, making it easier to identify and consolidate all of the relevant knowledge. This process is similar to "relevance feedback", which combines search with explicit supervision from users to indicate relevant or useful results (Rocchio, 1971;Agichtein, Brill and Dumais, 2006;States et al, 2009;Yu et al, 2009;Alatrash et al, 2012;Ji et al, 2016). Explicit relevance feedback was not widely adopted in traditional text search applications (Spink, Jansen and Ozmultu, 2000;Anick, 2003), but we are tackling a different problem (knowledge mapping, not search) that inherently involves user interaction and exploration in a manner that is more organic than previous attempts with text search. Critically, the end user (an expert biomedical investigator) is the ultimate arbiter of quality and accuracy, deciding at each step whether to include a particular entity or relationship in their map. 4) Rapid progress in biomedical R&D requires an interdisciplinary approach but knowledge and expertise are siloed. For example, the domain of relevant knowledge for COVID-19 is vast and interdisciplinary, including virology, immunology, pulmonology, molecular biology, and so on. In spite of incredible progress with vaccines, we only have a limited understanding of disease progression and its long-term consequences. Only a minority of investigators have the breadth of knowledge and interdisciplinary mindset to generate hypotheses about the consequences and possible treatment for long-COVID (i.e., the effects of COVID-19 infections that continue for weeks or months beyond the initial illness). And those highly interdisciplinary investigators may lack the depth to see it through. Researchers are often required to work with incomplete, ambiguous, anomalous or sometimes even deceptive data. Moreover, as with the current environment with COVID-19, time constraints require quick decisions that can conspire with natural human biases, leading to inaccurate or incomplete judgements. The Epistemic AI platform enables researchers to overcome these problems by helping them to identify and refute as many possible hypotheses as possible given a full range of data, information, assumptions and gaps pertinent to the problem at hand. The platform enables investigators to obtain a complete map of the relevant literature and knowledge, including links to related fields and topics. Such an interdisciplinary viewpoint is a key driver to discovery, increasing the likelihood of gaining critical insights. Solutions Our scientific knowledge mapping platform addresses these challenges with four technical innovations. Although the technology underlying each of these four innovations has a long history of development, their combined usage represents a major advancement on the state-of-the-art. All together, this aggregate, combining all four, is transformative. 1) Knowledge graphs and knowledge mapping. Knowledge graphs are not new, and the concept of knowledge mapping is not new (investigators map knowledge in their brains and/or with the help of spreadsheets or reference managers every second of the day), but our novel approach that uses a knowledge graph in combination with BioNLP, relevance feedback, and network analysis, for an interactive knowledge mapping platform, has not been previously developed and commercialized. Knowledge mapping is quite different from conventional search, and solves a critical problem for users. For example, the conventional process of literature search becomes exponentially cumbersome with more documents in a collection because there are more leads to chase down. The investigator has to look through all the references in all the documents in their collection, identify those that they have not yet read, choose a subset to read, and then repeat the process ad infinitum. With knowledge mapping, this process is radically simplified. Relevant knowledge is tightly connected (in close proximity) to entities in a knowledge map, so identifying additional relevant knowledge becomes easier (more robust and reliable) as the investigator adds entities to their map. In addition, the Epistemic AI knowledge graph contains information ingested from myriad databases in addition to documents, so a biomedical researcher may draw on all of these sources seamlessly during the knowledge mapping process. Provenance (a link to the original source) is always provided. 2) Information extraction. The Epistemic AI platform uses state-of-the-art and novel information extraction algorithms to create a network of biomedical knowledge by extracting entities and relations from documents and databases. Though information extraction from text is far from perfect, users of the Epistemic AI platform have validated that the technology is a huge improvement compared to previous workflows. As the technology improves over time, its application in this interactive platform will scale accordingly. 3) Proximity ranking. We have developed a novel machine learning algorithm for ranking documents and biomedical entities by calculating conceptual proximity from language and network features. This ranking algorithm is embedded in each step of an iterative process with a human-in-the-loop. Results We developed a human-in-the-loop responsive research platform that allows the user to specify the most important biomedical entities and parameters and to examine connections between biomedical entities. To illustrate how the platform works, we considered the question "Are Alzheimer's disease and dementia patients more at risk for poor outcomes with COVID-19?" We constructed a knowledge map (KM) by adding "COVID-19", "Alzheimer's Disease", and "Dementia" as Disease entities and began by exploring publications and clinical trials (Fig 1A). Specific publications related to dementia and COVID-19 were surfaced by the Epistemic AI platform and an interactive feature permitted us to select publications we deemed most relevant to our query. This was achieved simply by clicking the star next to the publication of interest to add it to the KM (Fig 1B-C). The platform then also retrieved other publications and clinical trials that were similar to our publication of interest (e.g., containing closely related entities) and re-ranked the results according to their relevance to the entire KM ( Fig 1D). As more publications and clinical trials were added to the KM, the results became increasingly refined, minimizing the need to scroll through irrelevant information and read non-related abstracts. Given the wealth of publications and clinical trials surfaced by the Epistemic AI platform, we also wanted to examine the relationships between entities within the KM more closely. To achieve this, we performed a deep dive to identify and add other associated entities to the KM. For instance, in this use case, the platform displayed a columnar "card" for each disease entity: COVID-19 (Fig 2A), Alzheimer's Disease, and Dementia. Each of these cards contained various sections with information specific to each individual disease but re-ranked according to the context of the entire KM. For example, the first related publication in the COVID-19 card concerned both COVID-19 and Dementia (Fig 2A). In addition, researchers also have access to, but not limited to, related clinical trials as well as associated genes ( Fig 2B) and drugs, any of which can be seamlessly added to the existing KM by clicking its star to explore further connections. In this use case, we starred one associated gene, IL-6, and presented an example of a gene card with its summary and related variants (Fig 2C). Because all the associated entities presented on the platform are ranked by relevance to the KM, the most relevant associations also appear at the top of each section of each card (Fig 2D). Using our platform, we determined which common bad outcome risk factors are highest for patients with dementia and COVID-19: CRP and IL-6. We then explored how IL-6 interacts with the immune system and discovered that it shares a common immunological response pathway with CRP. In addition, the platform revealed a new research opportunity: there is an open question of whether COVID-19 infection increases the risk of dementia but a lack of sufficient long-term data to reach a conclusion. The example provided here illustrates how any biomedical researcher can easily create a knowledge map starting with a simple query about the increased severity of COVID-19 for dementia patients. The Epistemic AI platform rapidly unearthed relevant publications, clinical trials, molecular connections, potential biomarkers, and interventions, alleviating the need to navigate between multiple web pages. Researchers can then not only explore existing connections effortlessly, but also identify new opportunities with unanswered research questions in the field. With the overwhelming body of research into COVID-19 and dementia spanning publications, clinical trials, and potential molecular mediators, the average researcher would have required hours, if not days, to weed through multiple sources to retrieve all the relevant knowledge on this topic using conventional methods. The Epistemic AI platform, by contrast, enables the user to acquire the same knowledge within minutes. The Epistemic AI platform automatically identifies relevant connections in the network of biomedical knowledge, so that the user does not have to stochastically locate them across multiple platforms or read dozens of papers to surface the same information. The Epistemic AI platform guides users to relevant information and connected entities, allowing them to seamlessly transition between different types of information, ultimately saving time and mental energy. Figure 1 . 1Exploring Publications and Clinical trials within the Epistemic AI platform. We entered COVID-19, Alzheimer's Disease and Dementia as Disease entities into the knowledge map (KM) (A). Upon querying, the platform surfaced relevant publications which could be starred and added to the KM by theuser (B). Once the user-selected publications were added to the KM (C), the Epistemic AI platform identified and re-ranked the results to provide even more relevant publications and clinical trials (D). Figure 2 . 2Performing a "deep dive" analysis on the Epistemic AI platform between entities in the KM. Every entity (e.g., COVID-19 disease) added to the KM has a columnar card containing various sections with information specific to that entity (e.g., related publications), but in the context of the rest of the KM (A). Entities within any section (e.g., IL-6 gene), can be added to the KM for further exploration (B). Once added, a new card for IL-6 is created. 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{'abstract': 'Epistemic AI accelerates biomedical discovery by finding hidden connections in the network of biomedical knowledge. The Epistemic AI web-based software platform embodies the concept of knowledge mapping, an interactive process that relies on a knowledge graph in combination with natural language processing (NLP), information retrieval, relevance feedback, and network analysis. Knowledge mapping reduces information overload, prevents costly mistakes, and minimizes missed opportunities in the research process. The platform combines state-of-the-art methods for information extraction with machine learning, artificial intelligence and network analysis. Starting from a single biological entity, such as a gene or disease, users may: a) construct a map of connections to that entity, b) map an entire domain of interest, and c) gain insight into large biological networks of knowledge. Knowledge maps provide clarity and organization, simplifying the day-to-day research processes.', 'arxivid': '2201.11331', 'author': ['Da Chen ', 'Emily Koo ', 'Heather Bowling ', 'Kenneth Ashworth ', 'David J Heeger ', 'Stefano Pacifico ', 'Epistemic Ai '], 'authoraffiliation': [], 'corpusid': 246294911, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11521, 'n_tokens_neox': 9420, 'n_words': 5814, 'pdfsha': '8c8912a90fef8daae3c828470d59a7008b2a4493', 'pdfurls': ['https://arxiv.org/pdf/2201.11331v3.pdf'], 'title': ['Epistemic AI platform accelerates innovation by connecting biomedical knowledge', 'Epistemic AI platform accelerates innovation by connecting biomedical knowledge'], 'venue': []}

arxiv

Feature Fusion Vision Transformer for Fine-Grained Visual Categorization Jun Wang jun.wang.3@warwick.ac.uk Xiaohan Yu xiaohan.yu@griffith.edu.au Griffith University Yongsheng Australia yongsheng.gao@griffith.edu.au Gao University of Warwick UK Griffith University Australia Feature Fusion Vision Transformer for Fine-Grained Visual Categorization The core for tackling the fine-grained visual categorization (FGVC) is to learn subtle yet discriminative features. Most previous works achieve this by explicitly selecting the discriminative parts or integrating the attention mechanism via CNN-based approaches. However, these methods enhance the computational complexity and make the model dominated by the regions containing the most of the objects. Recently, vision transformer (ViT) has achieved SOTA performance on general image recognition tasks. The self-attention mechanism aggregates and weights the information from all patches to the classification token, making it perfectly suitable for FGVC. Nonetheless, the classification token in the deep layer pays more attention to the global information, lacking the local and low-level features that are essential for FGVC. In this work, we propose a novel pure transformer-based framework Feature Fusion Vision Transformer (FFVT) where we aggregate the important tokens from each transformer layer to compensate the local, low-level and middle-level information. We design a novel token selection module called mutual attention weight selection (MAWS) to guide the network effectively and efficiently towards selecting discriminative tokens without introducing extra parameters. We verify the effectiveness of FFVT on four benchmarks where FFVT achieves the state-of-the-art performance. Code is available at this link. Introduction Fine-grained visual categorization (FGVC) aims to solve the problem of differentiating subordinate categories under the same basic-level category, e.g., birds, cars and plants. FGVC has wide real-world applications, such as autonomous driving and intelligent agriculture. Some FGVC tasks are exceedingly hard for human beings due to the small inter-class variance and large intra-class variance, e.g., recognizing 200 subordinate plant leaves and 200 subordinate birds. Therefore, FGVC is an important and highly challenging task. Owing to the decent designed networks and large-scale annotated datasets, FGVC has gained steady improvements in recent years. Current methods on FGVC can be roughly divided into localization-based methods and attention-based methods. The core for solving FGVC is to learn the discriminative features in images. Early localization-based methods [1,16,41] achieve this by directly annotating the discriminative parts in images. However, it is costly and time-consuming to build bounding box annotations, hindering the applicability of these methods on real-world applications. To alleviate this problem, recent localizationbased methods normally integrate the region proposal network (RPN) to obtain the potential discriminative bounding boxes. These selected proposals are then fed into the backbone network to gain the local features. After that, most methods often adopt a rank loss [2] on the classification outputs for all local features. However, [14] argues that RPN-based methods ignore the relationships among selected regions. Another problem is that this mechanism drives the RPN to propose large bounding boxes as they are more likely to contain the foreground objects. Confusion occurs when these bounding boxes are inaccurate and cover the background rather than objects. Besides, some discriminative regions, e.g., leaf vein in plant leaves, cannot be simply annotated by a rectangular [35]. Attention-based [40,50,54] methods automatically detect the discriminative regions in images via self-attention mechanism. These methods release the reliance on manually annotation for discriminative regions and have gained encouraging results. Recently, vision transformer has demonstrated potential performance on general image classification [6], image retrieval [9] and semantic segmentation [54]. This great success shows that the innate attention mechanism of a pure transformer architecture can automatically search the important parts in images that contribute to image recognition. However, few study investigate the performance of vision transformer in FGVC. As the first work to study the vision transformer on FGVC, [14] proposed to replace the inputs of the final transformer layer with some important tokens and gained improved results. Nonetheless, the final class token may concern more on global information and pay less attention to local and low-level features, defecting the performance of vision transformer on FGVC since local information plays an important role in FGVC. Besides, previous works focus on FGVC benchmarks containing more than ten thousands of annotated images, and no study explores the capability of vision transformer on small-scale and ultra-fine-grained visual categorization (ultra-FGVC) settings. In this paper, we propose a novel feature fusion vision transformer (FFVT) for FGVC. FFVT aggregates the local information from low-level, middle-level and high-level tokens to facilitate the classification. We present a novel important token selection approach called Mutual Attention Weight Selection (MAWS) to select the representative tokens on each layer that are added as the inputs of the last transformer layer. In addition, we explore the performance of our method on four FGVC datasets to comprehensively verify the capability of our proposed FFVT on FGVC. In conclusion, our work has four main contributions. 1. To our best knowledge, we are the first study to explore the performance of vision transformer on both small-scale and ultra-FGVC settings. The two small-scale datasets in this paper are highly challenging due to the ultra-fine-grained inter-category variances and few training data available. Some examples are visualized in Figure 1. 2. We propose FFVT, a novel vision transformer framework for fine-grained visual categorization tasks that can automatically detect the distinguished regions and take advantage of different level of global and local information in images. 3. We present a novel important token selection approach called Mutual Attention Weight Selection (MAWS). MAWS can effectively select the informative tokens that are having high similarity to class token both in the contexts of the class token and the token itself without introducing extra parameters. 4. We verify the effectiveness of our method on four fine-grained benchmarks. Experimental results demonstrate that FFVT achieves state-of-the-art performance on them, offering an alternative to current CNN-based approaches. Ablation studies show that our proposed method boost the performance of the backbone model by 5.42%, 4.67% and 0.80% on CottonCultivar80, SoyCultivarLocal and CUB datasets, respectively. Fine-Grained Visual Categorization Methods on FGVC can be coarsely divided into two groups: localization-based methods and attention-based methods. Similar to object detection task, localization-based methods often detect the foreground objects and perform classification based on them. Early works [1,16,41] achieve this by taking advantage of part annotation to supervise the learning of the detection branch. However, bounding box annotation requires large manual labor, hampering their real-world applications. To alleviate above problem, recent localization-based methods introduce the weakly supervised object detection (WSOD) technique to predict the potential discriminative regions with only image-level label. Ge et al. [13] used WSOD and instance segmentation techniques to obtain the rough object instances, and then selected the important instances to perform classification. He et al. [15] presented two spatial constraints to select the discriminative parts obtained by the detection branch. Wang et al. [38] utilized correlations between regions to select distinguished parts. However, these methods require a well designed WSOD branch to propose potential discriminative regions. Moreover, the selected parts sent to the classification head often cover the whole object instead of the truly discriminative parts. Alternatively, attention-based methods automatically localize the discriminative regions via self-attention mechanism without extra annotations. Zhao et al. [50] proposed a diversified visual attention network which uses the diversity of the attention to collect dicriminative information. Xiao et al. [40] presented a two-level attention mechanism to steadily filter out the trivial parts. Similar to [40], Zheng et al. [54] proposed a progressive-attention to progressively detect discriminative parts at multiple scales. However, these methods often suffer from huge computational cost. Transformer Transformer has achieved huge success in natural language processing [4,30,31]. Motivated by this, researchers try to exploit the transformers in computer vision. Recent work ViT [6] achieves the state-of-the-art performance on image classification by employing a pure transformer architecture on a number of fix-sized image patches. Later, researchers explore the performance of the pure transformer in other computer vision tasks. Zheng [54] et al. developed a pure transformer SETR on semantic segmentation task. Alaaeldin et al. [9] exploited a transformer to generate the image descriptor for image retrieval task. Nonetheless, few studies explore the vision transformer on FGVC. The most similar to our work is TransFG [14] which is the first study to extend the ViT into FGVC, while there are two notable differences between FFVT and TransFG. First, TransFG selects the discriminative tokens and directly send them to the last transformer layer (no feature fusion), while FFVT aims to aggregate the local and different level information from each layer to enrich the feature representation capability via feature fusion. Second, our proposed token selection strategy is totally different from that of TransFG which requires the attention information from all transformer layer to generate the selected token indexes via matrix multiplication. In contrast, our proposed MAWS utilize attention information from only one transformer layer to produce the corresponding indexes. Hence, MAWS is simple and efficient. Our work is also in accordance with the spirit of recent research [11,33,34,35,36,43,44,45,46,47,48,49,51,52], which focuses on localizing subtle yet vital regions. Methods To better comprehend our method, we first briefly review the knowledge of vision transformer ViT in Section 3.1. Our proposed methods are then elaborately described in the following subsections. ViT For Image Recognition ViT follows the similar architecture of transformer in natural language processing with minor modification. Transformer in natural language processing takes a sequence of tokens as inputs. Similarly, given an image with resolution H * W , vision transformer first processes the image into N = H P * W P fix-sized patches x p . where P is the size for each patch. The patches x p are then linearly projected into a D-dimensional latent embedding space. To introduce the positional differences, a learnable vector called position embedding with the same size of patch embedding is directly added to patch embedding. Similar to the class token in BERT [4], an extra class token is added to interact with all patch embeddings and undertakes the classification task. The procedure is shown in Eq (1): z z z 0 = [x x x class ; x x x 1 p E E E; x x x 2 p E E E; ...; x x x N p E E E] + E E E pos(1) Where E E E ∈ R (P 2 ·C)×D is the patch embedding projection and C is the number of the image channels. E E E pos denotes the position embedding. After that, these patch embeddings are fed into the transformer encoder containing several multi-head self-attention (MSA) and multi-layer perceptron (MLP) blocks. Note that all layers retain the same latent vector size D. The outputs of the l-th layer are calculated by Eqs (2) to (3): z z z l = MSA(LN(z z z l−1 )) + z z z l−1 (2) z z z l = MLP(LN(z z z l )) + z z z l .(3) Where LN(·) is the layer normalization operation and z l denotes the encoded image representation. Eventually, a classification head implemented by a MLP block is applied to the class token z z z 0 l to obtain the predicted category. [14] suggests that the ViT cannot capture enough local information required for FGVC. To cope with this problem, we propose to fuse the low-level features and middle-level features to enrich the local information. We present a novel token selection approach called mutual attention weight selection (MAWS) to determine the tokens to be aggregated in the deep layer. This section introduces the details of our proposed FFVT. The overall architecture of FFVT is illustrated in Fig 2. Linear Projection of Flattened Patches FFVT Architecture Feature Fusion Module The key challenge of the FGVC is to detect the discriminative regions that significantly contribute to figuring out the subtle differences among subordinate categories. Previous works often achieve this by manually annotating the discriminative regions or integrating the RPN module. However, these methods suffer from some problems discussed in Section 1&2, limiting their performance on real-world applications. The MSA mechanism in vision transform can perfectly meet the above requirement, whereas MSA in deep layer is likely to pay more attention to the global information. Therefore, we propose a feature fusion module to compensate local information. As shown in figure 2, given the important tokens (hidden features) from each layer selected by MAWS module, we replace the inputs (except for the class token) of the last transformer layer with our selected tokens. In this way, the class token in the last transformer layer fully interacts with the low-level, middle level and high-level features from the previous layers, enriching the local information and feature representation capability. Specifically, we denote the tokens selected by MAWS module in l-layer as: z z z local l = [z 1 l , z 2 l , ..., z K l ](4) Where K is the number of selected features. The fused features along with the classification token fed into the last transformer layer L are: z z z f f = [z z z 0 L−1 ; z z z local 1 ; z z z local 2 ; ...; z z z local L−1 ](5) Eventually, following the ViT, the classification token of the final transformer layer is sent to the classification head to perform categorization. The problem turns to how to select the important and discriminative tokens. To that end, we propose an effective and efficient token selection approach described in the next section. Mutual Attention Weight Selection Module Since an image is split into many patches, token selection turns to be an important problem. Noise is added when the background patches are frequently selected, while discriminative patches can boost the model performance. Hence, we propose a token selection approach which directly utilizes the attention scores generated by multi-head self-attention module. To be specific, an attention score matrix for one attention head A ∈ R (N+1)×(N+1) is denoted as: A A A = [a a a 0 ; a a a 1 ; a a a 2 ; ...; a a a i ; ...; a a a N ] a a a i = [a i,0 , a i,1 , a i,2 , ..., a i, j , ..., a i,N ] Where a i, j is the attention score between token i and j in the context of token i, i.e., dotproduct between the query of token i and the key of token j. One simple strategy is to pick the tokens having the higher attention scores with the classification token as the classification token contains rich information for categorization. We can do this by sorting the a a a 0 and picking the K tokens with the bigger value. We denote this strategy as single attention weight selection (SAWS). However, SAWS may introduce noisy information since the selected tokens could aggregate much information from noisy patches. Taking a three-patch attention score matrix γ shown below as an example: Token three is selected as it has the biggest value in the attention score vector for classification token. However, token three aggregates much information from token one (the maximum attention score in a a a 3 ) thus may introduce noises assuming token one is a noisy token. To cope with this problem, we develop a mutual attention weight selection module which requires the selected tokens to be similar to the classification token both in the contexts of the classification token and the tokens themselves. γ =     In particular, we denote the first column in the attention score matrix as b b b 0 . Note that b b b 0 is the attention score vector between classification token and other tokens in the context of other tokens compared with a a a 0 in the context of classification token. The mutual attention weight m m ma a a i between the classification token and token i is then calculated by Eqs (8) to (9): m m ma a a i = a 0,i * b i,0 (8) a 0,i = e a 0,i ∑ N j=0 e a 0, j , b i,0 = e b i,0 ∑ N j=0 e b j,0(9) For multi-head self-attention, we first average the attention scores of all heads. After obtaining the mutual attention weight, the indexes of important tokens are collected according to the mutual attention values. Our approach does not introduce extra learning parameters. It is simple and efficient compared with the matrix multiplication in [14]. Experiments Datasets We explore the effectiveness of FFVT on two widely used FGVC dataset and two smallscale ultra-fine-grained datasets, i.e., CUB-200-2011 [32], Stanford Dogs [18], SoyCultivar-Local [46] and CottonCultivar80 [46]. The SoyCultivarLocal and CottonCultivar80 are two highly challenging datasets as they further reduce the granularity of categorization, e.g. from species to cultivar, and with few training data available. The statistics of four datasets are shown in Table 1. Implementation Details The same as the most current transformer-based approaches, the backbone network (ViT) of FFVT is pretrained on the ImageNet21K dataset. Following the same data augmentation methods on most existing works, input images are first resized to 500 × 500 for Soy.Loc and Cotton datasets, and 600 × 600 for CUB and Stanford Dogs. We then crop the image into 384 × 384 for Soy.Loc and Cotton, and 448 × 448 for CUB and Stanford Dogs (Random cropping in training and center cropping in testing). Random horizontal flipping is adopted and an extra color augmentation is applied for CUB. K in Eq (4) is set to 12 for CUB, Soy.Loc and Cotton, and 24 for Stanford Dogs. We select the SGD optimizer to optimize the network with a momentum of 0.9. The initial learning rate is 0.02 with the cosine annealing scheduler for FFVT on CUB, Soy.Loc Cotton datasets, and 0.003 on the Stanford Dogs dataset. The batch size is set to 8 for all datasets except for the Stanford Dogs with a batch size of 4. For fair comparisons, we reimplement the experiments of ViT and TransFG on the Stanford Dogs benchmark with their default settings and the same batch size as FFVT. Experiments are conducted on four Nvidia 2080Ti GPUs using PyTorch deep learning framework. Comparison with the State-Of-The-Art Here, we demonstrate the experimental results on four datasets and compare our method with a number of state-of-the-art works. As shown in Table 2, FFVT obtains the second best-performed method on CUB with an accuracy of 91.6%, beating other methods by a large margin except for the most recent state-of-the-art fine-grained method TransFG (-0.1%). Note that FFVT achieves a comparable accuracy against TransFG with much less computation cost and GPU memory consumption since the overlapping strategy of TransFG significantly increases the number of the input patches from 784 to 1296. Besides, limited by our computation resources, the batch size of TransFG on the experiment of CUB dataset is two times larger than FFVT. This may also account for the relative performance differences. FFVT outperforms all the listed approaches on Stanford Dogs with an accuracy of 91.5%, strongly exceeding the second best-performed TransFG by 0.9%. Table 2: Comparison of different methods on CUB-200-2011 datasets. The best accuracy is highlighted in bold and the second best accuracy is underlined. Method Backbone Accuracy ResNet50 [15] ResNet50 84.5 GP-256 [39] VGG16 85.8 MaxEnt [8] DenseNet161 86.6 DFL-CNN [37] ResNet50 87.4 NTS-Net [42] ResNet50 87.5 Cross-X [23] ResNet50 87.7 DCL [3] ResNet50 87.8 CIN [12] ResNet101 88.1 DBTNet [53] ResNet101 88.1 ACNet [17] ResNet50 88.1 S3N [5] ResNet50 88.5 FDL [22] DenseNet161 89.1 PMG [7] ResNet50 89.6 API-Net [55] DenseNet161 90.0 StackedLSTM [13] GoogleNet 90.4 ViT [6] ViT-B_16 90.8 TransFG [14] ViT-B_16 91.7 FFVT ViT-B_16 91.6 SoyCultivarLocal and CottonCultivar80 are two extremely challenging ultra-fine-grained datasets. The difficulty lies in two folds, i.e., super-subtle inter-class differences and few training images (three for each category). Some examples are visualized in figure 1. Therefore, locating the discriminative regions plays an essential role in accurate classification. ViT-B_16 90.2 TransFG [14] ViT-B_16 90.6 (92.3) FFVT ViT-B_16 91.5 The results of experiments on SoyCultivarLocal and CottonCultivar80 are shown in Table 3. FFVT obtains the highest accuracy of 57.92% on CottonCultivar80, outperforming the second best-performed method by a large margin (+4.17%). Similarly, our proposed FFVT beats all methods with an accuracy of 44.17% on SoyCultivarLocal. Ablation Studies We perform the ablation studies on CottonCultivar80, SoyCultivarLocal and CUB to further validate the effectiveness of our proposed methods. SAWS is the single attention weight selection strategy designed in Section 3.2.2. As shown in Table 5, even the simple SAWS strategy can remarkably boost the performance by 4.58%, 3.50% and 0.64% on CottonCulti-var80, SoyCultivarLocal and CUB, respectively. The results confirm the necessity of aggregating the local and different level information for vision transformer on FGVC. A bigger improvement can be seen when applying the MAWS strategy (+6.67%, 4.84% and 0.80% on CottonCultivar80, SoyCultivarLocal and CUB, respectively), showing that MAWS better exploits the attention information. MAWS explicitly selects the most useful tokens thus forces the model to learn from these informative parts. We then investigate the influence of the hyper-parameter K. Table 6 summarizes the results of FFVT on the SoyCultivarLocal dataset with the value of K ranging from 10 to 14. FFVT achieves the best performance when there are 12 tokens selected for each layer. One possible reason is that the tokens focused by each attention head are selected by the proposed MAWS module and contribute positively to the classification since this value (12) is in accordance with the number of the attention heads. As K increases from 10 to 12, the accuracy steadily enhances from 43.17% to 44.17%. A different pattern can be seen when K continues increasing to 14, where the accuracy slightly reduces to 42.5%. The performance drop may due to that large K introduces the noisy tokens while small K value lead to insufficient discriminative information for classification. Note that results of all K settings show a significant improvements over that backbone ViT (39.33%), indicating that FFVT is not very sensitive to the value of K. Conclusion This paper proposes a novel fine-grained visual categorization architecture FFVT and achieves state-of-the-art performance on four benchmarks. To facilitate the performance of vision transformer in FGVC, we propose a feature fusion approach to enrich the local, low-level and middle-level information for the classification token. To select the discriminative tokens that to be aggregated, we develop a novel token selection module MAWS which explicitly takes advantage of the attention scores produced by self-attention mechanism. Experimental results show that FFVT significantly improve the classification accuracy of standard ViT on different fine-grained settings, i.e., normal-scale, small-scale and ultra-fine-grained settings. We observe that FFVT is very effective on the challenging datasets, confirming its capability of capturing subtle differences and discriminative information. Based on our encouraging results, we believe that the pure-transformer model has the huge potential on different FGVC settings, even in the small-scale datasets without the induction bias like convolutional neural networks. Figure 1 : 1Examples of images in SoyCultivarLocal and Cotton datasets. Images in the first row come from four species of Soy.Loc, while examples in the second row are selected from four categorizes of Cotton. Figure 2 : 2The overall architecture of the proposed FFVT. Images are split into a sequence of fix-sized patches which are then linearly projected into the embedding space. Combined with the position embedding, the patch embeddings are fed into the Transformer Encoder to learn the patch features. Feature fusion is exploited before the last transformer layer to aggregate the important local, low-level and middle level information from previous layers. This is implemented by replacing the inputs (exclude classification token) of the last transformer layer with the tokens selected by the MAWS Module. Table 3 : 3Comparison of different methods on Stanford Dogs (Dogs) dataset. The best accuracy is highlighted in bold and the second best accuracy is underlined. Values in parentheses are reported results in their papers.Method Backbone Dogs MaxEnt [8] DenseNet161 83.6 FDL [22] DenseNet161 84.9 RA-CNN [10] VGG19 87.3 DB [27] ResNet50 87.7 SEF [24] ResNet50 88.8 Cross-X [23] ResNet50 88.9 API-Net [55] DenseNet161 90.3 ViT [6] Table 4 : 4Comparison of different methods on SoyCultivarLocal (Soy.Loc) and CottonCulti-var80 (Cotton) datasets. The best accuracy is highlighted in bold and the second best accuracy is underlined.Method Backbone Cotton Soy.Loc AlexNet [19] AlexNet 22.92 19.50 VGG16 [26] VGG16 50.83 39.33 ResNet50 [15] ResNet50 52.50 38.83 InceptionV3 [28] GoogleNet 37.50 23.00 MobileNetV2 [25] MobileNet 49.58 34.67 Improved B-CNN [21] VGG16 45.00 33.33 NTS-Net [42] ResNet50 51.67 42.67 fast-MPN-COV [20] ResNet50 50.00 38.17 ViT [6] ViT-B_16 51.25 39.33 DeiT-B [29] ViT-B_16 53.75 38.67 TransFG [14] ViT-B_16 45.84 38.67 FFVT ViT-B_16 57.92 44.17 Table 5 : 5Ablation studies on CottonCultivar80 (Cotton), SoyCultivarLocal (Soy.Loc), and CUB datasets. The best accuracy is highlighted in bold.Method Cotton Soy.Loc CUB ViT [6] 51.25 39.33 90.85 ViT+Feature Fusion+SAWS 55.83 42.83 91.49 FFVT(ViT+Feature Fusion+MAWS) 57.92 44.17 91.65 Table 6 : 6Ablation studies of the hyper-parameter K on SoyCultivarLocal benchmark. The best accuracy is highlighted in bold. 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{'abstract': 'The core for tackling the fine-grained visual categorization (FGVC) is to learn subtle yet discriminative features. Most previous works achieve this by explicitly selecting the discriminative parts or integrating the attention mechanism via CNN-based approaches. However, these methods enhance the computational complexity and make the model dominated by the regions containing the most of the objects. Recently, vision transformer (ViT) has achieved SOTA performance on general image recognition tasks. The self-attention mechanism aggregates and weights the information from all patches to the classification token, making it perfectly suitable for FGVC. Nonetheless, the classification token in the deep layer pays more attention to the global information, lacking the local and low-level features that are essential for FGVC. In this work, we propose a novel pure transformer-based framework Feature Fusion Vision Transformer (FFVT) where we aggregate the important tokens from each transformer layer to compensate the local, low-level and middle-level information. We design a novel token selection module called mutual attention weight selection (MAWS) to guide the network effectively and efficiently towards selecting discriminative tokens without introducing extra parameters. We verify the effectiveness of FFVT on four benchmarks where FFVT achieves the state-of-the-art performance. Code is available at this link.', 'arxivid': '2107.02341', 'author': ['Jun Wang jun.wang.3@warwick.ac.uk ', 'Xiaohan Yu xiaohan.yu@griffith.edu.au ', 'Griffith University ', 'Yongsheng Australia yongsheng.gao@griffith.edu.au ', 'Gao ', '\nUniversity of Warwick\nUK\n', '\nGriffith University\nAustralia\n'], 'authoraffiliation': ['University of Warwick\nUK', 'Griffith University\nAustralia'], 'corpusid': 235742913, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15230, 'n_tokens_neox': 12874, 'n_words': 7331, 'pdfsha': '64d8af9153d68e9b50f616d227663385bece93b9', 'pdfurls': ['https://arxiv.org/pdf/2107.02341v3.pdf'], 'title': ['Feature Fusion Vision Transformer for Fine-Grained Visual Categorization', 'Feature Fusion Vision Transformer for Fine-Grained Visual Categorization'], 'venue': []}

arxiv

Dicke superradiance requires interactions beyond nearest-neighbors Wai-Keong Mok Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543Singapore California Institute of Technology 91125PasadenaCAUSA Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543Singapore California Institute of Technology 91125PasadenaCAUSA Ana Asenjo-Garcia Department of Physics Columbia University 10027New YorkNew YorkUSA Department of Physics Columbia University 10027New YorkNew YorkUSA SumTze Chien Division of Physics and Applied Physics School of Physical and Mathematical Sciences Nanyang Technological University 637371Singapore Division of Physics and Applied Physics School of Physical and Mathematical Sciences Nanyang Technological University 637371Singapore Leong-Chuan Kwek Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543Singapore MajuLab CNRS-UNS-NUS-NTU International Joint Research Unit Singapore UMI 3654 Singapore National Institute of Education Nanyang Technological University 637616SingaporeSingapore Quantum Science and Engineering Centre (QSec) Nanyang Technological University SuperradianceSingapore Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543Singapore MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit 3654Singapore UMI, Singapore National Institute of Education Nanyang Technological University 637616SingaporeSingapore Quantum Science and Engineering Centre (QSec) Nanyang Technological University Singapore CONTENTS Ben Grossmann Mccoy Lim Chris Chen Jasen Zion Kishor Bharti Davit Aghamalyan Lewis Ruks Thi Ha Kyaw Tobias Haug Steven Touzard Klaus Mølmer Stuart Masson Dicke superradiance requires interactions beyond nearest-neighbors Photon-mediated interactions within an excited ensemble of emitters can result in Dicke superradiance, where the emission rate is greatly enhanced, manifesting as a high-intensity burst at short times. The superradiant burst is most commonly observed in systems with long-range interactions between the emitters, although the minimal interaction range remains unknown. Here, we put forward a new theoretical method to bound the maximum emission rate by upper-bounding the spectral radius of an auxiliary Hamiltonian. We harness this tool to prove that for an arbitrary ordered array with only nearest-neighbor interactions in all dimensions, a superradiant burst is not physically observable. We show that Dicke superradiance requires minimally the inclusion of nextnearest-neighbor interactions. For exponentially-decaying interactions, the critical coupling is found to be asymptotically independent of the number of emitters in all dimensions, thereby defining the threshold interaction range where the collective enhancement balances out the decoherence effects. Our findings provide key physical insights to the understanding of collective decay in many-body quantum systems, and the designing of superradiant emission in physical systems for applications such as energy harvesting and quantum sensing.Introduction.-Collective spontaneous emission of N initially-inverted atoms with identical all-to-all interactions mediated by the electromagnetic vacuum results in a burst of light with intensity scaling as N 2 [1-3]. This phenomenon is commonly referred to as "Dicke superradiance" or "superradiant burst". Over the past decades, this many-body phenomenon has attracted a lot of interest in both theoretical and experimental studies[25,26]using a multitude of physical platforms such as trapped ions [27], molecular aggregates [28-31], solidstate emitters [32-36], cold atoms and molecules [37-40], and superconducting qubits [41-43], with wide-ranging applications including the generation of multi-photon states with improved metrological properties [18, 44-47], energy harvesting [48-50], ultrabright LEDs [51] and quantum sensing [52, 53].The atoms in Dicke's original model were assumed to be confined within a spatial extent smaller than the emission wavelength λ. Consequently, the atoms become indistinguishable with respect to the absorption or emission of photons, such that their quantum state |j = N/2, m⟩ (with −N/2 ≤ m ≤ N/2) is permutation-invariant. This permutation symmetry greatly reduces the complexity of the problem, as it constrains the dynamics to N + 1 states, instead of exploring the full Hilbert space (which scales as 2 N ). Recently, there has been substantial research progress with extended systems where atoms are distributed over a region larger than λ, thus breaking this symmetry. Of particular interest are ordered atomic arrays[54,55], in which the superradiant properties can be greatly affected by the geometry and dimensionality of the lattice[16,19,20,22,24,56]. The interactions between the emitters are typically modelled by long-range dipole-dipole interactions mediated via the electromagnetic vacuum[57,58].A long-standing fundamental question is the minimal interaction range required for the occurrence of a superradiant burst. Intuitively, superradiance can be thought of as a competition between (transient) phase synchronization, which leads to the buildup of atomic correlations, and decoherence[59]. Both effects stem from the same dissipative interactions[8,22]. Since synchronization of nonlinear classical phase oscillators has been demonstrated with nearest-neighbor (NN) coupling [60], one may expect the atomic phases to synchronize for sufficiently strong NN interactions resulting in a superradiant burst[59]. Moreover, for a fixed interaction range, higher dimensionality was reported to result in stronger superradiance due to long-range order[19,24]. On the flip side, it could also be argued that for short-range interactions, the buildup of correlations is not strong enough to overcome decoherence, thereby preventing superradiance.In this Letter, we prove that superradiant burst is impossible in an arbitrary D-dimensional array with only nearest-neighbor interactions, for arbitrary times and initial states. That is, we show that, in all cases, the emission rate is upper bounded by that of independent emitters, resulting in no enhancement from collective dynamics. Including next-nearest-neighbor interactions, we show that a superradiant burst can be physically observed for certain values of the interaction strengths, arXiv:2211.00668v2 [quant-ph] 25 May 2023 2 Photon emission rate Time Superradiant burst Monotonic decay / 0 0 1 Gap p / 0 s / 0 FIG. 1. Dynamics of the photon emission rate R(t) for emitter arrays with only nearest-neighbor interactions of strength γ, normalized by the individual emitter decay rate γ0. For γ/γ0 < γs,Ṙ(0) < 0 and the photon emission rate decays monotonically without a superradiant burst (blue). Superradiance occurs for γ/γ0 > γs (red). The physically-valid regime is defined by 0 < γ/γ0 ≤ γp. For nearest-neighbor interactions, γp < γs (with a finite gap between γp and γs) for any arbitrary emitter array in all dimensions, rendering Dicke superradiance physically impossible.thereby defining a minimal interaction range for superradiance. Another question is the threshold interaction range, which we define to be such that the critical coupling required for a burst becomes independent of the number of emitters, for any D. We show that exponentially-decaying interactions lie on the threshold interaction range for which the synchronization of the dipoles arising from the emission balances the decoherence effects.Model.-The dynamics of an undriven ensemble of N emitters can be described by the Lindblad master equation (setting ℏ = 1)with J ij = J * ji and γ ij = γ * ji to ensure Hermiticity. The raising and lowering operators for the j th emitter are denoted as σ + j ≡ |e i ⟩ ⟨g i | and σ − j ≡ |g i ⟩ ⟨e i | which describe transitions between the ground state |g i ⟩ and excited state |e i ⟩. The first term contains the coherent Hamiltonian interactions between the emitters, while the second term captures processes such as collective and local dissipation of the emitters via the superoperatorWe assume J ij and γ ij to be time-independent, such that the superoperator L generates a dynamical semigroup describing the dynamics of a Markovian open quantum system.For a physically valid evolution (i.e., a completely positive and trace-preserving map), the matrix Γ containing the elements γ ij (which we will refer to as the decoherence matrix ) must be positive semi-definite[61][62][63]. The decoherence matrix can be diagonalized to yield N decay rates Γ ν ≥ 0, with ν ∈ {1, . . . , N } and the corresponding collective jump operatorsĉ ν . The total photon emission rate of the emitters, integrated over all emission directions, is defined for any state ρ asFor independent emitters with γ ij = γ 0 δ ij , the total emission rate has a maximum of N γ 0 (saturated by the fullyexcited state), and R(t) ≡ R ρ(t) decays exponentially. However, interactions between the emitters can cause R(t) to increase beyond its initial value. This speedup in emission is commonly referred to as the superradiant burst, first discovered by Dicke [1] (seeFig. 1). Throughout this work, we refer to superradiant burst as the increase in the total emission rate beyond N γ 0 , but the peak intensity need not scale as N 2 . In general, characterizing the burst at arbitrary times can be difficult, hence one typically useṡevaluated at the fully-excited initial state ρ(0), witḣ R(0) ≡Ṙ ρ(0) > 0 a sufficient condition for a superradiant burst. While we consider the burst at t = 0, we will provide physical justification on why this is sufficient.Here, we put forward a new (and complementary) criterion to preclude any possibility of a burst: by a simple change of basis, one can write Eq. (2) as the expectation value of an auxiliary spin Hamiltonianwith R ρ = tr(H Γ ρ). The maximum photon emission rate can thus be calculated by bounding the spectral radius of the auxiliary spin Hamiltonian. If the upper bound is equal or smaller than N γ 0 , no burst can occur for all times and arbitrary initial states. While finding the largest eigenvalue of H Γ may be non-trivial, this criterion allows one to definitively prove the absence of a burst for arbitrary times, thus going beyond the con-ditionṘ(0) ≡Ṙ ρ(0) > 0. Furthermore, this approach opens up the possibility of finding theoretical limits for the emission rate arising from superradiant dynamics, as we show below and in the Supplementary Information[64].No superradiance for nearest-neighbor coupling.-Let us consider a hypercube array of N emitters with arbitrary dimension D (N = n D ). For the case of NN interactions, γ ii ≡ γ 0 = 1 and γ ij = γ if emitters i and 3 j are nearest-neighbor (γ ij = 0 otherwise). The coupling γ ∈ [0, 1] is required for the matrix Γ to be positive semidefinite. Without loss of generality, we have assumed γ ij to be real and positive. We prove that for this model, superradiant burst cannot occur for any t > 0, for any arbitrary initial state and for any Hamiltonian coupling J ij . To determine the physically valid regime, we impose the condition that Γ is positive semidefinite. Notice that the decoherence matrix can be expressed as Γ = I N +γA, where I N is the N × N identity matrix, and A is the adjacency matrix of a n × n grid graph. Using the fact that the grid graph is the Cartesian product of D path graphs P n □· · ·□P n , it can be shown that the smallest eigenvalue of Γ is [64]which gives the physically valid regime as γ ≤ γ p ,This rate reduces to γ p = 1/(2D) in the N → ∞ limit, or when imposing periodic boundary conditions for a finite N . This can be regarded as coming from the coordination number for each emitter, which approaches 2D in the infinite-array limit. We now state our main result.Theorem 1 Let Γ be the decoherence matrix for a nearest-neighbor interaction model, with γ ij = δ ij +γδ ⟨ij⟩ , where γ ∈ [0, 1], and δ ⟨ij⟩ = 1 if the emitters indexed by i and j are nearest-neighbor on the D−dimensional regular lattice, and 0 otherwise. For γ ≤ (2D) −1 , the emission rate R ρ is maximized by the fully-excited state |e⟩ ⊗N withWe provide a sketch of the proof here, while the details can be found in theSupplementary Information [64]. By expressing H Γ in the product-state basis and using the Gershgorin circle theorem [65], we can upper bound max t R(t) ≤ N in the physically valid regime γ < 1/(2D). This is saturated by N independent emitters in the fully-excited state, with eigenvalue N . Hence, Theorem 1 implies that superradiant burst is impossible at all times. To gain a deeper physical understanding, we evaluate the superradiant regime γ > γ s for the fullyexcited initial state, characterized by the transition aṫ R(0) = 0, for which [64]For all 2 < N 1/D < ∞, it can be shown that γ p < γ 2 s and therefore γ p < γ s . Hence, the superradiant regime does not overlap with the physically valid regime. Generalization to the hyper-rectangle configuration where the number of sites along each dimension can be different is Observable superradiance I II III FIG. 2. Region of superradiant burst in the γ2−γ1 plane. The physically valid (superradiant) regime is contained within the blue (red) boundary lines, with the conditions stated in the main text. Blue shaded region: Physically valid, but not superradiant. Regions I, II and III are defined in the main text. Red shaded region: Physically valid with superradiant burst. Grey shaded region: unphysical regime. The red shaded region requires a minimum of γ2 ≈ 0.185. All shaded regions here are obtained from numerical calculations for N = 100, which agree very well with the analytical results obtained in the infinite-array limit.straightforward, and the same conclusion is obtained[64]. While our analysis of the NN model is valid for any initial state, we consider a fully-inverted initial state for the next two sections: the analysis of next-nearest neighbor and exponentially decaying interactions. Next-nearest neighbor coupling.-Including the NNN interactions, we now show that a superradiant burst is indeed possible. For simplicity, let us consider a 1D ring of N emitters with periodic boundary conditions. In this configuration, Γ turns out to be a circulant matrix with the first column given by (1, γ 1 , γ 2 , 0, . . . , 0, γ 2 , γ 1 ) T with 0 ≤ {γ 1 , γ 2 } ≤ 1. The subsequent columns are simply cyclic permutations of the first column. Diagonalizing Γ exactly yields the eigenvaluesfor ν = 0, . . . , N − 1. In the infinite-array limit N → ∞, the eigenvalues form a continuous band in momentum space Γ(k) = 1 + 2γ 1 cos(k) + 2γ 2 cos(2k), with the dimensionless wavevector 0 ≤ k < 2π. At the turning points where ∂ k Γ = 0, we have: Γ(0) = 1 + 2(γ 1 + γ 2 ) which is always positive, Γ(π) = 1 − 2(γ 1 − γ 2 ) and Γ(k * ) = 1 − (γ 2 1 + 8γ 2 2 )/4γ 2 where cos k * = −γ 1 /4γ 2 . Demanding that Γ(k) > 0 thus produces the physically valid regimes: (I) γ 1 − γ 2 ≤ 1 2 , γ 1 > 4γ 2 and (II) γ 2 1 + 8γ 2 2 ≤ 4γ 2 , γ 1 ≤ 4γ 2 , together with the bounds 4 NNN NN (b) (a) NN Dicke Differential emission rate Δ 0 t 0 t FIG. 3. Differential emission rate ∆R = R(t)/R(0) − 1 against time (in units of emitter lifetime), for N = 9 emitters. ∆R > 0 indicates superradiance. (a) Dynamical behavior of ∆R for the Dicke model (red), Next-nearest neighbor 1D ring (NNN, orange), Nearest-neighbor 1D ring (NN, blue) and Nearest-neighbor 2D square (NN, green) (see labels in(b)). The coupling parameters are chosen to maximize g (2) (0). (b) Short-time behavior obtained by zooming into the grey region of (a). Only the Dicke and the next-nearest neighbor models exhibit superradiance. The curve for the Dicke model is scaled down by a factor of 10 for visualization purposes.γ 1 , γ 2 ∈ [0, 1] (blue regions inFig. 2). The superradiant condition can be obtain fromṘ(0) = 0 as (III) γ 2 1 + γ 2 2 > 1/2. There is an overlap region with the physically valid regime, as shown by the red shaded region inFig. 2. For certain values of γ 1 , γ 2 , superradiant burst can occur. Moreover, the fact that this overlap region requires γ 2 > (4 − √ 2)/14 ≈ 0.185 is consistent with our previous conclusion of no superradiance using only NN coupling (i.e., γ 2 = 0). Superradiance is also forbidden by having only NNN coupling (i.e., γ 1 = 0). Results from numerical simulations of N = 9 emitters are presented inFig. 3, which show that the NNN model has a small superradiant burst compared to the Dicke model, and no superradiance for NN models. We remark that this superradiance arises from destructive interference leading to dark decay channels with suppressed decay rates Γ ν ≈ 0 while the dominant decay channel has a rate that does not scale with N . This mechanism is generally true for all models with a sharp interaction cutoff beyond a certain range.Threshold interaction range for a superradiant burst.-In many previous works[16,19,20,22,24], Γ is obtained from a realistic modelling of the atomic interactions mediated by electromagnetic vacuum using the appropriate Green's function. Our goal here, however, is to shed light on the essential physics of superradiance by considering analytically tractable models that still exhibit interesting behaviors. Consider an interaction which decays exponentially with the separation r ij between the emitters: γ ij ∝ e −κrij , where κ controls the decay of the interaction strength with emitter separation. We set the diagonal elements of Γ as 1, and define γ ≡ e −κd with d the emitter NN separation such that γ ij = γ |⃗ xi−⃗ xj | , where ⃗ x i ∈ Z D is the position vector of the i th lattice site. Physically, this model describes exponentially-decaying interactions 1D 2D 3D 2 3 4 5 6 0.3 0.2 ! " ∼ $ %&.()* D Critical coupling, " Emitter number, N s ⇠ D 0.793 FIG. 4. Critical coupling γs for exponentially-decaying interactions in a 1D chain, a 2D square array and a 3D cubic array with N emitters. Superradiance occurs for γ > γs. For all dimension D, γs becomes independent of N for large N . (Inset) Log-log plot of γs against D for N ≈ 10 6 . γs decreases as D increases with a power-law scaling γs ∼ D −0.793 .between the atoms. For a sufficiently large N in D dimensions such that γ N ≪ 1,Ṙ(0) is approximately given by the asymptotic forṁfor some constant C [64]. Interestingly, this suggests that the critical coupling parameter γ s for superradiance is independent of N as N → ∞ for all dimension, agreeing with the numerical results shown inFig. 4. This is in stark contrast with previous results (primarily using long-range power-law interactions such as γ ij ∝ 1/r ij ), which predict that the critical emitter separation increases with N in 2D and 3D arrays[19,24].Figure 45 also shows that for large N , γ s ∼ D −0.793 exhibits a power-law scaling with the spatial dimension. This is intuitive as the average coupling per emitter increases with D which in turn lowers the critical coupling required for superradiance[24]. The N -independence of γ s for our short-range exponential model can be physically interpreted as the threshold interaction range where the synchronization effects due to collective interactions scales similarly with N as the local decoherence, such that adding more emitters do not affect the onset of the superradiant regime. For even shorter-range interactions such as the NN model, the local decoherence dominates which prevents superradiance. Longer-range models such as power-law interactions favor synchronization and thus enhance superradiance as N increases.Scaling of the peak emission rate with number of emitters-Eq. (4) shows that the problem of calculating the emission rate is equivalent to finding the average energy of a state under the Hamiltonian H Γ . This enables us to find upper bounds on the scaling of the peak emission rate with N , for arbitrary geometries and types of interactions. As we have shown before in Theorem 1, the maximum emission rate for arbitrary NN models is N γ 0 . For 1D arrays with an exponentially-decaying interaction, the upper bound on the emission rate is found to scale as O(N ) for γ < 1 [64]. This bound increases to O(N log N ) for 1D arrays with a power-law interaction of the form 1/r[64]. This latter scaling is consistent with the numerical results obtained in the literature which, in contrast to our bound, have only been obtained for relatively small systems and under certain approximations[20,22,24]. While finding exact bounds may be exponentially hard, one could in principle upper-bound other models, as well as tighten the currently-obtained bounds.Discussion.-In this Letter, we addressed the fundamental problem of the minimal interaction range required for superradiance. Crucially, we proved that nearestneighbor interactions cannot induce emitter correlations faster that the decoherence, resulting in the impossibility of superradiance. As shown, the minimal interaction range is therefore next-nearest neighbor, and longerrange interactions generally lead to stronger superradiance. We also found that the short-range exponential interaction marks the threshold interaction range in all dimensions where the emitter correlations and local decoherence scale similarly with the number of emitters such that the critical coupling required for superradiant burst becomes independent of the number of emitters, in stark contrast with previous conclusions using longerrange power-law interactions. We stress that, apart from the nearest-neighbor model, our classification of a superradiant burst is strictly speaking only valid at short times up to O((γ 0 t) 2 ) (ifR(0) < 0 which is true for the models considered here [64]), where the dynamics of the fullyexcited emitters do not depend on the Hamiltonian. This can be physically justified for later times using secondorder mean field theory[64].The techniques used in this work have broader applications in determining the theoretical bounds for the emission rate of different models, thereby exposing the ultimate limitations of superradiance beyond the NN model. Beyond providing fundamental insights to the physics of superradiance, our results can also motivate the design of atomic lattices in engineered baths such as nanophotonic crystals with engineered interactions or superconducting resonator arrays for qubits. Moreover, hypercube geometries should be within reach of state-of-the-art quantum simulators, given the recent advances in generating arbitrary networks in cavity [66] and circuit [67] quantum electrodynamics platforms.The authors are grateful to Photon-mediated interactions within an excited ensemble of emitters can result in Dicke superradiance, where the emission rate is greatly enhanced, manifesting as a high-intensity burst at short times. The superradiant burst is most commonly observed in systems with long-range interactions between the emitters, although the minimal interaction range remains unknown. Here, we put forward a new theoretical method to bound the maximum emission rate by upper-bounding the spectral radius of an auxiliary Hamiltonian. We harness this tool to prove that for an arbitrary ordered array with only nearest-neighbor interactions in all dimensions, a superradiant burst is not physically observable. We show that Dicke superradiance requires minimally the inclusion of nextnearest-neighbor interactions. For exponentially-decaying interactions, the critical coupling is found to be asymptotically independent of the number of emitters in all dimensions, thereby defining the threshold interaction range where the collective enhancement balances out the decoherence effects. Our findings provide key physical insights to the understanding of collective decay in many-body quantum systems, and the designing of superradiant emission in physical systems for applications such as energy harvesting and quantum sensing. Introduction.-Collective spontaneous emission of N initially-inverted atoms with identical all-to-all interactions mediated by the electromagnetic vacuum results in a burst of light with intensity scaling as N 2 [1][2][3]. This phenomenon is commonly referred to as "Dicke superradiance" or "superradiant burst". Over the past decades, this many-body phenomenon has attracted a lot of interest in both theoretical and experimental studies [25,26] using a multitude of physical platforms such as trapped ions [27], molecular aggregates [28][29][30][31], solidstate emitters [32][33][34][35][36], cold atoms and molecules [37][38][39][40], and superconducting qubits [41][42][43], with wide-ranging applications including the generation of multi-photon states with improved metrological properties [18,[44][45][46][47], energy harvesting [48][49][50], ultrabright LEDs [51] and quantum sensing [52,53]. The atoms in Dicke's original model were assumed to be confined within a spatial extent smaller than the emission wavelength λ. Consequently, the atoms become indistinguishable with respect to the absorption or emission of photons, such that their quantum state |j = N/2, m⟩ (with −N/2 ≤ m ≤ N/2) is permutation-invariant. This permutation symmetry greatly reduces the complexity of the problem, as it constrains the dynamics to N + 1 states, instead of exploring the full Hilbert space (which scales as 2 N ). Recently, there has been substantial research progress with extended systems where atoms are distributed over a region larger than λ, thus breaking this symmetry. Of particular interest are ordered atomic arrays [54,55], in which the superradiant properties can be greatly affected by the geometry and dimensionality of the lattice [16,19,20,22,24,56]. The interactions between the emitters are typically modelled by long-range dipole-dipole interactions mediated via the electromagnetic vacuum [57,58]. A long-standing fundamental question is the minimal interaction range required for the occurrence of a superradiant burst. Intuitively, superradiance can be thought of as a competition between (transient) phase synchronization, which leads to the buildup of atomic correlations, and decoherence [59]. Both effects stem from the same dissipative interactions [8,22]. Since synchronization of nonlinear classical phase oscillators has been demonstrated with nearest-neighbor (NN) coupling [60], one may expect the atomic phases to synchronize for sufficiently strong NN interactions resulting in a superradiant burst [59]. Moreover, for a fixed interaction range, higher dimensionality was reported to result in stronger superradiance due to long-range order [19,24]. On the flip side, it could also be argued that for short-range interactions, the buildup of correlations is not strong enough to overcome decoherence, thereby preventing superradiance. In this Letter, we prove that superradiant burst is impossible in an arbitrary D-dimensional array with only nearest-neighbor interactions, for arbitrary times and initial states. That is, we show that, in all cases, the emission rate is upper bounded by that of independent emitters, resulting in no enhancement from collective dynamics. Including next-nearest-neighbor interactions, we show that a superradiant burst can be physically observed for certain values of the interaction strengths, For γ/γ0 < γs,Ṙ(0) < 0 and the photon emission rate decays monotonically without a superradiant burst (blue). Superradiance occurs for γ/γ0 > γs (red). The physically-valid regime is defined by 0 < γ/γ0 ≤ γp. For nearest-neighbor interactions, γp < γs (with a finite gap between γp and γs) for any arbitrary emitter array in all dimensions, rendering Dicke superradiance physically impossible. thereby defining a minimal interaction range for superradiance. Another question is the threshold interaction range, which we define to be such that the critical coupling required for a burst becomes independent of the number of emitters, for any D. We show that exponentially-decaying interactions lie on the threshold interaction range for which the synchronization of the dipoles arising from the emission balances the decoherence effects. Model.-The dynamics of an undriven ensemble of N emitters can be described by the Lindblad master equation (setting ℏ = 1) ρ = −i N i,j=1 J ij σ + i σ − j , ρ + N i,j=1 γ ij D[σ − i , σ − j ]ρ ≡ L[ρ],(1) with J ij = J * ji and γ ij = γ * ji to ensure Hermiticity. The raising and lowering operators for the j th emitter are denoted as σ + j ≡ |e i ⟩ ⟨g i | and σ − j ≡ |g i ⟩ ⟨e i | which describe transitions between the ground state |g i ⟩ and excited state |e i ⟩. The first term contains the coherent Hamiltonian interactions between the emitters, while the second term captures processes such as collective and local dissipation of the emitters via the superoperator D[σ − i , σ − j ]ρ = σ − i ρσ + j − {σ + j σ − i , ρ}/2. We assume J ij and γ ij to be time-independent, such that the superoperator L generates a dynamical semigroup describing the dynamics of a Markovian open quantum system. For a physically valid evolution (i.e., a completely positive and trace-preserving map), the matrix Γ containing the elements γ ij (which we will refer to as the decoherence matrix ) must be positive semi-definite [61][62][63]. The decoherence matrix can be diagonalized to yield N decay rates Γ ν ≥ 0, with ν ∈ {1, . . . , N } and the corresponding collective jump operatorsĉ ν . The total photon emission rate of the emitters, integrated over all emission directions, is defined for any state ρ as R ρ ≡ N ν=1 Γ ν ⟨ĉ † νĉν ⟩ = N ν=1 Γ ν Tr(ĉ † νĉν ρ).(2) For independent emitters with γ ij = γ 0 δ ij , the total emission rate has a maximum of N γ 0 (saturated by the fullyexcited state), and R(t) ≡ R ρ(t) decays exponentially. However, interactions between the emitters can cause R(t) to increase beyond its initial value. This speedup in emission is commonly referred to as the superradiant burst, first discovered by Dicke [1] (see Fig. 1). Throughout this work, we refer to superradiant burst as the increase in the total emission rate beyond N γ 0 , but the peak intensity need not scale as N 2 . In general, characterizing the burst at arbitrary times can be difficult, hence one typically useṡ R ρ = i ν ⟨[H,ĉ † νĉν ]⟩ − µ,ν Γ µ Γ ν ⟨ĉ † µ [ĉ µ ,ĉ † ν ]ĉ ν ]⟩(3) evaluated at the fully-excited initial state ρ(0), witḣ R(0) ≡Ṙ ρ(0) > 0 a sufficient condition for a superradiant burst. While we consider the burst at t = 0, we will provide physical justification on why this is sufficient. Here, we put forward a new (and complementary) criterion to preclude any possibility of a burst: by a simple change of basis, one can write Eq. (2) as the expectation value of an auxiliary spin Hamiltonian H Γ = N j,k=1 γ kj σ + j σ − k ,(4) with R ρ = tr(H Γ ρ). The maximum photon emission rate can thus be calculated by bounding the spectral radius of the auxiliary spin Hamiltonian. If the upper bound is equal or smaller than N γ 0 , no burst can occur for all times and arbitrary initial states. While finding the largest eigenvalue of H Γ may be non-trivial, this criterion allows one to definitively prove the absence of a burst for arbitrary times, thus going beyond the con-ditionṘ(0) ≡Ṙ ρ(0) > 0. Furthermore, this approach opens up the possibility of finding theoretical limits for the emission rate arising from superradiant dynamics, as we show below and in the Supplementary Information [64]. No superradiance for nearest-neighbor coupling.-Let us consider a hypercube array of N emitters with arbitrary dimension D (N = n D ). For the case of NN interactions, γ ii ≡ γ 0 = 1 and γ ij = γ if emitters i and j are nearest-neighbor (γ ij = 0 otherwise). The coupling γ ∈ [0, 1] is required for the matrix Γ to be positive semidefinite. Without loss of generality, we have assumed γ ij to be real and positive. We prove that for this model, superradiant burst cannot occur for any t > 0, for any arbitrary initial state and for any Hamiltonian coupling J ij . To determine the physically valid regime, we impose the condition that Γ is positive semidefinite. Notice that the decoherence matrix can be expressed as Γ = I N +γA, where I N is the N × N identity matrix, and A is the adjacency matrix of a n × n grid graph. Using the fact that the grid graph is the Cartesian product of D path graphs P n □· · ·□P n , it can be shown that the smallest eigenvalue of Γ is [64] Γ min = 1 − 2Dγ cos π N 1/D + 1 ,(5) which gives the physically valid regime as γ ≤ γ p , γ p = 2D cos π N 1/D + 1 −1 .(6) This rate reduces to γ p = 1/(2D) in the N → ∞ limit, or when imposing periodic boundary conditions for a finite N . This can be regarded as coming from the coordination number for each emitter, which approaches 2D in the infinite-array limit. We now state our main result. Theorem 1 Let Γ be the decoherence matrix for a nearest-neighbor interaction model, with γ ij = δ ij +γδ ⟨ij⟩ , where γ ∈ [0, 1], and δ ⟨ij⟩ = 1 if the emitters indexed by i and j are nearest-neighbor on the D−dimensional regular lattice, and 0 otherwise. For γ ≤ (2D) −1 , the emission rate R ρ is maximized by the fully-excited state |e⟩ ⊗N with R ρ = N . We provide a sketch of the proof here, while the details can be found in the Supplementary Information [64]. By expressing H Γ in the product-state basis and using the Gershgorin circle theorem [65], we can upper bound max t R(t) ≤ N in the physically valid regime γ < 1/(2D). This is saturated by N independent emitters in the fully-excited state, with eigenvalue N . Hence, Theorem 1 implies that superradiant burst is impossible at all times. To gain a deeper physical understanding, we evaluate the superradiant regime γ > γ s for the fullyexcited initial state, characterized by the transition aṫ R(0) = 0, for which [64] γ s = 2D(1 − N −1/D ) −1/2 .(7) For all 2 < N 1/D < ∞, it can be shown that γ p < γ 2 s and therefore γ p < γ s . Hence, the superradiant regime does not overlap with the physically valid regime. Generalization to the hyper-rectangle configuration where the number of sites along each dimension can be different is straightforward, and the same conclusion is obtained [64]. While our analysis of the NN model is valid for any initial state, we consider a fully-inverted initial state for the next two sections: the analysis of next-nearest neighbor and exponentially decaying interactions. Next-nearest neighbor coupling.-Including the NNN interactions, we now show that a superradiant burst is indeed possible. For simplicity, let us consider a 1D ring of N emitters with periodic boundary conditions. In this configuration, Γ turns out to be a circulant matrix with the first column given by (1, γ 1 , γ 2 , 0, . . . , 0, γ 2 , γ 1 ) T with 0 ≤ {γ 1 , γ 2 } ≤ 1. The subsequent columns are simply cyclic permutations of the first column. Diagonalizing Γ exactly yields the eigenvalues Γ ν = 1 + 2γ 1 cos 2πν N + 2γ 2 cos 4πν N(8) for ν = 0, . . . , N − 1. In the infinite-array limit N → ∞, the eigenvalues form a continuous band in momentum space Γ(k) = 1 + 2γ 1 cos(k) + 2γ 2 cos(2k), with the dimensionless wavevector 0 ≤ k < 2π. At the turning points where ∂ k Γ = 0, we have: Γ(0) = 1 + 2(γ 1 + γ 2 ) which is always positive, Γ(π) = 1 − 2(γ 1 − γ 2 ) and Γ(k * ) = 1 − (γ 2 1 + 8γ 2 2 )/4γ 2 where cos k * = −γ 1 /4γ 2 . Demanding that Γ(k) > 0 thus produces the physically valid regimes: (I) γ 1 − γ 2 ≤ 1 2 , γ 1 > 4γ 2 and (II) Fig. 2). The superradiant condition can be obtain fromṘ(0) = 0 as (III) γ 2 1 + γ 2 2 > 1/2. There is an overlap region with the physically valid regime, as shown by the red shaded region in Fig. 2. γ 2 1 + 8γ 2 2 ≤ 4γ 2 , γ 1 ≤ 4γ 2 , For certain values of γ 1 , γ 2 , superradiant burst can occur. Moreover, the fact that this overlap region requires γ 2 > (4 − √ 2)/14 ≈ 0.185 is consistent with our previous conclusion of no superradiance using only NN coupling (i.e., γ 2 = 0). Superradiance is also forbidden by having only NNN coupling (i.e., γ 1 = 0). Results from numerical simulations of N = 9 emitters are presented in Fig. 3, which show that the NNN model has a small superradiant burst compared to the Dicke model, and no superradiance for NN models. We remark that this superradiance arises from destructive interference leading to dark decay channels with suppressed decay rates Γ ν ≈ 0 while the dominant decay channel has a rate that does not scale with N . This mechanism is generally true for all models with a sharp interaction cutoff beyond a certain range. Threshold interaction range for a superradiant burst.-In many previous works [16,19,20,22,24], Γ is obtained from a realistic modelling of the atomic interactions mediated by electromagnetic vacuum using the appropriate Green's function. Our goal here, however, is to shed light on the essential physics of superradiance by considering analytically tractable models that still exhibit interesting behaviors. Consider an interaction which decays exponentially with the separation r ij between the emitters: γ ij ∝ e −κrij , where κ controls the decay of the interaction strength with emitter separation. We set the diagonal elements of Γ as 1, and define γ ≡ e −κd with d the emitter NN separation such that between the atoms. For a sufficiently large N in D dimensions such that γ N ≪ 1,Ṙ(0) is approximately given by the asymptotic forṁ γ ij = γ |⃗ xi−⃗ xj | , where ⃗ x i ∈ Z D isR(0) ∼ N 2Dγ 2 1 − γ 2 − 1 + C (− ln γ) D(9) for some constant C [64]. Interestingly, this suggests that the critical coupling parameter γ s for superradiance is independent of N as N → ∞ for all dimension, agreeing with the numerical results shown in Fig. 4. This is in stark contrast with previous results (primarily using long-range power-law interactions such as γ ij ∝ 1/r ij ), which predict that the critical emitter separation increases with N in 2D and 3D arrays [19,24]. Figure 4 also shows that for large N , γ s ∼ D −0.793 exhibits a power-law scaling with the spatial dimension. This is intuitive as the average coupling per emitter increases with D which in turn lowers the critical coupling required for superradiance [24]. The N -independence of γ s for our short-range exponential model can be physically interpreted as the threshold interaction range where the synchronization effects due to collective interactions scales similarly with N as the local decoherence, such that adding more emitters do not affect the onset of the superradiant regime. For even shorter-range interactions such as the NN model, the local decoherence dominates which prevents superradiance. Longer-range models such as power-law interactions favor synchronization and thus enhance superradiance as N increases. Scaling of the peak emission rate with number of emitters-Eq. (4) shows that the problem of calculating the emission rate is equivalent to finding the average energy of a state under the Hamiltonian H Γ . This enables us to find upper bounds on the scaling of the peak emission rate with N , for arbitrary geometries and types of interactions. As we have shown before in Theorem 1, the maximum emission rate for arbitrary NN models is N γ 0 . For 1D arrays with an exponentially-decaying interaction, the upper bound on the emission rate is found to scale as O(N ) for γ < 1 [64]. This bound increases to O(N log N ) for 1D arrays with a power-law interaction of the form 1/r [64]. This latter scaling is consistent with the numerical results obtained in the literature which, in contrast to our bound, have only been obtained for relatively small systems and under certain approximations [20,22,24]. While finding exact bounds may be exponentially hard, one could in principle upper-bound other models, as well as tighten the currently-obtained bounds. Discussion.-In this Letter, we addressed the fundamental problem of the minimal interaction range required for superradiance. Crucially, we proved that nearestneighbor interactions cannot induce emitter correlations faster that the decoherence, resulting in the impossibility of superradiance. As shown, the minimal interaction range is therefore next-nearest neighbor, and longerrange interactions generally lead to stronger superradiance. We also found that the short-range exponential interaction marks the threshold interaction range in all dimensions where the emitter correlations and local decoherence scale similarly with the number of emitters such that the critical coupling required for superradiant burst becomes independent of the number of emitters, in stark contrast with previous conclusions using longerrange power-law interactions. We stress that, apart from the nearest-neighbor model, our classification of a superradiant burst is strictly speaking only valid at short times up to O((γ 0 t) 2 ) (ifR(0) < 0 which is true for the models considered here [64]), where the dynamics of the fullyexcited emitters do not depend on the Hamiltonian. This can be physically justified for later times using secondorder mean field theory [64]. The techniques used in this work have broader applications in determining the theoretical bounds for the emission rate of different models, thereby exposing the ultimate limitations of superradiance beyond the NN model. Beyond providing fundamental insights to the physics of superradiance, our results can also motivate the design of atomic lattices in engineered baths such as nanophotonic crystals with engineered interactions or superconducting resonator arrays for qubits. Moreover, hypercube geometries should be within reach of state-of-the-art quantum simulators, given the recent advances in generating arbitrary networks in cavity [66] and circuit [67] quantum electrodynamics platforms. The collective jump operators can be constructed aŝ c ν = N j=1 v jν σ − j (S1) where {(v 1ν . . . v N ν ) T } are the normalized eigenvectors of the decoherence matrix Γ, which are mutually orthogonal. In this new basis, the master equation can be rewritten in the standard Lindblad form aṡ ρ = −i   N i,j=1 H ij σ + i σ − j , ρ   + N ν=1 Γ ν D[ĉ ν ]ρ (S2) where we have defined the Lindblad superoperator as D[Ô]ρ =ÔρÔ † − {Ô †Ô , ρ}/2. In Ref. [? ], Masson and Asenjo-Garcia proposed to use the zero-delay second-order correlation function g (2) (0), defined here as g (2) (0) = N µ,ν=1 Γ µ Γ ν ⟨ĉ † µĉ † νĉνĉµ ⟩ N µ=1 Γ µ ⟨ĉ † µĉµ ⟩ 2 ,(S3) as the minimal superradiance condition, where g (2) (0) > 1 for the fully-excited initial state is the signature for superradiant burst. Intuitively, in a superradiant burst, the emission of the first photon enhances the emission of the second photon, which leads to photon bunching, i.e., g (2) (0) > 1. The same effect can also be interpreted as the synchronization of the emitters at short times, which manifests as nonzero correlations ⟨σ + j σ − k ⟩ between them. For pedagogical reasons, we derive an explicit formula that relatesṘ(0) to g (2) (0). Using the master equation (S2), we can calculateṘ(0) asṘ (0) = N µ,ν=1 Γ µ Γ ν ⟨ĉ † µĉ † νĉνĉµ ⟩ − ⟨ĉ † µĉµĉ † νĉν ⟩ + i N µ=1 Γ µ ⟨[H,ĉ † µĉµ ]⟩ ,(S4) where H is the Hamiltonian in Eq. (S2). Using the definition in Eq. (S3), the first sum is simply R 2 (0)g (2) (0). The second and third sums do not give simple results for arbitrary initial states. However, for the fully-excited initial state |e⟩ ⊗N , it can be shown that the second sum factorizes exactly to N µ,ν=1 Γ µ Γ ν ⟨ĉ † µĉµĉ † νĉν ⟩ = µ=1 Γ µ ⟨ĉ † µĉµ ⟩ 2 = R 2 (0) (S5) and the third sum vanishes for any arbitrary H ij . We can justify both of these claims by direct calculations: ⟨ĉ † µĉµĉ † νĉν ⟩ = N m,n,p,q=1 v * mµ v nµ v * pν v qν ⟨e|σ + m σ − n σ + p σ − q |e⟩ = N m,n,p,q=1 v * mµ v nµ v * pν v qν δ mn δ pq ⟨e|σ + m σ − n σ + p σ − q |e⟩ = N m,p=1 v * mµ v mµ v * pν v pν ⟨e|σ + m σ − m σ + p σ − p |e⟩ = ⟨ĉ † µĉµ ⟩ ⟨ĉ † νĉν ⟩ . (S6) Note that this is not true in general for arbitrary product states. Next, note that the Hamiltonian contributes the terms like i ⟨[H,ĉ † µĉµ ]⟩ inṘ(0). For a generic Hamiltonian, H = i,j H ij σ + i σ − j , expanding the commutator yields ⟨[H,ĉ † µĉµ ]⟩ = N i,j,k,l=1 H ij v * kµ v lµ (⟨e|σ + i σ − j σ + k σ − l |e⟩ − ⟨e|σ + k σ − l σ + i σ − j |e⟩) = N i,j,k,l=1 H ij v * kµ v lµ (δ ij δ kl − δ kl δ ij ) = 0, (S7) which shows that H does not affect the presence or absence of a superradiant burst in the system. This provides a simple explanation for the observation in Fig. 3(a) of Ref. [? ] which shows no significant effect of the Hamiltonian on superradiance. Thus, we obtain the relatioṅ R(0) = R 2 (0) g (2) (0) − 1 ,(S8) which reveals the close connection between superradiance and photon bunching, thereby justifying the results in Ref. [? ]. A more detailed analysis can be found in a recent work by F. Robicheaux [? ], which includes additional considerations beyond the scope of this work such as directional emission as well as arbitrary product initial states. B. Efficient calculation of g (2) (0) Evaluating Eq. (S3) using the initial state |e⟩ ⊗N yields the a simple expression [? ? ] Although the diagonalization of Γ is not necessary to compute g (2) (0), there are several reasons why it is still useful. First, the eigenvectors allow us to construct the jump operatorsĉ ν in Eq. (S1), which provide insights about the dominant decay channels in the system. Moreover, since the eigenvalues Γ ν must be non-negative, calculating the eigenvalues allow us to determine the physically valid regime for the generic Γ that we use throughout the paper. g (2) (0) = 1 − 2 N + N ν=1 Γ 2 ν N ν=1 Γ ν 2 = 1 − 2 N + Tr (Γ 2 ) (Tr (Γ)) 2 ,(S9) II. EXPLICIT CALCULATIONS FOR NEAREST-NEIGHBOR EMITTER ARRAYS A. 1D chain To begin, let us consider a chain of N dissipatively-coupled emitters with open boundary conditions, initialized in the fully-excited state. We neglect the Hamiltonian interactions between the emitters since they do not affect the existence of superradiant burst as explained in Sec. I A. Furthermore, we restrict to the case of NN interactions, leading to the master equation with H ij = 0 and γ ij = δ ij + γ(δ i,j+1 + δ i,j−1 ) for arbitrary 0 ≤ γ ≤ 1. Note that without loss of generality, we have assumed γ ij to be real and positive, since g (2) (0) is invariant under the transformation γ → γe iϕ for any ϕ ∈ R. Physically, if γ = e −κd where κ > 0 is the attenuation coefficient and d is the constant separation between each emitter, then this model describes the exponentially-decaying interactions which can arise from evanescent coupling in photonic crystals [? ]. The decoherence matrix Γ in this case is a tridiagonal Toeplitz matrix with all diagonal elements 1 and all super-/sub-diagonal elements γ, which can be exactly diagonalized to yield the eigenvalues Γ ν = 1 + 2γ cos νπ N + 1 , ν = 1, . . . , N.(S10) The g (2) (0) function thus reads g (2) (0) = 1 − 1 N + 2(N − 1) N 2 γ 2 .(S11) Let us denote γ s as the critical γ for a superradiant transition, where g (2) (0) = 1. Applying the condition g (2) (0) > 1 directly gives the superradiant condition γ > γ s , where γ s = N 2N (N − 1) . (S12) γ p = 1 2 sec π N + 1 . (S13) It is easy to show that for a positive integer N , γ p < γ s for all N > 2, with equality at exactly N = 2. This implies that no superradiance is possible within the physically valid regime, except in the trivial case of N = 2 where the system reduces to the original Dicke model for two emitters with the collective jump operatorĉ 1 = σ − 1 + σ − 2 . This example of a 1D NN model demonstrates a potential pitfall of naively applying the g (2) (0) > 1 condition for an arbitrary decoherence matrix Γ, where the predicted superradiant regime falls outside of the physically valid regime and is thus not possible. This is not a concern for previous works [? ? ? ] where the electromagnetic interactions are already physically valid by construction. While obtaining g (2) (0) can be done in O(N ) steps for ordered arrays, verifying that Γ is positive-semidefinite typically requires O(N 3 ) steps, such as by performing a Cholesky decomposition or by finding the smallest eigenvalue of Γ. B. Generalization to non-identical nearest-neighbor coupling In the main text, we have assumed that all the nearest-neighbor coupling are the same, γ ij = γ for |i − j| = 1. Now, we can lift the assumption and consider a more general case where the nearest-neighbor coupling in a 1D chain are not the same, so γ ij =      1, i = j γ min(i,j) , |i − j| = 1 0, else. (S14) Let us also define A = Γ − I N which has zero diagonal elements. The superradiance condition g (2) (0) > 1 becomes ||A|| F > √ N (S15) where ||A|| F is the Frobenius norm. Recall that for a physically valid model, the eigenvalues of Γ must be non-negative. Moreover, since A is a symmetric tridiagonal matrix with zero on the main diagonal, the spectrum of A is symmetrical with respect to zero. Hence, the spectral radius ρ λ min (Γ) ≥ 0 =⇒ ||A|| 2 = σ max (A) ≤ 1 =⇒ ||A|| F ≤ √ N , N even √ N − 1, N odd (S16) which is a necessary condition for physical validity. Comparing with the superradiance condition above, we once show that the superradiance regime is unphysical. C. D-dimensional hypercube array with nearest-neighbor interactions We can generalize the above calculations by considering an arbitrary array dimensionality D. The number of pairwise interactions is just Dn D−1 (n − 1) where N = n D , which means that the number of nonzero off-diagonal elements γ is simply twice of that: 2Dn D−1 (n − 1). Thus, Tr (Γ 2 ) = N i,j=1 |γ ij | 2 = N + 2Dn D−1 (n − 1)γ 2 = N + 2D(N − N 1−1/D )γ 2 ,(S17) which leads to g (2) (0) = 1 − 2 N + 1 N 2 (N + 2Dn D−1 (n − 1)γ 2 ) = 1 − 1 N + 2D(N − N 1−1/D )γ 2 N 2 = 1 + 2Dγ 2 − 1 N − 2Dγ 2 N 1+1/D . (S18) In the infinite-dimensional case D → ∞ for a fixed N , g (2) (0) → 1 − 1/N < 1 for N > 1 and the emitters behave independently. The D-dimensional hypercube configuration of emitters with nearest-neighbor coupling can be mapped onto a generalized grid graph G n D with adjacency matrix A. The decoherence matrix is simply Γ = I N + γA. G n D is the Cartesian product of D path graphs of length-n P n . The eigenvalues of the adjacency matrix of P n are 2 cos jπ n+1 , j = 1, . . . , n. Hence, the n D eigenvalues of A are given by the sum a {jα} = 2 D α=1 cos j α π n + 1 , j α = 1, . . . , n. (S19) The minimum value occurs when all j α = n, such that min {jα} a {jα} = 2D cos nπ n + 1 = −2D cos π n + 1 . (S20) From this, we obtain the minimum eigenvalue of Γ as Γ min = 1 − 2Dγ cos π n + 1 = 1 − 2Dγ cos π N 1/D + 1 . (S21) For the smallest non-trivial array, we have n = 2 with Γ min = 1 − Dγ. For an infinite-array, we have Γ min = 1 − 2Dγ. In both cases, the coefficient of γ is simply the coordination number of each emitter in the array (ignoring boundaries when N → ∞). D. Generalization to unequal edge lengths The generalization to the 'hyperrectangle' configuration where n can be different for each dimension is straightforward. The grid graph for N = j n j emitters is now G N = P n1 □ · · · □ P n D , leading to Γ min = 1 − 2γ D j=1 cos π n j + 1 (S22) and hence γ p =   2 D j=1 cos π n j + 1   −1 . (S23) In this generalized array, there are 2N (D − j n −1 j ) nearest-neighbor interactions which results in g (2) (0) = 1 + 2Dγ 2 − 1 N − 2γ 2 N D j=1 1 n j (S24) and the superradiant transition at γ s =   2   D − D j=1 1 j     −1/2 =   2 D j=1 1 − 1 n j   −1/2 .(S25) Since cos(π/(n j + 1)) > 1 − 1/n j for n j > 2 (with equality at n j = 2), we conclude that γ p ≤ γ 2 s and therefore no superradiant burst can be observed. E. Generalization to product initial states So far, we have considered the fully-excited initial state |e⟩ ⊗N . We can show that our main result of 'no superradiance for NN interactions' is valid even for product initial states n (cos θ/2 |g n ⟩ + e iϕ sin θ/2 |e n ⟩). Let us also consider the extreme case of the all-to-all Hamiltonian H = J m,n σ + m σ − n . For the dissipative coupling, we choose the phase convention γ mn ≥ 0, γ mn = γ nm . The time-derivative of the emission rate at time t = 0 is given by [? ]: R(0) = −N sin 2 θ 2 − 1 2 sin 2 θ ′ m,n γ mn + 2 sin 2 θ 2 sin 2 θ 2 − 1 2 ′ m,n γ 2 mn + 1 4 sin 2 θ sin 2 θ 2 − 1 2 ′ l,m,n γ mn (γ ml + γ nl ),(S26) where the notations ′ m,n and ′ l,m,n indicate that the dummy indices must all be different. Notice thatṘ(0) is independent of J, which means that the Hamiltonian does not contribute to the superradiance burst (at initial time). This is also valid even if we use a more general spin-spin Hamiltonian. As a check, we can set θ = π to recover the result for the fully-excited initial state:Ṙ(0) = −N + Choosing γ = 1/(2D), we havė R(0) N = sin 2 θ 2 8D [9 − 26D − 2(6D − 1) cos θ − (2D − 1) cos 2θ] ≤ 0,(S28) independent of ϕ and J. Since D ≥ 1, we haveṘ(0) < 0 except forṘ(0) = 0 at θ = 0 (corresponding to all emitters in the ground state). III. DERIVATIONS OF g (2) (0) FOR VARIOUS MODELS Here, we derive explicitly the various g (2) (0) expressions used throughout the paper. From the main text, the only calculation needed is Tr (Γ 2 ) which is equal to the Frobenium norm ∥Γ∥ 2 F = i,j |γ ij | 2 using the Hermiticity of Γ. Without loss of generality, we consider γ ij to be real and positive, and also rescale Γ appropriately such that Tr (Γ) = N . Since we assume that the atoms decay identically, this amounts to setting the diagonal elements to γ ii = 1. A. 1D chain with exponentially-decaying interactions For open boundary conditions, we label the emitters sequentially along the chain by the index i, starting from one end of the chain. Without loss of generality, we align the 1D chain to the x-axis, giving the spatial coordinate x i = id for the i th emitter. The matrix elements of Γ are then given by γ ij = e −κd|i−j| ≡ γ |i−j| . In the short-range limit, where we neglect terms like γ 2 or smaller, the results agree with the nearest-neighbor model up to O(γ). Without diagonalizing Γ analytically, we can check that it has a Cholesky decomposition for all 0 ≤ γ ≤ 1 which asserts that Γ is positive semidefinite and thus physically valid for any d ≥ 0. To calculate the superradiant transition, we first work out the g (2) (0) function. Starting from γ ij = e −κd|i−j| ≡ γ |i−j| , we have 2(N − m) off-diagonal matrix elements with the value γ m , m = 1, . . . , N − 1, and N diagonal elements with the value 1. Thus, Tr (Γ 2 ) = N i,j=1 |γ ij | 2 = N + 2 N −1 m=1 (N − m)γ 2m = N − 2γ 2 (1 − γ 2N − N (1 − γ 2 )) (1 − γ 2 ) 2 (S29) and g (2) (0) = 1 − 2 N + 1 N 2 N − 2γ 2 (1 − γ 2N − N (1 − γ 2 )) (1 − γ 2 ) 2 = 1 − 1 N − 2γ 2 (1 − γ 2N − N (1 − γ 2 )) N 2 (1 − γ 2 ) 2 . (S30) For a fixed N , we can obtain the superradiant transition point γ s numerically by solving g (2) (0) = 1. We can proceed further analytically by considering the regime where N ≫ 2/κd such that γ 2N = e −2N κd ≪ 1. This gives the asymptotic expansion g (2) (0) = 1 + 3γ 2 − 1 N (1 − γ 2 ) + O 1 N 2 ,(S31) which leads to γ s ∼ 1/ √ 3. In terms of κd, this means that superradiance occurs when At first glance, by comparing the large-N expressions for g (2) (0) in Eq. (S31) with Eq. (S11), there appears to be an inconsistency. However, this can be easily resolved by recalling that the two models only agree up to O(γ), while the discrepancy only arises in the O(γ 2 ) terms. It is worth emphasizing that the nearest-neighbor interaction model in Sec. II A and the NN interaction model here are fundamentally different: With nearest neighbor interactions, physical constraints require γ < γ p ∼ 1/2. On the other hand, adding long-range interactions ensure that the model is physically valid for all 0 ≤ γ ≤ 1, which allows for superradiance. B. 1D ring with exponentially-decaying interactions Next, we consider the case of periodic boundary conditions where the emitters are arranged in a ring configuration with a constant separation d. For simplicity, we consider the case where N is odd. The decoherence matrix Γ for the 1D ring turns out to be a circulant matrix with the first column given by (1, γ, . . . , γ (N −1)/2 , γ (N −1)/2 , . . . , γ) T , or γ i1 = max{γ i−1 , γ N −i+1 }. The subsequent columns are simply cyclic permutations of the first column. Thus, Tr (Γ 2 ) = N i,j=1 |γ ij | 2 = N (1+γ 2 +. . .+γ (N −1) +γ (N −1) +. . .+γ 2 ) = N +2N (N −1)/2 m=1 γ 2m = N +2N γ 2 (1 − γ N −1 ) 1 − γ 2 (S33) and we obtain g (2) (0) = 1 − 1 N + 2u 2 (1 − γ N −1 ) N (1 − γ 2 ) = 1 + 3γ 2 − 1 N (1 − γ 2 ) + O 1 N 2 (S34) for N ≫ 1/κd. Note that for sufficiently large N , the superradiant transition γ s = 1/ √ 3 is the same for both the chain and ring configurations, which physically means that the boundary contributions become negligible. This again provides a simple explanation for the observations in Ref. [? ], where the significant discrepancy between the 1D chain and ring only occurs for small N < 10. Alternatively, we can exploit the nice properties of circulant matrices to diagonalize Γ exactly, which yields the eigenvalues Γ j = (1 − γ) 1 + γ − 2γ (N +1)/2 cos(jπ) cos(jπ/N ) 1 + γ 2 − 2γ cos(2jπ/N ) ≈ 1 − γ 2 1 + γ 2 − 2γ cos(2jπ/N ) , j = 0, . . . , N − 1 (S35) for large N . The sum j Γ 2 j can be approximated by integral N −1 j=0 Γ 2 j ≈ N 2πˆ2 π 0 dθ 1 − γ 2 1 + γ 2 − 2γ cos θ 2 = N (1 + γ 2 ) (1 − γ 2 ) ,(S36) which gives the same limiting value for g (2) (0) as Eqs. (S31) and (S34). From the exact eigenvalues in Eq. (S35) we can argue that Γ is positive semidefinite for all 0 ≤ γ ≤ 1: It can be easily verified that 1 + γ ≥ 2γ (N +1)/2 for N > 1, with equality achieved at exactly γ = 1. This implies that the numerator of Γ j is positive. From the law of cosines, the denominator represents geometrically the length-squared of the triangular side opposite the angle 2jπ/N , subtended by the other two sides of lengths 1 and γ, which is also positive. Hence, the superradiant regime γ > γ s is physically valid. C. D-dimensional hypercube array with exponentially-decaying interactions Assuming a large number of emitters, each emitter approximately contributes 1 N Tr Γ 2 ≈ 1 + 2D(γ 2 + γ 4 + γ 6 + . . .) + 2 D n1,n2,...,n D =1 γ 2 √ n 2 1 +...+n 2 D ,(S37) where the second term comes from the emitters displaced by exactly integer multiples of the Cartesian unit vectors, and the third term comes from the rest of the emitters which can be divided into 2 D equal contributions by symmetry. The final sum can be approximated by converting it to an integral in hyperspherical coordinates and exploiting the rotational symmetry to get n1,n2,...,n D =1 γ 2 √ n 2 1 +...+n 2 D ∼ Cˆ∞ 0 dr r D−1 γ 2r = C (− ln γ) D (S38) for some constant C. Note that this is asymptotically independent of N . D. 1/r-interactions in 1D, 2D and 3D arrays Next, we consider dissipative long-range interactions that decay as 1/r, with γ ij ∝ 1 κrij . Although an accurate modelling of dipole interactions require additional short-and medium-range contributions (like 1/r 3 and 1/r 2 terms), we will show that including only the 1/r interactions results in qualitative agreement with previous numerical simulations for 1D, 2D and 3D arrays in the large-N regime [? ? ]. 1D chain Here, γ ij = δ ij + (1 − δ ij ) γ |i−j| , and thus the second order correlation function is readily found to be g (2) (0) = 1 − 2 N + 1 N 1 + 2γ 2 ∞ n=1 1 n 2 ≈ 1 + 1 N π 2 3 γ 2 − 1 ,(S39) which gives the superradiant transition point γ s = √ 3/π which is once again independent of N . 2D square array Labelling the emitters' Cartesian coordinates on the square grid as ⃗ r i = (x i , y i ), where x i , y i are integers, we have γ ij = δ ij + (1 − δ ij ) γ |⃗ ri−⃗ rj | . Therefore, Tr(Γ 2 ) ≈ N γ 2 1 + 2π 2 3 + 4 n x,y=1 1 x 2 + y 2 ≈ N γ 2 (A + B ln N ),(S40) where N = n 2 , and A and B are real constants which can be determined numerically. From this, we can deduce that superradiance condition takes the form γ > 2 √ A + B ln N ∼ 1 √ ln N . (S41) 3D cubic array The calculations are similar to the 2D case, but with ⃗ x i = (x i , y i , z i ). We have Tr(Γ 2 ) ≈ N γ 2 1 + π 2 + 8 n x,y,z=1 1 x 2 + y 2 + z 2 ≈ N γ 2 (A + BN 1/3 ),(S42) where N = n 3 . This gives the superradiance condition of the form γ > 2 √ A + BN 1/3 ∼ 1 N 1/6 .(S43) Note that these couplings are consistent with those found in Ref. [? ]. E. 1D chain with infinite-range interactions On the other extreme, we consider the case of infinite-range interactions between emitters on a 1D array, i.e. |γ ij | > 0 in the limit |x i −x j | → ∞. This can be realized for example in waveguide quantum electrodynamics by coupling atoms to a 1D photonic crystal or a nanophotonic waveguide, where the photons in the waveguide mediate the infinite-range interactions between the spins [? ? ]. We also consider the more general case of a chiral waveguide, where the photon emission rate is direction-dependent [? ]. Denoting the left(right) emission rate by γ L (γ R ) and neglecting the coherent interactions, we can model the dissipative emitter dynamics (after tracing out the environment) with the master equation [? ]ρ = −i[H, ρ] + γ L D[ĉ L ]ρ + γ R D[ĉ R ]ρ,(S44) whereĉ L/R = j e ±ikxj σ − j is the collective spin operator. H contains the individual emitter Hamiltonian as well as coherent interactions between the emitters. We will not specify the form of H here since it is not relevant for the purposes of calculating g (2) (0). The decoherence matrix elements are γ jl = cos((kd(j − l)) − iχ sin(kd(j − l)) (S45) where χ ≡ (γ R − γ L )/(γ R + γ L ) is the chirality parameter. Normalizing the decoherence matrix by dividing γ L + γ R throughout, we obtain the matrix elements γ jl = γ L γ L + γ R e ik(xj −x l ) + γ R γ L + γ R e −ik(xj −x l ) = cos(k(x j − x l )) − iχ sin(k(x j − x l )).(S46) We now assume a constant emitter separation of d, and calculate in a similar fashion as the case of exponentiallydecaying interactions Tr (Γ 2 ) = N i,j=1 |γ ij | 2 = 1 4 csc 2 (kd) (1 − χ 2 )(1 − cos(2N kd)) + N 2 (1 + χ 2 )(1 − cos(2kd))(S47) and g (2) (0) = 1 2 3 + χ 2 − 4 N + 1 − χ 2 2N 2 csc 2 (kd) sin 2 (N kd).(S48) Note that since ∂ |χ| g (2) (0) = |χ| csc 2 (kd) 2N 2 N 2 (1 − cos(2kd)) + cos(2N kd) − 1 ≥ 0 (S49) we conclude that chirality always enhances the superradiance effect, except at points when g (2) (0) is already at maximal value of 2(1 − 1/N ). For a consistency check, the limit kd → 0 yields the well-known result g (2) (0) = 2(1 − 1/N ) for the original Dicke model [? ? ], where g (2) (0) > 1 for all N > 2 can be interpreted as a spontaneous superradiance [? ]. Physically, when the emitters become sufficiently close together (compared to the wavelength) such that they become indistinguishable, and the only decay channel is that associated with the collective jump operatorĉ 1 = j σ − j at a rate N times of the individual decay rate. Noting that ∂ |χ| g (2) (0) ≥ 0, we find that a larger chirality |χ| always results in a stronger superradiance effect. The limiting case |χ| = 1 corresponds to a perfectly unidirectional emission, where the dynamics become independent of the emitter separation [? ? ? ]. When N kd = mπ where m ∈ Z and m/N / ∈ Z, g (2) (0) is minimal with the value (3 + χ 2 − 4/N )/2. For N > 3, superradiance always occurs regardless of chirality. However, for N = 3, we require a minimum chirality of |χ| = 1/ √ 3 such that g (2) (0) > 1, as confirmed via numerical simulations shown in Fig. S1. Of course, with only N = 3 emitters, the superradiance effect just above the threshold chirality will be quite weak. F. All-to-all interactions The all-to-all dissipatively-coupled system of emitters can be described a complete graph G with N vertices. The decoherence matrix is Γ = I N + γA, where γ is the interaction strength and A is the corresponding adjacency matrix. This leads to Tr (Γ 2 ) = N + 2γ 2 E(G) = N + γ 2 N (N − 1) (S50) FIG. S1. Total emission rate R(t) for N = 3 emitters coupled to a 1D chiral waveguide with various chirality parameters χ. Analytical calculations predictṘ(0) > 0 (superradiant burst) for χ > 1/ √ 3 ≈ 0.577. Inset shows the short-time behavior of R(t). The black horizontal line at R = 3 is added for reference. The numerical results demonstrate that a minimum chirality is required for superradiance. and g (2) (0) = 1 + γ 2 (N − 1) − 1 N . (S51) The case of γ = 1 is exactly the original Dicke model which always results in superradiance. For arbitrary γ, the transition point is γ > γ s , where γ s = 1/ √ N − 1. Such a model is always physically valid, i.e. γ p = 1. To see this, we note that Γ corresponds to the master equatioṅ ρ = γD j σ − j ρ + (1 − γ) j D[σ − j ]ρ,(S52) which is in the standard Lindblad form. Physically, this corresponds to the Dicke model with local dissipation serving as an imperfection. The condition γ > γ s therefore sets an upper bound to the amount of local dissipation such that superradiance is preserved. Intuitively, superradiance becomes more robust to local dissipation for large N . This also agrees with the exact solution of Eq. (S52) obtained recently in the large-N limit [? ]. IV. THIRD-ORDER CORRELATION FUNCTION, g (3) (0) We can characterize the superradiant burst beyond the initial rateṘ(0) by calculating the third-order correlation function g (3) (0) [? ], g (3) (0) = 1 − 6 N + 12 N 3 + 3 1 − 4 N Tr Γ 2 (Tr Γ) 2 + 2 Tr Γ 3 (Tr Γ) 3 .(S53) From Ref. [? ], the second derivative of R(t) evaluated at the initial time was found to bë R(0) = 8 N 2 (Tr Γ) 3 − 8 N (Tr Γ)(Tr Γ 2 ) + (Tr Γ 3 ) (S54) which can be expressed in terms of the correlation functions as This can then be used to estimate the peak time and intensity of the superradiant burst. In Ref. R(0) = (Tr Γ) 3 1 + 2 N − 2 N 2 − 3 2 + 2 N g (2) (0) + 1 2 g (3) (0) .( [? ], it was observed that for the fully excited emitters, the third-order correlation function g (3) (0) cannot be greater than 1 without superradiance, i.e., g (2) (0) > 1. The physical meaning is that the three-photon emission cannot be enhanced unless the two-photon emission is also enhanced. However, as we will see, this is not true in general and depends greatly on the dissipative model used. Without loss of generality, let Tr Γ = N . Calculating g (3) (0) subject to the constraint g 2 (0) = 1 yields the simple condition g (3) (0) > 1 =⇒ Tr Γ 3 > 6N.(S56) By the Cauchy-Schwarz inequality, we must have Tr Γ 3 ≤ (Tr Γ)(Tr Γ 2 ) = 2N 2 , which allows for the condition in Eq. (S56) to hold. Let us now test this for some of the previously introduced models: • 1D ring with exponentially-decaying interactions: At the superradiant transition γ = γ s = 1/ √ 3 and using the decay rates in Eq. (S35), we have Tr Γ 3 ≈ 11N/2 < 6N . Thus, g (3) (0) < 1, which agrees with our intuition. • 1D ring with NN and NNN interactions: At the superradiant transition γ 2 = (1 − 2γ 2 1 )/2 and γ 1 ∈ [0, 1/ √ 2], we have Tr Γ 3 = N (4 + 3 √ 2γ 2 1 1 − 2γ 2 1 ) < 6N . Thus, g (3) (0) < 1, which agrees with our intuition. • All-to-all interactions: At the superradiant transition γ = γ s = 1/ √ N − 1, we have Tr Γ 3 = N (N − 2 + 4 √ N − 1) √ N − 1 (S57) which can exceed 6N for N > 2(2 + √ 2) ≈ 6.8. Thus, for a sufficiently large number of emitters, we can enhance three-photon emission (g (3) (0) > 1) while suppressing two-photon emission (g (2) (0) < 1) (see Fig. S2). The range of γ for which this occurs decreases as 1/3N for large N , such that g (2) (0) and g (3) (0) both cross 1 at the same value of γ as N → ∞. The all-to-all interaction model, which is a Dicke model with local dissipation, provides a simple counterexample to the intuitive notion that in the absence of a superradiant burst, three-photon emission is suppressed. The pertinent question, however, is whether the simultaneous two-photon suppression and three-photon enhancement results in a 'delayed superradiance', that is, R(t) first decreases for a short time before rising above its initial value. Surprisingly, such a phenomenon is indeed possible, but we found that the resulting superradiant burst is extremely weak. (0) = N 2 γ 2 − N (1 + γ 2 ) (S58) andR (0) = N 3 γ 3 + N 2 γ 2 (3γ + 5) + N (1 + γ 2 (2γ + 5)). (S59) For a suitable choice of γ, one can satisfyṘ(0) < 0 andR(0) > 0 simultaneously. WhenṘ(0) = 0,R(0) > 0 for N > 2(5 + 2 √ 5) ≈ 18.9. However, havingṘ(0) < 0 andR(0) > 0 alone does not guarantee a delayed superradiance, since the total emission rate R(t) may not exceed the initial value R(0). From numerical simulations, we observe delayed superradiance for a narrow parameter regime, but the resulting superradiant burst is very weak, with the peak emission rate only very slightly above the initial rate (fractional increase on the order of 10 −6 ). We also remark that this effect is not present in the nearest and next-nearest interaction models, and therefore does not affect the validity of our claims in the main text. Generically, in the presence of local dissipation (where the physical origin of the local term is different from that of the interactions) one has to be mindful about how to properly account for the distinct decay channels. This can be done by defining a "directional" second order correlation function, as done in Ref. We can obtain an alternative physical interpretation of the correlation functions g (2) (0) and g (3) (0) by connecting them with quantum jumps. Let us define the normalized K-jump state by applying K collective jump operatorsĉ ν on the fully-excited initial state |e⟩ |Φ K (⃗ ν)⟩ = 1 N K j=1ĉ νj |e⟩ ,(S60) where ⃗ ν = (ν 1 , . . . , ν K ) and N is the normalization factor. For an arbitrary 1-jump state |Φ 1 (ν)⟩, using the expansion c ν = j v νj σ − j , we have the emission rate R ν = i,j γ ij ⟨Φ 1 (ν)|σ + i σ − j |Φ 1 (ν)⟩ = N − 1 + i,j i̸ =j γ ij v * νj v νi .(S61) Taking average of all the 1-jump states weighted by their respective decay rates givē R 1 = 1 N ν Γ ν R ν = N − 1 + 1 N (Tr(Γ 2 ) − N ) = N g (2) (0),(S62) where we have used γ jk = γ kj = ν Γ ν v * νj v νk from the spectral decomposition of Γ. Thus, we can interpret g (2) (0) as proportional to the average emission rate of the 1-jump states, and g (2) (0) < 1 implies that these states do not give rise to superradiance. Now, let us consider the normalized 2-jump states |Φ 2 (ν, µ)⟩ =ĉ µĉν |e⟩ 1 + δ µν − 2 j |v νj | 2 |v µj | 2 . (S63) Taking the weighted average of the rates of all 2-jump states, we havē R 2 = 1 N 2 ν,µ Γ ν Γ µ χ Γ χ ⟨e|ĉ † νĉ † µĉ † χĉχĉµĉν |e⟩ 1 + δ µν − 2 j |v νj | 2 |v µj | 2 .(S64) We now make the assumption that j |v νj | 2 |v µj | 2 vanishes as N → ∞. This is true for lattices with periodic boundary conditions in which v νj are the plane wave coefficients. As an example, we have v νj = exp (2πijν/N )/ √ N for the 1D ring which simply comes from the eigenvectors of a circulant matrix. Hence, as N → ∞, R 2 ≤ 1 N 2 ν,µ,χ Γ ν Γ µ Γ χ ⟨e|ĉ † νĉ † µĉ † χĉχĉµĉν |e⟩ = N g (3) (0).(S65) Thus, we can interpret N g (3) (0) as the upper bound for the average emission rate of the 2-jump states, and g (3) (0) < 1 implies that these states do not give rise to superradiance. V. UPPER BOUNDS ON THE EMISSION RATE The superradiance condition g (2) (0) > 1 is strictly speaking only valid for classifying superradiance at short times up to O(γ 0 t). 'Delayed superradiance' is thus technically possible, hence it is not true in general that R(t) > R(0) for some t =⇒ g (2) (0) > 1. Nonetheless, if we can upper bound R(t) and compare it with R(0) = N , we can obtain a regime in which the fully-excited initial state maximizes R which guarantees no superradiant burst at arbitrary times and also for arbitrary system Hamiltonian. Proposition 1 The emission rate R can be written as the average of some spin Hamiltonian H Γ , where the interaction strengths between the i th and j th spins are given by the decoherence matrix elements γ ij . This is straightforward to show: R = N ν=1 Γ ν N j,k=1 v * νj v νk ⟨σ + j σ − k ⟩ = N j,k=1 ⟨σ + j σ − k ⟩ N ν=1 Γ ν v * νj v ν k = N j,k=1 ⟨σ + j σ − k ⟩ N ν=1 Γ ν ⃗ v ν (⃗ v * ν ) T k,j = N j,k=1 γ kj ⟨σ + j σ − k ⟩ ≡ ⟨H Γ ⟩ ,(S66) where we have used the spectral decomposition of Γ in the last step. Hence, the problem of calculating R is equivalent to finding the average energy of a state under the Hamiltonian H Γ . We remark that H Γ introduced here is only used for calculation purposes, and should not be interpreted as a physical Hamiltonian of our system. With this, we now prove that for our NN interaction model in arbitrary dimensions, R is maximized by the fully excited state |e⟩ ⊗N . A. D-dimensional nearest-neighbor interactions Theorem 1 Let Γ describe a nearest-neighbor interaction model, with γ ij = δ ij + γδ ⟨ij⟩ , where γ ∈ [0, 1]. δ ⟨ij⟩ = 1 if the emitters indexed by i and j are nearest-neighbor on the D−dimensional regular lattice, and 0 otherwise. For γ ≤ 1/(2D), the emission rate R is maximized by the fully-excited state |e⟩ ⊗N with R = N . Proof. Let us construct the auxiliary Hamiltonian as H Γ = N j=1 σ + j σ − j + γ ⟨j,k⟩ (σ + j σ − k + σ + k σ − j ),(S67) which is the XY Hamiltonian with transverse magnetic field. For D = 1, this can be solved exactly by mapping to free fermions using the Jordan-Wigner transformation. However, such an approach is not necessary for the proof since we only require the highest-energy (dominant) eigenstate of H Γ . One can easily check that |e⟩ ⊗N is an eigenstate of H Γ with eigenenergy N . It remains to be shown that this is in fact the dominant eigenstate for γ ≤ 1/(2D). To this end, let us consider the matrix representation of H Γ in the standard tensor-product basis {|e⟩ , |g⟩} ⊗N . Note that H Γ conserves the total excitation j σ + j σ − j , and hence can be block-diagonalized into sub-blocks with different excitation numbers. In D-dimensions, each emitter can only at most interact with 2D nearest-neighbors. For the basis state with N − m excitations (0 ≤ m ≤ N ), the number of interactions is at most 2Dm, corresponding to the basis states where the m ground-state emitters are not nearest-neighbors with one another. Hence, the row sum of the matrix H Γ for that particular basis state is at most (N − m) + 2Dmγ, with the first term coming from the diagonal element. Using the Gershgorin circle theorem, we know that all the eigenvalues of H Γ must lie within the largest Gershgorin disc. More specifically, we can bound the largest eigenvalue as λ max ≤ max m,γ (N − m + 2Dmγ) ≤ max m N − m + 2Dm × 1 2D = N.(S68) Hence, |e⟩ ⊗N maximizes ⟨H Γ ⟩ and thus maximizes R. □ The condition γ ≤ 1/(2D) is within the physically valid regime. In fact, this is exactly the physically valid regime for either the N → ∞ limit or if the lattice obeys periodic boundary conditions. For open boundary conditions, the exact physically valid regime contains a correction factor of 1/ cos(π/(N 1/D + 1)) > 1 which makes it slightly larger than 1/(2D). This factor tends to 1 as ∼ π 2 /(2(N 1/D ) 2 ), hence the small discrepancy would not matter if the number of emitters per spatial dimension N 1/D is sufficiently large. Relating back to our original problem of superradiance, we can write down an obvious consequence of Theorem 1. Corollary 1 For a nearest-neighbor interaction model with γ ≤ 1/(2D), assuming a fully-excited initial state |ψ(0)⟩ = |e⟩ ⊗N , superradiance is impossible for any system Hamiltonian. In other words, R(t) ≤ R(0) ∀t > 0. For the nearest-neighbor model, the upper bound on R(t) is tight which allows us to rigorously prove the impossibility of superradiance at all times. However, this is not true in general for other interaction models ∀γ ∈ [0, γ p ]. Nonetheless, we can still use the bound to find a regime where superradiance is definitely not possible. For simplicity, we assume periodic boundary conditions for the rest of the section. B. 1D exponentially-decaying interactions Consider the 1D exponential model with N emitters. For simplicity, we take N to be odd and assume periodic boundary conditions. For the basis state with N − m excitations, the maximum number of couplings between the excited and ground states in the 1D ring is 2 × m ′ × (N − 1)/2 = m ′ (N − 1) where m ′ = min{m, N − m} ranges from 0 to (N − 1)/2. In general, all the couplings will have a different contribution γ |i−j| to the H Γ matrix. Thus, for a fixed γ, λ max ≤ max m ′ N − m ′ + 2m ′ γ + γ 2 + . . . + γ (N −1)/2 (S69) For γ = 1, λ max ≤ N + (N − 2)(N − 1)/2 ∼ N 2 /2 recovers the quadratic scaling of the Dicke model. For γ < 1, we can bound λ max as λ max ≤ max m ′ N − m ′ + 2m ′ γ 1 − γ = N, γ ≤ 1/3 N + N −1 2 3γ−1 1−γ , 1/3 < γ < 1(S70) from which we see that |e⟩ ⊗N maximizes R for γ ≤ 1/3 < 1/ √ 3 = γ s . For γ < 1, the bound for the peak emission rate has a linear scaling with N . dissipative interactions. The periodic boundary condition assumption is also not in the large-N limit, which is usually the case of interest. The assumption on the interaction Hamiltonian, including the reality assumption, is required to ensure that the state remains translationally-invariant which is a key part of the proof. The most extreme example of such a Hamiltonian is the coherent all-to-all interactions with a common arbitrary interaction strength J. Theorem 2 implies that even such a strong Hamiltonian would not significantly affect the emission rate, which we have verified numerically by simulating the master equation exactly for the 1D exponential model with N = 13 emitters. The translational symmetry used in the proof is also reasonable since one expects superradiance to be stronger for symmetric states (with the strongest superradiance achieved for the permutation-symmetric Dicke state). The strength of this argument thus boils down to the validity of the second-order mean field theory. It is well-known that the second-order mean field approach breaks down when calculating the strength of the two-body correlations particularly at long times t ≫ γ −1 0 , where γ 0 is the spontaneous emission rate of each emitter. Previous numerical simulations have also shown that the mean-field approach (at least second-order and above) is rather accurate in terms of the emission rate (see Fig. 3 of [? ] and Fig. 2(c) of [? ]). We point out two reasons why we believe that the inadequacies of mean-field theory do not greatly affect the validity of Theorem 2: 1. At long times t ≫ γ −1 0 , the total excitation of the emitters is close to zero. Note that the total excitation must decrease monotonically since the Hamiltonian conserves total excitation, and we do not introduce any pumping in the model. Hence, this makes the emission rate small. Another physical interpretation is that the subradiant states are populated at large times which is also why the mean field assumption breaks down. 2. Our argument is based on the translation symmetry of the state which is of course still true in the exact dynamics. Also, our second-order approach already accounts for short-range correlations which are even more so appropriate for the short-range interaction models we use in the manuscript (NNN, exponential), especially in the large-N limit. A. No superradiant burst for nearest-neighbor interactions, translation-invariant initial state We now argue that in the second-order mean-field theory, superradiant burst for NN interactions cannot occur at any time t ≥ 0, using a translation-invariant initial state. From Theorem 2, it suffices to only consider dissipative processes. For the D-dimensional NN interaction model, the decay rates are Γ k = 1 + γ D i=1 cos k i ,(S75) where k = (k 1 , . . . , k D ) T , and k i ∈ [−π, π]. The corresponding jump operators are c k = 1 √ N x e −ik·x σ − x (S76) where x ∈ Z D . We consider the thermodynamic limit N → ∞. For a translation-invariant state |ψ⟩, R = − k,k ′ Γ k Γ k ′ ⟨c † k [c k , c † k ′ ]c k ′ ⟩ = 1 N 2 k,k ′ x1,x2,x3 (1 + γ i cos k i )(1 + γ i cos k ′ i ) exp(ik · (x 1 − x 2 )) exp(ik ′ · (x 2 − x 3 )) ⟨σ † x1 σ z x2 σ − x3 ⟩ ≈ 1 (2π) 2D x1,x2,x3 ⟨σ † x1 σ z x2 σ − x3 ⟩ˆdk(1 + γ i cos k i ) exp(ik · (x 1 − x 2 ))ˆdk ′ (1 + γ i cos k ′ i ) exp(ik ′ · (x 2 − x 3 )) = x1,x2,x3 ⟨σ † x1 σ z x2 σ − x3 ⟩ δ x1x2 + γ 2 δ ⟨x1x2⟩ δ x2x3 + γ 2 δ ⟨x2x3⟩(S77) where the Kronecker delta δ ⟨x1x2⟩ is 1 if x 1 and x 2 are NN sites, and vanishes otherwise. Since the state remains translation-invariant under time evolution, we can denote the population of each site as p, NN and NNN correlations of the form ⟨σ + x σ − x ′ ⟩ as c 1 and c 2 respectively. Since we are only interested in the possibility of superradiance, and FIG. 1 . 1Dynamics of the photon emission rate R(t) for emitter arrays with only nearest-neighbor interactions of strength γ, normalized by the individual emitter decay rate γ0. FIG. 3 . 3Differential emission rate ∆R = R(t)/R(0) − 1 against time (in units of emitter lifetime), for N = 9 emitters. ∆R > 0 indicates superradiance. (a) Dynamical behavior of ∆R for the Dicke model (red), Next-nearest neighbor 1D ring (NNN, orange), Nearest-neighbor 1D ring (NN, blue) and Nearest-neighbor 2D square (NN, green) (see labels in (b)). The coupling parameters are chosen to maximize g (2) (0). (b) Short-time behavior obtained by zooming into the grey region of (a). Only the Dicke and the next-nearest neighbor models exhibit superradiance. The curve for the Dicke model is scaled down by a factor of 10 for visualization purposes. γ 1 , γ 2 ∈ 12[0, 1] (blue regions in FIG. 4 . 4the position vector of the i th lattice site. Physically, this model describes exponentially-decaying interactions Critical coupling γs for exponentially-decaying interactions in a 1D chain, a 2D square array and a 3D cubic array with N emitters. Superradiance occurs for γ > γs. For all dimension D, γs becomes independent of N for large N .(Inset) Log-log plot of γs against D for N ≈ 10 6 . γs decreases as D increases with a power-law scaling γs ∼ D −0.793 . Third-order correlation function, g (3) (0) 10 A. Delayed superradiance in Dicke model with local dissipation 12 B. Physical interpretation of g (2) (0) and g (3) (0) 12 V. Upper bounds on the emission rate 13 A. D-dimensional nearest-neighbor interactions 13 B. 1D exponentially-decaying interactions 14 C. 1D power-law interactions 15 VI. Effect of Hamiltonian on emission rate 15 A. No superradiant burst for nearest-neighbor interactions, translation-invariant initial state 16 arXiv:2211.00668v2 [quant-ph] (A) = −λ min (A) = λ max (A) = σ max (A) = ||A|| 2 , where λ max (A) and σ max (A) are the maximum eigenvalue and singular value of A respectively. For N even, we have rank(A) = N while for N odd we have rank(A) = N − 1 (due to 1 zero eigenvalue). Using the inequality ||A|| F ≤ rank(A)||A|| 2 , we obtain + . Now, we consider the nearest-neighbor dissipative interactions γ ⟨m,n⟩ = γ ∈ [0, 1/(2D)]. This simplifies the expres-Dγ(8 − (15 + 2D)γ)) cos θ + Dγ(4 cos 2θ + (2D − 1)γ cos 3θ)].(S27) FIG . S2. Second-and third-order correlation functions for the all-to-all interaction model with N = 10 emitters. By increasing the collective coupling u, g (3) (0) > 1 is achieved (at u = u3) before g (2) (0) > 1 (at u = γ2). The inset shows the gap γ2 − u3 against N , with the asymptotic behavior 1/3N as illustrated by the gray dashed line. S3. Fractional change in total emission rate with respect to the initial rate R(0), for N = 20 emitters and couplingparameter u = (1 − ϵ)/ √ N − 1, ϵ = 10 −4 .The peak emission rate has a very small fractional increase of approximately 10 −6 , and the superradiant burst is thus negligible.A. Delayed superradiance in Dicke model with local dissipationFor the Dicke model with local dissipation,Ṙ [? ? ]. B. Physical interpretation of g (2) (0) and g (3) (0) written purely in terms of the eigenvalues of the decoherence matrix Γ. Diagonalization can be avoided by taking traces of Γ and Γ 2 . Further reduction in complexity can be achieved by harnessing the Hermiticity of Γ which yields Tr (Γ 2 ) = ∥Γ∥ 2 F [? ]. For ordered arrays, the g (2) (0) function can be efficiently computed in O(N ) steps due to the periodicity of the array [? ? ]. Interestingly, the superradiant transition point is approximately independent of N for sufficient large N , which is fully consistent with the numerical simulations in Refs. [? ? ? ? ] which used a more realistic interaction model. Thus, our result provides a good qualitative explanation for the previous observations on superradiant burst in 1D arrays despite its simplicity.κd < 1 2 ln 3 ≈ 0.549. (S32) Supplementary Material: Dicke superradiance requires interactions beyondThis is similar to the calculations for the exponential model, but with the replacement γ |i−j| → γ/|i − j|, i ̸ = j. Performing the same analysis, we havewhere ζ ≈ 0.577 is the Euler-Mascheroni constant.VI. EFFECT OF HAMILTONIAN ON EMISSION RATEFor the fully-excited initial state, it can be shown that R(t) is independent of the Hamiltonian up to quadratic order in time[? ]. The situation is more complicated for other initial states, but this is beyond the scope of our work as we have noted throughout the manuscript. This is intuitive since the coherent spin-spin interactions are negligible at early times when the emitters are fully excited.An exact argument that the effect of Hamiltonian is small (at least insufficient to induce a superradiant burst) beyond a time of O t 2 requires one to solve the master equation analytically for arbitrary times, which is of course a difficult task. Nonetheless, we can provide a reasonable justification of this claim by using a second-order mean-field approach, also known as a second-order cumulant expansion [? ? ]. Consider the Hamiltonianwe can write the dynamics for the emission rate as (assuming γ ij = γ ji )Using this, we can now make the following claim:In the second-order mean-field theory, if H and Γ are translation-invariant with periodic boundary conditions, the emission rate R(t) is independent of H for any translation-invariant initial state and t ≥ 0.Proof. Notice from Eq. (S72) that the real part of J nl only couples to the imaginary part of the spin-spin correlation function ⟨σ + m σ − l ⟩. For any translation-invariant state, we must have ⟨σ + m σ − l ⟩ = ⟨σ + l σ m ⟩ since the correlations only depend on the geometrical separation between the spins l and m. Hence, the correlations must be real, and the effect of H on R vanishes. □ Another simple consequence of a real Hamiltonian is that it does not affect the rate of the K−jump statesup to a normalization factor.ĉ 1 . . .ĉ K are arbitrary jump operators obtained by diagonalizing Γ. This result is easy to prove. From the Heisenberg equation,Since Γ is real and symmetric, the expansion coefficients v νj inĉ ν can be chosen to be real. Hence |ψ K ⟩ is real, and any real Hamiltonian H will thus contributeṘ = 0. We now comment on the assumptions used in Theorem 2. Firstly, the assumption that Γ is translation-invariant already applies to the models used in the manuscript, where we assume that all the emitters experience the same c 1 , c 2 are real (by translation invariance), we only need to consider the case of c 1 > 0, c 2 > 0. Furthermore, since the interactions are NN, it is also reasonable to assume c 1 > c 2 . Hence, we can simplify the above expression to givėMaximizing the dissipative interactions by setting γ = 1/(2D), we havėwhich shows that SR is not possible for any translation-invariant state |ψ⟩. This also holds for the evolved (mixed) state at arbitrary time since it can be written as a convex mixture of translation-invariant pure states. Hence, we conclude thatṘ(t) ≤ 0 ∀t. 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{'abstract': 'Photon-mediated interactions within an excited ensemble of emitters can result in Dicke superradiance, where the emission rate is greatly enhanced, manifesting as a high-intensity burst at short times. The superradiant burst is most commonly observed in systems with long-range interactions between the emitters, although the minimal interaction range remains unknown. Here, we put forward a new theoretical method to bound the maximum emission rate by upper-bounding the spectral radius of an auxiliary Hamiltonian. We harness this tool to prove that for an arbitrary ordered array with only nearest-neighbor interactions in all dimensions, a superradiant burst is not physically observable. We show that Dicke superradiance requires minimally the inclusion of nextnearest-neighbor interactions. For exponentially-decaying interactions, the critical coupling is found to be asymptotically independent of the number of emitters in all dimensions, thereby defining the threshold interaction range where the collective enhancement balances out the decoherence effects. Our findings provide key physical insights to the understanding of collective decay in many-body quantum systems, and the designing of superradiant emission in physical systems for applications such as energy harvesting and quantum sensing.Introduction.-Collective spontaneous emission of N initially-inverted atoms with identical all-to-all interactions mediated by the electromagnetic vacuum results in a burst of light with intensity scaling as N 2 [1-3]. This phenomenon is commonly referred to as "Dicke superradiance" or "superradiant burst". Over the past decades, this many-body phenomenon has attracted a lot of interest in both theoretical and experimental studies[25,26]using a multitude of physical platforms such as trapped ions [27], molecular aggregates [28-31], solidstate emitters [32-36], cold atoms and molecules [37-40], and superconducting qubits [41-43], with wide-ranging applications including the generation of multi-photon states with improved metrological properties [18, 44-47], energy harvesting [48-50], ultrabright LEDs [51] and quantum sensing [52, 53].The atoms in Dicke\'s original model were assumed to be confined within a spatial extent smaller than the emission wavelength λ. Consequently, the atoms become indistinguishable with respect to the absorption or emission of photons, such that their quantum state |j = N/2, m⟩ (with −N/2 ≤ m ≤ N/2) is permutation-invariant. This permutation symmetry greatly reduces the complexity of the problem, as it constrains the dynamics to N + 1 states, instead of exploring the full Hilbert space (which scales as 2 N ). Recently, there has been substantial research progress with extended systems where atoms are distributed over a region larger than λ, thus breaking this symmetry. Of particular interest are ordered atomic arrays[54,55], in which the superradiant properties can be greatly affected by the geometry and dimensionality of the lattice[16,19,20,22,24,56]. The interactions between the emitters are typically modelled by long-range dipole-dipole interactions mediated via the electromagnetic vacuum[57,58].A long-standing fundamental question is the minimal interaction range required for the occurrence of a superradiant burst. Intuitively, superradiance can be thought of as a competition between (transient) phase synchronization, which leads to the buildup of atomic correlations, and decoherence[59]. Both effects stem from the same dissipative interactions[8,22]. Since synchronization of nonlinear classical phase oscillators has been demonstrated with nearest-neighbor (NN) coupling [60], one may expect the atomic phases to synchronize for sufficiently strong NN interactions resulting in a superradiant burst[59]. Moreover, for a fixed interaction range, higher dimensionality was reported to result in stronger superradiance due to long-range order[19,24]. On the flip side, it could also be argued that for short-range interactions, the buildup of correlations is not strong enough to overcome decoherence, thereby preventing superradiance.In this Letter, we prove that superradiant burst is impossible in an arbitrary D-dimensional array with only nearest-neighbor interactions, for arbitrary times and initial states. That is, we show that, in all cases, the emission rate is upper bounded by that of independent emitters, resulting in no enhancement from collective dynamics. Including next-nearest-neighbor interactions, we show that a superradiant burst can be physically observed for certain values of the interaction strengths, arXiv:2211.00668v2 [quant-ph] 25 May 2023 2 Photon emission rate Time Superradiant burst Monotonic decay / 0 0 1 Gap p / 0 s / 0 FIG. 1. Dynamics of the photon emission rate R(t) for emitter arrays with only nearest-neighbor interactions of strength γ, normalized by the individual emitter decay rate γ0. For γ/γ0 < γs,Ṙ(0) < 0 and the photon emission rate decays monotonically without a superradiant burst (blue). Superradiance occurs for γ/γ0 > γs (red). The physically-valid regime is defined by 0 < γ/γ0 ≤ γp. For nearest-neighbor interactions, γp < γs (with a finite gap between γp and γs) for any arbitrary emitter array in all dimensions, rendering Dicke superradiance physically impossible.thereby defining a minimal interaction range for superradiance. Another question is the threshold interaction range, which we define to be such that the critical coupling required for a burst becomes independent of the number of emitters, for any D. We show that exponentially-decaying interactions lie on the threshold interaction range for which the synchronization of the dipoles arising from the emission balances the decoherence effects.Model.-The dynamics of an undriven ensemble of N emitters can be described by the Lindblad master equation (setting ℏ = 1)with J ij = J * ji and γ ij = γ * ji to ensure Hermiticity. The raising and lowering operators for the j th emitter are denoted as σ + j ≡ |e i ⟩ ⟨g i | and σ − j ≡ |g i ⟩ ⟨e i | which describe transitions between the ground state |g i ⟩ and excited state |e i ⟩. The first term contains the coherent Hamiltonian interactions between the emitters, while the second term captures processes such as collective and local dissipation of the emitters via the superoperatorWe assume J ij and γ ij to be time-independent, such that the superoperator L generates a dynamical semigroup describing the dynamics of a Markovian open quantum system.For a physically valid evolution (i.e., a completely positive and trace-preserving map), the matrix Γ containing the elements γ ij (which we will refer to as the decoherence matrix ) must be positive semi-definite[61][62][63]. The decoherence matrix can be diagonalized to yield N decay rates Γ ν ≥ 0, with ν ∈ {1, . . . , N } and the corresponding collective jump operatorsĉ ν . The total photon emission rate of the emitters, integrated over all emission directions, is defined for any state ρ asFor independent emitters with γ ij = γ 0 δ ij , the total emission rate has a maximum of N γ 0 (saturated by the fullyexcited state), and R(t) ≡ R ρ(t) decays exponentially. However, interactions between the emitters can cause R(t) to increase beyond its initial value. This speedup in emission is commonly referred to as the superradiant burst, first discovered by Dicke [1] (seeFig. 1). Throughout this work, we refer to superradiant burst as the increase in the total emission rate beyond N γ 0 , but the peak intensity need not scale as N 2 . In general, characterizing the burst at arbitrary times can be difficult, hence one typically useṡevaluated at the fully-excited initial state ρ(0), witḣ R(0) ≡Ṙ ρ(0) > 0 a sufficient condition for a superradiant burst. While we consider the burst at t = 0, we will provide physical justification on why this is sufficient.Here, we put forward a new (and complementary) criterion to preclude any possibility of a burst: by a simple change of basis, one can write Eq. (2) as the expectation value of an auxiliary spin Hamiltonianwith R ρ = tr(H Γ ρ). The maximum photon emission rate can thus be calculated by bounding the spectral radius of the auxiliary spin Hamiltonian. If the upper bound is equal or smaller than N γ 0 , no burst can occur for all times and arbitrary initial states. While finding the largest eigenvalue of H Γ may be non-trivial, this criterion allows one to definitively prove the absence of a burst for arbitrary times, thus going beyond the con-ditionṘ(0) ≡Ṙ ρ(0) > 0. Furthermore, this approach opens up the possibility of finding theoretical limits for the emission rate arising from superradiant dynamics, as we show below and in the Supplementary Information[64].No superradiance for nearest-neighbor coupling.-Let us consider a hypercube array of N emitters with arbitrary dimension D (N = n D ). For the case of NN interactions, γ ii ≡ γ 0 = 1 and γ ij = γ if emitters i and 3 j are nearest-neighbor (γ ij = 0 otherwise). The coupling γ ∈ [0, 1] is required for the matrix Γ to be positive semidefinite. Without loss of generality, we have assumed γ ij to be real and positive. We prove that for this model, superradiant burst cannot occur for any t > 0, for any arbitrary initial state and for any Hamiltonian coupling J ij . To determine the physically valid regime, we impose the condition that Γ is positive semidefinite. Notice that the decoherence matrix can be expressed as Γ = I N +γA, where I N is the N × N identity matrix, and A is the adjacency matrix of a n × n grid graph. Using the fact that the grid graph is the Cartesian product of D path graphs P n □· · ·□P n , it can be shown that the smallest eigenvalue of Γ is [64]which gives the physically valid regime as γ ≤ γ p ,This rate reduces to γ p = 1/(2D) in the N → ∞ limit, or when imposing periodic boundary conditions for a finite N . This can be regarded as coming from the coordination number for each emitter, which approaches 2D in the infinite-array limit. We now state our main result.Theorem 1 Let Γ be the decoherence matrix for a nearest-neighbor interaction model, with γ ij = δ ij +γδ ⟨ij⟩ , where γ ∈ [0, 1], and δ ⟨ij⟩ = 1 if the emitters indexed by i and j are nearest-neighbor on the D−dimensional regular lattice, and 0 otherwise. For γ ≤ (2D) −1 , the emission rate R ρ is maximized by the fully-excited state |e⟩ ⊗N withWe provide a sketch of the proof here, while the details can be found in theSupplementary Information [64]. By expressing H Γ in the product-state basis and using the Gershgorin circle theorem [65], we can upper bound max t R(t) ≤ N in the physically valid regime γ < 1/(2D). This is saturated by N independent emitters in the fully-excited state, with eigenvalue N . Hence, Theorem 1 implies that superradiant burst is impossible at all times. To gain a deeper physical understanding, we evaluate the superradiant regime γ > γ s for the fullyexcited initial state, characterized by the transition aṫ R(0) = 0, for which [64]For all 2 < N 1/D < ∞, it can be shown that γ p < γ 2 s and therefore γ p < γ s . Hence, the superradiant regime does not overlap with the physically valid regime. Generalization to the hyper-rectangle configuration where the number of sites along each dimension can be different is Observable superradiance I II III FIG. 2. Region of superradiant burst in the γ2−γ1 plane. The physically valid (superradiant) regime is contained within the blue (red) boundary lines, with the conditions stated in the main text. Blue shaded region: Physically valid, but not superradiant. Regions I, II and III are defined in the main text. Red shaded region: Physically valid with superradiant burst. Grey shaded region: unphysical regime. The red shaded region requires a minimum of γ2 ≈ 0.185. All shaded regions here are obtained from numerical calculations for N = 100, which agree very well with the analytical results obtained in the infinite-array limit.straightforward, and the same conclusion is obtained[64]. While our analysis of the NN model is valid for any initial state, we consider a fully-inverted initial state for the next two sections: the analysis of next-nearest neighbor and exponentially decaying interactions. Next-nearest neighbor coupling.-Including the NNN interactions, we now show that a superradiant burst is indeed possible. For simplicity, let us consider a 1D ring of N emitters with periodic boundary conditions. In this configuration, Γ turns out to be a circulant matrix with the first column given by (1, γ 1 , γ 2 , 0, . . . , 0, γ 2 , γ 1 ) T with 0 ≤ {γ 1 , γ 2 } ≤ 1. The subsequent columns are simply cyclic permutations of the first column. Diagonalizing Γ exactly yields the eigenvaluesfor ν = 0, . . . , N − 1. In the infinite-array limit N → ∞, the eigenvalues form a continuous band in momentum space Γ(k) = 1 + 2γ 1 cos(k) + 2γ 2 cos(2k), with the dimensionless wavevector 0 ≤ k < 2π. At the turning points where ∂ k Γ = 0, we have: Γ(0) = 1 + 2(γ 1 + γ 2 ) which is always positive, Γ(π) = 1 − 2(γ 1 − γ 2 ) and Γ(k * ) = 1 − (γ 2 1 + 8γ 2 2 )/4γ 2 where cos k * = −γ 1 /4γ 2 . Demanding that Γ(k) > 0 thus produces the physically valid regimes: (I) γ 1 − γ 2 ≤ 1 2 , γ 1 > 4γ 2 and (II) γ 2 1 + 8γ 2 2 ≤ 4γ 2 , γ 1 ≤ 4γ 2 , together with the bounds 4 NNN NN (b) (a) NN Dicke Differential emission rate Δ 0 t 0 t FIG. 3. Differential emission rate ∆R = R(t)/R(0) − 1 against time (in units of emitter lifetime), for N = 9 emitters. ∆R > 0 indicates superradiance. (a) Dynamical behavior of ∆R for the Dicke model (red), Next-nearest neighbor 1D ring (NNN, orange), Nearest-neighbor 1D ring (NN, blue) and Nearest-neighbor 2D square (NN, green) (see labels in(b)). The coupling parameters are chosen to maximize g (2) (0). (b) Short-time behavior obtained by zooming into the grey region of (a). Only the Dicke and the next-nearest neighbor models exhibit superradiance. The curve for the Dicke model is scaled down by a factor of 10 for visualization purposes.γ 1 , γ 2 ∈ [0, 1] (blue regions inFig. 2). The superradiant condition can be obtain fromṘ(0) = 0 as (III) γ 2 1 + γ 2 2 > 1/2. There is an overlap region with the physically valid regime, as shown by the red shaded region inFig. 2. For certain values of γ 1 , γ 2 , superradiant burst can occur. Moreover, the fact that this overlap region requires γ 2 > (4 − √ 2)/14 ≈ 0.185 is consistent with our previous conclusion of no superradiance using only NN coupling (i.e., γ 2 = 0). Superradiance is also forbidden by having only NNN coupling (i.e., γ 1 = 0). Results from numerical simulations of N = 9 emitters are presented inFig. 3, which show that the NNN model has a small superradiant burst compared to the Dicke model, and no superradiance for NN models. We remark that this superradiance arises from destructive interference leading to dark decay channels with suppressed decay rates Γ ν ≈ 0 while the dominant decay channel has a rate that does not scale with N . This mechanism is generally true for all models with a sharp interaction cutoff beyond a certain range.Threshold interaction range for a superradiant burst.-In many previous works[16,19,20,22,24], Γ is obtained from a realistic modelling of the atomic interactions mediated by electromagnetic vacuum using the appropriate Green\'s function. Our goal here, however, is to shed light on the essential physics of superradiance by considering analytically tractable models that still exhibit interesting behaviors. Consider an interaction which decays exponentially with the separation r ij between the emitters: γ ij ∝ e −κrij , where κ controls the decay of the interaction strength with emitter separation. We set the diagonal elements of Γ as 1, and define γ ≡ e −κd with d the emitter NN separation such that γ ij = γ |⃗ xi−⃗ xj | , where ⃗ x i ∈ Z D is the position vector of the i th lattice site. Physically, this model describes exponentially-decaying interactions 1D 2D 3D 2 3 4 5 6 0.3 0.2 ! " ∼ $ %&.()* D Critical coupling, " Emitter number, N s ⇠ D 0.793 FIG. 4. Critical coupling γs for exponentially-decaying interactions in a 1D chain, a 2D square array and a 3D cubic array with N emitters. Superradiance occurs for γ > γs. For all dimension D, γs becomes independent of N for large N . (Inset) Log-log plot of γs against D for N ≈ 10 6 . γs decreases as D increases with a power-law scaling γs ∼ D −0.793 .between the atoms. For a sufficiently large N in D dimensions such that γ N ≪ 1,Ṙ(0) is approximately given by the asymptotic forṁfor some constant C [64]. Interestingly, this suggests that the critical coupling parameter γ s for superradiance is independent of N as N → ∞ for all dimension, agreeing with the numerical results shown inFig. 4. This is in stark contrast with previous results (primarily using long-range power-law interactions such as γ ij ∝ 1/r ij ), which predict that the critical emitter separation increases with N in 2D and 3D arrays[19,24].Figure 45 also shows that for large N , γ s ∼ D −0.793 exhibits a power-law scaling with the spatial dimension. This is intuitive as the average coupling per emitter increases with D which in turn lowers the critical coupling required for superradiance[24]. The N -independence of γ s for our short-range exponential model can be physically interpreted as the threshold interaction range where the synchronization effects due to collective interactions scales similarly with N as the local decoherence, such that adding more emitters do not affect the onset of the superradiant regime. For even shorter-range interactions such as the NN model, the local decoherence dominates which prevents superradiance. Longer-range models such as power-law interactions favor synchronization and thus enhance superradiance as N increases.Scaling of the peak emission rate with number of emitters-Eq. (4) shows that the problem of calculating the emission rate is equivalent to finding the average energy of a state under the Hamiltonian H Γ . This enables us to find upper bounds on the scaling of the peak emission rate with N , for arbitrary geometries and types of interactions. As we have shown before in Theorem 1, the maximum emission rate for arbitrary NN models is N γ 0 . For 1D arrays with an exponentially-decaying interaction, the upper bound on the emission rate is found to scale as O(N ) for γ < 1 [64]. This bound increases to O(N log N ) for 1D arrays with a power-law interaction of the form 1/r[64]. This latter scaling is consistent with the numerical results obtained in the literature which, in contrast to our bound, have only been obtained for relatively small systems and under certain approximations[20,22,24]. While finding exact bounds may be exponentially hard, one could in principle upper-bound other models, as well as tighten the currently-obtained bounds.Discussion.-In this Letter, we addressed the fundamental problem of the minimal interaction range required for superradiance. Crucially, we proved that nearestneighbor interactions cannot induce emitter correlations faster that the decoherence, resulting in the impossibility of superradiance. As shown, the minimal interaction range is therefore next-nearest neighbor, and longerrange interactions generally lead to stronger superradiance. We also found that the short-range exponential interaction marks the threshold interaction range in all dimensions where the emitter correlations and local decoherence scale similarly with the number of emitters such that the critical coupling required for superradiant burst becomes independent of the number of emitters, in stark contrast with previous conclusions using longerrange power-law interactions. We stress that, apart from the nearest-neighbor model, our classification of a superradiant burst is strictly speaking only valid at short times up to O((γ 0 t) 2 ) (ifR(0) < 0 which is true for the models considered here [64]), where the dynamics of the fullyexcited emitters do not depend on the Hamiltonian. This can be physically justified for later times using secondorder mean field theory[64].The techniques used in this work have broader applications in determining the theoretical bounds for the emission rate of different models, thereby exposing the ultimate limitations of superradiance beyond the NN model. Beyond providing fundamental insights to the physics of superradiance, our results can also motivate the design of atomic lattices in engineered baths such as nanophotonic crystals with engineered interactions or superconducting resonator arrays for qubits. Moreover, hypercube geometries should be within reach of state-of-the-art quantum simulators, given the recent advances in generating arbitrary networks in cavity [66] and circuit [67] quantum electrodynamics platforms.The authors are grateful to', 'arxivid': '2211.00668', 'author': ['Wai-Keong Mok \nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n', 'Ana Asenjo-Garcia \nDepartment of Physics\nColumbia University\n10027New YorkNew YorkUSA\n\nDepartment of Physics\nColumbia University\n10027New YorkNew YorkUSA\n', 'SumTze Chien \nDivision of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n637371Singapore\n\nDivision of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n637371Singapore\n', 'Leong-Chuan Kwek \nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n\nMajuLab\nCNRS-UNS-NUS-NTU International Joint Research Unit\nSingapore\n\nUMI 3654\nSingapore\n\nNational Institute of Education\nNanyang Technological University\n637616SingaporeSingapore\n\nQuantum Science and Engineering Centre (QSec)\nNanyang Technological University\nSuperradianceSingapore\n\nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n\nMajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit\n3654Singapore UMI, Singapore\n\nNational Institute of Education\nNanyang Technological University\n637616SingaporeSingapore\n\nQuantum Science and Engineering Centre (QSec)\nNanyang Technological University\nSingapore CONTENTS\n', 'Ben Grossmann ', 'Mccoy Lim ', 'Chris Chen ', 'Jasen Zion ', 'Kishor Bharti ', 'Davit Aghamalyan ', 'Lewis Ruks ', 'Thi Ha Kyaw ', 'Tobias Haug ', 'Steven Touzard ', 'Klaus Mølmer ', 'Stuart Masson '], 'authoraffiliation': ['Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore', 'California Institute of Technology\n91125PasadenaCAUSA', 'Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore', 'California Institute of Technology\n91125PasadenaCAUSA', 'Department of Physics\nColumbia University\n10027New YorkNew YorkUSA', 'Department of Physics\nColumbia University\n10027New YorkNew YorkUSA', 'Division of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n637371Singapore', 'Division of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n637371Singapore', 'Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore', 'MajuLab\nCNRS-UNS-NUS-NTU International Joint Research Unit\nSingapore', 'UMI 3654\nSingapore', 'National Institute of Education\nNanyang Technological University\n637616SingaporeSingapore', 'Quantum Science and Engineering Centre (QSec)\nNanyang Technological University\nSuperradianceSingapore', 'Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore', 'MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit\n3654Singapore UMI, Singapore', 'National Institute of Education\nNanyang Technological University\n637616SingaporeSingapore', 'Quantum Science and Engineering Centre (QSec)\nNanyang Technological University\nSingapore CONTENTS'], 'corpusid': 253255485, 'doi': '10.1103/physrevlett.130.213605', 'github_urls': [], 'n_tokens_mistral': 36395, 'n_tokens_neox': 31970, 'n_words': 18942, 'pdfsha': '05c8a15ee8553a528cae3a34546c5b11adc87a2d', 'pdfurls': ['https://export.arxiv.org/pdf/2211.00668v2.pdf'], 'title': ['Dicke superradiance requires interactions beyond nearest-neighbors', 'Dicke superradiance requires interactions beyond nearest-neighbors'], 'venue': []}

arxiv

Students Parrot Their Teachers: Membership Inference on Model Distillation Matthew Jagielski Google Research Milad Nasr Google Research Katherine Lee Google Research Christopher Choquette-Choo Google Research Nicholas Carlini Google Research Students Parrot Their Teachers: Membership Inference on Model Distillation Model distillation is frequently proposed as a technique to reduce the privacy leakage of machine learning. These empirical privacy defenses rely on the intuition that distilled "student" models protect the privacy of training data, as they only interact with this data indirectly through a "teacher" model. In this work, we design membership inference attacks to systematically study the privacy provided by knowledge distillation to both the teacher and student training sets. Our new attacks show that distillation alone provides only limited privacy across a number of domains. We explain the success of our attacks on distillation by showing that membership inference attacks on a private dataset can succeed even if the target model is never queried on any actual training points, but only on inputs whose predictions are highly influenced by training data. Finally, we show that our attacks are strongest when student and teacher sets are similar, or when the attacker can poison the teacher set. Introduction Model distillation (Hinton et al., 2015) is a common framework for knowledge transfer, where knowledge learned by a "teacher model" is transferred to a "student model" via the teacher's predictions. Distillation is helpful because the teacher's predictions are a more useful guide for the student model than hard labels; this phenomenon has been explained by the teacher's predictions containing some useful "dark knowledge". Variants of model distillation have been proposed for, e.g., model compression (Hinton et al., 2015;Ba & Caruana, 2014;Polino et al., 2018;Kim et al., 2018;Sun et al., 2019) or training more accurate models (Zagoruyko & Komodakis, 2016;Xie et al., 2020). Within the privacy-preserving machine learning community, distillation has been adapted to protect the privacy of a training dataset (Papernot et al., 2016;Tang et al., 2022;Shejwalkar & Houmansadr, 2021;Mazzone et al., 2022). Many of these approaches rely on the intuition that distilling the teacher model serves as a privacy barrier that protects the teacher's training data. Informally, restricting the student to learn only from the teacher's predictions is a form of data minimization, which should result in less private information being fed into, and memorized by, the student. This privacy barrier around the teacher also allows the teacher model to be trained with strong, non-private, training approaches, improving both the teacher model's and student model's accuracy. Because model distillation does not provide a rigorous privacy guarantee (such as those offered by differential privacy (Dwork et al., 2006)), in our work we evaluate the empirical privacy provided by these Our membership inference attack uses only the student model's predictions on the ten images on the right to reach 73% accuracy at predicting membership of the target image. Interestingly, while the target example is a bird, only two of the most informative student queries are birds. schemes. We show that distillation is vulnerable to membership inference attacks-a well-studied class of privacy attacks on machine learning (Shokri et al., 2017;Yeom et al., 2018). We adapt the state-of-the-art Likelihood Ratio Attack (LiRA) (Carlini et al., 2022a) to the distillation setting, and find that this attack works surprisingly well at inferring membership of the teacher's training data. For some training examples, model distillation fails to appropriately protect against membership inference. To explain this finding, we show for the first time that the membership presence of some examples can be inferred based on the model's predictions on other, seemingly unrelated examples. This observation provides new insights into how membership information is transmitted from the teacher to the student. Figure 1 gives an example of such cross-example leakage. A teacher model trained on the red parrot in Figure 1a (labeled as "bird"), never seeing the student queries on the right in Figure 1b, encodes membership information about the parrot in predictions on these student queries. Interestingly, the most informative student queries are not birds-they are images of other classes with similar red hues that the model confuses as being "bird-like" because of the influence of the parrot in the teacher's training set. In other words, the student model manages to "parrot" its teacher using the peculiarities transferred through distillation. We systematically evaluate a number of factors which impact the empirical privacy of model distillation. We find that similarity between the training datasets of the teacher and student models lead to increased privacy risk, along with higher temperature parameters. Moreover, an adversary capable of poisoning the teacher training set can amplify privacy risk for teacher examples, in line with recent work (Tramèr et al., 2016;Chen et al., 2022). We also find that distillation provides little privacy protection to student examples. We hope our attacks can assist experts in properly evaluating the privacy risks resulting from model distillation. Our work also highlights two possible approaches for mitigating privacy risks in distillation: deduplicating the teacher and student datasets, and reducing the leakage from the teacher model, e.g., using provable guarantees such as differential privacy (Dwork et al., 2006). Background and Related Work Machine Learning Privacy Machine learning models are known to be vulnerable to a variety of privacy attacks, including membership inference attacks (Shokri et al., 2017), attribute inference attacks (Fredrikson et al., 2015), property inference attacks (Ganju et al., 2018), and training data extraction . Each of these captures a different type of leakage about the training data or individual examples. In this work, we focus on membership inference, as it is the most widely studied privacy attack. In a membership inference attack, an adversary tries to determine whether or not a particular example was used to train a model. There have been a number of membership inference attacks proposed in the Figure 2: Knowledge distillation is a multi-step training process. A teacher model f T is first trained on a teacher dataset D T . A dataset of images annotated by the teacher D Q is generated by querying the teacher f T on an (unlabeled) student training set D S . Finally, a student model f S is trained on D Q . literature, which generally compute some "membership score", which is designed to be informative of a target example's membership. Most of these scores make use of the model's prediction on the target example, perhaps to compute the example's loss (Yeom et al., 2018;Sablayrolles et al., 2019) or to compute some other score (Watson et al., 2021;Shokri et al., 2017;Carlini et al., 2022a). Other attacks rely on querying the model with examples nearby or derived from the target (Jayaraman et al., 2020;Long et al., 2018;Li & Zhang, 2021;Choquette-Choo et al., 2021;Wen et al., 2022). One of our contributions is to design an attack which performs well despite relying on the model's predictions on entirely different examples from the target. Our attacks are based on the state-of-the-art Likelihood Ratio Attack (LiRA) (Carlini et al., 2022a). In LiRA, the adversary first trains many shadow models, such that half of these models will contain a given target example (x, y) (the IN models), and half will not (the OUT models). Next, the adversary queries each shadow model f , to compute the logits corresponding to the correct class, f (x) y , and fits a Gaussian (µ in , σ 2 in ) to the logits for all models containing the example (and similarly for the OUT models). To attack a new model f , the adversary computes the probability density function (PDF) of each Gaussian (which we write as p in ( f (x) y ) and p out ( f (x) y )), and then computes the likelihood ratio, which serves as the membership score. Our attacks in this paper will be derived from LiRA but adapted to the variety of distillation settings we consider. Knowledge Distillation Knowledge distillation (Hinton et al., 2015;Ba & Caruana, 2014) is a technique for transferring knowledge from a "teacher" model to a "student" model. There are two datasets in knowledge distillation: the teacher dataset D T = {x i , y i } n t i=1 ,= {x i , S( f T (x i ))} n s i=1 , where the softmax function S converts logits into a probability vector. Training on this query dataset then produces a student model f S , the output of distillation. Sometimes, distillation includes "temperature" scaling, where the logits f T (x i ) are scaled by a temperature parameter H, before the softmax is applied. The query dataset will instead be D Q = {x i , S( f T (x i )/H)} n s i=1 ; setting H = 1 recovers the original distillation procedure. We illustrate this process in Figure 2. Knowledge distillation in private machine learning. Prior work has suggested that distillation mitigates prevent privacy attacks. Perhaps the most well known example is the PATE framework (Papernot et al., 2016), where distillation is used to reduce an ensemble of teacher models into a single model, in such a way that the final model has provable differential privacy guarantees (Dwork et al., 2006). Zheng et al. (2021) and Tang et al. (2022) construct ensembles of models that are designed to be private and use distillation to condense these models. Mireshghallah et al. (2022) enforces a differential privacy guarantee by training the teacher and student models with differential privacy. In each of these approaches, distillation is only one component; an underlying ensemble, or provable differential privacy guarantees, may also improve the privacy of the overall approach. In our work, we focus on the distillation procedure itself, and leave to future work the task of designing attacks on these more complicated approaches. Indeed, other prior work has already suggested that using distillation alone to protect privacy. She-jwalkar & Houmansadr (2021) propose a defense that relies on a sufficient 2 distance between teacher and student examples and limited entropy of the queries. Mazzone et al. (2022) consider using repeated distillation to prevent membership inference attacks. In our work, we will design stronger membership inference attacks to adaptively evaluate the privacy provided by distillation. Threat Model and Experimental Setting Threat Model We investigate the ability of distillation to protect against membership inference attacks in three threat models: 1. Private Teacher. The teacher dataset D T is sensitive and the student dataset D S is nonsensitive. We assume the adversary has knowledge of the student dataset. This threat model is used in most private machine learning approaches. We evaluate this threat model in Section 4. 2. Private Student. The teacher dataset is nonsensitive and the student dataset is sensitive. We assume the adversary has access to the teacher dataset. This threat model may be used to transfer knowledge from a foundation model used as a teacher to a sensitive downstream task (Liang et al., 2022). We evaluate this threat model in Section 5.1. 3. Self-Distillation. Self-distillation is commonly used to refer to the setting where the teacher and student datasets are identical. Self-distillation is commonly used when distillation is used to improve model performance or during model compression. We evaluate this threat model in Section 5.2. For each of these threat models, we measure the success of the adversary at performing membership inference on the sensitive dataset, both by measuring the true positive rate (TPR) and false positive rate (FPR) as suggested by Carlini et al. (2022a), and also by investigating the membership inference accuracy on each individual example, as done in Carlini et al. (2022b). In general, we will report this per-example membership inference accuracy when presenting the performance of one or two attacks, due to the amount of information this metric conveys, and use ROC curves to compare between more than two. Beyond distillation's potential uses in privacy, our attacks on distillation have implications for private information leaked during learning-based model extraction attacks (Tramèr et al., 2016;Orekondy et al., 2019;Pal et al., 2020;Jagielski et al., 2020), which often resemble model distillation. In model extraction, an adversary uses API access to a target model to reproduce its functionality into the weights of a local model. The target model is analogous to the teacher model in distillation, and the local model is analogous to the student. Then attacks in the Private Teacher threat model can be cast instead as allowing an adversary to inspect their local model to perform membership inference attacks on the target model's training data, without directly querying it. The Private Student threat model has implications for a defensive setting in model extraction, where the target model's owner wants to link queries made to the target model with a model they believe has been extracted from their model. Experimental Setting We study four standard datasets for our analysis: CIFAR-10, WikiText103, Purchase100, and Texas100. On CIFAR-10, we start with code from the DAWNBench benchmark (Coleman et al., 2017) which trains an accurate ResNet-9 model in under 15 seconds; we adapt this code to support model distillation and the subsampling required for LiRA variants. We remove 5275 duplicates from CIFAR-10, using the imagededup library (Jain et al., 2019), and split the remaining dataset into a teacher set of 30,000 examples and a student set of 14,725 examples. On WikiText103, we used the GPT-2 architecture (with a context window of 256 tokens, 4 heads, 4 layers and an embedding size of 256 dimensions). We split WikiText103 into a teacher set of 500,000 records, and use the remaining records to train the student models. On Purchase100 and Texas100, we train single-layer neural networks with hidden layer sizes of 256 and 512, respectively, and subsample the datasets to produce teacher and student sets of 20000 examples each. On all datasets, we train our models with the cross entropy loss: teacher models are trained with the standard sparse cross entropy loss on the teacher dataset, and student models are trained with a dense cross entropy loss to mimic the soft labels predicted by the teacher. Unless otherwise stated, all LiRA-based attacks use 100 shadow models for calibration, and all figures are produced by running the attack on over 1000 models. Needing to train a large number of shadow models is one limitation of our attacks, and any shadow model-based membership inference attacks; we comment on training efficiency and code for our experiments in Appendix A. Privacy of the Teacher Training Set The most common way prior work in privacy has used distillation is to improve the privacy of the teacher set. Intuitively, distillation protects the teacher set because the adversary can only interact with the student model, which never sees any teacher examples. In fact, because the student model can be seen as a "post-processing" of the teacher model, the data processing inequality provably implies that attacks can be no more powerful on the student model than they are on the teacher. Stepping through the distillation process can help us anticipate how distillation can impact the privacy of the teacher set. The first step, training the teacher model, is the most well-understood from a privacy perspective, as the large literature on membership inference applies to the teacher model. In particular, recent work has found that state-of-the-art membership inference attacks are better at attacking some "outlier" examples than other "inlier" examples (Carlini et al., 2022a), which we expect to be true in the teacher dataset as well, making these "inlier" examples less vulnerable in later steps, as well. Subsequent steps of distillation are less well-explored by the privacy literature. The second step of distillation creates the query dataset, which can be seen as "compressing" the teacher model into its responses on these queries. Intuitively, this step is the most important at reducing private information leakage. However, we hypothesize that some queries will capture information about teacher examples, perhaps due to some similarity between the queries and teacher examples. In the final step, the student model is trained on the query set. While this step cannot contain more sensitive information than the queries themselves, it is possible that this step makes that information easier to discover, perhaps by interpolating between the queries. Membership Inference on Teacher Examples We now show that membership inference attacks work surprisingly well on distilled student models. We first propose the "Transfer LiRA" attack, a simple extension of LiRA to the distillation setting. In this attack, we train shadow teacher models. We calibrate the IN and OUT Gaussians for LiRA on these shadow teacher models, and then run the attack by applying it directly to the student models. Notice that in this attack, distillation never happens in training shadow models-the success of this attack relies on the similarity between teacher and student models' predictions. To capture the information loss because Transfer LiRA does not use distillation in any way, we also propose a second attack, which we call "End-to-End LiRA", where we instead train shadow models with the entire distillation procedure, first training shadow teacher models, and then distilling these teacher models into shadow student models. We then calibrate LiRA using these shadow student models. Note that performing this attack requires knowledge of the student training set to query the shadow teacher models and train the shadow student models. This is in contrast to Transfer LiRA which does not, although it still requires access to in-distribution data to train teacher models. On the x axis, we plot the vulnerability of each teacher example to membership inference before distillation, using teacher models. On the y axis, we plot the vulnerability of each teacher example to attack after distillation, using the End-to-End LiRA strategy. Observe that many data points lie near the y = x line, which indicates no reduction in vulnerability from distillation. Distillation provides limited privacy. In Figure 3, we plot the change in per-example attack success rate on distilled models. We compare each example's vulnerability to LiRA on the teacher model (i.e. vulnerability without distillation) on the x-axis, to each example's vulnerability to End-to-End LiRA (i.e. vulnerability after distillation) on the y-axis. In the Appendix, we provide plots for Transfer LiRA (Figure 7), and ROC curves for each strategy (Figure 8). We also find our attack significantly outperforms a simple logit threshold baseline, similar to an attack used in prior work to evaluate distillation (Figure 12). For a large fraction of teacher examples on each dataset, we find End-to-End LiRA achieves nearly the same membership inference performance as directly attacking the teacher model. In other words, many teacher examples do not observe any privacy benefits from distillation, despite student models never directly seeing these teacher examples! While the average membership inference accuracy (and TPR at low FPR) do decrease, 5% of examples' vulnerability drop by less than 8 percentage points on CIFAR-10, 5 points on Purchase-100, and 4 points on Texas-100. Because privacy is a worst-case guarantee, distillation provides limited privacy benefits. We note two other interesting takeaways from these results. First, the per-example student attack success rate has a high variance. This variance is partly due to statistical uncertainty (although each example's attack success rate is computed with over 1000 models, giving each coordinate a standard deviation of 1.5 percentage points for both the x and y axes), but more interestingly, some examples do see significant privacy benefits from distillation, even controlling for the original model's vulnerability. We investigate duplication as one potential cause for this variance in Section 4.3. Second, each plot has a positive correlation, meaning that examples that are more vulnerable to attack on the teacher model attack also tend to be more vulnerable after distillation. As a result, reducing the vulnerability of the teacher model, perhaps using techniques such as differential privacy, are likely to improve the student's privacy. Privacy Leakage Through Student Queries It is surprising that membership inference still performs well on distilled student models, despite these models never directly using any of the teacher data. We now investigate why. For a membership inference attack to be successful on the student model, it must be the case that the student queries reveal some membership information about the teacher examples. However, to rigorously evaluate how this information is encoded in the student queries, we will turn our attention to designing a new attack that only has access to the student query dataset D Q (and, in particular, has no knowledge of the teacher model's predictions on the teacher examples). To the best of our knowledge, this attack is also the only membership An indirect attack using student queries. We adapt LiRA to a setting where only the student query scores are available to the adversary. Because of this limitation, the adversary can only rely on information about teacher examples that is indirectly contained in the student query scores. To adapt the attack to this setting, we use the same approach that LiRA uses to combine queries on multiple "augmentations" of an example: multiplying their likelihood ratios to produce an aggregate membership score. Concretely, for each teacher example z T j = (x T j , y T j ), we fit a Gaussian distribution to the logits of each student example z S i = (x S i , y S i ), when z T j is either IN or OUT. We write the PDFs of these distributions as p I N j,i and p OU T j,i , so that the joint PDF of the IN Gaussians is p I N j = Π i p I N j,i and similarly for the OUT Gaussians. This natural adjustment allows us to infer membership of the teacher examples using the student examples. However, we find that this direct adaptation of LiRA tends to have poor performance, so we propose two modifications which significantly improve the attack. First, we find that a teacher example tends to have more influence on the logit corresponding to the teacher label y T j than the student label y S i , and so we choose to calibrate LiRA using the teacher label logit rather than the student label logit. Second, there tend to be many uninformative student examples for each teacher example, so we filter the student queries for all teacher examples. We filter these student queries by selecting only those with the largest mean gap |µ I N j,i − µ OU T j,i |, which are most informative about the teacher. This removes most of the noise from our LiRA adaptation, improving it especially when we train few shadow models. Figure 4, we plot the per-example membership inference accuracy after distillation, using our student query attack with 4000 shadow models. We consider all datasets except WikiText103, which we omit due to the high computational cost of our attack. Our results are similar to when we attacked the student models: MIA vulnerability reduces on average, but many examples maintain nearly identical membership inference attack success rate using only the predictions on these indirect student queries. Cases where this attack achieves high membership inference accuracy indicate that a large amount of information about the teacher set is encoded in the student queries. Student queries leak private information. In Observations we noticed for our student model attacks persist for this query-based attack. Examples which are more vulnerable to attack in the teacher model also tend to be more vulnerable using teacher model predictions. This is true on all datasets, although we also find the decay in vulnerability with queries to be dataset-dependent and example-dependent. Within each dataset, there is a high variance in student model membership inference accuracy. We also remark on the counterintuitive fact that our prior attack with access to the student models outperformed this attack, despite the fact that the student model is just a post-processing of these student queries. This implies that the process of training a student model makes it easier to extract private information from these student queries, by interpolating between the student queries. In the following subsections, we will investigate some factors that can impact the success of the student query attack, and what this tells us about distillation's dark knowledge. Ablation. In Figure 5a, we ablate the modifications we made for our student query attack. We show both LiRA scores (logits for the student label) and Label scores (logits for the teacher label). We also show attacks with "All", without filtering, and "Filtered" scores, with filtering to 10 student examples. Our modifications significantly improve membership inference: without either modification, the attack is no better than random chance. Applying both modifications achieves a TPR of 10 −3 at a FPR of 10 −4 . Qualitative analysis. Our ablations also shed light on where membership information is contained in distillation's dark knowledge: predominantly in the logit corresponding to the teacher label within only a few student examples. This corroborates our intuition from the parrot in Figure 1. There, only two student queries were also in the bird class, emphasizing the importance of label scores. We inspect other vulnerable examples in Figure 9 in Appendix B, finding some teacher examples whose most informative students mostly belong to their same class. Figure 5b (p < 10 −15 using a Chow Test). This indicates that deduplication significantly reduces privacy risk, although indirect queries can still carry membership information. In the Appendix, we show student attacks are also worsened by duplication in Figure 11. Tramèr et al. (2022) and Chen et al. (2022) find that data poisoning attacks added into a training set can amplify membership inference vulnerability for other examples in the training set. Here, we investigate how distillation interacts with this effect; i.e., we measure whether poisoning in the teacher set can increase an example's vulnerability in the teacher predictions. In Figure 5c, we evaluate this using our student query attack on CIFAR-10 using the label flipping poisoning strategy from prior work. With higher poisoning counts, many examples have higher teacher membership inference accuracy, i.e., they shift to the right on the x-axis; this is exactly in line with prior work. However, interestingly, we see that poisoning does not impact the relationship between teacher vulnerability and student vulnerability-they move upward on the y-axis identically to examples which were not poisoned. Student-Teacher Similarity Teacher Set Poisoning We remark that our poisoning attack does not specifically target the student query set; it is an interesting open question whether poisoning attacks exist that could increase membership leakage on the student model, but with less (or more!) impact on the teacher model's vulnerability. Section 4.3 hints at such a strategy, if the adversary can poison the student set instead: if an adversary adds duplicates of target examples to the student set, the resulting student queries will increase risk on those target teacher examples. Temperature Scaling When introducing knowledge distillation, Hinton et al. (2015) proposed a modification known as temperature scaling. Temperature scaling makes two simple changes to distillation: (1) introducing a temperature hyperparameter H, which when increased, rescales the logits and flattens the resulting probability distribution of student queries, and (2) rescaling gradients by a 1/H 2 factor. Though normally set to 1, modifying this parameter can improve performance. It is natural to consider whether such manipulation of the teacher outputs might improve empirical privacy; indeed, Shejwalkar & Houmansadr (2021) evaluate their distillation-based defense with various temperatures, and find that high temperature reduces private information. We use our strong End-to-End LiRA to evaluate the empirical privacy of student models trained with H ∈ [0.1, 4], shown in Figure 5d. We find lower temperatures are mildly less vulnerable than higher temperatures, reducing TPR by a factor of roughly 4 at an FPR of 10 −3 when H decreases from 4 to 0.1; model accuracy at all of these temperatures is similar to the baseline of H = 1. While this trend is small, it is also intuitive: in Section 4.2, we found that all logit values carry important membership information. When H is small, the entropy of query probabilities decreases, reducing information captured by these probabilities. We hypothesize that the reverse effect observed by Shejwalkar & Houmansadr (2021) may be a result of the accuracy decrease they found resulted from high temperatures (i.e. up to H = 10), which we do not observe. Finally, we remark on an interesting gap between an adversary with access to student queries and one who only has access to a student model. With access to temperature-scaled student logits, an adversary can multiply by H to reverse the scaling-temperature scaling cannot make attacks harder in this threat model. However, it does appear that low temperatures lead to weaker attacks on the student model, which we do not know how to reverse. Indeed, a natural way to adaptively attack a temperature-scaled student model is to rescale the student model's logits, but our LiRA accounts for this, as LiRA is scale-invariant. Adaptively attacking temperature-scaled models is an interesting open question. Privacy of Student Training Set Having evaluated the Private Teacher threat model, we now turn to the Private Student and Self-Distillation threat models, which we will consider simultaneously. The Private Student threat model can be used to perform knowledge transfer from large, general purpose models to task-specific models, by querying on (sensitive) task-specific student data. Self-distillation is often used in applications of distillation to compress models and improve their performance. Private Student The private student threat model does not involve data minimization, unlike the private teacher threat model; the empirical privacy we investigate here comes instead from an adversary having limited knowledge of the specifics of the teacher model. That is, the question we investigate is: how much does the adversary need to know about the teacher model to get reliable attacks on the private student dataset? We consider three levels of adversarial knowledge: Known Teacher, where the adversary knows the precise teacher model used to query the student examples; Unknown Teacher, where the adversary knows the teacher model is one of a small subset of models; and Surrogate, where the adversary can only collect similar data, to train their own surrogate teacher models. Both the Known and Unknown Teacher settings reflect a world where the teacher model is one of a small number of general purpose public models, such as a large language model. The Surrogate setting requires the adversary to train their own copy. We run the LiRA variants in a number of these settings on the CIFAR-10 dataset, calibrated to the knowledge the adversary has (for example, in the Surrogate threat model, the adversary trains their own teacher models, and trains a number of shadow student models to calibrate LiRA). We plot our results in Figure 6a, and find that, as expected, less knowledge about the teacher model reduces the adversary's success at membership inference. However, even the weakest threat model, Surrogate, allows for powerful attacks, with a TPR as large as 10 −2 at a FPR of 10 −3 . Privacy of Self-Distillation Having considered the privacy of the student and teacher datasets independently, we now investigate the common self-distillation setting (Furlanello et al., 2018;Xie et al., 2020), where the student and teachers are identical. Given that duplicate examples in the student set carry membership information of teacher examples (Section 4.3), and student examples themselves are not well protected by distillation (Section 5.1), we do not expect self-distillation to reduce privacy risk significantly. However, a common technique in self-distillation is to train the student on a loss function which combines the cross entropy loss on the query dataset Q with the cross entropy loss on the student examples' original "hard labels" S . We write α = α Q + (1 − α) S , so that α = 1 recovers the standard distillation objective, while α = 0 recovers the standard cross entropy loss (as if there was never a teacher model). To evaluate self-distillation, we run LiRA by training shadow student models with the entire selfdistillation algorithm, using identical datasets for each pair of teacher and student shadow models. We perform calibration on these shadow student models, and plot our results at a range of α values in Figure 6b. While we don't observe a large effect, it appears that larger α (that is, heavier reliance on the distillation loss function) results in better attacks. This is likely because relying on the distillation loss function reinforces the memorization from the teacher even further in the second round of training on the student. Conclusion In this work, we have used membership inference attacks to empirically evaluate to what extent knowledge distillation protects the privacy of training data. For those interested in using model distillation to improve privacy, our work offers three main design considerations. First, deduplicating the teacher set with respect to the student set is necessary to reduce risk on duplicated examples. Second, while average-case privacy can significantly improve after distillation, the worst-case vulnerable examples often see only marginal benefits. As a result, techniques which make the teacher model more private, such as differentially private training, should be seen as complementary to distillation. And third, the student training set should not be seen to have improved privacy as a result of model distillation. Our work also offers insights into the privacy properties of distillation's dark knowledge, which may be of broader interest. Our results also imply privacy attacks on model extraction attacks (Tramèr et al., 2016;Jagielski et al., 2020) which rely on similar algorithms to knowledge distillation. With the x axis, we plot the vulnerability of each teacher example to attack before distillation, using teacher models. With the y axis, we plot the vulnerability to attack after distillation, using the Transfer LiRA strategy to attack student models. Observe that many data points lie near the y = x line, which indicates no reduction in vulnerability from distillation. A More Experiment Details All of our results on CIFAR-10 make use of fewer than 30000 trained models. While a very large number of models, the fast, publicly available training code we use allows us to train this number of models in fewer than 1 GPU-week (although we decrease the wall-clock time by parallelizing over 4 GPUs). Our results on Purchase-100 and Texas-100 also use simple models, taking under 1 minute to train (we train all models for 20 epochs with SGD with a learning rate of 0.01 and momentum parameter of 0.99, which we found to maximize performance over our hyperparameter sweep). We train 8000 of these models for our analysis, taking fewer than 1 GPU-week for each of these datasets. Our most expensive attack, relying on only student queries, starts to outperform random guessing with as few as 100 models, which can be trained on 1 GPU in two hours on all three of these datasets. Unfortunately, we are unable to make our code public at this time due to organizational constraints. B Extended Results We plot the effectiveness of Transfer LiRA in Figure 7. ROC curves for our student attacks are found in Figure 8. Further qualitative examples can be found in Figure 9. Ablation of score information with and without duplicates is plotted in Figure 10. Per-example student attack success rates for CIFAR-10 with duplicates are found in Figure 11. In Figure 12, we compare our student model attacks against a simple logit threshold baseline, similar to the loss thresholding attack designed by Yeom et al. (2018), which was used to evaluate distillation privacy in Shejwalkar & Houmansadr (2021). The only exception is the eighth student query in (d) for the yellow truck in (c), which is an airplane. The filtered attack using the displayed student queries reaches 78% accuracy on the yellow automobile in (a), and 74% accuracy on the yellow truck in (c). Figure 1 : 1We can predict the membership status of a target example (a) in the teacher model's training set by querying the teacher model on different student training examples (b). Figure 3 : 3Many data points get no privacy benefits from distillation. Figure 4 : 4Per-example membership inference accuracy using only student queries, compared to LiRA accuracy on the teacher model. Membership inference accuracies are non-trivial, and remain high for some examples. Effectiveness is dataset-and example-dependent, with Texas-100 having the most vulnerable examples in the teacher model remaining vulnerable to attacks based on teacher predictions. inference attack in the literature which does not require querying a model directly on an example (or on algorithmically derived examples) to predict its membership status. Figure 5 : 5Additional investigations into the success of our attacks reveal: a) label scoring and score filtering are important for improving attack success; b) duplication between the teacher and student datasets increases privacy risk; c) data poisoning attacks amplify the performance of our indirect attack; d) temperature scaling causes mild changes in privacy vulnerability. All results are on CIFAR-10. Figure 6 : 6Distillation has limited ability to prevent membership inference either a) on sensitive student examples, or b) in self-distillation. However, reducing the knowledge available to the adversary seems to help in the Private Student threat model. Results for both on CIFAR-10. Figure 7 : 7Many data points do not get privacy benefits from distillation. Figure 8 :Figure 9 : 89ROC curves for our attacks on student models. Two examples of target examples for which the most informative student queries are predominantly in the same class. Figure 10 : 10The impact of denoising on duplicated and deduplicated teacher attacks. Figure 11 : 11Duplication also has an impact on CIFAR-10 student attacks. Compare withFigure 3a. Figure 12 : 12Our attacks outperform a simple logit threshold baseline attack, used by prior work. with n t examples, and the student datasetD S = {x i , y i } i=1 , with n s examples.Distillation begins by training a teacher model f T on the teacher dataset D T . The teacher model is used to generate (soft) labels for the student dataset, producing a query dataset D Qn s Zagoruyko, S. and Komodakis, N. Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer. arXiv preprint arXiv:1612.03928, 2016. Zheng, J., Cao, Y., and Wang, H. Resisting membership inference attacks through knowledge distillation. Neurocomputing, 452:114-126, 2021. ISSN 0925-2312. doi: https://doi.org/10.1016/j.neucom.2021. 04.082. 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Stealing machine learning models via prediction {APIs}. In 25th USENIX security symposium (USENIX Security 16), pp. 601-618, 2016. Truth serum: Poisoning machine learning models to reveal their secrets. F Tramèr, R Shokri, A San Joaquin, H Le, M Jagielski, S Hong, N Carlini, 10.1145/3548606.3560554Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, CCS '22. the 2022 ACM SIGSAC Conference on Computer and Communications Security, CCS '22New York, NY, USA, 2022Association for Computing Machinery356554Tramèr, F., Shokri, R., San Joaquin, A., Le, H., Jagielski, M., Hong, S., and Carlini, N. Truth serum: Poisoning machine learning models to reveal their secrets. In Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, CCS '22, pp. 2779-2792, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 9781450394505. doi: 10.1145/3548606.3560554. URL https://doi.org/1 .1145/35486 6.356 554. On the importance of difficulty calibration in membership inference attacks. L Watson, C Guo, G Cormode, A Sablayrolles, arXiv:2111.08440arXiv preprintWatson, L., Guo, C., Cormode, G., and Sablayrolles, A. On the importance of difficulty calibration in membership inference attacks. arXiv preprint arXiv:2111.08440, 2021. Y Wen, A Bansal, H Kazemi, E Borgnia, M Goldblum, J Geiping, T Goldstein, arXiv:2210.10750Better membership inference with ensembled adversarial queries. arXiv preprintWen, Y., Bansal, A., Kazemi, H., Borgnia, E., Goldblum, M., Geiping, J., and Goldstein, T. Canary in a coalmine: Better membership inference with ensembled adversarial queries. arXiv preprint arXiv:2210.10750, 2022. Self-training with noisy student improves imagenet classification. Q Xie, M.-T Luong, E Hovy, Q V Le, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. the IEEE/CVF conference on computer vision and pattern recognitionXie, Q., Luong, M.-T., Hovy, E., and Le, Q. V. Self-training with noisy student improves imagenet classification. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10687-10698, 2020. Privacy risk in machine learning: Analyzing the connection to overfitting. S Yeom, I Giacomelli, M Fredrikson, S Jha, 2018 IEEE 31st computer security foundations symposium (CSF). IEEEYeom, S., Giacomelli, I., Fredrikson, M., and Jha, S. Privacy risk in machine learning: Analyzing the connection to overfitting. In 2018 IEEE 31st computer security foundations symposium (CSF), pp. 268-282. IEEE, 2018.

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{'abstract': 'Model distillation is frequently proposed as a technique to reduce the privacy leakage of machine learning. These empirical privacy defenses rely on the intuition that distilled "student" models protect the privacy of training data, as they only interact with this data indirectly through a "teacher" model. In this work, we design membership inference attacks to systematically study the privacy provided by knowledge distillation to both the teacher and student training sets. Our new attacks show that distillation alone provides only limited privacy across a number of domains. We explain the success of our attacks on distillation by showing that membership inference attacks on a private dataset can succeed even if the target model is never queried on any actual training points, but only on inputs whose predictions are highly influenced by training data. Finally, we show that our attacks are strongest when student and teacher sets are similar, or when the attacker can poison the teacher set.', 'arxivid': '2303.03446', 'author': ['Matthew Jagielski \nGoogle Research\n\n', 'Milad Nasr \nGoogle Research\n\n', 'Katherine Lee \nGoogle Research\n\n', 'Christopher Choquette-Choo \nGoogle Research\n\n', 'Nicholas Carlini \nGoogle Research\n\n'], 'authoraffiliation': ['Google Research\n', 'Google Research\n', 'Google Research\n', 'Google Research\n', 'Google Research\n'], 'corpusid': 257378537, 'doi': '10.48550/arxiv.2303.03446', 'github_urls': ['https://github.com/idea'], 'n_tokens_mistral': 14590, 'n_tokens_neox': 12607, 'n_words': 8327, 'pdfsha': '1aa9c50a52537bdac9b1edf9f9399155ec16eee0', 'pdfurls': ['https://export.arxiv.org/pdf/2303.03446v1.pdf'], 'title': ['Students Parrot Their Teachers: Membership Inference on Model Distillation', 'Students Parrot Their Teachers: Membership Inference on Model Distillation'], 'venue': []}

arxiv

CREPES: Cooperative RElative Pose Estimation System Robot B Robot A Other Robots Zhiren Xun IR LEDs Jian Huang IR LEDs Zhehan Li IR LEDs Zhenjun Ying IR LEDs Yingjian Wang IR LEDs Chao Xu IR LEDs Fei Gao IR LEDs Yanjun Cao IR LEDs CREPES: Cooperative RElative Pose Estimation System Robot B Robot A Other Robots ESKF For Relative Pose Estimation Relative orientation Raw Relative Pose Estimation Relative Position Robot B Robot A PGO IR LEDs Camera UWB IMU Camera UWB IMU ID Extraction Relative Direction ID Extraction Relative Direction Fig. 1: Hardware design and software architecture of CREPES. The hardware consists of an IMU, an UWB, IR LEDs and a Camera.The Raw Relative Pose Estimation software module collects neighbors' ID (from ID Extraction), directional measurement (from Relative Direction), distance (from UWB) and IMU information to get the raw estimation. Then an ESKF module filters the result using the IMU information. When multiple neighbors are around, a PGO module is employed to further improve the performance.Abstract-Mutual localization plays a crucial role in multirobot cooperation. CREPES, a novel system that focuses on six degrees of freedom (DOF) relative pose estimation for multi-robot systems, is proposed in this paper. CREPES has a compact hardware design using active infrared (IR) LEDs, an IR fish-eye camera, an ultra-wideband (UWB) module and an inertial measurement unit (IMU). By leveraging IR light communication, the system solves data association between visual detection and UWB ranging. Ranging measurements from the UWB and directional information from the camera offer relative 3-DOF position estimation. Combining the mutual relative position with neighbors and the gravity constraints provided by IMUs, we can estimate the 6-DOF relative pose from a single frame of sensor measurements. In addition, we design an estimator based on the error-state Kalman filter (ESKF) to enhance system accuracy and robustness. When multiple neighbors are available, a Pose Graph Optimization (PGO) algorithm is applied to further improve system accuracy. We conduct enormous experiments to demonstrate CREPES' accuracy between robot pairs and a team of robots, as well as performance under challenging conditions. Fig. 1 : Hardware design and software architecture of CREPES. The hardware consists of an IMU, an UWB, IR LEDs and a Camera. The Raw Relative Pose Estimation software module collects neighbors' ID (from ID Extraction), directional measurement (from Relative Direction), distance (from UWB) and IMU information to get the raw estimation. Then an ESKF module filters the result using the IMU information. When multiple neighbors are around, a PGO module is employed to further improve the performance. Abstract-Mutual localization plays a crucial role in multirobot cooperation. CREPES, a novel system that focuses on six degrees of freedom (DOF) relative pose estimation for multi-robot systems, is proposed in this paper. CREPES has a compact hardware design using active infrared (IR) LEDs, an IR fish-eye camera, an ultra-wideband (UWB) module and an inertial measurement unit (IMU). By leveraging IR light communication, the system solves data association between visual detection and UWB ranging. Ranging measurements from the UWB and directional information from the camera offer relative 3-DOF position estimation. Combining the mutual relative position with neighbors and the gravity constraints provided by IMUs, we can estimate the 6-DOF relative pose from a single frame of sensor measurements. In addition, we design an estimator based on the error-state Kalman filter (ESKF) to enhance system accuracy and robustness. When multiple neighbors are available, a Pose Graph Optimization (PGO) algorithm is applied to further improve system accuracy. We conduct enormous experiments to demonstrate CREPES' accuracy between robot pairs and a team of robots, as well as performance under challenging conditions. I. INTRODUCTION mutual relative localization is the key to accomplishing tasks cooperatively. Stable, accurate and fast relative pose estimation between robots can significantly improve the quality of collaboration. For instance, robots can continuously transform neighbors' perceptions into their frames to acquire a robust collaborative perception. One common practice for relative localization is using robots' odometry in a global reference frame, like satellitebased global positioning system (GPS) [4], motion capture system (MCS) [5] and UWB system with multiple anchors [6]. The relative poses can be calculated from the subtraction between agents' global states. However, these systems rely on pre-installed infrastructure or require time-consuming calibration, and not applicable to robots in unknown environments. Simultaneous localization and mapping (SLAM) can provide each robot with the odometry in its global reference frame. Relative transformations between multiple robots can be estimated by matching common features in their maps, either centralized or distributed. Nevertheless, they usually need high computational resources and communication bandwidth. By equipping robots with specially designed structures, such as AprilTags [7] and LEDs [8], the relative pose can be estimated from direct robot-to-robot observations in many systems. However, short detection range, strict viewpoint requirements and sensitivity to ambient light limit their application in multi-robot systems. To overcome difficulties of dependence on the infrastructure and environment, high computational cost and low adaptability, we design a Cooperative RElative Pose Estimation System (CREPES) for multi-robot systems. CREPES, which can obtain instant 6-DOF relative poses to all the neighbors in a large-scale environment, consists of a novel hardware design and supported software. The hardware consists of active IR LEDs, an IR fisheye camera, an IMU and an UWB. The system can produce a raw estimation of the relative pose between two robots from one single frame of mutual observations. To cope with multiple sources of sensor noise, we establish a relative motion model and apply an adapted ESKF [9] where the reference frame is in motion. A pose graph optimization strategy is applied when two or more neighbors are around to further improves the accuracy. In summary, our contributions are as follows: 1) We propose CREPES, a novel relative pose estimation system that produces accurate relative position and orientation within one-shot mutual observations. 2) We design and implement the hardware prototype consisting of active IR LEDs, an IR fisheye camera, an IMU and an UWB. 3) We design a relative pose estimator based on ESKF, where the reference frame is in motion. 4) We propose a PGO-based algorithm to improve the accuracy when multiple neighbors are around. II. RELATED WORK While many infrastructure-based systems (e.g. GPS, MCS, UWB with anchors [5,10]) circumvent the mutual localization by using global poses, we focus on real-time relative pose estimation for autonomous navigation in an unknown environment. We classify current systems into direct and indirect methods depending on whether the relative pose can be estimated instantaneously. A. Indirect Methods Multi-robot SLAM is a typical indirect method in which agents estimate the relative transformation between robots' map frames by matching common features in their maps, either in a centralized or distributed fashion. Centralized works [11], [12], [13] usually require a powerful central server to collect keyframes from all agents and optimize their trajectories through the global bundle adjustment in a common coordinate frame. The relative information between agents can be acquired from the server directly. Distributed methods [14], [15] rely on inter-robot loops to estimate relative poses between robots' coordinate frames in a distributed manner. In these works, robots need to exchange map feature descriptors for inter-robot loop detection. More importantly, the feature descriptors should be generated from similar viewpoints to improve accuracy. Mutual observations, such as relative ranging or bearing, are applied to help reduce the high dependency on the environment and inter-loop detection. Cao [16] proposes an efficient method by combining the visual inertial odometry (VIO) system with mutual UWB ranging measurements between robots and an anchor. Wang [17] utilizes trajectories of Unmanned Aerial Vehicles (UAVs) and anonymous bearing measurements to formulate mutual localization as a mixed-integer quadratically constrained quadratic problem and obtain a certifiably global optimum. Xu [18] fuses omnidirectional visual inertial SLAM and UWB measurements with global graph-based optimization. Overall, these methods may suffer from degeneration due to the reliance on SLAM systems in the feature-less environment. Researchers also explored ranging/bearing only systems to further reduce environmental dependency. Zhou [20] provides theoretical proof that the minimum number of distance constraints required for 3-DOF relative pose estimation is five. Guo [21] proposes an infrastructure-free cooperative 3-DOF relative localization system with UWB measurements and applies it to real-world UAV formation control. Trawny [19] puts forth an algebraic algorithm using ten range measurements to estimate 6-DOF relative pose. However, these methods usually require enough motion excitation over long trajectories in practice. B. Direct Methods Although direct methods typically need customized hardware, the self-sufficiency, stability, efficiency and accuracy still attract enormous attention. Cutler [22] proposes a lightweight solution for estimating ranging and bearing relative to a known marker, which consists of three IR LEDs in a fixed pattern. Faessler [23] utilizes four infrared LEDs structures following certain rules and uses the Perspective-n-Point (PnP) algorithm to calculate the relative pose between a quadrotor and a ground robot. When coming to the multirobot scenario, active markers [24] or active LEDs coded board [8] is designed to encode ID information either by pulsating capabilities or LEDs arrangements. However, since the utilization of the PnP algorithm, these methods usually work at a short distance to keep the LED light spots distinguishable in the image. By using Ultraviolet LEDs and estimating the bearing vector and distance, Walter [25] has dramatically improved the detected range, with a maximum working distance of 15 meters. UWB is getting popular in multi-robot systems due to its low cost and good ranging accuracy. Fishberg [26] provides an inter-agent 3-DOF relative pose estimation system for robots in a 2D plane, where each agent is equipped with four UWB modules. The relative pose is calculated by modeling observed ranging biases and systematic antenna obstructions in a nonlinear least squares optimization. Cossette [27] presents a method for computing optimal formations for relative pose estimation, during which both the relative position and relative heading of the agents with two UWB modules are locally observable. An on-manifold gradient descent procedure is used to determine optimal formations for improving estimation. Since the noise property of UWB, the baseline distance between multiple UWB modules should be far to acquire good performance, which limits the platform size to use these systems. III. RELATIVE POSE ESTIMATION SYSTEM Our novel system includes a compact hardware design and supported software. As shown in Fig. 1, the hardware system includes IR LEDs, an IR fisheye camera, an IMU, and an UWB, and the software contains the ID Extraction module, raw relative pose estimation module, ESKF filter module, and PGO module. ID Extraction module establishes the data association between ranging measurement and directional information by using the IR camera and a disc-shaped IR LED board. The raw relative pose estimation module gets direct relative position and orientation. Then, we design a relative movement model and adapt ESKF to filter raw estimations. For systems with more than two robots, the PGO module launches to further improve accuracy. A. Hardware Fig. 1 shows an overview of sensor components and their physical settings. A disc-shaped board with six 950nm IR LEDs is designed to transmit ID information. We program an ARM Cortex-M3 STM32 microcontroller to control the flickering of LEDs for ID encoding. Correspondingly, we use a MV-SUA133GM camera made by MindVision, equipped with a 950 nm IR filter, to decode the ID information. The camera has a fisheye lens with a 185 degrees field of view (FOV) and is set to a frame rate of 200 Hz (maximum 245 Hz), with a global shutter. We use a DW1000-based UWB module from NoopLoop to provide mutual ranging. It uses a dongle antenna to get relatively good omnidirectional ranging and a maximum range of 500 meters, with a standard deviation of 5 centimeters. In addition, a 6-DOF low-cost MEMS IMU module is used to provide accelerations and angular velocities at a frequency of 100 Hz. The accelerometer noise density is 183.3µg/ √ Hz and the gyroscope noise density is 0.021 • /s/ √ Hz. In practice, we use imu-tk [28] to perform calibration to correct imprecise scaling factors and axes misalignments. Lastly, we use an Intel NUC with i5 processor as the computation platform and its onboard WiFi as the communication medium. Note that we configure the WiFi network card into a self-organizing MESH mode (BATMAN network 1 ) to remove the dependence on a central router. B. ID Extraction The ID information for each node is encoded into the LED pulsating control. IR LED boards are programmed with designed duty rates using a 50ms period. Benefitting from the infrared filter, all lighted IR LED boards are easily distinguished from natural features in a captured image. First, we convert the image into binary with a threshold and perform circle detection using Hough transform to obtain pixel coordinates of the centers of detected spots. Next, the detected spots are associated with previous ones according to a distance constraint, and the duty rates are calculated. Finally, IDs are decided by comparing the calculated duty rates with elements in an ID library. At the same time, the pixel coordinates of the detected spots' centers work as the directional measurement for pose estimation. C. Raw Relative Pose Estimation The raw relative pose estimation uses mutual directional measurement from the camera (described in the above section), UWB ranging, own IMU and neighbor's IMU measurements, as shown in Fig. 1. The working pipeline of how we estimate raw mutual relative pose is shown in Fig. 2, taking robots A and B as an example (B as the observer). Firstly, when two robots observe each other in their image frame, they extract ID and pixel coordinates from the ID extraction module. Based on the Double Sphere (DS) projection model of the fisheye camera [29], we get the unit directional vector B p uA ∈ R 3 for detected robot A in robot B's frame. The relative position of robot A in B's frame B p A can be calculated by A B � A � B � A � ̅ X Y Z � p � p Camera FOV Detected Point p pB p A = d AB * B p uA(1) where d AB is the ranging measurement from UWB between robots A and B. To estimate the relative orientation, we introduce intermediate framesĀ andB. We use Z − Y − X Euler angles to define roll, pitch and yaw, and extract roll and pitch angles using the gravity constraint. As shown in Fig. 2 (b), by rotating the roll and pitch angles to align the z-axis of robot B's body frame opposite gravity's direction, we get the new B frame. The unit directional vector B p uA in theB frame is expressed asBp uA , wherē B p uA = R B pitch R B roll B p uA(2) R B pitch and R B roll are obtained from robot B's IMU measurements. As shown in Fig. 2 (c), we projectBp uA to the X-O-Y plane to get an angleBψ A between the projected vector and the positive direction of the x-axis. In the same way, we take robot A as the observer and can also getĀψ B . Therefore the relative yaw angle ψ can be defined as, ψ =Bψ A −Āψ B + π(3) After all, we calculate the relative orientation matrix B R A by B R A = R B roll T R B pitch T R yaw {ψ} R A pitch R A roll (4) where R yaw {ψ} is the rotation matrix corresponding to ψ, R A pitch and R A roll are extracted from robot A's IMU measurements. Similarly, by taking robot A as the observer, we can also calculate the relative position A p B and relative orientation A R B in robot A's frame. D. ESKF Filter To improve the estimation quality, we adapt an ESKF to filter the raw relative pose estimations. Compared to the typical state estimation in a normal inertial system, our ESKF model takes extra consideration of the reference frame motion. Same as the above section, we keep robot B as the observer. 1) Prediction Model: In a static reference frame W , we define W q (·) , W p (·) and W v (·) as the quaternion, position and velocity of the robot (·), respectively. For robots A and B, the relative state can be calculated by B p A = R T W q B ( W p A − W p B ) (5a) B v A = R T W q B ( W v A − W v B ) (5b) B q A = W q * B ⊗ W q A (5c) where R {q} and R {θ} are the rotation matrices associated with the quaternion q and the angular vector θ, respectively, R T {·} is the inverse matrix of R {·} and ⊗ represents the quaternion product. It should be noted that B v A is not the time rate of the change of B p A and the relationship is revealed in equation (9a). For simplicity, we write B p A , B v A , and B q A as p, v, and q, respectively. We define that x is the nominal state, x t is the true state and δx is the error state, x =   p v q   x t =   p t v t q t   δx =     δp δv δθ A δθ B     (6) where δθ (·) is the small local angular error used to parameterize an error quaternion of the robot (·), δq (·) ≈ 1 δθ (·) 2 . The true state can be computed with nominal-state and errorstate by x t = x ⊕ δx (7) p t = R T {δθ B } (p + δp) (7a) v t = R T {δθ B } (v + δv) (7b) q t = δq * B ⊗q⊗δq A(7c) where δq * B is the conjugate quaternion of δq B . We take the robot (·) IMU acceleration measurements a m(.) and gyroscope measurements w m(·) as the ESKF filter input u m . The input noise vector u n consists of acceleration noise a n(.) and gyroscope noise w n(·) . a n(·) and w n(·) are modeled by white Gaussian processes. u m =     a mA w mA a mB w mB     u n =     a nA w nA a nB w nB     (8) We have the system model of the nominal-state as p ← R T {w mB ∆t} (p + v∆t + 1 2 (R {q} a mA − a mB )∆t 2 ) (9a) v ← R T {w mB ∆t} (v + (R {q} a mA − a mB )∆t) (9b) q ← q * {w mB ∆t} ⊗ q ⊗ q {w mA ∆t} (9c) where ← stands for a discrete time update, ∆t is the discrete time interval and q {θ} is the quaternion associated with the angular vector θ. We write the differential equations of the error-state as δx ← f (x, δx, u m , u n ) = F x (x, u m )δx + F i u n (10) δp ← R T {w mB ∆t} (δp + δv∆t) (10a) δv ← R T {w mB ∆t} (δv + α∆t) (10b) δθ A ← R T {w mA ∆t} δθ A − w nA ∆t (10c) δθ B ← R T {w mB ∆t} δθ B − w nB ∆t (10d) where α = −R {q} [a mA ] × δθ A + [a mB ] × δθ B − R {q} a nA + a nB and the definition of the cross-product matrices [ ] × can be found in [9]. We define F x and F i are the Jacobians of f with respect to δx and u n , F x and F i are calculated by F x = ∂f ∂δx x,um , F i = ∂f ∂δu n x,um(11) Then the prediction equations can be written aŝ δx ← F x (x, u m )δx (12) P ← F x PF x T + F i Q i F i T(13) where δx ∼ N (δx, P) and Q i is the covariance matrix of u n . When IMU data is received, we follow equations (9a) ∼ (9c), equations (12) and (13) to update the nominal-state, error-state and error-state covariance matrix, respectively. 2) Measurement Model: We take the raw calculation results in section III-C as measurements z, z =   B p A A p B B q A  (14) where B q A corresponds to B R A . Since the same sensors' observations are used to compute B R A and A R B , we only need to select one of them as the measurements. The relationship between the measurements z and the true-state is written as z = h(x t ) + v (15) B p A = p t + p nA (15a) A p B = −R T {q t } p t + p nB (15b) B q A = q t + q n (15c) where v = [p nA , p nB , q n ] T ∼ N (0, V) is a white Gaussian noise with the covariance V. The true state estimation can be calculated byx t = x⊕δx. As the error-state meanδx = 0, we havex t = x. Therefore, we take x as the evaluation point and the Jacobian matrix of the measurement model H is H = ∂h ∂δx x = ∂h ∂x t x ∂x t ∂δx x(16) The correction equations can be written as K = PH T (HPH T + V) −1 (17) δx ← K(z − h(x t )) (18) P ← (I − KH)P(19) We use equation (18) and equation (19) to compute the observed error and update the error-state covariance matrix, respectively. 3) Error Injection and Reset: When the measurements update is finished, we add the observed error to the nominal state by x ← x ⊕δx After the error injection step, we reset the error state for the next iteration by δx ← g(δx) = δx δ x (21) δp ← R T δ θ B (δp −δp) (21a) δv ← R T δ θ B (δv −δv) (21b) δθ A ← −δθ A + I − 1 2δ θ A × δθ A (21c) δθ B ← −δθ B + I − 1 2δ θ B × δθ B (21d) We define G as the jacobian matrix of the error reset function (21), G can be computed by G = ∂g ∂δx δ x(22) Finally, we update the error-state meanδx and its covariance matrix P byδ x ← 0(23)P ← GPG T(24) E. Pose Graph Optimization From the above sections, we get the refined relative pose estimation between two robots. When multiple robots are around, we propose a PGO-based algorithm to further improve the mutual localization accuracy. Unlike the classic PGO using multi-frame measurements in the continuous time domain, our PGO formulation is for any single frame of mutual measurements. As shown in Fig. 3, each robot represents a node in the graph and the edge is the mutual relative pose between two robots. Currently, each robot runs the PGO algorithm in a distributed manner after receiving all the available mutually measured poses from the neighbors. For an arbitrary robot, we denote its coordinate frame as C, the pose of robot i in C as X i = (R i , t i ) ∈ SE(3), the measured relative pose between robot i and robot j asT ij = (R ij ,t ij ) ∈ SE(3), i = j, and the PGO problem can be formulated as follows min X∈O (i,j)∈L ρ(r ij (X i , X j ,T ij ))(25) where O is the set of robots, L is the set of robot couples, ρ() is the kernel function, and r ij (X i , X j ,T ij ) is defined as r ij (X i , X j ,T ij ) = T ij · (X −1 j · X i ) − I 2 F(26) We use open-sourced GTSAM [30] to solve the graph optimization. IV. EXPERIMENT To show the accuracy and features of our system, we design a series of experiments with UAVs and Unmanned Ground Vehicles (UGVs). MCS and RTK are introduced as ground truth. We use the error evaluation method in [31] to demonstrate the mutual localization accuracy of our system. Our experiments contain two parts: 1). Accuracy comparison in two-robot and multi-robot scenarios. 2). Feature validation experiments. Considering the computation time firstly, the ID extraction takes less than 2 ms per image, raw relative pose estimation and ESKF iteration takes less than 1 ms, and the PGO needs around 3 ms. The selected computation platform, Intel NUC-i5, is proven to have sufficient computation resources. A. Accuracy Comparison 1) Two-Robot Mutual Localization: Experiments are implemented indoors, and MCS is used as ground truth. As shown in Fig.4, to demonstrate the robustness of our system, we conduct one manual control and multiple autonomous control experiments with two UAVs. To effectively visualize the comparison, we calculate the estimated trajectory of UAV1 by adding the relative pose estimations on the ground truth of UAV0, the observer. Experiments show that our system can consistently and stably provide relative pose estimations. We can see from 2) Multi-Robot Mutual Localization: Four robots, two UGVs and two UAVs, are prepared to prove the multi-robot mutual localization accuracy, as shown in Fig.6 (a). Different from the above experiments, we add PGO to improve multirobot mutual localization performance. Fig.6 (b) shows the estimated trajectories of other robots with respect to the UAV0 frame (as the observer). Table I shows the average Absolute Trajectory Error (ATE) [31] for n robots in the UAV0 body frame, under the condition of with and without PGO. Results show that PGO improves the accuracy of position and orientation estimation, albeit by a small amount in the clear condition. B. Feature Validation 1) Dark Scenario: As shown in Fig.7 (a), we test the system in an almost totally dark environment, which is challenging for VIO-based indirect methods. Similar to the two-UAV experiments, the observer can consistently estimate the relative pose of the other UAV as shown in Fig.7 (c). This experiment verifies that the proposed system can work in environments with low-light conditions. 2) Long-Range Scenario: As shown in Fig.7 (b), we conduct the long-range experiment outdoors with a UAV and a UGV. During the experiment, the UGV is static, and the UAV is flying under manual control. Since MCS can't be deployed in the outdoor environment, we use RTK GPS as our ground truth for comparison. Fig.7 (d) shows the system can stably estimate peer poses far from 27.5 meters, which shows better support in large areas than works [8] [22]- [24] using active LEDs (relative pose estimated within 6 meters distances). 3) Aggressive Motion Scenario: To testify our system under extreme conditions, we perform an aggressive motion experiment with large roll/pitch/yaw angular changes. The experiment is conducted via two handheld devices as it is difficult to control UAVs/UGVs to perform such large rotation angular excursions, either manually or autonomously. Two people walk around a circle (with a diameter of 4 meters) and move the two devices' attitudes randomly. As the absolute poses shown in Fig. 8(a) and Fig. 8(b), the maximum changing range for pitch angle reaches 87 degrees, for roll angle reaches 138 degrees and for yaw reaches 360 degrees. The relative poses also vary over large angles as shown in Fig. 8(c). From Fig. 8(d), we see the relative position error is below 0.4 meters with a median of 0.174 meters and the angle error is below 6 degrees with a median of 1.48 degrees. 4) Cooperative Localization in Occluded Scenario: Occlusion can not be avoided when deploying robot teams in real-world applications, which could fail the relative pose estimation between two robots as the lack of mutual visual measurements. In this situation, the PGO-based algorithm could be used to recover the relative poses through cooperative localization in certain conditions. Firstly we compare the accuracy under occluded and non-occluded conditions. We move a group of five robots in an environment with an isolated obstacle, where robot0 and robot1 happen to be occluded by the obstacle at some points as shown in the left of Fig.9 (a). The right of Fig.9 (a) shows the estimation error and we can see the error is slightly larger when occlusion happens, which is acceptable considering there's no direct measurements between robot1 and robot0. Secondly, we conduct the other experiment in a much more complex environment with many obstacles as shown in Fig.9 (b). We can see the estimated trajectories match well with ground truth trajectories. V. CONCLUSION This paper introduces CREPES, a novel, robust and accurate solution for multi-robot mutual localization. We have conducted extensive experiments to show the performance of CREPES in two-robot and multi-robot situations, even in dark or large-scale environments, with aggressive motion or under occlusion conditions. Our relative pose estimation system can achieve a median of 0.13 meters position accuracy and a median of 1.16 degrees orientation accuracy in clear conditions. Although we testify our system under occlusion conditions and show similar accuracy, we find there are much more outliers than in clear conditions. In future research, we will tackle this challenge from the perspective of multirobot cooperative optimization in continuous time domain. In addition, we will iterate the hardware to make the system smaller for robotic applications. Fig. 2 : 2The pipeline of the raw relative pose estimation algorithm. (a) We use the DS fisheye camera reprojection model to compute two-unit directional vectors B puA and A puB. (b) By using the gravity alignment, we get the expressionsBpuA andĀpuB of the two-unit directional vectors under the two intermediate frames A andB. (c) we projectBpuA andĀpuB onto the x-o-y plane and compute the angles between each projected vector and the corresponding x-axis. Fig. 3 : 3An illustration of PGO of 4 robots Fig. 4 :Fig. 5 : 45Two robots mutual localization. (a) shows our experimental scenario. (b) shows trajectories in 3D while UAVs are in manual control. Trajectories in (c)(d)(e) are shown in the top view while UAVs are programmed to fly autonomously. Boxplot of the position error (a) and orientation error (b) for experiments shown in Fig 6. Fig.5 that our system achieves high relative pose estimation accuracy. For all experiments, the median of mutual position estimation errors is in the range of 0.102 to 0.161 meters, and the median of mutual orientation errors is in the range of 0.733 to 1.517 degrees. Fig. 7 : 7Dark and Long-Range Scenarios Experiments. Fig. 8 : 8Aggressive Motion Experiment. Sub-figure (a),(b) show the ground truth orientation changes of robot0 and robot1. (c) shows the ground truth relative pose between the two robots. From (a),(b), and (c), we can see aggressive motion between robot0 and robot1. Then (d) shows the relative pose estimation errors. Fig. 9 : 9Occlusion experiment. We conduct experiments with five robots in two scenarios with different levels of occlusion. (a) shows the position and orientation errors of robot1 in robot0 frame under an isolated obstacle occlusion condition. (b) shows the remapped trajectories of robot0'neighbors into its own frame in a complex environment, compared with neighbors' ground truth trajectories. 1 https://www.open-mesh.org/projects/open-mesh/wiki/BATMANConceptgravity gravity gravity gravity � B (a) (b) (c) Hardware Coordinate System TABLE I : IAccuracy comparison of four-robot mutual localization experiment in UAV0 body framewithout with Improvement PGO PGO Traj. Lengths of UAV1, UGV0,1 (22.0, 17.7, 11.6) m AT Epos(X 0 i )/n 0.089m 0.073m 0.016m AT Erot(X 0 i )/n 0.884 • 0.879 • 0.005 • Meeting-merging-mission: A multi-robot coordinate framework for large-scale communication-limited exploration. 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{'abstract': "IR LEDs Camera UWB IMU Camera UWB IMU ID Extraction Relative Direction ID Extraction Relative Direction Fig. 1: Hardware design and software architecture of CREPES. The hardware consists of an IMU, an UWB, IR LEDs and a Camera.The Raw Relative Pose Estimation software module collects neighbors' ID (from ID Extraction), directional measurement (from Relative Direction), distance (from UWB) and IMU information to get the raw estimation. Then an ESKF module filters the result using the IMU information. When multiple neighbors are around, a PGO module is employed to further improve the performance.Abstract-Mutual localization plays a crucial role in multirobot cooperation. CREPES, a novel system that focuses on six degrees of freedom (DOF) relative pose estimation for multi-robot systems, is proposed in this paper. CREPES has a compact hardware design using active infrared (IR) LEDs, an IR fish-eye camera, an ultra-wideband (UWB) module and an inertial measurement unit (IMU). By leveraging IR light communication, the system solves data association between visual detection and UWB ranging. Ranging measurements from the UWB and directional information from the camera offer relative 3-DOF position estimation. Combining the mutual relative position with neighbors and the gravity constraints provided by IMUs, we can estimate the 6-DOF relative pose from a single frame of sensor measurements. In addition, we design an estimator based on the error-state Kalman filter (ESKF) to enhance system accuracy and robustness. When multiple neighbors are available, a Pose Graph Optimization (PGO) algorithm is applied to further improve system accuracy. We conduct enormous experiments to demonstrate CREPES' accuracy between robot pairs and a team of robots, as well as performance under challenging conditions.", 'arxivid': '2302.01036', 'author': ['Zhiren Xun \nIR LEDs\n\n', 'Jian Huang \nIR LEDs\n\n', 'Zhehan Li \nIR LEDs\n\n', 'Zhenjun Ying \nIR LEDs\n\n', 'Yingjian Wang \nIR LEDs\n\n', 'Chao Xu \nIR LEDs\n\n', 'Fei Gao \nIR LEDs\n\n', 'Yanjun Cao \nIR LEDs\n\n'], 'authoraffiliation': ['IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n', 'IR LEDs\n'], 'corpusid': 257772024, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12916, 'n_tokens_neox': 11328, 'n_words': 6909, 'pdfsha': 'c4af024a014707775b308891bcc3612843d7a629', 'pdfurls': ['https://export.arxiv.org/pdf/2302.01036v2.pdf'], 'title': ['CREPES: Cooperative RElative Pose Estimation System Robot B Robot A Other Robots', 'CREPES: Cooperative RElative Pose Estimation System Robot B Robot A Other Robots'], 'venue': []}

arxiv

STOPPING OF FUNCTIONALS WITH DISCONTINUITY AT THE BOUNDARY OF AN OPEN SET 28 Apr 2011 Jan Palczewski Łukasz Stettner STOPPING OF FUNCTIONALS WITH DISCONTINUITY AT THE BOUNDARY OF AN OPEN SET 28 Apr 2011arXiv:1006.4283v3 [math.OC]optimal stoppingFeller Markov processdiscontinuous functionalpenalty method We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O. The stopping horizon is either random, equal to the first exit from the set O, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of O. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or ε-optimal stopping times. Introduction The problem of optimal stopping of Markov processes has received continuous attention for last fourty years and produced diverse approaches for its solution. Foundations and general existence results can be found, e.g., in Bismut and Skalli [4], El Karoui [8], El Karoui et al. [9], Fakeev [10], and Mertens [16]. From 1980s functional analytic methods gave way to a more explicit approach initiated by Bensoussan and Lions [3]: value function was characterized as a solution to a variational inequality, which could be solved analytically or numerically. The main limitation of this method is the requirement of a particular differential form of the generator of the underlying Markov process. This paper belongs to another strand of literature which initially aimed at studying smoothness of the value function but also provides a different approach for the numerical approximation to the value function for a more general class of Markov processes (see Zabczyk [22] for a survey). These methods are not constrained by the form of generators and the development of the theory of PDEs. Specifically, we build on the penalty method introduced by Robin [19] and generalized by Stettner and Zabczyk [20] (see also [21]), which originates in ideas developed for partial differential equations but follows a purely probabilistic route. Of interest to numerical methods discussed in this paper is also a time-discretization technique explored by Mackevicius [14] and further applied by Kushner and Dupuis [12] for numerical algorithms; see also Palczewski and Stettner [18] for its application to stopping of time-discontinuous functionals. We assume that the state of the world is described by a standard Markov process X(t) defined on a locally compact separable space E endowed with a metric ρ with respect to which every closed ball is compact (see the Appendix for the definition and properties of standard Markov processes). The Borel σ-algebra on E is denoted by E. The process X(t) satisfies the weak Feller property: P t C 0 ⊆ C 0 , where C 0 is the space of continuous bounded functions E → R vanishing in infinity, and P t is the transition semigroup of the process X(t) , i.e., P t h(x) = E x h X(t) for any bounded measurable h : E → R. Let O ⊂ E be an open set and τ O = inf{t : X(t) O} -the first exit time from O. We study maximization of several classes of functionals: (1) Stopping is allowed up to time τ O . The payoff is described by a function G before τ O and by a function H at τ O : where the payoff function F is continuous apart from a possible discontinuity on [0, ∞) × ∂O. Optimal stopping problems of the first type were studied by Bensoussan and Lions [3] for non-degenerate diffusion processes under assumptions that G ≤ H and O is bounded with a smooth boundary ∂O. They used penalization techniques similar to ours but applied them on the level of variational inequalities. Generalizations were attempted by many authors in two directions: to extend the class of processes for which this approach applies and to relax assumptions on the functional; see, e.g., Menaldi [15] for the removal of restrictions on degeneracy of the diffusion, and Fleming, Soner [7] for relaxation of many assumptions regarding the functional and the coefficients of the diffusion via viscosity solutions approach. Functionals of the third type recently gained a lot of attention. Lamberton [13] obtained continuity and variational characterization of the value function for stopping of one-dimensional diffusions with bounded and Borelmeasurable payoff function F. His result, however, cannot be extended to multidimensional diffusions. Bassan and Ceci studied stopping of semi-continuous payoff functions F for diffusions and certain jumpdiffusions in one dimension ( [1,2]). They proved that value function for a functional with lower/upper semi-continuous function F is lower/upper semi-continuous. The existence of optimal stopping times was also shown but without an explicit construction. This paper complements existing theory in two aspects. Firstly, it provides results for a far larger family of Markov processes (in particular, in dimensions higher than 1) and enables numerical treatment of the value function. Secondly, it relaxes constraints on the region O, which can be unbounded and with nonsmooth boundary. Our main assumption is that the mapping x → E x {1 {τ O <t} h(X t )} is continuous for any t > 0 and a continuous bounded function h. This assumption is non-restrictive as we show in Section 5. It is usually satisfied by solutions to nondegenerate stochastic differential equations driven by Brownian or Levy noise. Consequently our results, based on probabilistic arguments, provide regularity of solutions to differential or integrodifferential variational inequalities, related to appropriate stopping problems, with various types of discontinuity. In our approach, the value function is approximated by a sequence of penalized value functions which are unique fixed points of contraction operators. These operators do not involve stopping or any other type of control, which makes them easier to compute numerically. Moreover, a discrete approximation of the state space can be used because we prove that the penalized functions are continuous. The remaining of the paper is organized in the following way. Section 2 introduces the penalty method for functionals of the first type. The following section explores the properties of the value function, in particular, its behaviour on the boundary ∂O. In Section 4 main results on optimal and ε-optimal stopping and the convergence of penalized value functions are obtained. Sufficient conditions for the main assumption (A1) are formulated in Section 5. Functionals of the second type are studied in Section 6. Section 7 extends these results to functionals of the third type with infinite time horizon. A finite time horizon setting is studied in Section 8. Important properties of Feller processes are listed in the Appendix. Penalty method We solve the stopping problem (1) using the penalty method introduced by Robin [19] and generalized by Stettner and Zabczyk [20]. For β > 0 consider a penalized equation (4) w β (s, x) = E x τ O 0 e −αu f s + u, X(u) + β G s + u, X(u) − w β s + u, X(u) + du + e −ατ O H s + τ O , X(τ O ) . LEMMA 2.1. Assume that g and h are bounded functions and α > 0. For any bounded progressively measurable process (b(t)), the following formulae z(s, x) = E x τ O 0 e −αu g(s + u, x(u))du + e −ατ O h(s + τ O , x(τ O )) ,(5)z(s, x) = E x τ O 0 e −αu− u 0 b(t)dt g(s + u, x(u)) + b(u)z(s + u, x(u)) du + e −ατ O − τ O 0 b(t)dt h(s + τ O , x(τ O )) (6) are equivalent in the following sense: z defined in (5) is a solution to (6); and any solution to (6) is of the form (5). Proof. We use similar arguments as in Lemma 1 of [21]. The only difference is that now we have τ O instead of the deterministic time T − s. Using this lemma, in a similar way as in Proposition 1 of [21], we show LEMMA 2.2. There is exactly one bounded measurable function w β that satisfies (4). Proof. By Lemma 2.1 the penalized function w β can be equivalently written as (7) w β (s, x) = E x τ O 0 e −(α+β)u f s + u, X(u) + β G s + u, X(u) − w β s + u, X(u) + + βw β s + u, X(u) du + e −(α+β)τ O H s + τ O , X(τ O ) . Hence, w β is a fixed point of the operator T defined for measurable bounded functions φ as follows: T φ(s, x) = E x τ O 0 e −(α+β)u f s + u, X(u) + β G s + u, X(u) − φ s + u, X(u) + + βφ s + u, X(u) du + e −(α+β)τ O H s + τ O , X(τ O ) . This operator is a contraction on the space of measurable bounded functions for any β > 0. Indeed, T φ is identically equal to H on [0, ∞) × O c , whereas the contraction property on [0, ∞) × O follows from the estimate T φ 1 − T φ 2 ≤ β α + β φ 1 − φ 2 ∞ . This implies that w β is a unique fixed point of T . We make the following assumption (A1) The stopped semigroup P τ O t h(x) = E x {1 {t<τ O } h(X(t)) } maps the space of continuous bounded functions into itself. The following three lemmas prove continuity results which, in particular, will be used to show that w β is continuous. LEMMA 2.3. Under (A1), for a continuous bounded function h : [0, ∞) × E → (−∞, ∞) the mapping (s, x) → P τ O t h(s, x) := E x {1 {t<τ O } h(s + t, X(t))} is continuous. Proof. Let (s n , x n ) → (s, x). By Proposition A.1 for a given ε > 0 there is a compact set K ⊂ E such that P x n ∃ u∈[0,s+t+1] X(u) K ≤ ε. For n large enough, i.e., such that |s − s n | ≤ 1, we have P τ O t h(s n , x n ) − P τ O t h(s, x) ≤ E x n {1 {t<τ O } h(s n + t, X(t))} − E x n {1 {t<τ O } h(s + t, X(t))} + E x n {1 {t<τ O } h(s + t, X(t))} − E x {1 {t<τ O } h(s + t, X(t))} ≤ ε h + E x n {1 {t<τ O } 1 {X(t)∈K} h(s n + t, X(t)) − h(s + t, X(t))} + E x n {1 {t<τ O } h(s + t, X(t))} − E x {1 {t<τ O } h(s + t, X(t))} = ε h + a n + b n . The sequence a n converges to 0 by uniform continuity of h is on [0, s + t + 1] × K. Assumption (A1) implies b n → 0, which completes the proof. LEMMA 2.4. Under assumption (A1) the mapping (s, x) → E x τ O 0 e −γu h(s + u, X(u))du is continuous for any function h ∈ C([0, ∞) × E) and γ > 0. Proof. Fubini's theorem implies [0, ∞) × E → (−∞, ∞) the mapping (s, x) → E x e −ατ O h(s + τ O , X(τ O )) is continuous. Proof. Assume first that (8) h(s, x) = E x ∞ 0 e −αuh s + u, X(u) du for a continuous bounded functionh. Using this decomposition we write H(s, x) = E x e −ατ O h(s + τ O , X(τ O )) = E x ∞ τ O e −αuh s + u, X(u) du = h(s, x) − E x τ O 0 e −αuh s + u, X(u) du . Hence, H is continuous by Lemma 2.4. By the weak Feller property of (X(t)) functions of the form (8) are dense in C 0 ([0, ∞) × E) (see Lemma 3.1.6 in [6]). Hence, H is continuous for h in C 0 . The extension of this result to continuous bounded functions h uses Proposition A.1 in the appendix. Fix a compact set K ⊆ E and S ≥ 0. For any T, ε > 0 there is a compact set L ⊆ E such that P x X(t) L for some t ∈ [0, T ] < ε, ∀ x ∈ K. Define r(s, x) = e −ρ(x,L)−(s−(S +T )) + h(s, x), where ρ(x, L) denotes the distance of x from the set L. Such r is in C 0 ([0, ∞) × E) and by preceding results R(s, x) = E x e −ατ O r(s + τ O , X(τ O )) is continuous. By definition r ≡ h on [0, S + T ] × L. Let A = {X(t) L for some t ∈ [0, T ]}. The distance of R and H is bounded in the following way: To establish convergence of w β to w as β → ∞ we introduce two additional representations of w β . H(s, x) − R(s, x) = E x e −ατ O (h − r)(s + τ O , X(τ O )) = E x e −ατ O ∧T 1 {A c } (h − r)(s + τ O ∧ T, X(τ O ∧ T )) + E x 1 {A} e −ατ O (h − r)(s + τ O , X(τ O )) + E x 1 {A c } e −ατ O (h − r)(s + τ O , X(τ O )) − e −ατ O ∧T (h − r)(s + τ O ∧ T, X(τ O ∧ T )) ≤ 0 + h − r ε + e −αT 2 h − r . This implies H(s, x) − R(s, x) ≤ 2 h (ε + 2e −αT ) for (s, x) ∈ [0, S ] × K. Since T LEMMA 2.7. Under assumption (A1), the function w β has the following equivalent representation: (9) w β (s, x) = sup τ J(s, x, τ) − E x 1 {τ<τ O } e −ατ G − w β + s + τ, X(τ) . Proof. Markov property implies that for any stopping time σ the following equality is satisfied: w β (s, x) = E x τ O ∧σ 0 e −αu f s + u, X(u) + β G s + u, X(u) − w β s + u, X(u) + du + e −α(τ O ∧σ) w β (s + τ O ∧ σ, X(τ O ∧ σ)) . This gives the lower bound: (10) w β (s, x) ≥ E x τ O ∧σ 0 e −αu f s + u, X(u) du + e −α(τ O ∧σ) w β (s + τ O ∧ σ, X(τ O ∧ σ)) . Further, (11) w β (s, x) ≥ E x τ O ∧σ 0 e −αu f s + u, X(u) du + 1 {σ<τ O } e −ασ G − (G − w β ) + s + σ, X(σ) + 1 {σ≥τ O } e −ατ O H(s + τ O , X(τ O )) , because w β = H on O c and G − (G − w β ) + ≤ w β . Define a stopping time σ * = inf{u ≥ 0 : w β (s + u, X(u)) ≤ G(s + u, X(u))}. Due to the continuity of G and w β (see Corollary 2.6) we have 1 {σ * <τ O } w β (s + σ * , X(σ * )) ≤ 1 {σ * <τ O } G(s + σ * , X(σ * )). This implies that for σ * the inequalities in (10) and (11) become equalities and (9) follows easily. LEMMA 2.8. The function w β has the following equivalent representation: (12) w β (s, x) = sup b∈M β E x τ O 0 e −αu− u 0 b(t)dt f s + u, X(u) + b(u)G s + u, X(u) du + e −ατ O − τ O 0 b(t)dt H s + τ O , X(τ O ) , where M β is the class of progressively measurable processes with values in [0, β]. Proof. By Lemma 2.1 the function w β has the following equivalent formulation: w β (s, x) = E x τ O 0 e −αu− u 0 b(t)dt f s + u, X(u) + β G s + u, X(u) − w β s + u, X(u) + + b(u)w β (s + u, x(u)) du + e −ατ O − τ O 0 b(t)dt h(s + τ O , x(τ O )) for any progressively measurable process b(t) with values in [0, β]. Since b(t) ≤ β we have β G s + u, X(u) − w β s + u, X(u) + + b(u)w β (s + u, X(u)) ≥ b(u)G s + u, X(u) , which implies w β (s, x) ≥ E x τ O 0 e −αu− u 0 b(t)dt f s + u, X(u) + b(u)G s + u, X(u) du + e −ατ O − τ O 0 b(t)dt h(s + τ O , X(τ O )) . This is an equality for b(t) given by b(t) =        β, G s + t, X(t) ≥ w β s + t, X(t) , 0, otherwise. Hence, formula (12) is proved. Proof. Equation (12) implies that the functions w β (s, x) are increasing in β. Hence the limit w ∞ (s, x) = lim β→∞ w β (s, x) exists. By (9) we have w β ≤ w and, therefore, w ∞ ≤ w. To prove that w ∞ = w we first show that w ∞ ≥ G. Let x ∈ O and, for η > 0, put b η (u) = 1 {u≤η} β. Then by (12) we have w β (s, x) ≥ E x τ O 0 e −αu− u 0 b η (t)dt f s + u, X(u) du + τ O ∧η 0 e −(α+β)u βG s + u, X(u) du + e −ατ O − τ O 0 b η (t)dt H τ O , X(τ O ) = E x (I) + (II) + (III) . Letting β → ∞ we can make (I) and (III) arbitrarily small and for sufficiently small η and large β the term (II) is arbitrarily close to G(s, x). Dominated Convergence Theorem implies w ∞ (s, x) ≥ G(s, x). From (9) for any stopping time τ we have w β (s, x) ≥ J(s, x, τ) − E x 1 {τ<τ O } e −ατ G − w β + s + τ, X(τ) . By letting β → ∞ we obtain Proof. The semicontinuity of w follows from Corollary 2.6 and Proposition 2.9. Dini's theorem implies uniform convergence on compact sets if w is continuous. (13) w ∞ (s, x) ≥ J(s, x, τ), because lim β→∞ (G − w β ) + (s, x) = 0 for x ∈ O. Since τ is arbitrary we conclude that w ∞ (s, x) = w(s, x) for x ∈ O. For x ∈ E \ O we have w β (s, x) = H(s, x) = w(s, x). Properties of the value function w In this section we explore the properties of the value function, in particular, its behaviour on the boundary of O. THEOREM 3.1. Under (A1), for x ∈ ∂O we have (14) lim y→x,y∈O w(s, y) = G ∨ H(s, x). The proof of this theorem consists of several steps which are of interest on their own. They are formulated and proved as separate results below. It is clear that w ≥ G on O and w = H on the complement of O. It is therefore natural to expect a discontinuity at the boundary of O if G > H. The following proposition shows that this discontinuity is constrained to the minimum: the absolute value of the difference between G and H. PROPOSITION 3.2. Assume (A1) and G ≥ H. For any x ∈ ∂O we have (15) lim y→x, y∈O w(s, y) = G(s, x) and the convergence is uniform in s and x from compact sets. Proof. Since w(s, x) ≥ G(s, x) for x ∈ O and G is continuous we obtain that lim inf y→x, y∈O w(s, y) ≥ G(s, x). In the remaining part of the proof we show that lim sup y→x, y∈O w(s, y) ≤ G(s, x), which implies that the limit in (15) exists and equals G(s, x). Fix a compact set K ⊆ E, T > 0 and ε > 0. First we make preparatory steps. By Proposition A.1 in the Appendix, there is a compact set L ⊆ E such that (16) sup x∈K P x ∃ t ∈ [0, T + 1] X(t) L ≤ ε. The extension of the time interval by one unit to [0, T + 1] is required to allow the initial time s to be in [0, T ] and leave time for the process (X(t)) to evolve. Notice that below δ and η are both bounded by 1. Let δ ∈ (0, 1) be such that for (s, x) ∈ [0, T ] × B(L, δ), y ∈ L, x − y ≤ δ and t ∈ [0, δ] (17) |G(s, x) − G(s + t, y)| ≤ ε, Proposition A.3 implies that there is η > 0, which, for convenience, is bounded by δ ∧ ε, such that (18) sup x∈L sup t≤η P x X(t) B(x, δ) ≤ ε. Fix x ∈ ∂O ∩ K and s ∈ [0, T ]. For any y ∈ O ∩ K we have w(s, y) = sup τ J(s, y, τ) ≤ sup τ E y τ∧τ O 0 e −αu f (s + u, X(u))su + e −α(τ∧τ O ) G(s + τ ∧ τ O , X(τ ∧ τ O )) ≤ P y (τ O > η) f α + G + P y (τ O ≤ η) η f + G(s, y) + sup τ E y 1 {τ O ≤η} e −α(τ∧τ O ) G(s + τ ∧ τ O , X(τ ∧ τ O )) − G(s, y) . Consider the last term. For any stopping time τ we have E y 1 {τ O ≤η} e −α(τ∧τ O ) G(s + τ ∧ τ O , X(τ ∧ τ O )) − G(s, x) ≤ G (1 − e −αη ) + E y G(s + τ ∧ τ O ∧ η, X(τ ∧ τ O ∧ η)) − G(s, x) ≤ αη G + E y G(s + τ ∧ τ O ∧ η, X(τ ∧ τ O ∧ η)) − G(s + η, X(η)) + E y G(s + η, X(η)) − G(s, y) = (I) + (II) + (III). The first term is bounded by αε G . The estimate of the second term requires conditioning on X(τ∧τ O ∧η), the use of the strong Markov property and inequalities (16), (18): E y G(s + τ ∧ τ O ∧ η, X(τ ∧ τ O ∧ η)) − G(s + η, X(η)) = E y E X(τ∧τ O ∧η) G(s + τ ∧ τ O ∧ η, X(0)) − G(s + η, X(η − τ ∧ τ O ∧ η)) ≤ 2 G P y {∃s ∈ [0, η] X(s) L} + 2 G P y ∀s ∈ [0, η] X(s) ∈ L and X(η) B X(τ ∧ τ O ∧ η), δ + ε ≤ 2 G ε + 2 G ε + ε = ε(1 + 4 G ). Term (III) is estimated similarly knowing that y is in L by assumption: (III) ≤ ε(1 + 2 G ). Combining these estimates we obtain w(s, y) ≤ h η (y) f α + G + (1 − h η (y)) η f + G(s, y) + ε 2 + (6 + α) G ≤ G(s, y) + h η (y) f α + 2 G + η f + ε 2 + (6 + α) G , where h η (y) = P y {τ O > η). Assumption (A1) implies that h η is continuous on E. Clearly, h η (x) = 0. Hence, lim sup y→x, y∈O w(s, y) ≤ G(s, x) + η f + ε 2 + (6 + α) G and the limit in the right-hand side is uniform in (s, x) ∈ [0, T ] × (∂O ∩ K). Since ε > 0 is arbitrary and η < ε this implies (15). COROLLARY 3.3. Under (A1), for any x ∈ O we have (19) lim sup y→x,y∈O w(s, y) ≤ G ∨ H(s, x) Proof. Notice that w(s, y) ≤ sup τ E y τ∧τ O 0 e −αu f (s + u, X(u))du + e −α(τ∧τ O ) G ∨ H(s + τ ∧ τ O , X(τ ∧ τ O )) and then continue as in the proof of Proposition 3.2 replacing G with G ∨ H. The following proposition explores the impact of the value of the functional on the complement of O on the value function close to the boundary of O. PROPOSITION 3.4. Assume (A1). For each ε > 0, T > 0 and a compact set K ⊆ E there is a compact set K ε ⊂ O such that for x ∈ K \ K ε , s ∈ [0, T ] and β > 0 we have (20) w β (s, x) ≥ H(s, x) − ε. Proof. Fix ε ′ > 0 and choose η > 0 such that (16)-(18) hold for the function H. By the definition of w β we have w β (s, x) ≥ − f η − f α P x {τ O > η} + E x e −ατ O H(s + τ O , X(τ O )) . Splitting the last term depending on whether τ O is greater or smaller than η and doing analogous estimates as in the proof of Proposition 3.2 we obtain the following lower bound w β (s, x) ≥ (1 − h η (x))H(s, x) − h η (x) f α + H − ε ′ (2 + 6 H + f ). By arbitrariness of ε ′ > 0 and continuity of h η we can choose K ε such that (20) is satisfied. According to Proposition 3.4, Assumption (A1) guarantees the "migration" of H into O, i.e., the function H provides a lower bound for w β when x approaches ∂O. As w β is the lower bound for w (see Proposition 2.9) this property is shared by the value function w. In particular, when H ≥ G the value function smoothly rises to the upper level H on the boundary of O. Proof of Theorem 3.1. From (20), letting first β → ∞ and then ε → 0 we obtain lim inf y→x,y∈O w(s, x) ≥ H(s, x). Since G is continuous and w(s, y) ≥ G(s, y) on O this extends to lim inf y→x,y∈O w(s, x) ≥ G ∨ H(s, x). Corollary 3.3 and the above inequality imply that the limit in (14) exists and equals G ∨ H. Continuity of w and existence of optimal stopping times Let D denote the set of functions ϕ(s, x) admitting the following decomposition: (21) ϕ(s, x) = E x τ O 0 e −αu ϕ 1 (s + u, X(u))du + e −ατ O ϕ 2 (s + τ O , X(τ O )) for ϕ 1 , ϕ 2 ∈ C 0 ([0, ∞) × E). LEMMA 4.1. The set D is a dense subset of C 0 ([0, ∞) × E). Proof. It follows immediately from the proof of Lemma 4 in [21]. LEMMA 4.2. If G has decomposition (21) with ϕ 1 = g 1 , ϕ 2 = g 2 ∈ C 0 ([0, ∞) × E) then w β (s, x) − G(s, x) ≥ − f − g 1 α + β − E x e −(α+β)τ O H − g 2 . If, moreover, G ≤ H then w β (s, x) − G(s, x) ≥ − f − g 1 α + β . Proof. Definew β (s, x) = w β (s, x) − G(s, x). Decomposition (21) of G and representation (4) of w β implȳ w β (s, x) = E x τ O 0 e −αu f − g 1 + β(w β ) − s + u, X(u) du + e −ατ O (H − g 2 ) s + τ O , X(τ O ) . By Lemma 2.1 we have the following equivalent form of the above equation w β (s, x) = E x τ O 0 e −(α+β)u f − g 1 + β(w β ) − + βw β s + u, X(u) du + e −(α+β)τ O (H − g 2 ) s + τ O , X(τ O ) . Since (w β ) − +w β ≥ 0, we obtain w β (s, x) ≥ E x τ O 0 e −(α+β)u ( f − g 1 ) s + u, X(u) du + e −(α+β)τ O (H − g 2 ) s + τ O , X(τ O ) . This implies the first statement of the lemma. Due to decomposition (21) we have g 2 (s, x) = G(s, x) for x O. Together with the condition G ≤ H this yields (H − g 2 ) s + τ O , X(τ O ) ≥ 0. THEOREM 4.3. Assume (A1) and G ≤ H. The value function w is continuous on E and an optimal stopping moment is given by (22) τ * (s) = inf{t ≥ 0 : w(s + t, X(t)) ≤ G(s + t, X(t)) or X(t) O}. Proof. Functions w β are continuous (by Lemma 2.2), increasing in β and dominated by w. Therefore, it suffices to estimate the difference w − w β . For functions G with decomposition (21), Lemma 4.2 and equation (9) give w β (s, x) ≥ w(s, x) − f − g 1 α + β . Since D is dense in C 0 ([0, ∞) × E) (Lemma 4.1) we also obtain the continuity of w for G ∈ C 0 ([0, ∞) × E). The extension of this result to continuous bounded G uses Proposition A.1 in the appendix. Fix a compact set K ⊆ E and S ≥ 0. For any T, ε > 0 there is a compact set L ⊆ E such that P x X(t) L for some t ∈ [0, T ] < ε, x ∈ K. DefineG(s, x) = e −ρ(x,L)−(s−(S +T )) + G(s, x), where ρ(x, L) denotes the distance of x from the set L. Letw be the value function corresponding toG. SinceG ∈ C 0 ([0, ∞) × E), preceding results imply thatw is continuous. We also have w(s, x) −w(s, x) ≤ (e −αT + ε) f /α + G + H ) for x ∈ K and s ∈ [0, S ]. Since T and ε are arbitrary this implies continuity of w on [0, S ] × K. By the arbitrariness of S , K the value function w is continuous on its whole domain. Define for ε > 0 (23) τ ε (s) = inf{t ≥ 0 : w(s + t, X(t)) ≤ G(s + t, X(t)) + ε or X(t) O} and (24) τ β (s) = inf{t ≥ 0 : w β (s + t, X(t)) ≤ G(s + t, X(t)) or X(t) O}. Fix δ > 0 and T > 0. By Proposition A.1 for a given x ∈ E there is a compact set K δ such that P (4), due to the Markov property of (X(t)), we obtain x {A δ } ≥ 1−δ, where A δ = {X(t) ∈ K δ ∀t ∈ [0, T ]}. Fromw β (s, x) ≤ P x {A c δ } + e −αT P x {A δ and τ β (s) ∨ τ ε (s) > T } G + H + f α + E x 1 {A δ } σ * ε,β,T (s) 0 e −αu f + β(G − w β ) + s + u, X(u) du + 1 {A δ } e −ασ * ε,β,T (s) w β s + σ * ε,β,T (s), X(σ * ε,β,T (s)) , where σ * ε,β,T (s) = τ β (s) ∧ τ ε (s) ∧ τ O ∧ T . Notice that by the uniform convergence on compact subsets of w β to w we have P x τ β (s) ∧ T < τ ε (s) ∧ T and A δ → 0 as β → ∞. Therefore letting β → ∞ we obtain by Dominated Convergence Theorem w(s, x) ≤ δ + e −αT G + H + f α + E x 1 {A δ } τ ε (s)∧τ O ∧T 0 e −αu f s + u, X(u) du + 1 {A δ } e −ατ ε (s)∧τ O ∧T w s + τ ε (s) ∧ τ O ∧ T, X(τ ε (s) ∧ τ O ∧ T ) . Proposition A.1 implies that (A δ ) form an increasing sequence of subsets when δ → 0 and lim δ→0 P x {A δ } = 1. Therefore, letting δ → 0 we get w(s, x) ≤ e −αT G + H + f α + E x τ ε (s)∧τ O ∧T 0 e −αu f s + u, X(u) du + e −ατ ε (s)∧τ O ∧T w s + τ ε (s) ∧ τ O ∧ T, X(τ ε (s) ∧ τ O ∧ T ) . Now taking the limit T → ∞ yields (25) w(s, x) ≤ E x τ ε (s)∧τ O 0 e −αu f s + u, X(u) du + e −ατ ε (s)∧τ O w s + τ ε (s) ∧ τ O , X(τ ε (s) ∧ τ O ) . Note that τ ε (s) → τ * (s), as ε → 0, and, further, by quasi-leftcontinuity of the process (X(t)) (see, e.g., [6]) we also have X(τ ε (s)) → X(τ * (s)), P x -a.s.. Consequently letting ε → 0 in (25) and using the continuity of w give w(s, x) ≤ E x τ * (s)∧τ O 0 e −αu f s + u, X(u) du + e −ατ * (s)∧τ O w s + τ * (s) ∧ τ O , X(τ * (s) ∧ τ O ) . This implies that there is an optimal stopping time dominating τ * (s). The optimality of τ * (s) is now obvious. To prove the continuity of the value function w without the requirement of an upward jump we introduce the following assumptions: Proof. Fix a compact set K ⊆ E and T, ε > 0. By Proposition A.1 there is a compact set L ⊆ E such that sup x∈K P x {X(h) L} < ε. Hence for (s, x) ∈ [0, T ] × K we have E x F s + h, X(h) − Φ(s, x) < F ε, where Φ(s, x) = E x 1 {X(h)∈L} F s + h, X(h) . Let (s n , x n ) → (s, x) such that (s n , x n ) ∈ [0, T ] × K for all n. By the continuity of F in s and by assumption (A3), for sufficiently large k, we have lim n→∞ Φ(s n , x n ) − Φ(s, x) ≤ lim n→∞ Φ(s k , x n ) − Φ(s, x n ) + lim n→∞ Φ(s, x n ) − Φ(s, x) = ε + 0. By the arbitrariness of ε this completes the proof. LEMMA 4.5. The mapping s → w(s, x) is continuous uniformly in x in compact subsets of E. Proof. Assume that s n → s and fix a compact set K ⊆ E. Since functions f , G and H are bounded and the discount rate α > 0, for any ε > 0 there is T > 0 such that |J(s n , x, τ) − J(s n , x, τ ∧ T )| ≤ ε for all x ∈ K and n = 1, 2, . . .. By Proposition A.1 there is a compact set L ⊆ E such that for all x ∈ K and τ ≤ T we have E x 1 {∃ t∈[0,T ] X(t) L} τ∧τ O 0 e −αu f s + u, X(u) du + 1 {τ<τ O } e −ατ G s + τ, X(τ) + 1 {τ≥τ O } e −ατ O H s + τ O , X(τ O ) ≤ ε. Uniform continuity of the functions f , G, and H on [0, T ] × L yields ρ(x, X(s)) ≥ ε = 0 uniformly in x from compact sets. E x 1 {∀ t∈[0,T ] X(t)∈L} τ∧τ O 0 e −αu f s + u, X(u) − f s n + u, X(u) du + 1 {τ<τ O } e −ατ G s + τ, X(τ) − G s n + τ, X(τ) + 1 {τ≥τ O } e −ατ O H s + τ O , X(τ O ) − H s n + τ O , X(τ O ) ≤ ε for τ ≤ T , a sufficiently large n and x ∈ K. Consequently w(s n , x) → w(s, x) as n → ∞ uniformly in x ∈ K. Such assumption is satisfied for a wide variety of Markov processes which are solutions to the stochastic differential equations with Levy noise with bounded coefficients. To prove this we simply use the Doob's maximal inequality to the martingale terms in the stochastic differential equation (see e.g. Theorem 1.3.8(iv) in [11]). Let for h > 0 (26) w h (s, x) = E x h 0 e −αu f s + u, X(u) du + e −αh w(s + h, X(h)) .τ ε (s) = inf{t ≥ 0 : w(s + t, X(t)) ≤ G(s + t, X(t)) + ε or X(t) O}. Proof. By Lemmas 4.4 and 4.5 the function w h is continuous in (s, x). Let τ h O = inf{t ≥ h : X(t) O} and J h (s, x, τ) = E x τ∧τ h O 0 e −αu f s + u, X(u) du + 1 {τ<τ h O } e −ατ G s + τ, X(τ) + 1 {τ≥τ h O } e −ατ h O H s + τ O , X(τ h O ) . By Theorem 3b of [16] applied to the Markov process consisting of a pair (s + t, X(t)) we have w h (s, x) = sup τ≥h J h (s, x, τ). Consider an auxiliary value functionw h (s, x) = sup τ≥h J(s, x, τ). We have the following inequalities ≤ f h + 2( G + H ) P x {τ O < h} + sup τ≤h E x {G(s + τ, X(τ))} − E x {G(s + h, X(h))}. It suffices to prove that as h → 0 imply their uniform convergence to w on compact sets. In an identical way as in Theorem 4.3 we prove that τ ε (s) is well-defined and ε-optimal (this last assertion follows directly from (25)). H(s, x) for some x ∈ ∂O) the function w has a discontinuity in this point. This follows from the observation that w ≥ G on the set [0, ∞) × O and w = H on [0, ∞) × O c . Therefore, the statement of the above theorem cannot be strengthened. This also implies that an optimal stopping time might not exist as the following example shows. EXAMPLE 4.9. Let E = R and X(t) be a Brownian motion. Take O = (−∞, 1) and α < 1/2. It is easy to see that assumptions (A1)-(A3) are satisfied. Put G(s, x) = min(e x , e) and H(s, x) = f (s, x) = 0. Notice that these functions do not depend on s, which implies that the value function is also time-independent. We shall, therefore, skip s in the notation. (1) for x < 1 and t ≥ 0 we have (29) sup τ≤h E x {G(s + τ, X(τ))} − E x {G(s + h, X(h))} →(30) l(t, x) := E x e −αt 1 {X(t)<1} e X(t) = e ( 1 2 −α)t+x Φ 1 − x − t √ t , where Φ is the standard normal cumulative distribution function, (2) w(x) ≥ l(t, x), for x < 1 and t ≥ 0, (3) w(x) > G(x), for x < 1. Sketch of the proof. The formula (30) can be calculated directly using the normality of X(t). To prove (2), define a sequence of stopping times τ n = inf{t : X(t) ≥ 1 − 1/n}. Clearly, w(x) ≥ E x e −α(τ n ∧t) e X(τ n ∧t) and lim n→∞ E x e −α(τ n ∧t) e X(τ n ∧t) ≥ l(t, x). The proof of the last assertion rests on the observation that G(x) = l(0, x) and ∂ ∂t l(0, x) > 0 for x < 1. Assume that there exists an optimal stopping moment τ * for some x * < 1, i.e., w(x * ) = E x * {e −ατ * 1 {τ * <τ O } G(X(τ * ))}. From the strong Markov property of the process X(t) we infer that G(X(τ * )) = w(X(τ * )), P x * -a.s. on {τ * < τ O }. Since w(x * ) ≥ e x * we have P x * (τ * < τ O ) > 0. This is a contradiction with assertion (3) of Lemma 4.10. REMARK 4.11. Penalty method offers a numerical procedure for solution of optimal stopping problems. Lemma 2.7 provides an estimate of the error: w − w β ≤ (G − w β ) + . This error decreases as β increases: by Proposition 2.9 w β forms a non-decreasing sequence of functions converging to w. Under (A1) functions w β are continuous (c.f. Corollary 2.6). Theorems 4.3 and 4.8 state assumptions under which w is continuous and is approximated by w β uniformly on compact sets. The continuity of w and w β implies that state space discretization methods can be safely applied. Following Lemma 2.2 function w β can be computed as a fixed point of a contraction operator T given by T φ(s, x) = E x τ O 0 e −(α+β)u f s + u, X(u) + β G s + u, X(u) − φ s + u, X(u) + + βφ s + u, X(u) du + e −(α+β)τ O H s + τ O , X(τ O ) for a bounded measurable function φ. This operator can be implemented via PDE or Kushner-Dupuis space-time discretization approach (see [12]). The fixed point is approximated by an iterative procedure with an exponential decrease of the error (due to the contraction property of T ). Sufficient conditions for (A1) Define for η > 0 h η (x) = P x {τ O > η}. Consider the following assumption: (A4) lim x→∂O, x∈O h η (x) = 0. This assumption ensures that when approaching the boundary of O the probability of crossing it in a short time converges to 1. It is clearly satisfied (by Chebyshev inequality) whenever the mapping x → E x {τ O } is continuous. It can be viewed as a complementary assumption to (A2). We will show that (A2)-(A4) imply (A1) and (A1) is sufficient for (A4). LEMMA 5.1. The function h η is continuous on E under assumptions (A2)-(A4). Proof. For δ ∈ (0, η) define r δ (x) = E x {h η−δ (X(δ))}. This function is continuous by (A3). The difference between r δ and h η can be bounded in the following way: 0 ≤ r δ (x) − h η (x) ≤ P x {τ O < δ}. Assumption (A2) states that the right-hand side of the above inequality converges to 0 as δ → 0 uniformly in x from compact subsets of O. Hence, h η is continuous in O. It is identically zero on E \ O. These two pieces fit continuously at the boundary of O because, due to (A4), h η (x) converges to 0 as x approaches the boundary of O. The continuity of h η implies uniformity of the limit in assumption (A4), which is formalized in the following corollary. COROLLARY 5.2. If h η is continuous then for any compact set L ⊆ E and constants η, ε > 0 there is an open set L η,ε such that L η,ε ⊂ O and for each x ∈ L \ L η,ε we have P x (τ O > η) ≤ ε. PROPOSITION 5.3. Under (A2)-(A3) the mapping x → P τ O t h(x) is continuous on E \ ∂O for any bounded measurable function h and t > 0. If additionally (A4) holds then P τ O t maps the space of bounded measurable functions into the space of continuous bounded functions and as a result condition (A1) is satisfied. Proof. Let h be a bounded measurable function. By the strong Feller property (A3), for s < t, the mapping x → E x E X(s) 1 {t−s<τ O } h(X(t − s)) is continuous. Furthermore, (31) E x {1 {t<τ O } h(X(t))} = E x 1 {s<τ O } E X(s) 1 {t−s<τ O } h(X(t − s)) = E x E X(s) 1 {t−s<τ O } h(X(t − s)) − E x 1 {τ O ≤s} E X(s) 1 {t−s<τ O } h(X(t − s)) . Therefore E x {1 {t<τ O } h(X(t))} − E x E X(s) 1 {t−s<τ O } h(X(t − s)) ≤ h P x {τ O ≤ s} ≤ h P x {τ O < 2s} → 0 uniformly on compact subsets of O as s → 0 by (A2). This shows the continuity of x → P τ O t h(x) for x ∈ O. For x in E \ O we clearly have E x {1 {t<τ O } h(X(t))} = 0. Now we prove continuous fit at the boundary of O. Assumption (A4) implies that |E x {1 {t<τ O } h(X(t))}| ≤ P x {t < τ O } h decreases to 0 as x approaches the boundary. Hence, P τ O t h is continuous on E. Proposition 5.3 states that (A2)-(A4) are sufficient for Assumption (A1). The following lemma shows that (A1) implies (A4). Recall that (A1) also implies (A2), see Lemma 4.6. LEMMA 5.4. Under (A1) the function h η is continuous on E, which, in particular, implies (A4). Proof. Follows from the identity h η = P τ O η 1, where 1 denotes a function identically equal 1. Stopping with discontinuities on O c In this section we explore a stopping problem with a more general payoff function F: J(s, x, τ) = E x τ∧τ O 0 e −αu f s + u, X(u) du + e −α(τ∧τ O ) F s + (τ ∧ τ O ), X(τ ∧ τ O ) , where f, F are measurable bounded functions that are continuous in s uniformly in x from compact sets and F is continuous on [0, ∞) × O. In particular, F can be of the form F(s, x) = 1 {x∈Ō} G(s, x) + 1 {x Ō } H(s, x), where G, H are continuous bounded functions. This is a complementary problem to the one described in preceding sections: a discontinuity of the payoff manifests itself only when the process (X(t)) jumps toŌ c at the time τ O . For a continuous process (X(t)) the form of F outside ofŌ is irrelevant and the problem simplifies to stopping with a continuous payoff function G. However, if (X(t)) jumps at τ O , the process migrates to the setŌ c and the value of the functional is given by H. Define a value function w(s, x) = sup τ J(s, x, τ). PROPOSITION 6.1. Under (A2) and (A3), the function w is continuous in O. Proof. As in Lemma 4.5, using continuity of s → f (s, x), F(s, x) uniform in x from compact sets we obtain that s → w(s, x) is continuous uniformly in x from compact sets. The rest of the proof follows similar lines as the proof of Theorem 4.8. Define a penalized equation (c.f. equation (4)): w β (s, x) = E x τ O 0 e −αu f + β(F − w β ) + s + u, X(u) du + e −ατ O F s + τ O , X(τ O ) . As previously, this function is a fixed point of a contraction operator (see the proof of Lemma 2.2). To establish the convergence of w β to w, we need the following technical lemma: LEMMA 6.2. Under (A1) and (A3) the mapping (s, x) → E x {F(s + τ O , X(τ O ))} is continuous in [0, ∞) × O for any bounded measurable function F that is continuous in s uniformly in x from compact sets. Proof. Lemma 4.4 implies that Φ h (s, x) = E x E X(h) F(s + h + τ O , X(τ O )) is continuous for h > 0. Note that E x {F(s + τ O , X(τ O ))} = E x E X(h) {F(s + h + τ O , X(τ O ))} + 1 {τ O <h} F(s + τ O , X(τ O )) − E X(h) {F(s + h + τ O , X(τ O ))} . Hence E x {F(s + τ O , X(τ O ))} − Φ h (s, x) ≤ 2 F P x {τ O < h}. The right-hand side converges to 0 uniformly in x from compact subsets of O, as h → 0, by virtue of Lemma 4.6. w β (s, x) = sup τ J(s, x, τ) − E x 1 {τ<τ O } e −ατ F − w β + s + τ, X(τ) . Proceeding as in Proposition 2.9 we prove the pointwise convergence of w β to w. By Proposition 6.3, functions w β are non-decreasing in β, which implies, by Dini's theorem, uniform convergence on compact sets of [0, ∞) × O. Infinite time horizon Consider an optimal stopping problem with infinite horizon (32) w ∞ (s, x) = sup τ J ∞ (s, x, τ), where (33) J ∞ (s, x, τ) = E x τ 0 e −αu f s + u, X(u) du + e −ατ F s + τ, X(τ) . Assume the process X(t) satisfies the strong Feller property (A3), α > 0 and functions f, F are measurable bounded and continuous in s uniformly in x from compact sets. The penalized equation has the following form: for β ≥ 0 (34) w β,∞ (s, x) = E x ∞ 0 e −αu f + β(F − w β,∞ ) + s + u, X(u) du .(35) w β,∞ (s, x) = sup b∈M β E x ∞ 0 e −αu− u 0 b(t)dt f s + u, X(u) + b(u)F s + u, X(u) du , where M β is the class of progressively measurable processes with values in [0, β]. Proof. (1) Similarly as in Lemma 2.2 the function w β,∞ is a fixed point of the operator T ∞ φ(s, x) = E x ∞ 0 e −(α+β)u f + βφ + β(F − φ) + s + u, X(u) du . This operator is a contraction on the space of measurable bounded functions, which implies that w β,∞ is a unique fixed point of T ∞ on this space. Lemma 4.4 implies that T ∞ maps the space of measurable bounded functions into the space of continuous bounded functions. Hence, w β,∞ is continuous. (2) The proof is similar to that of Lemma 4.5. We use the continuity of s → f (s, x), F(s, x) uniform in x from compact sets. (3) This assertion follows immediately from Lemma 2.8 with O = E. LEMMA 7.2. Assume there is A ⊆ E such that for x ∈ A F(s, x) = R α φ(s, x) := E x ∞ 0 e −αu φ(s + u, X(u))du , where φ : [0, ∞) × A → R is measurable and bounded. Then (F − w β,∞ ) + (s, x) ≤ f − φ E x ∞ 0 e −αu−β u 0 1 {X(t)∈A} dt du , (s, x) ∈ [0, ∞) × A. Proof. First notice that for any bounded measurable function ζ the following representations are equivalent (36) v(s, x) = E x ∞ 0 e −αu ζ s + u, X(u) du , v(s, x) = E x ∞ 0 e −αu− u 0 b(t)dt ζ s + u, X(u) + b(u)v s + u, X(u) du for any bounded progressively measurable process b(t) (compare to Lemma 2.1 with O = E). Define (37)ŵ β,∞ (s, x) = E x ∞ 0 e −αu f + β(w β,∞ ) − − φ s + u, X(u) du , wherew β,∞ = w β,∞ − F. Notice thatŵ β,∞ coincides withw β,∞ on [0, ∞) × A. Applying equivalence (36) for ζ = f + β(w β,∞ ) − − φ and b(u) = β1 {X(u)∈A} yieldŝ w β,∞ (s, x) = E x ∞ 0 e −αu−β u 0 1 {X(t)∈A} dt f + β(w β,∞ ) − − φ s + u, X(u) + β1 {X(u)∈A}ŵ β,∞ s + u, X(u) du . Since (w β,∞ ) − +ŵ β,∞ ≥ 0 on [0, ∞) × A we havê w β,∞ (s, x) ≥ E x ∞ 0 e −αu−β u 0 1 {X(t)∈A} dt f − φ s + u, X(u) du ≥ − f − φ E x ∞ 0 e −αu−β u 0 1 {X(t)∈A} dt du . This completes the proof sinceŵ β, ∞ = w β,∞ − F on [0, ∞) × A. We impose the following assumptions (38) w β,∞ (s, x) ≥ sup τ J ∞ (s, x, τ) − E x e −ατ F − w β,∞ + s + τ, X(τ) , and if F ≥ G ∨ H on ∂O, i.e., F is upper semicontinuous, then w β,∞ has the following equivalent representation: (39) w β,∞ (s, x) = sup τ J ∞ (s, x, τ) − E x e −ατ F − w β,∞ + s + τ, X(τ) . Proof. We follow the proof of Lemma 2.7. For any stopping time τ we have w β,∞ (s, x) = E x τ 0 e −αu f + β(F − w β,∞ ) + (s + u, X(u))du + e −ατ w β,∞ (s + τ, X(τ) . Since w β,∞ ≥ F − (F − w β,∞ ) + we obtain (38). Let σ = inf{u : w β,∞ (s + u, X(u)) ≤ F(s + u, X(u))}. On the set {σ < ∞} the upper semicontinuity of F and the continuity of w β,∞ implies w β,∞ (s + σ, X(σ)) ≤ F(s + σ, X(σ)). Combining this with a trivial result on the set {σ = ∞} yields w β,∞ (s, x) = J(s, x, σ), E x e −ασ F − w β,∞ + s + σ, X(σ) = 0. This, together with (38), implies representation (39). In what follows we shall need the following two assumptions: (A5) For any x ∈ ∂O we have lim ε→0 σ ε = 0 P x -a.s., and lim ε→0 σ c ε = 0 P x -a.s., where σ ε = inf{u ≥ 0 : X(u) ∈ E \ (O ∪ Γ ε )}, σ c ε = inf{u ≥ 0 : X(u) ∈ E \ (O c ∪ Γ ε )}, and Γ ε is the ε-neighbourhood of ∂O: Γ ε = {x ∈ E : inf y∈∂O x − y < ε}. (A6) P x {X(T ) ∈ ∂O} = 0 for any x ∈ E and T > 0. Take x ∈ ∂O. Its regularity means that T O = 0 P x -a.s.. Therefore, lim ε→0 σ c ε = 0 P x -a.s.. The convergence of σ ε to 0 can be proved in an analogous way. Assumption (A6) is satisfied whenever Markov process (X(t)) has a density at time T with respect to a measure which puts zero weight on the set ∂O. ( 1) w ∞ is continuous on [0, ∞) × (E \ ∂O). (2) w ∞,∞ (s, x) := lim β→∞ w β,∞ (s, x) is lower semicontinuous (l.s.c.) with values in R ∪ {∞}. (3) if F is l.s.c., then w ∞ is l.s.c. and w ∞,∞ ≥ w ∞ .(4Proof. Letw ∞ h (s, x) = sup τ≥h J ∞ (s, x, τ) . Theorem 3b of [16] implies w ∞ h (s, x) = E x e −αh w ∞ (s + h, X(h)) + h 0 e −αu f s + u, X(u) du . By Lemmas 4.4 and 7.1 the functionw ∞ h is continuous for each h > 0. Under (A2), which follows by Remark 4.7 from (A2'), in the same way as in Theorem 4.8 we prove thatw ∞ h → w ∞ , as h → 0, uniformly on compact subsets of [0, ∞) × (E \ ∂O). This implies assertion (1). Assertion (2) follows from Lemma 7.1. Indeed, w β,∞ is non-decreasing in β and continuous for each β. Hence, the limit w ∞,∞ is well defined and lower semicontinuous. Definew ∞ (s, x) = sup τ>0 J ∞ (s, x, τ). Notice thatw ∞ h (s, x) ↑w ∞ (s, x) as h → 0, which impliesw ∞ is l.s.c. [16, Theorem 3b] implies w ∞ (s, x) = max {w ∞ (s, x), F(s, x)}. Hence, if F is l.s.c., then the mapping w ∞ (s, x) is l.s.c. as maximum of two l.s.c. functions. Applying (35) with b(u) = β1 {u≤h} and a sufficiently small h yields w ∞,∞ ≥ F. Letting β → ∞ in (38) and using w ∞,∞ ≥ F we obtain w ∞,∞ ≥ w ∞ . This completes the proof of assertion (3). Last assertion is the most demanding. We assume first that F = G ∨ H on ∂O. We will relax this assumption later. By Lemma 7.3 we obtain w ∞,∞ ≤ w ∞ . The proof of the opposite inequality is divided into several steps. Assertion (4) will then follow from Dini's theorem. Step 1. Assume G = R α g and H = R α h, where the functions g, h : [0, ∞) × E → R are continuous bounded and the resolvent R α is defined in Lemma 7.2. It is sufficent to consider g, h ∈ C 0 ([0, ∞) × E), but it does not simplify the reasoning in any way. Lemmas 7.2 and 7.3 imply the following bound: w β,∞ (s, x) ≥ sup τ J ∞ (s, x, τ) − E x 1 {X(τ)∈O} f − g φ β O X(τ) − E x 1 {X(τ)∈O c \∂O} f − h φ β O c \∂O X(τ) − E x 1 {X(τ)∈∂O} f − g ∨ f − h α , where, for an open set A ⊂ E, we define φ β A (x) = E x ∞ 0 e −αu−β u 0 1 {X(t)∈A} dt du . By dominated convergence theorem, lim β→∞ φ β A (x) = 0 for x ∈ A. Taking a limit as β → ∞ yields (40) w ∞,∞ ≥ sup τ J ∞ (s, x, τ) − MP x {X(τ) ∈ ∂O} , where M = ( f − g ∨ f − h )/α. Step 2. We will show that the supremum in (40) can be restricted to stopping times satisfying P x {X(τ) ∈ ∂O} = 0. Fix a stopping time τ and define for ε > 0 τ ε =              τ + (σ ε • θ τ ), if X(τ) ∂O, τ + (σ ε • θ τ ), if X(τ) ∈ ∂O, F s + τ, X(τ) = H s + τ, X(τ) , τ + (σ c ε • θ τ ) , if X(τ) ∈ ∂O, F s + τ, X(τ) = G s + τ, X(τ) , whereσ ε = inf{u ≥ 0 : X(u) Γ ε } and σ ε , σ c ε are defined in assumption (A5). The stopping time τ ε might attain the value ∞, in which case the functional J is also well defined due to discounting. Notice the difference betweenσ ε and σ ε (σ c ε ): the former is the first exit time from the ε-neighbourhood Γ ε of ∂O, whereas the latter is the first exit time from O ∪ Γ ε (O c ∪ Γ ε , resp.). The stopping time τ ε equals τ for appropriately small ε if X(τ) ∂O. Otherwise, i.e., when X(τ) ∈ ∂O, it follows from assumption (A5) that lim ε→0 τ ε → τ P x -a.s. If F = H at the time τ then X(τ ε ) ∈ O c (if it is finite) and by the continuity of H we obtain lim ε→0 e −ατ ε H τ ε , X(τ ε ) = e −ατ H τ, X(τ) P x -a.s. We proceed similarly when F = G at the time τ and get lim ε→0 e −ατ ε F τ ε , X(τ ε ) = e −ατ F τ, X(τ) P x -a.s. Dominated convergence theorem implies lim ε→0 J ∞ s, x, τ ε = J ∞ (s, x, τ). We also have P x {X(τ ε ) ∈ ∂O} = 0 for each ε > 0. Hence, J ∞ (s, x, τ) = lim ε→0 J ∞ (s, x, τ ε ) − MP x {X(τ ε ) ∈ ∂O} ≤ sup τ J ∞ (s, x,τ) − MP x {X(τ) ∈ ∂O} . Combining this result with (40) yields J ∞ (s, x, τ) ≤ w ∞,∞ (s, x), which, due to arbitrariness of τ, gives the required inequality w ∞ ≤ w ∞,∞ . Step 3. Using standard methods we extend above result to continuous bounded G and H in a similar way as in Theorem 4.3. We relax now the assumption F = G ∨ H on ∂O. Let F be as in the statement of the theorem and F(s, x) =        F(s, x), x ∂O, (G ∨ H)(s, x), x ∈ ∂O. Denote byw β,∞ andw ∞ the value functions corresponding toF. Fubini's theorem and assumption (A6) imply that E x ∞ 0 e −αu 1 {X(u)∈∂O} du = 0. Using this equality we obtain w β,∞ (s, x) = E x ∞ 0 e −αu f s + u, X(u) + 1 {X(u) ∂O} β F − w β,∞ + s + u, X(u) + 1 {X(u)∈∂O} β F − w β,∞ + s + u, X(u) du = E x ∞ 0 e −αu f s + u, X(u) + 1 {X(u) ∂O} β F − w β,∞ + s + u, X(u) + 1 {X(u)∈∂O} β F − w β,∞ + s + u, X(u) du = E x ∞ 0 e −αu f s + u, X(u) + β F − w β,∞ + s + u, X(u) du , where the second equality follows from the fact that F coincides withF on E \ ∂O and the third term under the integral integrates to zero. Sincew β,∞ is a unique solution of the penalized equation (34) with functioñ F (see Lemma 7.1) we conclude thatw β,∞ = w β,∞ . Denote byJ ∞ the functional J ∞ with the functionF. Fix any stopping time τ and define for ε > 0 τ ε =              τ +σ ε • θ τ , if X(τ) ∂O, τ + σ ε • θ τ , if X(τ) ∈ ∂O,F s + τ, X(τ) = H s + τ, X(τ) , τ + σ c ε • θ τ , if X(τ) ∈ ∂O,F s + τ, X(τ) = G s + τ, X(τ) . Similarly, as in Step 2 we obtain lim ε→0 J ∞ (s, x, τ ε ) =J ∞ (s, x, τ), which impliesw ∞ ≤ w ∞ . Opposite inequality is obvious asF ≥ F. In the first part of the proof of assertion (4)w β,∞ was shown to converge tow ∞ uniformly on compact sets in [0, ∞) × (E \ ∂O). Sincew β,∞ coincides with w β,∞ andw ∞ coincides with w ∞ this uniform convergence holds for w β,∞ and w ∞ . REMARK 7.6. The complexity of the proof of assertion (4) in Theorem 7.5 is caused by the incompatibility of the continuity conditions that one has to impose on the function F. On the one hand, we need to prove that w ∞,∞ ≥ F, which requires that F is lower semicontinuous. On the other hand, the inequality w β,∞ ≤ w ∞ is true under the condition that F is upper semicontinuous (see Lemma 7.3). Finite time horizon Methods from previous section can be applied to optimal stopping of the following functional: J T (s, x, τ) = E x τ∧(T −s) 0 e −αu f s + u, X(u) du + e −α(τ∧(T −s)) F (s + τ) ∧ T, X(τ ∧ (T − s)) , where α ≥ 0, the function f is measurable bounded and continuous in s uniformly in x from compact sets, and F has the following form: The first term vanishes as n → ∞. Assumption (A2') implies that the second term converges to zero. The convergence of the third term follows from the continuity of the mapping (see Proposition A.2): (s, x, h) → E x {e −αh G(s + h, X(h))}. Convergence to x ∈ O c \ ∂O can be proved in an analogous way. Representation (42) is obtained in an analogous way as in the proof of Lemma 2.8. Assertion (3) follows immediately from (42). where M = ( f − g ∨ f − h )/α. Step 2. Fix δ > 0. For any T > 0, s ∈ [0, T ) and a stopping time τ ≤ T − s define for ε > 0 τ ε =              τ + (σ ε • θ τ ) ∧ δ, if X(τ) ∂O, τ + (σ ε • θ τ ) ∧ δ, if X(τ) ∈ ∂O, F s + τ, X(τ) = H s + τ, X(τ) , τ + (σ c ε • θ τ ) ∧ δ, if X(τ) ∈ ∂O, F s + τ, X(τ) = G s + τ, X(τ) , whereσ ε = inf{u ≥ 0 : X(u) Γ ε } and σ ε , σ c ε are defined in assumption (A5). Contrary to the proof of Theorem 7.5 the difference τ ε − τ is bounded by δ. This, however, does not affect the limits as ε → 0. If F = H at the time τ then X(τ ε ) ∈ O c for appropriately small ε and by continuity of H we obtain lim ε→0 H τ ε , X(τ ε ) = H τ, X(τ) P x -a.s. We proceed similarly when F = G at the time τ and get lim ε→0 F τ ε , X(τ ε ) = F τ, X(τ) P x -a.s. Recalling that τ ε ≤ T − s + δ we obtain J ∞ (s, x, τ ε ) − MP x {X(τ ε ) ∈ ∂O} ≤ sup τ≤T −s+δ J ∞ s, x, τ − MP{X(τ) ∈ ∂O} . As ε → 0 the left-hand side converges to J ∞ (s, x, τ). The arbitrariness of τ implies This completes the proof of both assertions in the case when G and H can be written in resolvent forms (assertion (4) follows from Dini's theorem and assertion (1)). Step 4. Using standard methods we extend above result to continuous bounded G and H in a similar way as in Theorem 4.3. Step 5. We relax the assumption that F = G ∨ H on [0, T ] × ∂O as in the proof of Theorem 7.5. ( 1 ) 1J(s, x, τ) = E x τ∧τ O 0 e −αu f s + u, X(u) du + 1 {τ<τ O } e −ατ G s + τ, X(τ) + 1 {τ≥τ O } e −ατ O H s + τ O , X(τ O ) , Date: Date: 2011-04-28 13:26:31 . Research of both authors supported in part by MNiSzW Grant no. NN 201 371836. 1 where (s, x) ∈ [0, ∞) × E, α > 0, τ ≥ 0 and f, G, H : [0, ∞) × E → R are continuous bounded functions. (2) Stopping is allowed up to time τ O and the payoff is given by a function F : [0, ∞) × E → R which is continuous on [0, ∞) × O and possibly discontinuous in the space variable on [0, ∞) × O c :(2) J(s, x, τ) = E x τ∧τ O 0 e −αu f s + u, X(u) du + e −α(τ∧τ O ) F s + (τ ∧ τ O ), X(τ ∧ τ O ) .This, in particular, covers a complementary problem to(1): with F continuous on [0, ∞) ×Ō and on [0, ∞) ×Ō c with a possible jump at the boundary [0, ∞) × ∂O. (3) Stopping is unconstrained (infinite horizon, T = ∞) or constrained by a constant T (finite horizon) with the following functional: (3) J(s, x, τ) = E x τ∧(T −s) 0 e −αu f s + u, X(u) du + e −α(τ∧T ) F (s + τ) ∧ T, X(τ ∧ (T − s)) , e −γu h(s + u, X(u))du = ∞ 0 e −γu ϕ(s, u, x)du, where ϕ(s, u, x) = E x {1 {u<τ O } h(s + u, X(u)}. This function is continuous in (s, x) for any fixed u ≥ 0 by Lemma 2.3. Dominated convergence theorem concludes. LEMMA 2.5. Under (A1) for α > 0 and a continuous bounded function h : PROPOSITION 2. 9 . 9Under (A1), the functions w β (s, x) increase pointwise to w(s, x) as β → ∞. COROLLARY 2. 10 . 10Under (A1), the value function w is lowersemicontinuous. Moreover, if w is continuous then w β approaches w uniformly on compact sets. A2) lim η→0 P x {τ O < η} = 0 uniformly in x from compact subsets of O. (A3) (X(t)) is strongly Feller, i.e., the mapping x → E x {h(X(t))} is continuous for any measurable bounded function h and t > 0.Before we formulate Theorem 4.8 we prove three auxiliary results.Lemma 4.4 shows that under (A3) the time-state process semigroup maps time-continuous bounded functions into functions continuous in both parameters. Lemma 4.5 states that the weak Feller continuity of the process X(t) is sufficient for the continuity of the value function w in the time parameter s. Lemma 4.6 shows that (A2) follows from (A1). LEMMA 4. 4 . 4Under assumption (A3), the mapping(s, x) → E x {F(s + h, X(h))} is continuous for h >0 and a bounded measurable function F, provided that the mapping s → F(s, x)is continuous uniformly in x in compact subsets of E. LEMMA 4. 6 . 6Assumption (A1) implies (A2).Proof. By Lemma 2.4 the function g γ (x) = E x {e −γτ O } is continuous on O for any γ > 0. By dominated convergence theorem g γ (x) converges to 0 when γ → ∞ and x ∈ O. This convergence is monotone and, due to Dini's theorem, uniform on compact subsets of O. Chebyshev's theorem yieldsP x {τ O < η} = P x {e −τ O /η > e −1 } ≤ e g 1/η (x).The right-hand side converges to 0, when η → 0, uniformly on compact subsets of O, which completes the proof. REMARK 4.7. Assumption (A2) holds if the process (X(t)) satisfies the following continuity condition:(A2') for any ε > 0 lim t→0 P x sup s∈[0,t] THEOREM 4. 8 . 8Under (A2) and (A3), the function w is continuous on [0, ∞) × O. Assume additionally (A1). The penalized functions w β are continuous and converge to w uniformly on compact subsets of [0, ∞) × O. An ε-optimal stopping time is given by ( 27 ) 27|w h (s, x) −w h (s, x)| ≤ C P x τ O < h and (28) 0 ≤ w(s, x) −w h (s, x) ≤ sup τ J(s, x, τ) − J(s, x, τ h ) =: I h (s, x), where τ h = τ ∨ hand C > 0. Assumption (A2) implies the difference |w h −w h | converges to 0 as h → 0 uniformly on compact subsets of [0, ∞) × O. The proof of uniform convergence of I h is more involved. First notice I h (s, x) = sup τ≤h J(s, x, τ) − J(s, x, h) the continuity of w in [0, ∞) × O. It is also clear that w is continuous on [0, ∞) × O c because on this set w coincides with H. However, if there is a downward jump on the boundary of O (G(s, x) > LEMMA 4 . 10 . 410In the setting of the example, PROPOSITION 6. 3 . 3Under (A1) and (A3): (1) There is a unique measurable bounded solution w β to the above penalized equation. This function is continuous on [0, ∞) × O. (2) Functions w β are non-decreasing in β.Proof. Existence of a unique bounded measurable solution follows from Lemma 2.2. Lemma 4.4 and 6.2 imply the continuity of w β . Assertion (2) follows from Lemma 2.8. PROPOSITION 6. 4 . 4Under (A1) and (A3) the sequence of functions w β converges to w and this convergence is uniform on compact subsets of [0, ∞) × O. Proof. Using continuity of F on [0, ∞) × O, in a similar way as in Lemma 2.7 we obtain LEMMA 7. 1 .( 1 ) 11Assume (A3). There is a unique (in the space of measurable bounded functions) solution w β,∞ of the penalized equation (34) and this solution is continuous. (2) The mapping s → w ∞ (s, x) is continuous uniformly in x from compact sets. (3) We have the following equivalent representation of w β,∞ : REMARK 7.4. Assumption (A5) is satisfied whenever each point of ∂O is regular for O and E\(O∪∂O) (see, e.g., Blumenthal and Getoor [5] for a definition and properties of regular points). Indeed, [5, Proposition 10.4] implies that T O ≥ lim ε→0 σ c ε , where T O is the first hitting time of O, i.e. T O = inf{t > 0 : X(t) ∈ O}. THEOREM 7. 5 . 5Assume (A2') and (A3). ) if assumptions (A5)-(A6) are satisfied and F ≤ G ∨ H on [0, ∞) × ∂O, then w β,∞ converges to w ∞ , as β → ∞, uniformly on compact subsets of [0, ∞) × (E \ ∂O). e F(s, x) = G(s, x) for x ∈ O and F(s, x) = H(s, x) for x ∈ O c \ ∂O for bounded continuous functions G and H. This functional is a finite time horizon version of the functional J ∞ (s, x, τ).Denote the value function by w T (s, x) = sup τ J T (s, x, τ). Notice that w T can be equivalently written asw(s, T ) = sup τ≤T −s J ∞ (s, x, τ).To enable numerical approximations of this value function we introduce a penalized equation:(41) w β,T (s, x) = E x T −s 0 e −αu f + β F − w β,T + s + u, X(u) du + e −α(T −s) F T, X(T − s) .LEMMA 8.1. Assume (A3). (1) There is a unique measurable bounded solution to (41) and this solution is continuous on [0, T )×E. Under Assumption (A2') the continuity extends to [0, T ) × E ∪ T × (E \ ∂O) . (2) This solution has an equivalent representation: −αu− u 0 b(t)dt f s + u, X(u) + b(u)F s + u, X(u) du + e −α(T −s)− T −s 0 b(t)dt F T, X(T − s) , where M β is the class of progressively measurable processes with values in [0, β]. (3) w β,T is increasing in β. Proof. Similarly as in Lemma 2.2 we show that there is a unique bounded measurable solution to (41). Lemma 4.4 implies that this solution is continuous on [0, T ) × E. The continuity on T × (E \ ∂O) is more delicate. Fix x ∈ O and a sequence (s n , x n ) ⊂ [0, T ] × O converging to (T, x). We have w β,T (s n , x n ) − w β,T (T, x) ≤ f + β (F − w β,T ) + (T − s n ) + 2 F P x n (τ O ≤ T − s n ) + E x n e −α(T −s n ) G(T, X(T − s n )) − G(T, x) . THEOREM 8. 2 . 2Assume (A2'), (A3). (1) The function w T is continuous on [0, T ] × (E \ ∂O). (2) w ∞,T (s, x) := lim β→∞ w β,T (s, x) is lower semicontinuous on [0, T ) × E with values in R ∪ {∞}. (3) if F is l.s.c., then w T is lower semicontinuous on [0, T ) × E and w ∞,T ≥ w T . J ∞ s, x, τ ε = J ∞ (s, x, τ), and lim ε→0 P x {X(τ ε ) ∈ ∂O} = 0. ( 45) w T (s, x) ≤ sup τ≤T −s+δ J ∞ s, x, τ − MP{X(τ) ∈ ∂O} .Step 3. Combining formulas (44) and (45) we get for any δ > 0w ∞,T (s, x) ≥ w T −δ (s, x), (s, x) ∈ [0, T − δ] × E.Take a stopping time τ ≤ T − s and defineτ δ = τ ∧ (T − s − δ). We have lim sup δ→0 J ∞ (s, x, τ δ ) − J ∞ (s, x, τ) ≤ lim sup δ→0 E x e −ατ δ F(s, x, τ δ ) − e −ατ F(s, x, τ) ≤ lim sup δ→0 E x 1 {τ<T −s} e −ατ δ F(s, x, τ δ ) − e −ατ F(s, x, τ) + lim sup δ→0 E x 1 {τ=T −s, X(τ)∈∂O} e −ατ δ F(s, x, τ δ ) − e −ατ F(s, x, τ) + lim sup δ→0 E x 1 {τ=T −s, X(τ) ∂O} e −ατ δ F(s, x, τ δ ) − e −ατ F(s, x, τ) = (1) + (2) + (3).Limit (1) equals 0 from dominated convergence theorem. By quasi-left continuity of the process X(t) and dominated convergence theorem limit (3) is 0 as well. Term (2) is dominated by 2 F P x {X(T − s) ∈ ∂O}, which by (A6) is equal to 0. Hence, lim δ→0 w T −δ (s, x) = w T (s, x). Since this operator is a contraction on the space of bounded measurable functions it is a contraction on the space of continous bounded functions. This implies that w β as a unique fixed point is continuous.and ε are arbitrary this implies continuity of H on [0, S ] × K. Hence, H is continuous on its whole domain by the arbitrariness of S , K. COROLLARY 2.6. Under (A1), the unique bounded solution w β of (4) is continuous. Proof. Lemmas 2.4 and 2.5 imply that the operator T introduced in the proof of Lemma 2.2 maps the space of continuous bounded functions into itself. \∂O, where G and H are bounded continuous functions. Notice that F can be arbitrary on [0, ∞)×∂O as long as it is continuous in s uniformly in x from compact sets. In particular, F can be equal to G or H on ∂O.on F: F(s, x) = G(s, x) for x ∈ O and F(s, x) = H(s, x) for x ∈ O c LEMMA 7.3. Under assumption (A3) we have J T (s, x, τ) − MP x {X(τ) ∈ ∂O} , Assume further (A5), (A6).(4) If F ≤ G ∨ H on [0, T ] × ∂O then w β,T converges to w T as β → ∞ uniformly on compact subsets of [0, T ] × (E \ ∂O). (5) The mapping (s, ∞) ∋ T → w T (s, x) is continuous for fixed s and x ∈ E \ ∂O.Proof. Similarly as in the proof of Theorem 7.5, we show that w T is continuous on [0, T ) × (E \ ∂O). The extension of the continuity to T × (E \ ∂O) follows an analogous route as in the proof of Lemma 8.1. Fix x ∈ O and a sequence (s n ,The first term vanishes as n → ∞. Assumption (A2') implies that the second term converges to zero. The convergence of the third term follows from the continuity of the mapping (see[18,Corollary 3Convergence to x ∈ O c \ ∂O can be proved in a similar way.Assertions (2)-(3) are proved in a similar way as in Theorem 7.5. Assertion (4) is trivial for s = T since w β,T (T, x) = w T (T, x) = F(T, x). In the following we will address the case s < T . In a similar way as in the proof of Theorem 7.5 we show that it suffices to prove both assertions for F = G ∨ H on ∂O. Under this condition, as in Lemma 7.3, we show that the function w β,T has the following equivalent representation:This implies that w ∞,T := lim β→∞ w β,T ≤ w T . The proof of the opposite inequality requires similar but slightly more delicate argument as in the proof of Theorem 7.5.Step 1. Assumefor continuous bounded functions g, h. Combining arguments from proofs of Lemmas 4.2 and 7.2 we obtainAbove estimates and identity (43) imply the following bound:Taking the limit as β → ∞ and recalling that w β,T (T, x) = F(T, x) yieldAppendix Appendix A Properties of weak Feller processesA Markov process defined on a locally compact separable space is called standard (see[6], p. 104, or [5, Definition 9.2]) if(1)it is a strong Markov process,(2)it is càdlàg and quasi-left-continuous, (3) the filtration is complete and right-continuous. Let X(t) be a càdlàg Markov process defined on a locally compact separable space (E, E) endowed with a metric ρ with respect to which every closed ball is compact. Assume that this process satisfies the weak Feller property: P t C 0 ⊆ C 0 , where C 0 is the space of continuous bounded functions E → R vanishing in infinity, and P t h(x) = E x h X(t) for any bounded measurable h : E → R. Right continuity of X(t) and Theorem T1, Chapter XIII in[17]implies that the semigroup P t satisfies the following uniform continuity property:Theorem 3.1 (p. 104) in[6]implies that there exists a standard Markov process on the state space E with the semigroup P t . In fact, it follows from the proof of the aforementioned theorem that the process (X(t)) satisfies the conditions of a standard process if its filtration is complete. The filtration of (X(t)) can be completed without changes to other properties of the process due to Proposition A.2 below and Theorem 3.3 and Subsection 3.6 in[6]. 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Control and Optimization 48:8, 4874-4909 M Robin, Controle impulsionnel des processus de Markov (Thesis). University of Paris IXRobin M (1978) Controle impulsionnel des processus de Markov (Thesis), University of Paris IX Optimal Stopping for Feller Markov Processes. Ł Stettner, J Zabczyk, IMPAN, WarsawPreprint No. 284Stettner Ł, Zabczyk J (1983) Optimal Stopping for Feller Markov Processes, Preprint No. 284, IMPAN, Warsaw Penalty method for finite horizon stopping problems, to appear in SIAM. Ł Stettner, J. Control and Optimization. Stettner Ł (2008) Penalty method for finite horizon stopping problems, to appear in SIAM J. Control and Optimization Stopping Problems in Stochastic Control. J Zabczyk, Proc. ICM-83. ICM-83PWN North HollandIIZabczyk J (1984) Stopping Problems in Stochastic Control, Proc. ICM-83, Vol. II, PWN North Holland, 1425-1437 Leeds LS2 9JT, UK E-mail address. J.Palczewski@mimuw.edu.pl Institute of Mathematics, Polish Academy of Sciences,Śniadeckich. 2Faculty of Mathematics, University of Warsaw ; Poland and School of Mathematics, University of Leeds ; Poland and Academy of FinancePoland E-mail address: stettner@impan.gov.plFaculty of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland and School of Mathematics, University of Leeds, Leeds LS2 9JT, UK E-mail address: J.Palczewski@mimuw.edu.pl Institute of Mathematics, Polish Academy of Sciences,Śniadeckich 8, 00-956 Warszawa, Poland and Academy of Finance, Warszawa, Poland E-mail address: stettner@impan.gov.pl

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{'abstract': 'We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O. The stopping horizon is either random, equal to the first exit from the set O, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of O. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or ε-optimal stopping times.', 'arxivid': '1006.4283', 'author': ['Jan Palczewski ', 'Łukasz Stettner '], 'authoraffiliation': [], 'corpusid': 22133374, 'doi': '10.1016/j.spa.2011.05.013', 'github_urls': [], 'n_tokens_mistral': 27118, 'n_tokens_neox': 24174, 'n_words': 13855, 'pdfsha': 'ee3bfc1e41ded478fe1f0c70f266e4d979d07b63', 'pdfurls': ['https://arxiv.org/pdf/1006.4283v3.pdf'], 'title': ['STOPPING OF FUNCTIONALS WITH DISCONTINUITY AT THE BOUNDARY OF AN OPEN SET', 'STOPPING OF FUNCTIONALS WITH DISCONTINUITY AT THE BOUNDARY OF AN OPEN SET'], 'venue': []}

arxiv

Sr-and Ni-doping in ZnO nanorods synthesized by simple wet chemical method as excellent materials for CO and CO 2 gas sensing 2016. 2016 Parasharam M Shirage paras.shirage@gmail.com*email:gneri@unime.it Department of Physics Centre of Materials Science and Engineering Indian Institute of Technology Indore Simrol Campus Khandwa Road453552IndoreIndia Amit Kumar Department of Physics Centre of Materials Science and Engineering Indian Institute of Technology Indore Simrol Campus Khandwa Road453552IndoreIndia Rana Yogendra Kumar Somaditya Sen Department of Physics Centre of Materials Science and Engineering Indian Institute of Technology Indore Simrol Campus Khandwa Road453552IndoreIndia S G Leonardi Department of Physics Centre of Materials Science and Engineering Indian Institute of Technology Indore Simrol Campus Khandwa Road453552IndoreIndia Department of Engineering University of Messina 98166MessinaItaly G Neri Department of Engineering University of Messina 98166MessinaItaly Sr-and Ni-doping in ZnO nanorods synthesized by simple wet chemical method as excellent materials for CO and CO 2 gas sensing RSC Advances 6862016. 20161 2 In this study, the effect of Sr-and Ni-doping on microstructural, morphological and sensing properties of ZnO nanorods has been investigated. Nanorods with different Sr and Ni loadings were prepared using a simple wet chemical method and characterized by means of scanning electron microscopy (SEM), X-ray diffraction (XRD) and photoluminescence (PL) analysis. XRD data confirmed that Sr-and Ni-doped samples maintainsthe wurtzite hexagonal structure of pure ZnO. However, unlikes Sr, Ni doping modifies the nanorod morphology, increases the surface area (SA) and decreases the ratio of I UV /I green photoluminescence peak to a greater extent. Sensing tests were performed on thick films resistive planar devices for monitoring CO and CO 2 , as indicators of indoor air quality.The effect of the operating temperature, nature and loading of dopant on the sensibility and selectivity of the fabricated sensors towards these two harmful gases were investigated. The gas sensing characteristics of Ni-and Sr-doped ZnO based sensors showed a remarkable enhancement (i. e. the response increased and shifted towards lower temperature for both gases) compared to ZnO-based one, demonstrating that these ZnO nanostructures are promising to fabricate sensor devices for monitoring indoor air quality. Introduction Metal oxide semiconductor (MOS) gas sensors have been intensively studied for applications in the detection of hazardous pollutant gases 1,2 . They have many advantageous features respect to other gas sensor devices, such as high sensitivity, low power consumption, low price, quick response and simple structure with high microelectronic processing compatibility. Actually, the research for improving the performance of MOS sensors is addressed towards novel syntheses and/or modification of nanostructured metal oxides with improved sensing characteristics 2 . Among these, ZnO is a well-known material for semiconductor device applications 3 . It has a direct and wide band gap in the near-UV spectral region 4 and has noticeable gas sensing properties 5 . At present, numerous efforts have been made to improve the sensing properties of MOS gas sensors based on ZnO. Low dimensional nanostructures such as nanorods, nanotube, nanoflower, etc., showed peculiar characteristics [6][7][8] . ZnO nanostructures demonstrated novel applications in optoelectronics, sensors, transducers and biomedical sciences 9,10 . With reduction in size, novel electrical, mechanical, chemical and optical properties are introduced, as a result of surface and quantum confinement effects and are of benefit for developing nano-devices of new generation with high performance. Doping of base metal oxide with various metallic elements, for example, noble metals, transition metal and metal oxides, had been proven also effective for this scope 11,12 . Indeed, doping enhances the performance gas sensor via controlling donor density, changing the acid-base properties and varying electronic properties so to change the interaction between gases surrounding the sensor and sensing layer. In this paper, we investigated ZnO nanorods doped with Sr and Ni. Very few papers regarding Ni-doped ZnO sensors are reported in the literature [13][14][15][16][17] , whereas, at the best of our knowledge, only one paper deals instead with sensing properties of Sr-ZnO sensors 18 . Therefore, we decided to evaluate the sensing capability of ZnO nanorods doped with different loading of Sr and Ni synthesized by a simple wet chemical method. The morphology and microstructure of the synthesized materials were first investigated. Then, they were used to fabricate resistive gas sensors for monitoring CO and CO 2 ,. Sensors that measure accurately CO and CO 2 concentration in air are highly required for indoor air quality control. CO is a concern in indoor environments where poorly ventilated appliances are present or where outdoor air intakes are located in areas subject to high concentrations of vehicle exhaust 19 . OSHA organization has established a permissible exposure limit for CO of 50 ppm as an 8-hour time-weighted average. CO 2 levels can be used as a rough indicator of the effectiveness of ventilation, and excessive population density in a structure. The eight-hour permissible limit for CO 2 is 5000 ppm 20 . Then, sensors with promising sensing characteristics to these two target gases are highly desired to be applied in electronic devices for monitoring indoor air quality and used to trigger an alarm, turn on a ventilation fan, or control a heating, ventilating and air conditioning (HVAC) system. Experimental details Pure, Ni-and Sr-doped ZnO nanorods were prepared by simple wet chemical method. High purity chemicals zinc acetate, nickel nitrate, and strontium nitrate (from Alfa Aesar) were used as the raw materials. First, the stoichiometric amount of zinc nitrate, and nickel nitrate or strontium nitrate were dissolved in 100 ml double distilled water and stirred for 30 minutes. Then, under constant stirring, aqueous ammonia solution was added continuously to maintain a pH ~ 11. Finally, a transparent solution is obtained in the case of pure ZnO and Sr-doped, whereas a blue solution is obtained in case of Ni-doped ZnO. In the next step, the solutions were kept at 100°C for 2 hours and then the precipitates were separated and annealed at 250°C (for 2 hrs.). The pure ZnO, Ni-(5% and 10%), and Sr-doped (4% and 8%) samples are named ZO, Ni5ZO, Ni10ZOand Sr4ZO, Sr8ZO respectively. The morphology of the prepared samples was investigated by field emission scanning electron microscopy (FESEM) using Supra 55 Zeiss microscope. The crystalline structure was verified by Bruker D8 Advance X-ray diffractometer with Cu-Kα radiation (λ=1.54 Å). The average grain size (D) was calculated from the XRD result by using the Scherer formula: D= K λ/β cosθ, where K is the shape factor (~0.9), λ is the X-rays wavelength, θ is the diffraction angle and β is the full width at half maximum of the XRD peak. The nitrogen adsorption and desorption isotherms were obtained by Quantumchrome Q 2 , and the specific surface area and the pore size distribution were calculated by the Brunauer-Emmett-Teller (BET) and the Barrett-Joyner-Halenda (BJH) methods, respectively. The room temperature photoluminescence (PL) measurements were carried out on Dongwoo Optron spectrophotometer. The excitation wavelength was fixed at 325 nm and emission spectra were scanned from 300 to 850 nm. Sensing tests were carried on resistive sensors having a planar configuration and based on alumina substrates (6 mm × 3 mm) with Pt interdigitated electrodes and a Pt heater located on the back. The devices for electrical and sensing tests were prepared by printing films (~10 μm thick) of the nanorods dispersed in water. A picture of the final device with the Ni10ZO printed film after annealing at 350°C is shown in Fig. 5. The sensors were then introduced in a Teflon test chamber for the sensing tests. The electrical measurements were carried out over the temperature range from RT to 350°C under a synthetic dry air (20% O 2 -80% N 2 ) stream of 100 sccm by collecting the electrical resistance of the sensitive films. An Agilent 34970A multimeter data acquisition unit was used for this purpose, while an Agilent E3632A dual-channel power supplier instrument was employed to bias the built-in heater of the device. Sensing tests were performed by injecting pulses of the analyte from certified bottles. The concentration of the target gas was varied by using mass flow controllers. The gas response is defined as the ratio R air /R gas , where R air is the electrical resistance of the sensor in dry air and R gas the resistance at different target gas concentrations. This paper published in RSC Advances 6 (86), 82733-82742 (2016) 5 Results and discussion Characterization X-ray diffraction (XRD) analysis was performed to examine the effect of Sr and Ni doping on the crystal structure of the ZnO nanorods. b) Ni5ZO (c) Ni10ZO (d) Sr4ZO (e) Sr8ZO. Relative shift in (002) peak of ZnO with respect to (f) Sr-doping and (g) Ni-doping. The nanocrystallites of the pure ZnO are preferentially oriented along the c-axis, [002] direction. The doping with Sr and Ni change the relative intensity of the diffraction peaks, indicating a modification of the axis orientation, in special way for Ni-doped samples. No This paper published in RSC Advances 6 (86), 82733-82742 (2016) 6 characteristic peak of any other possible phases, such as NiO, has been observed, indicating the easy substitution of Ni in ZnO, likely due to similar ionic radius of Ni 2+ ion (0.68 Å) with Zn 2+ (0.74 Å) 21 . The (002) peak was used for the calculation of average grain sizes and results for all samples are shown in Table 1 For Sr-doped samples, there is increment in both a and c parameters because of large ionic radii of Sr 2+ (1.18 Å) than Zn 2+ . It is also observed that small change in lattice parameter is due the lower solubility of Sr 2+ in ZnO because of its large ionic radii. A similar result has been also observed by L. Xu et al. 23 and Water et al. 24 b) Ni5ZO (c) Ni10ZO (d) Sr4ZO (e) Sr8ZO (inset images at 20 KX). The growth mechanism of ZnO was already discussed elsewhere 6 In order to evaluate the textural effects induced by the addition of dopants, the nitrogen adsorption-desorption isotherm and the corresponding Barrett-Joyner-Halenda (BJH) pore size distribution plot (see inset) of all samples are shown in Figure S1. Using the BJH method and the desorption branch of the nitrogen isotherm, the BET surface area (SA) have been calculated and are shown in Table 1. Ni doping increase strongly the surface area (more than tenfold), unlikely Sr doping which lead instead to a decrease of SA. Interestingly, the pore size distribution of pure and Ni-doped samples revealed a prominent peak at very small pore radius, whereas the opposite was found for the Sr-doped samples. On the basis of SEM and XRD characterizations, it appears that the SA enhancement of SA is mainly linked to change of nanorod morphology, rather than a change in their size. The lower SA area of Ni-doped samples could be instead due to the increase of pore radius with Sr doping. increases. From data reported in the inset of Fig. 4 it seems that Ni doping decreases the ratio of I UV /I green photoluminescence peak to a greater extent than Sr. CO and CO 2 sensing tests CO and CO 2 are major air pollutants whose concentration has to be controlled as per international indoor air quality standards at industrial and public work places. Then, there is a high interest to fabricate efficient CO and CO 2 sensors. Resistive sensors having a planar configuration It is noteworthy that at lower temperature (200°C), the pure ZnO and Sr doped sensors exposed to CO pulses present an inversion of the response (see Fig. S2 in Supporting Information). respectively. At all temperatures, the sensors are little influenced by oxygen variation. For the two target gases, at temperature higher than 250°C, all sensors exhibit greater response to CO than CO 2 . Examining the trend of responses to target gases, it is observed that, for Ni-doped sensors, the response to CO is higher with respect the undoped and Sr-doped ZnO sensors and increase with decreasing the temperature. At the temperature of 200-250°C, their response is more than three times than that observed for the ZnO sensor. Hence, doping ZnO with Ni allows a remarkable enhancement of the gas sensing characteristics towards CO. Vice-versa, Sr-doped sensors display a CO sensing behavior similar to undoped ZnO sensor. In fact, as concerns CO gas, these sensors show a maximum of the response above 350°C, while a drastic loss of response is registered below 250°C. The possible reason for higher CO gas sensing response above 350°C is assumed to higher thermal energy with increase in operating temperature which helps to overcome the activation energy barrier to the reaction and a significant increase in the electron concentration for the sensing reaction to happen 31 . It has been reported that the stable oxygen ions were O 2below 100°C, Obetween 100°C and 300°C and O 2above 300°C. The relevant surface reactions can be listed as, 32 2 + 2 − → 2 2 + − -(4) + − → 2 + − -(5)+ 2− → 2 + 2 − -(6) Combining aforementioned facts and corresponding surface reactions, equ. (6) turns out as most desirable for the gas sensing, because the reaction could release more number of electrons to follow a generalized sensing pattern and increase in the gas response. This is only possible when the operating temperature is maintained above 300°C but less than 400°C above which possibly degradation of doped ZnO causes decrease in CO response. In accordance to the equations it can also be explainable that the drastic loss of response for CO gas below 250°C is due to catalytic oxidation of CO by O 2 -(equ.(4)) 31 . Instead, the response to CO 2 is negligible at temperature higher than 250°C, and increases with decreasing the temperature. This opposite trend, indicate that Sr-doped ZnO sensors have the potential as CO 2 selective sensors at low operating temperatures. The dopant load has a great effect on the response against the tested gases. As shown in Figure 7, the 5% Ni doped sensor shows higher response than the sample with higher Ni loading. This agrees with the reports of other authors. For example, Wang and co-workers 33 reported that the response to C 2 H 2 has been greatly enhanced by Ni doping ZnO at the optimal doping concentration of 5%. The samples with the excess Ni concentration deteriorate with worse response. Ni5ZO sensor was also tested at different low CO concentrations (5-80 ppm) at the best operating temperature of 250°C. The dynamic responses and related calibration curve of the sensor are reported in Figure 8. The sensor response shows a linear behavior in the log-log plot with CO concentration. The good response allows to reaching a lower detection limit of less than 2 ppm. Figure 8. a) Dynamic response towards different CO concentrations and b) calibration curve at 250°C related to Ni5ZO sensor. The same finding has been observed for the CO 2 , being the Sr4ZO sensor, with an intermediate loading of Sr, the most responsive. In Figure 9 is shown the dynamic response to pulses of different CO 2 concentration, in the range of 0.25-2% in volume, at a temperature of 250°C. In these operative conditions the sensor is able to detect CO 2 with fast dynamic response. The response of Sr4ZO sensor to both gases at different temperatures is also shown, evidencing as the selectivity towards CO 2 for this sensor is maximized at the lower temperature tested (200°C). Overall results indicated that the Ni5ZO sensor, operating at the temperature of 250°C, is highly sensitive to CO and inherently selective with respect to CO 2 . As regards the Sr4ZO sensor, although decreasing the temperature the response to CO decrease and that to CO 2 increases, we cannot achieve the required selectivity towards this latter specie. In this case, statistical and signal processing techniques are necessary, for example, developing adaptive searching algorithms to retrieve each gas concentrations with improved precision, training data set evaluated by the twosensor array 34 . A comparison with the sensing performance of some recent CO sensors based on ZnO nanorods reported in the recent literature is shown in In case of CO response on Ni-ZnO sensors, a good correlation has been found taking into account the lattice distortion (see Fig. 10). Lattice distortion is advantageous for creating defects which are the sites for the interaction between test gas molecule and sensor surface 43 . For Sr-doped sensor, the decrease of operative temperature leads instead to a decrease of CO response and correspondingly to an enhancement of the response to CO 2 . As suggested by us for Ca-doped ZnO sensors, CO 2 sensing follows a mechanism relying on the adsorption of CO 2 on the sensing layer 44,45 . On this basis, the CO 2 response enhancement for Sr-doped ZnO could be related to the basicity of the dopant. In fact, as an acid molecule, the interaction of CO 2 with a solid material to form surface carbonates and hydroxycarbonates is favored if the surface basicity is increased. These intermediate species react subsequently with adsorbed oxygen species releasing electrons and consequently changing the resistance. Then, it seems that the improvement of the sensor response may be attributed to the higher adsorption of CO 2 provided by Sr-doped surface which is able to adsorb more strongly the acid CO 2 gas molecules compared to a undoped and Nidoped ZnO surface. Conclusion Sr-and Ni-doped ZnO nanorods were synthesized by simple wet chemical method. Characterization measurements revealed that they maintain the wurtzite hexagonal structure of pure ZnO. However, the decrease of I UV /I green photoluminescence peak ratio and XRD data indicate the presence of lattice distortion, due to the successfully incorporation of dopants within the crystalline structure of ZnO. The gas sensing characteristics of the chemo-resistive sensors developed with Sr-and Nidoped ZnO nanorods as sensing layer towards CO and CO 2 as target gases were found enhanced with respect to undoped ZnO-based sensor. In the specific, Ni-doped sensors exhibited high response to low concentration of CO in air. The enhanced CO response was attributed to the increased lattice distortion introduced by Ni which favors the interaction between CO gas molecules and the sensor surface. The CO 2 response enhancement for Sr-doped ZnO could be instead related to the basicity of the dopant. On the basis of this report, these simple doped ZnO nanostructures are promising to be applied in electronic devices for monitoring indoor air quality. Fig. 1 shows the XRD diffraction pattern of the pure, Sr-and Ni-doped ZnO samples. The diffraction peaks in the pattern of pure ZnO can be indexed to hexagonal wurtzite structured ZnO (space group: P63mc (186); a = 0.3253 nm, c = 0.5211 nm) in agreement with JCPDS card for ZnO (JCPDS 036-1451). The high intensity of the peaks demonstrates the good crystallinity of the samples. Figure 1 . 1XRD patterns of the powder sample (a) ZO ( . No remarkable difference in the grain size among the different samples was evinced. Instead, a shift of XRD peaks of doped ZnO to lower 2θ value relative to those of undoped ZnO is observed for the incorporation of Ni ions into ZnO. The (002) reflection peak in Ni5ZO show a slight shift towards a lower value relative to undoped ZnO. Further, (100) and (101) planes diffraction peaks are prominent than (002) plane, which indicates the change in the orientation/morphology with increasing Ni doping 6 . Figure 2 . 2Variation of lattice parameters a and c with (a) Ni doping and (b) Sr doping in pure ZnO. Figure 3 . 3SEM images of the powder sample (a) ZO ( . In pure ZnO average length and diameter of nanorods are about 1-2 µm and 200-400 nm respectively. In case of 5% Ni doping the average length and diameter of nanorods increase to 2-4 µm and 600-800 nm respectively. Again with higher Ni concentration there is no major change in length and diameter but with 10% Ni doping there is a change in morphology from rods into flakes, which is due to the incorporation of Ni ions in ZnO structure. In Sr 4% doping the range of length and diameter are 4-6 µm and 600-800 nm respectively while in Sr 8% there is no significant change in dimension and morphology of nanorods as compare to Ni doping ZnO. It has been observed that in higher Sr doping there is formation of bunches of rods as compared to lower doping where we are getting separate nanorods. Another interesting fact observed is the formation of pores in Sr-ZnO nanorods with increasing Sr concentration, due to large difference in ionic radii between the host matrix and the dopant. Fig. 4 4shows the PL spectra of Sr-and Ni-doped ZnO nanorods with different doping concentrations. Two peaks are observed in all samples, an ultraviolet (UV) emission (around 385 nm), attributed to near band-edge emission -namely, the recombination of free excitons through an exciton-exciton collision process, and a broad green emission peak (around 580 nm) arising from the intrinsic defects in the ZnO nanorods such as the Zn interstitials and the oxygen vacanciesrelated donor defects 17, 25-27 . Figure 4 . 4PL spectra at room temperature with excitation energy 325 nm in the range 330 nm to UV emission strongly relate to crystallite quality of ZnO. As clearly shown inFig. 4, where the PL curves have been normalized at the maximum of green emission peak at 580 nm, a decrease of UV emission follows the increase in Sr-and Ni-doping. By doping ZnO with high amount of Sr and Ni, these atoms go into lattice sites substituting for Zn atom causing lattice distortion and leading to decrease of UV emission peak. Green emission, centered at around 580 nm, is attributed to ZnO surface defect in the sample. Sr-and Ni-doping can also introduce impurity defects, such as interstitial and substitution defects, which contribute to the increase of visible (green) emission in PL, contributing to decrease the I UV /I green PL peak ratio. Has been reported in literature that decreasing I UV /I green PL peak ratio (i.e. corresponding to an increase in concentration of oxygen vacancies) can improve the sensing properties28 . In fact with increasing the concentration of oxygen vacancies as electron donors in ZnO 1−x lattice, the amount of active absorbed oxygen species ( Fig. 5) have been fabricated by printing thick films of the synthesized samples on the Pt interdigitated electrodes of an alumina support. Figure 5 . 5Photograph of the sensor device and micrograph of the sensing film surface. Inset shows a high magnification of the highly porous sensing film surface.The surface of the sensing layer shows a highly porous structure. Before sensing tests all the sensors prepared were heated up to 350°C and left to stabilize for a time of about 2 hours in order to stabilize the printed films. InFigure 6is shown a panoramic view of the dynamic response of some samples investigated towards CO and CO 2 at different temperatures. The response to a decrease of oxygen, from 20% to 10% (v/v), is also shown. Operating temperature was found to plays an important role in gas response of the investigated sensor. In the temperature range between 200°C and 350°C all samples show responses to CO and CO 2 . In accordance with n-type semiconductor behavior of the samples, after pulses at different concentration of CO and CO 2 in synthetic air, a decrease of the electric resistance is observed. Indeed, when a reducing gas such as CO is added, theinteraction of this gas with the surface chemisorbed oxygen, O 2-, can take place. The reducing gas readily releasing electrons back to the conduction band, according eq. (1), and the electrical resistance of the semiconductor decreases. CO + O 2-(ads) → CO 2 (g) + 2e -(1) Different sensing mechanisms are responsible of the electrical behavior of ZnO-based materials towards CO2. Among these, the reactions of CO2 with adsorbed OH − on the surface to form carbonates and hydroxycarbonates are the most important. These intermediate species react with adsorbed oxygen on the ZnO surface, releasing electrons and, consequently decreasing the resistance. Furthermore, at removal of the target gas and subsequent exposure again to synthetic air, all the samples are able to recover the baseline. For example, even at the low operating temperature of 200°C, the dynamic response shown by the Ni-ZnO sensor is rather fast ensuring rapid response times of about 15 seconds and recovery less than 90 seconds. -the unusual sensing behavior can be ascribed to the chemical reaction of CO molecule by the adsorption to the grain boundary of oxygen gas molecules which plays the key role in physical barrier for carrier's movement and increase the resistance 30 . The inversion of response indicate further that, at low temperature, the pure ZnO and Sr doped sensors behave as p-type semiconductors. In these materials, majority carriers are holes, so the electron released in equations(2) and (3) contribute to decrease their concentration and consequently the conductance decreases (resistance increases). Due to this fact, the response of the cited sensors is deteriorated. Instead, Nidoped samples work well also at lower temperature (up to 150°C for 10% Ni doped), showing no response inversion and are able to recover the baseline although with longer times. Figure 6 . 6Dynamic responses of sensors: a) ZO; b) Sr8ZO; c) Ni10ZO; d) Ni5ZO. Figure 7 7summarizes the responses recorded at different temperatures for pure and doped sensors when exposed to 1% CO 2 , 200 ppm CO and 10% variation of O 2 concentration, Figure 7 . 7Sensor responses a) 1% CO 2 ; b) 200 ppm CO; c) 10% O 2 at different temperatures. Figure 9 . 9a) Dynamic response towards different CO 2 concentrations and b) calibration curve at 250°C related to Sr4ZO sensor. c) Comparison of the response of Sr4ZO sensor to CO and CO 2 , at different temperatures. Figure 10 . 10Sensor response as a function of lattice distortion for Ni-doped ZnO. . This verifies that Sr atoms are really incorporating into ZnO. In Sr doped samples the lattice orientation is still along c-axis as compared to Ni doped samples. Hence, we conclude that Sr doping is maintaining c-axis orientation of pure ZnO. SEM images of the various samples reported inFig. 3 (a-e)reveal that Ni doping causes a change in the nanorods morphology, while Sr doping create pores in nanorods. Table 2 . 2It can be noted the higher response of our sensor compared to others where ZnO nanorods are doped with different dopants. The search has produced instead no literature data about CO 2 sensing using ZnO nanorods-based sensors.Table 2. Comparison of ZnO nanorods-based CO sensors.The above reported results demonstrate that the gas sensing characteristics of Ni-and Srdoped ZnO based sensors towards CO and CO 2 are remarkably enhanced compared to ZnO-based one. Indeed, both Sr and Ni increase the response and also shift the maximum response at lowertemperatures. An attempt is here made to correlate this behavior with the morphological and microstructural characteristics of the materials used as sensing layers.Characterization results give indication of the presence of a higher amount of crystal defects in the doped samples. The key role of crystal defects in gas sensing is highly accepted. Hu et al.proposed that the high sensing properties towards of transition-metal doped ZnO nanorods are related to a higher content of donor related defects and a lower content of acceptor related defects, because the donors would provide electrons for the adsorbed oxygen to produce the active ionosorbed oxygen41 . PL data here reported show a decrease of I UV /I green PL peak ratio with increase of Sr-and Ni-loading, suggesting an increase of the concentration of oxygen vacancies, which can improve the sensing properties42 . However, in our case, this factor cannot fully explain the decrease of response found on the Sr-and Ni-doped sensors with the higher dopant loading. This means that other factors, such as non-stoichiometry, lattice distortion and smaller grain size, are involved. On the basis of XRD characterization we can exclude the effects due to a decrease of particle size and to the formation of local p-n junction because in our case all Sr 2+ and Ni 2+ ions substitute the Zn 2+ atoms in ZnO lattice and no SrO or NiO phase is found13 . Also, the sensing characteristics reported for both Ni-and Sr-doped ZnO sensors do not correlate in any way with the surface area of the corresponding samples.Sensing material Temperature (°C) CO (ppm) Response (R a /R g ) Response/ppm × 100 Ref. Ni-ZnO nanorods 250 200 5 2.5 This work Ni-ZnO hexagonal plates 300 300 2.5 0.83 35 Co-ZnO electrodeposited nanorods Nanotubes 350 150 1.3 0.86 36 Au/ZnO nanorods 300 200 3 1.5 37 Al-ZnO nanorods 350 100 1.7 1.7 38 Ga-ZnO nanorods 75 200 1.1 0.55 39 CuO-ZnO nanorods 300 200 3.5 1.75 40 Acknowledgments Semiconductor gas sensors. R Jaaniso, O K Tan, ElsevierAmsterdamR. Jaaniso and O. K. Tan, Semiconductor gas sensors, Elsevier, Amsterdam, 2013. . G Neri, Chemosensors. 3G. Neri, Chemosensors, 2015, 3, 1-20. Zinc oxide bulk, thin films and nanostructures: processing, properties, and applications. C Jagadish, S J Pearton, ElsevierAmsterdamC. Jagadish and S. J. Pearton, Zinc oxide bulk, thin films and nanostructures: processing, properties, and applications: Elsevier, Amsterdam, 2011. . D C Reynolds, D C Look, B Jogai, C W Litton, G Cantwell, W C Harsch, Physical Review B. 602340D. C. Reynolds, D. C. Look, B. Jogai, C. W. Litton, G. Cantwell, and W. C. 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{'abstract': 'In this study, the effect of Sr-and Ni-doping on microstructural, morphological and sensing properties of ZnO nanorods has been investigated. Nanorods with different Sr and Ni loadings were prepared using a simple wet chemical method and characterized by means of scanning electron microscopy (SEM), X-ray diffraction (XRD) and photoluminescence (PL) analysis. XRD data confirmed that Sr-and Ni-doped samples maintainsthe wurtzite hexagonal structure of pure ZnO. However, unlikes Sr, Ni doping modifies the nanorod morphology, increases the surface area (SA) and decreases the ratio of I UV /I green photoluminescence peak to a greater extent. Sensing tests were performed on thick films resistive planar devices for monitoring CO and CO 2 , as indicators of indoor air quality.The effect of the operating temperature, nature and loading of dopant on the sensibility and selectivity of the fabricated sensors towards these two harmful gases were investigated. The gas sensing characteristics of Ni-and Sr-doped ZnO based sensors showed a remarkable enhancement (i. e. the response increased and shifted towards lower temperature for both gases) compared to ZnO-based one, demonstrating that these ZnO nanostructures are promising to fabricate sensor devices for monitoring indoor air quality.', 'arxivid': '1708.09577', 'author': ['Parasharam M Shirage paras.shirage@gmail.com*email:gneri@unime.it \nDepartment of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia\n', 'Amit Kumar \nDepartment of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia\n', 'Rana ', 'Yogendra Kumar ', 'Somaditya Sen \nDepartment of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia\n', 'S G Leonardi \nDepartment of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia\n\nDepartment of Engineering\nUniversity of Messina\n98166MessinaItaly\n', 'G Neri \nDepartment of Engineering\nUniversity of Messina\n98166MessinaItaly\n'], 'authoraffiliation': ['Department of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia', 'Department of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia', 'Department of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia', 'Department of Physics\nCentre of Materials Science and Engineering\nIndian Institute of Technology Indore\nSimrol Campus\nKhandwa Road453552IndoreIndia', 'Department of Engineering\nUniversity of Messina\n98166MessinaItaly', 'Department of Engineering\nUniversity of Messina\n98166MessinaItaly'], 'corpusid': 100046473, 'doi': '10.1039/c6ra15891a', 'github_urls': [], 'n_tokens_mistral': 12147, 'n_tokens_neox': 10040, 'n_words': 6224, 'pdfsha': '0d25d347ab763396b964708fd3e2c8b7dc3ba410', 'pdfurls': ['https://export.arxiv.org/pdf/1708.09577v1.pdf'], 'title': ['Sr-and Ni-doping in ZnO nanorods synthesized by simple wet chemical method as excellent materials for CO and CO 2 gas sensing', 'Sr-and Ni-doping in ZnO nanorods synthesized by simple wet chemical method as excellent materials for CO and CO 2 gas sensing'], 'venue': ['RSC Advances']}

arxiv

W-potentials in nonlinear biophysics of microtubules Slobodan Zdravković Institut za Nuklearne Nauke Vinča Univerzitet u Beogradu Laboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia Dragana Ranković Institut za Nuklearne Nauke Vinča Univerzitet u Beogradu Laboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia W-potentials in nonlinear biophysics of microtubules In the present article we investigate the nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton, and rely on the known general model. A crucial interaction among constitutive particles is modelled using W-potential. Three kinds of this potential are studied, symmetrical and two non-symmetrical. We demonstrate an advantage of the latter ones. Introduction There are two kinds of cells. These are eukaryotic, having a membranebound nucleus, and much simpler prokaryotic cells, without the nucleus. In the eukaryotes, an intracellular protein filament network exists. Microtubules (MTs), studied in this article, are the basic components of this cytoskeleton. They play essential roles in the shaping and maintenance of cells and in cell division. Also, MTs represent a traffic network for motor proteins moving along them. Information about their structure and function can be found in many references [1][2][3]. Here, we mention some basic pieces of information only. MT is a long hollow cylinder spreading between the nucleus and cell membrane. Its surface is usually formed of 13 long structures called protofilaments (PFs), representing a series of heterodimers, as shown in Fig. 1. A key point is that the heterodimer, or dimer for short, is an electric dipole. This means that MT behaves as ferroelectric [4], which is crucial for many models of MTs. In this paper, we rely on the so-called general model (GM) [5]. The lengths of MTs vary from a few hundred nanometers up to meters in long nerve axons [6]. For most of the models, a dimer is a constitutive unit, which means that its internal structure is not taken into consideration. Its mass and length are kg 10 , respectively [9]. Hence, z p is in the direction of MT. Fig. 1. Microtubule In Section 2, we very briefly explain the used model. For one of the interactions between the dimers, we use W-potential energy, or potential for short. We study three types of them and obtain three dynamic equations of motion. These are crucial equations, solved in Section 3, while Section 4 is devoted to some concluding remarks. W-potential within the general model of MTs It was mentioned above that MT behaves like a ferroelectrics. Based on this fact is the first model [4] in a series of models describing MT nonlinear dynamics. As the used coordinate is a longitudinal one, the model belongs to the group of longitudinal models, as well as its improved ancestor [10]. There are a few degrees of freedom for each dimer, but the essential is the angular one. Hence, angular models represent a crucial group of models describing MT dynamics. The first in this series is the so-called u-model [11], a perhaps somewhat naive but rather important step in the evolution of the models. The next one is the general model (GM), mentioned above [5], which we rely on in this paper. We conclude the series with the model introduced recently [12]. This two-component angular model has been under active investigation and is not relevant for this work. A starting point for all these models is Hamiltonian. For GM, it is [5]   where I is a moment of inertia of the single dimer, k is an inter-dimer stiffness parameter, 0  E is the intrinsic electric field, and 0  p stands for electric dipole moment. Notice that E is the internal electric field, which means that a particular dimer exists in the field of all other dimers. The angle n  describes the dimer`s oscillation and n is its position. We recognize the kinetic and potential energies of the interaction of the two neighbouring dimers belonging to the same PF. A term () W  represents the interaction of a single dimer with all other ones that do not belong to the same PF. It is called W-potential as it looks like a letter W, which will be clear later on. The very last term is coming from the fact that the electric dipole is in the field of all other ones. For this paper, the most important is the W-potential. We study the following three cases: Case 1. W-potential is a symmetric function:                ,(1)4 2 1 4 2 n n B A W      , 0 A  , 0 B  .(2) Case 2. W-potential is a non-symmetric function: n n n C B A W        4 2 2 4 2 .(3) Case 3. W-potential is a non-symmetric function: 3 4 2 3 3 4 2 n n n D B A W        .(4) We use Hamilton's dynamical equation n n H I         , a continuum approximation ( ) ( , ) n t x t   , series expansion of the cosine term, as well 10], and obtain the following dynamical equations of motion: as 2 2 2 1 2 1 l x l x n             [Case 1.   0 6 3 2 2 2 2 2                     t pE B pE A x kl t I      ,(5)Case 2.   0 6 3 2 2 2 2 2                      t C pE B pE A x kl t I      , (6) Case 3.   0 6 2 3 2 2 2 2 2                      t D pE B pE A x kl t I       ,(7) where  is a viscosity parameter. Hence, we obtained partial differential equations. It is well known that, for a given wave equation, a travelling wave     is a solution that depends upon x and t only through a unified variable t x      ,(8) where  and  are constants. According to Eqs. (5)-(8), we straightforwardly obtain ordinary differential equations (ODE): Case 1. 0 3              ,(9)Case 2. 0 3                ,(10)Case 3. 0 2 3                ,(11) where,    K pE B pE A     6 , pE A kl I    2 2 2    , pE A      ,(12)  pE A K C    ,            6 pE B pE A D  ,(13) and dd      . Experimental values of the involved parameters do not exist but our estimations strongly suggest pE A  and 6 pE B  [5], which was used above. Let us point out the great importance of the parameter  . From Eqs. (1), (5)-(7), and (12), we conclude that its negative sign means that the elastic term is larger than the inertial one, and vice versa. Also, Eq. (12) can be written as pE A c v I pE A I kl I pE A kl I          ) ( ) / / ( 2 2 2 2 2 2 2 2 2 2        ,(14) where v is the velocity of the solitary wave, while c is the speed of sound. This means that the sign of  shows if the wave is subsonic or supersonic. Solutions of equations (9) -(11) There are many mathematical procedures for solving Eqs. (9)- (11). One of the simplest is a procedure that we call the tangent hyperbolic function method (THFM). According to THFM, we expect the solution  as [13]           M i i i i i b a a 1 0  ,(15) where  is the solution of the well-known Riccati equation [10,13]. The one having physical sense is 2     b ,(16)   b b      tanh ,(17)                K K K K K K K ,(19) which is satisfied if all the coefficients i K are simultaneously equal to zero. Of course, for 0 1  b , the system is simplified. In what follows, we solve ODEs for all three cases, i.e., Eqs. (9)-(11). Case 1. This case was solved in Ref. [5], where the model we rely on was introduced. Using Mathematica, we easily obtain the following two solutions: 2 1 ) ( 0    a ,3 2 )(     a , 2 16 9    b , 9 2 2     ,(20) which yields                      4 3 tanh 1 2 1 ) ( 1 .(21) The functions 2    a  ,(22) where a is an arbitrary constant introduced in Eq. (18). This function is similar to the one shown in Fig. 2 were studied in Ref. [5]. No solution having physical sense, relevant for this article, was obtained. Case 2. Eq. (10) exists in Ref. [10], even though different models were established. Solutions, corresponding to Eq. (20), are [10]                 2 0 0 2 0 0 2 3 1 3 tanh 3 1 ) ( i i i i i a a a a , 3 , 2 , 1  i ,(26) shown in Fig. 4. According to Eqs. (13) and (23), we see that the approximation 0   yields to 0 0  a and 0   and, consequently, to the symmetric potential. 2 ) 1 ( 16  K b   ,(28), 4 0 ) 2 ( 0 K a    0 ) 2 ( 3 4 K a      ,     2 32 3 0 2 2 2 0 ) 2 (      K K b     (29) , 4 0 ) 3 ( 0 K a    0 ) 3 ( 3 4 K a      ,     2 0 2 2 0 ) 3 ( 128 2 3          K K b ,(30)where 4 2 0    K .(31) It may be useful to keep in mind that the different analytical procedures, or different kinds of software, can bring about the expressions for the parameter that looks different from those given by Eqs. (29)      K K b       (32) and         4 0 2 0 2 2 4 2 2 ) 3 ( 3 3 2 9 10 2 9 2 K K b                .(33)                4 tanh 2 2 ) ( 0 0 31 K K ,(34)                    4 3 tanh 1 4 ) ( 0 0 32 K K ,(35) Discussions and future research Each of the three potentials given by Eqs. (red). Both transitions can be seen in Fig. 4, but there is one more. This is L R    (black), representing a transition from a deeper minimum to a shallower one. An obvious conclusion is that the non-symmetric potential 2 W is better than the symmetric one. Also, the same conclusion is suggested by the geometry of MTs. The potentials 2 W and 3 W bring about Figs. 4 and 5. These figures are basically of the same value, i.e., describe all three transitions. This means that it is difficult to state which potential is better for MT modelling. In the case of 3 W , the transition L R    goes slowly in comparison to 2 W (black lines), but this depends on the values of the parameters  and  . This might mean that 3 W is better but we are not ready for any suggestions without further research. We should notice that 0 M   in cases 1 W and 3 W . This indicates that the unstable orientations of the dimers are exactly in the direction of PF. Unfortunately, the orientation of the dimers has not been experimentally determined yet. We can state that the potentials 2 W and 3 W are more convenient for MT modelling than the symmetric one. This issue should be studied within a new model [12]. The equations (5)- (7) were solved using the continuum approximation. The question of whether MT is a discrete or continuum system was studied in Ref. [14]. we are not interested in diverging solutions in this work, and Eq. (15) becomes. Fig. 2 . 2Kink a . This case corresponds to the positive b , which brings about the diverging solution [10]. The finite solutions, i.e., the functions i 2  , are determined by Eqs. (17), (18), and (23)-(25). They are Fig. 3 .Fig. 4 . 34The Kink Case 3 . 3It was explained above that Eq. (19) gives a system of equations from which we obtain values of the parameters determining the function  . ( 2 ) 2-(4) has two minima and one maximum. In terms of the functions  , we can talk of the right and left minima, and of the maximum, denoted as R  , L  , and M  , respectively. Let us compare Figs. 2 and 4. The solutions in Fig. Fig. 5 . 5Kink e-mail address: szdjidji@vin.bg.ac.rs † e-mail address: dragana.rankovic@pharmacy.bg.ac.rs  , can be obtained according to Eqs. (17), (18), and (28)-(31). They areTherefore, all the three expressions are identical, which might not be obvious at first glance. The final solutions, i.e., the functions ) 1 ( 3  , ) 2 ( 3  , and ) 3 ( 3 . P Dustin, Microtubules, Springer, BerlinP. Dustin, Microtubules, Springer, Berlin, 1984. . S Zdravković, J. Serb. Chem. Soc. 825469S. Zdravković, J. Serb. Chem. Soc. 82 (5) (2017) 469. Mechanical Models of Microtubules. S Zdravković, In Complexity in Biological and Physical Systems. Ricardo Lopez-RuizS. Zdravković, Mechanical Models of Microtubules, In Complexity in Biological and Physical Systems, Edited by Ricardo Lopez-Ruiz, Chapter 1, IntechOpen, 2018. . M V Satarić, J A Tuszyński, R B Žakula, Phys. Rev. E. 48589M. V. Satarić, J. A. Tuszyński and R. B. Žakula, Phys. Rev. E 48 (1993) 589. . S Zdravković, M V Satarić, V Sivčević, Nonlinear Dyn, 92479S. Zdravković, M. V. Satarić and V. Sivčević, Nonlinear Dyn. 92 (2018) 479. . S Hameroff, R Penrose, Phys. Life Rev. 1139S. Hameroff and R. Penrose, Phys. Life Rev. 11 (2014) 39. . S Sahu, S Ghosh, K Hirata, D Fujita, A Bandyopadhyay, Appl. Phys. Lett. 102123701S. Sahu, S. Ghosh, K. Hirata, D. Fujita and A. Bandyopadhyay, Appl. Phys. Lett. 102 (2013) 123701. . D Havelka, M Cifra, O Kučera, J Pokorný, J Vrba, J. Theor. Biol. 28631D. Havelka, M. Cifra, O. Kučera, J. Pokorný, J. Vrba, J. Theor. Biol. 286 (2011) 31. . J E Schoutens, J. Biol. Phys. 3135J. E. Schoutens, J. Biol. Phys. 31 (2005) 35. . S Zdravković, L Kavitha, M V Satarić, S Zeković, J Petrović, Chaos Solitons Fract. 451378S. Zdravković, L. Kavitha, M. V. Satarić, S. Zeković and J. Petrović, Chaos Solitons Fract. 45 (2012) 1378. . S Zdravković, M V Satarić, A Maluckov, A Balaž, Appl. Math. Comput. 237227S. Zdravković, M. V. Satarić, A. Maluckov and A. Balaž, Appl. Math. Comput. 237 (2014) 227. . S Zdravković, S Zeković, A N Bugay, J Petrović, Chaos Soliton Fract. 152111352S. Zdravković, S. Zeković, A. N. Bugay and J. Petrović, Chaos Soliton Fract. 152 (2021) 111352. . S A El-Wakil, M A Abdou, Chaos Solitons Fract. 31840S. A. El-Wakil and M. A. Abdou, Chaos Solitons Fract. 31 (2007) 840. . S Zdravković, A Maluckov, M Đekić, S Kuzmanović, M V Satarić, Appl. Math. Comput. 242353S. Zdravković, A. Maluckov, M. Đekić, S. Kuzmanović and M. V. Satarić, Appl. Math. Comput. 242 (2014) 353.

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{'abstract': 'In the present article we investigate the nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton, and rely on the known general model. A crucial interaction among constitutive particles is modelled using W-potential. Three kinds of this potential are studied, symmetrical and two non-symmetrical. We demonstrate an advantage of the latter ones.', 'arxivid': '2301.08724', 'author': ['Slobodan Zdravković \nInstitut za Nuklearne Nauke Vinča\nUniverzitet u Beogradu\nLaboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia\n', 'Dragana Ranković \nInstitut za Nuklearne Nauke Vinča\nUniverzitet u Beogradu\nLaboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia\n'], 'authoraffiliation': ['Institut za Nuklearne Nauke Vinča\nUniverzitet u Beogradu\nLaboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia', 'Institut za Nuklearne Nauke Vinča\nUniverzitet u Beogradu\nLaboratorija za Atomsku Fiziku (040)11001, 11221Beograd, BeogradSerbia, Serbia'], 'corpusid': 256080821, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6508, 'n_tokens_neox': 5347, 'n_words': 2822, 'pdfsha': '875c1de00854e6474e848912f29f2c8924724305', 'pdfurls': ['https://export.arxiv.org/pdf/2301.08724v1.pdf'], 'title': ['W-potentials in nonlinear biophysics of microtubules', 'W-potentials in nonlinear biophysics of microtubules'], 'venue': []}

arxiv

Micromagnetic structures: flexomagnetoelectric coupling point symmetry and Neumann's principle 7 Apr 2015 B M Tanygin b.m.tanygin@gmail.comb.m.tanygin Faculty of Radiophysics Taras Shevchenko National University of Kyiv Acad. Glushkov Ave 4G, 03187KyivUkraine Micromagnetic structures: flexomagnetoelectric coupling point symmetry and Neumann's principle 7 Apr 2015Preprint submitted to Physica B: Physics of Condensed Matter April 9, 2015arXiv:1501.03140v2 [cond-mat.str-el]Micromagnetic structuresFlexomagnetoelectric interactionPoint symmetryNeumann's principle Recent series of articles (B.M. Tanygin, 2011(B.M. Tanygin, -2012 has been aimed to study flexomagnetoelectric properties of magnetically ordered crystals by means of a magnetic point symmetry description. Besides its fundamental interest (complete symmetry classification), applications to a classical micromagnetic variational problem of a thermodynamically equilibrium state calculation had been provided without a proof. It was shown that the magnetic point symmetry group determination of the given micromagnetic structure could be used to predict qualitatively the electric polarization distribution induced by a flexomagnetoelectric coupling. This article is an important addendum which provides a formal mathematical explanation why and how a symmetry-based description of flexomagnetoelectric phenomena conforms to a quantitative calculation of an equilibrium state of a magnetization and electrical polarization vectors spatial distribution. An application of the Neumann's principle and a group-theoretical methodology to real micromagnetic structures is clarified as well. The well-known role of a symmetry-based approach in new interactions and phenomena discoveries has been discussed. Introduction A group-theoretical approach was originally well applied to ferroelectric domain structures [1]. A magnetic point symmetry classification of different micromagnetic structures had been established [2][3][4][5][6][7][8][9][10][11][12][13]: the complete list of magnetic point groups is obtained for magnetic domain walls, Bloch lines, Bloch points, magnetic vortex and skyrmion structures. An important practical feature of this symmetry classification is its ability to predict qualitatively [3,5,9,10] a magnetization M and/or electric polarization P vector equilibrium spatial distribution: an even/odd/zero/other type of components M i (r j ) and P i (r j ) had been identified for each point group. Hence, a knowledge of the magnetic point group G P of a crystal sample simultaneous paramagnetic and paraelectric phase [14] allows to describe (bypassing calculations) equilibrium spatial distributions of magnetization and polarization with [9] or without [8] their magnetoelectric coupling. Although this statement is very natural in the framework of the group-theoretical consideration, it requires a proof in the scope of the micromagnetics [15] and the Ginzburg -Landau theory [16]. Also, quantitative parameters cannot be obtained from a symmetry theory. The quantitative calculation of an equi-librium state of a magnetization and electrical polarization vectors spatial distribution is a solution of micromagnetic problem, which is based on a free energy functional minimization [15,16]. On the other hand, the inhomogeneous magnetoelectric coupling [17] (also known as flexomagnetoelectric (FME) coupling [18]) free energy density expression is defined by a crystal symmetry [19]. Hence, two approaches are obviously related: a variational problem solution and a symmetry-based prediction. This relation was shown only for some micromagnetic structures without a general proof. The derivation of this proof as well as a methodological analysis is the purpose of this work. As a constructive example, only Bloch lines in a planar domain wall has been considered. A generalization to other micromagnetic structures is straightforward. Theory A calculus of variations of a total free energy functional Φ is given by: δ [F M (r, M, ∇ x M, ∇ y M, ∇ z M) + +F P (r, P, ∇ x P, ∇ y P, ∇ z P) + +F ME (r, M, P, ∇ x M, ∇ y M, ∇ z M)]d 3 r = 0,(1) where XYZ is the reference Cartesian crystallophysical coordinate system [20]; the ferromagnetic F M , ferroelectric F P , and flexomagnetoelectric F ME [19] components of the local free energy density is integrated over the whole crystal sample volume. The equation (1) could be supplemented by additional conditions, like the equality |M| = M S for a ferromagnetic material. This problem conventionally reduces to a set of Euler-Lagrange equations which can be solved numerically in the general case [21]. The FME interaction free energy density (here and hereinafter, Einstein notation is being used): F ME = γ ijkl P i M j ∇ k M l(2) is an invariant of a magnetic point group of a spatially unlimited crystal in a paramagnetic and paraelectric phase G ∞ P [17,19]: g · γ ijkl P i M j ∇ k M l = γ ijkl P i M j ∇ k M l ,(3) Here and hereinafter, a magnetic point symmetry transformation g ∈ G ∞ P is applying to individual vectors and an operator ∇, which is treated as a polar time-even vector. The FME interaction tensor γ ijkl is a material constant. A total free energy minimization in scope of the variational problem (1) leads to the following expression [22,23]: P i (r) = −χ e (r) γ ijkl M j (r) ∇ k M l (r) ,(4) where χ e (r) is a dielectric susceptibility of a crystal paramagnetic phase. In case of a spatially restricted crystal sample, the magnetic point group G P of a crystal sample in the simultaneous paramagnetic and paraelectric phase [14] is derived by means of the Curie symmetry principle: G P = G ∞ P ∩ G S ,(5) where G S is a magnetic point group of a crystal sample shape. In case of a thin film or bulk plate, it is given by the limit Curie group supplemented by a time-reversal operation 1 ′ : G S = ∞/mmm1 ′(6) The symmetry transformationsg ∈ U l of a Bloch line magnetic point group U l : U l ⊂ G P(7) correspond to functional dependencies of order parameters [3,4,13]: g · M (r) = M (g · r) (8) g · P (r) = P (g · r)(9) The detailed algorithm and the table data of the magnetic point group derivation depending on the boundary conditions and the domain wall plane as well as Bloch line orientation had been published before [8,11,14]. According to (5) and (7 -9), the crystal sample free surfaces symmetry (G S ) perturbs the spatial distribution of electric polarization and magnetization vectors. Although a magnetic point symmetry of this perturbation can be the same for the same experimental sample shape like (6), an ab initio mechanism of this impact can be different: the strain-induced magnetocrystalline anisotropy, the demagnetization and depolarization phenomenon, the surface anisotropy [24], the boundary conditions of inhomogeneous magnetostrictive and piezoelectric effects, the non-local magnetoelectric coupling [4], etc. As a rule, each dependency M i (r j ) and P i (r j ) (any combination: i = x, y, z and j = x, y, z) is a sum (F ) of an even (S) and an odd (A) function (the notation [2,11] is being used) in the triclinic-pedial (G ∞ P = 1 ′ ) magnetically ordered crystal. In case of an arbitrary crystal, aside of the transformation 1 ′ , additional crystallographic magnetic point symmetry transformationsg apply sequentially restrictions (8,9) to the components of the dependencies M (r) and P (r) compare to the lowest symmetry case. Possible transformationsg are n, n ′ ,n, andn ′ ; where n ∈ {1, 2, 3, 4, 6}. This leads to the dependencies M i (r j ) and P i (r j ) symmetry increase: (F ) → (S) (F ) → (A)(10) Let us consider (here and hereinafter) only classical planar micromagnetic structures (scope of [11]), i.e. a planar domain wall with a Bloch line inside: [13]. Then, the group U l consists of the transformationsg with can be x, y, or z in the crystallophysical coordinate system [20]), both sides of the equation (4) changes sign by the transformationg only simultaneously due to the requirement (3) taking into account (7). U l ⊂ mmm1 ′ Hence, the equation (4) satisfies the components functional dependence parity requirements (8,9). Former one is a predictive capability in scope of a magnetic symmetry consideration of the given crystal. Latter one is a solution of the FME coupling variational problem (1): a magnetization is treated as a known function and a polarization is treated as an unknown one, which means that a stable micromagnetic structure induces a polarization by FME coupling. Important to note, that any requirements satisfaction by the equation (4) does not mean that a solution of (1) cannot have other symmetries, which will be considered in the next section. The Neumann's principle application If some component of M (r) is zero in terms of a total free energy minimization and its components type of a spatial parity cannot be found in the tables with the magnetic point groups list [11] then the Bloch line group U l should be selected in a way taking into account the rule that a zero functional dependence is a particular case of an even or odd dependency: (0) → (A) or (0) → (S) or (0) → (F ) for any component M i (r j ) and P i (r j ). In this case, the group U l is a medium symmetry after a spontaneous symmetry breaking G P → U l through a magnetic ordering; and a magnetic point group V of the variational problem solution [M (r) , P (r)] represents a physical properties symmetry (cf. Neumann's principle [25,26]): V ⊇ U l ⊂ G P(11) Here, the group V describes completely a magnetic point symmetry of equilibrium magnetization and polarization spatial distributions in a framework of a given model (1) independently on a crystal sample symmetry G P . As an example of this concept: a magnetic vortices and skyrmion point symmetry has been described by a rotational infinity-fold symmetry axis ∞ z [12] even though this symmetry element does not exist in crystallographic groups [20]. Same methodology should be followed in the case when a particular set of even and odd functions M i (r j ) and P i (r j ) parity types cannot be found in the tables published in the Ref. [11,13]. According to Neumann's principle, It is important to note, that as it was shown in the Ref. [3], if a variational problem solution provides a too highly symmetrical solution then it means that a some free energy invariant has been omitted. Hence, described here methodology of the magnetic symmetry-based predictions application to the real variational problem solutions can be used as a pointer towards the new material properties search. Let us consider a similar situation in case of a FME coupling. The wellknown expression of the FME coupling [22,27,28] is given by a Lifshitz invariant-like single-constant coupling term: F ME =γ ijkl P i (M j ∇ k M l − M l ∇ k M j ) ,(12) which can be derived from the equation (2) P = −Γ [e 12 × [S 1 × S 2 ]] ,(13) where Γ is a material constant; The described type of the FME coupling (12) requires two non-zero magnetization components. According to the term (12), magnetic domain walls or spin-density-wave with a collinear magnetic ordering cannot induce an electric polarization. Oppositely, according to the [11], if at least one magnetization components M α (r) = 0 then nonzero polarization components can be induced by the FME coupling by means of the free energy density term [19]: f 1 = γ iαiα P i ∇ i M 2 α /2(14) in case of G ∞ P belonging to a cubic, orthorhombic system, or tetragonal system (dihedral groups only) [19]. A formation of multiferroic phases at the magnetic commensurability transitions of YMn 2 O 5 has been explained by the provided FME coupling type (14) as a phase dislocation related mechanism [23], which is suppressed by the well-known expression (12) as a total spatial derivative term 14. Hence, a progress of an experimental and theoretical research of a novel FME coupling mechanism leads to the same conclusion as a symmetry based one (4) published originally [4,17]. Generally, according to the Neumann's principle, a zero value of a specific component P β (r) is possible in the variational problem solution with given model assumptions. Example is a saturation magnetization M (r) = const ≡ M S constancy in a case of a ferromagnetic material leading to the suppression of the invariant (14) in crystals of a cubic system [19]. Also, it is possible in a case of a given measurement tolerance and a macroscopic level of consideration corresponding to a physically infinitesimal volume selection in a research program. Here, if a symmetry-based prediction allows P β (r) = 0 then a physical mechanism of the FME coupling should be searched: an increase of a variational problem solution accuracy; a considering of all nonzero (allowed by G ∞ P ) values of material tensor components γ ijkl ; looking for a new integro-differential [4,19] expressions of a total free energy additionally to the (3). The Bloch line always contains direction of a continuous translational symmetry. Ifg · (e y r) = − (e y r) andg · M i = −M i org · P i = −P i then corresponding components should be zero [11]. This is not the case for Bloch points [13], which are more general micromagnetic structures compare to Bloch lines. In other words, a Bloch line is a particular case of a Bloch point. The Neumann's principle application takes place here as well. Magnetic point group U l of the Bloch point never requires M α (r) = 0 or P β (r) = 0. It was shown that that the set of all possible groups U l (magnetic point symmetry classification) of Bloch points is identical to the one of the Bloch lines [13]. There are 48 magnetic point groups of the Bloch points including 22 (11 time-invariant and 11 time-noninvariant ones) enantiomorphic and 26 non-enantiomorphic groups. Conclusion Hence, symmetry-based predictions of the FME phenomena correspond to a variational problem solution. Neumann's principle corresponds to removal of negligible interaction mechanisms or a model related coarse-grained approximation. It can be treated as a designation for further discovery of new physical interactions in magnetically and electrically ordered media including a coupling of these two subsystems. n = 1 or 2 only. Moduli of the vector components M i and P i are not changing by such symmetry transformations. For each multi-index (ijkl) (each index following symmetry reduction is possible in case of group U l selection: (A) → (F ) or (S) → (F ) for any component M i (r j ) or P i (r j ) in the given Bloch line. It means that an even and odd dependency is a particular case of an arbitrary dependency (F ). by a removal of total spatial derivatives ∇ k M l M j . It was shown [10] that this step cannot be justified by the medium symmetry invariance requirements (3). Hence, a reason of a wide usage of the expression (12) is related to an experimental experience; i.e. a predominance of a specific microscopical mechanism in case of some published experimental observations of the FME phenomena. A microscopic nature of an exchange-relativistic [4] FME interaction has been suggested based on the spin supercurrent model with the similar expression for the cases (different constants) of the superexchange and double-exchange interactions [29]: S 1 and S 2 2are spins of transition metal ions in the M − O − M three-atom triad. The e 12 is a polar time-even unit vector directed along the line segment connected metal ions. A symmetry approach to domain structures. V Janovec, Ferroelectrics. 12V. Janovec, A symmetry approach to domain structures, Ferroelectrics 12 (1976) 43-53. Magnetic symmetry of the domain walls in magnetically ordered crystals. V G Bar&apos;yakhtar, V A L&apos;vov, D A Yablonskiy, Sov. Phys. JETP. 60V. G. Bar'yakhtar, V. A. L'vov, D. A. Yablonskiy, Magnetic symmetry of the domain walls in magnetically ordered crystals, Sov. Phys. JETP 60 (1984) 1072-80. Magnetic symmetry of the domain walls with Bloch lines in ferromagnets and ferrites. V G Bar&apos;yakhtar, E B Krotenko, D A Yablonskiy, Sov. Phys. JETP. 64V. G. Bar'yakhtar, E. B. Krotenko, D. A. Yablonskiy, Magnetic sym- metry of the domain walls with Bloch lines in ferromagnets and ferrites, Sov. Phys. JETP 64 (1986) 542-8. Theory of electric polarization of domain boundaries in magnetically ordered crystals. V G Bar&apos;yakhtar, E B Krotenko, D A Yablonskiy, Problems in solid-state physics. V. G. Bar'yakhtar, E. B. Krotenko, D. A. Yablonskiy, Theory of electric polarization of domain boundaries in magnetically ordered crystals, in: Problems in solid-state physics, 1984, pp. 56-80. Spontaneous polarization and/or magnetization in non-ferroelastic domain walls: symmetry predictions. J Privratska, V Janovec, Ferroelectrics. 222J. Privratska, V. Janovec, Spontaneous polarization and/or magneti- zation in non-ferroelastic domain walls: symmetry predictions, Ferro- electrics 222 (1999) 23-32. Magnetic Point Group Symmetries of Spontaneously Polarized and/or Magnetized Domain Walls. J Pívratská, B Shaparenko, V Janovec, D B Litvin, Ferroelectrics. 269J. Pívratská, B. Shaparenko, V. Janovec, D. B. Litvin, Magnetic Point Group Symmetries of Spontaneously Polarized and/or Magnetized Do- main Walls, Ferroelectrics 269 (2002) 39-44. Possible Appearance of Spontaneous Polarization and/or Magnetization in Domain Walls Associated with Non-Magnetic and Non-Ferroelectric Domain Pairs. J Pívratská, Ferroelectrics. 353J. PÍvratská, Possible Appearance of Spontaneous Polarization and/or Magnetization in Domain Walls Associated with Non-Magnetic and Non-Ferroelectric Domain Pairs, Ferroelectrics 353 (2007) 116-23. Magnetic symmetry of the plain domain walls in ferro-and ferrimagnets. B Tanygin, O Tychko, Physica B: Condensed Matter. 404B. Tanygin, O. Tychko, Magnetic symmetry of the plain domain walls in ferro-and ferrimagnets, Physica B: Condensed Matter 404 (2009) 4018-22. Symmetry theory of the flexomagnetoelectric effect in the magnetic domain walls. B Tanygin, Journal of Magnetism and Magnetic Materials. 323B. Tanygin, Symmetry theory of the flexomagnetoelectric effect in the magnetic domain walls, Journal of Magnetism and Magnetic Materials 323 (2011) 616-9. On the free energy of the flexomagnetoelectric interactions. B Tanygin, Journal of Magnetism and Magnetic Materials. 323B. Tanygin, On the free energy of the flexomagnetoelectric interactions, Journal of Magnetism and Magnetic Materials 323 (2011) 1899-902. Symmetry theory of the flexomagnetoelectric effect in the Bloch lines. B Tanygin, Journal of Magnetism and Magnetic Materials. 324B. Tanygin, Symmetry theory of the flexomagnetoelectric effect in the Bloch lines, Journal of Magnetism and Magnetic Materials 324 (2012) 1659-63. Symmetry theory of the flexomagnetoelectric interaction in the magnetic vortices and skyrmions. B Tanygin, Physica B: Condensed Matter. 407B. Tanygin, Symmetry theory of the flexomagnetoelectric interaction in the magnetic vortices and skyrmions, Physica B: Condensed Matter 407 (2012) 868-72. Magnetoelectric properties and symmetry classification of micromagnetic structures. B Tanygin, KyivTaras Shevchenko National University of Kyiv (Faculty of Radiophysics) and Institute of Magnetism NASU and MESYSUPh.D. ThesisPh.D. thesisB. Tanygin, Ph.D. Thesis "Magnetoelectric properties and symme- try classification of micromagnetic structures", Ph.D. thesis, Taras Shevchenko National University of Kyiv (Faculty of Radiophysics) and Institute of Magnetism NASU and MESYSU, Kyiv, 2012. Magnetic symmetry of the plain domain walls in the plates of cubic ferro-and ferrimagnets. B M Tanygin, O V Tychko, Acta Physica Polonica-Series A General Physics. 117215B. M. Tanygin, O. V. Tychko, Magnetic symmetry of the plain do- main walls in the plates of cubic ferro-and ferrimagnets, Acta Physica Polonica-Series A General Physics 117 (2010) 215. . W F Brown, Micromagnetics , Interscience Publishers New York18W. F. Brown, Micromagnetics, 18, Interscience Publishers New York, 1963. A Landau primer for ferroelectrics. P Chandra, P B Littlewood, Physics of ferroelectrics. SpringerP. Chandra, P. B. Littlewood, A Landau primer for ferroelectrics, in: Physics of ferroelectrics, Springer, 2007, pp. 69-116. Inhomogeneous magnetoelectric effect. V G Bar&apos;yakhtar, V A L&apos;vov, D A Yablonskiy, JETP Letters. 37V. G. Bar'yakhtar, V. A. L'vov, D. A. Yablonskiy, Inhomogeneous mag- netoelectric effect, JETP Letters 37 (1983) 673-5. Flexomagnetoelectric interaction in multiferroics. A P Pyatakov, A K Zvezdin, The European Physical Journal B. 71A. P. Pyatakov, A. K. Zvezdin, Flexomagnetoelectric interaction in multiferroics, The European Physical Journal B 71 (2009) 419-27. Flexomagnetoelectric interaction in cubic, tetragonal and orthorhombic crystals. B Tanygin, Journal of Magnetism and Magnetic Materials. 324B. Tanygin, Flexomagnetoelectric interaction in cubic, tetragonal and orthorhombic crystals, Journal of Magnetism and Magnetic Materials 324 (2012) 1878-81. I Zheludev, Symmetry and its applications. MoscowEnergoatomizdatI. Zheludev, Symmetry and its applications, Energoatomizdat, Moscow, 1983. A Systematic Approach to Multiphysics Extensions of Finite-Element-Based Micromagnetic Simulations: Nmag. T Fischbacher, M Franchin, G Bordignon, H Fangohr, IEEE Transactions on Magnetics. 43T. Fischbacher, M. Franchin, G. Bordignon, H. Fangohr, A System- atic Approach to Multiphysics Extensions of Finite-Element-Based Mi- cromagnetic Simulations: Nmag, IEEE Transactions on Magnetics 43 (2007) 2896-8. Ferroelectricity in spiral magnets. M Mostovoy, Physical Review Letters. 9667601M. Mostovoy, Ferroelectricity in spiral magnets, Physical Review Letters 96 (2006) 067601. Multiferroicity Induced by Dislocated Spin-Density Waves. J Betouras, G Giovannetti, J Van Den, Brink, Physical Review Letters. 98257602J. Betouras, G. Giovannetti, J. van den Brink, Multiferroicity Induced by Dislocated Spin-Density Waves, Physical Review Letters 98 (2007) 257602. Surface anisotropy in micromagnetics. A Aharoni, Journal of Applied Physics. 613302A. Aharoni, Surface anisotropy in micromagnetics, Journal of Applied Physics 61 (1987) 3302. Generalized symmetry and Neumanns principle. S Bhagavantam, P V Pantulu, Proceedings of the Indian Academy of Sciences -Section A. 66S. Bhagavantam, P. V. Pantulu, Generalized symmetry and Neumanns principle, Proceedings of the Indian Academy of Sciences -Section A 66 (1967) 33-9. Magnetic symmetry, improper symmetry, and Neumann's principle. E J Post, Foundations of Physics. 8E. J. Post, Magnetic symmetry, improper symmetry, and Neumann's principle, Foundations of Physics 8 (1978) 277-94. Electric-field effects on the spindensity wave in magnetic ferroelectrics. A Sparavigna, A Strigazzi, A Zvezdin, Physical Review B. 50A. Sparavigna, A. Strigazzi, A. Zvezdin, Electric-field effects on the spin- density wave in magnetic ferroelectrics, Physical Review B 50 (1994) 2953-7. . A P Pyatakov, D A Sechin, A S Sergeev, A V Nikolaev, E P , A. P. Pyatakov, D. A. Sechin, A. S. Sergeev, A. V. Nikolaev, E. P. Magnetically switched electric polarity of domain walls in iron garnet films. A S Nikolaeva, A K Logginov, Zvezdin, Europhysics Letters). 9317001EPLNikolaeva, A. S. Logginov, A. K. Zvezdin, Magnetically switched electric polarity of domain walls in iron garnet films, EPL (Europhysics Letters) 93 (2011) 17001. Spin Current and Magnetoelectric Effect in Noncollinear Magnets. H Katsura, N Nagaosa, A V Balatsky, Physical Review Letters. 9557205H. Katsura, N. Nagaosa, A. V. Balatsky, Spin Current and Magne- toelectric Effect in Noncollinear Magnets, Physical Review Letters 95 (2005) 057205.

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{'abstract': "Recent series of articles (B.M. Tanygin, 2011(B.M. Tanygin, -2012 has been aimed to study flexomagnetoelectric properties of magnetically ordered crystals by means of a magnetic point symmetry description. Besides its fundamental interest (complete symmetry classification), applications to a classical micromagnetic variational problem of a thermodynamically equilibrium state calculation had been provided without a proof. It was shown that the magnetic point symmetry group determination of the given micromagnetic structure could be used to predict qualitatively the electric polarization distribution induced by a flexomagnetoelectric coupling. This article is an important addendum which provides a formal mathematical explanation why and how a symmetry-based description of flexomagnetoelectric phenomena conforms to a quantitative calculation of an equilibrium state of a magnetization and electrical polarization vectors spatial distribution. An application of the Neumann's principle and a group-theoretical methodology to real micromagnetic structures is clarified as well. The well-known role of a symmetry-based approach in new interactions and phenomena discoveries has been discussed.", 'arxivid': '1501.03140', 'author': ['B M Tanygin b.m.tanygin@gmail.comb.m.tanygin \nFaculty of Radiophysics\nTaras Shevchenko National University of Kyiv\nAcad. Glushkov Ave\n4G, 03187KyivUkraine\n'], 'authoraffiliation': ['Faculty of Radiophysics\nTaras Shevchenko National University of Kyiv\nAcad. Glushkov Ave\n4G, 03187KyivUkraine'], 'corpusid': 118617934, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7133, 'n_tokens_neox': 6102, 'n_words': 3656, 'pdfsha': '670f2f031435b8f52ed6774ad93f98955c6745ec', 'pdfurls': ['https://arxiv.org/pdf/1501.03140v2.pdf'], 'title': ["Micromagnetic structures: flexomagnetoelectric coupling point symmetry and Neumann's principle", "Micromagnetic structures: flexomagnetoelectric coupling point symmetry and Neumann's principle"], 'venue': []}

arxiv

Characterization of Excited States in Time-Dependent Density Functional Theory Using Localized Molecular Orbitals Souloke Sen Division of Theoretical Chemistry Faculty of Sciences Vrije Universiteit Amsterdam De Boelelaan 10831081 HVAmsterdamThe Netherlands Bruno Senjean ICGM Université de Montpellier CNRS ENSCM MontpellierFrance Lucas Visscher Division of Theoretical Chemistry Faculty of Sciences Vrije Universiteit Amsterdam De Boelelaan 10831081 HVAmsterdamThe Netherlands Characterization of Excited States in Time-Dependent Density Functional Theory Using Localized Molecular Orbitals (Dated: 14 December 2022) Localized molecular orbitals are often used for the analysis of chemical bonds, but they can also serve to efficiently and comprehensibly compute linear response properties. While conventional canonical molecular orbitals provide an adequate basis for the treatment of excited states, a chemically meaningful identification of the different excited-state processes is difficult within such a delocalized orbital basis. In this work, starting from an initial set of supermolecular canonical molecular orbitals, we provide a simple one-step top-down embedding procedure for generating a set of orbitals which are localized in terms of the supermolecule, but delocalized over each subsystem composing the supermolecule. Using an orbital partitioning scheme based on such sets of localized orbitals, we further present a procedure for the construction of local excitations and charge-transfer states within the linear response framework of time-dependent density functional theory (TDDFT). This procedure provides direct access to approximate diabatic excitation energies and, under the Tamm-Dancoff approximation, also their corresponding electronic couplings -quantities that are of primary importance in modelling energy transfer processes in complex biological systems. Our approach is compared with a recently developed diabatization procedure based on subsystem TDDFT using projection operators, which leads to a similar set of working equations. Although both of these methods differ in the general localization strategies adopted and the type of basis functions (Slaters vs. Gaussians) employed, an overall decent agreement is obtained. I. INTRODUCTION Time-dependent density functional theory [1][2][3][4][5] (TDDFT) based on the Kohn-Sham 6 (KS) ground-state determinant is a popular method for the calculation of low-lying excited states of medium to large-sized molecules, owing to its favourable accuracy to cost ratio. 4,5,[7][8][9] After the advent of range-separated density functionals that are better suited for describing charge-transfer excitations, [10][11][12][13][14][15][16][17] TDDFT has become the dominant approach for the study of photo-induced processes in biochromophoric systems. 7,[18][19][20][21][22] In such processes, excitation energy transfer (EET) and photo-induced electron transfer (ET) pathways play a crucial role. 23,24 A prominent application is found in the study of antenna complexes of photosynthetic systems, responsible for the lightharvesting in higher plants. 25,26 To understand how the initial energy transfer and subsequent charge separation happen, phenomenological models employ a simplified diabatic picture [27][28][29][30] such as the Frenkel-Davydov exciton theory, in which the excited-state wavefunction is considered to be a superposition of local excitations (LEs) 31,32 and for which several extensions have been proposed to better describe shortrange interactions. [33][34][35][36][37][38] To connect TDDFT results to these models, it is necessary to express the supermolecular wave function in terms of localized molecular orbitals (LMOs) and to distinguish between LEs and charge-transfer (CT) excitations. Within this local or (approximate) diabatic picture, [39][40][41][42][43] ET and EET processes can be analyzed in a conceptually clear manner. 38,41,42,44,45 There are two dominant strategies to obtain LMOs and LEs in a supermolecular system. 7 One may either (1) a priori partition the supermolecular system into several subsystems and obtain LMOs from individual calculations on each subsystem, or (2) partition the system after a set of supermolecular orbitals is obtained. The first class of approaches to obtain all relevant supermolecular states in a diabatic picture belongs to the extensive family of embedding techniques. [46][47][48][49][50][51][52][53][54][55][56] In particular, subsystem DFT, 50,57-64 along with its time-dependent extension in the linear response regime, has long been an efficient workhorse for DFT-in-DFT embedding 23,[65][66][67][68][69] where different levels of approximations can be used for different subsystems. 70,71 In the second class of approaches, LMOs are obtained by localizing the supermolecular canonical molecular orbitals (CMOs). [72][73][74][75] The resulting LMOs can then be used as a starting point for embedding calculations, [76][77][78][79][80] and/or to directly construct (approximate) diabatic states. 45,[81][82][83] As shown by the Neugebauer group, subsystem TDDFT using projectionbased embedding techniques can exactly reproduce supermolecular results and capture the complete range of LE/CT interactions. 45,70,79,84,85 An elaborate discussion of the relations between these two approaches to localization of orbitals and approximate diabatization of electronic states is beyond the scope of this work, and we refer the reader to Refs. 70 and 86 for more details on these methods. To complete this brief list of relevant related methods, we also note the multistate fragmentation excitation difference-fragment charge difference (multistate FED-FCD) method that can also be used to build approximate diabatic states starting from supermolecular CMOs. 44,[87][88][89][90][91] By exploiting the spatial locality of LMOs, computationally efficient implementations of TDDFT can be achieved, as shown by Wu and co-workers who demonstrated linear scaling for TDDFT in combination with "fragment LMOs". 92 Wu and co-workers also nicely summarized the choices that need to be made in terms of different kinds of locality. Supermolecular CMOs are "local in energy and delocalized in space" while atomic orbitals (AOs) are "local in space but delocalized in energy". Both types of locality can furthermore be exploited by partially re-canonicalizing the orbital space. 93 Such a partial re-canonicalization step is useful to improve the convergence of the iterative solution of the TDDFT equations as well as for interpretation purposes, as the orbital energies resulting from the re-canonicalization can be readily interpreted as perturbed subsystem energies. In this work, we build on these previous developments and define a simple and computationally efficient strategy involving a one-step localization procedure to treat LE and CT states. Our approach can be regarded as an extension of our recent work on intrinsic localized fragment (molecular or atomic) orbitals, 94 by adding a re-canonicalization procedure. This step results in a blocked Fock matrix that is similar to the one obtained with the embedding procedure by Tölle et al. 45 Since our subsystem orbitals are orthogonal by construction, our procedure does, however, not require the construction of any projection operators. 46,86,95 The procedure allows for a partition of the full orbital space into LE and CT subspaces that are subsequently utilized in a TDDFT calculation to generate approximate diabatic states. Using the Tamm-Dancoff approximation (TDA), 96 explicit expressions for the electronic couplings can be derived between these states. Since our procedure is similar to projection-based subsystem DFT/TDDFT, we compare our result to Ref. 45. Note that the general framework presented here is particularly appealing as it can easily be extended to any excited-state method. The paper is organized as follows. In the first section, we provide the TDDFT working equations and then outline the construction of the re-canonicalized LMOs, followed by their application as a reference basis in the calculation of excitation energies and electronic couplings in the TDDFT framework. In the next section, a proof of principle application of our method on small model systems is provided, followed by an illustrative example of a chlorophyll dimer in which we also consider the use of additional tight-binding approximations. Conclusions and perspectives are then given in the final section. II. THEORY A. Time Dependent Density Functional Theory In the conventional formulation of TDDFT with real spatial orbitals in the adiabatic approximation, 97-99 excitation energies are determined by solving the non-Hermitian eigenvalue problem, A B −B −A X Y = ω X Y ,(1) where the matrix elements of the A and B matrices are written as A ia, jb = δ i j F ab − δ ab F i j + K ia, jb ,(2) and B ia, jb = K ia,b j .(3) In the above equation, indices i, j denote occupied orbitals and a, b denote virtual orbitals. F pq is an element of the KS Fock matrix, and the coupling matrix K is defined as K ia, jb = d 3 r d 3 r φ i ( r)φ a ( r) f Hxc [ρ]( r, r ) φ j ( r )φ b ( r ) − c X d 3 r d 3 r φ i ( r)φ j ( r) 1 | r − r | φ a ( r )φ b ( r ),(4) with the kernel (5) consisting of the Coulomb term and the second derivative of the exchange-correlation energy functional E xc [ρ]. The second line of (4) allows for use of hybrid DFT approaches that mix in a fraction c X of exact exchange. f Hxc [ρ GS ]( r, r ) = 1 | r − r | + (1 − c X ) δ 2 E xc [ρ] δ ρ( r)δ ρ( r ) ρ GS In the CMO basis, the KS Fock matrix F CMO pq = δ pq ε p is diagonal so that different excitation amplitudes X ia and X jb are only coupled by the coupling matrix K, i.e., A CMO ia, jb = δ i j δ ab (ε a − ε i ) + K ia, jb ,(6) where ε p denotes the energy of orbital p. The difference between canonical and non-canonical orbitals does not affect the B matrix that describes the coupling between excitations and de-excitations. For the sake of simplicity, we will employ the TDA in the following and include excitations only. The equation then reads AX = ωX,(7) where the lowest eigenvalues and eigenvectors can be obtained by iterative diagonalization techniques. In this work, we first partition the matrix A into subsystems to be treated separately. B. Construction of Re-canonicalized Intrinsic Localized Molecular Orbitals In a previous work 94 we introduced the concept of (polarized) intrinsic fragment orbitals (IFOs), a minimal basis in which the supermolecular KS determinant can be expressed exactly. These are constructed from a basis B 2 of reference fragment orbitals (RFOs) that are usually selected as the core and valence orbitals of each fragment. After expressing the supermolecular CMO basis B 1 in terms of these IFOs, both the occupied and virtual valence spaces can be localized to generate a final set of so-called intrinsic localized molecular orbitals (ILMOs), using for instance Pipek-Mezey (PM) localization. 74 The IFOs and ILMOs span the occupied space exactly but contain only N B 2 − N occ virtual orbitals. Such a truncation of the virtual space is undesirable for the current application to TDDFT, in which we want to retain a sufficient number of virtual orbitals to describe the electronically excited states accurately. A straightforward possibility is to increase the size of the RFO basis (N B 2 ) by selecting more orbitals per fragment, but this will also affect the definition of the valence virtual orbitals that are convenient for interpretation purposes. Another possibility is to keep the minimal valence space as it is, but add a selected number of localized virtual orbitals from the complementary virtual space that is not spanned by the valence virtual orbitals. For this purpose, numerous methods have been developed 100 , which either explicitly rotate these virtual orbitals to increase their localization 73,74 or apply projections of the complementary virtual space onto a predefined set of "proto-hard" atomic virtual orbitals 101 . In this work, we explore an approach in which the localization of the complementary virtual space that consists of socalled "hard virtual" (hv) orbitals is carried out with the same algorithm as is used for the occupied space. This allows us to work with atomic or molecular fragments, or with a combination of both. Another advantage of our approach is that there is no bias for a particular fragment in defining the initial localization, the procedure can work with an arbitrary number of fragments and allocation of orbitals that are bonding two (or more) fragments together is done after the localization procedure. The key idea is to separate the supermolecular basis into four subspaces: B 1 = B 1c ∪B 1m ∪B 1v ∪B 1d , with B 1c indicating core orbitals, B 1m indicating the minimal valence basis (comprising both occupied and virtual orbitals), B 1v indicating energetically low-lying virtual orbitals, and B 1d indicating energetically high-lying virtual orbitals. The core orbitals in space B 1c are straightforwardly identified from the orbital energies of the supermolecular calculation and are removed from the orbital set before the localization procedures commences, i.e. we therein use space B 1 = B 1m ∪ B 1v ∪ B 1d . This is a simple extension of our original implementation in which the (usually already very local) core orbitals were also localized. This is now an optional step with as advantage that the core-valence separation is kept exactly as it was in the CMO basis. For the RFOs that comprise reference basis B 2 we similarly partition the relevant part of the orbital space on each fragment as B 2 = B 2m ∪ B 2v . The procedure is then as follows: 1. Remove core orbitals from B 1 (basis B 1 ). Steps 2 through 4 are identical to the procedure described in our previous work, reference 94, to which we refer for details about the algorithm and definitions of the valence RFOs, IFOs and ILMOs. Define valence Step 5 can also be done before step 4 (as it does not require information about the ILMOs) and is easily implemented by taking the eigenvectors corresponding to the zero eigenvalues of the singular value decomposition (Eq. 15 of reference 94) to define the complementary virtual space B 1v ∪ B 1d . Steps 9 and 10 are carried out with exactly the same algorithm as steps 3 and 4 and are implemented by simply changing the arguments to the subroutines that perform these operations. The number of desired virtual orbitals N v thereby takes the place of the number N occ that is used when forming the occupied ILMOs. The advantage of this scheme is that mixing-in of high-energy virtual orbitals into the localized orbital set is strictly prohibited as rotations are only carried out between the lowest N v virtual orbitals. In this way we can effectively combine energy selection and orbital localization. Turning back towards our current application to partitioned TDDFT, we note that all localization transformations are unitary and do not mix the occupied and virtual orbitals that are obtained in the supermolecular calculation. They do therefore not change the structure of the TDDFT (Eq.1) and TDA (Eq.7) equations. The main difference compared to the CMO basis is the appearance of off-diagonal elements in the ILMO basis. In our previous work we observed that the diagonal elements of the transformed Fock matrix can provide an effective energy, but this pragmatic approach is not invariant for rotations that do not affect the localization extent (i.e. rotations between orbitals on the same fragment). It is therefore not well-suited for TDDFT where it is better to have a diagonally dominant A matrix. In the current work, we therefore introduce a final re-canonicalization 93 of the occupied and virtual orbitals of each fragment that restores the diagonal dominance and also provides an unambiguous definition of the effective orbital energies. To perform this re-canonicalization, we first sort the ILMO basis such that each orbital is assigned to the fragment on which it has its major contribution, according to Mulliken population analysis. 102 This automatic assignment can optionally be fine-tuned for typical embedding cases in which dangling bonds should always be assigned to the fragment of primary interest. For brevity of notation, we assume from now on a partitioning of the supermolecule into two fragments I and J, each with an integer number of occupied and virtual orbitals. After transformation of the supermolecular Fock matrix to this sorted ILMO basis, the subblocks of this Fock matrix, each in a basis of either an occupied or a virtual set of ILMOs, are diagonalized separately as shown in Fig. 1. The resulting eigenvectors form a unitary matrix U RILMO that transforms the ILMO basis to the final re-canonicalized ILMO (RILMO) basis. The Fock matrix in RILMO basis is partly di-agonal, F RILMO p∈I,q∈I =ε p δ pq ,(8)F RILMO i∈I,a∈J = 0,(9)F RILMO i∈I, j∈J =I = 0,(10)F RILMO a∈I,b∈J =I = 0,(11) withε p the eigenvalues obtained in this final recanonicalization step. These eigenvalues can readily be interpreted as orbital energies of the fragments, perturbed by the interaction with the other fragment(s). By choosing chemically relevant fragments, it is possible to identify π and π * orbitals of conjugated aromatic systems that are delocalized over a single fragment only. Given the similarity in structure of the Fock matrix in the RILMO basis with those arising in projection-based embedding, our re-canonicalization procedure can be seen as a onestep embedding where the resulting RILMOs are comparable to a set of polarized orthonormal subsystem orbitals re-sulting from a top-down embedding scheme. 45,54,79,[84][85][86] The main advantage of the current procedure is its simplicity and efficiency: the generation of ILMOs is done via an iterative Jacobi algorithm that typically shows a rapid convergence, whereas the diagonalizations needed for the recanonicalizations are done in subspaces of the full CMO space. The whole procedure therefore takes only a fraction of the time needed for the supermolecular DFT calculation and is carried out by a dedicated program requiring only the definition of the supermolecular and fragment orbitals and the atomic basis in which these are expressed. Within the RILMO basis, characterization of LE and CT excitations for TDDFT calculations is easy. Choosing again two subsystems, 1 and 2, and labeling the excitations as either local (L 1 and L 2 ) or as electron transfer from system 1 to 2 (CT 1 ) or vice versa (CT 2 ) we obtain the same structure for the TDA equation (7) as found in projection-based embedding 45,54,79,84 , but without explicit use of projectors.     A L 1 A L 1 /L 2 A L 1 /CT 1 A L 1 /CT 2 A L 2 /L 1 A L 2 A L 2 /CT 1 A L 2 /CT 2 A CT 1 /L 1 A CT 1 /L 2 A CT 1 A CT 1 /CT 2 A CT 2 /L 1 A CT 2 /L 2 A CT 2 /CT 1 A CT 2         X L 1 X L 2 X CT 1 X CT 2     = ω     X L 1 X L 2 X CT 1 X CT 2     ,(12) We can now identify couplings between local and charge-transfer excitations as off-diagonal blocks of the A matrix. 79,86 Thanks to the re-canonicalization and equation (8), the diagonal blocks have the same structure as in the CMO basis, i.e. Eq. (6). For the off-diagonal blocks, we need to employ the more general expression (2) to account for Eqs. (10) and (11). Without truncation the results in the RILMO basis will be identical to those obtained in the CMO basis as we will numerically verify in section IV A. Obviously, this case is not of practical interest, as one aims to reduce the computational cost of the method. Hence, we consider two subsequent truncations that will each affect the results: (i) reduction of the size of the RILMO basis by limiting the total number of virtual orbitals N v that are localized and (ii) reduction of the number of couplings between subsystems that are accounted for by restricting the solution of the TDA equation to only one or a few subsets of excitations. In the following, we refer to applying (ii) as the 'reduced diagonalization' (RD), as opposed to the 'complete diagonalization' (CD) when (ii) is not applied. One should keep in mind that the truncation (i) can be so severe that even the CD results will significantly deviate from the full CMO treatment. We will therefore first study the effect of this initial truncation (i), referred to as the "RILMO truncation". C. Reduced diagonalization As the number of solutions sought is typically far smaller than the dimension of the A matrix, it is advantageous to use a Davidson diagonalization algorithm. 103 We can thereby make use of the partitioning, and solve each particular subset S of excitations individually, initially without accounting for couplings between the subsets, A S V S = ω S V S .(13) For a partitioning into two subsystems, the eigenvectors V S γ associated with the excitation energies ω S γ can be interpreted as approximate diabatic states corresponding to L 1 , L 2 , CT 1 and CT 2 . They are subsequently coupled to form adiabatic states viaĀW =ωW,(14) with the matrix elements ofĀ defined as A S γδ = δ γδ ω S γ (15) A S/T γδ = ∑ ia∈S ∑ jb∈T V S ia,γ A S/T ia, jb V T jb,δ ,(16) yielding the coefficient vectorW which expresses the adiabatic state in terms of the diabatic states. This approach can be generalized to many subsystems, with the efficiency arising from the fact that only a small number M S of lowest eigenvectors for each excitation subset S is used. The resulting M = ∑ S M S dimensional matrix problem for the full system is then easily solvable. The calculation of the electronic couplings between subsystems and subsequent diagonalization in a reduced space has a similar structure as encountered in the subsystem TDA approach, as discussed in Refs. 67 and 45. Hence, RD consists in constructing and subsequently diagonalizing (see Eq. 14) a basis of approximate diabatic states, where only a limited set of the lowest diabatic states per subspace is considered. The RD pathway for a case of two subsystems is schematically depicted in Fig. 2 and can be summarized as: 1. Starting from the set of RILMOs, partition the occupied and virtual orbitals and perform four separate TDA calculations to obtain sets of eigenvectors {V S γ } for each of the L 1 , L 2 , CT 1 and CT 2 sub-spaces. 2. Calculate electronic couplings between the diabatic states using the sets of eigenvectors from the previous step and construct matrixĀ, 3. DiagonalizeĀ to obtain the adiabatic states. Note that in the limit of including all the diabatic states for each subspace, the CD and RD procedures yield identical results. D. Evaluation of the coupling elements The most time-consuming steps in the calculation are the matrix-vector multiplications, in which trial vectors are contracted with a block of the A matrix. We will briefly summarize our implementation, which is based on the pair-fitting approach of the ADF engine of the Amsterdam Modelling Suite (AMS), as described in Ref. 5. We first define the RILMOs in terms of atomic orbitals χ as φ l (r) = ∑ κ C RILMO κl χ κ (r),(17) and introduce the density functions f p to fit the transition den- sities ρ T δ (r) of state δ of subspace T ρ T δ (r) = ∑ jb∈T φ j (r)φ b (r)V T jb,δ ≈ ∑ p f p (r)c T pδ .(18) The fit coefficients c T pδ can be expressed in terms of atomic orbital fit coefficients c AO f it and the (trial) V matrices as c T pδ = ∑ jb∈T ∑ κ,λ C RILMO κ j C RILMO λ b c AO f it κλ ,p V T jb,δ .(19) Using Eq. (18), the induced potential δ v T ind,δ for state δ of subspace T can then be written as δ v T ind,δ (r)= ∑ p g p (r) + f xc (r) f p (r) c T pδ ,(20) with g p (r) the Coulomb potential of the fit function f p (r) and f xc (r) the exchange-correlation kernel. The matrix representation of the induced potential in the RILMO excitation basis of subspace S can subsequently be obtained by numerical integration [δ v T ind,δ ] S ia = ∑ k w k φ i (r k )δ v T ind,δ (r k )φ a (r k ),(21) with w k the weight associated to the grid point r k . Because the functions φ i and φ a are local, detection and neglect of small contributions to the integrand can be used to speed up this step. For hybrid functionals, we also need to consider the exchange contribution. Here, density fitting can be used as well, using the implementations reported in Refs. 104 and 105. The exchange contribution to the induced potential is given by [δ v T exc,δ ] S ia = ∑ jb∈T (i j|ab)V T jb,δ ,(22) which is first expressed in the AO basis and evaluated using density fitting as [δ v T exc,δ ] κλ = ∑ µν (κ µ|λ ν)V T µν,δ ≈ ∑ µν ∑ p,q c AO f it κ µ,p c AO f it λ ν,q ( f p | f q )V T µν,δ , ,(23) with V T µν,δ = ∑ jb∈T C RILMO µ j C RILMO νb V T jb,δ .(24) After a transformation to the excitation subspace S [δ v T exc,δ ] S ia = ∑ κλ C RILMO κi C RILMO λ a [δ v T exc,δ ] κλ ,(25) this term can be combined with the other contributions. The total electronic coupling in Eq. (16) between two excitations γ and δ belonging to the S and T subspaces then becomes A S/T γδ = ∑ ia∈S ∑ jb∈T V S ia,γ (F RILMO i j δ ab − F RILMO ab δ i j )V T jb,δ + ∑ ia∈S V S ia,γ [δ v T ind,δ ] S ia − c x [δ v T exc,δ ] S ia .(26) E. Tight-binding approximations to the coupling matrix The above-mentioned scheme can also be easily applied to approximate TDDFT approaches. An example is Grimme's simplified TDA 106 in which matrix elements of the coupling matrix K are given as K ia, jb = N atoms ∑ A,B (2q A ia γ J AB q B jb − q A i j γ K AB q B ab ),(27) with atomic transition charges q A pq obtained as q A pq = ∑ µ∈A C µ p C µq .(28) The parameters γ J AB and γ K AB depend on the interatomic distance and the chemical hardness of the atoms as well as on a few empirical parameters. The matrix C denotes MO coefficients in the orthogonal AO basis obtained from C = S 1/2 C with C the coefficients in the original AO basis and S the AO overlap matrix. This approach is easily adapted to the RILMO partitioning by just replacing the CMOs by the RILMOs in Eqs. (27) and (28). III. COMPUTATIONAL DETAILS All DFT and TDDFT calculations using the RILMOs were carried out using a locally modified version of the Amsterdam Modelling Suite of Programs 107,108 (AMS). For all TDDFT calculations in AMS, the ALDA kernel and TDA were used throughout this work (denoted as RILMO-TDA). The localization and re-canonicalization of the molecular orbitals were carried out separately using a recently developed stand-alone program, so-called Reduction of Orbital Space Extent (ROSE) that has an interface to AMS 94,109 . All calculations for ROSE and AMS were automated using the Python Library for Automating Molecular Simulation 110 (PLAMS) of AMS. The sorting of the orbitals before the re-canonicalization step was done based on a Mulliken population analysis of the LMOs over each fragment. In this work we used the same level of basis sets, B 1 and B k , for the supermolecule and the k fragments respectively (see reference 94 for the notations). For the localization step, we used the Pipek-Mezey localization for both the occupied and the virtual orbitals. 74 IV. RESULTS AND DISCUSSIONS A. RILMO truncation and Complete versus Reduced Diagonalization In order to demonstrate our procedure, we choose two test systems: the ethylene-tetrafluoroethylene dimer and the adenine-thymine DNA base pair. While the former presents an adequate model for studying LE and CT type transitions, the latter provides a biologically relevant model system where such quantities are of importance to model the stability of biomolecules under irradiation 115,116 . The geometry of both of these structures were taken from Ref. 45 and Ref. 115 and is displayed in Fig. 3a and 3b, respectively. The CAM-B3LYP 13 exchange-correlation functional along with the Slater-type basis set TZ2P as employed in AMS was used for both these systems. The uncoupled LE (uLE) and uncoupled CT (uCT) states considered for both systems consist of π → π * type transitions. Fig. 4 (middle column) and relevant π and π * RILMOs associated with the C 2 H 4 − C 2 F 4 dimer and the adenine-thymine dimer, respectively. The corresponding supermolecular CMOs are also shown. The CMOs are delocalized over both fragments, and the localization and re-canonicalization procedures restore the typical π and π * character in the RILMOs. For comparison, we show the PbE-sDFT orbitals that yield a similar localized picture. For the ethylene-tetrafluoroethylene dimer we consider two uLE states for C 2 H 4 , one uLE for C 2 F 4 and one uCT state from C 2 F 4 → C 2 H 4 . For C 2 H 4 we have a π → π * state and a π → σ * state which is near-degenerate with the former. These two states mix upon increasing the size of the virtual space in the localization procedure, so that we will denote them as uLE 1 and uLE 1 below. In the limit of the full virtual space, the uLE 1 retains mostly the π → π * character (∼ 71 %). For C 2 F 4 we have a π → π * state denoted as uLE 2 and for the CT transition we have a C 2 F 4 , π → C 2 H 4 , π * state denoted as uCT. For the adenine-thymine dimer, we label the two near-degenerate adenine π → π * states (originally denoted L a and L b in Ref. 115) as uLE 1a and uLE 1b , while denoting the thymine π → π * state as uLE 2 . Besides these three uLE states, there is one uCT state: adenine, π → thymine, π * . It is important to note that the above set of four excitations for both of these dimers does not constitute the lowest set of excitations but selects the primary valence π → π * type transitions that are often used to model LE and CT type excitations in the literature. As a numerical verification of our implementation, we first show that the supermolecular excitation spectrum arising from the CD (see Fig. 2) of RILMO-TDA converges to the CMO results in the limit of using the full virtual space. Fig. 6 plots the error in the energy of the lowest RILMO-TDA states from the CMOs as a function of the virtual space size, i.e from severe to no RILMO truncation. Besides the expected convergence, the plot also indicates that RILMO truncation is a viable way to restrict the excitation space. A modest amount of virtual orbitals suffices to obtain adiabatic state energies close to the reference supermolecular CMO results. Next, we consider combining RILMO truncation with truncation in the number of diabatic states employed in the RD scheme. For this purpose, we couple for both dimers only the aforementioned sets of four uLE and uCT states. Fig. 7 shows the effect of RILMO truncation combined with RD. From Fig. 7, we observe a decrease in the uLE energies for both the C 2 H 4 − C 2 F 4 and adenine-thymine dimers, whereas the uCT energies are quite stable upon increasing the number of virtual orbitals. This stability can be explained by the fact that the CT excitations are well described by single orbital transitions, 117 so that moving from a minimal valence space to the full virtual space causes little to no change in the corresponding excitation energies. The LE excitations, on the other hand, are mixtures of local single orbital transitions and are thus more sensitive to the number of virtual orbitals. The uLE excitation energies exhibit a smooth convergence upon increasing the number of virtual orbitals, similar to what is seen for isolated or well-separated fragments (see SI, Figs. S1 and S2). Fig. 8 plots the corresponding coupled adiabatic states using the above four diabatic states for both of these dimers. As can be seen, the curves follow closely the uncoupled energies and converge smoothly upon increasing the total number of virtual orbitals. Figs. 7 and 8 show that it is sufficient to work with a modest number of virtual orbitals to construct the local states in the RD framework. Having thus shown that RILMO truncation in combination with RD is viable, we also consider the RD procedure itself in more detail. To do so, we couple the above minimal set of four diabatic states with additional low-lying states and investigate how this changes the excitation energies. Fig. 9 plots the error in the excitation energies of the lowest RILMO-TDA adiabatic states from their respective CMO states. We thereby chose a simple set up, taking either the lowest 5 or the lowest 20 diabatic states in each of the four excitation subspaces, yielding a total of 20 and 80 adiabatic states, respectively. A direct comparison of all resulting states with CMO states is complicated, as the coupling of adiabatic states also gives rise to high-energy states that can not be readily matched with CMO states. For this reason, we focus on the lowest energy states up to the above-mentioned bright π → π * excitations. As can be seen for both of these dimers, increasing the total number of states from 20 to 80 shows a reduction in the overall deviation of the lower-lying states to well within one-tenth of an eV, again already with a modest amount of virtual orbitals (i.e. with strong RILMO truncation). From Fig. 9, we conclude that for the C 2 H 4 − C 2 F 4 and adenine-thymine dimers, a set of five states for each of the subspaces is sufficient to closely reproduce the lowest CMO states, including the characteristic π → π * transitions. As an additional proof of principle, we have further checked the smooth convergence of the lowest 10 RILMO-TDA adiabatic states towards the reference supermolecular CMO states, with respect to the M number of diabatic states used in the RD pathway on a smaller waterammonia test system (see Fig. S4 in SI). B. Comparison with projection-based embedding subsystem DFT Having shown the convergence towards the supermolecular CMO results for the CD and RD pathways, we now focus on a more detailed comparison of the above approximate diabatic uLE and uCT states as well as their electronic couplings with those obtained from PbE-sDFT/sTDA. For the sake of consistency, we retain the full virtual space in the RILMO-TDA calculation of the above dimers. Both of these systems have been studied in Ref. 45 and their diabatic LE and CT states were obtained using the multistate-FCD-FED approach and PbE-sTDA by employing different orbital partition schemes for varying displacements d between the corresponding monomers. In Ref. 45, an overall decent agreement was obtained between the two methods in spite of their differences in the construction of the diabatic states. We refer the reader to Ref. 45 for more details, and only the necessary PbE-sDFT/sTDA calculations are repeated here for comparison. For the purpose of this work, the Gaussian type def2-TZVP basis set was chosen in SERENITY in order to compare with the Slater-type basis set TZ2P as employed in AMS. While the two basis sets are reasonably large, the effect of basis set truncation is still nonnegligible as can be seen when comparing the supermolecular CMO results of the two codes. In Fig. 10 (top panel) we plot the differences in the lowest 10 excitation energies and see that these can amount to a few tenths of eV, with the most pronounced differences occurring for the C 2 H 4 − C 2 F 4 dimer. While for the Adenine-Thymine dimer our states of interest (the characteristic π → π * transitions) lies within the lowest 6 supramolecule CMO states (states 2,3,5 and 6), such an identification for the C 2 H 4 −C 2 F 4 dimer becomes difficult due to the delocalized nature of the CMOs as shown in Fig. 4. To compare the ground-state embedded orbitals, we again refer to Fig. 4 (right column) and Fig. 5 (bottom row) for the relevant π and π * orbitals of C 2 H 4 − C 2 F 4 and the adeninethymine dimer. We see that despite differences in the formalism, and type of basis functions employed, both embedding procedures lead to very similar local orbitals for both systems. Focusing first on the lowest uCT state in each dimer, Fig. 11 plots the uCT energy from RILMO-TDA along with the corresponding uCT energy from PbE-sTDA as a function of the separation distance between the monomers. Both methods agree very well at longer distances, displaying the asymptotic decay of −R −1 characteristic for CT states. At shorter distances, the uLE energies from RILMO-TDA are consistently lower compared to those of PbE-sTDA (see Table I for uLE 1 , Table S1 in SI for the energies of uLE 1 , and Table II). Besides the already mentioned difference in basis sets, the deviation observed between the uCT energies could also arise from the differences in the two methods, in particular the different localization of the virtual orbitals. In the PbE approach, the virtual orbitals are subject to a SVD of the MO overlap matrix computed with only the AO basis functions located on the fragment of interest to obtain virtual LMOs for that fragment 45,85 . In our scheme, on the other hand, the virtual orbitals are subject to two separate PM localizations, resulting in a small set of 'valence' virtual orbitals and a larger set of 'hard' virtual orbitals. The resulting differences in virtual orbital energies, which in the PbE case rise more steeply upon reducing the monomer distance, appear to be the main cause for this difference (see Tables S2 and S3 in SI). The differences observed at short distances and agreements at long distances are also reflected in the electronic couplings between the respective uLE and uCT states. Fig. 12 plots their absolute values as a function of the separation distance between the monomers for each of these dimers. Only a few illustrative couplings are shown, see Fig. S5 and S6 in SI for all the couplings. The couplings between the uLE and uCT states calculated from both of the two methods display an exponential decay (e −r ) at long range, which is typical for exchange interactions. The most significant difference between the methods is seen in the uLE 1 /uCT coupling of the C 2 H 4 − C 2 F 4 dimer, where the coupling is stronger for PbE-sTDA at short distances and weaker at long distances, as compared to RILMO-TDA. For the other couplings, the two methods give qualitatively similar results. As both the PbE-sTDA and our approach converge to the supermolecular CMO results when sufficient states are coupled, we also show in Fig. 10 (bottom panel), the excitation energies obtained when a total of 80 diabatic states are cou- pled via RD. This gives a similar picture as seen for the CMO calculations, with relatively small deviations for the adeninethymine dimer and larger deviations for the C 2 H 4 − C 2 F 4 dimer. C. Application to a chlorophyll a dimer in Light Harvesting Complex II After this assessment of the RILMO-TDA approach, we now provide an application to molecules of biological importance. We choose a dimer consisting of chlorophyll molecules, that has been studied previously by some of us 118 using subsystem DFT. In this work, we used the traditional version with non-additive kinetic energy functionals as opposed to projection-based embedding. This particular dimer is known to play a crucial role in light-induced energy transfer processes in light-harvesting complexes in higher plants 118,119 . In this previous work, we observed an increase in the mixing of charge-transfer states with the local states upon a decrease in the distance between the two chromophores. A direct manifestation of this mixing is the red-shifting of the lowest excited states and a decrease in the total oscillator strength. To study this mixing with the current method that offers an easy analysis of the effect of CT, we seek to calculate the uLE and uCT states along with their corresponding electronic couplings for two representative structures. In conformation 1 the distance between the two chlorophylls is relatively large, while in conformation 2 the two chlorophylls are stacked at relatively close distance (see Fig. 3c). The CAM-B3LYP exchange-correlation functional along with the Slater-type basis set DZP was used for this system, while we performed Gaussian-type def2-SVP with SEREN-ITY for comparison. Here again, the full virtual space was retained (no RILMO truncation) for comparison purposes. The RILMOs and orbitals from PbE-sDFT corresponding to the HOMO (π) and LUMO (π * ) for the two chlorophyll molecules in conformation 2 are shown in Fig. 13. The supermolecular CMOs are also shown for reference. As can be seen, the canonical HOMOs are delocalized over both fragments, and localization (followed by re-canonicalization) helps to retain the delocalized nature of the π and π * orbitals localized over each fragment. Employing these localized orbitals, we carried out RILMO-TDA calculations for the uLE and uCT states. In order to demonstrate the flexibility of such an approach for this system, we also used Grimme's simplified TDA 106 approach (RILMO-simTDA) to calculate the uncoupled excitation energies and the electronic couplings given by Eq. (27). The transitions considered in the uLE and uCT subspaces for the dimer consist of two lowest π → π * transitions of LE type for each of the chlorophyll, CLA1 and CLA2 (denoted as uLE 1 and uLE 2 respectively), and two lowest π → π * of CT type -CLA1 → CLA2 and CLA2 → CLA1 (denoted as uCT 1 and uCT 2 respectively). Table III lists these energies for two of the conformations shown in Fig. 3c, using both of these methods and PbE-sTDA. Although the full TDA and simplified TDA methods present some differences in the excitation energies, they both predict a distinct drop in the CT energies for conformation 2, although the latter yields a more drastic change probably owing to the approximations introduced in the calculation of the coupling elements of the A matrix for simplified TDA. The effect of the truncation of the size of the virtual space on the uLE and the uCT energies for both the conformations showed a similar trend as before, with the uLE energies displaying a gradual convergence behaviour with the increase in the number of virtual orbitals, and the uCT energies showing little to no change (see Fig. S7 in SI). Thus, overall, the RILMO-TDA excitation energies are in reasonable agreement with those of PbE-sTDA. Table IV also lists the corresponding couplings between the uLE and uCT states calculated from RILMO-TDA and RILMO-simTDA approaches. Although there are minor differences between these two methods, the couplings of RILMO-TDA are in excellent agreement with that of PbE-sTDA. Furthermore, the truncation of the size of the virtual space shows no discernible changes in the couplings (see Fig. S8 in the SI). V. CONCLUSIONS In this work, we presented a simple one-step embedding approach for generating a set of re-canonicalized intrinsic localized molecular orbitals (RILMOs) for each fragment starting from a set of canonical molecular orbitals (CMOs). As an extension to the valence ILMOs discussed in our previous work 94 , these orbital sets include also the so-called hard virtual orbitals that are needed for a quantitative description of electronically excited states. While fully localized on individual fragments, RILMOs are delocalized within a fragment and can thus retain the intrinsic π and π * character encountered in isolated conjugated systems. Since the ILMOs are orthogonal by construction, our procedure eliminates the need for the construction of any projection operators. Nevertheless, the orbitals are comparable to a set of polarized orthonormal orbitals arising from a top-down projection-based embedding procedure. We furthermore outlined a TDDFT strategy for obtaining LE and CT states and investigated the effect of truncating the size of the virtual space in the localization procedure on the energies of these states. While the diabatic LE energies displayed a relatively slow convergence behaviour upon increasing the total number of virtual orbitals in the localization step, the diabatic CT energies were found to converge very quickly. The excitation energies and electronic couplings obtained in this approximate diabatic basis of the LE and CT states are in reasonable agreement with those from subsystem DFT using projection-based embedding. Our observations are in accordance with previous studies that showed a similar dependence of the diabatic excitation energies and electronic couplings on the diabatization procedure employed. 41,45 . Nonetheless, we believe that our procedure will be useful due to its simple implementation and possibilities for easy extension to other excited state methods (DFTB, BSE, equation-of-motion coupled cluster, etc). The current scheme can for instance be used as an analysis toolkit for studying ET and EET processes in photo-induced systems. SUPPLEMENTARY MATERIAL See the supplementary material for 1) effect of truncation of virtual space on the uLE energies of isolated C 2 H 4 − C 2 F 4 and adenine-thymine dimers, 2) effect of truncation of virtual space on the electronic couplings of both of these dimers, 3) Convergence of the RILMO-TDA states in the CD and RD pathway for a water-ammonia dimer, 4) electronic couplings and orbital energies for both of these dimers with varying distances and 5) effect of truncation of virtual space on the uLE and uCT energies and electronic couplings of the chlorophyll dimer ACKNOWLEDGMENTS The authors would like to thank Dr. Johannes Tölle for valuable discussions and Prof. Johannes Neugebauer for providing access to the ethylene-tetrafluoroethylene structure used in this work. S.S. and L.V. acknowledge support from NWO via the CSER program and for access to the National Computing Facilities. FIG. 13: HOMOs and LUMOs associated with the lowest π → π * transition for each of the chlorophyll molecules in conformation 2(denoted as CLA1 and CLA2) from different set of orbitals: CMOs (top row), RILMOs (middle row) and orbitals from PbE-sDFT (bottom row). AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions Souloke Sen: Data curation (lead); Investigation (equal); Methodology (equal); Validation (equal); Writing -original draft (lead). Bruno Senjean: Methodology (equal); Validation (equal); Writing -review & editing (equal). Lucas Visscher: Conceptualization (lead); Methodology (equal); Supervision (lead); Investigation (equal); Validation (equal); Writing -review & editing (equal). DATA AVAILABILITY The data used in this work are available upon reasonable request. 1. Effect of truncating the virtual space on the uLE energies of the isolated C 2 H 4 − C 2 F 4 and adenine-thymine dimers Figure Figure S5: Absolute electronic couplings between the uLE and uCT states of the C 2 H 4 −C 2 F 4 dimer with respect to the separation distance, using the RILMO-TDA method. S6 5. uLE and orbital energies for the C 2 H 4 − C 2 F 4 dimer HOMO(1) -9.37 -9.33 -9.31 -9.29 -9.28 -9.24 HOMO(2) -9.04 -9.07 -9.09 -9.11 -9.12 -9.16 S7 6. LE/LE and LE/CT electronic couplings for the adeninethymine dimer Figure S6: Absolute electronic couplings between the uLE/uLE states (left panel) and the uLE/uCT states (right panel) of the adenine-thymine dimer with respect to the separation distance, using the RILMO-TDA method. S8 7. Orbital energies for the adenine-thymine dimer (1) and (2) refer to fragments adenine and thymine respectively. All orbital energies are in eV. Figure S7: Excitation energies of the uLE and uCT states of conformation 1 (left panel) and conformation 2 (right panel) with respect to the number of hard virtual orbitals used in the localization procedure, using the RILMO-TDA method. Figure S8: Absolute electronic couplings between the uLE and uCT states of conformation 1 (left panel) and conformation 2 (right panel) with respect to the number of hard virtual orbitals used in the localization procedure, using the RILMO-TDA method. S10 FIG. 1 : 1Representation of the Fock matrix in different bases. Colored blocks represent non-zero matrix elements, where the smaller squares are single matrix elements. FIG. 2 : 2RD procedure for obtaining the LE and CT states and electronic couplings using RILMOs. For TDDFT calculations with the Grimme's simplified Tamm-Dancoff 106 approximation (denoted as RILMO-simTDA), the parameters from Ref. 106, corresponding to the standard settings in AMS, were used. All projection-based embedding subsystem DFT (PbE-sDFT) and TDDFT/TDA calculations (denoted as PbE-sTDA) calculations were carried out using SERENITY version 1.4.0 111 with the LevelShift projection 112 operator of the top-down embedding 79,85,86 procedure as outlined in Ref. 86. Additional localization of the virtual orbitals was carried out using the approach outlined in Ref. 85 as implemented inside SERENITY. In all the calculations performed in SERENITY, the resolution of identity (RI) 113 approximation was used for the evaluation of the Coulomb contribution in all the DFT and TDDFT/TDA 114 calculations. Fig. 5 ( 5middle row) plot the FIG. 3: Geometries of the a) tetrafluoroethylene-ethylene dimer, b) adenine-thymine base pair, and c) chlorophyll dimer. FIG. 4 :FIG. 6 : 46The occupied and virtual orbitals associated with the uLE and uCT states for the C 2 H 4 −C 2 F 4 dimer obtained from CMOs (left column), RILMOs (middle column) and PbE-sDFT (right column).FIG. 5: The occupied and virtual orbitals associated with the uLE and uCT states for the adenine-thymine dimer obtained from CMOs (top row), RILMOs (middle row) and PbE-sDFT (bottom row). Error in excitation energy (as compared to the supermolecular CMO reference energies) due to RILMO truncation of hv orbitals. Left panels: lowest 20 RILMO-TDA (CD) states for the C 2 H 4 −C 2 F 4 dimer. Right panels: lowest 10 RILMO-TDA (CD) states of the adenine-thymine dimer. M denotes the number of diabatic states (top panels: M = 20, bottom panels: M = 80). The color of the marker indicates the energy ordering of states. Excitation energy, eV Excitation energy, eV Total number of virtuals Total number of virtuals FIG. 7: Left panel: Excitation energy of the uLE and uCT states using RILMO-TDA (RD) for the C 2 H 4 −C 2 F 4 dimer with respect to the number of hv orbitals used in the localization procedure. Right panel: same as the left panel for the adenine-thymine dimer. FIG. 8 : 8Excitation energies of the coupled states using the minimal set of four diabatic states for the C 2 H 4 −C 2 F 4 (left panel) and adenine-thymine (right panel) dimers with respect to the number of hv orbitals used in the localization procedure. RILMO-TDA(RD,20) RILMO-TDA(RD,80) FIG. 9: Error in excitation energy (as compared to the supermolecular CMO reference energies) due to RILMO truncation of hv orbitals. Left panels: lowest 15 RILMO-TDA (RD,M) states for the C 2 H 4 −C 2 F 4 dimer. Right panels: lowest 6 RILMO-TDA (RD,M) states of the adenine-thymine dimer. M denotes the number of diabatic states (top panels: M = 20, bottom panels: M = 80). The color of the marker indicates the energy ordering of states. FIG. 10 : 10Top panel: Excitation energies of the lowest 10 supermolecular states in the CMO basis from AMS (full lines) and SERENITY (dashed lines) using TDA with respect to the separation distance of the C 2 H 4 −C 2 F 4 (left panel) and adenine-thymine (right panel) dimers. Bottom panel: Excitation energies of the lowest 10 adiabatic states obtained by RD pathway with M = 80 diabatic states from RILMO-TDA (full lines) and PbE-sTDA (dashed lines) for both dimers. Separation distance, Å Excitation energy, eV Separation distance, Å Excitation energy, eV FIG. 11: uCT energies obtained from RILMO-TDA (full lines) and PbE-sTDA (dashed lines) as a function of the separation distance for the C 2 H 4 −C 2 F 4 (left panel) and adenine-thymine (right panel) dimers. Separation distance, Å Absolute electronic couplings, meV Separation distance, Å Absolute electronic couplings, meV FIG. 12: Absolute electronic couplings between the uLE and uCT states from RILMO-TDA (full lines) and PbE-sTDA (dashed lines) as a function of the separation distance for the C 2 H 4 −C 2 F 4 (left panel) and adenine-thymine (right panel) dimers. S1: Effect of truncation of the number of virtual orbitals on the uLE 1 (left panel) and uLE 2 (right panel) excitation energies for the isolated fragments and the C 2 H 4 − C 2 F 4 dimer at d = 4 Å. Figure S2 :Figure S3 :Figure S4 : S2S3S4Effect of truncation of the number of virtual orbitals on the uLE 1a , uLE 1b (left panel) and uLE 2 (right panel) excitation energies for the isolated fragments and the adeninethymine dimer at d = 4 Å.S32. Effect of truncating the virtual space on electronic couplings of the C 2 H 4 − C 2 F 4 and adenine-thymine dimers Absolute electronic couplings between the uLE and uCT states of the C 2 H 4 −C 2 F 4 (left panel) and adenine-thymine (right panel) dimers with respect to the number of hard virtual orbitals used in the localization procedure, using the RILMO-TDA method.S4 3. Convergence of the RILMO-TDA states in the CD and RD pathways for a water-ammonia dimer Left panel: Error in excitation energy (as compared to the supermolecular CMO reference energies) due to RILMO truncation of hard virtual orbitals for the lowest 10 RILMO-TDA (CD) states. Right panel: Error in excitation energy (as compared to the supermolecular CMO reference energies) due to truncation to M diabatic states for the lowest 10 RILMO-TDA (RD,M ) states, while keeping all the hard virtual orbitals in the localization procedure. Calculations were carried out at CAMB3LYP/DZP level of theory. S5 4. LE/LE and LE/CT electronic couplings for the C 2 H 4 − C 2 F 4 dimer S2: Orbital energies associated with the uLE and uCT transitions of the C 2 H 4 − C 2 F 4 dimer obtained from RILMO and PbE-sDFT methods at varying distances d (Å). Also shown are the isolated fragment energies. The uCT and uLE states in the main text comprises of : 1) uLE 1 : HOM O(1) → LU M O(1), 2) uLE 2 : HOM O(2) → LU M O(2)/LU M O + 1(2) and 3) uCT : HOM O(2) → LU M O(1). (1) and (2) refer to fragments C 2 H 4 and C 2 F 4 respectively. All units are in eV. d = 0 d = 0.5 d = 1.0 d = 1.5 d = 2.0 isolated LUMO(1) S3: Orbital energies associated with the uLE and uCT transitions of adenine-thymine dimer obtained from RILMO and PbE-sDFT methods at varying distances d (Å). Also shown are the isolated fragment energies. The uCT and uLE states in the main text comprises of : 1) uLE 1a : HOM O(1) → LU M O(1), 2) uLE 1b : HOM O(1) → LU M O+1(1)/LU M O+2(1) , 3) uLE 2 : HOM O(2) → LU M O(2) and uCT : HOM O(1) → LU M O(2) . RFOs (basis B 2m ).3. Compute valence IFOs (basis B 1m ). 4. Localize valence orbitals expressed in the IFO basis to obtain valence ILMOs. 5. Define the complementary virtual space B 1v ∪ B 1d . 6. Re-canonicalize B 1v ∪ B 1d to obtain semicanonical su- permolecular virtual orbitals with effective energies ε . 7. Select N v orbitals with lowest values of ε (basis B 1v ). 8. Define hv RFOs (basis B 2v ). 9. Compute hv IFOs (basis B 1v ). 10. Localize the hv orbitals expressed in the IFO basis to obtain hv ILMOs. TABLE I : IuLE energies of the C 2 H 4 −C 2 F 4 dimer obtained from RILMO-TDA and PbE-sTDA methods at varying distances d (Å). Also shown are the isolated fragment energies. All units are in eV. d = 0 d = 0.5 d = 1.0 d = 1.5 d = 2.0 isolated uLE 1 8.33 8.31 8.30 8.29 8.28 8.28 RILMO-TDA uLE 2 9.03 9.02 9.00 9.00 8.99 8.99 uLE 1 8.49 8.39 8.34 8.31 8.29 8.29 PbE-sTDA uLE 2 9.03 8.97 8.94 8.92 8.91 8.91 TABLE II : IIuLE energies of the adenine-thymine dimer obtained from RILMO-TDA and PbE-sTDA methods at varying distances d (Å). Also shown are the isolated fragment energies. All units are in eV. d = 0 d = 1.0 d = 2.0 d = 3.0 d = 4.0 isolated uLE 1a 5.72 5.69 5.68 5.67 5.67 5.67 RILMO-TDA uLE 1b 5.64 5.62 5.61 5.61 5.61 5.61 uLE 2 5.51 5.48 5.47 5.47 5.47 5.48 uLE 1a 5.75 5.67 5.65 5.65 5.65 5.65 PbE-sTDA uLE 1b 5.69 5.60 5.59 5.59 5.59 5.59 uLE 2 5.55 5.47 5.45 5.45 5.45 5.46 TABLE III : IIIuLE and uCT energies of the chlorophyll dimer obtained from PbE-sTDA, RILMO-TDA and RILMO-simTDA methods for both conformations. All units are in eV.States PbE-sTDA RILMO-TDA RILMO-simTDA uLE 1 2.21 2.22 1.93 uLE 2 2.25 2.26 1.96 Conformation 1 uCT 1 2.82 2.80 2.36 uCT 2 2.97 2.97 2. 54 uLE 1 2.31 2.27 1.98 uLE 2 2.25 2.25 1.96 Conformation 2 uCT 1 2.70 2.67 2.22 uCT 2 2.30 2.24 1.79 TABLE IV : IVAbsolute electronic couplings between the uLE/uCT states of the chlorophyll dimer obtained from PbE-sTDA, RILMO-TDA and RILMO-simTDA methods for both conformations. All units are in meV.States PbE-sTDA RILMO-TDA RILMO-simTDA uLE 1 /uLE 2 27 26 35 uLE 1 /uCT 1 7 8 9 Conformation 1 uLE 1 /uCT 2 20 20 20 uLE 2 /uCT 1 19 17 17 uLE 2 /uCT 2 7 8 9 uLE 1 /uLE 2 36 36 44 uLE 1 /uCT 1 27 29 28 Conformation 2 uLE 1 /uCT 2 44 44 45 uLE 2 /uCT 1 31 29 37 uLE 2 /uCT 2 23 21 19 Table S1 : S1uLE energies of the C 2 H 4 − C 2 F 4 dimer using RILMO-TDA and PbE-sTDA methods at varying distances, d (Å). 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{'abstract': 'Localized molecular orbitals are often used for the analysis of chemical bonds, but they can also serve to efficiently and comprehensibly compute linear response properties. While conventional canonical molecular orbitals provide an adequate basis for the treatment of excited states, a chemically meaningful identification of the different excited-state processes is difficult within such a delocalized orbital basis. In this work, starting from an initial set of supermolecular canonical molecular orbitals, we provide a simple one-step top-down embedding procedure for generating a set of orbitals which are localized in terms of the supermolecule, but delocalized over each subsystem composing the supermolecule. Using an orbital partitioning scheme based on such sets of localized orbitals, we further present a procedure for the construction of local excitations and charge-transfer states within the linear response framework of time-dependent density functional theory (TDDFT). This procedure provides direct access to approximate diabatic excitation energies and, under the Tamm-Dancoff approximation, also their corresponding electronic couplings -quantities that are of primary importance in modelling energy transfer processes in complex biological systems. Our approach is compared with a recently developed diabatization procedure based on subsystem TDDFT using projection operators, which leads to a similar set of working equations. Although both of these methods differ in the general localization strategies adopted and the type of basis functions (Slaters vs. Gaussians) employed, an overall decent agreement is obtained.', 'arxivid': '2212.06189', 'author': ['Souloke Sen \nDivision of Theoretical Chemistry\nFaculty of Sciences\nVrije Universiteit Amsterdam\nDe Boelelaan 10831081 HVAmsterdamThe Netherlands\n', 'Bruno Senjean \nICGM\nUniversité de Montpellier\nCNRS\nENSCM\nMontpellierFrance\n', 'Lucas Visscher \nDivision of Theoretical Chemistry\nFaculty of Sciences\nVrije Universiteit Amsterdam\nDe Boelelaan 10831081 HVAmsterdamThe Netherlands\n'], 'authoraffiliation': ['Division of Theoretical Chemistry\nFaculty of Sciences\nVrije Universiteit Amsterdam\nDe Boelelaan 10831081 HVAmsterdamThe Netherlands', 'ICGM\nUniversité de Montpellier\nCNRS\nENSCM\nMontpellierFrance', 'Division of Theoretical Chemistry\nFaculty of Sciences\nVrije Universiteit Amsterdam\nDe Boelelaan 10831081 HVAmsterdamThe Netherlands'], 'corpusid': 254591490, 'doi': '10.1063/5.0137729', 'github_urls': [], 'n_tokens_mistral': 32836, 'n_tokens_neox': 27928, 'n_words': 16298, 'pdfsha': 'daed43b40dbeee0c230a84586ca16cc45bf09b7f', 'pdfurls': ['https://export.arxiv.org/pdf/2212.06189v1.pdf'], 'title': ['Characterization of Excited States in Time-Dependent Density Functional Theory Using Localized Molecular Orbitals', 'Characterization of Excited States in Time-Dependent Density Functional Theory Using Localized Molecular Orbitals'], 'venue': []}

arxiv

Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor 25 May 2023 Naman K Gupta Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Ronny Sutarto Canadian Light Source University of Saskatchewan S7N 2V3SaskatoonSaskatchewanCanada Rantong Gong Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Stefan Idziak Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Hiruy Hale Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Young-June Kim Department of Physics University of Toronto M5S 1A7TorontoCanada David G Hawthorn Department of Physics and Astronomy University of Waterloo N2L 3G1WaterlooOntarioCanada Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor 25 May 2023 Unidirectional spin and charge density wave order in the cuprates is known to compete with superconductivity. In the stripe order (La,M)2CuO4 family of cuprates, spin and charge order occur as unidirectional order that can be stabilized by symmetry breaking structural distortions, such as the low temperature tetragonal (LTT) phase. Here we examine the interplay between structure and the formation of charge density wave (CDW) order in the LTT phase of La1.475Nd0.4Sr0.125CuO4 by applying uniaxial stress to distort the structure and influence the formation of CDW order. Using resonant soft x-ray scattering to measure both the CDW order and (0 0 1) structural-nematic Bragg peaks, we find that the application of uniaxial stress along the Cu-O bond direction suppresses the (0 0 1) peak and has the net effect of reducing CDW order, but does so only for CDW order propagating parallel to the applied stress. We connect these observations to previous work showing an enhanced superconducting transition temperature under uniaxial stress; providing insight into how CDW, superconductivity, nematicity, and structure are related and can be tuned relative to one another in cuprates. In the cuprate superconductors, superconductivity, spin and charge density wave orders [1][2][3], and electronic nematicity [4][5][6][7][8] are intertwined, often co-existing and competing [9][10][11]. How these orders manifest and relate to each other depends on the crystalline structure (tetragonal or orthorhombic, single layer or bilayer,..) and symmetries that result from the structure [12]. A powerful avenue to explore and tune these relationships is via applying uniaxial stress to controllably manipulate the structure and examine the response of key properties related to superconductivity present in different families of cuprate superconductors. Application of uniaxial stress in the cuprates has been shown to result in significant changes in the superconducting transition temperature and charge density wave order [13][14][15][16][17][18][19][20]. In the stripe-ordered cuprate, La 1.74 Eu 0.2 Sr 0.16 CuO 4 , Takeshita et al. showed that uniaxial stress applied in the ab plane enhances the superconducting transition temperature, T C [13]. More recent work in La 1.885 Ba 0.115 CuO 4 (LBCO), showed a similar enhancement of T C , along with an accompanying suppression of magnetic order, as measured from µSR [15,20]. This result is indicative of a competition between magnetic order and superconductivity that can be tuned by uniaxial stress. Other studies have examined the impact of uniaxial stress on charge density wave order. In YBa 2 Cu 3 O 6+x (YBCO), ∼1% compressive uniaxial strain enhances both 2D CDW order and results in a unidirectional 3D CDW order, characterized by long in-plane and c-axis correlation lengths along with a strong competition with superconductivity [16,17]. In La 1.875 Sr 0.125 CuO 4 (LSCO), uniaxial stress was found to enhance CDW order propagating perpendicular to the applied stress and sup-press CDW order propagating parallel to the applied stress [18,21]. How uniaxial stress impacts CDW order depends on the structure of the crystal. For instance, in LSCO, the structure is in the low-temperature orthorhombic ( [22,23]. Consequently, the application strain along [100] was interpreted to detwin the CDW order to favour only domains of unidirectional CDW order propagating along [100] for stress applied along [010] [18]. However, in many stripe-ordered cuprates, the application of strain may play a different role than in LSCO. In the (La,M) 2 CuO 4 , substituted with larger rare earth ions Nd, Eu or Ba, a low temperature tetragonal (LTT) phase can be induced that is characterized by octahedral tilts parallel to the Cu-O bond [24,25]. Notably, it is in these materials that spin and charge stripe order was first observed [1] and where, in 1/8th doped LBCO, stripe order is found to be most robust and compete strongly with superconductivity [26]. Early on it was recognized that there was a strong relationship between the LTT structure and stabilizing stripe order, with several compounds exhibiting stripe order to onset at the LTO to LTT phase transition [1,12,26]. The reason for this association is that in these compounds, the LTT structure, shown in Figure 1A and B, induces anisotropy in the electronic structure, such as the hopping between Cu and O (t pd ), the nearest neighbour exchange interaction, J, or the charge transfer energy ∆ pd . [ 1. A The crystal structure of (La,M)2CuO4 in the high temperature tetragonal (HTT) phase. Two CuO2 planes at z = 0 and z = 0.5 are present within a single unit cell. B In the low temperature tetragonal (LTT) phase CuO6 octahedra tilt along the Cu-O bond direction, with the tilt axis alternating between a and b for neighbouring layers. Uniaxial stress along a will be parallel to the tilt axis for half the layers and perpendicular to the tilt axis for the other half of the layers. C The orientation of unidirectional stripe order alternates between neighbouring layers. Uniaxial stress along a will be parallel or perpendicular to the CDW propagation wavevector, depending on the layer. D Schematic of the uniaxial stress device. Rotation of a differential screw is used to apply uniaxial stress to a sample that is mounted to a BeCu plate. The device can be actuated in vacuum and at low temperatures in order to change the applied stress or to rotate the device azimuthally to orient the strain parallel or perpendicular to the horizontal scattering plane of the diffractometer. The sample shown in the inset is set in position using silver epoxy. this anisotropy that is thought to stabilize unidirectional stripe order that within an individual layer is oriented parallel or perpendicular to the octahedral tilt axis. Importantly, although each individual layer breaks C 4 rotational symmetry due to the octahedral tilts, the axis by which the octahedra tilt rotates by 90 • between neighbouring CuO 2 planes such that the crystal structure remains globally tetragonal, as depicted in Figure 1B. Consequently, unidirectional spin and charge order in neighbouring layers is alternates between propagating along [100] and [010] [1,28], as depicted in Figure 1C. Unlike in LSCO without Nd, Eu or Ba substitution, the structural distortion of LTT phase serves to detwin the CDW order within an individual layer. Consequently, the application of modest uniaxial stress may be used to explore the role of modifying the anisotropy of the electronic structure on CDW order, rather than the balance of [100] and [010] domains within a layer. As shown in Figure 1(A-C), the application of a uniaxial stress along the [100] direction will act along the tilt axis for half of the layers and perpendicular to the tilt axis for the other half of the layers, affecting the anisotropy of the electronic structure differently for the two orientations of layers. The conjugate (strain) field provides a local as well as a global orientating potential to the free energy stabilizing the electronic symmetry breaking phases. In this study we utilize resonant soft x-ray scatter-ing to study the impact of uniaxial strain on both the structural phase transition and the CDW order, in La 1.475 Nd 0.4 Sr 0.125 CuO 4 (LNSCO). We find that the application of strain along [100] reduces the intensity of the CDW peak at Q CDW = (-0.24 0 1.5), propagating parallel to applied stress, by a factor of ∼ 2, while having little impact on its correlation length or temperature dependence. In contrast, the applied strain has only modest impact on the intensity of the CDW peak at (0 -0.24 1.5), propagating perpendicular to applied stress. The overall suppression of CDW order due to uniaxial stress is consistent with competition between CDW order and superconductivity, namely that uniaxial stress along [100] modifies the anisotropy of the electronic structure in a manner that suppresses CDW order, which in turn enables the enhancement of the superconducting transition temperature [13,15,20]. Resonant x-ray scattering measurements were performed at the REIXS beamline at the Canadian Light Source [29], on a sample of La 1.475 Nd 0.4 Sr 0.125 CuO 4 , fragments of which have been studied in past reports of x-ray absorption spectroscopy [30] and resonant x-ray scattering [6,31]. The CDW (-0.24 0 1.5) and (0 -0.24 1.5) Bragg peaks were investigated at a photon energy corresponding to the peak of the Cu-L 3 absorption edge (931.3 eV). Note, measurements of the CDW peaks are shown as raw data, normalized only to the incident beam . Scattering intensity is normalized at base temperature (20 K). The inset shows a theta-2theta scan of the (0 0 1) Bragg peak on-resonance at the apical O-K edge, comparing the strained (red and green) and the unstrained (black) case. Notably, the peak intensity drops upon applying uniaxial stress. intensity, but otherwise free of any background subtraction. In addition, to examine the LTO to LTT structural phase transition, we measured the (0 0 1) Bragg peak beak at the Cu-L edge (931.3 eV) and at an energy associated with the apical oxygen (533.3 eV). The sample was orientated by measuring the (0 0 4), (-1 0 3) and (0 -1 3) structural Bragg peaks at 2200 eV, with all Bragg peaks indexed to the high temperature tetragonal phase. The sample was cut and polished to form a rectangular ab face sample of dimensions 2 x 0.5 x 0.2 mm. Unixial stress was applied parallel to the longest axis of the sample, which is oriented parallel to [100]. Subsequent to polishing, the sample's surface was etched with bromine. The application of uniaxial stress was applied by mounting the sample on a custom mechanically-actuated stage, depicted in Figure 1D. The sample is mounted on a BeCu plate that has been machined with notch flexures to enable motion along a single axis, but provide a degree of stiffness to bending or tilting. The sample is positioned in a groove in the BeCu plate, with the edges encapsulated above and below with EPO-TEK H2OE silver epoxy in an effort to apply uniform stress. The BeCu plate is affixed to two Cu blocks that can be translated relative to one another by actuating a differential screw that enables small controlled displacements between the plates, resulting in uniaxial compressive or tensile stress on the sample. A novel feature of the strain device is the capability to rotate the device azimuthally under stress, in-vacuum and at low temperature in order to orient the direction of the applied stress either parallel or perpendicular to the scattering plane of the diffractometer. As such, stress can be applied to the sample along the a-axis and CDW peaks can be investigated along both (-0.24 0 1.5) and (0 -0.24 1.5) without needing to thermally cycle or remount the sample. Unfortunately, our device does not measure the magnitude of the applied stress. However, by measuring the structural Bragg peaks before and after the application of uniaxial stress, we can identify that the peak positions do not shift by amounts larger than the accuracy of our measurements. This limits the in-plane (or caxis) compressive strain to be less than 0.2%. Given the C 11 = 267 GPa in related materials [32], this strain limit corresponds to a uniaxial stress along [100] of < 0.5 GPa. As shown in Figure 2, the temperature dependence of the (0 0 1) peaks measured at the apical O-K edge, exhibits a sharp transition to the LTT phase at 70 K, consistent with previous unstrained measurements on this sample [6,33]. The application of stress along [100] results in a decrease in the (0 0 1) peak intensity at low temperatures, as well as, a more gradual temperature dependence between 70 K and 40 K, with a plateau occurring below 40 K. At this photon energy, the (0 0 1) peak results from a difference in the orbital symmetry of apical O atoms between neighbouring (La,M) 2 O 2 layers [6,34,35]. Consequently, the more gradual temperature dependence of the (0 0 1) peak under applied stress is understood to be associated with a reduction in the difference in orbital symmetry between neighbouring layers. This reduction may result from changes in the angles by which CuO 6 octahedra tilt out of the ab plane and/or changes in the octahedral tilt axis away from the Cu-O bond direction. Notably, the strain dependence of the (0 0 1) Bragg peak in Figure 2 is in qualitative agreement with recent reports hard x-ray scattering study of the LTO-LTT transition under unaxial stress in another stripeordered cuprate La 1.885 Ba 0.115 CuO 4 [20]. In that material, Guguchia et al. show the LTT transition to be completely suppressed for compressive uniaxial stress above ∼0.06 GPa. However, for lower stress values (∼0.017 GPa), the LTT phase remains, but with the LTT Bragg peak intensity suppressed, the onset temperature is only weakly dependent on stress, and the LTO-LTT transition is broadened [20], qualitatively similar to the dependence on uniaxial stress dependence of the (0 0 1) we observe in LNSCO for the range of pressure applied. Regarding CDW order, in the unstrained situation, we observe comparable peak intensities for the (-0.24 0 1.5) and (0 -0.24 1.5) peaks. As shown in Figure 3, the response of CDW order to uniaxial stress is asymmetric -having a different impact of the (-0.24 0 1.5) and (0 -0.24 1.5) peaks. For the largest strain applied, the (-0.24 0 1.5) Bragg peak, corresponding to CDW order . A) and B) Intensity at 20 K and at 90 K, above the CDW ordering temperature, for the unstrained and strained case. In A) measurements are shown through the Bragg peak (-0.24 0 1.5) peak position, corresponding to CDW order propagating parallel to the applied stress. In B measurements are shown through the Bragg peak (0 -0.24 1.5) peak position, corresponding to CDW order propagating perpendicular to the applied stress. C) Measurements through (-0.24 0 1.5) in the unstrained configuration, before (UNS 1) and after releasing (UNS 2) applied stress to the sample. D) The temperature dependence at (-0.24 0 1.5) with and without strain. The onset temperature of the CDW order is similar in the strained and unstrained configurations. All panels show raw data, only normalized to incident beam intensity, free from any background subtractions. propagating parallel to the applied stress, is reduced in intensity by a factor of ∼ 2 at 20 K. However, as shown in Figure 3B, this same strain has minimal to no impact on the intensity of the (0 -0.24 1.5) Bragg peak, corresponding to CDW order propagating perpendicular to the applied stress. Moreover, although the intensity of the (-0.24 0 1.5) decreases with the application of stress, the width of the Bragg peak, associated with the CDW correlation length along the a-axis, is unaffected. In both cases, the application of uniaxial stress does not change the background measured at 90 K, which is dominated by x-ray fluorescence from the sample. The temperature dependence of the intensity at (-0.24 0 1.5) in strained and unstrained configurations is shown in Figure 3D. This shows that the onset temperature is not strongly impacted by strain, with the main impact of strain being the reduction in peak intensity in the region where CDW order exists. This reduction in intensity without a substantive change in the CDW order onset temperature matches well with how the spin order measured by muon spin rotation responds to compressive uniaxial stress in LBCO [20]. To verify that the strain-induced suppression in the (-0.24 0 1.5) peak intensity is not associated with an irreversible degradation of the sample (cracking, buckling, mosaic domains...) upon straining, we released the strain and remeasured the sample in an unstrained configuration (UNS 2). These measurements show a recovery of the original CDW peak intensity (see UNS 1 in Figure 3C), as well as a recovery of the (0 0 1) peak temperature dependence. Notably, our results with uniaxial compressive stress differ from reported measurements in LNSCO with uniaxial tensile stress by Boyle et al. [19]. They also observe a reduction in the CDW peak intensity at low T under strain. However, where we find T LT T and T CDW are unchanged by the compressive strain we applied, they report that a tensile strain of ε a = +0.046 ± 0.026% reduced both T CDW and T LT T by 29 K. Moreover, they do not observe an appreciable broadening of the LTT transition or reduction in the (0 0 1) peak intensity at low temperatures, as we observed in our study. This may indicate a marked difference in the impact of tensile versus compressive strain or the magnitudes of strain applied, warranting further investigation. The compressive strain dependence observed in our study is also in contrast to that of reports in LSCO, where CDW order occurs within the LTO structural phase. In LSCO, CDW propagating along the a and b axes would be degenerate in an unstrained crystal, giving rise to domains of unidirectional CDW order that run along both a and b within an individual layer [18]. As such, even a small uniaxial strain along a or b can break the degeneracy, detwinning the CDW such that only a or b oriented CDW order occur, consistent with the measurements of Choi et al. [18]. Notably, in LSCO application of strain beyond that required to detwin the CDW order did not further enhance the CDW order [18]. Connecting the observations to our results in LNSCO, the application of uniaxial strain here does not have a similar effect of detwinning CDW order -enhancing CDW order along b and suppressing CDW order along a. This is likely due to octahedral tilts along a and b in the LTT structure already being effective at detwinning the CDW order within individual layers. As such, the strain dependence CDW order observed here is likely associated with how anisotropic strain impacts the in-plane anisotropy of the electronic structure, such as the nearest-neighbour hopping and exchange interactions. This electronic anisotropy, in turn, can affect the amplitude of CDW order. Structural refinements of CDW order in the LTT phase of LBCO have identified that CDW order within an individual layer propagates along a direction parallel to the bent O-Cu-O bonds, as opposed to the straight O-Cu-O bonds, as depicted in Figure 1 [28]. While it is not clear how the electronic anisotropy of the individual layers changes in response to anisotropic strain, one might expect that the electronic anisotropy increases for the layers with the applied stress parallel to the straight O-Cu-O bonds and decreases for layers with the applied stress perpendicular to the straight O-Cu-O bonds. If this is indeed the case, our findings of a suppression of CDW order along H (parallel to the applied stress) indicate that a reduction in electronic anisotropy of a layer results in a reduction of CDW order, but the converse need not be the case. Ultimately, the net effect of applying uniaxial stress along the Cu-O bond direction is to suppress CDW order. This is in accordance with observed increases in the superconducting transition temperature by applying stress in the [100] direction in related systems La [15,20] that share the LTT structure with LNSCO. As such, at least part of the increase in T c may be attributed to the competition of CDW order with superconductivity. Namely, uniaxial stress affects the anisotropy of the electronic structure in a manner that suppresses CDW order, resulting in an enhancement of superconductivity. In addition to investigating CDW order, we also probe the temperature dependence of the Q x = Q y = 0 electronic anisotropy by measuring the temperature dependence of the (0 0 1) Bragg peak under uniaxial strain at the Cu-L edge, in addition to O-K edge (at an energy corresponding to apical oxygen). As shown in Figure 4, the temperature dependence of the (0 0 1) peak inten-sity is different and more gradual when measured at the Cu L relative to the O-K edge, consistent with previous studies without strain [6,33]. However, uniaxial stress appears to eliminate this difference in the T dependence. Achkar et al. [6] argued that this difference in temperature dependence is due to electronic nematic order that is coupled to CDW order and is in addition electronic asymmetry that directly results from the structure distortions. Moreover, although the (0 0 1) peak measures the difference in orbital symmetry between neighbouring layers, in the unstrained case the symmetry of the crystal structure is such that this difference in the interlayer orbital asymmetry maps to a measure of nematicity within a single layer. However, because of the inequivalence of neighbouring layers under anisotropic strain, this mapping is not valid, complicating the interpretation of the (0 0 1) peak T -dependence under uniaxial stress. The lack of a difference in the T -dependence between Cu-L and apical O-K measurements may indicate that uniaxial stress suppresses electronic nematic order within individual layers. However, it may also indicate that nematic order remains strong and is perhaps saturated. In this scenario, uniaxial stress may align electronic nematic order between neighbouring layers, such that signatures of it cancel in measurements of the (0 0 1) Bragg peak. Future work will be required to differentiate the origin of this anomalous result. In conclusion, we find that in stripe-ordered La 1.475 Nd 0.4 Sr 0.125 CuO 4 , uniaxial stress along [100] suppresses the magnitude of CDW order parallel to the applied stress, but has little impact on CDW order propagating perpendicular to the applied stress. We attribute this suppression to modification of the electronic asymmetry within the CuO 2 planes rather than a detwinning of CDW order, seen in LSCO. Signatures of nematic order, as observed via the relative temperature dependence of the (0 0 1) peak at the Cu-L edge and O-K edge, are also observed to be suppressed by uniaxial strain. This suppression of CDW order is likely linked to the enhancement of superconductivity under uniaxial stress along [100] direction. We gratefully acknowledge Christopher Keegan, Erica Carlson, Steve Kivelson, and Louis Taillefer for useful discussions. We also acknowledge assistance from Rafael Mirabal and Michael Boulaine during the sample etching process. This research was supported by the Natural Sciences and Engineering Research Council (NSERC). N.K. Gupta acknowledges support from the Waterloo Institute of Nanotechnology (WIN). Part of the research described in this paper was performed at the Canadian Light Source, a national research facility of the University of Saskatchewan, which is supported by the Canada Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council (NSERC), the National Research Council Canada (NRC), the Canadian Institutes of Health Research (CIHR), the Government FIG. 2 . 2Temperature dependence of the (0 0 1) Bragg peak at apical O-K edge under varying strain (increasing as shown by an arrow) FIG. 3 . 3Response of CDW Bragg peaks to applied uniaxial stress along [100] FIG. 4 . 4Temperature dependence of the structural-nematic (0 0 1) Bragg peaks for planar and apical atoms. A Temperature evolution of the theta-2theta scans of the (0 0 1) Bragg peak at the Cu-L edge under in-plane, compressive uniaxial strain. Raw data, only normalized to incident beam intensity, free from any background subtractions. B Comparison of temperature dependence of the (0 0 1) peak measured at the Cu-L (931.3 eV) and apical O-K edge (533.3 eV) energies for unstrained and strained case. Scattering intensity is normalized at base temperature(20 K). The temperature evolution of the (0 0 1) peak at the two energies differs in the unstrained case,[36], but behaves similarly under uniaxial stress. LTO) phase, characterized by tilts of CuO 6 octahedra about an axis diagonal to the Cu-O bond. 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{'abstract': 'Unidirectional spin and charge density wave order in the cuprates is known to compete with superconductivity. In the stripe order (La,M)2CuO4 family of cuprates, spin and charge order occur as unidirectional order that can be stabilized by symmetry breaking structural distortions, such as the low temperature tetragonal (LTT) phase. Here we examine the interplay between structure and the formation of charge density wave (CDW) order in the LTT phase of La1.475Nd0.4Sr0.125CuO4 by applying uniaxial stress to distort the structure and influence the formation of CDW order. Using resonant soft x-ray scattering to measure both the CDW order and (0 0 1) structural-nematic Bragg peaks, we find that the application of uniaxial stress along the Cu-O bond direction suppresses the (0 0 1) peak and has the net effect of reducing CDW order, but does so only for CDW order propagating parallel to the applied stress. We connect these observations to previous work showing an enhanced superconducting transition temperature under uniaxial stress; providing insight into how CDW, superconductivity, nematicity, and structure are related and can be tuned relative to one another in cuprates.', 'arxivid': '2305.16499', 'author': ['Naman K Gupta \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n', 'Ronny Sutarto \nCanadian Light Source\nUniversity of Saskatchewan\nS7N 2V3SaskatoonSaskatchewanCanada\n', 'Rantong Gong \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n', 'Stefan Idziak \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n', 'Hiruy Hale \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n', 'Young-June Kim \nDepartment of Physics\nUniversity of Toronto\nM5S 1A7TorontoCanada\n', 'David G Hawthorn \nDepartment of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Canadian Light Source\nUniversity of Saskatchewan\nS7N 2V3SaskatoonSaskatchewanCanada', 'Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Department of Physics\nUniversity of Toronto\nM5S 1A7TorontoCanada', 'Department of Physics and Astronomy\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada'], 'corpusid': 258947740, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14928, 'n_tokens_neox': 12750, 'n_words': 6916, 'pdfsha': '2464622f1ea566de05f90432ff3adc05a3f9f97c', 'pdfurls': ['https://export.arxiv.org/pdf/2305.16499v1.pdf'], 'title': ['Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor', 'Tuning charge density wave order and structure via uniaxial stress in a stripe-ordered cuprate superconductor'], 'venue': []}

arxiv

TAUBERIAN THEOREMS FOR ORDINARY CONVERGENCE 6 Dec 2020 Paolo Leonetti TAUBERIAN THEOREMS FOR ORDINARY CONVERGENCE 6 Dec 2020 We show that a real sequence x is convergent if and only if there exist a regular matrix A and an F σδ -ideal I on N such that the set of subsequences y of x for which Ay is I-convergent is of the second Baire category. This includes the cases where I is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. 33 (1958), 121-123]. Our proofs are of a different nature and rely on recent results in the context of I-Baire classes and filter games.As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix A, there exists a {0, 1}-valued sequence x such that Ax is not statistically convergent.2010 Mathematics Subject Classification. Primary: 40A35, 40G15. Secondary: 54A20, 40A05. Introduction Given an infinite real matrix A = (a n,k ), we say that A sums a real sequence x if Ax = ( k a n,k x k ) is well defined and convergent. The matrix A is regular if it maps convergent sequences into convergent sequences, preserving the corresponding limits. The classical Steinhaus' theorem states that a regular matrix A cannot sum all {0, 1}valued sequences, see e.g. [10]. This can be rewritten equivalently as follows: a regular matrix A cannot sum all subsequences of a divergent {0, 1}-valued sequence x. We recall also that Hahn's theorem [4,Theorem 2.4.5] states that a matrix A sums all {0, 1}-valued sequences if and only if it sums all bounded sequences. On this direction, Buck [7,8] showed that a regular matrix A cannot sum all subsequences of a given divergent sequence x. Let Σ be the set of strictly increasing functions σ : N → N, so that a subsequence of a sequence x is uniquely identified by some σ ∈ Σ, and write σ(x) := (x σ(n) ). Accordingly, Buck's result tells us that, if A is regular and x is divergent, then the set Σ x,A := {σ ∈ Σ : Aσ(x) ∈ c} (1) cannot be equal to Σ. Finally, Keogh and Petersen [25] proved the following: Theorem 1.1. Let x be a divergent sequence and A be a regular matrix. Then Σ x,A is meager. This is meaningful since Σ is a G δ -subset of the Polish space N N , hence it is a Polish space as well by Alexandrov's theorem; in particular, Σ is not meager in itself. Special instances of Theorem 1.1 have been recently rediscovered in [5,6,35]. Related results can be found in [22,Theorem 2.3] and [13,23,30,33]. Note that, if x is convergent and A is regular, then Σ x,A is equal to the whole Σ. Therefore, by Theorem 1.1, x is convergent if and only if there exists a regular matrix A such that Σ x,A is not meager. The aim of this work is to show that Theorem 1.1 holds even if the space of convergent sequences c in the definition of Σ x,A in (1) is replaced by a much bigger set, e.g., the space of statistically convergent sequences (see Section 2). Our methods are different from those employed by Keogh and Petersen, and rely on more recent result on filter games and ideal Baire classes. Main result Let I be an ideal on the positive integers N, that is, a hereditary family of subsets of N closed under finite unions; moreover, it is assumed that I contains the family of finite sets Fin and it is different from P(N). Let I ⋆ := {S ⊆ N : S c ∈ I} be its dual filter. Every subset of P(N) is endowed with the relative Cantor-space topology. In particular, we may speak about F σ -subsets of P(N), meager ideals, etc. In addition, let Z be the ideal of asymptotic density zero sets, that is, Z := {S ⊆ N : lim n |S ∩ [1, n]|/n = 0} . For our purposes, it will be important also to recall the definition of the ideal Fin × Fin, which can be represented as an ideal on N by Fin × Fin := {S ⊆ N : ∀ ∞ k ∈ N, {n ∈ S : ν 2 (n) = k} ∈ Fin} , where ν 2 (n) stands for the 2-adic valuation of n ∈ N (that is, ν 2 (n) = max{k ≥ 0 : 2 k divides n}). We recall that Z is a F σδ -ideal, while Fin × Fin is F σδσ . Given ideals I, J on N, we say that I contains an isomorphic copy of J if there exists an injection ι : N → N such that S ∈ J if and only if ι −1 [S] ∈ I for all S ⊆ N; see e.g. [16]. Let ω be the vector space of all real sequences, together with its classical subspaces ℓ ∞ , c, c 0 , ℓ 1 , c 00 of bounded, convergent, convergent to 0, absolutely summable, and eventually 0 sequences, respectively. In addition, given an ideal I on N, we define also the following subspaces of ω: Note that c = c(Fin) = c b (Fin) and ℓ ∞ = ℓ ∞ (Fin). Here, a sequence x ∈ ω is I-bounded if there exists k such that {n ∈ N : |x n | ≥ k} ∈ I. Moreover, x is I-convergent to η ∈ R, shortened as Ilim x = η, provided that {n ∈ N : |x n − η| > ε} ∈ I for all ε > 0. Every subspace of ℓ ∞ will be endowed with the topology induced by the supremum norm. In the literature, Z-convergence is usually called statistical convergence. At this point, for each infinite real matrix A = (a n,k ), let ω A be the set of sequences x such that Ax is well defined, so that ω A := {x ∈ ω : k a n,k x k converges for all n} . For all nonempty X, Y ⊆ ω, let (X, Y ) be the matrix class of matrices A which map sequences in X into sequences in Y , i.e., (X, Y ) := {A : x ∈ ω A and Ax ∈ Y for all x ∈ X}. Given ideals I, J on N, a matrix A is (I, J )-regular if it is maps I-convergent bounded sequences into J -convergent bounded sequences, preserving the corresponding ideal limits, that is, A ∈ (c b (I), c b (J )) and I-lim x = J -lim Ax for all x ∈ c b (I). Note that (Fin, Fin)-regular matrices are simply the classical regular matrices. Probably the most important regular matrix is the Cesàro matrix C 1 = (a n,k ) defined by a n,k = 1 n if k ≤ n and a n,k = 0 otherwise. Classes of (I, J )-regular matrices have been recently used and characterized in [12]. In particular, the following result extends the classical Silverman-Toeplitz characterization, see [ (R1) sup n k |a n,k | < ∞; (R2) Ilim n a n,k = 0 for all k ∈ N; (R3) Ilim n k a n,k = 1. Finally, for each real sequence x, matrix A, and ideal I, define Σ x,A (I) := {σ ∈ Σ : Aσ(x) ∈ c(I)} . Note that Σ x,A = Σ x,A (Fin). Our main result follows (for the proof, see Section 4): Theorem 2.2. Let x be a divergent sequence, I be a Borel ideal on N which does not contain an isomorphic copy of Fin×Fin, and A be a (Fin, I)-regular matrix. Then Σ x,A (I) is meager. Some remarks are in order. First of all, Theorem 2.2 holds for (Fin, I)-regular matrices, hence in particular it holds for regular matrices. Secondly, all F σδ -ideals satisfy the above hypothesis, see Remark 3.2 below. In particular, Theorem 2.2 holds for the ideal Z of asymptotic density zero sets, for the ideals generated by nonnegative regular matrices (see [3,Proposition 13]), for the ideal Fin (hence, providing another proof of Theorem 1.1), and for the ideal of Banach density zero sets. Lastly, if x is convergent then Σ x,A (I) = Σ. Putting everything together, we obtain: Corollary 2.3. A real sequence x is convergent if and only if there exists a regular matrix A such that {σ ∈ Σ : Aσ(x) is statistically convergent } is not meager. In particular, letting x be the sequence (0, 1, 0, 1, . . .), we obtain that {σ(x) : σ ∈ Σ} = {0, 1} N and Σ x,A (Z) = Σ, which give us a stronger version of Steinhaus' theorem: Corollary 2.4. For each regular matrix A, there exists a {0, 1}-valued sequence x such that Ax is not statistically convergent. On a similar direction, letting A be the infinite identity matrix, it follows that, a real sequence x is convergent if and only if {σ ∈ Σ : σ(x) ∈ c(I)} is not meager, provided that I is an ideal on N as in Theorem 2.2. For the latter class of ideals, this gives us a generalization of [1, Corollary 2.7], which states that lim x = η if and only if {σ ∈ Σ : Ilim σ(x) = η} is not meager, provided that I is a meager ideal and Ilim x = η (note that η is explicit). Other Tauberian theorems related to statistical convergence can be found in [2,9,20,21,31,32]. Finally, note that Theorem 2.2 does not hold without any restriction on the ideal I. Indeed, if A is a (Fin, I)-regular matrix, then it satisfies condition (R1) of Theorem 2.1, so that A ∈ (ℓ ∞ , ℓ ∞ ). Thus, if x ∈ ℓ ∞ \ c and I is a maximal ideal, then every subsequence σ(x) is bounded, hence Aσ(x) ∈ c b (I). In particular, for each bounded divergent sequence x, there exist an ideal I and a (Fin, I)-regular matrix A such that Σ x,A (I) is not meager. Preliminaries Given an ideal I, let G(I) be the following game introduced by Laflamme in [28]: at stage n ∈ N player I chooses a set A n ∈ I ⋆ and, then, player II chooses a nonempty finite set F n ⊆ A n . At the end of the game, player II is declared the winner if n F n / ∈ I. Moreover, a sequence (F k ) of nonempty finite sets is said to be a I ⋆ -universal set if each A ∈ I ⋆ contains some F k . We say that I ⋆ is ω-diagonalizable by I ⋆ -universal sets if there exists an infinite matrix (F n,k ) of nonempty finite sets such that each row is a I ⋆ -universal set and, moreover, for each A ∈ I ⋆ there exist n, m ∈ N such that F n,k ∩ A = ∅ for all k > m. (c1) I ⋆ is ω-diagonalizable by I ⋆ -universal sets; (c2) player II has a winning strategy in the game G(I). If, in addition, I is a Borel ideal, then they are also equivalent to: (c3) I does not contain an isomorphic copy of Fin × Fin; (c4) I is F σ -separated from its dual filter I ⋆ (that is, there exists an F σ -set K such that I ⊆ K and K ∩ I ⋆ = ∅). Proof. See [28, Theorem 2.16(ii)] for (c1) ⇐⇒ (c2). For the other equivalences, see [27]. At this point, for each ideal I and topological space X, define the I-Baire one class B I 1 (X) := f ∈ R X : ∃(f n ) ∈ C(X) N , ∀x ∈ X, f (x) = I-lim f n (x) , where C(X) stands for the space of real-valued continuous functions on X. Note that B I 1 (X) coincides with the classical Baire one class B 1 (X) if I = Fin. However, it has been shown in [27,Proposition 8], the same holds if X is a complete metric space and I is an ideal for which player II has a winning strategy in the game G(I), cf. also [14] for related results. Some years later, Filipów and Szuca proved in [18], in particular, that the hypothesis of completeness on X is unnecessary to obtain the latter conclusion: This has been also obtained in [27,Corollary 12] for nonpathological analytic P-ideals I and arbitrary topological spaces X. We recall also the classical Baire classification theorem. Finally, we will use the well-known characterization of meager ideals due to Talagrand. Theorem 3.5. Let I be an ideal on N. Then the following are equivalent: (m1) I is meager ; (m2) there exists σ ∈ Σ such that A / ∈ I whenever N ∩ [σ(n), σ(n + 1)) ⊆ A for infinitely many n ∈ N; (m3) I is F σ -separated from the Fréchet filter Fin ⋆ (that is, there exists a sequence (F n ) of closed sets such that I ⊆ n F n and F n ∩ Fin ⋆ = ∅ for all n ∈ N). Proof At this point, we split the intermediate results into two cases: the first one assumes that x is a bounded divergent sequence, the second one that x is unbounded. Bounded case. Lemma 3.6. Fix two sequences x ∈ ℓ ∞ and a ∈ ℓ 1 . Then the map Σ → R : σ → a · σ(x) is uniformly continuous. Proof. The claim holds trivially for x = 0, hence suppose hereafter that x = 0, so that x > 0. Note that the topology on Σ is metrizable by d : Σ × Σ → R defined as ∀σ 1 , σ 2 ∈ Σ, d(σ 1 , σ 2 ) = i∈ Im(σ 1 )△ Im(σ 2 ) 1 2 i . Fix ε > 0. Since a ∈ ℓ 1 , there exists k 0 ∈ N such that k>k 0 |a k | < ε 2 x . At this point, define δ := 1/2 k 0 and note that, if there exists k ≤ k 0 such that k ∈ Im(σ 1 ) △ Im(σ 2 ) then d(σ 1 , σ 2 ) ≥ 1/2 k ≥ δ. Therefore, for each σ 1 , σ 2 ∈ Σ with d(σ 1 , σ 2 ) < δ, we obtain that the least element of Im(σ 1 ) △ Im(σ 2 ) is greater than k 0 , which implies that |a · σ 1 (x) − a · σ 2 (x)| = k>k 0 a k (x σ 1 (k) − x σ 2 (k) )) ≤ 2 x k>k 0 |a k | < ε. This concludes the proof. Corollary 3.7. Fix a matrix A ∈ (c 0 , ℓ ∞ ) and a sequence x ∈ ℓ ∞ . Then the map Σ → R : σ → k a n,k x σ(k) is uniformly continuous for each n ∈ N. Proof. Thanks to [4, Theorem 2.3.5], A belongs to (c 0 , ℓ ∞ ) if and only if sup n k |a n,k | < ∞. The claim follows by Lemma 3.6. is everywhere discontinuous. Proof. Let B ⊆ Σ an arbitrary open ball. Observe that there exist σ 1 , σ 2 ∈ B such that lim σ 1 (x) = α and lim σ 2 (x) = β, where α := lim sup n x n and β := lim inf n x n are finite and distinct since x ∈ ℓ ∞ \ c. Considering that A is (Fin, I)-regular, we obtain that α = Ilim Aσ 1 (x) and β = Ilim Aσ 2 (x). In particular, B ∩ Σ x,A (I) = ∅, hence Σ x,A (I) is dense. In addition, sup σ,σ ′ ∈B∩Σ x,A (I) |T (σ) − T (σ ′ )| ≥ α − β for every open ball B ⊆ Σ, therefore T cannot be continuous at any point of Σ x,A (I). Remark 3.9. With the same hypotheses of Lemma 3.8, assume that I is a Borel ideal. Then Σ x,A (I) is Borel and the map T defined in (2) is Borel measurable. For, note that necessarily A ∈ (c 0 , ℓ ∞ ). Hence the map T is the I-pointwise limit of the sequence of functions T n : σ → k a n,k x k restricted to Σ x,A (I), which are continuous for each n ∈ N thanks to Corollary 3.7. In particular, each of them is Borel measurable. The claim follows by [26,Lemma 1 and Lemma 2]. However, as remarked in [26,Example 1], some assumptions on I are needed to ensure that the I-pointwise limit of measurable functions is still measurable. Theorem 3.10. Fix a bounded divergent sequence x ∈ ℓ ∞ \ c, an ideal I on N such that I ⋆ is ω-diagonalizable by I ⋆ -universal sets, and a (Fin, I)-regular matrix A. Then Σ x,A (I) is meager. Proof. Set X := Σ x,A (I) and let T be the map defined in (2). Thanks to Corollary 3.7, T ∈ B I 1 (X). Since X is metrizable and I ⋆ is ω-diagonalizable by I ⋆ -universal sets, then T ∈ B 1 (X) by Theorem 3.3. At this point, it follows by Theorem 3.4 that the set of continuity points of T is a comeager subset of X. Hence, by Lemma 3.8, X is a dense subset of Σ which is meager in itself. We conclude that X is meager in Σ. Unbounded case. Theorem 3.11. Fix sequences x ∈ ω \ ℓ ∞ and a ∈ ω \ c 00 . Then σ ∈ Σ : k≤n a k x σ(k) ∈ ℓ ∞ is meager. Proof. Let E be the claimed set. Observe that E = m E m , where ∀m ∈ N, E m := σ ∈ Σ : k≤n a k x σ(k) ≤ m for all n . We are going each E m is nowhere dense. Note that each E m is closed. Thus, let us assume for the sake of contradiction that there exists m 0 such that E m 0 has nonempty interior, hence there exist positive integers t 1 < · · · < t j such that σ ∈ E m 0 whenever σ(s) = t s for all s = 1, . . . , j. Since a / ∈ c 00 , there exists a minimal integer i 0 ≥ j + 1 such that a i 0 = 0. In addition, since x / ∈ ℓ ∞ , there exists an integer t 0 ≥ t j + i 0 such that |a i 0 x t 0 | ≥ m 0 + 1 + k≤j a k x t k .(3) Finally, define σ 0 : N → N as follows: (i) σ 0 (s) = t s for all s = 1, . . . , j; (ii) σ 0 (j + s) = t j + s for all s = 1, . . . , i 0 − j − 1; (iii) σ 0 (i 0 + s) = t 0 + s for all integers s ≥ 0. Note that σ 0 is strictly increasing, hence σ 0 ∈ Σ. Moreover, thanks to (i), σ 0 ∈ E m 0 . On the other hand, by (iii) we have σ 0 (i 0 ) = t 0 . And by the minimality of i 0 we get a j+s = 0 for all s = 1, . . . , i 0 − j − 1. Putting everything together with (3), we obtain that k≤i 0 a k x σ 0 (k) = a i 0 x t 0 + k≤j a k x t k ≥ m 0 + 1, which would imply that σ 0 / ∈ E m 0 . This contradiction concludes the proof. A matrix A is said row finite if every row is in c 00 , that is, for each n there exists k 0 ∈ N such that a n,k = 0 for all k ≥ k 0 . Corollary 3.12. Let A be a matrix which is not row finite, and fix x ∈ ω \ ℓ ∞ . Then {σ ∈ Σ : σ(x) ∈ ω A } is meager. Proof. By hypothesis, there exists r 0 ∈ N such that a r 0 ,k = 0 for infinitely many k. Hence {σ ∈ Σ : σ(x) ∈ ω A } = r σ ∈ Σ : k≤n a r,k x σ(k) ∈ c ⊆ σ ∈ Σ : k≤n a r 0 ,k x σ(k) ∈ ℓ ∞ . The conclusion follows by Theorem 3.11. Theorem 3.13. Let I be a meager ideal on N. Fix an unbounded sequence x ∈ ω \ ℓ ∞ and a row finite matrix A such that ∀w ∈ N, Z w := {n ∈ N : a n,k = 0 for all k ≥ w} ∈ I. (4) Then {σ ∈ Σ : Aσ(x) ∈ ℓ ∞ (I)} is meager. Proof. Thanks to Theorem 3.5, there exists a sequence (F n ) of closed sets in P(N) such that I ⊆ n F n and F n ∩ Fin ⋆ = ∅ for all n ∈ N. Fix w ∈ N. Since Z w ∈ I by condition (4), it follows that the family J w := {U \ Z : U ∈ I} is an ideal on T w := N \ Z w such that J w ⊆ n G n and G n ∩ Fin ⋆ = ∅ for all n ∈ N, where G n := F n ∩ P(T w ) is closed in P(T w ). It follows, again by Theorem 3.5, that J w is a meager ideal on T w . Hence there exists a partition {I w,1 , I w 2 , . . .} of T w into nonempty finite subsets such that a set U ⊆ T w does not belong to J w whenever I w,n ⊆ U for infinitely many n. At this point, let S be the claimed set. Since T w ∈ I ⋆ for each w ∈ N, we obtain that ∀w ∈ N, S = m {σ ∈ Σ : {n ∈ N : | k a n,k x σ(k) | ≥ m} ∈ I} = m {σ ∈ Σ : {n ∈ T w : | k a n,k x σ(k) | ≥ m} ∈ J w } ⊆ m {σ ∈ Σ : {n ∈ I w,q : | k a n,k x σ(k) | ≥ m} = I w,q for all large q} = m p q≥p {σ ∈ Σ : {n ∈ I w,q : | k a n,k x σ(k) | ≥ m} = I w,q } = m p q≥p J Iw,q {σ ∈ Σ : {n ∈ I w,q : | k a n,k x σ(k) | ≥ m} = J}. Note that all the inner sets are closed, so that also each S w (m, p) := q≥p {σ ∈ Σ : ∀q ≥ p, ∃n ∈ I w,q , | k a n,k x σ(k) | < m} is closed. Since S ⊆ m,p S w (m, p), it is sufficient to show that there exists w ∈ N such that each S w (m, p) has empty interior: this would imply that S is contained in a countable union of nowhere dense sets. To this aim, suppose for the sake of contradiction that there exist m 0 , p 0 ∈ N and positive integers t 1 < · · · < t j 0 such that σ ∈ S(m 0 , p 0 ) whenever σ(s) = t s for all s = 1, . . . , j 0 . Set w 0 := j 0 + 1. By condition (4), the set T w 0 belongs to I ⋆ , hence it is nonempty. Note that, by construction, r n ≥ w 0 for all n ∈ T w 0 . Define n 0 := min T w 0 , let p 1 be the positive integer for which n 0 ∈ I w 0 ,p 1 , and finally set α := min{|a n,k | : a n,k = 0 and n ∈ I w 0 ,q 0 }, where q 0 := max{p 0 , p 1 + 1}. Note that α is well defined since A is row finite, and set k 0 := max{r n : n ∈ I w 0 ,q 0 }. Finally, define σ 0 : N → N recursively as follows: (i) σ 0 (s) = t s for all s = 1, . . . , j 0 ; (ii) for each s = j 0 + 1, . . . , k 0 , if σ 0 (1) < · · · < σ 0 (s − 1) are already defined, then σ 0 (s) is an integer h > σ 0 (s − 1) such that |x h | ≥ 1 α m 0 + max k<s a n,k x σ 0 (k) : n ∈ I w 0 ,q 0 . (iii) σ 0 (k 0 + s) = σ 0 (k 0 ) + s for all s ∈ N. Considering that σ 0 is strictly increasing by construction, it follows by (i) that σ 0 ∈ S w 0 (m 0 , p 0 ). At the same time, since j 0 < r n ≤ k 0 for all n ∈ I w 0 ,q 0 , we obtain by (ii) that ∀n ∈ I w 0 ,q 0 , | k a n,k x σ 0 (k) | = | k≤rn a n,k x σ 0 (k) | ≥ |a n,rn x σ 0 (rn) | − k<rn a n,k Secondly, suppose that x is unbounded. If A is not row finite, then Σ x,A (I) ⊆ {σ ∈ Σ : σ(x) ∈ ω A }, which is meager by Corollary 3.12. Otherwise, suppose hereafter that A is a row finite (Fin, I)-regular matrix. Note that I is a Borel ideal, hence it is meager. Thanks to Theorem 2.1, A satisfies conditions (R1)-(R3). For each w ∈ N, it follows by (R2) and (R3) that Ilim n k≥w a n,k = 1. In particular, Z w is contained in {n ∈ N : k≥w a n,k = 0}, which belongs to I. This implies that condition (4) in Theorem 3.13 holds. It follows that Σ x,A (I) is contained in {σ ∈ Σ : Aσ(x) ∈ ℓ ∞ (I)}, which is meager by Theorem 3.13. x σ 0 (k) ≥ α|x σ 0 (rn) | − max k<rn a i,k x σ 0 (k) : i ∈ I w 0 ,q 0 ≥ m 0 . This implies that σ 0 / ∈ S w 0 (m 0 , p 0 ), Concluding remarks It doesn't come as a surprise that, under suitable hypotheses on the matrix A and the ideal I, the set Σ x,A (I) is either meager or the whole Σ. Indeed, by a known topological 0-1 law, see e.g. [24,Theorem 8.47], a tail subset of Σ with the Baire property is either meager or comeager. This applies also in our case. For, let I be a Borel ideal and A be a (Fin, I)-regular matrix. Then, it follows by Remark 3.9 that Σ x,A (I) is Borel, hence it has the Baire property. Moreover, if σ 1 (n) = σ 2 (n) for all but finitely many n, then y ∈ c 0 where y n := x σ 2 (n) − x σ 1 (n) for all n. Hence, if σ 1 ∈ Σ x,A (I) then Ilim Aσ 2 (x) = Ilim Aσ 1 (x) + Ilim Ay = Ilim Aσ 1 (x). This implies that also σ 2 ∈ Σ x,A (I), proving that Σ x,A (I) is also a tail set. In the same direction of [13,19,23], we leave as open question for the interest reader to check whether the analogues of Theorem 2.2 hold for permutations and strechings of the sequence x. Finally, our main result seems to be related also to a conjecture of DeVos [15] which can be reformulated as follows: if E is an FK-space (that is, a locally convex vector space of ω which is also Fréchet and with continuous coordinates) containing c 00 , then {0, 1} N ⊆ E if and only if E ∩ {0, 1} N is not meager. However, it seems quite unlikely that spaces of the type {x ∈ ω : Ax ∈ c(I)} may provide a counterexample to the latter conjecture. Indeed, it has been shown by Connor in [11,Theorem 3.3] that, even for the well-behaved ideal I = Z, the unique FK-space containing c(Z) is ω. ℓ ∞ (I) := { I-bounded sequences }, c(I) := { I-convergent sequences }, c b (I) := { I-convergent bounded sequences }. Theorem 3 . 1 . 31Let I be an ideal on N. Then the following are equivalent: Remark 3. 2 . 2As it has been shown in [34, Corollary 1.5], all F σδ -ideals satisfy condition (c4). Related results on condition (c4) can be found in [17, Proposition 3.6] and [29, Theorem 2.1]. Theorem 3. 3 . 3Let I be an ideal on N such that I ⋆ is ω-diagonalizable by I ⋆ -universal sets. Moreover, let X be a perfectly normal topological space. Then B I 1 (X) = B 1 (X). Proof. See[18, Theorem 3.2]. Theorem 3. 4 . 4Let X be a metrizable space and fix a Baire one function f ∈ B 1 (X). Then the set of points of continuity of f is a comeager G δ -set.Proof. See[24, Theorem 24.14]. Lemma 3 . 8 . 38Fix a bounded divergent sequence x ∈ ℓ ∞ \ c, an ideal I on N, and (Fin, I)matrix A. Then Σ x,A (I) is dense and the map T : Σ x,A (I) → R : σ → Ilim n k a n,k x σ(k) 12, Theorem 1.2 and Theorem 1.3]: Theorem 2.1. 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Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. A universal Tauberian theorem. F R Keogh, G M Petersen, J. London Math. Soc. 33F. R. Keogh and G. M. Petersen, A universal Tauberian theorem, J. London Math. Soc. 33 (1958), 121-123. Pointwise I -convergence and I -convergence in measure of sequences of functions. A Komisarski, J. Math. Anal. Appl. 3402A. Komisarski, Pointwise I -convergence and I -convergence in measure of sequences of functions, J. Math. Anal. Appl. 340 (2008), no. 2, 770-779. Ideal limits of sequences of continuous functions. M Laczkovich, I Recław, Fund. Math. 2031M. Laczkovich and I. Recław, Ideal limits of sequences of continuous functions, Fund. Math. 203 (2009), no. 1, 39-46. C Laflamme, Filter games and combinatorial properties of strategies, Set theory. Boise, ID; Providence, RIAmer. Math. Soc192C. Laflamme, Filter games and combinatorial properties of strategies, Set theory (Boise, ID, 1992-1994), Contemp. Math., vol. 192, Amer. Math. Soc., Providence, RI, 1996, pp. 51-67. Limit points of subsequences. P Leonetti, Topology Appl. 263P. Leonetti, Limit points of subsequences, Topology Appl. 263 (2019), 221-229. A Tauberian theorem for subsequences. I J Maddox, Bull. London Math. Soc. 2I. J. Maddox, A Tauberian theorem for subsequences, Bull. London Math. Soc. 2 (1970), 63-65. A Tauberian theorem for statistical convergence. I J Maddox, Math. Proc. Cambridge Philos. Soc. 1062I. J. Maddox, A Tauberian theorem for statistical convergence, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 2, 277-280. A measure theoretical subsequence characterization of statistical convergence. H I Miller, Trans. Amer. Math. Soc. 3475H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1811-1819. A characterization of the matrix class. P N Natarajan, Bull. London Math. Soc. 233P. N. Natarajan, A characterization of the matrix class (l ∞ , C 0 ), Bull. London Math. Soc. 23 (1991), no. 3, 267-268. Filters and sequences. S Solecki, Fund. Math. 1633S. Solecki, Filters and sequences, Fund. Math. 163 (2000), no. 3, 215-228. Summability of subsequences of a divergent sequence. C Stuart, Rocky Mountain J. Math. 441C. Stuart, Summability of subsequences of a divergent sequence, Rocky Mountain J. Math. 44 (2014), no. 1, 289-295. Compacts de fonctions mesurables et filtres non mesurables. M Talagrand, Studia Math. 671M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), no. 1, 13-43. Italy Email address: leonetti.paolo@gmail. MilanDepartment of Statistics, Università Bocconivia Roentgen 1Department of Statistics, Università Bocconi, via Roentgen 1, Milan 20136, Italy Email address: leonetti.paolo@gmail.com URL: https://sites.google.com/site/leonettipaolo/

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{'abstract': "We show that a real sequence x is convergent if and only if there exist a regular matrix A and an F σδ -ideal I on N such that the set of subsequences y of x for which Ay is I-convergent is of the second Baire category. This includes the cases where I is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. 33 (1958), 121-123]. Our proofs are of a different nature and rely on recent results in the context of I-Baire classes and filter games.As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix A, there exists a {0, 1}-valued sequence x such that Ax is not statistically convergent.2010 Mathematics Subject Classification. Primary: 40A35, 40G15. Secondary: 54A20, 40A05.", 'arxivid': '2012.03311', 'author': ['Paolo Leonetti '], 'authoraffiliation': [], 'corpusid': 227348397, 'doi': '10.1016/j.jmaa.2022.126798', 'github_urls': [], 'n_tokens_mistral': 11618, 'n_tokens_neox': 10048, 'n_words': 6059, 'pdfsha': '40bfb6570561e03187e32c283d683851d5b57b5c', 'pdfurls': ['https://arxiv.org/pdf/2012.03311v1.pdf'], 'title': ['TAUBERIAN THEOREMS FOR ORDINARY CONVERGENCE', 'TAUBERIAN THEOREMS FOR ORDINARY CONVERGENCE'], 'venue': []}

arxiv

Three Parton Corrections in B → P P decays 13 Feb 2008 Tsung-Wen Yeh Department of Science Application And Dissemination National Taichung University 403TaichungTaiwan Three Parton Corrections in B → P P decays 13 Feb 2008numbers: 1325Hw Keywords: power correctionsQCD factorizationB decays * Electronic address: twyeh@ms3 The 1/m b corrections from the three parton qqg Fock state of the final state light meson in B → P P decays are evaluated by means of a collinear expansion method. The impacts of these corrections on the CP averaged branching ratios of the B → πK decays are analyzed. I. INTRODUCTION The QCD factorization [1,2,3] has been widely used to investigate the charmless hadronic B decays. For an operator O i of the weak effective Hamiltonian, the matrix element for B → M 1 M 2 decays under the QCD factorization is found to be expressible as proaches for processes other than hadronic decays, such as the calculation scheme for the inclusive hard scattering processes [5,6,7] or the method for the exclusive hard scattering processes [29,30,31,32,33]. However, a systematic Feynman-diagram approach for charmless hadronic B decays is still inaccessible. On the other hand, the effective-theory approaches for charmless hadronic B decays have been extensively investigated in recent years [34,35,36,37,38,39]. M 1 M 2 |O i |B = j F B→M 1 j (m 2 2 ) 1 0 duT I ij (u)Φ M 2 (u) + (M 1 ↔ M 2 ) + 1 0 dξdudvT II i (ξ, u, v)Φ B (ξ)Φ M 1 (u)Φ M 2 (v) ,(1) In this paper, a calculation scheme based on the Feynman-diagram approach for charmless hadronic B decays will be developed. We will concentrate on the construction of this calculation scheme and apply the constructed method to calculate the tree level three parton corrections. The O(α s ) three parton corrections is also desirable to understand their factorization properties. Since the related analysis is tedious, we plan to present the relevant calculations in our another preparing work [44]. The organization of this paper is as following. In Section II, the calculation scheme will be constructed. The factorization of the tree level three parton corrections into the partonic and hadronic parts will be outlined. In Section III, the analysis on how the tree level three parton corrections can be factorized into its partonic and hadronic parts will be described in detail. In Section IV, we will apply the results of the Section III to make predictions for the branching ratios of B → πK decays. The last section devotes for discussion and conclusion. II. COLLINEAR EXPANSION AT TREE LEVEL In this section, we will generalize the collinear expansion method [5,6,7] There exist other types of power corrections, such as the power corrections from soft gluons or renormalons. We identify these as non-partonic power corrections. For these power corrections, our proposed scheme may not be useful. However, to include these non-partonic power corrections requires further assumptions beyond the factorization. For example, the soft gluon power corrections are better determined by nonperturbative theories, such as the QCD sum rules or lattice QCD. In this work, we only investigate how the partonic (or the dynamic) power corrections can be included into the QCD factorization in a consistent way. The collinear expansion method arises from a motivation of generalizing the leading twist factorization theorem for the hard scattering processes to include the corrections from high Fock states of the target hadrons. The original idea of the collinear expansion method was proposed by Polizer at 1980 [4]. The systematical method was developed by Ellis, Furmanski and Petronzio (EFP) in their pioneer works [5,6]. Using the collinear expansion, the EFP group showed that, for the DIS processes, the twist-4 power suppressed corrections can be factorized into short distance and long distance parts, which are in a similar factorized form as the leading twist contributions. However, in the EFP's approach, the parton interpretation for the twist-4 corrections are lost. To recover the parton model picture, Qiu then introduced a Feynman-diagram approach [7] to re-formula the EFP's method. In this Feynman-diagram language, a parton model interpretation for the twist-4 corrections becomes trivial. To begin with, we first express the matrix element of an operator O i of the weak effective Hamiltonian H ef f of the standard model for the hadronic decaysB → M 1 M 2 in terms of parton model amplitudes M 1 M 2 )|O i |B = j=+,0 F B→M 1 j (m 2 M 2 ) d 4 l (2π) 4 Tr[T I ij (l)Φ M 2 (l)] + (M 1 ↔ M 2 ) + j=+,0 F B→M 1 j (m 2 M 2 ) d 4 l 1 (2π) 4 d 4 l 2 (2π) 4 Tr[T I ij,µ (l 1 , l 2 )Φ µ M 2 (l 1 , l 2 )] + (M 1 ↔ M 2 ) + d 4 l B (2π) 4 d 4 l M 1 (2π) 4 d 4 l M 2 (2π) 4 Tr[T II i (l N , l M 1 , l M 2 )Φ B (l B )Φ M 1 (l M 1 )Φ M 2 (l M 2 )] ,(3) where F B→M 1 j denote the form factors forB → M 1 lν transition. The form factor F j are defined as M 1 (p)|qγ µ b|B(P b ) = F + (q 2 )(p + P b ) µ + F + (q 2 ) − F 0 (q 2 ) q 2 q µ(4) where q = P b − p and F + (q 2 ) = F 0 (q 2 ) under the limit q 2 → 0. The parton amplitudes T I ij (l), T I ij,µ (l 1 , l 2 ) and T II i (l B , l M 1 , l M 2 ) are defined to describe the hard scattering center involving four parton, five parton, and six parton interactions corresponding to those diagrams depicted in Fig. 1(a)-(c), respectively. Note that there are also other types of parton amplitudes involving five or six parton interactions not being presented in Fig. 1, which can be attributed to either the physical form factors, or to be of higher twist than three. We have neglected these contributions in Eq. (3). The Tr symbol denotes the trace operation applied on the color and spin indices. For convenience, we employ the light-cone coordinate system such that P µ B = (p µ + q µ ) with two light-like vectors q µ = (q + , q − , q i ⊥ ) = (Q, 0, 0) and The last one is important for us to derive a meaningful perturbation theory beyond the leading twist. p µ = (p + , p − , p i ⊥ ) = (0, Q, 0) with Q = m B / √ 2,Φ M 2 (l) = d 4 ye il·y M 2 |q(y)q(0)|0 ,(5)Φ µ M 2 (l 1 , l 2 ) = d 4 y d 4 ze il 1 ·y e i(l 2 −l 1 )·z M 2 |q(y)(−gA µ (z))q(0)|0 .(6) For latter uses, we define the soft, collinear and hard loop parton momenta. We let the soft momentum scale as (l + , l − , l ⊥ ) ∼ (λ, λ, λ), the collinear momentum scale as (l + , l − , l ⊥ ) ∼ (Q, λ 2 /Q, λ), and the hard momentum scale as (l + , l − , l ⊥ ) ∼ (Q, Q, Q). The scale variables are defined as Q ∼ m b and λ ∼ Λ QCD . For a collinear loop parton, it is convenient to parametrize its momentum l µ into its components proportional to the meson momentum q µ , the light-cone vector n ν , and the transversal directions l µ = n · lq µ + l 2 + l 2 ⊥ 2n · l n µ + l µ ⊥ ,(7) where the vector n µ satisfies n · q = 1, n · l ⊥ = 0, and n 2 = 0. For convenience, we further define the collinear component,l µ , the on-shell component, l µ L , and the off-shell component, l µ S of the momentum l µ asl µ = n · lq µ , l µ L =l µ + l 2 ⊥ 2n · l n µ + l µ ⊥ , l µ S = l 2 2n · l n µ .(8) In the above expansion of the parton momentum into different parts, we have assumed According to the parametrization in Eq. (7), a parton propagator can be separated into its long distance part and short distance part (the special propagator). If we write the loop parton propagator as [7] F (y, m M i = 0, i = 1z) = d 4 l (2π) 4 e il·(y−z) [F L (l) + F S (l)] = F L (y, z) + F S (y, z) ,(9) where F L (l) = i/ l L l 2 , F S (l) = i/ n 2n · l .(10) The F L (l) propagator corresponds to the long distance part of the propagator, since F L (y, z) ∝ θ(y − z) . The F S (l) propagator represents the short distance part because F S (y, z) ∝ δ(y − z) . We now describe one important property of the long distance propagator F L (l). In a parton amplitude, the F L (l) may contact with a / qn µ component of a vertex γ µ . Whenever this happens, the / qn µ vertex will extract one short distance propagator F S (l) and one interaction vertex iγ ν from the relevant hadron amplitude [7] i/ l L l 2 / q = i/ l L l 2 (iγ ν ) i/ n 2n · l / q(l −l) ν .(11) The momentum factor (l −l) ν is then absorbed by the hadron amplitude due to the Ward identity [7]. We now explain how the identity Eq. (11) can be obtained by a simple manipulation. We first insert an identity 1 = (/ l 2 )/l 2 into the left hand side of Eq. (11) and expresse each / l into / l L + / l S to obtain i/ l L l 2 / l/ l l 2 / q = i/ l L l 2 (/ l L + / l S )(/ l L + / l S ) l 2 / q .(12) Since (/ l L ) 2 = 0 = (/ l S ) 2 , the above equation then becomes i/ l L l 2 / l/ l l 2 / q = i/ l L l 2 (/ l L / l S + / l S / l L ) l 2 / q ,(13) where the first term / l L / l S in the right hand side leads to a vanishing result as it contacts with the term i/ l L /l 2 term. The only contribution can only come from the second term / l S / l L in the right hand side of Eq. (13). In addition, the / l L can be expanded in the terms proportional to / q, / n, / l ⊥ . This gives / l S / l L / q = l 2 / n 2n · l (n · l/ q + l 2 ⊥ / n 2n · l + / l ⊥ )/ q . Due to / q 2 = / n 2 = 0, it further reduces to l 2 / n 2n · l (/ l ⊥ )/ q . By substituting the above back into Eq. (12), Eq. (11) is then obtained by noting that / l L (iγ α )(i/ n)/ q(l −l) α = / l L / n/ l ⊥ / q . Using Eq. (11), one can systematically include the effects from the non-collinearity and the off-shellness of the collinear partons. This property of the long distance part of the parton propagator plays an important role in our following analysis, and its meaning will become more clear after we have investigated real cases lately. According to the parton model, the hadron amplitudes are defined as the probability for finding the on-shell partons inside the hadron. The parton amplitudes are then required to contain only the on-shell components of the external parton momenta. However, according to Eq. (8), either the on-shell momentum l L or the collinear momentuml can be assigned for an on-shell parton. Therefore, there arise two factorization schemes, the collinear factorization [8,9,10,11,12,13](QCD factorization) and the k T factorization [14,15,16,17] (PQCD factorization [18,19,20,21,22,23,24,25,26,27,28]). In the k T factorization scheme, an onshell parton carries a momentum l L . On the other hand, in the collinear factorization scheme, an on-shell parton carries a momentuml. In this work, we follow the QCD factorization to use the collinear factorization scheme as our basics. Our proposed collinear expansion method is composed of following steps: 1. Use scale analysis for the parton amplitudes according to the scales of parton momenta to find out the leading regions of the parton momentum configuration. 2. The parton amplitudes are expanded into a Taylor series with respect to the leading regions of parton momenta. 3. The expanded parton amplitudes are substituted back into the contraction with the hadron amplitudes to extract relevant contributions up to specific twist order. The factorization of parton momentum integrals is accomplished by means of integral transformations (See, for example, Eq. (24)). 5. The color structure of the parton amplitude is extracted to be attributed to the hadron amplitudes to complete the color factorization. 6. The factorization of spin indices is completed by means of Fierz transformation. 7. The property of the long distance parton propagator is used to extract higher twist contributions. We are now ready to discuss the collinear expansion. First, we order the parton ampli- tudes in α s T I ij (l) = T I(0) ij + T I(1) ij (l) + O(α 2 s ) ,(14) T I ij,µ (l 1 , l 2 ) = T I(0) ij,µ (l 1 , l 2 ) + T I(1) ij,µ (l 1 , l 2 ) + O(α 2 s ) ,(15)T II i (l B , l M 1 , l M 2 ) = T II(1) i (l B , l M 1 , l M 2 ) + O(α 2 s ) ,(16) where the superscription (0) and (1) We first expand the parton amplitudes T I(0) ij,µ with respect to the collinear components of their relevant parton momenta as O i = (q 1 Γ i b)(q 2Γi q 3 ) with Γ i (Γ i )T I(0) ij,µ (l 1 , l 2 ) = T I(0) ij,µ (l 1 ,l 2 ) + 2 k=1 ∂T I(0) ij,µ ∂l ν k l k =l k (l k −l k ) ν + · · · ,(17) where T I(0) ij,µ (l 1 ,l 2 ) = ((iγ µ ) i/ l 2 l 2 2Γ i +Γ i −i/ l 1 l 2 1 (−iγ µ ))δ ij ,(18) ∂T I(0) ij,µ ∂l ν k (l 1 ,l 2 ) = ((iγ µ ) i/ l 2 l 2 2 (iγ ν ) i/ l 2 l 2 2Γ i δ k2 −Γ i −i/ l 1 l 2 1 (−iγ ν ) −i/ l 1 l 2 1 (−iγ µ )δ k1 )δ ij .(19) The expansion series are then substituted back into the convolution integrals with the hadron amplitudes for further analysis. The reason why one can expand the parton amplitudes with respect to the relevant collinear momenta will be explained in detail below. A. Expansion with T I(0) ij The expression for the contributions associated with T I(0) ij is written as j=+,0 F B→M 1 j (m 2 M 2 ) d 4 l (2π) 4 Tr[T I(0) ij Φ M 2 (l)] ,(20) where the loop momentum l is carried by the loop parton in Fig. 1(a). Since the parton amplitude T I(0) ij is independent of l, we propose to use the following integral identity to transform the expression into a form consistent with the parton model picture 1 0 dxδ(x − n · l) = 1 0 dx ∞ 0 dλ 2π e iλ(x−n·l) = 1 .(21) The transformed result appears as 1 0 dxTr[T I(0) ij Φ M 2 (x)](22) where Φ M 2 (x) = ∞ 0 dλ 2π e iλx M 2 |q(λn)q(0)|0 .(23) The following integral transformation has been used in the above to simplify the expression d 4 l (2π) 4 d 4 ye il·(y−λn) G(y, 0) = d 4 yδ (4) (y − λn)G(y, 0) = G(λn, 0) ,(24) where G(y, 0) denotes any function of the coordinates. Two comments for the above integral transformations Eqs. (21) and (24) Because the parton amplitude T I(0) ij is equal toΓ i δ ij , the integral over x is then associated with Φ M 2 (x). The factorization of spin indices depends on the structure ofΓ i . For (V − A)(V ± A) operators,Γ i = (V ± A) and can be expanded into / q(1 ± γ 5 ), / n(1 ± γ 5 ) and γ ⊥ γ 5 . If M 2 is a pseudo-scalar meson, only the axial vector part can contribute. However, only / nγ 5 leads to leading twist contributions. The / qγ 5 will result in twist-4 contributions and γ ⊥ γ 5 will not contribute. For other types of meson, similar considerations can be made. We now explain how the / qγ 5 part can contribute. The long distance part of the parton propagator can interact with vertex / qγ 5 to have i/ l L l 2 / q = i/ l L l 2 (iγ α ) i/ n 2n · l / q(l −l) α .(25) It is also applicable for the other parton propagator of the anti-quark line. The short distance part of the parton propagator i/ n/(2n · l) and the vertex iγ α are absorbed into the parton amplitude. This results in 1 0 dxTr[T ij,αβ (x, x, x)w α α ′ w β β ′ Φ α ′ β ′ M 2 ,∂ (x, x, x)](26) where w α α ′ = g α α ′ − q α n α ′ , T ij,αβ (x, x, x) ≡ (iγ α ) i/ n 2n · l T I(0) ij −i/ n 2n ·l (−iγ β ) , Φ α ′ β ′ M 2 ,∂ (x, x, x) = ∞ 0 dλ 2π e iλx M 2 |q(λn)i∂ α ′ (λn)i∂ β ′ (λn)q(0)|0 .(27) There are corresponding contributions from the two gluon insertion diagrams depicted in Fig. 6, whose expression is written as d 4 l 1 (2π) 4 d 4 l 2 (2π) 4 d 4 l 3 (2π) 4 Tr[T I(0) ij,αβ (l 1 , l 2 , l 3 )w α α ′ w β β ′ Φ αβ M 2 ,A (l 1 , l 2 , l 3 )](28) where we have employed the light-cone gauge n · A = 0 for the gluon fields, and the parton amplitude and hadron amplitude are expressed as T I(0) ij,αβ (l 1 , l 2 , l 3 ) = (iγ α ) i/ n 2n · l 2 T I(0) −i/ n 2n ·l 2 (−iγ β ) Φ αβ M 2 ,A (l 1 , l 2 , l 3 ) = d 4 z d 4 y d 4 we il 1 ·y e i(l 2 −l 1 )·z e i(l 3 −l 2 )·w × M 2 |q(y)(−gA α (z))(−gA β (w))q(0)|0(29) Since T I(0) ij,αβ (l 1 , l 2 , l 3 ) can be replaced by T I(0) ij,αβ (x 1 , x 2 , x 3 ) straightforwardly, the momentum integrations over l 1 , l 2 , l 3 can be transformed into the integrations over x 1 , x 2 , x 3 . We then obtain dx 1 dx 2 dx 3 Tr[T I(0) ij,αβ (x 1 , x 2 , x 3 )Φ αβ M 2 ,A (x 1 , x 2 , x 3 )] .(30) The combination of Eq. (26) and Eq. (30) gives dx 1 dx 2 dx 3 Tr[T I(0) ij,αβ (x 1 , x 2 , x 3 )w α α ′ w β β ′ Φ α ′ β ′ M 2 ,D (x 1 , x 2 , x 3 )] (31) where Φ αβ M 2 ,D (x 1 , x 2 , x 3 ) = dλ 2π dη 2π dω 2π e iλx 1 e iη(x 2 −x 1 ) e iω(x 3 −x 2 ) × M 2 |q(λn)(iD α (ηn))(iD β (ωn))q(0)|0 (32) with iD α = i∂ α −gA α being the covariant derivative. Since / n is of O(Q −1 ), T I(0) ij,αβ is of O(Q −2 ) as the scale of T I(0) ij being of O(1). The relevant contributions are of higher than twist-4. The above example is to show that, using the collinear expansion, one can calculate the tree level higher twist corrections from the dynamical partons in a systematic way. Because we only intend to calculate the twist-3 corrections, we will not further explore the contributions of twist order higher than three. For −2(S − P )(S + P ) operators,Γ i = γ 5 . Up to twist-3, the expression appears as Tr[/ nγ 5 Φ M 2 (x)] = −if M 2 φ tw2 M 2 (x) ,(34)Tr[γ 5 Φ M 2 (x)] = −if M 2 µ χ φ tw3 M 2 ,P (x) ,(35) where µ M 2 χ = m 2 M 2 /(m q +mq) withm q andmq the current quark masses and m M 2 the meson mass, we recover the naive factorization result up to twist-3 order. ij,µ is counted as Q −1 . We conclude that the region composed of two collinear loop parton momenta is dominant. Let's, first, consider the light-cone gauge n · A = 0. The expansion series of T I(0) ij,µ (l 1 , l 2 ) with respect tol i , i = 1, 2, are written as T I(0) ij,µ (l 1 , l 2 ) = T I(0) ij,µ (l 1 ,l 2 ) + k=1,2 T I(0) ijk,µν (l 1 ,l k ,l 2 )(l k −l k ) ν + · · · ,(36) where T I(0) ijk,µν (l 1 ,l k ,l 2 ) are defined by assuming the low energy theories T I(0) ijk,µν (l 1 ,l k ,l 2 ) = ∂T I(0) ij,µ ∂l ν k l 1 =l 1 ,l 2 =l 2 .(37) The expansion series are then substituted back into the convolution integrals. Since the gauge condition n · A = 0 with a light-cone vector n µ satisfying n · q = 1, n 2 = 0, and n · l ⊥ = 0, the first term T I(0) ij,µ (l 1 ,l 2 ) leads to the result d 4 l 1 (2π) 4 d 4 l 2 (2π) 4 Tr[T I(0) ij,µ (l 1 ,l 2 )w µ α Φ α M 2 (l 1 , l 2 )](38) where we have introduced w µ α = g µ α − q µ n α . Similarly, we employ the following integral identities to simplify the momentum integrations 1 0 dx 1 1 0 dx 2 ∞ 0 dλ 2π ∞ 0 dη 2π e iλ(x 1 −n·l 1 ) e iη(x 2 −n·l 2 ) = 1(39) The expression appears as 1 0 dx 1 1 0 dx 2 Tr[T I(0) ij,µ (x 1 , x 2 )w µ α Φ α M 2 (x 1 , x 2 )](40) in which T I(0) ij,µ (x 1 , x 2 ) = T I(0) ij,µ (l 1 ,l 2 ) l 1 =x 1 ,l 2 =x 2 ,(41)Φ α M 2 (x 1 , x 2 ) = ∞ 0 dλ 2π ∞ 0 dη 2π e iλx 1 e iη(x 2 −x 1 ) × M 2 |q(λn)(−gA µ (ηn))q(0)|0 .(42) In the above equations, we have used the following transformation for any function G(y, z, 0) of coordinates y µ and z µ d 4 l 1 (2π) 4 d 4 l 2 (2π) 4 d 4 y d 4 ze il 1 ·(y−z−λn) e il 2 ·(z−ηn) G(y, z, 0) = G(λn, ηn, 0) .(43)l 2 1 → i/ n 2n · l 1 .(44) This is because the vertex iγ µ in T I(0) ij,µ σ αβ γ 5 ]w µ µ ′ Tr[σ αβ γ 5 Φ µ ′ M 2 (x 1 , x 2 )] .(45) The explicit expression for Eq. (45) is left to the next section. We now consider the expansion with covariant gauge ∂ · A = 0. Since the factorizations of the momentum integrals and color indices are independent of gauge condition, we can go through to consider the factorization of spin indices. The first term in the expansion of T I(0) ij,µ (x 1 , x 2 ), under covariant gauge, is related to the gauge invariant phase factor of the related two parton amplitudes. The gluon fields A µ in Φ µ M 2 can be expanded as A µ = n · Aq µ + q · An µ + d µ α A α . The contraction Tr[T I(0) ij,µ (x 1 , x 2 )q µ n · Φ M 2 (x 1 , x 2 )] leads to dx 1 dx 2 Tr[Γ i 2 n · k n · Φ M 2 (x 1 , x 2 )] ,(46) where k = l 2 − l 1 being the gluon momentum and n · Φ M 2 (x 1 , x 2 ) = ∞ 0 dλ 2π ∞ 0 dη 2π e iλx 1 e iη(x 2 −x 1 ) M 2 |q(λn)(−gn · A(ηn))q(0)|0 .(47) The terms with q · An µ vanish since the covariant gauge condition ∂ · A = 0. The terms with the contraction Tr[T I(0) ij,µ (x 1 , x 2 )d µ α Φ α M 2 (x 1 , x 2 )] are of higher twist than twist-3 and will be neglected. With the above considerations, the contraction Tr[T I(0) ij,µ (x 1 , x 2 )Φ µ M 2 (x 1 , x 2 ) ] leads to contributions of twist-2 or higher than twist-3. We next consider the contraction Tr[T I(0) ijk,µν (x 1 , x 2 )(l k −l k ) ν Φ µ M 2 (x 1 , x 2 )] , which can be rewritten as Tr[T I(0) ijk,µν (x 1 , x k , x 2 )w ν ν ′ Φ ν ′ µ M 2 (x 1 , x k , x 2 )] (48) with Φ ν ′ µ M 2 (x 1 , x k , x 2 ) ≡ dλ 2π dη 2π e iλx 1 e iη(x 2 −x 1 ) M 2 |q 2 (λn)igG ν ′ µ (ηn)q 3 (0)|0 .(49) Note that only transversal part d ν ⊥,β (l k −l k ) β of the (l k −l k ) ν can contribute at twist-3. For (V − A)(V ± A) operators, the contributions are of twist-4. For −2(S − P )(S + P ) operators, the result appears as 1 8 1 0 dx 1 x 1 0 dx 2 dx k Tr[T I(0) ijk,µν (x 1 , x k , x 2 )d ν ⊥,ν ′ σ αβ γ 5 ][σ αβ γ 5 Φ ν ′ µ M 2 (x 1 , x k , x 2 )] ×(δ(x k − x 1 ) + δ(x k − x 2 )) .(50) The reader may have noticed that the terms in the expansion series of T ij,µ (l 1 , l 2 ). Since the parton amplitude and hadron amplitude under the collinear expansion are required to be gauge invariant, respectively, this feature of the collinear expansion method can be used as a guiding principle for calculations. III. TWIST-3 CORRECTIONS In this section, we make a more detail descriptions for the twist-3 contributions from the three parton Fock state qqg of the M 2 meson. The amplitude for the three parton qqg of M 2 interacting with the operator O 6 at the tree level forB → M 1 M 2 decays is written as M 1 |q 1 (0)(1 − γ 5 )b(0)|B × d 4 y d 4 z d 4 l (2π) 4 d 4 k (2π) 4 e il·z e ik·(y−z) M 2 |q 2 (z)[(−ig/ A(y)) i(/ l + / k) (l + k) 2 + iǫ (1 + γ 5 ) + −i(/ q − / l + / k) (q − l + k) 2 + iǫ (+ig/ A(y))(1 + γ 5 )]q 3 (0)|0 .(51) The l and k denote the momenta carried by the q 2 quark and g gluon fields in Fig. 3(a) and (b). We first employ the light-cone gauge n · A(y) = 0. The gluonic fields A α (y) represents A α,a (y)T a with the color matrix T a in the fundamental representation T a T b = δ ab /2. To relate to the previous introduced collinear expansion, we recast the convolution integration part of Eq. (51) into the form d 4 l (2π) 4 d 4 k (2π) 4 Tr[T I(0) µ (k, l)w µ µ ′ Φ µ ′ (k, l)](52) where the parton amplitude T I(0) µ (k, l) is defined as T I(0) µ (k, l) = [(iγ µ ) i(/ l + / k) (l + k) 2 + iǫ + −i(/ q − / l + / k) (q − l + k) 2 + iǫ (−iγ µ )](1 + γ 5 )(53) and the meson amplitude Φ µ ′ (k, l) Φ µ ′ (k, l) = d 4 y d 4 ze il·z e ik·(y−z) M 2 |q 2 (z)(−gA µ ′ (y))q 3 (0)|0 .(54) The tensor w µ µ′ = g µ µ′ − q µ n µ′ has been introduced. Note that, for convenience, we have made a change of variables for the loop parton momenta l = l µ 1 and k µ = (l 2 − l 1 ) µ . We assume that the emitted M 2 meson is highly energetic. As shown in last section, the dominante configuration is composed of collinear l 1 and l 2 . This allows us to expand the parton amplitude T I(0) ij,α (k, l) with respect tol = xq andk = (x ′ − x)q T I(0) ij,µ (k, l) = T I(0) µ (k,l) + ∂T I(0) ij,µ (k, l) ∂l ν l=l,k=k (l −l) ν + ∂T I(0) ij,µ (k, l) ∂k ν l=l,k=k (k −k) ν + · · · .(55) Substituting the first term back into the convolution integrations gives dx dx ′ Tr[T I(0) ij,µ (x, x ′ )w µ µ ′ Φ µ ′ (x, x ′ )] (56) where Φ µ ′ (x ′ , x) = d 4 l (2π) 4 δ(x − l · n) d 4 k (2π) 4 δ(x ′ − x − k · n) d 4 y d 4 z ×e il·z e ik·y M 2 |q 2 (z)(−gA µ ′ (y))q 3 (0)|0 . (57) In the above collinear limit step T I(0) ij,µ (l, k) → T I(0) ij,µ (x, x ′ ) , there arises an infrared divergence as x ′ → 0, which is from the denominators of virtual quark propagators ix ′ / q (x ′ q) 2 + iǫ .(58) We regularize this divergence by the following method. Since the full quark propagator with momentum l ′ = l + k can be decomposed into its long distance part and short distance part as i/ l (l + k) 2 + iǫ = i/ l ′ L (l ′ ) 2 + iǫ + i/ n 2n · l ′ .(59) The long distance part gives vanishing result upto twist-3. The short distance part is absorbed by the parton amplitude. The divergence is then regularized by replacing the quark propagators with its corresponding special propagators [7,40] ix ′ / q (x ′ q) 2 + iǫ → i/ n 2x + iǫ x ′ − x x ′ − x + iǫ .(60) The introduction of a special propagator for an on-shell fermion propagator is due to the fact that the fermion propagators in Fig. 3(a) and 3(b) become on-shell and divergent after the collinear expansion. The divergent part of these propagators leads to long distance contributions that should be included into the twist-2 distribution amplitude for the M 2 meson. However, there are also finite contact part of these propagators, which leads to contributions of one twist higher. The more detailed explanation about the meaning of the special propagator refers to [7,40]. Under light-cone gauge n · A(y) = 0, it is convenient to transform the gluon fields A µ (y) into its field strength G νµ (y) by using following replacement A µ ′ (y) → in ν G νµ ′ (y) (x ′ − x) ,(61) and Φ µ ′ M 2 (x, x ′ ) → in ν x ′ − x Φ νµ ′ M 2 (x, x ′ ) .(62) The factor in ν /(x ′ − x) is then absorbed by T I(0) µ (x, x ′ ) into the form T I(0) µν (x, x ′ ) ≡ T I(0) µ (x, x ′ ) in ν x ′ − x .(63) The factorization of the spin indices gives 1 8 dx dx ′ Tr[T I(0) µν (x, x ′ )σ αβ γ 5 ]w µ µ ′ Tr[σ αβ γ 5 Φ νµ ′ (x, x ′ )] + · · · ,(64) in which other spin decompositions give higher twist contributions. The numerators in the contraction Tr[T I(0) µν σ αβ γ 5 ] can give terms proportional to n ν n µ (q α n β − n α q β ), n ν d ⊥,µµ ′′ (q α n β − n α q β ), and n ν d ⊥,µµ ′′ ǫ ⊥,αβ . The transversal tensors d ⊥,αβ and ǫ ⊥,αβ are defined as d ⊥,αβ = q α n β + q β n α − g αβ and ǫ ⊥,αβ = ǫ αβηλ q η n λ . The trace of d ⊥,αβ is defined to be negative d α ⊥,α = −2. Since ν and µ indices in Φ νµ M 2 are antisymmetric under µ ↔ ν, the terms proportional to n ν n µ (q α n β − n α q β ) then vanish. For those terms proportional to n ν d ⊥,µµ ′′ (q α n β − n α q β ), as they are contracted with Tr[σ αβ γ 5 Φ νµ ′ (x, x ′ )], the q α factor in n ν d ⊥,µµ ′′ (q α n β − n α q β ) results in twist-4 contributions by using the property of the long distance propagator of the quark fields. The terms proportional to n ν d ⊥,µµ ′′ ǫ ⊥,αβ lead to twist-3 contributions. The final result appears as dx dx ′ G β µ (x, x ′ )n µ n β (x ′ − x)x ,(65) where the function G β µ (x, x ′ ) is defined as G β µ (x, x ′ ) = d 4 l (2π) 4 δ(x − l · n) d 4 k (2π) 4 δ(x ′ − x − k · n) d 4 y d 4 z ×e il·z e ik·y M 2 |q 2 (z)σ µα γ 5 w α α ′ gG βα ′ (y)q 3 (0)|0 .(66) Note that we have used the G-parity symmetry x ↔x ′ to simplify the above result. This assumption is valid for π mesons, but may not be appropriate for the K or η mesons. Therefore, it is noted that, in the above result, there exist symmetry breaking effects for K and η mesons. However, we will ignore such a corrections from the symmetry breaking in the following calculations. By referring to the definition [41] M 2 |q 2 (z)σ µν γ 5 gG αβ (y)q 3 (0)|0 = −i f M 2 m 2 M 2 m q 2 + mq 3 (q α q µ d ⊥,νβ − q α q ν d ⊥,µβ − q β q µ d ⊥,να + q β q ν d ⊥,αµ )T (z, y) + · · · ,(67) where T (z, y) = 1 0 dx x 0 dx ′ e −ixq·z e −i(x−x ′ )q·y T (x, x ′ ) ,(68) we can arrive at the result dx dx ′ G β µ (x, x ′ )n µ n β (x ′ − x)x = − 2if M 2 m 2 M 2 m q 2 + mq 3 dx dx ′ T (x, x ′ ) (x ′ − x)x .(69) By using the normalization for M 1 |q 1 (0)(1 − γ 5 )b(0)|B , it is easy to derive the tree level three parton contributions for operator O 6 as O 6 1−gluon = 2A G3 M 2 m 2 M 2 m b (m q 2 + mq 3 ) O 1 f (70) with A G3 M 2 = 2 1 0 dx x 0 dx ′ T M 2 (x ′ , x) (x ′ − x)x .(71) We now explain the expansion with the covariant gauge ∂ · A = 0. We first decompose A µ (y) into its longitudinal and transversal components as A µ (y) = n · A(y)q µ + d µ ⊥,µ ′ A µ ′ (y). The transversal part d µ ⊥,µ ′ A µ ′ (y) results in contributions of higher than twist-3 . The longitudinal part n · A(y)q α gives twist-3 contributions. Similar to the light-cone gauge, we need to transform the gluon fields into its field strength. Here, it needs one transversal momentum k ⊥ factor from expansion of the parton amplitude T I(0) µ (l, k) in Eq. (55). The contraction of T I(0) µ (x, x ′ ) with q µ n · Φ M 2 (x, x ′ ) leads to two parton gauge phase factor Tr[T I(0) µ (x, x ′ )q µ n · Φ M 2 (x, x ′ )] = Tr[ T I(0) (x ′ ) − T I(0) (x) x ′ − x n · Φ M 2 (x, x ′ )](72) It is convenient to write (k −k) ρ = d ρ ⊥ρ ′ (k −k) ρ ′ + q · kn ρ . Only transversal part k ρ ⊥ = d ρ ⊥ρ ′ (k −k) ρ ′ contributes at twist-3. This is because the term ∂T I(0) µ /∂k ν can have terms proportional to g µν and σ µν . The terms related to q · kn ν leads to twist-4 contributions. For the transversal part k ⊥ , only σ µν terms can contribute. Let the k ρ ⊥ factor absorbed into Φ µ (l, k) and use the replacement k ν ⊥ A µ (y) → −iG νµ (y), we can derive the result O 6 t=3 1−gluon = −2 dx dx ′ Tr[ ∂T I(0) µ (x, x ′ ) ∂k ν G νµ (x, x ′ )] × M 1 |q 1 (0)(1 − γ 5 )b(0)|B ,(73) where ∂T I(0) µ (x ′ , x) ∂k ν = −iσ µν (x ′ − x)xq 2 (1 + γ 5 )(74) and G νµ (x, x ′ ) = d 4 l (2π) 4 δ(x − l · n) d 4 k (2π) 4 δ(x ′ − x − k · n) d 4 y d 4 z ×e il·z e ik·y M 2 |q 2 (z)igG νµ (y)q 3 (0)|0 .(75) The contraction of σ µν with G νµ (x, x ′ ) gives Tr[iσ µν G νµ (x, x ′ )] = −2if M 2 m 2 M 2 q 2 (m q 2 + mq 3 ) T (x, x ′ ) .(76) Note that the q 2 factor in the denominator of ∂T I(0) µ (x ′ , x)/M 1 |q 1 2p ν + γ ν / k 2p · k Γ i − Γ i 2P bν − / kγ ν 2P b · k b|B ,(77) where k is the momentum of the gluon from M 2 and the equation of motions for b and q 1 quarks have been used. After taking the collinear limit, k → x ′ q, we write the expression as Aq ν + B µν q µ ,(78) where corrections. As mentioned previously, we plan to discuss these contributions in other places [44]. A ≡ 1 x ′ Q 2 M 1 |q 1 Γ i |B ,(79)B µν ≡ 1 2x ′ Q 2 M 1 |q 1 (γ ν γ µ Γ i + Γ i γ µ γ ν )b|B .(80) The total twist-3 contribution from operator O 6 is then equal to O 6 t=3 = 2(1 + A G3 M 2 )m 2 M 2 m b (m q 2 + mq 3 ) O 1 f .(81)A(B − → π −K 0 ) = λ p (a p 4 − 1 2 a p 10 ) + r K χ (a p 6 − 1 2 a p 8 ) A πK +(λ u b 2 + (λ u + λ c )(b 3 + b EW 3 ))B πK , − √ 2A(B − → π 0 K − ) = [λ u a 1 + λ p (a p 4 + a p 10 ) + λ p r K χ (a p 6 + a p 8 )]A πK +[λ u a 2 + λ p 3 2 (−a 7 + a 9 )]A Kπ +(λ u b 2 + (λ u λ c )(b 3 + b EW 3 ))B πK , −A(B 0 → π + K − ) = [λ u a 1 + λ p (a p 4 + a p 10 ) + λ p r K χ (a p 6 + a p 8 )]A πK +((λ u + λ c )(b 3 − 1 2 b EW 3 ))B πK √ 2A(B 0 → πK 0 ) = A(B − → π −K 0 ) + √ 2A(B − → π 0 K − ) − A(B 0 → π + K − ) (82) where λ p = V pb V * ps , a i ≡ a i (πK), and λ p a p i = λ u a u i + λ c a c i . The CP conjugation of decay amplitudes are obtained by replacing λ p → λ * p for the above amplitudes. The factorized matrix elements are defined as A πK = i G F √ 2 (m 2 B − m 2 π )F B→π 0 (m 2 K )f K , A Kπ = i G F √ 2 (m 2 B − m 2 K )F B→K 0 (m 2 π )f π .(83) The form factors are defined P (p)|qγ µ b|B = F B→P + (q 2 )(P µ B + p µ ) + [F B→P 0 (q 2 ) − F B→P + (q 2 )] m 2 B − m 2 P q 2 q µ .(84) The form factors coincide as q 2 = 0, F B→P + (0) = F B→P 0 (0). The expressions for the parameters a i are referred to [2,3]. For numerical calculations, we will use the following input For λ u and λ c , we take the following convention for their parametrization λ u λ c = tan 2 θ c R b e −iγ(86) where tan 2 θ c = λ 2 1 − λ 2 , R b = 1 − λ 2 /2 λ | V ub V cb | , λ = |V us | .(87) The value of λ is taken as 0.22. By using previous input parameters, we list the values of a i , i = 1, · · · , 10, and b j , j = 1, · · · , 3, calculated at the scale m b = 4.2GeV as below a 1 = 0.995 + 0.018i , a 2 = 0.209 − 0.104i , a 3 = −0.003 + 0.003i , a u 4 = −0.031 − 0.013i a c 4 = −0.030 + 0.027i , a 5 = 0.007 − 0.004i , r K χ a u 6 = −0.050 − 0.015i , r K χ a c 6 = −0.047 − 0.005i , a 7 /α = 0.007 + 0.006i , r K χ a u 8 /α = 0.087 − 0.043i , r K χ a c 8 /α = 0.094 − 0.021i , a 9 /α = −1.135 − 0.024i , a u 10 /α = −0.175 + 0.093i , a c 10 /α = −0.175 + 0.093i , r A b 1 = 0.021 , r A b 2 = −0.008 , r A b 3 = −0.006 , r A b EW 3 /α = −0.018 ,(88) in which r A = B πK A πK = f B f π m 2 B F B→π 0 (0) ,(89) and r K χ = 2m 2 K m b (m q + m s ) .(90) The tree level three parton contributions modify the parameters r K χ as r K χ (1 + A G 3 M 2 ) with parameter A G 3 M 2 defined in Eq. (71) . The value of A G 3 M 2 depends on the model of the three parton distribution amplitude T M 2 (x, x ′ ). Here we employ the model derived from the lightcone sum rule [41] T (x, x ′ ) = 360ηxx ′ (x − x ′ ) 2 (1 + ω 2 (7(x − x ′ ) − 3)),(91) where the parameters are assumed to be η = 0.015 and ω = −3.0 for M 2 = π, K, or η. This give us A G 3 M 2 = 0.585. The branching ratio for aB → πK decay is given by this expressions Br(B → πK) = τ B 16πm B |A(B → πK)| 2 .(92) We can use the above formula to predict CP averaged branching ratios for B → πK decays. The predictions with three parton corrections in units of 10 −6 are given as Br(B − → π −K 0 ) = 19.0 , Br(B − → π 0 K − ) = 10.0 , Br(B 0 → π + K − ) = 16.1 , Br(B 0 → π 0K 0 ) = 7.7 .(93) For comparison, we also list the predictions with only two parton contributions in units of 10 −6 in the following, Br(B − → π −K 0 ) = 11.2 , Br(B − → π 0 K − ) = 6.1 , Br(B 0 → π + K − ) = 9.4 , Br(B 0 → π 0K 0 ) = 4.4 .(94) For reference, we enlist the experimental data in units of 10 −6 summarized by the HFAG group [42] Br(B − → π −K 0 ) = 23.1 ± 1.0 , Br(B − → π 0 K − ) = 12.8 ± 0.6 Br(B 0 → π + K − ) = 19.7 ± 0.6 Br(B 0 → π 0K 0 ) = 10.0 ± 0.6 . By comparing the predictions with or without the three parton corrections, one may notice that the the predicted branching ratios are significantly enhanced by about 1.65 ∼ 1.75 times in their magnitudes. The two parton predictions for theB → πK decays made here are consistent with the findings of previous studies using QCD factorization approach [3,43]. The two parton predictions made in [43] are much lower than the experimental data under the QCD factorization approach. Because the calculations of the three parton corrections were inaccessible in their studies, this led them to conclude that the QCD factorization is impossible to explain the penguin dominantB → πK decays. The two parton predictions made in [3] for B → πK decays are still lower than the data. Only extending the predictions by using extreme limits of input parameters can make the predictions to be consistent with the measurements. This seems not a reasonable solution from the theoretical point of view. On the other hand, as shown in the above, our approach has shown that the predictions with three parton contributions are more close to the data than the two parton predictions. Although the predictions with three parton contributions are still lower than the experimental data, the O(α s ) corrections can improve the predictions. V. DISCUSSIONS AND CONCLUSIONS The significance of the three parton contributions for penguin dominantB → πK decays can also be seen from a phenomenological point of view. By appropriate arrangement, the parameters a i can be calculated for B → ππ decays. For the pure penguin B − → π −K 0 , its dominant contributions arise from the |a c 4 (πK) + r K χ a c 6 (πK)| term. The uncertainty due to the form factors can be eliminated by considering the ratio between the decay rates of B − → π −K 0 and B − → π − π 0 as the following a c 4 (πK) + r K χ a c 6 (πK) a 1 (ππ) + a 2 (ππ) = |V ub | |V cb | f π f K Γ(B − → π −K 0 ) 2Γ(B − → π − π 0 ) 1/2 = 0.105 ± 0.001 ,(96) where the error comes from the branching ratios. In the above, we have used the branching ratio Br(B − → π − π 0 ) = 5.7 ± 0.4. According to QCD factorization calculations, |a 1 (ππ) + a 2 (ππ)| = 1.17 and |a c 4 (πK) + r K χ a c 6 (πK)| = 0.08 with only two parton contributions. This gives the prediction of the ratio to be 0.066, which is lower than the experimental value. By adding the tree level three parton contributions, the factor r K χ then becomes r K χ (1 + A G3 K ) and makes |a c 4 (πK) + r K χ (1 + A G3 K )a c 6 (πK)| = 0.104. The predicted ratio is changed to be 0.089, which is closer to the two parton prediction. The above fact may also indicate that the three parton corrections could be important for understanding the penguin dominant B → πK decays under the QCD factorization approach. In order to make sure that the three parton contributions are indeed significant and also compatible with the QCD factorization, an important task is to finish O(α s ) calculations for the three parton contributions. The related work in this direction has been proceeded and will be reported in our another preparing paper [44]. There are similar three parton contributions having been calculated under the lightcone sum rule [45,46]. In [45], the twist-3 three parton contributions associated with soft gluons are calculated in the framework of light-cone sum rule. The contributions are shown negligible inB → ππ decays. In [46], the contributions from the diagrams similar to Fig. 3(c) and 3(d) were calculated for B → πω decays. They are found to be twist-4 and vanishing. In addition, significant effects were found due to the three parton Fock state of the π in the B → πω decays. Since they are dominated by soft gluons, it is better determined by QCD sum rule. As shown in [46], in the Euclidean region of (p + q) 2 , the relevant contributions are from the twist-3 and twist-4 three parton LCDAs of the π. As mentioned before, we identify these power corrections as non-partonic ones. From the theoretical point of view, we suggest that the partonic and non-partonic power corrections should be distinquished under the QCD factorization, although they may be equally important in phenomenology. For comparison, we employ the replacing rules for the a i coefficients [46] to account for the three parton effects from the M 1 meson. The rule is a 2i → a 2i + [1 + (−1) δ 3i +δ 4i ]C 2i−1 f 3 /2 , a 2i−1 → a 2i−1 + (−1) δ 3i +δ 4i C 2i f 3 , where i = 1, · · · , 5, and C i are the Wilson coefficients calculated at the scale µ h = 1. 45 GeV, and f 3 = 0.12, which is assumed to be universal. With these three parton corrections, the predicted CP averaged branching ratios forB → πK in units of 10 −6 are Br(B − → π −K 0 ) = 9.5 , Br(B − → π 0 K − ) = 5.9 , Br(B 0 → π + K − ) = 8.4 , Br(B 0 → π 0K 0 ) = 3.4 ,(98) which becomes smaller than those predictions in Eq. (94). where T I(II) denote the parton amplitudes and Φ B , Φ M 1 , Φ M 2 represent the light-cone distribution amplitudes (LCDAs) for the initial stateB meson and the final state M 1 and M 2 mesons, respectively. The parton amplitudes T I(II) contain short distance interactions involved in the decay processes. The LCDAs Φ B , Φ M 1 , and Φ M 2 are introduced to account for the long distance interactions. The F B→M 1 j (m 2 2 ) with j = +, 0 are the B → M 1 transition form factors. The meson state vector |M i , i = 1, 2, for the meson M i is composed of Fock states with different number of partons|M i = |qq M i + |qqg M i + · · · .(2)So far, most applications of the factorization formula Eq. (1) are limited to leading Fock state |qq M i of the light mesons. However, the three parton Fock state |qqg M i of the M i meson can also contribute. The corrections related to the higher Fock state are usually classified as subleading twist contributions, since their contributions are suppressed by factors of O(1/m n b ) with n ≥ 1 in comparison with the leading ones. Here, m b denotes the b quark mass. Within QCD factorization, one can employ the Feynman-diagram approach or the effective-theory approach for studies of subleading twist contributions. There exist established Feynman-diagram ap- to calculate the three parton corrections from the Fock state |qqg of the meson M 2 in the decayB → M 1 M 2 . 3 . 3The contributions of the same twist order from the expansion series of each Feynman diagram with the same α s order are added up together. 4. The factorization properties of the final expression with a specific twist and a specific α s order are analyzed. , 2 , 2and q 2 = 0. The loop partons except of the bottom quark are assumed massless for simplicity. The contributions from non-vanishing light quark masses are taken as corrections. Because the light quark mass contributions are relatively negligible as compared to the bottom quark mass, we also neglect the light quark mass effects in the following calculations. are used to denote the relevant parton amplitude of zeroth and first order in α s , respectively. The parton amplitude T I(0) ij is just the tree vertex Γ i δ ij in the diagram as depicted in Fig. 2. There are vertex and penguin diagrams for T I(1) ij amplitudes as depicted in Fig. 4. The parton amplitude T I(0) ij,µ describes the tree diagrams as depicted in Fig. 3, in which two quark partons and one gluon parton from the meson M 2 are interacting with a local four fermion operator the Dirac gamma matrix for the operator. The parton amplitude T II i starts from O(α s ) diagrams as depicted in Fig. 5 and is denoted as T II(1) . are necessary. First, the momentum fraction x for the parton of the meson M 2 is introduced. Second, the quark fieldq(λn) is ordered in light-cone direction n. This implies that the hadron amplitude Φ M 2 (x) is defined on the light-cone n 2 = 0 where n µ is a null light-cone vector. By using the above integral transformations, the parton amplitude and the hadron amplitude are only related by the momentum fraction x. γ 5 ] 5qγ 5 ]Tr[/ nγ 5 Φ M 2 (x)] Tr[γ 5 Φ M 2 (x)] . 0 0We show the light-cone gauge n · A = 0 and the covariant gauge ∂ · A = µ . Because the analysis is tedious, we outline the procedure, here, and leave the details for next section. The first step is to take a power counting for the parton amplitude TI(0) ij,µ . There are three interesting regions. The first region is composed of either two soft loop parton momenta, or one soft loop parton momentum and one collinear loop parton momentum. The T I(0) ij,µ in the first region is counted as λ −1 . The second region is composed of two collinear loop parton momenta. The T I(0) ij,µ in the second region is counted as Qλ −2 . The third region is composed of either one collinear loop parton momentum and one hard loop parton momentum, or two hard loop parton momenta. In the third region, For tree diagrams, only color singlet operators can contribute at twist-3. The remaining task is to finish the factorization of spin indices. For (V − A)(V ± A) operators , the related contributions are of twist-4, which are beyond our accuracy. For −2(S −P )(S +P ) operators, the related contributions are of twist-3. To obtain the collinear limitl i → x i , µ (x 1 , x 2 ) , we have used the following substitution for the parton propagators i/ l 1 µ (x 1 , x 2 ) is transversal and only the off-shell part of the parton propagators can contribute. See further explanations in the next section. The terms associated with T I(0) ijk,µν are of twist-4 and higher. They are neglected accordingly. The factorization of spin indices results µ (l 1 , l 2 ) in Eq. (36) are of different twist order under the covariant or the light-cone gauge. For example, the twist-3 contributions are from the the first term in the expansion series of T I(0) ij,µ (l 1 , l 2 ) under the light-cone gauge. On the other hand, under the covariant gauge, the twist-3 contributions are from the the second term in the expansion series of T I(0) +, 0 . 0∂k ν is cancelled by the q 2 factor in the numerator of Tr[iσ νµ G νµ (x, x ′ )]. It is easy to see that Eq. (73) is equal to the result derived from the light-cone gauge. This explicitly shows the gauge invariance of the three parton contributions. There are related diagrams, such as those in Fig. 3(c) and 3(d). Because the spectator quark of theB meson can carry only a soft momentum, this makes the relevant contributions associated with Fig. 3(c) and 3(d) dominated by soft gluons as the form factors F B→M 1 In addition, the relevant contributions are of O(m −2 b ) with respect to the leading twist amplitude. It can be understood as following. The sum of the lower parts of the diagarms in Fig. 3(c) and 3(d) is proportional to For light-cone gauge, only A term contributes. As for the covariant gauge, only B µν term contributes. The only contributions come from (V − A)(V ± A) operators. This implies that the upper parts of the diagarms in Fig. 3(c) and 3(d) are proportional to the twist-4 LCDA of M 2 . The combination of the upper and the lower parts gives a O(m −2 b ) contributions with respect to the leading twist amplitude. There are possibilities that the additional gluon of the M 2 meson can interact with the spectator quark of theB meson. Since the spectator quark carries a soft momentum, the momentum conservation at the interaction vertex prevents the momentum of the gluon from being collinear to the M 2 meson's momentum. Therefore, there require additional radiative gluons interacting between the other parton lines and the spectator quark line to make the momentum of the gluon to be collinear to the M 2 meson's momentum. This results in contributions of order O(α s ). We identify the relevant contributions as O(α s ) three parton = 0.225GeV , m b (m b ) = 4.2GeV m c (m b ) = 1.3GeV , m s (2GeV) = 0.090GeV , |V cb | = 0.41 , |V ub /V cb | = 0.09 , γ = 70 • , τ (B − ) = 1.67(ps) , τ (B d ) = 1.54(ps) , f π = 131MeV , f K = 160MeV , f FIG. 1 :FIG. 3 : 13The parton topologies correspond to the parton amplitudes of four, five and six parton interactions, respectively. FIG. 2: The Feynman diagram describes the tree level four parton amplitude, T I(0) ij . The square symbol represents the vertex of weak interactions. The Feynman diagrams describe the tree level five parton amplitude, T I(0) ij,µ . The square symbol represents the vertex of weak interactions. FIG. 4 : 4The Feynman diagrams describe the O(α s ) four parton amplitude, T I(1) ij . The square symbol represents the vertex of weak interactions. FIG. 5 : 5The Feynman diagrams for the O(α s ) six parton amplitude, T II(1) . The square symbol represents the vertex of weak interactions. FIG. 6: The Feynman diagram for the six parton amplitude with |qqgg Fock state. The square symbol represents the vertex of weak interactions. which are defined as the momenta carried by the final state M 2 and M 1 mesons, respectively. The M 1 meson is defined to receive the spectator quark of the bottom meson. The M 2 meson is defined as the emitted mesonproduced from the hard scattering center. The hadron amplitudes Φ M 2 and Φ µ M 2 (l 1 , l 2 ) are defined as ner. As the loop corrections are considered, the twist expansion then interplays with the expansion in α s . To be specific, we choose the following expansion strategy.1. All possible Feynman diagrams ordered in α s are first drawn.2. 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{'abstract': 'The 1/m b corrections from the three parton qqg Fock state of the final state light meson in B → P P decays are evaluated by means of a collinear expansion method. The impacts of these corrections on the CP averaged branching ratios of the B → πK decays are analyzed.', 'arxivid': '0802.1855', 'author': ['Tsung-Wen Yeh \nDepartment of Science Application And Dissemination\nNational Taichung University\n403TaichungTaiwan\n'], 'authoraffiliation': ['Department of Science Application And Dissemination\nNational Taichung University\n403TaichungTaiwan'], 'corpusid': 118440305, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 22244, 'n_tokens_neox': 18769, 'n_words': 11046, 'pdfsha': 'a15f66b860fbdfbec679989063b8b60c1620fafa', 'pdfurls': ['https://arxiv.org/pdf/0802.1855v1.pdf'], 'title': ['Three Parton Corrections in B → P P decays', 'Three Parton Corrections in B → P P decays'], 'venue': []}

arxiv

Slidephononics: Tailoring Thermal Transport Properties by van der Waals Sliding Linfeng Yu Institute of Materials Science Technical University of Darmstadt 64287DarmstadtGermany College of Mechanical and Vehicle Engineering State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University 410082ChangshaP. R. China Chen Shen chenshen@tmm.tu-darmstadt.de Institute of Materials Science Technical University of Darmstadt 64287DarmstadtGermany Guangzhao Qin College of Mechanical and Vehicle Engineering State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University 410082ChangshaP. R. China Hongbin Zhang Institute of Materials Science Technical University of Darmstadt 64287DarmstadtGermany Slidephononics: Tailoring Thermal Transport Properties by van der Waals Sliding 1 * Author to whom all correspondence should be addressed. E- (Chen Shen), gzqin@hnu.edu.cn (Guangzhao Qin) 2Slidephononicsboron nitridethermal conductivityvan der Waals sliding By interlayer sliding in van der Waals (vdW) materials, the switching electric polarization of ultrathin ferroelectric materials leads to the widely studied slidetronics. In this work, we report that such sliding can further tailor anharmonic effects and hence thermal transport properties due to the changed intrinsic coupling between atomic layers. And we propose an unprecedented concept dubbed as slidephononics, where the phonons and associated physical properties can be controlled by varying the intrinsic stacking configurations of slidetronic vdW materials. Based on the state-of-the-art firstprinciples calculations, it is demonstrated that the thermal conductivity of boron nitride (BN) bilayers can be significantly modulated (by up to four times) along the sliding pathways. Detailed analysis reveals that the variation of thermal conductivities can be attributed to the tunable (de-)coupling of the out-of-plane acoustic phonon branches with the other phonon modes, which is induced by the interlayer charge transfer. Such strongly modulated thermal conductivity via interlayer sliding in vdW materials paves the way to engineer thermal management materials in emerging vdW electronic devices, which would shed light on future studies of slidephononics. Introduction The outstanding physical properties of two-dimensional (2D) materials have attracted intensive attention in the last two decades, as found in quite a few 2D materials such as graphene-like materials 1 , carbon monolayer materials 2 , transition metal chalcogenides (or halides) 3,4 , and layered MY2X4 materials 5,6 . Such materials, particularly their vdW heterostructures, can be applied to engineering high-performance electronic devices. For such ultrathin devices working in the quantum limit, it is essential to embrace efficient thermal management to maintain their optimal performance and lifetime. The approaches to regulate the thermal conductivity of the thermal management materials can be divided into two categories: firstly crystal structure modification or functionalization, including techniques such as doping 7,8 , straining [9][10][11] , hydrogenation [12][13][14] , alloying 15,16 , and defects 17,18 , and secondly electric field-induced thermal management techniques 19,20 . One significant distinction between these two classes of approaches is that the former class relies mostly on irreversible modification of the intrinsic properties of materials, whereas the second class enables a non-destructive and reversible regulation of heat transport. As the electric fields may intervene with the functioning of the vdW devices, it is appealing to develop a novel approach to control the thermal conductivity of 2D semiconductor materials, which can further promote the application of such materials in micro-nano electronic devices. Recently, slidetronics has made its debut in the 2D materials community 21,22 , where the lateral shifts between vdW layers give rise to sign changes in their out-of-plane electric polarization [23][24][25][26][27][28][29][30][31] . That is, the physical properties can also be tuned by interlayer sliding or different stacking configurations. This is another degree of freedom beyond the twisting in vdW bilayers or multilayers, which lead to intriguing twistronics [32][33][34][35][36] . It was first theoretically demonstrated that the polarization in boron nitride (BN) bilayers can be switched by interlayer sliding 30 . Moreover, the electronic properties of relevant vdW materials can also be tuned for various stacking configurations 21,27,30 . For example, in interlayerslip ferroelectrics, interlayers lead to unequal upper and lower layers of a 2D material bilayer, leading to a net vertical charge transfer between layers, prompting a reversal of the vertical ferroelectric polarization. The experimentally demonstrated widespread existence of sliding ferroelectricity in van der Waals layered materials 27,30,[37][38][39][40] , where the sliding barrier between layers is several orders of magnitude lower than in conventional ferroelectricity, is anticipated to reduce the energy needed for ferroelectric switching significantly. Additionally, for 2D materials, such as BN and GaX monolayers 41,42 , the out-of-plane phonon mode dominates the thermal conductivity and exhibits a potentially large sensitivity to vertical potential fields. So, it's reasonable to wonder whether interlayer sliding may be applied to tailor the phononic thermal transport properties. In this work, by solving the Boltzmann phonon transport equation based on first-principles calculations, we conduct a comprehensive investigation of the phonon transport properties of BN bilayers in different stacking configurations driven by interlayer sliding. In BN bilayers, we observe that interlayer sliding significantly affects the lattice thermal conductivity without changing the intrinsic bonding between the monolayers. Fundamental insights into sliding-modulated thermal conductivity are obtained by analyzing the phonon anharmonicity, phonon renormalization, and charge transfer. The intrinsic potential field between the BN bilayers is renormalized during sliding as a consequence of the redistribution of the free interlayer charge transfer, which leads to phonon renormalization to modulate thermal conductivity. Dubbing such an effect as slidephononics, our study establishes a novel reversible approach to tailor the thermal transport properties of vdW materials, unlocking a fascinating subject for future research and applications. Results and Discussion Tunable and strong slidephononics As shown in Fig. 1 Sliding-driven phonon renormalization Heat transfer in solids is described by phonon interactions, and the phonon dispersion essentially describes the functional relationship between phonon energy and momentum. Traditionally, applying strain breaks the quadratic dispersion relation of the ZA acoustic phonon branch of 2D materials to tune the phonon properties due to strong mechanical stress 9 . Slightly differently, interlayer sliding depends on van der Waals interactions, which are substantially weaker than intrinsic bonding. We have observed that the intrinsic potential field of BN bilayers has redistributed, which may affect the anharmonic interaction between phonons, although the intrinsic bond itself has not changed significantly. Higher-order variables take on greater significance in anharmonic systems because the interaction between phonons is no longer linear. This nonlinearity develops as a result of departures from the ideal harmonic behavior brought on by modifications to the potential energy pattern. The redistribution of the potential well impacts the behavior of the phonons when adjusting the intrinsic energy potential field in a bilayer system. Because phonons are vulnerable to local potential energy, variations in the potential well can affect how they spread and behave. The anharmonic interaction between the phonon and the phonon changes due to the redistribution of the potential well, considering that the intrinsic energy potential field in the bilayer system is dramatically adjusted, as seen in Fig. 1(g) and (h). To explore the phonon-phonon interaction under different slip potential fields, we will divide the phonon-phonon scattering into three states based on energy conservation: supersaturated (w1 + w2 > w3), saturated (w1 + w2 = w3), and undersaturated states (w1 + w2 < w3). Phonon-phonon scattering may be separated into emission and absorption processes based on energy conservation. In the instance of three phonons, the absorption process results in the merger of two lower-energy phonons into one higher-energy phonon, whereas the emission process results in splitting a higher-energy phonon into two lower-energy phonons. As shown in Fig. 2(a), the supersaturation (and undersaturation) state does not satisfy the energy law in phonon-phonon scattering, where phonons in the two low-frequency regions usually scatter to form higher-frequency (and lower-frequency) phonons that are higher (and lower) than the system phonons energy levels, leading to suppressed phonon-phonon scattering and high thermal conductivity, while phonon-phonon scattering in a saturated state contributes a strong scattering rate. This also still holds true for the process of splitting a phonon into two phonons, i.e., the emission process. Phonons in supersaturated and undersaturated states are less prone to participate in scattering, which enhances the thermal conductivity of the overall system. Whereas in BN bilayers, the sliding potential field can realize phonon state switching in the aforementioned states via the (de-)coupling of phonon branches, as discussed below. To explore the phonon state switching induced by the slidephononics, specific phonon dispersions of BN bilayers as examples are extracted along the armchair (0%, 10%, and 33%) and zigzag (0%, 20%, and 50%) directions, as shown in Fig. 2 Competing phonon scattering and scattering channels We further monitor the real-time phonon properties of different slide configurations due to the phonon renormalization induced by the transition of the saturation regime. As shown in Fig. 3(a) and (d), the difference in thermal conductivity during the sliding process is mainly contributed by the lowfrequency phonons of 0-15 THz, corresponding to the ZA, ZA' branches. Interestingly, the change of phonon group velocity is not obvious, even the group velocity of 10% (20 %) sliding lattice along the armchair (zigzag) direction is slightly lower than that of 0% and 33% (50%) sliding lattice as shown in Fig. 3(b) and (e), but it has higher thermal conductivity. The interlayer sliding mainly affects the interlayer van der Waals force, and has little effect on the strong covalent bond in the layer, and does not change the chemical composition of the system. The phonon group velocity reflecting the harmonic nature is mainly determined by the strength of chemical bonds and atomic mass 43 . Thus, the group velocity exhibits a weak modulating effect on the thermal conductivity. However, the interlayer sliding breaks the out-of-plane symmetry, leading to an anharmonic change in the potential energy. Hence, the phonon relaxation time, representing an anharmonic feature, will change significantly due to phonon renormalization. As shown in Fig. 3(c) and (f), the relaxation time of the 10% (20%) sliding lattice along the armchair (zigzag) direction is significantly larger than that of the 0% and 33% (50%) sliding lattice. Large relaxation times indicate weak phonon-phonon scattering dominated by moderate ZA-ZA' bandgap [ Fig. 2(e)]. To further reveal the relationship between phonon branch coupling and phononphonon scattering, we decompose different phonon-phonon scattering channels based on the energy conservation process. In the low-frequency region in the initial state, the phonon-phonon scattering of BN bilayers is mainly contributed by ZA phonons, i.e., ZA+ZAZA and ZAZA+ZA, as shown in Fig. 4(a) and (d). Taking the high symmetry point M as an example, a bandgap of (0.059) 0.199 THz is found between the ZA and ZA' branches when sliding along the armchair (zigzag) direction. Such a narrow phonon band gap is insufficient to satisfy the fusion of two ZA phonons into a relatively high-frequency ZA' phonon due to the energy conservation of "oversaturation", as shown in Fig. 2(e). When sliding into the intermediate state, the narrow phonon channel between ZA and ZA' at point M is slightly opened to 0.359 (armchair) and 0.697 (zigzag) THz. The slightly moderately wide bandgap harmonizes the phonon-phonon scattering of ZA and ZA', suppressing the strong triple ZA phonon-scattering channel as shown in Fig. 4(b) and (e). At this time, the phonon channels participated by ZA and ZA' are undersaturated, and they strongly do not satisfy energy conservation and contribute a large thermal conductivity, as shown in Fig. 2(e), i.e., the saturated state is transformed into 20% and (f) 50% along zigzag direction. Sliding-modulated interlayer charge interaction Phonon transport is essentially achieved through charge interactions, thus fundamental insights into the thermal transport of interlayer slip can be obtained through interlayer charge interaction transitions. Intrinsic bonding is almost unaffected during sliding, resulting in little change in the group velocity dominated by harmonic properties. Relative to the fixed bonding electrons, the non-bonding charges 13 are free around the atom and are significantly affected by the slip order. Fig. 5(a) and (b) demonstrate the in-plane charge variation along the vacuum layer. Significant charge transitions originate from interlayer charges, ~10Å, while relatively few transitions are inherent to bonding electrons between layers, revealing a strong tunable effect of slip on interlayer charge. In thermal vibrations, atomic anharmonicity originates from the nonlinear Coulomb repulsion between free electrons and bonded electrons. 44,45 As shown in Fig. 5(c), the three-dimensional charge differential density reveals that interlayer charge transfer plays a dominant role. As revealed by the electron localization function in Fig. 5(d), slidephononics are generally non-intrusive, with bonding electrons within the layer barely affected. Although the intrinsic bonding is barely changed, the slip redistribution of interlayer charges enhances the well anharmonicity of atomic vibrations, as revealed in Fig. 1(e-h). The force of the nonlinear force interaction is caused by the van der Waals interaction, which is equivalent to applying a vdW force field in the out-of-plane direction. When the vdW force field is regulated by the slip order, the interlayer interaction charges will be redistributed to cause fluctuations in the potential energy surface of the lattice, resulting in phonon renormalization. In BN bilayers, significant fluctuations are found along the acoustic phonon branch of out-of-plane vibrations. Therefore, the critical feature of slidephononics is that it does not change or destroy the intrinsic bonding of materials, reflecting the non-destructive properties of phonon transport regulation. should be mentioned that the ELF in the 50% slip configuration has a larger mismatch in the same section, but the intrinsic bonding is hardly affected. Conclusion In summary, we propose the concept of slidephononics, which is significant as it provides a novel and controllable approach to tuning the phonon properties of materials. Based on first-principles calculations, sliding phononics demonstrates a wide modulation ratio of thermal conductivity in BN bilayers. The deep fundamental insight is that sliding the layers renormalizes the interlayer charge and causes the phonon renormalization. It achieves the tuning of thermal conductivity by decoupling the acoustic phonon branch, thereby transforming the phonon-phonon scattering mechanism competitively. Compared with traditional methods such as doping or applying an external forcing field, sliding phononics does not change the chemical composition of the material structure but instead by sliding material layers. This regulation is highly controllable and reversible, making it a promising technique for applications in fields such as materials science and electronics. This technique could be used to design devices with improved stability and durability and to study novel quantum effects and spinphonon phenomena. Computational Methodology The basic theory to perform the calculation is the density functional theory, with the projected augmented wave (PAW) pseudopotential, which is implemented in the Vienna ab initio simulation package (VASP) 46 , and the exchange-correlation functional was approximated by the Perdew-Burke-Ernzerh of generalized gradient approximation (GGA-PBE). The kinetic energy cutoff was set as 400eV. In the self-consistent calculation, the k-mesh was set to be 24 × 24 × 1 in Monkhorst-Pack 47 to sample the Brillouin Zone (BZ), and the convergence energy threshold is 10 -8 eV. Van der Waals interactions are corrected via the optB86b parameter 48 . To perform interatomic force constant calculation, a 6 × 6 × 1 supercell with 72 atoms was generated, and a Monkhorst-Pack k-mesh of 4 × 4 × 1 was used to sample the Brillouin Zone. In order to reduce computational resources, it is necessary to ignore the interactions between atoms at a certain large distance, so the cutoff radius (rcutoff) is introduced. The cutoff interactions was chose up to 13 th based on the convergence test κ v.s. rcutoff. 49 The κ is calculated by solving the linearized BTE 50 . (a) and (b), the BN bilayers are slid along both the armchair and zigzag directions to investigate the tunability of the thermal transport properties. For the armchair (zigzag) interlayer sliding, the upper or lower layer of BN moves along the axial (perpendicular) direction of the bonding axis, respectively. The AA-stacked BN configuration, where the upper boron atom corresponds to the lower nitrogen atom, is considered the pristine configuration. The ratio of the thermal conductivity after slip ( ) to the initial thermal conductivity ( ) is defined as the regulation ratio, R = ⁄ , which indicates the modulation effect strength of the thermal conductivity. When R > 1, it indicates that the thermal conductivity has an enhanced effect, otherwise it is weakened.Fig. 1(c) and (d) depict sliding configurations with relative local peak and valley thermal conductivities along two sliding paths. High thermal conductivities are found in atomic misalignment (armchair) and bonding misalignment (zigzag) configurations, respectively, while locally low thermal conductivities occur when atoms (armchair) or bonds (zigzag) overlap. For instance, the lowest and highest thermal conductivities along the armchair path are 460 and 90 W/mK at 300 K, respectively [Fig. 1(e)]. The trend of R is consistent with that of thermal conductivity, and the highest R of ~2 is found in the configuration with 80% slide along the armchair direction, while the lowest R of ~0.5 was found at33% sliding. The wide modulation ratio reveals the high efficiency of the sliding-regulated phonon transport. For sliding along the armchair direction, the peaks of thermal conductivities when sliding along the armchair direction are found in the case of boron atoms (or nitrogen atoms) located in the middle of the B-N bond, typically at 10 %, 50 %, and 80 % sliding, i.e., atomic misalignment. That is, when the atoms in the upper and lower layers are aligned with 0 % and 33 % sliding, the corresponding thermal conductivities are reduced. Slightly differently, for the zigzag sliding, the peaks of thermal conductivities occur when the B-N bonds are mutually misaligned, i.e., 20~30 % and 70~80 % sliding, while the valleys are in the slip configurations of 0% and 50% with bonding overlap. The overall variations of the thermal conductivities along the armchair and zigzag sliding pathways are shown in Fig. 1(e) and (f), respectively, where significant modulations are clearly visible. It is noted that the sliding barrier along both sliding pathways is marginal, e.g., 35.70 and 17.54 meV per unit cell for the armchair and zigzag cases, as shown inFig. 1(g)and (h), respectively. Such an energy barrier can be attributed to the vdW interaction between the layers. The formation of a centrosymmetric configuration is lower in energy than other metastable stacking configurations, and such a tendency is in good agreement with previous first-principles predictions for a two-layered system21 . Interestingly, the valleys of thermal conductivities exist in both energy valleys and peaks of the migration barrier, regardless of sliding along the armchair or zigzag direction. This indicates that the sliding configuration with high thermal conductivities can be obtained between the potential valley and the potential peak, i.e., the configuration of atoms (armchair direction) and B-N bonding mismatch (zigzag direction). Figure 1 . 1The slidephononics of BN bilayers. Schematic diagram of sliding along the (a) armchair and (b) zigzag direction. Schematic diagram of slip configurations with high and low thermal conductivities along the (c) armchair and (d) zigzag direction. Variation of thermal conductivities for slip along the (e) armchair and (f) zigzag directions. Schematic of the slidephononics of BN bilayers along (g) armchair and (h) zigzag directions, where the percentages indicate how much the atomic positions slide in fractional coordinates. 8 Figure 2 . 82(b), (c) and (d). In both the armchair and zigzag directions, it comes to light that the two out-of-plane vibration branches (ZA and ZA') in the BN bilayers go through analogous transforms: ZA-ZA' decouples with sliding. As shown in Fig. 2(e), these three energy-conserving saturation mechanisms compete with each other in the initial state, intermediate state, and final state of the acoustic phonon branch decoupling, which intuitively reflects the phonon renormalization induced by the decoupling of the acoustic phonon branch. In the BN bilayer, the decoupling process of the acoustic branch of the out-of-plane vibration is further directly captured in Fig. 2(f) and (g), which regulates phonon-phonon scattering and ultimately thermal conductivity. Phonon dispersion and energy conservation. (a) Energy conservation in phonon-phonon scattering. (b) Initial configuration of BN bilayers, which corresponds to the initial sliding configuration. (c) The bilayer configurations, with 10% slip along the armchair direction and 20% along the zigzag direction, correspond to an intermediate sliding configuration state. (d) The bilayer configurations with 33% slip along the armchair direction and 50% along the zigzag direction, correspond to the sliding configuration's final state. (e) Schematic diagram of slip-regulated phononphonon scattering. The phonon dispersion along (f) the armchair direction and along (g) the zigzag direction. supersaturated and undersaturated states. When sliding further to the final state, the bandgap is opened to the maximum, i.e., 1.059 (0.850) THz for the armchair (zigzag) direction, implying the maximum decoupling of the acoustic phonon branch. The scattering channels participated by ZA and ZA' phonons both reach the strongest saturation state. At this time, the wide band gap allows ZA to cross the band gap and scatter with higher-frequency ZA' phonons. The transition of the scattering channel reveals a significant regulatory effect of slip modulation on phonon-phonon scattering through the decoupling of the acoustic phonon branch. Figure 3 . 3Mode-level phonon properties. (a) Cumulative thermal conductivity, (b) group velocity, and (c) relaxation time along the armchair direction. (d) Cumulative thermal conductivity, (e) group velocity, and (f) relaxation time along the zigzag direction. Figure 4 . 4Transition of the slip-regulated scattering channel. Mode level scattering channel for slip (a) 0%, (b) 10% and (c) 33% along armchair direction. Mode level scattering channel for slip (d) 0%, (e) Figure 5 . 5Charge interaction renormalization. (a) The transition of the planar charge density along armchair direction. (b) The transition of the planar charge density along zigzag direction. (c) Threedimensional charge density transfer image, where charge transfer mainly occurs between layers at surface 1x10 -4 . 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{'abstract': 'By interlayer sliding in van der Waals (vdW) materials, the switching electric polarization of ultrathin ferroelectric materials leads to the widely studied slidetronics. In this work, we report that such sliding can further tailor anharmonic effects and hence thermal transport properties due to the changed intrinsic coupling between atomic layers. And we propose an unprecedented concept dubbed as slidephononics, where the phonons and associated physical properties can be controlled by varying the intrinsic stacking configurations of slidetronic vdW materials. Based on the state-of-the-art firstprinciples calculations, it is demonstrated that the thermal conductivity of boron nitride (BN) bilayers can be significantly modulated (by up to four times) along the sliding pathways. Detailed analysis reveals that the variation of thermal conductivities can be attributed to the tunable (de-)coupling of the out-of-plane acoustic phonon branches with the other phonon modes, which is induced by the interlayer charge transfer. Such strongly modulated thermal conductivity via interlayer sliding in vdW materials paves the way to engineer thermal management materials in emerging vdW electronic devices, which would shed light on future studies of slidephononics.', 'arxivid': '2306.03562', 'author': ['Linfeng Yu \nInstitute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany\n\nCollege of Mechanical and Vehicle Engineering\nState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body\nHunan University\n410082ChangshaP. R. China\n', 'Chen Shen chenshen@tmm.tu-darmstadt.de \nInstitute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany\n', 'Guangzhao Qin \nCollege of Mechanical and Vehicle Engineering\nState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body\nHunan University\n410082ChangshaP. R. China\n', 'Hongbin Zhang \nInstitute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany\n'], 'authoraffiliation': ['Institute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany', 'College of Mechanical and Vehicle Engineering\nState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body\nHunan University\n410082ChangshaP. R. China', 'Institute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany', 'College of Mechanical and Vehicle Engineering\nState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body\nHunan University\n410082ChangshaP. R. China', 'Institute of Materials Science\nTechnical University of Darmstadt\n64287DarmstadtGermany'], 'corpusid': 259088778, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 11121, 'n_tokens_neox': 9059, 'n_words': 5417, 'pdfsha': '1396fe129a38bdff6b278b5b92e7d8a024b3fca8', 'pdfurls': ['https://export.arxiv.org/pdf/2306.03562v1.pdf'], 'title': ['Slidephononics: Tailoring Thermal Transport Properties by van der Waals Sliding', 'Slidephononics: Tailoring Thermal Transport Properties by van der Waals Sliding'], 'venue': []}

arxiv

arXiv:physics/0610054v1 [physics.hist-ph] 9 Oct 2006 arXiv:physics/0610054v1 [physics.hist-ph] 9 Oct 2006 Jul i an Schwi nger:Nucl earPhysi cs,the Radi ati on Laboratory,Renorm al i zed Q ED , Source Theory,and Beyond K i m bal lA .M i l ton H om erL.D odge D epartm entofPhysi cs and A stronom y U ni versi ty ofO kl ahom a,N orm an,O K 73019 U SA D ecem ber 8,2021A bstract Jul i an Schw i nger' s i n uence on twenti eth century sci ence i s profound and pervasi ve. O fcourse,he i s m ost fam ous for hi s renorm ali zati on theory of quantum el ectrodynam i cs,for w hi ch he shared the N obelPri ze w i th R i chard Feynm an and Si n-i ti ro Tom onaga. B ut although thi stri um ph wasundoubtedl y hi sm ostheroi c work,hi sl egacy l i ves on chi e y through subtl e and el egant work i n cl assi calel ectrodynam i cs,quantum vari ati onalpri nci pl es,proper-ti m e m ethods,quantum anom al i es, dynam i cal m ass generati on, parti al sym m etry, and m ore. Starti ng as just a boy,he rapi dl y becam e the pre-em i nent nucl ear physi ci st i n the l ate 1930s, l ed the theoreti cal devel opm ent of radar technol ogy at M IT duri ng W orl d W ar II,and then,soon after thewar,conquered quantum el ectrodynam i cs,and becam ethel eadi ng quantum el d theori st for two decades, before taki ng a m ore i conocl asti c route duri ng hi s l ast quarter century.K eywords: Jul i an Schw i nger, nucl ear physi cs, wavegui des, quantum el ectrodynam i cs,renorm al i zati on,quantum acti on pri nci pl e,source theory,axi al -vector anom al y K . A .M i l ton i sProfessorofPhysi csattheU ni versi ty ofO kl ahom a.H ewasa Ph. D .stu-dentofJul i an Schw i ngerfrom 1968{71,and hi spostdoc atU C LA forthe restofthe 1970s. H e has w ri tten a sci enti c bi ography ofSchw i nger,edi ted two vol um es ofSchw i nger' s sel ected works,and co-authored two textbooks based on Schw i nger' s l ectures. 1 Introduction G i ven Jul i an Schw i nger' s com m andi ng stature i n theoreti calphysi cs forhal f a century, i t m ay seem puzzl i ng w hy he i s rel ati vel y unknow n now to the educated publ i c,even to m any younger physi ci sts,w hi l e Feynm an i s a cul t gure w i th hi s photograph needi ng no m ore i ntroducti on than Ei nstei n' s. y T hi s rel ati ve obscuri ty i s even m ore rem arkabl e, i n vi ew of the enorm ous num berofem i nentphysi ci sts,aswel lasotherl eadersi n sci ence and i ndustry, w ho recei ved thei r Ph. D . ' s under Schw i nger' s di recti on,w hi l e Feynm an had practi cal l y none. In part,the answer l i es i n Schw i nger' s reti ri ng nature and reserved dem eanor. Sci ence, research and teachi ng, were hi s l i fe, and he detested the l i m el i ght. G eneral l y, he was not cl ose to hi s students, so few knew hi m wel l . H e was a graci ous host and a good conversati onal i st, and had a broad know l edge ofm any subjects,but he was never one to i ni ti ate a rel ati onshi p or aunt hi s erudi ti on. H i s styl e ofdoi ng physi cs was al so di cul t to penetrate. O ppenhei m er once sai d that others gave tal ks to show others how to do the cal cul ati on, w hi l e Schw i nger gave tal ks to show that onl y he coul d do i t. A l though a com m onl y shared vi ew ,thi sw i tti ci sm i sunki nd and untrue. H e was,i n fact, a superb teacher,and generati ons ofphysi ci sts, students and facul ty al i ke, l earned physi cs at hi s feet. O n the one hand he was a form al i st,i nventi ng form al i sm sso powerfulthatthey coul d l ead,atl easti n hi shands,unerri ngl y to the correct answer. H e di d not,therefore,di spl ay the i ntui ti ve vi sual i zati ons,for exam pl e, that Feynm an com m anded, w hi ch eventual l y took over the popul ar and sci enti c cul ture. But,m ore profoundl y,he was a phenom enol ogi st. Ironi cal l y,even som e ofhi sow n students cri ti ci zed hi m i n hi sl ateryears forhi sphenom enol ogi cal ori entati on,notrecogni zi ng thathe had,from hi s earl i est experi ences i n nucl earphysi cs,i nsi sted i n groundi ng theoreti calphysi csi n the phenom ena and data ofexperi m ent. Isi dorR abi ,w ho di scovered Schw i ngerand broughthi m to C ol um bi a U ni versi ty, general l y had a poor opi ni on of theoreti calphysici sts. ButR abiwasal waysvery i m pressed w i th Schw i nger because i n nearl y every paper,he \gotthe num bersout" to com pare w i th experi m ent. Even i n y A n exam pl ei stheseri esofpostersproduced by theA m eri can Physi calSoci ety i n w hi ch the i m pressi on i s gi ven that Feynm an was the chi efi nnovator i n quantum el ectrodynami cs. In contradi cti on to thi s,N orm an R am sey has stated that \i t i s m y i m pressi on that Schw i nger overw hel m i ngl y deserved the greatest credi t for Q ED .I don' t thi nk Feynm an had an expl anati on ofthe anom al oushyper nestructure beforethe [ 1948 A PS]m eeti ng. " 1 hi sm ostel aborate el d-theoreti c papershe wasal waysconcerned w i th m aki ng contact w i th the realworl d,be i t quantum el ectrodynam i cs,or strongl y i nteracti ng hadrons. A l though hi s rst,unpubl i shed,paper,w ri tten at the age of16,was on the subject ofthe then poorl y-understood quantum el ectrodynam i cs,Jul i an Schw i ngerwasal m ostexcl usi vel y a nucl earphysi ci stunti lhejoi ned theR adiati on Laboratory atM IT i n 1943.T here,faced w i th cri ti caldeadl i nesand the di cul ty ofcom m uni cati ng w i th el ectri calengi neers,he perfected vari ati onal techni ques for sol vi ng the cl assi calel ectrodynam i c probl em s ofwavegui des. A s the W ar wound dow n,he started thi nki ng about radi ati on produced by el ectrons i n betatronsand synchrotrons,and i n so doi ng recogni zed thatthe reacti ve and resi sti ve porti ons of the el ectrom agneti c m ass of the el ectron are uni ted i n a i nvari ant structure. R ecrui ted by H arvard,he started teachi ng there i n 1946,and at rst conti nued research i n nucl ear physi cs and i n cl assi caldi racti on. T he Shel ter Isl and conference of1947 changed al lthat. H e and W ei sskopf suggested to Bethe that el ectrodynam i c processes were responsi bl e for the Lam b shi ft,w hi ch had been know n for som e ti m e as the Pasternack e ect. Im m edi atel y, however, Schw i nger saw that the m ost direct consequence ofquantum el ectrodynam i cs l ay i n the hyper ne anom al y reported for the rst ti m e at Shel ter Isl and. H e anti ci pated that the e ect was due to an i nduced anom al ous m agneti c m om ent of the el ectron. T he actualcal cul ati on had to wai t three m onths, w hi l e Schw i nger took an extended honeym oon,but by D ecem ber 1947 Schw i nger had hi s fam ous resul t for the gyrom agneti c rati o. In the process he i nvented renorm al i zati on of m ass and charge,onl y di m l y pre gured by K ram ers. T hi s rst form ul ati on ofQ ED was rather crude,bei ng noncovari ant; to obtai n the correct Lam b shi ft,a rel ati vi sti c form ul ati on was requi red,w hi ch fol l owed the next year. A com edy oferrorsensued: Both Feynm an and Schw i ngerm ade an i ncorrect patch between hard and softphoton processes,and so obtai ned i denti cal ,but i ncorrect,predi cti ons forthe Lam b shi ft,and the wei ghtofthei rreputati ons del ayed the publ i cati on ofthe correct,i fpedestri an, cal cul ati on by French and W ei sskopf. z By 1950 Schw i nger had started hi s thi rd reform ul ati on of quantum el ectrodynam i cs,i n term s ofthe quantum acti on pri nci pl e. A t the sam e ti m e he w rote hi s rem arkabl e paper,\O n G auge Invari ance and Vacuum Pol ari zati on, " form ul ated i n a com pl etel y gauge-covari ant way,w hi ch z Schw i nger l ater cl ai m ed that hi s rst noncovari ant approach had yi el ded the correct resul t,except that he had not trusted i t. anti ci pated m any l ater devel opm ents,i ncl udi ng the axi alvector anom al y. H i sstrong phenom enol ogi calbenteventual l y l ed hi m away from them ai nstream ofphysi cs. A l though he had gi ven the basi s for w hat i s now cal l ed the Standard M odelofel em entary parti cl es i n 1957,he never coul d accept the exi stence ofquarksbecause they had no i ndependentexi stence outsi de of hadrons.(A secondary consi derati on wasthatquarkswere i nvented by M urray G el l -M ann,w i th w hom a l ong-runni ng feud had devel oped. ) H e cam e to appreci ate the noti on ofsupersym m etry,but he rejected noti ons of\G rand U ni cati on" and of\Superstri ngs" notbecauseofthei rstructurebutbecause he saw them aspreposterous specul ati ons,based on the noti on thatnothi ng new rem ai ns to be found i n the enorm ous energy range from 1 TeV to 10 19 G eV .H e wassure thattotal l y new ,unexpected phenom ena were wai ti ng just around the corner. T hi s seem s a reasonabl e vi ew ,but i t resul ted i n a sel fi m posed i sol ati on,i n contrast,agai n,to Feynm an,w ho contri buted m i ghtl y to the theory ofpartons and quantum chrom odynam i cs up to the end. A com pl ete bi ography ofJul i an Schw i nger was publ i shed si x years ago. 2 T he present paper draw s upon that book,as wel las on l ater i ntervi ew s and research by the author. Q uotati ons of Schw i nger not otherw i se attri buted arebased on an extended i ntervi ew conducted forthatbook by m y co-author Jagdi sh M ehra i n 1988. E arly Y ears Jul i an Schw i nger was born i n M anhattan,N ew York C i ty,on February 12, 1918,to rather wel l -o m i ddl e-cl ass parents. H i s father was a wel l -know n desi gner ofwom en' s cl othes. H e had a brother H arol d ten years ol der than hi m sel f,w hom Jul i an i dol i zed aschi l d. H arol d cl ai m ed thathe taughtJul i an physi cs unti l he was 13. A l though Jul i an was recogni zed as i ntel l i gent i n school ,everyone thought H arol d was the bri ght one. (H arol d i n facteventual l y becam e a respected l aw yer,and hi sm otheral waysconsi dered hi m asthe successfulone,even after Jul i an recei ved the N obelPri ze. ) T he D epressi on cost Jul i an' s father hi s busi ness,but he was su ci entl y appreci ated that he was o ered em pl oym ent by other desi gners;so the fam i l y survi ved,but not so com fortabl y as before. T heD epressi on di d m ean thatJul i an woul d haveto rel y on freeeducati on, w hi ch N ew York wel l -provi ded i n those days: A year or two at Tow nsend H arri s H i gh School , a publ i c preparatory schoolfeedi ng i nto C i ty C ol l ege, w here Jul i an m atri cul ated i n 1933. Jul i an had al ready di scovered physi cs, rst through H arol d' s Encycl opedia Britannica at hom e,and then through the rem arkabl e i nsti tuti on ofthe N ew York Publ i c Li brary. Larry C ranberg was a student at Tow nsend H arri s at the sam e ti m e as Jul i an Schw i nger. 3 T hey had som e cl asses together,and both graduated i n January 1933, w i th a di pl om a that stated that graduates were enti tl ed to autom ati c entry to C C N Y .H e recal l ed thatJul i an was\very,very qui et. H e never gave reci tati ons. H e sat i n the l ast row , unsm i l i ng and unspeaki ng, and was a reall oner. But the scuttl ebutt was that he was our star. H e very earl y showed prom i se, " butC ranberg saw nothi ng overt. \R um orswere that he was not very good outsi de m ath and physi cs, and that he was unki ng G erm an. " A m ong the teachers at Tow nsend H arri s,C ranberg parti cul arl y rem embers A l fred Bender, x w ho was apparentl y not on the regul ar facul ty. Ei l een Lebow ,w ho recentl y w rote a hi story ofTow nsend H arri sH i gh School , 4 does notrecal lBender' snam e.C ranberg sai d thatBender\ xed m eon thecourse to be a physi ci st. H e was di l i gent,passi onate,and m eti cul ous i n hi s reci tati ons. H e was a great guy,one ofthe best teachers at Tow nsend H arri s. " It seem s very l i kel y that i t was Bender to w hom Schw i nger referred to as an anonym ous i n uence: I took m y rst physi cs course i n H i gh School . T hat i nstructorshowed unl i m i ted pati ence i n answeri ng m y endl ess questi ons about atom i c physi cs,after the cl ass peri od was over. A l though Itry,Icannot l i ve up to that l ofty standard. 5 A t C i ty C ol l ege Jul i an was readi ng and di gesti ng the l atest papers from Europe,and starti ng to w ri te papersw i th i nstructorsw ho were,atthe sam e ti m e,graduate students atC ol um bi a and N Y U .Joseph W ei nberg,w ho went on to becom e a wel l -know n rel ati vi st,was hi s cl osest fri end at C i ty C ol l ege. W ei nberg recal l ed hi s rst m eeti ng w i th Jul i an. 6 Because ofhi s outstandi ng l aboratory reports,W ei nberg had been granted the pri vi l ege ofenteri ng the cl osed l i brary stacksatC i ty C ol l ege. O ne day he wasseeki ng a m athem ati cs book, 7 w hi ch had been m enti oned attheM ath C l ub theday before,and w hi l e hereached fori t,anotheryoungsterwastryi ng to geti t.T hey had both heard the tal k,on functi ons w hi ch are conti nuous but now here di erenti abl e,and x B ender was the father ofphysi ci st C arlB ender. C arl ' s uncl e A bram B ader was the physi cs teacher ofR i chard Feynm an at Far R ockaway H i gh School . so they shared the book between them ,bal anci ng the heavy vol um e on one kneeeach.T heotherfel l ow kept ni shi ng thepagebeforeW ei nberg,w ho was a very fast reader. O fcourse,hi s i m pati ent co-reader was Jul i an Schw i nger. Both were 15.W ei nberg m enti oned thathe usual l y spenthi sti m e,noti n the m athem ati cs secti on ofthe l i brary,but i n the physi cs secti on,w hi ch turned out to be Jul i an' s base as wel l . W ei nberg com pl ai ned that D i rac' s book on quantum m echani cs 8 wasvery i nteresti ng and exci ti ng,butdi cul tto fol l ow . Jul i an concurred,and sai d i twasbecause i twaspol i shed too hi ghl y;he sai d thatD i rac' sori gi nalpaperswere m uch m ore accessi bl e. W ei nberg had never concei ved ofconsul ti ng the ori gi nall i terature,so thi sopened a doorforhi m . T hi sadvi ce aboutover-re nem entSchw i ngerhi m sel fforgotto fol l ow i n l ater l i fe. Jul i an no l onger had the ti m e to spend i n the cl assroom attendi ng l ectures.In physi csand m athem ati cshewasabl eto ski m thetextsand work out the probl em s from rst pri nci pl es, frequentl y l eavi ng the professors ba ed w i th hi s ori gi nal ,unorthodox sol uti ons,but i t was not so si m pl e i n hi story, Engl i sh, and G erm an. C i ty C ol l ege had an enorm ous num ber of requi red courses then i n al lsubjects. H i s m arks were not good,and he woul d have unked out i fthe C ol l ege had not al so had a rather forgi vi ng pol i cy toward grades. JoeW ei nberg recal l ed anothervi vi d i nci dent.A m ong therequi red courses were two yearsofgym nasi um . O ne had to passexam si n hurdl i ng,chi nni ng, paral l elbars,and sw i m m i ng. Because W ei nberg and Jul i an had nearby l ockers,they often fel li nto physi csconversati onshal fdressed,and fai l ed thecl ass forl ack ofattendance. W ei nberg rem em bered seei ng Jul i an' shurdl i ng exam . Jul i an ran up to the bar,but cam e to a standsti l lw hen he was supposed to jum p over si deways. T he i nstructor repri m anded hi m ,atw hi ch poi ntJul i an sai d,sotto voce,\there' s notenough ti m e to sol ve the equati ons ofm oti on. " Edward G erjuoy wasanotherofJul i an' scl assm atesatC i ty C ol l ege. 9 \M y m ai n cl ai m to fam e i s that Jul i an and I took the sam e course i n m echani cs together, taught by a m an nam ed Shea, and I got an A and Jul i an a B, " because Jul i an di d notdo the work. \Ittook abouta week before the peopl e i n the cl ass real i zed we were deal i ng w i th som ebody of a di erent order of m agni tude. " A t a ti m e w hen know l edge of a bi t of vector al gebra was consi dered com m endabl e,\Jul i an coul d m ake i ntegral s vani sh| he was very, very i m pressi ve. T he onl y person i n the cl assroom w ho di dn' t understand thi s about Jul i an was the i nstructor hi m sel f. " \H e was unki ng out ofC i ty C ol l ege i n everythi ng except m ath and physi cs. H e was a phenom enon. H e di dn' t l ead the conventi onall i fe ofa hi gh schoolstudent before he cam e to C i ty C ol l ege"| unl i ke G erjuoy and Si dney Borow i tz he wasnoton the m ath team i n hi gh schoolso they had notknow n hi m earl i er| \w hen he appeared he was just a phenom enon. " M orton H am erm esh recal l ed another di sastrous course. 10 W e were i n a cl ass cal l ed M odern G eom etry. Si dney Borow i tz,anothercl assm ate ofJul i an' s,recal l ed that\we had the pl easure ofseei ng Jul i an attack a probl em de novo,and thi s used to dri ve R eynol ds crazy. " 12 { In addi ti on, he was al so apparentl y a notori ous anti sem i te. H e used to di scourage Jew i sh students from studyi ng m athem ati cs,w hi ch worked to the advantage ofphysi cs. 11 3 P aper N um ber Zero N otonl y wasJul i an al ready readi ng the l i terature atC i ty C ol l ege,he qui ckl y started to do ori gi nal research. Jul i an had studi ed a paper by C hri sti an M l l er 13 i n w hi ch he had cal cul ated the two-el ectron scatteri ng crosssecti on by usi ng a retarded i nteracti on potenti al . O fcourse,Schw i nger read al lof D i rac' spaperson quantum el d theory,and wasparti cul arl y i m pressed by the one on \R el ati vi sti c Q uantum M echani cs, " 14 \i n w hi ch D i rac went through hi s attem pt to recreate an el ectrodynam i cs i n w hi ch the parti cl es and l i ght were treated di erentl y. " In a paper ofD i rac,Fock,and Podol sky, 15 i t was recogni zed that thi s was si m pl y a uni tary transform ati on ofthe H ei senberg-Paul itheory, 16 i n w hi ch the uni tary transform ati on was appl i ed to the el ectrom agneti c el d. A nd I sai d to m ysel f, ' W hy don' t we appl y a si m i l ar uni tary transform ati on to the second-quanti zed el ectron el d?' I di d that and worked outthe l owest approxi m ati on to the scatteri ng am pl i tudesi n unrel ati vi sti c notati on. It was a rel ati vi sti c theory but i t was not covari ant. T hat was i n 1934,and I woul d use i t l ater;[ the noti on,cal l ed the interaction representation,]i s al ways ascri bed to Tom onaga,but Ihad done i t m uch earl i er. In deri vi ng hi sresul t,Schw i ngerhad to om i ta term w hi ch \representsthe i n ni te sel f-energy ofthechargesand m ustbedi scarded. " T hi sheeventual l y cam e to see as a m i stake: \T he l ast i njuncti on m erel y parrots the w i sdom of m y el ders, to be l ater rejected, that the theory was fatal l y awed, as w i tnessed by such i n ni te term s, w hi ch at best, had to be di scarded, or subtracted. T hus,the ' subtracti on physi cs'ofthe 1930s. " 17 T hi s paperwas never subm i tted to a journal ,but was recentl y publ i shed i n a sel ecti on ofSchw i nger' s works. 18 C olum bia U niversity ItwasLl oyd M otz,onethei nstructorsatC i ty C ol l ege,w ho had l earned about Jul i an from H arol d, and w i th w hom Jul i an was w ri ti ng two papers, w ho i ntroduced hi m to Isi dor I.R abi . T hen,i n a conversati on between R abiand M otz overthe fam ousEi nstei n,Podol sky,and R osen paper, 19 w hi ch had just appeared,Jul i an' s voi ce appeared w i th the resol uti on ofa di cul ty through the com pl eteness pri nci pl e, and Schw i nger' s career was assured. R abi ,not w i thoutsom e di cul ty,had Schw i ngertransferred to C ol um bi a,and by 1937 he had 7 papers publ i shed,m ostl y on nucl ear physi cs,w hi ch consti tuted hi s Ph. D .thesi s,even though hi s bachel or' s degree had not yet been granted. Schw T he papers w hi ch Jul i an w rote at C ol um bi a were on both theoreti cal and experi m entalphysi cs,and R abipri zed Jul i an' s abi l i ty to \get the numbers out" to com pare w i th experi m ent. T he form alawardi ng ofthe Ph. D . had to wai t ti l l1939 to sati sfy a U ni versi ty regul ati on. In the m eanti m e, Schw i nger was busy w ri ti ng papers (one,for exam pl e,was fundam entalfor the theory ofnucl ear m agneti c resonance, 22 ) and spent a som ew hat l onel y, but producti ve w i nter (1937) i n W i sconsi n, k w here he l ai d the groundwork for hi s predi cti on that the deuteron had an el ectri c quadrupol e m om ent, 23 i ndependentl y establ i shed by hi sexperi m entalcol l eaguesatC ol um bi a a year l ater. 24 W i sconsi n con rm ed hi s predi l ecti on for worki ng at ni ght,so as not to be \overw hel m ed" by hi s hosts,Eugene W i gner and G regory Brei t. R abil ater am usi ngl y sum m ari zed Schw i nger' s year i n W i sconsi n. But the theory seem s val i dated. T w o Y ears in B erkeley By 1939,R abifel t Schw i nger had outgrow n C ol um bi a,so w i th a N RC Fell ow shi p,he was sent to J.R obert O ppenhei m er i n Berkel ey. T hi s exposed hi m to new el ds: quantum el ectrodynam i cs (al though as we recal l ,he had w ri tten an earl y,unpubl i shed paperon thesubjectw hi l ejust16)and cosm i cray physi cs,but he m ostl y conti nued to work on nucl ear physi cs. H e had a num ber of col l aborati ons; the m ost rem arkabl e was w i th W i l l i am R ari ta, w ho was on sabbati calfrom Brookl yn C ol l ege;R ari ta was Schw i nger' s \calcul ati ng arm " on a seri es ofpapers extendi ng the noti on ofnucl ear tensor forces w hi ch he had concei ved i n W i sconsi n over a year earl i er. R ari ta and Schw i nger al so w rote the presci ent paper on spi n-3/2 parti cl es, 25 w hi ch was to be i n uenti aldecades l ater w i th the bi rth ofsupergravi ty. Ed G erjuoy,w ho had been an undergraduate w i th Schw i ngeratC i ty C oll ege i n 1934,now wasone ofO ppenhei m er' s graduate students. H e recal l ed 9 Left-l eani ng Joe W ei nberg accused Jul i an ofexpl oi ti ng R ari ta,but Jul i an responded that these papers establ i shed R ari ta' s reputati on. an am usi ng i nci dent w hi ch happened one day w hi l e he, Schw T hey had sol ved the probl em ,and they covered the w hol e board w i th the el aborate sol uti on. O ppenhei m er l ooked ati t,sai d i tl ooked reasonabl e,and then asked, \Jul i e, di dn' t you tel lm e you worked thi s cross secti on out?" Schw i nger pul l ed the yel l owed, crum pl ed bl anks from hi s pocket,stared at them a m om ent,and then pronounced the students'sol uti on okay apartfrom a factoroftwo.O ppenhei m ertol d them to nd thei rerror,and they shu ed out,di spi ri ted. Indeed, Schw i nger was ri ght,they found they had m ade a m i stake,and they publ i shed the paper, 26 but they were su ci entl y crushed that both sw i tched to experi m entalphysi cs. A t the ti m e, Schw i nger and G erjuoy were col l aborati ng on a paper 27 fol l ow i ng from Schw i nger' s tensor theory ofnucl ear forces. T he work i nvol ved cal cul ati ng about200 fai rl y com pl i cated spi n sum s,w hi ch sum s Jul i an and Iperform ed i ndependentl y and then com pared. To have the pri vi l ege ofworki ng w i th Jul i an m eant I had to accom m odate m ysel fto hi s worki ng habi ts,as fol l ow s. Except on daysw hen Jul i an had to com e i nto the uni versi ty duri ng conventi onalhours to confer w i th O ppenhei m er, I woul d m eet hi m at 11: 45 pm i n the l obby ofhi sresi dence,the Berkel ey Internati onal H ouse. H e woul d then dri ve us to som e Berkel ey al l -ni ght bi stro w here he had breakfast,after w hi ch we drove to LeC onte H al l , the Berkel ey physi cs bui l di ng,w here we worked unti labout 4: 00 am ,Jul i an' s l unchti m e. A fter l unch i t was back to LeC onte H al l unti labout8: 30 am ,w hen Jul i an wasready to thi nk aboutdi nner and poor TA m e woul d m eet m y 9: 00 am reci tati on cl ass. Si nce I had just gotten m arri ed,and sti l lwas young enough to badl y need m y sl eep,these m onths ofworki ng w i th Jul i an were tryi ng, to put i t m i l dl y. W hat m ade i t even m ore tryi ng i s the fact that w hen Jul i an and Icareful l y worked outtogetherthe 20 orso spi n sum sw here ouri ndependentcal cul ati onsdi sagreed,Jul i an proved to be ri ght every ti m e! I accepted the fact that Jul i an was a m uch better theori st than I, but I fel t I was at l east pretty good, and was i nfuri ated by hi s apparent consti tuti onali nabi l i ty to m ake a si ngl e errori n 200 com pl i cated spi n sum cal cul ati ons. T hi si nabi l i ty stood Schw i nger wel lw hen he em barked on the cal cul ati ons that earned hi m the N obelPri ze. ...[ A l ] though Jul i an certai nl y reali zed how extraordi nari l y tal ented he was,he never gl oated about hi s error free cal cul ati ons or i n any other way put m e dow n. 28 T heyearoftheN RC Fel l ow shi p wasfol l owed by a second yearatBerkel ey asO ppenhei m er' s assi stant. T hey w rote an i m portantpapertogetherw hi ch woul d prove cruci alnearl y a decade l ater:A l though O ppenhei m erwashappy to i m agi ne new i nteracti ons,Schw i nger showed that an anom al y i n uori ne decay coul d be expl ai ned by the exi stence ofvacuum pol arization,thati s,by the vi rtualcreati on ofel ectron-posi tron pai rs. 29 T hi s gave Schw i nger a head start over Feynm an,w ho for years suspected that vacuum pol ari zati on di d not exi st. T he W ar and the R adiation Laboratory A fter two years at Berkel ey,O ppenhei m er and R abiarranged a realjob for Schw i nger: H e becam e rst an i nstructor, then an A ssi stant Professor at Purdue U ni versi ty,w hi ch had acqui red a num ber ofbri ght young physi ci sts under the l eadershi p ofK arlLark-H orow i tz. But the war was i m pi ngi ng on everyone' s l i ves,and Schw i nger was soon recrui ted i nto the work on radar. T he m ove to the M IT R adi ati on Laboratory took pl ace i n 1943. T here Schw i ngerrapi dl y becam e the theoreti call eader,even though he wassel dom seen, goi ng hom e i n the m orni ng just as others were arri vi ng. H e developed powerfulvari ati onalm ethods for deal i ng w i th com pl i cated m i crowave ci rcui ts,expressi ng resul ts i n term s ofquanti ti es the engi neers coul d understand,such as i m pedance and adm i ttance. A t rst i t seem s strange that Schw i nger, by 1943 the l eadi ng nucl ear theori st,shoul d nothave gone to LosA l am os,w here nearl y al lhi scol l eagues eventual l y settl ed for the durati on. T here seem to be at l east three reasons w hy Schw i nger stayed at the R adi ati on Laboratory throughout the war. T hereason hem ostoften ci ted l ateri n l i fewasoneofm oralrepugnance. W hen he real i zed the destructi ve powerofw hatwasbei ng constructed at Los A l am os, he wanted no part of i t. In contrast, the radi ati on l ab was devel opi ng a pri m ari l y defensi ve technol ogy,radar,w hi ch had al ready saved Bri tai n. H e bel i eved that the probl em s to sol ve at the R adi ati on Laboratory were m ore i nteresti ng. Both l aboratori eswere i nvol ved l argel y i n engineeri ng,yet al though M axwel l ' s equati ons were certai nl y wel lknow n, the process ofappl yi ng them to wavegui des requi red the devel opm ent ofspeci altechni ques that woul d prove i nval uabl e to Schw i nger' s l ater career. A nother factor probabl y was Schw i nger' s fear of bei ng overw hel m ed. In C am bri dge he coul d l i ve hi s ow n l i fe,worki ng atni ght w hen no one wasaround the l ab. T hi spri vacy woul d have been m uch m ore di cul t to m ai ntai n i n the m i croworl d ofLos A l am os. Si m i l arl y,the worki ng condi ti ons at the R ad Lab were m uch freer than those at Los A l am os. Schw i ngerneverwascom fortabl e i n a team setti ng,asw i tnesshi sl ater aversi on to the atm osphere at the Insti tute for A dvanced Study. T he m ethods and the di scoveri es he m ade at the R ad Lab concerni ng the real i ty ofthe el ectrom agneti c m ass woul d be i nval uabl e for hi s work on quantum el ectrodynam i cs a few years l ater. A s the war wound dow n,physici sts started thi nki ng about new accel erators, si nce the pre-war cycl otrons had been defeated by rel ati vi ty,and Schw i nger becam e a l eader i n thi s devel opm ent: H e proposed a m i crotron, yy an accel erator based on accel erati on through m i crowave cavi ti es,devel oped the theory ofstabi l i ty ofsynchrotron orbi ts,and m osti m portantl y,worked outi n detai lthe theory ofsynchrotron radi ati on, zz ata ti m e w hen m any thoughtthatsuch radi ati on woul d be negl i gi bl e because ofdestructi ve i nterference. Schw i nger never properl y w rote up the work he conducted i n hi s one and one-hal fyears at the R ad Lab,an om i ssi on that has now be recti ed i n part by publ i cati on ofa vol um e based yy T he m i crotron i s usual l y attri buted to Veksl er. zz T hi s was rst ci rcul ated as a prepri nt i n 1945. T he paper 30 publ i shed i n 1949 was substanti al l y di erent. on hi sl ecturesthen and l ater,and i ncl udi ng both publ i shed and unpubl i shed papers. 31 A l though he never real l y publ i shed hi s consi derati ons on sel f-reacti on, he vi ewed that understandi ng as the m ost i m portant part of hi s work on synchrotron radi ati on: It was a usefulthi ng for m e for w hat was to com e l ater i n el ectrodynam i cs, because the techni que I used for cal cul ati ng the el ectron' s cl assi calradi ati on was one ofsel f-reacti on,and Idi d i t rel ati vi sti cal l y,and i twasa si tuati on i n w hi ch Ihad to take seriousl y thepartofthesel f-reacti on w hi ch wasradi ati on,so w hy not takeseri ousl y thepartofthesel f-reacti on thati sm asschange? In otherwords,thei deasofm assrenorm al i zati on and rel ati vi sti cal l y handl i ng them were al ready present at thi s cl assi call evel . JustaftertheTri ni ty atom i cbom b test,Schw i ngertravel ed to LosA l am os to speak about hi s work on wavegui des,el ectrom agneti c radi ati on,and hi s i deasaboutfuture accel erators. T here he m etR i chard Feynm an forthe rst ti m e. Feynm an recal l ed that at the ti m e Schw i nger 33 had al ready a great reputati on because he had done so m uch work ...and I was very anxi ous to see w hat thi s m an was l i ke. I' d al ways thought he was m uch ol der than Iwas [ they were the sam e age] because he had done so m uch m ore. A t the ti m e I hadn' t done anythi ng. Q E D In 1945 H arvard o ered Schw i nger an A ssoci ate Professorshi p, w hi ch he prom ptl y accepted,partl y becausei n them eanti m ehehad m ethi sfuturew i fe C l ari ce C arrol . C ountero ers qui ckl y appeared, from C ol um bi a, Berkel ey, and el sew here,and H arvard shortl y m ade Schw i nger the youngest ful lprofessoron the facul ty to thatdate.T here Schw i ngerqui ckl y establ i shed a pattern that was to persi st for m any years| he taught bri l l i ant courses on cl assi calel ectrodynam i cs,nucl ear physi cs,and quantum m echani cs,surrounded hi m sel fw i th a devoted coteri e ofgraduate students and post-doctoralassi stants,and conducted i nci si veresearch thatsetthetonefortheoreti calphysi cs throughout the worl d. W ork on cl assi cal di racti on theory, begun at the R adi ati on Lab, conti nued for severalyears l argel y due to the presence ofH arol d Levi ne,w hom Schw i ngerhad broughtal ong asan assi stant. Vari ati onalm ethods,perfected i n the el ectrodynam i c wavegui de context,were rapi dl y appl i ed to probl em s i n nucl ear physi cs. But these were ol d probl em s,and i t was quantum el ectrodynam ics that was to de ne Schw i nger' s career. But i t took new experi m entaldata to catal yze thi s devel opm ent. T hat data was presented at the fam ous Shel ter Isl and m eeti ng hel d i n June 1947, a week before Schw i nger' s weddi ng. T here he,Feynm an,V i ctor W ei sskopf, H ans Bethe, and the other parti ci pants l earned the detai l s of the new experi m ents ofLam b and R etherford 32 that con rm ed the pre-war Pasternack e ect,show i ng a spl i tti ng between the 2S 1=2 and 2P 1=2 states ofhydrogen, that shoul d be degenerate accordi ng to D i rac' s theory. In fact,on the way to the conference,W ei sskopf and Schw i nger specul ated that quantum el ectrodynam i cs coul d expl ai n thi s e ect,and outl i ned the i dea to Bethe there, w ho worked out the detai l s,nonrel ati vi sti cal l y,on hi s fam ous trai n ri de to Schenectady after the m eeti ng. 34 But the other experi m ent announced there was unexpected: T hi s was the experi m ent by R abi ' s group and others 35 N orm an R am sey added an am usi ng footnote to the story about LaM er, the chem i st w ho unked Jul i an. 39 In 1948 Schw i nger had to repeat hi s bri ll i antl ecture on quantum el ectrodynam i csthree ti m esatthe A m eri can Physi calSoci ety m eeti ng at C ol um bi a,i n successi vel y l arger room s. y It was a superb l ecture. W e were i m pressed. A nd as we wal ked back together| R abiand Iweresi tti ng togetherduri ng thel ecture | R abii nvi ted m e to the C ol um bi a Facul ty C l ub for l unch. W e goti n theel evator[ i n the Facul ty C l ub]w hen w ho shoul d happen to wal k i n the el evator w i th us but LaM er. A nd as soon as R abi saw that,a m i schi evous gl eam cam e i nto hi seye and he began by sayi ng that was the m ost sensati onalthi ng that' s ever happened i n the A m eri can Physi cal Soci ety. T he rst ti m e there' s been 42 It i s worth rem arki ng that Schw i nger' s approach was conservati ve. H e took el d theory at face val ue,and fol l owed the conventi onalpath ofPaul i , H ei senberg,and D i rac. H i s geni us was to recogni ze that the wel l -know n divergences of the theory, w hi ch had stym i ed al lpre-war progress, coul d be consi stentl y i sol ated i n renorm al i zati on ofcharge and m ass. T hi s bore a super ci alresem bl ance to the i deas ofK ram ers advocated as earl y as 1938, 43 but K ram ers proceeded cl assi cal l y. H e had i nsi sted that rst the cl assi cal theory had to be rendered ni te and then quanti zed. T hat i dea was a bl i nd al l ey. R enorm al i zati on ofquantum el d theory i sunquesti onabl y the di scovery ofSchw i nger. Feynm an was m ore i nterested i n ndi ng an al ternati ve to el d theory, el i m i nati ng enti rel y the photon el d i n favor of acti on at a di stance. H e was, by 1950,qui te di sappoi nted to real i ze that hi s approach was enti rel y equi val entto theconventi onalel ectrodynam i cs,i n w hi ch el ectron and photon el ds are treated on the sam e footi ng. A s earl y as January 1948, w hen Schw i nger was expoundi ng hi s noncovari ant Q ED to over ow crow ds at the A m eri can Physi calSoci ety m eeti ng at C ol um bi a U ni versi ty, he l earned from O ppenhei m er of the exi stence of the work of Tom onaga carri ed out i n Tokyo duri ng the terri bl e condi ti ons of warti m e. Tom onaga had i ndependentl y i nvented the \Interacti on R epresentati on" w hi ch Schw i nger had used i n hi s unpubl i shed 1934 paper,and had com e up w i th a covari ant versi on of the Schr odi nger equati on as had Schw i nger,w hi ch upon i ts W estern redi scovery was dubbed the Tom onaga-Schw i nger equati on. 44 Both Schw i nger and Tom onaga i ndependentl y w rote the sam e equati on,a general i zati on ofthe Schr odi nger equati on to an arbitrary spacel i ke surface,usi ng nearl y the sam e notati on.T he form al i sm found by Tom onaga and hi s schoolwas essenti al l y i denti calto that devel oped by Schw i nger ve years l ater;yet they at the ti m e cal cul ated nothi ng,nor di d they di scoverrenorm al i zati on.T hatwascertai nl y no re ecti on on the abi l i ty oftheJapanese;Schw i ngercoul d nothave carri ed theform al i sm to i tsl ogi cal concl usi on w i thoutthe i m petusofthe postwarexperi m ents,w hi ch overcam e prewar paral ysi s by show i ng that the quantum correcti ons \were nei ther i nni te nor zero,but ni te and sm al l ,and dem anded understandi ng. " 17 H owever,at rstSchw i nger' scovari antcal cul ati on ofthe Lam b shi ftcontai ned another error,the sam e as Feynm an' s. 45 By thi s ti m e I had forgotten the num ber I had gotten by just arti ci al l y changi ng thew rong spi n-orbi tcoupl i ng.Because Iwas now thoroughl y i nvol ved w i th thecovari antcal cul ati on and i twas the covari ant cal cul ati on that betrayed m e, because som ethi ng went w rong there as wel l . T hat was a hum an error ofstupi di ty. N obody thought that i fyou gi ve the photon a ni te m ass i t w i l l al so a ect the l ow energy probl em . T here are no l onger the two transverse degreesoffreedom ofa m assl ess photon,there' sal so a l ongi tudi naldegreeoffreedom .Isuddenl y real i zed thi sabsol utel y stupi d error,that a photon of ni te m ass i s a spi n one parti cl e, not a hel i ci ty one parti cl e. Feynm an was m ore forthri ght and apol ogeti c i n acknow l edgi ng 45 hi s error w hi ch substanti al l y del ayed the publ i cati on ofthe French and W ei sskopfpaper. 50 PaulM arti n presented an entertai ni ng account ofthe prehi story ofthei r work together. 51 D uri ng the l ate 1940s and earl y 1950sH arvard was the hom e of a school of physi cs w i th a speci al outl ook and a di sti ncti ve set of ri tual s. Som ew hat before noon three ti m es each week, the m aster woul d arri ve i n hi s bl ue chari ot and,i n forcefuland beauti ful l ectures, reveal profound truths to hi s C antabri dgi an fol l owers, H arvard and M . I. T .students and facul ty. z C ast i n a l anguage m ore powerfuland generalthan any ofhi sl i stenershad everencountered,thesecerem oni algatheri ngshad som esacri ci al overtones| i nterrupti onswere di scouraged and si nce the serm ons usual l y l asted past the l unch hour, fasti ng was often requi red. Fol l ow i ng a m i d-afternoon break,pri vateaudi encesw i th them asterwere perm i tted and,i n uncertai n anti ci pati on,studentswoul d gather i n l ong l i nes to seek counsel . Q uantum A ction P rinciple D uri ng thi s peri od the rel i gi on had i ts ow n gol den rul e| the acti on pri nci pl e| and i tsow n crypti ctestam ent| ' O n theG reen' s Functi onsofQ uanti zed Fi el ds. ' 50 M astery ofthi spaperconferred on fol l owers a hi gh pri est status. x T he testam ent was couched i n term sthatcoul d notbequesti oned,i n a l anguagew hoseel em ents z In a l aterrecol l ecti on, 52 M arti n el aborated:\Speaki ng el oquentl y,w i thoutnotes,and w ri ti ng w i th both hands, he expressed w hat was al ready know n i n new , uni ed ways, i ncorporati ng ori gi nalexam pl es and resul ts al m ost every day. Interrupti ng the ow w i th questi ons was l i ke i nterrupti ng a theatri calperform ance. T he l ectures conti nued through H arvard' sreadi ng peri od and then the exam i nati on peri od. In one course we attended,he presented the l astl ecture| a novelcal cul ati on ofthe Lam b Shi ft| duri ng C om m encem ent W eek. T he audi ence conti nued com i ng and he conti nued l ecturi ng. " 57 Schw i nger recal l ed l ater that he vi ewed thi s paper,i n part,as a reacti on to the \i nvari ant regul ari zati on" ofPaul iand V i l l ars. 58 It was thi s paper,w i th i ts m athem ati calm ani pul ati on,w i thout physi cali nsi ghtparti cul arl y aboutquesti onssuch asphoton m ass and so forth,w hi ch was the di rect i nspi rati on for ' G auge Invariance and Vacuum Pol ari zati on. ' T he w hol e poi nt i s that i fyou have a propagati on functi on,i thasa certai n si ngul ari ty w hen the two poi nts coi nci de. Suppose you pretend that there are several parti cl es of the sam e type w i th di erent m asses and w i th coupl i ng constants w hi ch can suddenl y becom e negati ve i nstead of posi ti ve. T hen,ofcourse,you can cancelthem . It' s cancel l ati on agai n,subtracti on physi cs,done i n a m ore sophi sti cated way,but sti l l ,thi ngs m ust be m ade to add up to zero. W ho needs i t? In thi spaper,Schw i ngerobtai ned a cl osed form forthe el ectron propagatori n an externalm agneti c el d,by sol vi ng proper-ti m eequati onsofm oti on, openi ng a el d w hi ch woul d befashi onabl enearl y threedecadesl aterw i th the di scovery ofpul sars; gave the de ni ti ve deri vati on of the Eul er-H ei senberg Lagrangi an descri bi ng the scatteri ng of l i ght by l i ght, a phenom enon sti l l not observed di rectl y;and gave the preci se connecti on between axi al -vector and pseudoscal ar m eson theori es, w hat becam e know n as the axi al -vector anom al y w hen i t was redi scovered nearl y two decades l ater by A dl er,Bel l , and Jacki w . 59 (W e w i l ldi scuss thi sthe anom al y l ateri n Sec. 17. ) T he paper i s not onl y a thi ng ofgreat beauty, but a powerfulstorehouse of practi cal techni que for sol vi ng gauge-theory probl em s i n a gauge-i nvari ant way. H arvard and Schw inger So i twasno surpri se thati n the l ate 1940sand earl y 1950sH arvard wasthe center ofthe worl d,as far as theoreti calphysi cs was concerned. Everyone, students and professors al i ke, ocked to Schw i nger' s l ectures. Everythi ng was reveal ed,l ong before publ i cati on; and not i nfrequentl y others recei ved the credi t because of Schw i nger' s rel uctance to publ i sh before the subject was ri pe. A case i n poi nt i s the so-cal l ed Bethe-Sal peter equati on, 60 w hi ch as G el l -M ann and Low noted, 61 \ rst appeared i n Schw i nger' s l ectures at H arvard. " A t any one ti m e, Schw i nger had ten or twel ve Ph. D .students, w ho typi cal l y saw hi m butrarel y. In part,thi swas because he was avai l abl e to see hi s l arge ock but one afternoon a week, but m ost saw hi m onl y w hen absol utel y necessary, because they recogni zed that hi s ti m e was too val uabl eto bewasted on tri vi alm atters.A studentm ay haveseen hi m onl y a handfulofti m esi n hi sgraduatecareer,butthatwasal lthestudentrequi red. W hen adm i tted to hi s sanctum , students were never rushed, were l i stened to w i th respect,treated w i th ki ndness, and gi ven i nspi rati on and practi cal advi ce. O ne m ustrem em berthatthe student' sprobl em swere typi cal l y qui te unrel ated to w hat Schw i nger hi m sel f was worki ng on at the ti m e; yet i n a few m om ents,he coul d com e up w i th am azi ng i nsi ghts that woul d keep the studentgoi ng forweeks,i fnotm onths.A few studentsgotto know Schw i nger fai rl y wel l ,and were i nvi ted to the Schw i ngers'house occasi onal l y;butm ost saw Schw i nger pri m ari l y as a vi rtuoso i n the l ecture hal l ,and now and then i n hi so ce.A few facul ty m em berswere a bi tm ore i nti m ate,butessenti al l y Schw i nger was a very pri vate person. C ustodian of F ield T heory Feynm an l eft the el d ofquantum el ectrodynam i cs i n 1950,regardi ng i t as essenti al l y com pl ete. Schw i nger never di d. D uri ng the next fteen years, he conti nued to expl ore quantum el d theory,tryi ng to m ake i t revealthe secrets of the weak and strong i nteracti ons. A nd he accom pl i shed m uch. In studyi ng the rel ati vi sti c structure of the theory, he recogni zed that al l the physi cal l y si gni cant representati ons of the Lorentz group were those thatcoul d bederi ved from the\attached" four-di m ensi onalEucl i dean group, w hi ch i s obtai ned by l etti ng the ti m e coordi nate becom e i m agi nary. 62 T hi s i dea was ori gi nal l y ri di cul ed by Paul i , but i t was to prove a m ost frui tful suggesti on. R el ated to thi s was the C PT theorem , rst gi ven a proof for i nteracti ng system s by Schw i nger i n hi s \Q uanti zed Fi el d" papers of the earl y 1950s,and el aborated l ater i n the decade. 63 By the end ofthe 1950s,Schw i nger,w i th hi sform erstudentPaulM arti n, was appl yi ng el d theory m ethods of m any-body system s, w hi ch l ed to a revol uti on i n that el d. 64 M ethods sui tabl e for descri bi ng system s out of equi l i bri um , usual l y associ ated w i th the nam e ofK el dysh, 65 were obtai ned som e fouryearsearl i erby Schw i nger. 66 A l ong the way,i n w hathe consi dered rather m odest papers, he di scovered Schw i nger term s, 67 anom al i es i n the com m utati on rel ati ons between el d operators,and the Schw i nger m odel , 68 sti l lthe onl y know n exam pl e ofdynam i calm ass generati on. T he begi nni ngs of a quantum el d theory for non-A bel i an el ds was m ade; 69 the ori gi nal exam pl e ofa non-A bel i an el d bei ng that ofthe gravi tati onal el d,he l ai d the groundwork for l ater canoni calform ul ati ons ofgravi ty. 70 M easurem ent A lgebra In 1950 or so, as we m enti oned, Schw i nger devel oped hi s acti on pri nci pl e, w hi ch appl i es to any quantum system , i ncl udi ng nonrel ati vi sti c quantum m echani cs. T wo years l ater,he reform ul ated quantum ki nem ati cs,i ntroduci ng sym bol sthatabstracted the essenti alel em entsofreal i sti c m easurem ents. T hi s was m easurem ent al gebra,w hi ch yi el ded conventi onalD i rac quantum m echani cs. But al though the resul t was as expected,Schw i nger saw the approach as ofgreat val ue pedagogi cal l y,and as provi di ng a i nterpretati on of quantum m echani cs that was sel f-consi stent. H e taught quantum m echani cs thi sway form any years,starti ng i n 1952 atthe LesH ouchessum m erschool ; butonl y i n 1959 di d hestartw ri ti ng a seri esofpapersexpoundi ng them ethod to the worl d. H e al ways i ntended to w ri te a de ni ti ve textbook on the subject,but onl y an i ncom pl ete versi on based on the Les H ouches l ectures ever appeared duri ng hi s l i feti m e. 71 Engl ert has now put hi s l ater undergraduate U C LA l ectures together i n a l ovel y book publ i shed by Spri nger. 72 O Butby thatpoi nt hi s extraordi nary com m and ofthe m achi nery ofquantum el d theory had convi nced hi m that i t was too el aborate to descri be the real worl d, at l east di rectl y. In hi s N obel Lecture, he appeal ed for a phenom enol ogi cal el d theory that woul d m ake i m m edi ate contact w i th the parti cl es experi enci ng the strong i nteracti on. W i thi n a year, he devel oped such a theory,Source T heory. E lectrow eak Synthesis Source T heory and U C LA E ngineering A pproach to P article T heory In 1969 Schw i ngergavetheStanl ey H .K l osk l ecture to theN ew York U ni versi ty SchoolofEngi neeri ng Sci ence. Because thatl ecture captureshi sphi l osophy underpi nni ng source theory so wel l ,atan earl y stage i n the devel opm ent ofthat approach,Iquote the transcri pti on ofi t i n ful l . 87 It i s a fam i l i ar si tuati on i n physi cs that w hen an experi m entaldom ai n i s to be codi ed,even though a fundam entaltheory m ay be avai l abl e,rarel y i s i t brought di rectl y to bear upon the experi m ental m ateri al . T he fundam ental theory i s too com pl icated,general l y too rem ote from the phenom ena that you want to descri be. Instead, there i s al ways an i nterm edi ate theory, a phenom enol ogi caltheory,w hi ch i s desi gned to dealdi rectl y w i th the phenom ena,and therefore m akes use ofthe l anguage ofobservati on.O n the otherhand,i ti sa genui ne theory,and em pl oys abstract concepts that can m ake contact w i th the fundam ental theory. T he true rol e of fundam ental theory i s not to confront the raw data, but to expl ai n the rel ati vel y few param eters of the phenom enol ogi caltheory i n term sofw hi ch the greatm assofraw data has been organi zed. Il T he engi neer' si nterm edi ate phenom enol ogi caltheory l ooksi n both di recti ons. Itcan beconnected to thefundam entaltheory at oneend,and attheotheri ti sappl i ed di rectl y to theexperi m ental data. T hi s i s an exam pl e of the engi neeri ng atti tude. It i s a pragm ati c approach that i s desi gned speci cal l y for use. It i s a nonspecul ati ve procedure. H ypotheses that go beyond w hat i s rel evant to the avai l abl e data are avoi ded. N ow ,w hen we com e to real m ofhi gh-energy physi cs,we are i n a new si tuati on. W e do not know the underl yi ng dynam i cs, the underl yi ng fundam entaltheory. T hat rai ses the questi on of ndi ng the beststrategy. T hati s,w hati s the m oste ecti ve way ofconfronti ng the data,oforgani zi ng i t,ofl earni ng l essons from resul ts w i thi n a l i m i ted dom ai n ofexperi m entalm ateri al ? Iwantto argue thatwe shoul d adopta pragm ati c engi neeri ng approach. W hat we shoul d not do i s try to begi n w i th som e fundam ental theory and cal cul ate. A s we saw , thi s i s not the best thi ng to do even w hen you have a fundam entaltheory,and i fyou don' t have one,i t' s certai nl y the w rong thi ng to do. H i stori cal l y,rel ati vi sti c quantum m echani cs had proved very successfuli n expl ai ni ng atom i c and nucl ear physi cs unti lwe got accel eratorssu ci entl y hi gh i n energy to createthestrongl y i nteracti ng parti cl es,w hi ch i ncl ude parti cl es that are hi ghl y unstabl e and decay through very strong forces.T heordi nary m ethodsthat had evol ved up to thi s poi ntwere si m pl y powerl ess i n the face of thi snew si tuati on. A tthe hi gher energi es,parti cl es can be| and are| created and destroyed w i th hi gh probabi l i ty. In otherwords,thei m m utabi l i ty oftheparti cl e| a foundati on ofordi nary physi cs| had di sappeared. Ifthei m m utabl e parti cl e hasceased to exi stasthefundam entalconcept i n term s ofw hi ch a si tuati on can be descri bed,w hat do we repl ace i t w i th? T here have been two di erent poi nts of vi ew abouthow to constructa fundam entaltheory forthe strong i nteracti ons. T he rst| the poi nt of vi ew of conventi onal operator el d theory| proposes to repl ace the parti cl e w i th three-di m ensi onal space i tsel f. In other words, you thi nk of energy, m om entum , el ectri c charge, and other properti es as di stri buted throughout space, and of sm al l vol um es of three-di m ensi onal space as the thi ngs that repl ace parti cl es. T hese vol um es are the carri ers of energy,m om entum ,and so on. Peopl e, i ncl udi ng m ysel f, have been acti vel y devel opi ng the el d i dea form any years.Ibel i evethatthi ski nd oftheory m ay be the ul ti m ate answer,butpl ease recogni ze thati ti sa specul ati on. Itassum esthatonei si ndeed abl eto descri bephysi calphenom ena dow n to arbi trari l y sm al ldi stance,and,ofcourse,that goes far beyond anythi ng we know at the m om ent. A l lwe are abl e to do experi m ental l y aswe go to hi gherand hi gherenergi esi sto pl um b to sm al l er and sm al l er di stances,but never to zero di stance. T hequesti on i s,shoul d you,i n di scussi ng thephenom ena that arepresentl y know n,m akeuseofa specul ati ve hypothesi sl i keoperator el d theory? C an we not di scuss parti cl e phenom enol ogy and handl e the correl ati onsand organi zati on ofdata w i thoutbecom i ng i nvol ved i n a specul ati ve theory? In operator el d theory you cannotseparate parti cl e phenom ena from specul ati onsabout the structure ofparti cl es. T he operators ofquantum -m echani cal el d theory conceptual l y m i x these together. To be abl e to di scuss anythi ng from the operator-el d-theory poi nt ofvi ew ,you m ust accept i ts fundam entalhypothesi s. You have to accepta specul ati on abouthow parti cl es are constructed before you can begi n to di scuss how parti cl es i nteract w i th each other. H i stori cal l y,thi s has proved to be a very di cul t program to appl y,and peopl e have,ofcourse,been anxi ous to dealdi rectl y w i th theexperi m entaldata,and so therehasbeen a reacti on.T he extrem e reacti on to operator el d theory i sto i nsi st thatthere i s nothi ng m orefundam entalthan parti cl esand that,w hen you have a num ber ofparti cl es col l i di ng w i th each other and the num ber ofparti cl esceasesto be constant,al lyou can do i scorrel ate w hat com es i nto a col l i si on w i th w hat goes out,and cease to descri be i n detai lw hat i s happeni ng duri ng the course ofthe col l i si on. T hi s poi nt ofvi ew i s cal l ed S-m atri x theory. T he quanti tati ve descri pti on i s i n term s ofa scatteri ng m atri x that connects the outgoi ng state w i th the i ncom i ng state. In thi s theory the parti cl es are basi c and cannot be anal yzed. T hen,ofcourse,the questi on com es up: w hat di sti ngui shes the parti cul ar set ofparti cl es that do exi st from any other concei vabl e set? T he onl y answerthathasbeen suggested i sthatthe observed parti cl es exi st as a consequence ofsel f-consi stency. G i ven a certai n set ofparti cl es,other parti cl es can be form ed as aggregates orcom posi tes ofthese. O n the other hand,i fparti cl es are unanal yzabl e,then thi s shoul d not be a new set ofparti cl es,but the very parti cl es them sel ves. T hat i s the second i dea, but I beg you to appreci ate that i t i s al so a specul ati on. W e do not know for a fact that our present i nabi l i ty to descri be thi ngs i n term s ofsom ethi ng m ore fundam entalthan parti cl es re ects an i ntri nsi c i m possi bi l i ty. So these are the two pol ari zed extrem es i n the search for a fundam entaltheory| the operator-el d-theory poi ntofvi ew and the S-m atri x poi nt ofvi ew . N ow m y reacti on to al lofthi s i s to ask agai n w hy we m ust specul ate,si nce the probabi l i ty offal l i ng on the ri ght specul ati on i s very sm al l . C an we notseparate the theoreti calprobl em ofdescri bi ng the properti es ofthese parti cl es from specul ati ons about thei r i nner structure? C an we notsetasi de the specul ati on ofw hetherparticl es are m ade from operator el ds orare m ade from nothi ng but them sel ves,and nd an i nterm edi ate theory,a phenom enol ogi cal theory thatdi rectl y confrontsthe data,butthati ssti l la creati ve theory? T hi s theory shoul d be su ci entl y exi bl e so thati tcan m ake contact w i th a future,m ore fundam entaltheory ofthe structure ofparti cl es,i fi ndeed any m ore fundam entaltheory everappears. T hi s i s the l i ne ofreasoni ng that l ed m e to consi der the theoreti calprobl em for hi gh-energy physi cs from an engi neeri ng poi nt ofvi ew . C l earl y I have som e i deas i n m i nd about how to carry out such a program ,and Iwoul d l i ke to gi ve you an enorm ousl y si m pl i ed account ofthem . W e want to el i m i nate specul ati on and take a pragm ati c approach. W e are not goi ng to say that parti cl es are m ade out of el ds,or that parti cl es sustai n each other. W e are si m pl y goi ng to say that parti cl es are w hat the experi m ental i sts say they are. Butwe w i l lconstructa theory and notan experi m enter' sm anual i n thatwe w i l ll ook atreal i sti c experi m entalproceduresand pi ck out thei r essence through i deal i zati ons. T here i sone characteri sti c thatthe hi gh-energy parti cl eshave i n com m on| they m ust be created. T hrough the act ofcreati on, we can de ne w hat we m ean by a parti cl e. H ow ,i n fact,do you create a parti cl e? By a col l i si on. T he experi m ental i st arranges for a beam ofparti cl es to fal lon a target. In the center-of-m ass system ,the target i s just another beam ,so two beam s ofparticl es are col l i di ng. O ut ofthe col l i si on,the parti cl e that we are i nterested i n m ay be produced. W e say that i t i s a parti cl e rather than a random bum p on an exci tati on curve because i ts properti es are reproduci bl e. W e sti l lrecogni ze the sam e parti cl e event though we vary a num ber ofexperi m entalparam eters,such as energy,angl es,and the ki nd ofreacti on. T he properti esofthe parti cl e i n questi on rem ai n the sam e| i t has the sam e m ass,the sam e spi n,the sam e charge. T hese cri teri a can be appl i ed to an object that m ay l ast for onl y 10 24 sec| w hi ch decayseven beforei tgetsoutofthenucl eus i n w hi ch i twas created. N everthel ess,i ti ssti l lusefulto cal lthi s ki nd ofobjecta parti cl e because i tpossesses essenti al l y al lofthe characteri sti cs that we associ ate w i th the parti cl e concept. W hat i s si gni cant i s that w i thi n a som ew hat control l abl e regi on ofspace and ti m e,the properti escharacteri sti c ofthe parti cl e have becom e transform ed i nto the parti cl e i tsel f. T he other parti cl es i n the col l i si on are there onl y to suppl y the net bal ance ofproperti es. T hey are i deal i zed as the source ofthe parti cl e. T hi s i s our new theoreti calconstruct. W e i ntroduce a quanti tati ve descri pti on of the parti cl e source i n term s of a source functi on,S(x),w here x refersto the space and ti m e coordi nates. T hi s functi on i ndi cates that,to som e extent,we can controlthe regi on the parti cl e com es from . Butwe do nothave to cl ai m thatwe can m ake the source arbi trari l y sm al lasi n operator el d theory. W e l eave thi squesti on open,to be tested by future experi m ent. A parti cul ar source m ay be m ore e ecti ve i n m aki ng particl es that go i n one di recti on rather than another,so there m ust be another degree of control expressed by a source functi on of m om entum , S(p). But from quantum m echani cs we know that the di m ensi on ofthe system and the degree ofdi recti onal i ty are cl osel y rel ated. T he sm al l er the system , the l ess di recti onal i t can be. A nd rel ati vi sti c m echani cs i s i ncorporated from the very begi nni ng i n that the energy and the m om entum are rel ated to i ts m ass i n the usualrel ati vi sti c way. N ow the experi m enter' s job onl y begi ns w i th the producti on ofa beam . A tthe otherend,he m ust detect the parti cl es. W hat i s detecti on? U nstabl e parti cl es eventual l y decay,and the decay processi sa detecti on devi ce.M oregeneral l y,any detecti on devi ce can be regarded asa ki nd ofcol l i si on thatanni hi l atesthe parti cl e and hands i ts properti es on i n a m ore usabl e form . T hus the source concept can agai n be i ntroduced as an abstracti on ofan anni hi l ati on col l i si on,w i th thesource acti ng negati vel y,asa si nk. W e now have a com pl ete theoreti calpi cture ofany physi cal si tuati on,i n w hi ch sources are used to create the i ni ti alparti cl e ofi nterest from the vacuum state,and sources are used to detect the nalparti cl esresul ti ng from som e i nteracti on,thusreturni ng to the vacuum state. [ Schw i nger then w rote dow n an expressi on that descri bes the probabi l i ty am pl i tude that the vacuum state before sources act rem ai ns the vacuum state after sources act,the vacuum persi stence am pl i tude. ] T he basi c thi ngsthatappeari n thi sexpressi on are the source functi ons and space-ti m e functi ons that represent the state i nto w hi ch the parti cl e i s em i tted and from w hi ch i t i s absorbed,thus descri bi ng the i nterm edi ate propagati on ofthe parti cl e. T hi ssi m pl eexpressi on can begeneral i zed to appl y to parti cl es that have charge,spi n,etc. ,and to si tuati ons w here m ore than one parti cl e i s present at a ti m e. Interacti ons between parti cl es are descri bed i n term s contai ni ng m ore than two sources. O ur starti ng poi nt accepts parti cl es as fundam ental | we use sources to i denti fy the parti cl es and to i ncorporate a si m pl i ed vi ew ofdynam i cs. From thatwe evol ve a m ore com pl ete dynami caltheory i n w hi ch we com bi ne si m pl e source arrangem ents l i ke bui l di ng bl ocks to produce descri pti ons ofsi tuati ons that can i n pri nci pl e be as com pl ex as we want. A rst test ofthi s approach woul d be to see i fwe can reproducetheresul tsofsom ewel l -establ i shed theory such asquantumel ectrodynam i cs. W hat i s the starti ng poi nt i n thi s attack on el ectrodynam i cs? It i s the photon,a parti cl e that we know has certai n stri ki ng properti es such as zero rest m ass and hel i ci ty 1. So we m ust i ncl ude al l these aspects of the photon i n the pi cture, and descri be how photons are em i tted and absorbed. In consequence,the source m ust be a vector,and i t m ust be di vergencel ess. T hi s approach l eads us to som ethi ng resem bl i ng a vector potenti al ,and w hen we ask w hat di erenti alequati ons i t sati s es we nd they are M axwel l ' s equati ons. W e startw i th the concept of the source as pri m ary and are l ed to M axwel l ' s di erenti al equati ons as deri ved concepts. T he descri pti on of i nteracti ons fol l ow s the tentati ve procedures of l i fe i n the real worl d. T he theory i s not stated once and for al l . It begi ns w i th si m pl e phenom ena| for exam pl e,accel erated charges radi ate. It then extrapol ates that i nform ati on outsi de i ts dom ai n,predi cts m ore com pl i cated phenom ena, and awai ts the test ofexperi m ent. W e do not begi n w i th a nalde-scri pti on of,say,el ectron scatteri ng. W e extrapol ate to i t from m ore el em entary si tuati ons,and thi si ssti l lnotthe naldescri pti on. A s the theory devel ops and becom es m ore encom passi ng,we go back to re ne the descri pti on of the scatteri ng process and obtai n a m ore quanti tati ve account of i t. A m orefundam entaltheory m ay com e i nto bei ng one day,but i tw i l lbetheoutcom eofconti nued experi m entalprobi ng to hi gher energi es,and w i l ldoubtl ess i nvol ve theoreti calconcepts that are now onl y di m l y seen. Butthatday w i l lbe greatl y speeded i fthe ood ofexperi m entalresul ts i s organi zed and anal yzed w i th the ai d ofa theory that does not have bui l t i nto i t a preconcepti on about the very questi on that i s bei ng attacked. T hi s theory i s source theory. 16.2 T he Im pact of Source T heory coupl i ng, w hi ch form al l y appeared be equi val ent, but cal cul ati ons i n the 1940s gave di screpant answers. Schw i nger resol ved thi s i ssue i n 1950 by show i ng that the two theori es were i ndeed equi val ent provi ded that proper care (gauge-i nvari ance) was used, and that the form alresul t was m odi ed by an addi ti onalterm . Probl em sol ved, and i t was then forgotten for the next 18 years. In the l ate 1960s A dl er, Bel l , and Jacki w redi scovered thi s sol uti on, 59 but the l anguage was a bi t di erent. T he extra term Schw i nger had found was now cal l ed an anom al y,but the form ofthe equati ons,and the predi cti on for the decay ofthe pi on,were i denti cal . In fact,at rst i t i s apparent that A dl er,Bel l ,and Jacki w were unaware ofSchw i nger' s m uch earl i er resul t, and i t was the addi ti on ofK en Johnson (one of Schw i nger' s m any bri l l i ant students) i nto the col l aborati on that corrected the hi stori cal record. 98 Shortl y thereafter,A dl er and Bardeen proved the \nonrenorm al i zati on" theorem , 99 that the anom al y i s exact,and i s not corrected by hi gher-order quantum e ects. T hi si si n contrastto m ostphysi calphenom ena,such asthe anom al ous m agneti c m om ent ofthe el ectron,w hi ch i s subject to correcti ons i n al lorders of perturbati on theory i n the strength of the el ectrom agneti c coupl i ng,the ne structure constant. T hi s seem ed surpri si ng to Schw i nger, so he suggested to hi s postdocs at U C LA that they work thi s out i ndependentl y,and they di d,publ i shi ng two papersi n 1972, 100 i n w hi ch they showed, usi ng two i ndependent m ethods,that there was i ndeed such a correcti on i n hi gher order. H owever,A dl er,w ho was the revi ewer ofthese papers forced them to tone dow n thei rconcl usi on,and to poi ntoutthatthe resul tdepends on the physi calpoi ntatw hi ch the renorm al i zati on i scarri ed out. N onrenorm al i zati on i ndeed can beachi eved by renorm al i zati on atan unphysi calpoi nt, w hi ch m ay be acceptabl e for the use of the theorem i n establ i shi ng renorm al i zabi l i ty ofgauge theori es,i tschi efappl i cati on,buti ti sneverthel esstrue that physi cal processes such as the ori gi nalprocess of pi on decay recei ves hi gher-order correcti ons. A s was typi cal , Schw i nger apparentl y took no noti ce of thi s di spute at the ti m e. But toward the end ofthe 1970s,w hi l e he was w ri ti ng the thi rd vol um e ofParticl es,Sources,and Fiel ds,he l ooked atthe questi onsofradi ati ve correcti ons to neutralpi on decay and found the sam e resul tasD eR aad, M i l ton,and T sai . H e w rote an expl i ci tl y confrontati onalpaper on the subject,w hi ch was the basi s for the above-m enti oned tal k at M IT .T he paper wasapparentl y de ni ti vel y rejected,and the tal k washarshl y cri ti ci zed,and on the basi s of these cl osed-m i nded attacks, Schw i nger l eft the el d. For-tunatel y for the record,Schw i nger' s paper exi sts as a chapter i n the nal l y publ i shed thi rd vol um e ofParticl es,Sources,and Fiel ds. H owever,the controversy l i ves on. In 2004 Steve A dl er w rote a hi storicalperspecti ve on hi s work on the axi al -vector anom al y. 101 H e devotes ve pagesofhi sretrospecti ve to attack the work ofSchw i nger and hi sgroup.H e even deni es thatSchw i nger wasthe rstto cal cul ate the anom al y,i n bl atant di sregard ofthe hi stori calrecord. O fcourse,physi calunderstandi ng had i ncreased i n the nearl y two decades between Schw i nger' s and A dl er' s papers, butto deny thatSchw i ngerwasthe rstperson to o erthe basi sforthe connecti on between the axi al -vector and pseudoscal ar currents,and the ori gi n ofthe photoni c decay ofthe neutralpi on,i s preposterous. T hom as-Ferm i A tom , C old Fusion, and Sonolum inescence 104 Fol l ow i ng the T hom as-Ferm i work, Schw i nger conti nued to col l aborate w i th Engl ert,and w i th M arl an Scul l y,on the questi on ofspi n coherence. If an atom i cbeam i sseparated i nto sub-beam sby a Stern-G erl ach apparatus,i s i tpossi bl e to reuni te thebeam s? Scul l y had argued thati tm i ghtbepossi bl e, but Jul i an was skepti cal ;the resul t was three joi nt papers,enti tl ed \Is Spi n C oherence Li ke H um pty D um pty?",w hi ch bore outJul i an' s i ntui ti on ofthe i m possi bi l i ty ofbeati ng the e ects ofquantum entangl em ent. 105 In M arch 1989 began one ofthe m ostcuri ousepi sodesi n physi calsci ence i n the l ast century, one that i ni ti al l y attracted great i nterest am ong the sci enti c as wel las the l ay com m uni ty,but w hi ch rapi dl y appeared to be a characteri sti c exam pl e of \pathol ogi calsci ence. " yy T he e ect to w hi ch we refer was the announcem ent by B. S. Pons and M . Fl ei schm ann 107 of the di scovery ofcol d fusi on. T hat i s,they cl ai m ed that nucl ear energy, i n the form ofheat,was rel eased i n a tabl e-top experi m ent,i nvol vi ng a pal l adi um cathode el ectrol yzi ng heavy water. So i twasa shock to m ostphysi ci sts zz w hen Schw i ngerbegan speaki ng and w ri ti ng aboutcol d fusi on,suggesti ng thatthe experi m entsofPonsand Fl ei schm ann were val i d,and that the pal l adi um l atti ce pl ayed a cruci alrol e. In one ofhi sl aterl ectures on the subject i n Sal tLake C i ty,Schw i nger recal l ed, \A partfrom a bri efperi od ofapostasy,w hen Iechoed the conventi onalw i sdom that atom i c and nucl ear energy scal es are m uch too di sparate,I have retai ned m y bel i ef i n the i m portance of the l atti ce. " 5 H i s rst publ i cati on on the subject was subm i tted to PhysicalReview Letters,but was roundl y rejected,i n a m annerthatSchw i ngerconsi dered deepl y i nsul ti ng. In protest, he resi gned asa m em ber(he was,ofcourse,a fel l ow )ofthe A m eri can Physi -calSoci ety,ofw hi ch PhysicalReview Letters i sthe m ostpresti gi ousjournal . (A t rsthe i ntended m erel y to w i thdraw thepaperfrom PR L,and hi sfel l owshi p,but then he fel t com pel l ed to respond to the referees'com m ents: O ne com m ent was som ethi ng to the e ectthatno nucl ear physi ci st coul d bel i eve such an e ect,to w hi ch Jul i an angri l y retorted,\Iam a nucl earphysi ci st! " 5 ) In thi s l etter to the edi tor (G .W el l s) i n w hi ch he w i thdrew the paper and resi gned from the A m eri can Physi calSoci ety,he al so cal l ed for the rem oval of the source theory i ndex category the A PS journal s used: \Inci dental l y, yy T hi s term was coi ned i n 1953 by Irvi ng Langm ui r,w ho gave a cel ebrated l ecture at G eneral El ectri c' s K nol l s A tom i c Power Laboratory (transcri bed from a di sc recordi ng by R obert H al l ) on the phenom enon w herei n reputabl e sci enti sts are l ed to bel i eve that an e ect, just at the edge of vi si bi l i ty, i s real , even though, as preci si on i ncreases, the e ect rem ai ns m argi nal . T he sci enti st becom es sel f-del uded, goi ng to great l engths to convi nce one and al lthat the rem arkabl e e ect i s there just on the m argi ns ofw hat can be m easured. G reat accuracy i s cl ai m ed neverthel ess,and fantasti c,ad hoc,theori es are i nvented to expl ai n the e ect. Exam pl esi ncl ude N -rays,the A l l i son e ect, yi ng saucers, and ESP. It was not a coi nci dence that Physics Today publ i shed the arti cl e, w i thout com m ent,i n the fal lof1989. 106 zz H owever, a few other em i nent physi ci sts spoke favorabl y of the possi bi l i ty of col d fusi on,notabl y Edward Tel l er and W i l l i s Lam b,w ho publ i shed three arti cl es i n the Proceedings ofthe U .S.N ationalAcadem y ofSciences on the subject. the PA C S entry (1987) 11. 10. m n can be del eted. T here w i l lbe no further occasi on to use i t. ' ; 5,108 A ratherstri ki ng actofhubri s:Ifhe coul dn' tpubl i sh source theory,nei ther coul d anybody el se. But the PhysicalReview obl i ged. (U nfortunatel y,Schw i nger fai l ed to real i ze that the PA C S i ndex system has becom e the predom i nant system for physi cs journal s worl dw i de,a re ecti on of the prem i er status of the A PS journal s. So he l argel y spi ted hi s ow n contri buti ons. ) N ot w i shi ng to use any other A PS venue,he turned to hi s fri end and col l eague,Berthol d Engl ert,w ho arranged that \C ol d Fusi on: A H ypothesi s" be publ i shed i n the Zeitschriftf ur N aturforschung,w here i tappeared i n O ctober of that year. 109 Schw i nger then went on to w ri te three substanti alpapers,enti tl ed \N ucl earEnergy i n an A tom i c Latti ce I,II,III, " to esh out these i deas. 5,110 T he rst was publ i shed i n the Zeitschrift f ur Physik D , 111 w here i twasaccepted i n spi te ofnegati ve revi ew s, 5 butdi rectl y preceded by an edi tori alnote,di scl ai m i ng any responsi bi l i ty for the the paper on the part of the journal . T hey subsequentl y refused to publ i sh the rem ai ni ng papers. Schw i nger' s l ast physi cs endeavor m arked a return to the C asi m i r e ect, ofw hi ch he had been enam ored nearl y two decadesearl i er.Itwassparked by therem arkabl e di scovery ofsi ngl e-bubbl e sonol um i nescence,i n w hi ch a sm al l bubbl e ofai ri n water,dri ven by a strong acousti c standi ng wave,undergoes a stabl e cycl e of col l apse and re-expansi on; at m i ni m um radi us an i ntense ash ofl i ght,consi sti ng ofa m i l l i on opti calphotons,i s em i tted. It was not coi nci dentalthat the l eadi ng l aboratory i nvesti gati ng thi s phenom enon was, and i s, at U C LA , l ed by erstw hi l e theori st Seth Putterm an, l ong a fri end and con dant. Putterm an and Schw i nger shared m any i nterests i n com m on, i ncl udi ng appreci ati on of ne w i nes,and they shared a si m i l ar i conocl asti c vi ew of the decl i ne of physi cs. So, of course, Schw i nger heard about thi s rem arkabl e phenom enon from the horse' sm outh,and wasgreatl y i ntri gued. Schw i nger i m m edi atel y had the i dea that a dynam i cal versi on of the C asi m i re ectm i ghtpl ay a key rol e.H esaw paral l el sbetween col d fusi on and sonol um i nescence i n that both dealw i th seem i ngl y i ncom m ensurate energy scal es,and both depend si gni cantl y on nonl i neare ects. Si nce by the earl y 1990s,col d fusi on was l argel y di scredi ted,he put hi s e orts to understandi ng sonol um i nescence,w hi ch undoubtedl y does exi st. U nfortunatel y nei ther Schw i nger,nor anyone subsequentl y, was abl e to get very far w i th dynam -Fora revi ew ofthe phenom ena,and a detai l ed eval uati on ofvari oustheoreti calexpl anati ons,see R ef. [ 112] . i calzero-poi nt phenom ena; he l argel y contented hi m sel f w i th an adi abati c approxi m ati on based on stati c C asi m i r energi es;and was abl e to obtai n sufci ent energy onl y because he retai ned the \bul k energy, " w hi ch m ost now bel i eve i s unobservabl e,bei ng subsum ed i n a renorm al i zati on ofbul k m ateri alproperti es. H i s work on the subject appeared as a seri es ofshort papers i n the PN A S,the l ast appeari ng 113 shortl y after hi s death i n June 1994. Iti s tti ng to cl ose thi s retrospecti ve w i th Schw i nger' s ow n words,del i vered som e si x m onths before hi s nali l l ness,w hen he recei ved an honorary degree from the U ni versi ty ofN otti ngham . 114y T he D egree C erem ony i s a m odern versi on ofa m edi evalri te that seem ed to confer a ki nd of pri esthood upon i ts reci pi ents, thereby excl udi ng al lothers from i ts i nner ci rcl e. But that w i l l y T hi s bri ef acceptance speech was fol l owed by a bri l l i ant l ecture on the i n uence of G eorge G reen on Schw i nger' s work. 114 not do for today. Sci ence, w i th i ts o shoot of Technol ogy, has an overw hel m i ng i m pact upon al l of us. T he recent events at W i m bl edon i nvi te m e to a som ew hat outrageous anal ogy. Very few of us, i ndeed, are qual i ed to step onto centre court. Yet thousands ofspectators gai n great pl easure from watchi ng these tal ented speci al i sts perform . Som ethi ng si m i l ar shoul d be, but general l y i s not,true forthe rel ati onshi p between the practi ti oners ofSci ence and the generalpubl i c. T hi s i s m uch m ore seri ous than not know i ng the di erence between 30 al land deuce. Science,on a bi g scal e,i s i nevi tabl y i ntertw i ned w i th pol i ti cs. A nd pol i ti ci anshavel i ttl epracti cei n di sti ngui shi ng between,say comm on l aw and N ew ton' s l aw . Iti s a sui tabl y educated publ i c that m ust step i nto the breach. T hi s has been underl i ned l atel y by M i ni ster W al degrave' s cry forsom eone to educate hi m aboutthe properti es of the H i ggs boson, to be rewarded w i th a bottl e of cham pagne. A ny m em ber ofthe educated publ i c coul d have tol d hi m thattheci ted parti cl ei san arti factofa parti cul artheoreti cal specul ati on,and the realchal l enge i s to enter uncharted waters to see w hati sthere. T he fai l ure to do thi sw i l li nevi tabl y putan end to Sci ence. A soci ety thatturn i n on i tsel fhassow n the seeds ofi tsow n dem i se. Earl y i n the 16th century,powerfulC hi na had sea-goi ng vessel s expl ori ng to the west. T hen a si gnalcam e from new m astersto return and destroy theshi ps.Itwasi n thoseyears that Portuguese sai l ors entered the Indi an O cean. T he outcom e was 400 years ofdom i nance ofthe East by the W est. C onclusion T here areotherthreatsto Sci ence. A recentbestsel l eri n Engl and,U nderstanding the present,hasthe subti tl e Science and the soulofM odern M an. Ishal lonl y touch on the w ri ter' s vi ew s toward quantum m echani cs,surel y the greatest i ntel l ectualdi scovery ofthe 20th century. Fi rst,he com pl ai nsthatthe new physi cs of quantum m echani cs tosses cl assi calphysi cs i n the trash bi n. T hi s I woul d di sm i ss as m ere techni cal i gnorance; the m anner i n w hi ch cl assi caland quantum m echani cs bl end i nto each other has l ong been establ i shed. Second,the author i s upset that i ts theori es can' t be understood by anyone not m athem ati cal l y sophi sti cated and so m ustbe accepted by m ostpeopl e on fai th.H e i s,i n short,sayi ng thatthere i sa pri esthood. A gai nstthi sIpose m y ow n experi ence i n presenti ng the basi c concepts ofquantum m echani cs to a cl ass of A m eri can hi gh school students. T hey understood i t;they l oved i t. A nd I used no m ore than a bi t of al gebra,a bi t ofgeom etry. So: catch them young;educate them properl y; and there are no m ysteri es, no pri ests. It al l com es dow n to a properl y educated publ i c. A cknow ledgem ents Iam greatfulto m any col l eaguesforthei ntervi ew sand conversati onsgranted m e i n w ri ti ng about Jul i an Schw i nger. I am parti cul arl y gratefulto R obert Fi nkel stei n and Edward G erjuoy for conversati ons i n the past few m onths. A gai n I m ust thank C harl otte Brow n, C urator of Speci al C ol l ecti ons at U C LA , for m aki ng the Schw i nger archi ves avai l abl e to m e on m any occasi ons. M y research over the year,not pri m ari l y hi stori cal ,has been funded by grants from the U S D epartm ent ofEnergy and the U S N ati onalSci ence Foundati on. Idedi cate thi s m em oi r to Jul i an' s w i dow ,C l ari ce. [ 3] Larry C ranberg,tel ephone i ntervi ew by K .A .M i l ton,2001. R eferences [ 4] Ei l een F.Lebow ,T he BrightBoys: A H istory ofTownsend H arrisH igh School(W estport C T :G reenwood Press,2000). [ 5] Jul i an Schw i nger Papers (C ol l ecti on 371),D epartm ent ofSpeci alC oll ecti ons,U ni versi ty R esearch Li brary,U ni versi ty ofC al i forni a,LosA ngel es. [ 6] Joseph W ei nberg,tel ephone i ntervi ew by K .A .M i l ton,Jul y 12,1999. [ 7] E.J.Tow nsend,Functions ofRealVariabl es (N ew York: H ol t,1928). i nger sti l lwas derel i ct i n attendi ng cl asses,and ran i nto troubl e i n a chem i stry course taughtby V i ctor LaM er. Itwas a dul lcourse w i th a dul l exam . A questi on on the nalexam was \Prove that d = d + d , " w here none ofthe vari abl es , ,or were de ned. R abirecal l ed, 20 LaM er was, for a chem i st, aw ful l y good. A great part of hi s l i fework was testi ng the D ebye-H uckeltheory 21 ratherbri l l i antl y. But he was thi s ri gi d,reacti onary type. H e had thi s m ean way about hi m . H e sai d,' You have thi s Schw i nger? H e di dn' t pass m y nalexam . ' Isai d,' H e di dn' t? I' l ll ook i nto i t. 'So Ispoke to a num ber ofpeopl e w ho' d taken the sam e course. A nd they had been greatl y assi sted i n that subject by Jul i an. So Isai d,I' l l x thatguy.W e' l lsee w hatcharacterhe has.' N ow V i cky,w hatsort ofguy are you anyway,w hat are your pri nci pl es? W hat' re you goi ng to do aboutthi s?'W el l ,he di d unk Jul i an,and Ithi nk i t' s qui te a badge ofdi sti ncti on forhi m ,and Iforone am notsorry at thi s poi nt,they have thi s bl ack m ark on Jul i an' s rather el evated record. But he di d get Phi Beta K appa as an undergraduate, som ethi ng Inever m anaged to do. 20 k 20Itwasa col d w i nteraswel l ,forhe fai l ed to unpack the trunk i n w hi ch hi sm otherhad pl aced a warm w i nter coat. Ithoughtthathe had abouthad everythi ng i n C ol um bi a thatwe coul d o er| by we,as theoreti calphysi cs i s concerned,[ I m ean] m e. So I got hi m thi s fel l ow shi p to go to W i sconsi n, w i th the general i dea that there were Brei t and W i gner and they coul d carry on. Itwas a di sastrous i dea i n one respect,because,before then,Jul i an was a regul ar guy. Present i n the dayti m e. So I' d ask Jul i an (I' d see hi m from ti m e to ti m e) ' H ow are you doi ng?' ' O h, ne, ne. ' ' G etti ng anythi ng out of Brei t and W i gner?' ' O h yes, they' re very good, very good. ' I asked them . T hey sai d,' W e never see hi m . ' A nd thi s i s m y ow n theory| I' ve never checked i tw i th Jul i an| that| there' sone thi ng aboutJul i an you al lknow | Ithi nk he' san even m ore qui etm an than D i rac.H e i s not a ghter i n any way. A nd I i m agi ne hi s i deas and W i gner' s and Brei t' s orthei r personal i ti es di d notagree. Idon' tfaul thi m for thi s, but he' s such a gentl e soul , he avoi ded the battl e by worki ng at ni ght. H e got thi s i dea ofworki ng ni ghts| i t' s pure theory,i t has nothi ng to do w i th the truth. i nger,and O ppenhei m er were tal ki ng i n O ppenhei m er' s o ce i n LeC onte H al l . T wo other students,C hai m R i chm an and Bernard Peters,cam e i n seeki ng a suggesti on for a research probl em from O ppenhei m er. Schw i nger l i stened w i th i nterest w hi l e O ppenhei m er proposed cal cul ati ng the cross secti on for the el ectron di si ntegrati on ofthe deuteron. T hat m i dni ght,w hen G erjuoy cam e to pi ck up Schw i ngerforthe l atter' sbreakfastbefore thei ral l -ni ghtwork sessi on,he noted thatSchw i nger,w hi l e wai ti ng forhi m i n the l obby ofthe Internati onal H ouse,w hereJul i an wasl i vi ng,had l l ed thebacksofseveraltel egram bl anks w i th cal cul ati onson thi sprobl em . Schw i ngerstu ed the sheetsi n hi spocket and they went to work. Si x m onths l ater, agai n G erjuoy and Schw i nger were i n O ppenhei m er' s o ce w hen R i chm an and Peters returned beam i ng. H e beat out H ans B ethe for the job. ofthe hyper ne anom al y that woul d prove to m ark the exi stence ofan anom al ousm agneti c m om entofthe el ectron,expressi ng thecoupl i ng ofthespi n oftheel ectron to an appl i ed m agneti c el d,devi ati ng from the val ue agai n predi cted by D i rac.Schw i ngeri mm edi atel y saw thi sasthe cruci alcal cul ati on to carry out rst,because i twas purel y rel ati vi sti c,and m uch cl eanerto understand theoreti cal l y,noti nvol vi ng the com pl i cati on ofbound states. H owever,he wasdel ayed three m onths i n begi nni ng the cal cul ati on because ofan extended honeym oon through the W est. H e di d return to i ti n O ctober,and by D ecem ber1947 had obtai ned a resul t 36 com pl etel y consi stentw i th experi m ent. H e al so saw how to com pute the rel ati vi sti c Lam b shi ft(al though he di d nothave the detai l squi te ri ght), and found the error i n the pre-war D anco cal cul ati on ofthe radi ati ve correcti on to el ectron scatteri ng by a C oul om b el d. 37 In e ect,he had sol ved al lthe fundam entalprobl em s thathad pl agued quantum el ectrodynam i cs i n the 1930s:T he i n ni ti es were enti rel y i sol ated i n quanti ti es w hi ch renorm ali zed the m ass and charge ofthe el ectron. Further progress,by hi m sel fand others,wasthusa m atteroftechni que. C oncerni ng Schw i nger' stechni que at the ti m e,Schweber w ri tes 38 T henotesofSchw i nger' scal cul ati on [ oftheLam b shi ft]areextant [ and]gi ve proofofhi s awesom e com putati onalpowers. ...T hese tracesoverphoton pol ari zati onsand i ntegrati onsoverphoton energi es ...were carri ed out fearl essl y and seem i ngl y e ortl essl y. ...O ften,i nvol ved stepswerecarri ed outm ental l y and theanswer wasw ri tten dow n.A nd,m osti m portant,thel engthy cal cul ati ons are error free! 8 C ovariant Q uantum E lectrodynam ics D uri ng the nexttwo yearsSchw i ngerdevel oped two new approachesto quantum el ectrodynam i cs. H i s ori gi nal approach, w hi ch m ade use of successi ve canoni cal transform ati ons, w hi l e su ci ent for cal cul ati ng the anom al ous m agneti c m om ent of the el ectron, was noncovari ant, and as such, l ed to i nconsi stent resul ts. In parti cul ar, the m agneti c m om ent appeared al so as part ofthe Lam b shi ft cal cul ati on,through the coupl i ng w i th the el ectri c el d i m pl i ed by rel ati vi sti c covari ance; but the noncovari ant schem e gave the w rong coe ci ent. (Ifthe coe ci ent were m odi ed by hand to the correct val ue,w hat turned out to be the correct rel ati vi sti c val ue for the Lam b shi ft em erged, but w hat that was was unknow n i n January 1948,w hen he announced hi s resul ts at the A m eri can Physi calSoci ety m eeti ng. ) y K .K .D arrow ,secretary ofthePhysi calSoci ety,w ho apparentl y had l i ttl eappreci ati on of theory, al ways schedul ed the theoreti cal sessi ons i n the sm al l est room . Schw i nger' s second l ecture was gi ven i n the l argest l ecture hal li n Pupi n Lab, and the thi rd i n the l argesttheatre on cam pus. thi sthreerepeats| i t' sa m arvel ousrevol uti on that' sbeen done| LaM er got m ore and m ore i nterested and nal l y sai d,' W ho di d thi s m arvel ous thi ng?' A nd R abisai d,' O h,you know hi m ,you gave hi m an F,Jul i an Schw i nger. ' So rst at the Pocono C onference i n A pri l1948, then i n the M i chi gan Sum m erSchoolthatyear,and nal l y i n a seri esofthree m onum entalpapers, \Q uantum El ectrodynam i cs I,II,and III, " 40 Jul i an set forth hi s covari ant approach to Q ED .A taboutthe sam e ti m e Feynm an wasform ul ati ng hi scovari antpath-i ntegralapproach;and al though hi spresentati on atPocono was notwel l -recei ved,Feynm an and Schw i ngercom pared notesand real i zed that they had cl i m bed the sam e m ountai n by di erentroutes. Feynm an' ssystemati c papers 41 were publ i shed onl y afterD yson had proved the equi val ence of Schw i nger' s and Feynm an' s schem es. French and W ei sskopf46 had gotten the ri ght answer, because they put i n the correct val ue of the m agneti c m om ent and used i t al lthe way through. I,at an earl i er stage,had done that,i n e ect,and al so got the sam e answer. But now he and Feynm an \fel li nto the sam e trap. W e were connecti ng a rel ati vi sti c cal cul ati on ofhi gh energy e ects w i th a nonrel ati vi sti c cal cul ati on ofl ow energy e ects,a l a Bethe. " Based on the resul t Schw i nger had presented at the A PS m eeti ng i n January 1948,Schw i nger cl ai m ed pri ori ty for the Lam b shi ft cal cul ati on: Ihad theansweri n D ecem berof1947.Ifyou l ook atthose[ other] papers you w i l l nd that on the cri ti cal i ssue of the spi n-orbi t coupl i ng,they appealto the m agneti c m om ent. T he de ci ency i n the cal cul ati on Idi d [ i n 1947]was[ thati twas]a non-covari ant cal cul ati on. French and W ei sskopf were certai nl y doi ng a noncovari ant cal cul ati on. W i l l i s Lam b 47 was doi ng a non-covari ant cal cul ati on. T hey coul d not possi bl y have avoi ded these sam e probl em s.T he error Feynm an and Schw i nger m ade had to do w i th the i nfrared probl em that occurred i n the rel ati vi sti c cal cul ati on,w hi ch was handl ed by gi vi ng the photon a cti ti ous m ass. Schw i nger l earned from hi s com peti tors, parti cul arl y Feynm an and D yson. Just as Feynm an had borrowed the i dea that henceforward woul d go by the nam eofFeynm an param etersfrom Schw i nger,Schw i ngerrecogni zed thatthe system ati c approach ofD yson and Feynm an was superi or i n hi gher orders. So by 1949 he repl aced the Tom onaga-Schw i nger approach by a m uch m ore powerfulengi ne,the quantum acti on pri nci pl e. T hi swasa l ogi caloutgrow th of the form ul ati on of D i rac, 48 as was Feynm an' s path i ntegral s; the l atter was an i ntegral approach, Schw i nger' s a di erenti al . T he form al sol uti on ofSchw i nger' s di erenti alequati ons was Feynm an' s functi onali ntegral ;yet w hi l e the l atter was i l l -de ned, the form er coul d be gi ven a preci se m eani ng,and forexam pl e,requi red the i ntroducti on offerm i oni c vari abl es,w hi ch i ni ti al l y gave Feynm an som e di cul ty. Itm ay be fai rto say thatw hi l e the path i ntegralform ul ati on to quantum el d theory recei ves al lthe press,the m ost preci se exegesi s of el d theory i s provi ded by the functi onaldi erenti alequati onsofSchw i nger resul ti ng from hi s acti on pri nci pl e. H e began i n the \T heory ofQ uanti zed Fi el ds I" 49 by i ntroduci ng a com pl ete set ofei genvectors \speci ed by a spacel i ke surface ...and the ei genval ues ...of a com pl ete set of com m uti ng operators constructed from el d quanti ti esattached to thatsurface. " T he questi on i show to com putethetransform ati on functi on from onespacel i kesurfaceto another. A fterrem arki ng thatthi sdevel opm ent,ti m e-evol uti on,m ustbedescri bed by a uni tary transform ati on,he assum ed that any i n ni tesi m alchange i n the transform ati on functi on m ust be gi ven i n term s ofthe i n ni tesi m alchange i n a quantum acti on operator,or ofa quantum Lagrange functi on. T hi s i s the quantum dynam i calpri nci pl e,a general i zati on ofthe pri nci pl e ofl east acti on,or ofH am i l ton' s pri nci pl e i n cl assi calm echani cs. Ifthe param eters ofthe system are notal tered,the onl y changesari se from those ofthe i ni ti al and nalstates,from w hi ch Schw i nger deduced the Principl e ofStationary Action,from w hi ch the el d equati onsm ay be deri ved. A seri esofsi x papers fol l owed w i th the sam e ti tl e,and the m ost i m portant \G reen' s Functi ons of Q uanti zed Fi el ds" publ i shed i n the Proceedi ngs ofthe N ati onalA cadem y. x Schw i ngerevi dentl y wasawareofthe m ysti que.In a l aterl etterrecom m endi ng M arti n for a perm anentappoi ntm ent atH arvard he stated that M arti n was\superi ori n i ntri nsi c abi l i ty and perform ance. Q uantum el d theory i s the new rel i gi on ofphysi cs,and Paul C .M arti n i s one ofi ts hi gh pri ests. " 5 H owever,as the l astparagraph ofthe presentessay dem onstrates,Schw i nger throughout hi s l i fe m ai ntai ned a tensi on between an el i ti st and a dem ocrati c vi ew ofsci ence.weretheval uesofrealphysi calobservabl esand thei rcorrel ati ons. T he l anguage wasenl i ghteni ng,butthe l ectureswere exci ti ng because they were m ore than m etaphysi cal . A l ong w i th structural i nsi ghts,succi nct and i m pl i ci t sel f-consi stent m ethods for generati ng true statem ents were reveal ed.R ecentl y, a percepti ve anal ysi s of Schw i nger' s G reen' s functi ons papers has been gi ven by Schweber53 . T here he concl udes that Schw i nger' s form ul ati on ofrel ati vi sti c Q FT s [ quantum el d theori es]i n term sofG reen' sfuncti onswasa m ajoradvance i n theoreti calphysi cs. It was a representati on i n term s ofel em ents (the G reen' s functi ons) that were i nti m atel y rel ated to realphysi cal observabl esand thei rcorrel ati on.Itgave deep structurali nsi ghts i nto Q FT s;i n parti cul ar,i tal l owed the i nvesti gati on ofthe structureofthe G reen' sfuncti onsw hen thei rvari abl esare anal yti cal l y conti nued to com pl ex val ues,thus establ i shi ng deep connecti ons to stati sti calm echani cs. 10 \G auge Invariance and V acuum P olarization"T he paper\O n G auge Invari ance and Vacuum Pol ari zati on" 54 ,subm i tted bySchw i nger to the PhysicalReview near the end ofD ecem ber 1950,i s nearl y uni versal l y accl ai m ed as hi s greatest publ i cati on. A s hi s l ectures have ri ghtful l y been com pared to the works ofM ozart,so thi sm i ghtbe com pared to a m i ghty constructi on ofBeethoven,the 3rd Sym phony,the Eroica,perhaps.It i s m ost rem arkabl e because i t stands i n spl endi d i sol ati on. It was w ri tten over a year after the l ast of hi s seri es of papers on hi s second, covari ant, form ul ati on of quantum el ectrodynam i cs was com pl eted: \Q uantum El ectrodynam i csIII.T he El ectrom agneti c Properti esofthe El ectron| R adi ati ve C orrecti ons to Scatteri ng" 40 was subm i tted i n M ay 1949. A nd barel y two m onths l ater, i n M arch 1951, Schw i nger woul d subm i t the rst of the seri es on hi s thi rd reform ul ati on of quantum el d theory, that based on the quantum acti on pri nci pl e, nam el y, \T he T heory of Q uanti zed Fi el ds I. " 49 But \G auge Invari ance and Vacuum Pol ari zati on" stands on i ts ow n, and has endued the rapi d changes i n tastes and devel opm ents i n quantum el d theory,w hi l e the papers i n the other seri es are m ostl y ofhi stori cali nterest now . A s Lowel lBrow n 55 poi nted out,\G auge Invari ance and Vacuum Pol ari zati on" sti l lhasover one hundred ci tati onsperyear,and i s farand away Schw i nger' sm ostci ted paper. { Yeteven such a m asterpi ece wasnotw i thout i ts cri ti cs. A braham K l ei n,w ho was ni shi ng hi s thesi s at the ti m e under Schw i nger' s di recti on,and woul d go on to be one ofSchw i nger' s second set of \assi stants" (w i th R obert K arpl us), as, rst, an i nstructor, and then a Juni orFel l ow ,recal l ed thatSchw i nger(and,i ndependentl y,he and K arpl us) ran afoulofa tem porary edi toratthe PhysicalReview.T hatedi torthought Schw i nger' s ori gi nalpaper repeated too m any com pl i cated expressi ons and that sym bol s shoul d be i ntroduced to represent expressi ons that appeared m ore than once. Schw i nger com pl i ed, but had hi s assi stants do the di rty work. H arol d Levi ne, w ho was sti l lshari ng Schw i nger' s o ce, worki ng on the never-to-be-com pl eted wavegui de book, typed the revi sed m anuscri pt, w hi l e K l ei n w rote i n the m any equati ons. K l ei n recal l ed that he took m uch m ore care i n w ri ti ng those equati ons than he di d i n hi s ow n papers. ne cannot concl ude a retrospecti ve ofSchw i nger' s work w i thout m enti oni ng two other m agni cent achi evem ents i n the quantum m echani caldom ai n. H e presented a de ni ti ve devel opm ent ofangul ar m om entum theory deri ved i n term s ofosci l l ator vari abl es i n \O n A ngul ar M om entum , " w hi ch was never properl y publ i shed; 73k and he devel oped a \ti m e-cycl e" m ethod ofcal cul ati ng m atri x el em ents w i thout havi ng to nd al lthe wavefuncti ons i n hi s beauti ful\Brow ni an M oti on ofa Q uantum O sci l l ator, " 66 w hi ch as we m enti oned above anti ci pated the work of K el dysh. 65 W e shoul d al so m enti on the fam ous Li ppm an-Schw i nger paper, 75 w hi ch i s chi e y rem em bered for w hat Schw i nger consi dered a standard exposi ti on ofquantum scatteri ng theory,not for the vari ati onalm ethods expounded there. In spi te ofhi s awesom e abi l i ty to m ake form al i sm work for hi m ,Schw i nger wasathearta phenom enol ogi st. H e wasacti ve i n the search forhi ghersymm etry;w hi l e he cam e up w i th W 3 ,G el l -M ann found the correctapproxi m ate sym m etry of hadroni c states, SU (3). Schw i nger' s greatest success i n thi s peri od was contai ned i n hi s m asterpi ece, hi s 1957 paper \A T heory ofthe Fundam entalInteracti ons". 76 A l ong w i th m any other i nsi ghts, such as the exi stence of two neutri nos and the V A structure of weak i nteracti ons, Schw i nger there l ai d the groundwork for the el ectroweak uni cati on. H e i ntroduced two charged i nterm edi ate vector bosons as partners to the photon, w hi ch coupl e to charged weak currents. A few years l ater,hi s form erstudent,Shel don G l ashow ,as an outgrow th ofhi sthesi s,woul d i ntroducea neutralheavy boson to cl osethesystem to the m odern SU (2) U (1)sym m etry group; 77 Steven W ei nberg 78 woul d com pl ete the pi cture by generati ng the m asses for the heavy bosons by spontaneous sym m etry breaki ng. Schw i nger di d not have the detai l s ri ght i n 1957, i n parti cul arbecause experi m ent then seem ed to di sfavor the V A theory hi s k T hi sand otherofSchw i nger' sm osti m portantpaperswere repri nted i n two sel ecti ons ofhi s work. 18,74 approach i m pl i ed,but there i s no doubt that Schw i nger m ust be counted as the grandfather ofthe Standard M odelon the basi s on thi s paper. 15 T he N obelP rize and R eaction R ecogni ti on ofSchw i nger' s enorm ous contri buti ons had com e earl y. H e recei ved the C harl es L.M eyer N ature ofLi ght A ward i n 1949 on the basi s of thepartl y com pl eted m anuscri ptsofhi s\Q uantum El ectrodynam i cs" papers. T he rstEi nstei n pri ze wasawarded to hi m ,al ong w i th K urtG odel ,i n 1951. T he N ati onalM edalofSci ence was presented to hi m by Presi dent Johnson i n 1964,and,ofcourse,the N obelPri ze wasrecei ved by hi m ,Tom onaga,and Feynm an from the K i ng ofSweden i n 1965. Itsurel y wasthe di cul ty ofi ncorporati ng strong i nteracti onsi nto el d theory that l ed to \Parti cl es and Sources, " recei ved by the Physical Review barel y si x m onths after hi s N obell ecture,i n Jul y 1966, 79 based on l ectures Schw i nger gave i n Tokyo that sum m er. O ne m ust appreci ate the m i l i eu i n w hi ch Schw i ngerworked i n 1966.Form orethan a decadeheand hi sstudents had been nearl y the onl y exponentsof el d theory,asthe com m uni ty sought to understand weak and strong i nteracti ons,and theprol i ferati on of\el em entary parti cl es, " through di spersi on rel ati ons, R egge pol es, current al gebra, and the l i ke, m ost am bi ti ousl y through the S-m atri x bootstrap hypothesi s ofG eo rey C hew and Stanl ey M andel stam . 80{83 W hat work i n el d theory di d exi stthen wasl argel y axi om ati c,an attem ptto turn the structure ofthe theory i nto a branch ofm athem ati cs,starti ng w i th A rthurW i ghtm an,84 and carri ed on by m any others,i ncl udi ng A rthur Ja e at H arvard.85 (T he nam e changed from axi om ati c el d theory to constructi ve el d theory al ong the way. ) Schw i nger l ooked on al lof thi s w i th consi derabl e di staste; not that he di d not appreci ate m any ofthe contri buti ons these techni ques o ered i n speci c contexts, but he coul d not see how they coul d form the basis of a theory. T he new source theory was supposed to supersede el d theory,m uch as Schw i nger' s successi ve covari ant form ul ati ons of quantum el ectrodynam i cs had repl aced hi s earl i er schem es. In fact, the revol uti on was to be m ore profound,because there were no di vergences,and no renorm al i zati on. T he concept of renorm al i zati on i s si m pl y forei gn to thi s phenom enol ogi caltheory. In source theory,we begi n by hypothesi s w i th the descri pti on ofthe actualparti cl es,w hi l e renorm al i zati on i s a el d theory concept i n w hi ch you begi n w i th the m ore fundam entaloperators,w hi ch are then m odi ed by dynam i cs. Iemphasi zethattherenevercan bedi vergencesi n a phenom enol ogi cal theory. W hat one m eans by that i s that one i s recogni zi ng that al lfurtherphenom ena are consequences ofone phenom enol ogi cal constant,nam el y the basi c charge uni t,w hi ch descri besthe probabi l i ty ofem i tti ng a photon rel ati veto theem i ssi on ofan el ectron. W hen one saysthatthere are no di vergencesone m eansthati ti s not necessary to i ntroduce any new phenom enol ogi calconstant.A l lfurtherprocessesascom puted i n term softhi spri m i ti ve i nteracti on autom ati cal l y em erge to be ni te,and i n agreem ent w i th those w hi ch hi stori cal l y had evol ved m uch earl i er.86 earned thi s l esson 25 years ago duri ng W orl d W ar II,w hen Ibecam e i nterested i n the probl em s ofm i crowave system s,wave gui des i n parti cul ar. Bei ng very nai ve, I started out sol vi ng M axwel l ' s equati ons. I soon l earned better. M ost of the i nform ati on i n M axwel l ' s equati ons i s real l y super uous. A s far as any parti cul ar probl em i s concerned,one i s onl y i nterested i n the propagati on ofjust a few m odes ofthe wave gui de. A l i mi ted num berofquanti ti esthatcan be m easured orcal cul ated tel l you how these few m odes behave and exactl y w hat the system i s doi ng.You are l ed di rectl y to a phenom enol ogi caltheory ofthe ki nd engi neers i nvari abl y use| a pi cture, say, i n term s of equi val ent transm i ssi on l i nes. T he onl y rol e of M axwel l ' s equati ons i s to cal cul ate the few param eters,the e ecti ve l um ped constantsthat characteri ze the equi val ent ci rcui ts. T hi s i s the concept of an i nteracti on skel eton. T he process i s there but i t i s not nal l y descri bed to startw i th,i ts exi stence i sm erel y recogni zed. T hi s si m pl i ed reconstructi on of el ectrodynam i cs i s com pl etel y successful . To i ndi cate the w i de sweep of the new approach,I m enti on that cl assi calgravi tati on theory (Ei nstei n) can be reconstructed and si m pl i ed i n a si m i l ar way by begi nni ng w i th the quantum rel ati vi sti c properti esofthe basi c parti cl e,the gravi ton,al though here i ndi rect evi dence for i ts properti es m ust be adduced.Butthe realprovi ng ground forsource theory com esfrom the dom ai n forw hi ch i twasi nvented,strong i nteracti ons. T he starti ng poi nt i s experi m entali nform ati on at l ow energi es. T he tentati ve extrapol ati ons are toward hi gher energi es. T he m ethod i s qui te el em entary com pared to othercurrenttechni ques. T he successfulcorrel ati ons that have been obtai ned em phasi ze the compl etel y phenom enol ogi calnature ofourpresent know l edge about parti cl es and refute attem pts to l end fundam ental credence to thi s or that parti cl e m odel . R obert Fi nkel stei n has o ered a percepti ve di scussi on ofSchw i nger' s source theory program :In com pari ng operator el d theory w i th source theory Jul i an reveal ed hi s pol i ti calori entati on w hen he descri bed operator el d theory asa tri ckl e dow n theory (aftera fai l ed econom i c theory)| si nce i tdescends from i m pl i ci t assum pti ons aboutunknow n phenom ena ati naccessi bl eand very hi gh energi esto m akepredi cti ons at l ower energi es. Source theory on the other hand he descri bed asanabati c (asi n X enophon' sA nabasi s)by w hi ch he m eantthat i tbegan w i th sol i d know l edge aboutknow n phenom ena ataccessi bl e energi es to m ake predi cti ons about physi calphenom ena at hi gherenergi es. A l though source theory was new ,i tdi d notrepresent a com pl ete break w i th the past but rather was a natural evol uti on ofJul i an' s work w i th operator G reen' s functi ons. H i s tri l ogy on source theory i snotonl y a stunni ng di spl ay ofJul i an' s powerasan anal ystbuti ti sal so total l y i n the spi ri tofthe m odest sci enti c goal s he had set i n hi s Q ED work and w hi ch had gui ded hi m earl i er as a nucl ear phenom enol ogi st.88 But the new approach was not wel lrecei ved. In part thi s was because the ti m es were changi ng;w i thi n a few years,' t H ooft 89 woul d establ i sh the renorm al i zabi l i ty ofthe G l ashow -W ei nberg-Sal am SU (2) U (1)el ectroweak m odel ,and el d theory wasseen by al lto bevi abl eagai n.W i th thedi scovery ofasym ptoti c freedom i n 1974, 90 a non-A bel i an gauge theory ofstrong i nteracti ons,quantum chrom odynam i cs,w hi ch was proposed som ew hat earl i er, 91 was prom ptl y accepted by nearl y everyone. A n al ternati ve to conventi onal el d theory di d notseem to be requi red afteral l .Schw i nger' si nsi stence on a cl ean break w i th the past,and hi srejecti on of\rul es" asopposed to l earni ng through servi ng as an \apprenti ce, " di d not encourage conversi ons.A l ready before the source theory revol uti on,Schw i nger fel t a grow i ng senseofuneasew i th hi scol l eaguesatH arvard.Butthechi efreason Schw i nger l eftH arvard forU C LA washeal th rel ated.Form erl y overwei ghtand i nacti ve, hehad becom e heal th consci ousupon theprem aturedeath ofW ol fgang Paul i i n 1958. (Ironi cal l y,both di ed ofpancreati c cancer. ) H e had been fond of tenni sfrom hi syouth,had di scovered ski i ng i n 1960,and now hi sdoctorwas recom m endi ng a dai l y sw i m for hi s heal th. So he l i stened favorabl y to the entreati es ofD avi d Saxon,hi s cl osest col l eague atthe R adi ati on Lab duri ng the war, w ho for years had been tryi ng to i nduce hi m to com e to U C LA . Very m uch agai nsthi s w i fe' s w i shes,he m ade the m ove i n 1971.H e brought al ong hi s three seni or students at the ti m e, Lester D eR aad, Jr. , W u-yang T sai ,and the present author,w ho becam e l ong-term \assi stants" at U C LA . H e and Saxon expected,asi n the earl y daysatH arvard,thatstudentswoul d ock to U C LA to work w i th hi m ;butthey di d not.Schw i ngerwasno l onger the center oftheoreti calphysi cs. T hi s i s not to say that hi s l i ttl e group at U C LA di d not m ake an heroi c attem pt to establ i sh a source-theory presence. Schw i nger rem ai ned a gi fted i nnovator and an awesom e cal cul ator. H e w rote 2-1/2 vol um es of an exhausti ve treati se on source theory,Particl es,Sources,and Fiel ds, 92 devoted pri m ari l y to the reconstructi on ofquantum el ectrodynam i cs i n the new l anguage;unfortunatel y,he abandoned the project w hen i t cam e ti m e to deal w i th strong i nteracti ons,i n partbecause he becam e too busy w ri ti ng papers on an \anti -parton" i nterpretati on ofthe resul ts ofdeep-i nel asti c scatteri ng experi m ents. 93 H e m ade som e si gni cantcontri buti onsto the theory ofm agneti c charge; parti cul arl y noteworthy was hi s i ntroducti on of dyons. 94 H e rei nvi gorated proper-ti m e m ethods of cal cul ati ng processes i n strong-el d el ectrodynam i cs; 95 and he m ade som e m ajor contri buti ons to the theory of the C asi m i re ect,w hi ch are sti l lhavi ng repercussi ons. 96 Buti twascl earhe wasreacti ng,notl eadi ng,asw i tnessed by hi squi tepretty paperon the\M ulti spi nor Basi s ofFerm i -Bose Transform ati on, " 97 i n w hi ch he ki cked hi m sel f fornotdi scoveri ng supersym m etry,fol l ow i ng a com m and pri vateperform ance by Stanl ey D eser on supergravi ty.17 T he A xial-V ectorA nom aly and Schw inger's D eparture from P article P hysicsIn 1980 Schw i ngergave a sem i naratM IT thatm arked hi sl astsci enti c vi si t to the East C oast, and caused hi m to abandon hi s attem pt to i n uence the devel opm ent of hi gh-energy theory w i th hi s source theory revol uti on. T he tal k was on a subject that he l argel y started i n hi s fam ous \G auge Invari ance and Vacuum Pol ari zati on" paper, 54 the tri angl e or axi al -vector anom al y. In i ts si m pl est and basi c m ani festati on,thi s \anom al y" descri bes how the neutral pi on decays i nto two photons. T he pi on coupl i ng coul d be regarded as occurri ng ei ther through a pseudoscal ar or an axi al vector T hi s does notcounta tal k he gave atM IT i n 1991 i n honorofbi rthdaysoftwo ofhi s students,w here he gave a \progressreport" on hi s work on col d fusi on and sonol um i nescence,excerpts ofw hi ch i s gi ven i n R ef.[ 2] . W hen the l astofhi s H arvard postdocs l eft U C LA i n 1979,and the ap over the axi al -vector anom al y ensued,Schw i nger abandoned hi gh-energy physi cs al together. In 1980, after teachi ng a quantum m echani cs course (a notunusual sequence of events), Schw i nger began a seri es of papers on the T hom as-Ferm i m odel of atom s. 102 H e soon hi red Berthol d-G eorg Engl ert, repl aci ng M i l ton as a postdoc,to hel p w i th the el aborate cal cul ati ons. T hi s endeavor l asted unti l1985. It i s i nteresti ng that thi s work not onl y i s regarded as i m portant i n i ts ow n ri ght by atom i c physi ci sts, but has l ed to som e si gni cant resul ts i n m athem ati cs. A l ong seri es ofsubstanti alpapers by C .Fe erm an and L.Seco 103 has been devoted to provi ng hi s conjecture about the atom i c num ber dependence of the ground state energy of l arge atom s. A s Seth Putterm an has rem arked, i t i s l i kel y that,ofal lthe work thatSchw i nger accom pl i shed atU C LA ,hi swork on the stati sti calatom w i l l prove the m ost i m portant. Iti si m possi bl e to do justi ce i n a few wordsto thei m pactofJul i an Schw i nger on physi cal thought i n the 20th C entury. H e revol uti oni zed el ds from nucl ear physi cs to m any body theory, rst successful l y form ul ated renorm al i zed quantum el ectrodynam i cs,devel oped the m ost powerfulfuncti onal form ul ati on ofquantum el d theory,and proposed new ways ofl ooki ng at quantum m echani cs,angul ar m om entum theory,and quantum uctuati ons. H i s l egacy i ncl udes \theoreti caltool s" such as the proper-ti m e m ethod,the quantum acti on pri nci pl e, and e ecti ve acti on techni ques. N ot onl y i s he responsi bl e for form ul ati ons beari ng hi s nam e: the R ari ta-Schw i nger equati on,the Li ppm ann-Schw i nger equati on,the Tom onaga-Schw i nger equati on, the D yson-Schw i nger equati on,the Schw i nger m echani sm ,and so forth,but som e attri buted to others, or know n anonym ousl y: Feynm an param eters, the Bethe-Sal peter equati on,coherent states,Eucl i dean el d theory;the l i st goeson and on.H i sl egacy ofnearl y 80 Ph. D .students,i ncl udi ng fourN obel l aureates,l i ves on. It i s i m possi bl e to i m agi ne w hat physi cs woul d be l i ke i n the 21st century w i thout the contri buti ons of Jul i an Schw i nger, a very pri vate yet wonderfulhum an bei ng. It i s m ost grati fyi ng that a dozen years afterhi sdeath,recogni ti on ofhi sm ani fol d i n uencesi sgrow i ng,and research projects he i ni ti ated are sti l lunderway. [ 1 ] 1N orm an R am sey,i ntervi ew by K .A .M i l ton,June 8,1999. [ 2] Jagdi sh M ehra and K i m bal lA .M i l ton,C l im bing the M ountain: T he Scienti c Biography ofJul ian Schwinger,(O xford: O xford U ni versi ty Press,2000). ass and ask m e to show hi m m y copy ofthi s book and he woul d ski m through i t fast and see w hat was goi ng on. A nd thi s fel l ow R eynol ds, al though he was a dodderer, was a very m ean character. { H e used to send peopl e up to the board to do a probl em and he was al ways sendi ng Jul i an to the board to do probl em s because he knew he' d never seen the course and Jul i an woul d get up at the board,and| ofcourse,projecti ve geom etry i s a very strange subject. T he probl em s are tri vi ali fyou thi nk aboutthem pi ctori al l y,butJul i an neverwoul d do them thi sway. H e woul d i nsi ston doi ng them al gebrai cal l y and so he' d getup at theboard atthebegi nni ng ofthe hourand he' d work through the w hol e hourand he' d ni sh the thi ng and by thatti m e the course wasoverand anyway,R eynol dsdi dn' tunderstand the proof,and that woul d end i t for the day.It was taught by an ol d dodderer nam ed Fredri ck B.R eynol ds. H e was head of the m ath departm ent. H e real l y knew absol utel y nothi ng. It was am azi ng. But he taught thi s course on M odern G eom etry. It was a course i n projecti ve geom etry from a m i serabl e book by a m an nam ed G raustei n from Pri nceton, and Jul i an was i n the cl ass, but i t was very strange because he obvi ousl y never coul d get to cl ass,at l east not very often,and he di dn' t ow n the book. T hat was cl ear. A nd every once i n a w hi l e,he' d grab m e before cl { In the 2005 Science C itation Index,i thad 105 ci tati ons,outofa totalof458 ci tati ons to al lofSchw i nger' s work.56 T hese num bers have rem ai ned rem arkabl y constantoverthe years. P A M , Principl es of Q uantum M echanics (O xford: O xford U ni versi ty Press. P. A . M . D i rac, Principl es of Q uantum M echanics (O xford: O xford U ni versi ty Press,1930). . G Edward, Erjuoy, K .A .M i l tonEdward G erjuoy,tel ephone i ntervi ew by K .A .M i l ton,June 25,1999. \R ecol l ecti ons" at Jul i an Schw i nger' s 60th bi rthday cel ebrati on. M , U C LAA IP A rchi veM . H am erm esh, \R ecol l ecti ons" at Jul i an Schw i nger' s 60th bi rthday cel ebrati on,U C LA ,1978 (A IP A rchi ve). tal k gi ven at the U ni versi ty of Pi ttsburgh and at G eorgi a Tech. G Edward, Erjuoy, pri vate com m uni cati onEdward G erjuoy, tal k gi ven at the U ni versi ty of Pi ttsburgh and at G eorgi a Tech,1994,pri vate com m uni cati on. 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J Schw I Nger, ibid.902105{2106;\C asi m i rLi ght:Photon Pai rs. 958{959; \C asi m i r Li ght: T he SourceJ.Schw i nger,\C asi m i rLi ght:A G l i m pse, " Proc.N atl .Acad.Sci.U SA 90 (1993), 958{959; \C asi m i r Li ght: T he Source, " ibid. 90 (1993), 2105{2106;\C asi m i rLi ght:Photon Pai rs, " ibid.90 (1993),4505{4507; Pi eces of the A cti on. \c Asi M I R Li Ght, 90\C asi m i r Li ght: Pi eces of the A cti on, " ibid. 90 (1993), 7285{7287; . \c Asi M I R Li Ght, Fi el d Pressure, " ibid.91\C asi m i r Li ght: Fi el d Pressure, " ibid.91 (1994),6473{6475. \Schw i nger' s R esponse to an H onorary D egree atN otti ngham. T he Physicist,the Teacher,and the M an. Y .J.N g (Si ngapore: W orl d Sci enti c\Schw i nger' s R esponse to an H onorary D egree atN otti ngham , " i n Ju- l ian Schwinger: T he Physicist,the Teacher,and the M an,ed.Y .J.N g (Si ngapore: W orl d Sci enti c,1996),p.11{12.

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{'abstract': "Jul i an Schwi nger:Nucl earPhysi cs,the Radi ati on Laboratory,Renorm al i zed Q ED , Source Theory,and Beyond K i m bal lA .M i l ton H om erL.D odge D epartm entofPhysi cs and A stronom y U ni versi ty ofO kl ahom a,N orm an,O K 73019 U SA D ecem ber 8,2021A bstract Jul i an Schw i nger' s i n uence on twenti eth century sci ence i s profound and pervasi ve. O fcourse,he i s m ost fam ous for hi s renorm ali zati on theory of quantum el ectrodynam i cs,for w hi ch he shared the N obelPri ze w i th R i chard Feynm an and Si n-i ti ro Tom onaga. B ut although thi stri um ph wasundoubtedl y hi sm ostheroi c work,hi sl egacy l i ves on chi e y through subtl e and el egant work i n cl assi calel ectrodynam i cs,quantum vari ati onalpri nci pl es,proper-ti m e m ethods,quantum anom al i es, dynam i cal m ass generati on, parti al sym m etry, and m ore. Starti ng as just a boy,he rapi dl y becam e the pre-em i nent nucl ear physi ci st i n the l ate 1930s, l ed the theoreti cal devel opm ent of radar technol ogy at M IT duri ng W orl d W ar II,and then,soon after thewar,conquered quantum el ectrodynam i cs,and becam ethel eadi ng quantum el d theori st for two decades, before taki ng a m ore i conocl asti c route duri ng hi s l ast quarter century.K eywords: Jul i an Schw i nger, nucl ear physi cs, wavegui des, quantum el ectrodynam i cs,renorm al i zati on,quantum acti on pri nci pl e,source theory,axi al -vector anom al y K . A .M i l ton i sProfessorofPhysi csattheU ni versi ty ofO kl ahom a.H ewasa Ph. D .stu-dentofJul i an Schw i ngerfrom 1968{71,and hi spostdoc atU C LA forthe restofthe 1970s. H e has w ri tten a sci enti c bi ography ofSchw i nger,edi ted two vol um es ofSchw i nger' s sel ected works,and co-authored two textbooks based on Schw i nger' s l ectures.", 'arxivid': 'physics/0610054', 'author': [], 'authoraffiliation': [], 'corpusid': 684471, 'doi': '10.1007/s00016-007-0326-6', 'github_urls': [], 'n_tokens_mistral': 55775, 'n_tokens_neox': 51804, 'n_words': 32421, 'pdfsha': 'd58c0912ea3249914fd5a09fbbf133abc11e6ce2', 'pdfurls': ['https://export.arxiv.org/pdf/physics/0610054v1.pdf'], 'title': ['arXiv:physics/0610054v1 [physics.hist-ph]', 'arXiv:physics/0610054v1 [physics.hist-ph]'], 'venue': []}

arxiv

CONVEX BODIES OF CONSTANT WIDTH WITH EXPONENTIAL ILLUMINATION NUMBER 20 Apr 2023 A Arman A Bondarenko A Prymak CONVEX BODIES OF CONSTANT WIDTH WITH EXPONENTIAL ILLUMINATION NUMBER 20 Apr 2023 We show that there exist convex bodies of constant width in E n with illumination number at least (cos(π/14)+o(1)) −n , answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in E n which cannot be covered by (2/ √ 3 + o(1)) n balls of diameter 1, improving a result by Introduction Let E n denote the n-dimensional Euclidean space and S n−1 := {x : x = 1} be the unit sphere in E n . Consider a convex body K in E n , i.e., a convex compact set with non-empty interior, and a point x from the boundary ∂K of K. We say that x is illuminated by a direction ξ ∈ S n−1 if the half-line {x + tξ : t ≥ 0} contains an interior point of K. The illumination number I(K) is the minimal number of directions sufficient to illuminate all points x ∈ ∂K. A convex body is said to be of constant width d if the distance between any two distinct parallel supporting hyperplanes of K is d. The best known upper bound for illumination number of arbitrary convex body K ⊂ E n of constant width was obtained by O. Schramm in [9], that is I(K) ≤ ( 3/2 + o(1)) n . However, it was not known if such bodies with I(K) ≥ (1 + ǫ) n exist for some ǫ > 0, see G. Kalai's survey [8,Problem 3.3]. (Note that in [8] the question is stated in terms of covering by smaller homothetic copies, but this is equivalent to illumination, for example see [1] or [2].) We prove Theorem 1. For every positive integer n there exists a convex body K ⊂ E n of constant width satisfying I(K) ≥ (cos(π/14) + o(1)) −n . To explain the main idea of the proof we need a few definitions. For non-zero x, y ∈ E n , we use the notation θ(x, y) := arccos( x·y x y ) for the angle between the directions of x and y. particular, if x, y ∈ S n−1 , then θ(x, y) is the geodesic distance between x and y. For x ∈ S n−1 and 0 < α < π the spherical cap of S n−1 centered at x of (angular) radius α is C(x, α) := {y ∈ S n−1 : θ(x, y) ≤ α}. Also, for fixed x ∈ S n−1 and 0 < α ≤ π/6 define Q(x, α) := {x} ∪ {y ∈ S n−1 : x − y = 2 cos α}. The convex hull of Q(x, α) is the right circular cone with angle α, apex x, and axis containing the origin, which is inscribed into S n−1 . Clearly, diam Q(x, α) = 2 cos α. Our main geometric observation is Lemma 1. Suppose 0 < α ≤ π/6, K is a convex body in E n such that diam K = 2 cos α and for some x ∈ S n−1 we have Q(x, α) ⊂ K. Then x ∈ ∂K and any direction ξ ∈ S d−1 illuminating x satisfies ξ ∈ C(−x, π 2 − α). In other words, under the hypotheses of the lemma, any direction illuminating x must belong to a certain "well-controlled" spherical cap. Fixing α = π 14 , we will choose a finite but "large" set X ⊂ S n−1 . The points from X will have sufficiently "separated" directions which will guarantee that the set W = W(X) := x∈X Q(x, α) satisfies diam W = 2 cos α. Next we will take an arbitrary convex body K of width 2 cos α that contains W (such a body always exists, see, e.g., [6, Th. 54, p. 126]). We will show that for a suitable randomly constructed X only O(n log n) points from X in K can be illuminated simultaneously by a direction ξ ∈ S n−1 . This will immediately imply the proof of Theorem 1. The choice of X ⊂ S n−1 will be provided by the following probabilistic lemma. Lemma 2. Suppose 0 < ψ < φ < π 2 are fixed. Then for every positive integer n there exists a collection X = {x 1 , . . . , x N } ⊂ S n−1 with N = 1+o(1) sin φ n such that (a) ψ ≤ θ(x i , x j ) ≤ π − ψ for all 1 ≤ i < j ≤ N. (b) every point of S n−1 is contained in at most O(n log n) spherical caps C(x i , φ), 1 ≤ i ≤ N. Note that in part (b) the probabilistic measure of each cap C(x, φ) on S n−1 is (sin φ + o(1)) n (see for example [3, Corollary 3.2(iii)]) . Hence the value of N in the lemma is optimal up to the o(1) terms. Lemma 2 has the same spirit as P. Erdös and C.A. Rogers's theorem [7] concerning the covering of the space (see also the work [3] by K. Böröczky and G. Wintschesee for the spherical case) and happens to be quite useful for constructing "spread" sets of fixed diameter. The lemma also readily implies the following result. Theorem 2. For every positive integer n there exists a finite set of diameter 1 in E n which cannot be covered by (2/ √ 3 + o(1)) n balls of diameter 1. This improves (note that 2/ √ 3 ≈ 1.1547) the result of J. Bourgain and J. Lindenstrauss [4] who established that one needs at least 1.0645 n balls for large n. An exponential lower bound of 1.003 n was originally obtained by L. Danzer [5]. Both problems, of illumination of convex bodies of constant width and of covering of finite sets by balls with the same diameter, are related to the Borsuk's conjecture on partitioning a set into pieces of smaller diameter. Namely, an upper bound in any of the former two problems implies the same upper bound in the latter. An interested reader is referred to [8] for more details. Proofs We begin with geometric arguments and then proceed to the probabilistic ones concluding with the proofs of the theorems. For geometric lemmas, we define for x ∈ S n−1 and α ∈ (0, π 6 ], R(x, α) := {y ∈ S n−1 : x − y = 2 cos α} = Q(x, α) \ {x}. Proof of Lemma 1. Clearly R(x, α) = ∅, hence x ∈ ∂K as otherwise diam K > 2 cos α. Assume by contradiction that for some ξ ∈ S n−1 \ C(−x, π 2 − α) the point x is illuminated by ξ. Consider a two-dimensional plane H containing x, ξ and the origin. Note that H ∩ R(x, α) consists of the two points y 1 , y 2 on S 1 such that θ(y j − x, −x) = α, j = 1, 2. On the other hand, as ξ ∈ S n−1 \ C(−x, π 2 − α), we have θ(ξ, −x) > π 2 − α. Therefore, θ(y k − x, ξ) > π 2 for some k ∈ {1, 2}. Suppose k = 1. The distance from any point of the open half-line ℓ := {x + tξ : t > 0} to y 1 is larger than y 1 − x = 2 cos α, see Fig. 1. So ℓ contains no points from K which contradicts our assumption. The case k = 2 is completely similar. We next show that some necessary "separation" conditions on X ⊂ S n−1 guarantee that diam W(X) ≤ 2 cos α (recall that W(X) = ∪ x∈X Q(x, α)). Lemma 3. Suppose 0 < α ≤ π/6 and X ⊂ S n−1 . (i) If θ(x, y) ≤ π − 2α for all x, y ∈ X, then diam X ≤ 2 cos α. (ii) If 4α ≤ θ(x, y) ≤ π − 6α for all x, y ∈ X, then diam W(X) ≤ 2 cos α. Proof. We can switch to the geodesic distance by observing that θ(x, y) ≤ π −2α for x, y ∈ S n−1 implies x − y = 2 sin θ(x,y) 2 ≤ 2 sin π−2α 2 = 2 cos α. Thus, (i) readily follows. For (ii), suppose x, y ∈ X, x = y. Clearly, θ(x, y) ≤ π − 6α < π − 2α. For any u ∈ R(y, α) we have θ(−y, u) = 2α, so θ(u, x) ≤ θ(u, −y) + θ(−y, x) = 2α + π − θ(x, y) ≤ π − 2α. Similarly, θ(v, y) ≤ π − 2α for any v ∈ R(x, α). Finally, suppose u ∈ R(y, α) and v ∈ R(x, α). Then θ(u, v) ≤ θ(u, −y) + θ(−y, −x) + θ(v, −x) = 4α + θ(x, y) ≤ π − 2α as required. Proof of Lemma 2. It is enough to consider the case of sufficiently large n. The main result of [3], Theorem 1.1, states that there is a covering of S n−1 with N spherical caps of a given radius φ ∈ (0, π 2 ) such that every point of S n−1 is covered at most 400n log n times. We start our proof with following the lines of the proof of Theorem 1.1. The proof of Theorem 1.1 [3] relies on a random approach. In particular, if Ω(θ) denotes the probabilistic measure of a cap C(x, θ) on S n−1 , then the value of N is chosen to be N = ⌈ 8n log n Ω((1− 1 2n )φ) ⌉. Using [3, Corollary 3.2(iii)], we get Ω (1 − 1 2n )φ = ((1 + o(1)) sin φ) n , and so N = 1+o(1) sin φ n . With such a choice of N, the authors of [3] show that a set X ′ consisting of N points on S n−1 , chosen independently and according to the uniform distribution on the sphere, satisfies the property (b) of Lemma 2 with probability 1 − o (1). See the proof of [3, Case 1, p. 241] for full details and exact calculations. Next, using the deletion method, we show that with a positive probability there exists a large subset X ⊂ X ′ satisfying property (a). Indeed, consider all possible N 2 = 1+o (1) sin φ 2n pairs of points from X ′ . Let B := B(X ′ ) := {(x i , x j ) : θ(x i , x j ) ∈ [ψ, π − ψ], 1 ≤ i < j ≤ N} be the set of pairs of points from X ′ not satisfying the property (a). Note that a pair (x i , x j ), 1 ≤ i < j ≤ N, belongs to B with probability at most p = ((1 + o(1)) sin ψ) n , where we again used the estimate on the measure of caps C(±x k , ψ), k ∈ {i, j}, provided by [3,Corollary 3.2(iii)]. Thus the expected number of such "bad" pairs is E(|B|) ≤ p · N 2 = (1 + o(1)) sin ψ sin φ n N. Since ψ < φ we can conclude that for large n, E(|B|) ≤ N 4 . Now, using the Markov's inequality, we deduce that with probability at least 1 2 the number of pairs in B does not exceed N 2 . To summarize, we showed that the random choice of X ′ satisfies property (b) with high probability, and |B(X ′ )| < N 2 with probability at least 1/2. Selecting such an X ′ , we remove a point from each pair in B(X ′ ) to obtain X ⊂ X ′ that has at least N 2 points. Clearly X still satisfies the property (b), and now X satisfies the property (a) as well. Finally, |X| ≥ N 2 = 1+o(1) sin φ n . Proof of Theorem 1. Let X be a collection of points guaranteed by Lemma 2 with parameters φ = 6π 14 + ǫ, ψ = 6π 14 , where ǫ > 0. By Lemma 3(ii) with α = π 14 and the definition of W(X), diam (W(X)) = 2 cos π 14 . By [6, Th. 54, p. 126], there exists a convex body K of constant width 2 cos π 14 which contains W(X). Clearly, diam K = 2 cos π 14 . Now, if an x i ∈ ∂K ∩ X is illuminated by a direction ξ, then ξ ∈ C(−x i , φ) by Lemma 1. Note that if property (b) of Lemma 2 holds for X, then it also holds for the symmetric set −X. Therefore no direction ξ can belong to more than O(n log n) caps C(−x i , φ). So every direction ξ illuminates at most O(n log n) points from X ⊂ ∂K and we need at least |X| O(n log n) = 1+o(1) sin( 6π 14 +ǫ) n = (cos( π 14 − ǫ) + o(1)) −n directions to illuminate ∂K. Since ǫ > 0 can be arbitrarily small, this completes the proof. Proof of Theorem 2. Let X be a collection of points guaranteed by Lemma 2 with parameters φ = π 3 + ǫ, ψ = π 3 , where ǫ > 0. By Lemma 3(i) with α = π 6 , diam X ≤ 2 cos π 6 = √ 3. Now, if a ball with center x of diameter √ 3 contains an x i ∈ X, then x x ∈ C(x i , φ). Every such ball cannot contain more than O(n log n) points of X, by Lemma 2(b). Therefore we need at least |X| O(n log n) = 1+o (1) sin( π 3 +ǫ) n balls of diameter √ 3 to cover the set X of diameter at most √ 3. We conclude the proof by taking an appropriate homothetic copy of X and noting that ǫ > 0 can be selected arbitrarily small. Figure 1 . 1Section in the plane H In Key words and phrases. Convex bodies of constant width, illumination number, sphere covering. The first author was supported by a postdoctoral fellowship of the Pacific Institute of Mathematical Sciences and the Department of Mathematics of the University of Manitoba. The second author was supported in part by Grant 275113 of the Research Council of Norway. The third author was supported by NSERC of Canada Discovery Grant RGPIN-2020-05357.Date: April 21, 2023. 2020 Mathematics Subject Classification. Primary 52C17; Secondary 52A20, 52A40, 52C35. The geometry of homothetic covering and illumination, Discrete geometry and symmetry. K Bezdek, M A Khan, Springer Proc. Math. Stat. 234SpringerK. Bezdek and M. A. Khan, The geometry of homothetic covering and illumination, Discrete geometry and symmetry, Springer Proc. Math. Stat., vol. 234, Springer, Cham, 2018, pp. 1-30. The problem of illuminating the boundary of a convex body. V Boltyanski, Izv. Mold. Fil. AN SSSR. 76V. Boltyanski, The problem of illuminating the boundary of a convex body, Izv. Mold. Fil. AN SSSR 76 (1960), 77-84. Covering the sphere by equal spherical balls, Discrete and computational geometry. K BöröczkyJr, G Wintsche, Algorithms Combin. 25SpringerK. Böröczky Jr. and G. Wintsche, Covering the sphere by equal spherical balls, Discrete and computational geometry, Algorithms Combin., vol. 25, Springer, Berlin, 2003, pp. 235-251. On covering a set in R N by balls of the same diameter. J Bourgain, J Lindenstrauss, Geometric aspects of functional analysis (1989-90). BerlinSpringer1469J. Bourgain and J. Lindenstrauss, On covering a set in R N by balls of the same diameter., Geometric aspects of functional analysis (1989-90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 138-144. On the k-th diameter in E d and a problem of Grünbaum. L Danzer, Proc. Colloquium on Convexity. Colloquium on ConvexityCopenhagen41L. Danzer, On the k-th diameter in E d and a problem of Grünbaum, Proc. Colloquium on Convexity, Copenhagen, 1965, pp. 41. . H G Eggleston, Convexity , Cambridge Tracts in Mathematics and Mathematical Physics. 47Cambridge University PressH. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. Covering space with convex bodies. P Erdős, C A Rogers, Acta Arith. 7P. Erdős and C. A. Rogers, Covering space with convex bodies, Acta Arith. 7 (1961/62), 281-285. Some old and new problems in combinatorial geometry I: around Borsuk's problem, Surveys in combinatorics. G Kalai, London Math. Soc. Lecture Note Ser. 424Cambridge Univ. PressG. Kalai, Some old and new problems in combinatorial geometry I: around Borsuk's problem, Surveys in combinatorics 2015, London Math. Soc. Lecture Note Ser., vol. 424, Cambridge Univ. Press, Cambridge, 2015, pp. 147-174. Illuminating sets of constant width. O Schramm, Mathematika. 352O. Schramm, Illuminating sets of constant width., Mathematika 35 (1988), no. 2, 180-189.

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{'abstract': 'We show that there exist convex bodies of constant width in E n with illumination number at least (cos(π/14)+o(1)) −n , answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter 1 in E n which cannot be covered by (2/ √ 3 + o(1)) n balls of diameter 1, improving a result by', 'arxivid': '2304.10418', 'author': ['A Arman ', 'A Bondarenko ', 'A Prymak '], 'authoraffiliation': [], 'corpusid': 258236233, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5075, 'n_tokens_neox': 4380, 'n_words': 2689, 'pdfsha': 'aef94406b09955f4a00b6bd276399d824546999b', 'pdfurls': ['https://export.arxiv.org/pdf/2304.10418v1.pdf'], 'title': ['CONVEX BODIES OF CONSTANT WIDTH WITH EXPONENTIAL ILLUMINATION NUMBER', 'CONVEX BODIES OF CONSTANT WIDTH WITH EXPONENTIAL ILLUMINATION NUMBER'], 'venue': []}

arxiv

Er, 12.15.Mm, 13.60.Hb 14.60.Cd 14.65.Bt 29.30.Aj 29 May 2013 R Subedi University of Virginia 22904CharlottesvilleVAUSA D Wang University of Virginia 22904CharlottesvilleVAUSA K Pan Massachusetts Institute of Technology 02139CambridgeMAUSA X Deng University of Virginia 22904CharlottesvilleVAUSA R Michaels Thomas Jefferson National Accelerator Facility 23606Newport NewsVAUSA P E Reimer Physics Division Argonne National Laboratory 60439ArgonneILUSA A Shahinyan Yerevan Physics Institute 0036YerevanArmenia B Wojtsekhowski Thomas Jefferson National Accelerator Facility 23606Newport NewsVAUSA X Zheng University of Virginia 22904CharlottesvilleVAUSA Dallas County Community College District 75243DallasTexasUSA Er, 12.15.Mm, 13.60.Hb 14.60.Cd 14.65.Bt 29.30.Aj 29 Ca85May 2013Preprint submitted to Elsevier Science 31 May 20131 6 27 28 1 Present address: Richland College,Jefferson LabHall APVDISDAQ 11An experiment that measured the parity-violating asymmetries in deep inelastic scatter-12 ing was completed at the Thomas Jefferson National Accelerator Facility in experimental 13 Hall A. From these asymmetries, a combination of the quark weak axial charge could be 14 extracted with a factor of five improvement in precision over world data. To achieve this, 15 asymmetries at the 10 −4 level needed to be measured at event rates up to 600 kHz and the 16 high pion background typical to deep inelastic scattering experiments needed to be rejected 17 efficiently. A specialized data acquisition (DAQ) system with intrinsic particle identifica-18 tion (PID) was successfully developed and used: The pion contamination in the electron 19 samples was controlled at the order of 2 × 10 −4 or below with an electron efficiency of 20 higher than 91% during most of the production period of the experiment, the systematic 21 uncertainty in the measured asymmetry due to DAQ deadtime was below 0.5%, and the 22 statistical quality of the asymmetry measurement agreed with the Gaussian distribution to 23 over five orders of magnitudes. The DAQ system is presented here with an emphasis on 24 its design scheme, the achieved PID performance, deadtime effect and the capability of 25 measuring small asymmetries. 26 53 q = u, d indicating an up or a down quark, g e A(V ) is the electron axial (vector) 54 coupling and g q V (A) is the quark vector (axial) coupling to the Z 0 boson. In the tree-55level Standard Model, the C 1,2q are related to the weak mixing angle θ W : C 1u = 56 − 1 2 + 4 3 sin 2 θ W , C 2u = − 1 2 + 2 sin 2 θ W , C 1d = 1 2 − 2 3 sin 2 θ W , and C 2d = 1 2 − 57 2 sin 2 θ W . Although the weak mixing angle and the quark weak vector charge C 1q 58 have been measured from various processes [5], the current knowledge of the quark 59 weak axial charge C 2q is poor and their deviations from the Standard Model value 60 would reveal possible New Physics in the quark axial couplings that could not be 61 accessed from other Standard Model parameters.62The goal of JLab E08-011 was to measure the PVDIS asymmetries to statistical 63 precisions of 3% and 4% at Q 2 = 1.1 and 1.9 (GeV/c) 2 , respectively, and under the 64 65 weak charge combination (2C 2u − C 2d ). In addition, the systematic uncertainty 66 goal was less than 3%. For this experiment, the expected asymmetries were 91 and 67 160 ppm respectively at the two Q 2 values [1]. To achieve the required precision, 68 an event rate capability of up to 600 kHz was needed.69The main challenge of deep inelastic scattering experiments is the separation of 70 scattered electrons from the pion background in the spectrometer and detector sys-71 tem. The neutral pions would decay into e + e − pairs from which the electrons pro-72 duced cannot be rejected by detectors. This pair production background was studied 73 by reversing the spectrometer magnet settings and measure the e + yield, and the ef-74 fect on the measured asymmetries was found to be negligible. Charged pions are 75 produced primarily from nucleon resonance decays and could carry a parity vio-76 lation asymmetry corresponding to the Q 2 at which the resonances are produced, 77 typically a fraction of the asymmetry of electrons with the same scattered momen-78 tum.Assuming that a fraction f π/e of the detected events are π − and 1 − f π/e are 79 electrons, the measured asymmetry iswhere A e is the desired electron scattering asymmetry and A π is the asymmetry of 81 the pion background. To extract A e to a high precision, one needs either to mini-82 mize the pion contamination f π/e to a negligible level, or to correct the measured 83 asymmetry for the asymmetry of pions, which itself needs to be measured precisely.84For the PVDIS experiment, the goal was to control f π/e to the 10 −4 level provided 85 that the pion asymmetries did not exceed those of electrons.86The experiment used a 100 µA electron beam with a polarization of approximately 87 90% and a 20-cm long liquid deuterium target. The two High Resolution Spec-88 trometers (HRS) [6] were used to detect scattered events. While the standard HRS 89 detector package and data acquisition (DAQ) system routinely provide a 10 4 pion 90 rejection with approximately 99% electron efficiency, they are based on full record-91 ing of the detector signals and are limited to event rates up to 4 kHz [6]. This is not 92 sufficient for the high rates expected for the experiment. (The HRS DAQ will be 93 referred to as "standard DAQ" hereafter.) 94 Recent parity violation electron scattering experiments, such as HAPPEX [8,9,10,11,12], 95 and PREX [13] at JLab, focused on elastic scattering from nuclear or nucleon tar-96 gets that are typically not contaminated by inelastic backgrounds. Signals from the 97 detectors can be integrated and a helicity dependence in the integrated signal can 98 be used to extract the physics asymmetry. An integrating DAQ was also used in 99 the preceding PVDIS measurement at SLAC [14,15] in which approximately 2% 100 of the integrated signal was attributed to pions. The SAMPLE experiment [7] at 101 MIT-Bates focused also on elastic scattering but the inelastic contamination was 102 more challenging to reject, and an air Cherenkov counter was used to select only 103 3 elastic scattering events. In the Mainz PVA4 experiment [16,17,18], particles were 104 detected in a total absorption calorimeter and the integrated energy spectrum was 105recorded. Charged pions and other background were separated from electrons in 106 the offline analysis of the energy spectrum, and the pion rejection was on the order 107 of 100:1 based on the characteristics of the calorimeter.108High performance particle identification can usually be realized in a counting-based 109 DAQ where each event is evaluated individually. In the G0 experiment [19,20,21,22,23] 110 at JLab, a superconducting spectrometer with a 2π azimuthal angle coverage was 111 used to detect elastically scattered protons at the forward angle and elastic elec-112 trons at the backward angle. At the forward angle, protons were identified using 113 time-of-flight. At the backward angle, pions were rejected from electrons using an 114 aerogel Cherenkov counter, and a pion rejection factor of 125 : 1 or better was 115 reported [23]. The deadtime correction of the counting system was on the order of 116 a few percent [22,23]. 117 While the PVDIS experiment could fully utilize existing spectrometers and detec-118 tors at JLab, examination of all existing techniques for PV measurements made it 119 clear that a custom electronics and DAQ were needed to keep the systematic uncer-120 tainties due to data collection to below 1%. In this paper we describe a scaler-based, 121 cost effective counting DAQ which limited the pion contamination of the data sam-122 ple to a negligible level of f π/e ≈ 10 −4 . Basic information on the detector package 123 and the DAQ setup will be presented first and followed by the analysis of electron 124 detection efficiency, pion rejection and contamination, corrections due to counting 125 deadtime, and the statistical quality of the asymmetry measurement. 126 2 Detector and DAQ Overview 127 The design goal of the DAQ is to record data up to 600 kHz with hardware-based 128 PID and well measured and understood deadtime effects. The following detectors 129 in the HRS [6] were used to characterize scattered particles: Two scintillator planes 130 provided the main trigger, while a CO 2 gas Cherenkov detector and a double-layer 131segmented lead-glass detector provided particle identification information. The ver-132 tical drift chambers (as the tracking detector) were used during calibration runs but 133 were turned off during production data taking because they were not expected to 134 endure the high event rates.135For the gas Cherenkov and the lead-glass detector, a full recording of their out-136 put ADC data was not feasible at the expected high rate. Instead their signals were 137 passed through discriminators and logic units to form preliminary electron and pion 138 triggers. These preliminary triggers were then combined with the scintillator trig-139 gers to form the final electron and pion triggers, which were sent to scalers to record 140 the event counts and used offline to form asymmetries A = (n R − n L )/(n R + n L ), 141 528 even higher event rates or backgrounds with this type of scaler-based DAQ.529Acknowledgments 530 Introduction 29 The Parity-Violating Deep Inelastic Scattering (PVDIS) experiment E08-011 was 30 completed in December 2009 at the Thomas Jefferson National Accelerator Facil-31 ity (JLab). The goal of this experiment [1,2,3] was to measure with high precision 32 the parity-violating asymmetry in deep inelastic scattering of a polarized 6 GeV 33 electron beam on an unpolarized liquid deuterium target. This asymmetry is sensi-34 tive to the quark weak axial charge C 2q which corresponds to a helicity dependence 35 in the quark coupling with the Z 0 boson. 36 For electron inclusive scattering from an unpolarized target, the electromagnetic 37 interaction is parity conserving and is insensitive to the spin flip of the incom- The PVDIS asymmetry from a deuterium target is [4] 46 A P V = − G F Q 2 4 √ 2πα 2g e A Y 1 F γZ 1 (x, Q 2 ) F γ 1 (x, Q 2 ) + g e V Y 3 F γZ 3 (x, Q 2 ) F γ 1 (x, Q 2 ) ,(1) where Q 2 is the negative of the four-momentum transfer squared, G F is the Fermi 47 weak coupling constant, α is the fine structure constant, Y 1 and Y 3 are kinematic 48 factors, x is the Bjorken scaling variable, and F γ(Z) 1,3 (x, Q 2 ) are deuteron structure 49 functions that can be evaluated from the parton distribution functions and the quark-50 Z 0 vector and axial couplings g q V,A . From this asymmetry one can extract the quark 51 weak vector and axial charges C 1,2q , where the quark weak vector charge is defined 52 as C 1q ≡ 2g e A g q V and the quark weak axial charge is given by C 2q ≡ 2g e V g q A with assumption that hadronic physics corrections are small, to extract the quark axial where n R(L) is the integrated rate of the triggers normalized to the integrated beam charge for the right(R) and left(L) handed spin (helicity) states of the incident 143 electron beam. The scalers that counted triggers and the beam charge were inte-144 grated over the helicity period, which was flipped pseudo-randomly at 30 Hz per 145 the experimental technique used by the HAPPEX experiments [12]. 146 For the HRS the two layers of the lead-glass detector are called "preshower" and 147 "shower" detectors, respectively. The preshower in the Right HRS (the spectrome- were split into two identical copies using passive splitters. 178 A schematic diagram of the DAQ electronics for the Right HRS is shown in Fig. 2. 179 Preliminary electron and pion triggers were formed by passing shower (SS) and The PID performance of the DAQ system was studied with calibration runs taken at 243 low beam currents using fbTDC signals along with ADC data of all detector signals 244 recorded by the standard DAQ. Events that triggered the DAQ would appear as a 245 Kine# Table 1 Overview of kinematics settings during the experiment, including: the beam energy E b , the spectrometer central angle setting θ 0 and central momentum setting E ′ 0 , the observed electron rate R e and the π − /e ratio R π − /R e . timing peak in the corresponding fbTDC spectrum of the standard DAQ, and a cut 246 on this peak can be used to select those events. Figure 3 shows the preshower vs. corrections were applied to the asymmetry data. in Fig. 2) as well as to fbTDCs. Figure 5 shows the Cherenkov ADC sum with and 264 without the fbTDC cut, which clearly shows the capability of rejecting pions. Table 2. The discriminator clearly selected electrons while rejecting pions. HRS E b (GeV) θ 0 E ′ 0 (GeV) R e (kHz) R π − /R e DIS#1f π/e,n(w) = R π η GC π η LG π R e η GC e η LG e + R π η LG π {R e η GC e [150 ns − τ n(w) ]} R e η GC e η LG e(3) where R e and R π are the input electron and the pion rates, respectively; η LG(GC) LG, n 101.5 ± 1.6 78.9 ± 0.9 72.7 ± 0.3 LG, w 103.9 ± 1.7 81.5 ± 1.0 74.3 ± 0.3 Pion contamination in the electron trigger f π/e , narrow path (×10 −4 ) f π/e,n Table 2 Average electron detection efficiency and pion rejection factor of electron triggers achieved for the DIS kinematics through the lead glass (LG) and the gas Cherenkov (GC) detectors, respectively, and the combined performance. The error bars of the efficiencies and the rejection factors are statistical only. The error bars for f π/e , ∆f π/e,n(w) , are shown separately for statistical uncertainties, systematic uncertainties due to our understanding of the rates, detector efficiencies and deadtimes, and day-to-day variations in the measured detector efficiencies. Table 3 Average electron detection efficiency and pion rejection factor of electron triggers achieved for the resonance kinematics through the lead glass (LG) and the gas Cherenkov (GC) detectors, respectively, and the combined performance. The error bars of the efficiencies and the rejection factors are statistical only. The error bars for f π/e , ∆f π/e,n(w) , are shown separately for statistical uncertainties, systematic uncertainties due to our understanding of the rates, detector efficiencies and deadtimes, and day-to-day variations in the measured detector efficiencies. To understand fully the effect of pion background on the measured electron asymmetry, it is important to extract asymmetries of the pion background to confirm that they are indeed smaller than the electron asymmetry. A complete PID analysis was carried out on the pion triggers of the DAQ where the electron contamination in the pion trigger f e/π was evaluated in a similar method as f π/e above, following f e/π,n(w) = R e ξ GC e ξ LG e R π ξ GC π ξ LG π + R e ξ LG e {R π ξ GC π [150 ns − τ n(w) ]} R π ξ GC π ξ LG π(4) where as before R e and R π are the electron and the pion rates incident on the de- 317 To illustrate the importance of the deadtime, consider its effect on the asymmetry A. to the deadtime as listed below: 325 (1) The "group" deadtime: deadtime due to discriminators and logical AND mod-326 ules used to form group triggers. 327 (2) The "GATE" deadtime: deadtime from the GATE circuit that used scintillators 328 and gas Cherenkov signals to form the GATE signals, which controlled the Table 4 Average pion detection efficiency and electron rejection factor of pion triggers achieved for DIS kinematics through the lead glass (LG) and the gas Cherenkov (GC) detectors, respectively, and the combined performance. The error bars of the efficiencies and the rejection factors are statistical only. The error bars for f e/π , ∆f e/π,n(w) , are shown separately for statistical uncertainties, systematic uncertainties, and day-to-day variations in the measured detector efficiencies. Table 5 Average pion detection efficiency and electron rejection factor of pion triggers achieved for resonance kinematics through the lead glass (LG) and the gas Cherenkov (GC) detectors, respectively, and the combined performance. The error bars of the efficiencies and the rejection factors are statistical only. The error bars for f e/π , ∆f e/π,n(w) , are shown separately for statistical uncertainties, systematic uncertainties, and day-to-day variations in the measured detector efficiencies. AND (OR) module of each group to form group electron (pion) triggers. 330 (3) The "OR" deadtime: deadtime due to the logical OR module used to combine 331 all group triggers into final global triggers. 332 The total deadtime is a combination of all three. In order to evaluate the DAQ 333 deadtime, a full-scale trigger simulation is necessary. This trigger simulation will 334 be described in the next section followed by results on the group, GATE, and OR 335 deadtime as well as on the total deadtime correction that was applied to the asym-336 metry data. Fig. 7. Top: A fraction of the group electron trigger. Each point corresponds to: 1 -Shower sum of the group; 2 -Total shower sum of the group; 3 -Total shower discriminator output (high threshold), narrow path; 4 -Preshower discriminator output (medium threshold), narrow path; 5 -group electron trigger, narrow path; 6 -Total shower discriminator output, wide path; 7 -Preshower discriminator output, wide path; 8 -group electron trigger, wide path. Bottom: Signals 1-8 as simulated by HATS. One can see that the second physical event is recorded by the narrow path group trigger (5) but not the wide path (8) due to deadtime loss. Group Deadtime Measurement 355 In order to study the group deadtime, a high rate pulser signal ("tagger") was mixed 356 with the Cherenkov and all preshower and total shower signals using analog sum- 384 The pileup effect can be measured using the delay between the tagger-385 trigger coincidence output and the input tagger. This is illustrated in Fig. 8 and 386 the pileup effect contributes to both I 1 and I 2 regions of the fbTDC spectrum. 387 The I 1 distribution is produced by PMT pulses that arrive after the delayed 388 tagger signal but before the tagger signal would propagate through the trig-389 ger electronics. Peak I 2 occurs when a PMT pulse arrives at the coincidence 390 module earlier than the delayed tagger signal but which forms a coincidence 391 with the delayed tagger signal, giving an output whose time is set by the lat- The relative loss of tagger events due to DAQ deadtime is evaluated as D = 1 − (1 − p)(R o /R i ),(5) where R i is the input tagger rate, R o is the output tagger-trigger coincidence rate, 398 and p = (I 1 + I 2 )/I 0 is a correction factor for pileup effects as defined in Fig. 8. 399 Results for the deadtime loss D are shown in Figs. 9 and 10 The two logical OR units immediately following the tagger input "B" serve as width adjusters. Bottom: fbTDC spectrum for the relative timing between tagger-trigger coincidence and the input tagger. The fbTDC module worked in a common stop and the multi-hit mode. Two different scenarios are shown: 1) Main peak I 0 (hollow peak): when there is no PMT signal preceding the tagger, the tagger triggers the DAQ and forms a tagger-trigger coincidence. 2) Pileup events I 1 (light-shaded region) and I 2 (heavy-shaded region): when there is a PMT signal preceding the tagger by a time interval shorter than w +t 1 , the PMT signal triggers the DAQ and forms a tagger-trigger coincidence signal with the delayed tagger. HRS and group 4 on the right HRS, respectively, and are compared with simula- These data were taken (or simulated) at kinematics DIS #2. The total group deadtime can be determined from the linear fit slope coefficients: tagger data narrow (71.1 ± 0.9) × 10 −9 s, wide (107 ± 1.2) × 10 −9 s, simulation narrow (73.9 ± 1.5) × 10 −9 s, wide (115 ± 1.5) × 10 −9 s. Group 4 is from the central blocks of the lead-glass detector and has the highest rate among all groups. See Fig. 9 caption for more details. Fig. 11. A zoom-in view of the GATE electronics for Left (top) and Right (bottom) spectrometers. shortly after. The fractional loss due to GATE deadtime can be estimated as data. Figure 12 shows spectra of the timing difference between the gas Cherenkov Timing difference between Gas Cherenkov and Scintillator signals in 5-ns channels. These data were taken with a beam current of 110 µA and at kinematics DIS#1 on the Left and DIS#2 on the Right HRS, respectively. The fractional loss of electron events due to GATE deadtime can be estimated using the ratio of N 1 /N 0 , where N 1 is the count difference between the two spectra in the dead zone, and N 0 is the counts under the main peak near 0 ns. See text for details. Figure 13 shows comparisons of the fractional losses due to GATE deadtime es-464 timated using trigger simulation, the analytic method Eq. (6), and FADC data extracted from Fig. 12 from the total deadtime, all three of which were direct results from the simulation. Gate Deadtime Evaluation 430 DT gate = R SC&GC (w SC,in − w SC,out ) + R GC&SC (w GC,in − w GC,out ),(6) 480 The difference between the analytic method and trigger simulation was used as the 481 systematic uncertainty of the OR deadtime. The simulated deadtime loss of the global electron triggers and its decomposition 484 into group, GATE, and OR are shown in Table 6, along with the total deadtime 485 correction at a beam current of 100 µA. The total deadtime loss not only increases 486 with higher electron rate R e , but also with higher pion to electron ratio R π /R e 487 (see Table 1) which would cause larger GATE deadtime. The deadtime loss is also 488 shown in Fig. 14 as a function of the total event rate. 489 Table 6 Simulated DAQ deadtime loss in percent for all kinematics and for both narrow (n) and wide (w) paths, along with the fractional contributions from group, GATE, and OR deadtimes. The fractional deadtime from OR was calculated as one minus those from group and GATE, and its uncertainty was estimated from the difference between simulation and the analytical results. The variation of group deadtime contribution among kinematics is due to changes in the rate distribution among different groups. The uncertainty of the total deadtime is the uncertainties from group, GATE and OR added in quadrature. Kine, Table 6 for final uncertainty evaluation of the total deadtime loss. Asymmetry Measurement 496 The physics asymmetries sought for in this experiment were expected to be in the 503 The asymmetries can be formed from event counts of each beam helicity pair, with 504 33-ms of helicity right and 33-ms of helicity left beam, normalized by the beam 505 charge. Figure 15 shows the pull distribution of these pair-wise asymmetries with 506 the "pull" defined as 507 p i ≡ (A i − A )/δA i ,(7) where A i is the asymmetry extracted from the i-th beam helicity pair with the HWP 508 states already corrected and δA i = 1/ N R i + N L i its statistical uncertainty with N R(L) i the event count from the right (left) helicity pulse of the pair, and A is the 510 asymmetry averaged over all beam pairs. One can see that the asymmetry spectrum Summary 514 A scaler-based counting DAQ with hardware-based particle identification was suc-515 cessfully implemented in the 6 GeV PVDIS experiment at Jefferson Lab to mea-516 sure parity-violating asymmetries at the 10 −4 level at event rates of up to 600 kHz. 517 Asymmetries measured by the DAQ followed Gaussian distributions as expected 518 from purely statistical measurements. Particle identification performance of the 519 DAQ was measured and corrections were applied to the data on a day-to-day basis. 520 The overall pion contamination in the electron sample was controlled to approxi-521 mately 2 × 10 −4 or lower, with an electron efficiency above 91% during most of the 522 data production period of the experiment. The DAQ deadtime was evaluated from a 38 ing electron beam. Only the weak interaction violates parity and causes a differ-39 ence between the right-and the left-handed electron scattering cross-sections σ R 40 and σ L . The dominant contribution to the parity violation asymmetry, A P V ≡ 41 (σ R − σ L )/(σ R + σ L ), arises from the interference between electromagnetic and 42 weak interactions and is proportional to the four momentum transfer squared Q 2 43 for Q 2 ≪ M 2 Z . The magnitude of the asymmetry is on the order of 10 −4 or 10 2 parts 44 per million (ppm) at Q 2 = 1 (GeV/c) 2 .45 ter located to the right side of the beamline when viewed along the beam direction)149 has 48 blocks arranged in a 2 × 24 array, with the longest dimension of the blocks 150 aligned perpendicular to the particle trajectory. For the two blocks in each row, only151 the ends facing outward are read out by photo-multiplier tubes (PMTs), while the 152 other ends of the two blocks are facing each other and not read out. Therefore, the 153 preshower detector has 48 output channels. All preshower blocks were individually 154 wrapped to prevent light leak. The shower detector in the Right HRS had 75 blocks 155 arranged in a 5 × 15 array with the longest dimension of the blocks aligned along 156 the trajectory of scattered particles. PMTs were attached to each block of the Right 157 shower detector on one end only, giving normally 75 output channels. However to 158 minimize the electronics needed for this experiment (see next paragraph), only 60 159 of the 75 shower blocks were used while signals from the 15 blocks on the edge 160 were not utilized by the DAQ. The reduction of the HRS acceptance due to not 161 using these side blocks was negligible. The preshower and the shower detectors in 162 the Left HRS are similar to the preshower detector on the Right HRS except that 163 for each detector there are 34 blocks arranged in a 2 × 17 array. 164 Because the lead-glass detectors in the Left and Right HRS are different, design of 165 the lead-glass-based triggers of the DAQ is also different, as shown in Fig. 1. As a 166 compromise between the amount of electronics needed and the rate in the front end 167 logic modules, the lead-glass blocks in both the preshower and the shower detectors 168 were divided into 6 (8) groups for the Left (Right) HRS, with each group consist-169 ing typically 8 blocks. Signals from the 8 blocks in each group were added using a 170 custom-made analog summing unit called the "SUM8 module", then passed to dis-171 criminators. The geometry and the position of each preshower group were carefully 172 chosen to match those of the corresponding shower group to maximize electron de-173 tection efficiency. On the Left HRS, adjacent groups in both preshower and shower 174 had overlapping blocks, while for the Right HRS only preshower groups were over-175 lapping. To allow overlap between adjacent groups, signals from preshower blocks 176 on the Right HRS and from both preshower and shower blocks on the Left HRS177 preshower (PS) signals and their sums, called total shower (TS) signals, through 181 discriminators with different thresholds. For electron triggers, logical ANDs of 182 the PS discriminator and the TS discriminator outputs were used. For pions, low 219 Full 219sampling of partial analog signals was done using Flash-ADCs (FADCs) at 220 low rates intermittently during the experiment. For one group on the Left and one 221 group on the Right HRS, the preshower and the shower SUM8 outputs, the inter-222 mediate logical signals of the DAQ, and the output electron and pion triggers were 223 recorded. These FADC data provided a study of pileup effects to confirm the dead-224 time simulation and to provide the input parameters for the simulation, specifically 225 the rise and fall times of the signals and their widths. the experiment data were taken at two deep inelastic scattering (DIS) kine-228 matics at Q 2 = 1.1 and 1.9 (GeV/c) 2 . These were the main production kinematics 229 and will be referred to as DIS#1 and DIS#2, respectively. Due to limitation of the230 spectrometer magnets, DIS#1 was taken only on the Left HRS, while DIS#2 was 231 taken on both Left and Right HRSs. In addition, data were taken at five kinematics 232 within or near the nucleon resonance region with their invariant mass W between 233 the ∆(1232) resonance and just above W = 2 GeV. These data were used for the 234 purpose of radiative corrections and will be referred to as RES I through V (al-235 though kinematics V was located slightly above W = 2 GeV). Data for each of 236 the resonance settings were taken only with one HRS because of the spectrometer 237 magnet limitations as well as to optimize the beam time allocation. The kinematic 238 settings are shown inTable 1along with the observed electron rate R e and the pion 239 to electron ratio R π − /R e in the HRS. The highest electron rate occurred at RES II 240 at approximately 600 kHz. 247shower signals for group 2 on the Left HRS. A comparison between no fbTDC 248 cut and with cut on the fbTDC signal of the electron wide trigger from this group 249 clearly shows the hardware PID cuts. Fig. 3 .Fig. 4 . 34Preshower vs. Shower ADC data (sum of 8 blocks each) for group 2 on the Left HRS, without the fbTDC cut (left panel) and with cut on the group 2 electron wide trigger fbTDC signal (right panel). This clearly shows the thresholds of the preshower and the total shower signals, indicating that the DAQ is selecting the correct events as electrons.250 Electron efficiency and pion rejection factors of the lead-glass detector on the Left 251 HRS during a one-hour run are shown in Fig. 4 as functions of the location of the 252 hit of the particle in the preshower detector. PID performance on the Right HRS is 253similar. Electron efficiency from wide groups is slightly higher than from narrow 254 groups because there is less event loss due to timing misalignment when taking 255 the coincidence between the preshower and the total shower discriminator outputs.256Variations in the electron efficiency across the spectrometer acceptance effectively 257 influence the Q 2 of the measurement. For this reason, low-rate calibration data 258 were taken daily during the experiment to monitor the DAQ PID performance, and Electron detection efficiency (left) and pion rejection factor (right) vs. vertical (dispersive) hit position of the particle in the preshower detector for the narrow electron triggers in the Left HRS. An 8-minute run with a reduced beam current of 2 µA at kinematics DIS #2 was used in this evaluation. For electron efficiencies, the total efficiency and the statistical error bars are shown as the curve, while the shaded area indicates events that were recorded by two adjacent groups. The average electron efficiency achieved by the lead glass detector alone for this run is [94.60 ± 0.11(stat.)]% and the average pion rejection factor is [76.2 ± 1.5(stat.)] : 1. PID performance for the wide path and the Right HRS are similar. 260 The gas Cherenkov detector signals were read out by 10 PMTs on both the Left and 261 the Right HRS. Signals from all 10 PMTs were summed in an analog-sum module 262 and sent to a discriminator. The discriminator output was sent to the DAQ (as shown 263 265Fig. 5 . 5As described in the Introduction, pion contamination in the electron trigger would 266 affect the measured electron asymmetry as A m = (1 − f π/e )A e + f π/e A π where A m 267 and A e are the measured and the true electron asymmetries, respectively, and A π is 268 the parity violation asymmetry of pion production. The pion contamination in the 269 electron trigger, f π/e , comes from two effects: There is a small possibility that a270 pion could trigger both the lead-glass and the gas Cherenkov detectors, causing a 271 false electron trigger output. This possibility is determined by the direct combina-272 tion of the pion rejection factors of the two detectors and is below 10 −4 . A larger 273 effect comes from the width of the electron GATE signal: Since each coincidence 274 between the gas Cherenkov and the scintillator signals would open the electron 275 counting GATE by 150 ns, while the DAQ deadtime of the lead-glass detector is 276 less than this value, pions that arrived after the DAQ deadtime but before the clos-277 ing of the electron GATE signal would cause a false electron trigger. The sum of 278the two effects can be written as [Color online] Gas Cherenkov ADC data (sum of 10 PMTs) for the Left HRS during a one-hour run at kinematics DIS #2, with a fbTDC cut on the Cherenkov discriminator output and without. The beam current during this run was about 100 µA, the incident electron rate on the detector was about 23 kHz with a pion to electron rate ratio of approximately 3.5. The electron efficiency achieved by the gas Cherenkov alone for this kinematics on the Left HRS was approximately 99% with a pion rejection of approximately 300:1, see electron detection efficiency of the lead-glass (gas Cherenkov) detectors, and 281 η LG(GC) π is the pion detection efficiency, i.e., the inverse of the rejection factor, 282 of the lead-glass (gas Cherenkov) detector. The DAQ group deadtime of the lead-283 glass detector for the narrow (wide) path, τ n(w) , is approximately 60 ns (100-110 284 ns) and the analysis obtaining these results will be presented in the next section.285The term R e η GC e [150 ns − τ n(w) ] gives the probability of a pion's arriving within 286 a valid electron GATE signal and thus such a pion can not be rejected by the gas 287 Cherenkov detector.288The electron detection efficiency and pion rejection factor averaged throughout the 289 data production period are shown inTables 2 and 3for DIS and resonance kine-290 matics, respectively, along with the resulting pion contamination f π/e evaluated 291 separately for the narrow and the wide paths.292As shown in Tables 2-3, the overall pion contamination was on the order of 2 × 293 10 −4 or lower throughout the experiment. Because pions are produced from nucleon 294 resonance decays, the parity violation asymmetry of pion production is expected to 295 be no larger than that of scattered electrons with the same momentum. This was 296 confirmed by asymmetries formed from pion triggers during this experiment. The 297 uncertainty in the electron asymmetry due to pion contamination is therefore on the 298 order of 2 ×10 −4 and is negligible compared with the 3 −4% statistical uncertainty. tectors, respectively; the detection efficiencies ξ are now defined for the pion trig-301 gers of the DAQ: ξ LG(GC) e is the electron detection efficiency of the lead-glass (gas 302 Cherenkov) detectors, and ξ LG(GC) π is the pion detection efficiency of the lead-glass 303 (gas Cherenkov) detector. Although the goal of the pion triggers is to collect pions, 304 only the gas Cherenkov played a role in rejecting electrons in the pion trigger, and 305 all electrons would form valid pion triggers in the lead-glass counters. Therefore 306 ξ LG e ≈ 1 and the electron contamination is high. Results for electron contamination 307 in the pion trigger are summarized in Tables 4 and 5.308 5 DAQ Deadtime 309 Deadtime is the amount of time after an event during which the system is unable 310 to record another event. Identifying the exact value of the deadtime is always a 311 challenge in counting experiments. By having a narrow and a wide path, we can 312 observe the trend in the deadtime: The wider path should have higher deadtime. By 313 matching the observed trend with our simulation we can benchmark and confirm 314 the result of our deadtime simulation. In addition, dividing lead-glass blocks into 315 groups greatly reduces the deadtime loss in each group compared with summing 316 all blocks together and forming only one final trigger. 318 For a simple 318system with only one contribution to the deadtime loss δ, the observed 319 asymmetry A O is related to the true asymmetry A according toA O = (1 − δ)A.In 320 this experiment δ was expected to be on the order of (1-2)%. Since the statistical 321 accuracy of the asymmetry is (3-4)%, it was desirable to know δ with a (10-20)% 322 relative accuracy so that it would become a negligible systematic error. The DAQ 323 used in this experiment, however, was more complex and had three contributions324 TheFig. 6 . 6Hall A Trigger Simulation (HATS) was developed for the purpose of dead-339 time study for this experiment. The inputs to HATS include the analog signals for 340 preshower, shower, scintillator and gas Cherenkov. The signal amplitudes were pro-341 vided by ADC data from low-current runs, and the signal rates were from high-342 current production runs. The rise and fall times for the preshower and shower 343 SUM8 outputs play an important role in HATS. The signal shape is simulated by 344 the function S(t) = Ate −t/τ , where A is related to the amplitude of the signal, and 345 the time constant τ was determined from FADC data, see Fig. 6. [Color online] Calibration of time constants τ for Preshower (left) and Shower (right) of the Right HRS. The FADC snapshots (histograms) is compared with the fit S(t) = Ate −t/τ (smooth dashed curves). The time constant τ was found to be approximately 11 ns for the Right HRS Preshower, and 21 − 22 ns for the Left HRS Preshower and Shower as well as for the Right HRS Shower.With the recorded DAQ electronics and delay cables, HATS first rebuilds the DAQ 347 system on the software level. At each nano-second, detector input signals are gen-348 erated randomly according to the actual event rates and signal shape, and HATS 349 simulates output signals from all discriminators, AND, and OR modules.Figure 7 shows a fraction of the DAQ electronics and the simulated results for a very short351 time period. By comparing output with input signals, HATS provides results on the 352 fractional loss due to deadtime for all group and global triggers with respect to the 353 input signal. 364 ( 1 ) 372 ( 2 ) 36413722ming modules, see Figs. 2 and 8. In the absence of all detector signals, a tagger 358 pulse produces without loss an electron trigger output and a "tagger-trigger coinci-359 dence" pulse between this output and the "delayed tagger" -the tagger itself with 360 an appropriate delay to account for the DAQ response time. When high-rate detec-361 tor signals are present, however, some of the tagger pulses would not be able to 362 trigger the DAQ due to deadtime. The deadtime loss in the electron trigger output 363 with respect to the tagger input has two components: The count loss R o /R i : When a detector PMT signal precedes the tagger signal 365 by a time interval δt shorter than the DAQ deadtime but longer than w + t 1 , 366 the tagger signal is lost and no coincidence output is formed. Here w is the 367 width of the electron trigger output and t 1 is the time interval by which the 368 delayed tagger precedes the tagger's own trigger output, seeFig. 8. During369 the experiment w was set to 15 ns for all groups, and t 1 was measured at the 370 end of the experiment and found to be between 20 and 40 ns for all narrow 371 and wide groups of the two HRSs. The pileup fraction p: When a PMT signal precedes the tagger signal by a time 373 interval δt shorter than w + t 1 , there would be a coincidence output between 374 the delayed tagger and the electron output triggered by the detector PMT sig-375nal. If furthermore δt is less than the DAQ deadtime (which is possible for 376 this experiment since the deadtime is expected to be as long as 100 ns for 377 the wide path), the tagger itself is lost due to deadtime, and the tagger-trigger 378 coincidence is a false count and should be subtracted. In the case where δt 379 is shorter than w + t 1 but longer than the DAQ deadtime (not possible for 380 this experiment but could happen in general), the tagger itself also triggers a 381 tagger-trigger coincidence, but in this case, there are two tagger-trigger coin-382 cidence events. Both are recorded by the fbTDC if working in the multi-hit 383 mode, and one is a false count and should be subtracted. ter. Fractions of I 1 and I 2 relative to I 0 are expected to be I 1 /I 0 = Rt 1 and 393 I 2 /I 0 = Rw, respectively, where R is the PMT signal rate. The pileup effect 394 was measured using fbTDC spectrum for electron narrow and wide triggers 395 for all groups. Data for I 1,2 extracted from fbTDC agree very well with the 396 expected values. Fig. 8 . 8[Color online] Top: schematic diagram for the tagger setup and signal timing sequence. Fig. 9 . 9tion. Different beam currents between 20 and 100 µA were used in this dedicated 402 deadtime measurement. In order to reduce the statistical fluctuation caused by the 403 limited number of trials in the simulation within a realistic computing time, simu-404 lations were done at higher rates than the actual measurement. [Color online] Deadtime loss vs. event rate from the tagger method for group 4 on the Left HRS. Top: actual deadtime loss from tagger measurements; Bottom: simulated deadtime loss of the tagger. The tagger fractional count loss 1 − R o /R i (fit by solid and dashed lines) and the pileup correction p (fit by dotted and dash-dotted lines) are combined to form the total group deadtime D (fit by dash-double-dotted and dash-triple-dotted lines). These data were taken (or simulated) at kinematics DIS #1. To minimize the statistical uncertainty while keeping the computing time reasonable, the simulation used higher event rates than the tagger measurement. The total group deadtime can be determined from the linear fit slope coefficients: tagger data narrow (61.5 ± 0.2) × 10 −9 s, wide (99.9 ± 0.3) × 10 −9 s, simulation narrow (62.5 ± 1.4) × 10 −9 s, wide (102 ± 1.3) × 10 −9 s. Group 4 is from the central blocks of the lead-glass detector and has the highest rate among all groups.The slope of the tagger loss vs. event rate, as shown in Figs. 9 and 10, gives the 406 value of group deadtime in seconds. One can see that the deadtime for the wide path 407 is approximately 100 ns as expected. The deadtime for the narrow path, on the other 408 hand, is dominated by the input PMT signal width (typically 60-80 ns) instead of 409 the 30-ns discriminator width. The simulated group deadtime agrees with the data 410 at a 10% level or better, for both HRSs and for both wide and narrow paths. 10. [Color online] Deadtime loss vs. group event rate from the tagger method for group 4 on the Right HRS. Top: tagger data; Bottom: simulation. 417 Figure 11 41711shows the GATE electronics for both spectrometers, with the bottom 418 panel reproducing the GATE portion of Fig. 2. It contributes to the total deadtime 419 as follows: When both the gas Cherenkov and the Scintillator are triggered by elec-420 trons, the two signals align in time and produce an electron GATE signal. However 421 the Scintillator can be triggered by pions and other backgrounds, most of which do 422 not trigger the Cherenkov. If an electron event arrives shortly after such background 423 events, it triggers the Cherenkov but may not trigger the PS755 module that first 424 processes the Scintillator signal because of the non-updating feature of PS755. In 425 this case, the Cherenkov signal triggered by the electron may miss the Scintillator 426 signal from the previous pion or background event and will not produce a valid 427 electron GATE signal. Likewise, if a background event triggers the Cherenkov but 428 not the Scintillator, it would also cause a loss to the electron events that follow 429 452 ( 452GC) and the Scintillator (SC) signals extracted from FADC data. Timing of the 453 GC signal should represent the timing of an electron event, while the SC signal can 454 be triggered by the same electron (as represented by the main peak near 0 ns), or 455 a pion event that preceded the electron (as represented by the region < 0 ns). The 456region beyond ±100 ns were pure random events since the SC signal input to the 457 GATE electronics was only 100 ns wide. As one can see, the region between -100 458 and ≈ −30 ns represents a "dead zone" where the preceding pion triggered the 459 PS755 unit that first processed the SC signal, and caused the electron events that 460 followed to not trigger the GATE circuit. The probability for the electron events to 461 not be recorded by the DAQ due to this GATE deadtime is thus the ratio of the dead 462 zone area (N 1 ) and the area of the main peak near 0 ns (N 0 ), seeFig. 12. Fig. 12 . 12[Color online] Fig. 14 . 14[Color online] Simulated deadtime loss of the global electron trigger for the three DIS spectrometer and kinematics combinations and the five resonance kinematics, for the narrow path (open triangles, fit by dashed lines) and the wide path (open squares, fit by solid lines). The error bars shown are due to statistical uncertainty of the simulation. See order of 10 2 ppm. The measured asymmetries were about 90% of the expected 498 values due to beam polarization. To understand the systematics of the asymmetry 499 measurement, a half-wave plate (HWP) was inserted in the beamline to flip the laser 500 helicity in the polarized source during half of the data taking period. The measured 501 asymmetries flipped sign for each beam HWP change and the magnitude of the 502 asymmetry remained consistent within statistical error bars. full-scale timing simulation and contributed an uncertainty of no more than 0.5% to524 the final asymmetry results. Systematic uncertainties from the pion contamination 525 and the counting deadtime therefore were both negligible compared to the (3 −4)% 526 statistical uncertainty and other leading systematic uncertainties. Results presented 527 Fig. 1. Grouping scheme (side-view) for the double-layer lead-glass detectors for the Left and the Right HRS. Scattered particles enter the detector from the left. The dashed vertical bars represent the range of each group. The Right HRS Shower blocks are labeled as 1 through 64 for historical reasons, but row 16 (blocks 16, 32, 48 and 64) was not present during this experiment.threshold discriminators on the TS signal alone were sent to logical OR modules 184 to produce preliminary triggers. Additional background rejection was provided by 185 the "GATE" circuit, which combined signals from the gas Cherenkov (GC) and the 186 "T1" signal [6] from the scintillators (SC). Each valid coincidence between GC and T1 would produce a 150-ns wide electron GATE signal that allowed an output to be 188 formed by the logical AND modules from the preliminary electron triggers. Each valid T1 signal without the GC signal would produce a 150-ns wide pion GATE signal that allowed an output to be formed by the logical OR modules from the191 preliminary pion triggers. The outputs of the logical AND and OR modules are 192 called group electron and pion triggers, respectively. All six (eight) group electron or pion triggers were then ORed together to form the global electron or pion trigger 194 for the Left (Right) HRS. All group and the final electron and pion triggers were 195 counted using scalers. Because pions do not produce large enough lead-glass signals to trigger the high threshold TS discriminators for the electron triggers, pions 197 do not introduce extra counting deadtime for the electron triggers. However, the 198 150-ns width of the electron GATE signal would cause pion contamination in the 199 electron trigger. This effect will be presented in Section 4.In order to monitor the counting deadtime of the DAQ, two identical paths of elec-Electronics diagram for the Right HRS DAQ used by the PVDIS experiment. The Sum8's, discriminators and logic modules for two groups are shown, as well as the location of tagger signal inputs, setup of the GATE circuit using scintillator (SC) and gas Cherenkov (GC) signals, the logic units for combining triggers from all eight groups into final triggers, the counting scalers, and the monitoring fastbus TDCs.1 2 3 4 5 6 7 8 10 11 9 12 13 14 15 16 17 Left HRS Preshower Shower (1,18) (2,19) (17,34) (1,18) (2,19) (17,34) group1 group1 group3 group4 group4 group3 group5 group6 group6 group5 group2 group2 25 26 27 28 29 30 31 32 33 34 35 36 37 39 38 40 41 42 43 44 45 46 47 48 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 18 19 17 20 21 22 23 24 Right HRS Shower Preshower S01,17,33,49 S02,18,34,50 S03,19,35,51 S04,20,36,52 S05,21,37,53 S06,22,38,54 S08,24,40,56 S07,23,39,55 S10,26,42,58 S11,27,43,59 S12,28,44,60 S09,25,41,57 S13,29,45,61 S14,30,46,62 S15,31,47,63 12 group1 group2 group3 group4 group5 group6 group7 group8 group8 group7 group6 group5 group4 group3 group2 group1 187 189 190 193 196 200 201 ps1h ts1h 757 NIM−ECL NIM−ECL NIM−ECL NIM−ECL NIM−ECL NIM−ECL narrow electron 20ns en OR 757 copy copy copy copy tagger 1/4cp1 tagger 1/4cp2 tagger 1/4cp4 tagger cp1 tagger cp2 p8wa NIM−ECL NIM−ECL e8wa e1wa NIM−ECL NIM−ECL p8na p1na NIM−ECL NIM−ECL e8na e1na scaler fbTDC scaler fbTDC ps1h NIM−ECL NIM−ECL ps8h scaler fbTDC NIM−ECL NIM−ECL ts1h ts8h S19 S35 S51 S03 S18 S34 S50 S02 S49 S33 S17 S01 tagger 1/4cp3 ADC FADC HRS TS2 DH TS2 30 TS2 DM 30 PS2 100 PS2 HRS ADC FADC FADC FADC TS1 TS1 DL TS1 30 PS1 DM 100 PS1 FADC B1 A1 A1 B1 Sum8 Sum8 Sum8 758 DL DH 100 30 ADC ADC HRS ADC 30ns electron narrow pion pion V electron 100ns TS1 DL 100 100ns pion wide 100ns DL TS2 100 TS2(DL−100ns) TS1(DL−20ns) pion 30ns pion 30ns V V V V 70ns 1 V 150ns 150ns OR & & & & OR (Total Sum1) TS1 TS2 2 1 2 & 706 758 758 706 706 706 706 706 755 757 757 OR OR 755 150ns 150ns P24Lb PS24 P23La P23Lb P23Ra P23Rb P24La P5,a P6,a Sum8 30 DH 100 DH 30 706 electron 30ns electron 100ns ts2h ps2h e1na e1wa e1nb e1wb e2nb e2wb e2na e2wa p1wa p2wa p1na p2na p1wb p2wb pw p2nb p1nb OR electron e1wb e2wb ew wide 100ns 20ns pn p1wa en ew pn pw Final production scalers e2nb e1nb Σ Σ HRS Σ Σ x8 x8 x8 x8 (BNC to LEMO) patch panel PS740 PS740 PS740 PS740 DM DM PS758 PS758 16ns C&S 755 20ns 32ns 757 C&S 755 SC(T1) passive splitter P1La P1Lb P1Ra P1Rb P2La P2Lb P2Ra P2Rb 1/2 1/2 1/2 1/2 PS47 PS48 1/2 1/2 1/2 1/2 48ns GC e8nb p8wb p8nb e8wb PS1 PS25 PS2 PS26 PS23 Group pion wide triggers Group electron wide triggers Group pion narrow triggers Group electron narrow triggers Preshower wide triggers Total shower wide triggers group electron narrow pion narrow group triggers wide global triggers GATE circuit triggers group pion wide triggers triggers electron group pion GATE electron GATE P24Ra P24Rb PS gr1 Shower gr1 Shower gr2 PS gr2 60ns 100ns 100ns 757 Fig. 2. [Color online] The dis- criminators had three different levels of threshold settings: low threshold (DL) was used on the total shower (TS) signals to form pion triggers; medium (DM) and high thresholds (DH) were used on preshower (PS) and TS signals respectively to form electron triggers. During the experiment the thresholds were approximately -20 mV for DL and DM, and in the range (−50, −70) mV for DH depending on the momentum setting of the spectrometer. Electronics for the Left HRS are similar except for the grouping scheme. tronics were constructed. The only difference between the two paths is in the PS 202 and the TS discriminator output widths, set at 30 ns and 100 ns for the "narrow" and 203 the "wide" paths, respectively. The scalers are rated for 250 MHz (4 ns deadtime) 204 and therefore do not add to the deadtime. In addition, the output width of all logic 205 modules was set to 15 ns, so the deadtime of the DAQ for each group is dominated 206 by the deadtime of the discriminators. Detailed analysis of the DAQ deadtime will 207 be presented in Section 5. 208 The SUM8 modules used for summing all lead-glass signals also served as fan-out 209 modules, providing exact copies of the input PMT signals. These copies were sent to the standard HRS DAQ for calibration. During the experiment, data were col- 211 lected at low rates using reduced beam currents with both DAQs functioning, such 212 that a direct comparison of the two DAQs could be made. Vertical drift chambers 213 were used during these low rate DAQ studies. Outputs from all discriminators, sig- 214 nals from the scintillator and the gas Cherenkov, and all electron and pion group 215 and global triggers were sent to Fastbus TDCs (fbTDC) and were recorded in the 216 standard DAQ. Data from these fbTDCs were used to align the amplitude spectrum 217 and timing of all signals. They also allowed the study of the Cherenkov and the 218 lead-glass detector performance for the new DAQ. Pion contamination in the electron trigger f π/e , narrow path (×10 −4 )Resonance Kinematics and Spectrometer combinations RES I RES II RES III RES IV RES V HRS Left Left Right Left Left Electron detection efficiency η e (%) GC 99.16 ± 0.09 99.17 ± 0.13 98.59 ± 0.11 99.41 ± 0.07 99.16 ± 0.11 LG, n 97.73 ± 0.07 97.13 ± 0.07 98.14 ± 0.06 84.71 ± 0.18 84.66 ± 0.21 LG, w 98.32 ± 0.07 97.83 ± 0.08 98.56 ± 0.06 85.31 ± 0.18 85.88 ± 0.23 GC+LG, n 96.91 ± 0.11 96.32 ± 0.15 96.76 ± 0.12 84.20 ± 0.20 83.95 ± 0.24 GC+LG, w 97.49 ± 0.11 97.02 ± 0.15 97.17 ± 0.13 84.80 ± 0.20 85.16 ± 0.26 Pion rejection 1/η π GC 82.8 ± 9.2 97.7 ± 10.5 195.0 ± 24.5 149.6 ± 10.2 151.4 ± 11.5 LG, n 43.6 ± 4.0 57.4 ± 5.4 37.0 ± 0.9 182.4 ± 15.1 207.2 ± 20.5 LG, w 39.4 ± 3.6 53.5 ± 5.1 33.9 ± 0.9 171.4 ± 14.1 201.1 ± 23.5 f π/e,n 0.79 2.40 3.82 0.26 0.45 (stat.) ±0.02 ±0.06 ±0.02 ±0.01 ±0.01 (syst.) ±0.11 ±0.19 ±0.23 ±0.02 ±0.03 (total) ±0.11 ±0.20 ±0.23 ±0.03 ±0.03 Pion contamination in the electron trigger f π/e , wide path (×10 −4 ) f π/e,w 0.54 1.50 2.14 0.22 0.32 (stat.) ±0.02 ±0.04 ±0.02 ±0.01 ±0.01 (syst.) ±0.14 ±0.25 ±0.48 ±0.03 ±0.04 (total) ±0.15 ±0.25 ±0.48 ±0.03 ±0.04 Electron contamination in pion triggers f e/π , narrow path329 Kinematics and Spectrometer Combinations DIS#1 DIS#2 HRS Left Left Right Pion detection efficiency ξ π (%) GC 99.52 ± 0.01 99.73 ± 0.01 99.74 ± 0.01 LG, n 21.67 ± 0.01 79.72 ± 0.02 15.61 ± 0.01 LG, w 21.67 ± 0.01 79.71 ± 0.02 15.60 ± 0.01 GC+LG, n 21.57 ± 0.01 79.70 ± 0.02 15.57 ± 0.01 GC+LG, w 21.57 ± 0.01 79.69 ± 0.02 15.56 ± 0.01 Electron rejection 1/ξ e GC 31.42 ± 0.78 89.44 ± 2.48 48.48 ± 1.55 LG, n 1.0468 ± 0.0003 1.0487 ± 0.0005 1.0271 ± 0.0002 LG, w 1.0469 ± 0.0003 1.0499 ± 0.0005 1.0279 ± 0.0002 f e/π,n 0.2653 0.0331 0.0103 (stat.) ±0.0029 ±0.0006 ±0.0002 (syst.) ±0.0602 ±0.0033 ±0.0013 (total) ±0.0603 ±0.0034 ±0.0013 Electron contamination in pion triggers f e/π , wide path f e/π,w 0.2176 0.0281 0.0091 (stat.) ±0.0029 ±0.0006 ±0.0002 (syst.) ±0.0573 ±0.0036 ±0.0012 (total) ±0.0573 ±0.0037 ±0.0013 The above tagger measurements were performed at kinematics DIS#1 on the Left and DIS#2 on the Right HRS. No tagger data was available for resonance kinematics. However since the group deadtime is expected to rely only on the signal width and the module width settings, as demonstrated by the tagger data, a 10% systematic uncertainty was used for group deadtime for all kinematics.412 413 414 415 416 70ns 150ns 150ns 2 1 & 755 755 150ns C&S 755 32ns 70ns 757 757 C&S 755 SC(T1) GC 60ns Left HRS GATE circuit pion GATE electron GATE tagger tagger cp1 cp2 100ns 150ns 70ns 150ns 150ns 2 1 & 755 755 150ns C&S 755 20ns 32ns 757 757 C&S 755 SC(T1) GC 100ns Right HRS GATE circuit pion GATE electron GATE tagger tagger cp1 cp2 100ns 150ns where R SC&GC (R GC&SC ) refers to the rate of events that triggered the Scintillator 431 (Cherenkov) but not the Cherenkov (Scintillator), w SC,in(out) and w GC,in(out) refer to the input (output) signal widths of the PS755 module that first processes the Scintillator and the Cherenkov signals in the GATE electronics, respectively. Note that if the electronics used to generate the Scintillator and the Cherenkov signals have intrinsic deadtimes themselves that are longer than w SC,in and w GC,in , these intrinsic deadtimes should be used in place of the measured w SC,in and w GC,in .In Eq. (6), each term on the right hand side is present only if w in > w out .From Fig. 11, the signal widths were measured to be: w SC,in,left = w SC,in,Right = 100 ns, w SC,out,left = w SC,out,Right = 32 ns, w GC,in,left = 60 ns, w GC,in,Right = 100 ns, 440 w GC,out,left = w GC,out,Right = 70 ns. However it was observed from the data that the Left HRS Cherenkov signal had an intrinsic deadtime of longer than 70 ns. In fact, data showed both HRSs had contributions from the two terms on the right hand side of Eq. (6).Because trigger rates from Scintillator and the gas Cherenkov were much higher than individual group rates, the GATE deadtime could dominate the total deadtime of the DAQ, and the difference in total deadtime loss between narrow and wide paths could be smaller than that in their group deadtimes.The GATE deadtime can be extracted from the trigger simulation HATS using the known signal widths and module settings, and be compared with the estimation of Eq. (6). In addition, evidence of the GATE deadtime can be extracted from FADC432 433 434 435 436 437 438 441 442 443 444 445 446 447 448 449 450 451 . The agreement between simulation and FADC was found to be better than 10% and this was used as the systematic uncertainty of the GATE deadtime. For resonance kinematics no FADC data was available. GATE deadtime for resonance data was obtained from trigger simulation and the same systematic uncertainty was used because the mechanism of the GATE deadtime was expected to remain the same throughout the experiment.Fig. 13. [Color online] Fractional loss due to GATE deadtime as a function of beam current obtained from trigger simulation (solid circles), the analytic method Eq. (6) (dashed line), and FADC data (open triangle). Error bars for simulations are statistical.5.4 OR Deadtime EvaluationThere is no direct measurement of the logical OR deadtime, but the effect of the logical OR module is straightforward and can be calculated analytically: When two electron triggers from different groups overlap in time as they arrive at the logical OR module, they generate only one output in the global trigger. This OR deadtime loss can be calculated using the recorded trigger rates and the known trigger signal widths. To confirm the analytic method results, the OR deadtime was evaluated from trigger simulation by subtracting the group and the GATE deadtimes466 467 468 469 470 471 Beam Current (uA) 0 50 100 Fractional loss due to GATE deadtime (%) 0.0 0.2 0.4 0.6 0.8 1.0 Left HRS DIS 1 Simulation Data (FADC) Theory Beam Current (uA) 0 50 100 Fractional loss due to GATE deadtime (%) 0.0 0.2 0.4 0.6 Right HRS DIS 2 Simulation Data (FADC) Theory 472 473 474 475 476 477 478 479 Fig. 15. [Color online] Pull distribution [Eq.(7)] for the global electron narrow trigger for the three DIS spectrometer and kinematics combinations, and the five resonance kinematics. 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{'abstract': '11An experiment that measured the parity-violating asymmetries in deep inelastic scatter-12 ing was completed at the Thomas Jefferson National Accelerator Facility in experimental 13 Hall A. From these asymmetries, a combination of the quark weak axial charge could be 14 extracted with a factor of five improvement in precision over world data. To achieve this, 15 asymmetries at the 10 −4 level needed to be measured at event rates up to 600 kHz and the 16 high pion background typical to deep inelastic scattering experiments needed to be rejected 17 efficiently. A specialized data acquisition (DAQ) system with intrinsic particle identifica-18 tion (PID) was successfully developed and used: The pion contamination in the electron 19 samples was controlled at the order of 2 × 10 −4 or below with an electron efficiency of 20 higher than 91% during most of the production period of the experiment, the systematic 21 uncertainty in the measured asymmetry due to DAQ deadtime was below 0.5%, and the 22 statistical quality of the asymmetry measurement agreed with the Gaussian distribution to 23 over five orders of magnitudes. The DAQ system is presented here with an emphasis on 24 its design scheme, the achieved PID performance, deadtime effect and the capability of 25 measuring small asymmetries. 26 53 q = u, d indicating an up or a down quark, g e A(V ) is the electron axial (vector) 54 coupling and g q V (A) is the quark vector (axial) coupling to the Z 0 boson. In the tree-55level Standard Model, the C 1,2q are related to the weak mixing angle θ W : C 1u = 56 − 1 2 + 4 3 sin 2 θ W , C 2u = − 1 2 + 2 sin 2 θ W , C 1d = 1 2 − 2 3 sin 2 θ W , and C 2d = 1 2 − 57 2 sin 2 θ W . Although the weak mixing angle and the quark weak vector charge C 1q 58 have been measured from various processes [5], the current knowledge of the quark 59 weak axial charge C 2q is poor and their deviations from the Standard Model value 60 would reveal possible New Physics in the quark axial couplings that could not be 61 accessed from other Standard Model parameters.62The goal of JLab E08-011 was to measure the PVDIS asymmetries to statistical 63 precisions of 3% and 4% at Q 2 = 1.1 and 1.9 (GeV/c) 2 , respectively, and under the 64 65 weak charge combination (2C 2u − C 2d ). In addition, the systematic uncertainty 66 goal was less than 3%. For this experiment, the expected asymmetries were 91 and 67 160 ppm respectively at the two Q 2 values [1]. To achieve the required precision, 68 an event rate capability of up to 600 kHz was needed.69The main challenge of deep inelastic scattering experiments is the separation of 70 scattered electrons from the pion background in the spectrometer and detector sys-71 tem. The neutral pions would decay into e + e − pairs from which the electrons pro-72 duced cannot be rejected by detectors. This pair production background was studied 73 by reversing the spectrometer magnet settings and measure the e + yield, and the ef-74 fect on the measured asymmetries was found to be negligible. Charged pions are 75 produced primarily from nucleon resonance decays and could carry a parity vio-76 lation asymmetry corresponding to the Q 2 at which the resonances are produced, 77 typically a fraction of the asymmetry of electrons with the same scattered momen-78 tum.Assuming that a fraction f π/e of the detected events are π − and 1 − f π/e are 79 electrons, the measured asymmetry iswhere A e is the desired electron scattering asymmetry and A π is the asymmetry of 81 the pion background. To extract A e to a high precision, one needs either to mini-82 mize the pion contamination f π/e to a negligible level, or to correct the measured 83 asymmetry for the asymmetry of pions, which itself needs to be measured precisely.84For the PVDIS experiment, the goal was to control f π/e to the 10 −4 level provided 85 that the pion asymmetries did not exceed those of electrons.86The experiment used a 100 µA electron beam with a polarization of approximately 87 90% and a 20-cm long liquid deuterium target. The two High Resolution Spec-88 trometers (HRS) [6] were used to detect scattered events. While the standard HRS 89 detector package and data acquisition (DAQ) system routinely provide a 10 4 pion 90 rejection with approximately 99% electron efficiency, they are based on full record-91 ing of the detector signals and are limited to event rates up to 4 kHz [6]. This is not 92 sufficient for the high rates expected for the experiment. (The HRS DAQ will be 93 referred to as "standard DAQ" hereafter.) 94 Recent parity violation electron scattering experiments, such as HAPPEX [8,9,10,11,12], 95 and PREX [13] at JLab, focused on elastic scattering from nuclear or nucleon tar-96 gets that are typically not contaminated by inelastic backgrounds. Signals from the 97 detectors can be integrated and a helicity dependence in the integrated signal can 98 be used to extract the physics asymmetry. An integrating DAQ was also used in 99 the preceding PVDIS measurement at SLAC [14,15] in which approximately 2% 100 of the integrated signal was attributed to pions. The SAMPLE experiment [7] at 101 MIT-Bates focused also on elastic scattering but the inelastic contamination was 102 more challenging to reject, and an air Cherenkov counter was used to select only 103 3 elastic scattering events. In the Mainz PVA4 experiment [16,17,18], particles were 104 detected in a total absorption calorimeter and the integrated energy spectrum was 105recorded. Charged pions and other background were separated from electrons in 106 the offline analysis of the energy spectrum, and the pion rejection was on the order 107 of 100:1 based on the characteristics of the calorimeter.108High performance particle identification can usually be realized in a counting-based 109 DAQ where each event is evaluated individually. In the G0 experiment [19,20,21,22,23] 110 at JLab, a superconducting spectrometer with a 2π azimuthal angle coverage was 111 used to detect elastically scattered protons at the forward angle and elastic elec-112 trons at the backward angle. At the forward angle, protons were identified using 113 time-of-flight. At the backward angle, pions were rejected from electrons using an 114 aerogel Cherenkov counter, and a pion rejection factor of 125 : 1 or better was 115 reported [23]. The deadtime correction of the counting system was on the order of 116 a few percent [22,23]. 117 While the PVDIS experiment could fully utilize existing spectrometers and detec-118 tors at JLab, examination of all existing techniques for PV measurements made it 119 clear that a custom electronics and DAQ were needed to keep the systematic uncer-120 tainties due to data collection to below 1%. In this paper we describe a scaler-based, 121 cost effective counting DAQ which limited the pion contamination of the data sam-122 ple to a negligible level of f π/e ≈ 10 −4 . Basic information on the detector package 123 and the DAQ setup will be presented first and followed by the analysis of electron 124 detection efficiency, pion rejection and contamination, corrections due to counting 125 deadtime, and the statistical quality of the asymmetry measurement. 126 2 Detector and DAQ Overview 127 The design goal of the DAQ is to record data up to 600 kHz with hardware-based 128 PID and well measured and understood deadtime effects. The following detectors 129 in the HRS [6] were used to characterize scattered particles: Two scintillator planes 130 provided the main trigger, while a CO 2 gas Cherenkov detector and a double-layer 131segmented lead-glass detector provided particle identification information. The ver-132 tical drift chambers (as the tracking detector) were used during calibration runs but 133 were turned off during production data taking because they were not expected to 134 endure the high event rates.135For the gas Cherenkov and the lead-glass detector, a full recording of their out-136 put ADC data was not feasible at the expected high rate. Instead their signals were 137 passed through discriminators and logic units to form preliminary electron and pion 138 triggers. These preliminary triggers were then combined with the scintillator trig-139 gers to form the final electron and pion triggers, which were sent to scalers to record 140 the event counts and used offline to form asymmetries A = (n R − n L )/(n R + n L ), 141 528 even higher event rates or backgrounds with this type of scaler-based DAQ.529Acknowledgments 530', 'arxivid': '1302.2854', 'author': ['R Subedi \nUniversity of Virginia\n22904CharlottesvilleVAUSA\n', 'D Wang \nUniversity of Virginia\n22904CharlottesvilleVAUSA\n', 'K Pan \nMassachusetts Institute of Technology\n02139CambridgeMAUSA\n', 'X Deng \nUniversity of Virginia\n22904CharlottesvilleVAUSA\n', 'R Michaels \nThomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA\n', 'P E Reimer \nPhysics Division\nArgonne National Laboratory\n60439ArgonneILUSA\n', 'A Shahinyan \nYerevan Physics Institute\n0036YerevanArmenia\n', 'B Wojtsekhowski \nThomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA\n', 'X Zheng \nUniversity of Virginia\n22904CharlottesvilleVAUSA\n', '\nDallas County Community College District\n75243DallasTexasUSA\n'], 'authoraffiliation': ['University of Virginia\n22904CharlottesvilleVAUSA', 'University of Virginia\n22904CharlottesvilleVAUSA', 'Massachusetts Institute of Technology\n02139CambridgeMAUSA', 'University of Virginia\n22904CharlottesvilleVAUSA', 'Thomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA', 'Physics Division\nArgonne National Laboratory\n60439ArgonneILUSA', 'Yerevan Physics Institute\n0036YerevanArmenia', 'Thomas Jefferson National Accelerator Facility\n23606Newport NewsVAUSA', 'University of Virginia\n22904CharlottesvilleVAUSA', 'Dallas County Community College District\n75243DallasTexasUSA'], 'corpusid': 119267814, 'doi': '10.1016/j.nima.2013.05.040', 'github_urls': [], 'n_tokens_mistral': 22540, 'n_tokens_neox': 17857, 'n_words': 10781, 'pdfsha': '9b091c42ca1284506fee0e81d5fc7325eb16be74', 'pdfurls': ['https://arxiv.org/pdf/1302.2854v2.pdf'], 'title': ['Er, 12.15.Mm, 13.60.Hb 14.60.Cd 14.65.Bt 29.30.Aj 29', 'Er, 12.15.Mm, 13.60.Hb 14.60.Cd 14.65.Bt 29.30.Aj 29'], 'venue': []}

arxiv

Constant-Modulus Waveform Design for Dual-Function Radar-Communication Systems in the Presence of Clutter Wenjun Wu Bo Tang Xuyang Wang Constant-Modulus Waveform Design for Dual-Function Radar-Communication Systems in the Presence of Clutter PREPRINT 1Index Terms DFRC systemsMIMO arraytarget detectionmulti-user communication We investigate the constant-modulus (CM) waveform design for dual-function radar communication systems in the presence of clutter. To minimize the interference power and enhance the target acquisition performance, we use the signal-to-interference-plus-noise-ratio as the design metric. In addition, to ensure the quality of the service for each communication user, we enforce a constraint on the synthesis error of every communication signals. An iterative algorithm, which is based on cyclic optimization, Dinkinbach's transform, and alternating direction of method of multipliers, is proposed to tackle the encountered nonconvex optimization problem. Simulations illustrate that the CM waveforms synthesized by the proposed algorithm allow to suppress the clutter efficiently and control the synthesis error of communication signals to a low level. A. Radar Model Denote the target direction by θ 0 , and the direction of the qth interference source by θ q (θ q = θ 0 , q = 1, 2, · · · , Q). The received signals at the lth instant (l = 1, 2, · · · , L, and L is the code length) is formulated by y l = α 0 a R (θ 0 )a T (θ 0 )x l + Q q=1 α q a R (θ q )a T (θ q )x l + n l , where α 0 , α 1 , · · · , α Q are the amplitudes of the target and the Q interference sources, a T (θ) and a R (θ) denote the transmit array steering vector and the receive array steering vector at θ, x l = [x l (1), x l (2), · · · , x l (N T )] , x l (n) denotes the code of the lth subpulse of the (baseband) waveforms in the nth transmitter (n = 1, 2, · · · N T ), and n l is the receiver noise (we assume that it is white Gaussian). Stacking y 1 , y 2 , · · · , y L in one column, we obtain y = α 0 A(θ 0 )x + Q q=1 α q A(θ q )x + n,(2) where y = [y 1 , y 2 , · · · , y L ] ∈ C LNR×1 , A(θ) = I L ⊗ (a R (θ)a T (θ)), x = [x 1 , x 2 , · · · , x L ] ∈ C LNT×1 , and n = [n 1 , n 2 , · · · , n L ] ∈ C LNR×1 . To detect the target, we pass y through a finite impulse response filter, denoted by w. The filter output z can be written as z = w † y = α 0 w † A(θ 0 )x Target + w † Q q=1 α q A(θ q )x Interference + w † n Noise .(3) According to (3), we define the output SINR as follows: SINR r (x, w) = σ 2 0 |w † A(θ 0 )x| 2 w † Q q=1 σ 2 q A(θ q )xx † A † (θ q ) w + σ 2 n w † w ,(4) where the subscript r stands for "radar", σ 2 0 is the target power, σ 2 q = E{|α q | 2 } is the average power of the qth interference source, q = 1, 2, · · · , Q, σ 2 n is the noise power level, and we have assumed that the interference sources are uncorrelated. B. Communication Model The signals received by the M users is given by Y = HX + Z,(5) where H = [h 1 , h 2 , · · · , h M ] ∈ C M ×NT is the channel matrix; X = [x 1 , x 2 , · · · , x L ] ∈ C NT×L denotes the transmit waveform matrix (x = vec(X)), and Z is the noise matrix in the M communication receivers. Let s m ∈ C L×1 denote the desired symbols for the mth user (m = 1, 2, · · · , M ), and let S = [s 1 , s 2 , · · · , s M ] ∈ C M ×L . Then we can rewrite (5) as where HX − S stands for the multi-user interference (MUI). In [31], it is shown that for a Gaussian broadcasting channel and a Gaussian input, the achievable rate of the mth communication user (m = 1, 2, · · · , M ) is given by Y = S + HX − S MUI +Z,(6)m = log 2 (1 + SINR m,c ),(7) where SINR m,c is the SINR of the mth communication receiver, defined by SINR m,c = E{|s m,l | 2 } E{|h m x l − s m,l | 2 } + σ 2 z,m ,(8) where the subscript c stands for "communication", s m,l is the lth symbol of s m (l = 1, 2, · · · , L), and σ 2 z,m is the noise power level of the mth communication receiver. Note that h m x l − s m,l is the (m, l)th element of HX − S. Thus, the minimization of ψ m (X) = h m X − s m 2 2 = H m x − s m 2 2 ≈ LE{|h m x l − s m,l | 2 },(9) which is the MUI for the mth communication receiver, results in the maximization of the achievable rate of the mth user, where H m = h m ⊗ I L . C. Problem Formulation To maximize the target detection performance and ensure the communication performance of each user, we formulate the waveform design problem as follows: max x,w SINR r (x, w) s.t. ψ m (X) ≤ ς m , m = 1, 2, · · · , M, x ∈ X ,(10) where ς m is the maximum allowed synthesis error of the mth user, and X denote the constraint set on the waveforms. In practice, to make the radio frequency (RF) amplifier work at maximum efficiency and avoid nonlinear distortions, CM waveforms are often used. Thus, we enforce the CM constraint on the transmit waveforms, i.e., |x(n)| = √ p s , n = 1, 2, · · · , LN T ,(11) where x(n) indicates the nth element of x, p s = e T /(LN T ), and e T is the total transmit energy. By combining the results in (4), (9), and (11), the joint design of transmit waveforms and receive filters for the DFRC system can be formulated as max x,w |w † A(θ 0 )x| 2 w † Q q=1 σ 2 q A(θ q )xx † A † (θ q ) w + σ 2 n w † w s.t. H m x − s m 2 2 ≤ ς m , m = 1, 2, · · · , M, |x(n)| = √ p s , n = 1, 2, · · · , LN T .(12) Remark 1: In the formulation of (12), we have assumed that the average power and the directions of the interference sources as well as the channel matrix are known a priori. Such an assumption is justified if the access to the estimates of them from the previous scans are available (see also in [30], [32] for similar assumptions). Remark 2: We note that the formulated problem in (12) is different from that in [30]. The studies in [30] controlled the total MUI energy indirectly by tuning the combination coefficient, while we constrain the MUI energy of each user directly to control their communication performance. More specifically, according to (9), if the per-user MUI energy constraint is satisfied, we can approximately obtain that E{|h m x l − s m,l | 2 } ≤ ς m /L, m = 1, 2, · · · , M.(13) As a result, the achievable information rate for the mth user satisfies m ≥ log 2 1 + p m ς m /L + σ 2 z,m ,(14) where p m = E{|s m,l | 2 } is the average power of the mth communication signals, m = 1, 2, · · · , M . On the other hand, if only the total MUI energy constraint is enforced, the initialization of the transmit waveforms will affect the communication performance of each user (i.e., the communication performance of each user cannot be guaranteed). III. ALGORITHM DESIGN Due to the CM constraint, it is evident that the fractional programming problem in (12) is non-convex. In this section, a cyclic optimization method is used to deal with this non-convex problem. Specifically, at the (k +1)th iteration, we optimize w (k+1) for fixed x (k) ; then we optimize x (k+1) for fixed w (k+1) . Next we present solutions to the two optimization problems. To lighten the notations, we omit the superscript if doing so does not result in confusions. A. Optimization of w (k+1) for fixed x (k) The corresponding optimization problem is given by max w |w † A(θ 0 )x| 2 w † Q q=1 σ 2 q A(θ q )xx † A † (θ q ) w + σ 2 n w † w .(15) It can be checked that the solution to (15) is given by w = γR −1 x A(θ 0 )x,(16) where γ = 0 is an arbitrary constant, and R x = Q q=1 σ 2 q A(θ q )xx † A † (θ q ) + σ 2 n I LNR .(17)B. Optimization of x (k+1) for fixed w (k+1) Define R 0 = A † (θ 0 )ww † A(θ 0 )(18) and R 1 = Q q=1 σ 2 q A † (θ q )ww † A(θ q ) + σ 2 n w † w/e T · I LNT .(19) Then the waveform design problem for fixed w (k+1) can be written as max x x † R 0 x x † R 1 x s.t. H m x − s m 2 2 ≤ ς m , m = 1, 2, · · · , M, |x(n)| = √ p s , n = 1, 2, · · · , LN T .(20) Next we resort to Dinkelbach's transform [33] to deal with the optimization problem in (20). To this end, we use g(x (k,l) ) to indicate the objective value of (20) at the (k, l)th iteration (Here we use the superscript k to indicate the outer iteration for cyclic optimization, and l to denote the inner iteration for Dinkelbach's transform). By using Dinkelbach's transform, the optimization problem at the (k, l + 1)th iteration is formulated as max x x † Tx s.t. H m x − s m 2 2 ≤ ς m , m = 1, 2, · · · , M, |x(n)| = √ p s , n = 1, 2, · · · , LN T ,(21) where T = R 0 − g(x (k,l) )R 1 . To proceed, we define T = T − βI LNT ,(22) where β ≤ λ min ( T), and λ min ( T) is the smallest eigenvalue of T. It is evident that T is positive semidefinite such that its square root exists. In addition, x † Tx = x † Tx − βe T .(23) Therefore, the optimization problem in (21) is equivalent to max x x † Tx s.t. H m x − s m 2 2 ≤ ς m , m = 1, 2, · · · , M, |x(n)| = √ p s , n = 1, 2, · · · , LN T .(24) Next we apply ADMM (we refer to [34] for a comprehensive survey of ADMM) to tackle (24). By using the variable splitting trick and introducing auxiliary variablesx and x m (m = 1, · · · , M ), the optimization problem in (24) is recast by max x,x, xx †x (25a) s.t.x = T 1/2 x, (25b) x m 2 2 ≤ ς m , x m = H m x − s m , m = 1, 2, · · · , M, (25c) |x(n)| = √ p s , n = 1, 2, · · · , LN T . (25d) The augmented Lagrange function of (25) is written as L µ (x,x, { x m }, ν, {υ m }) = −x †x + µ 2 x − T 1/2 x + ν 2 2 − ν 2 2 + µ 2 M m=1 x m − H m x + s m + υ m 2 2 − υ m 2 2 ,(26) where µ is the penalty parameter; ν and υ m (m = 1, · · · , M ) are the Lagrange multipliers associated with the constraints in (25b) and (25c), respectively. In the (t + 1)th iteration of the ADMM algorithm, we carry out the following steps sequentially: x (t+1) = argmin x∈X L µ (x,x (t) , { x (t) m }, ν (t) , {υ (t) m }),(27)x (t+1) = argmin x L µ (x (t+1) ,x, { x (t) m }, ν (t) , {υ (t) m }),(28)March 1, 2023 DRAFT x (t+1) m = argmin xm L µ (x (t+1) ,x (t+1) , { x m }, ν (t) , {υ (t) m }),(29)ν (t+1) = ν (t) +x (t+1) − T 1/2 x (t+1) ,(30)υ (t+1) m = υ (t) m + x (t+1) m − H m x (t+1) + s m .(31) Next we derive the solutions to (27), (28), and (29). (27): The optimization problem in (27) can be recast as 1) Solution to min x x − T 1/2 x + ν 2 2 + M m=1 x m − H m x + s m + υ m 2 2 s.t. |x(n)| = √ p s , n = 1, 2, · · · , LN T .(32) Let B = T + M m=1 H † m H m ,(33)and b = T 1/2 (x + ν) + M m=1 H † m ( x m + s m + υ m ).(34) Then, we can rewrite the optimization problem in (32) as min x x † Bx − 2Re(b † x) s.t. |x(n)| = √ p s , n = 1, 2, · · · , LN T .(35) Note that (35) is a standard unimodular quadratic programming (UQP) problem. A number of algorithms have been proposed to tackle the UQP problem, including the power-method like (PML) iterations [35], [36], the coordinate descent method (CDM) [37], [38], the gradient projection (GP) method [39], and the majorization-minimization (MM) method [15], [38], [40]- [42]. We find that these methods have similar performance but the MM method is usually the fastest. Therefore, we use the MM method to tackle the problem in (35). 2) Solution to (28): The optimization problem in (28) is formulated by min x −x †x + µ 2 x − T 1/2 x + ν 2 2 .(36) Define q = T 1/2 x − ν.(37) Then, we can recast (36) as min x µ 2 x − q 2 2 −x †x ,(38) Assume that µ > 2. Then the quadratic optimization problem in (38) is convex. Taking the derivative of (38) with respect tox and setting it equal to zero, we can acquire the solution to (38): x = µ µ − 2 q.(39) 3) Solution to (29): Define p m = H m x − υ m − s m .(40) Then, we can rewrite (29) as min xm x m − p m 2 2 s.t. x m 2 2 ≤ ς m .(41) It can be verified that the solution to (41) is x m =      p m , p m 2 2 ≤ ς m , √ ς m p m / p m , p m 2 2 > ς m .(42) We summarize the proposed ADMM algorithm in Algorithm 1, where we stop the proposed ADMM method when the norm of the primal residual r (t) m 2 ≤ primal (m = 1, 2, · · · , M + 1) and the norm of the dual residual d (t) m 2 ≤ dual (m = 1, 2, · · · , M + 2), where r (t) m =      H m x (t) − s m − x (t) m , 1 ≤ m ≤ M, T 1/2 x (t) −x (t) , m = M + 1,(43)d (t) m =              x (t) m − x (t−1) m , 1 ≤ m ≤ M, x (t) −x (t−1) , m = M + 1, x (t) − x (t−1) , m = M + 2,(44) primal > 0 and dual > 0 are the feasible tolerances of the primal and dual conditions, respectively. C. Algorithm Summary and Computational Complexity Analysis We summarize the proposed constant-modulus waveform design algorithm for the DFRC systems in Algorithm 2, where we terminate the algorithm if |SINR (k) r − SINR (k−1) r | SINR (k−1) r < ϑ O ,(45) and we terminate the inner loop (for Dinkelbach's transform) if |g(x (k,l) ) − g(x (k,l−1) )| g(x (k,l−1) ) < ϑ I ,(46) Algorithm 1: ADMM algorithm for the problem in (24). Input: T, p s , {H m , s m , ς m } M m=1 . Output: x (k,l+1) . 1 Initialize: 2 Compute B by (33). 3 t = 0, x (t) = x (k,l) , ν (t) = 0, υ (t) m = 0. 4 Computex (t) , { x (t) m } M m=1 . 5 repeat 6 b (t) = T 1/2 (x (t) + ν (t) ) + M m=1 H † m ( x (t) m + s m + υ (t) m ). 7 Update x (t+1) through solving (35) by MM. 8 q (t) = T 1/2 x (t+1) − ν (t) . 9x (t+1) = µ µ − 2 q (t) . 10 p (t) m = H m x (t+1) − υ (t) m − s m . 11 x (t+1) m = min( √ ς m / p (t) m , 1) · p (t) m . 12 ν (t+1) = ν (t) +x (t+1) − T 1/2 x (t+1) . 13 υ (t+1) m = υ (t) m + x (t+1) m − H m x (t+1) + s m . 14 t = t + 1. 15 until convergence; 16 x (k,l+1) = x (t) . where ϑ O and ϑ I are predefined small values (e.g., 10 −5 ). The computational complexity of Algorithm 2 is determined by the number of outer iterations and the complexity at each outer iteration. We present the computational complexity for each outer iteration in which deals with a non-convex optimization problem, remains unknown (for some recent progress on this problem, we refer to [43]). Therefore, it is non-trivial to prove the convergence property of Algorithm 2. 2 H m = h m ⊗ I L M m=1 , A(θ) = I L ⊗ (a R (θ)a T (θ)). 3 k = 0, x (0) . 4 repeat 5 // Optimization of w. 6 Compute R (k) x by (17). 7 Update w (k+1) by (16). 8 // Optimization of x. 9 Compute R (19). 11 l = 0, x (k,l) = x (k) . 12 repeat 13 Compute g(x (k,l) ) = (x (k,l) ) † R (k) 0 x (k,l) (x (k,l) ) † R (k) 1 x (k,l) . 14 Compute T (k,l) = R (k) 0 − g(x (k,l) )R (k) 1 . 15 Compute T (k,l) = T (k,l) − βI LNT . 16 Compute x (k,l+1) by Algorithm 1. 22 x = x (k) , w = w (k) . Fortunately, we have not encountered any convergence problems in our extensive numerical studies. D. Algorithm Extension We note that the proposed algorithm can be extended to deal with the peak-to-average-power-ratio (PAPR) constraint. To this purpose, we only need to replace the optimization problem in (35) R −1 x O(L 3 N 3 R ) w O(L 2 NRNT) Update x R0, R1 O(L 2 NRNT) g(x) O(NDL 2 N 2 T ) Compute x (k,l+1) by Algorithm 1 O(NDNA(2M + 3)L 2 N 2 T ) Total O(L 3 N 3 R + O(NDNA(2M + 3)L 2 N 2 T ) following: min x x † Bx − 2Re(b † x) s.t. x(n) † x(n) = e T /N T , PAPR(x(n)) ≤ ρ, n = 1, 2, · · · , N T .(47) where x(n) denotes the waveforms of the nth transmitter, 1 ≤ ρ ≤ L, PAPR(x(n)) = max l |x l (n)| 2 1 L L l=1 |x l (n)| 2 ,(48) and we have assumed that the transmit energy across the antenna elements are uniform. Similarly, we can use the MM method to tackle the optimization problem in (47) [40]. In addition to the PAPR constraint, we can also extend the proposed algorithm to design waveforms under a similarity constraint [44], [45] as well as both similarity and constant-modulus constraints. However, we skip the details due to space limitations. IV. NUMERICAL RESULTS In and ς 2 = 5 × 10 −3 , respectively. In addition, the upper bound 1 and the SINR curve associated with the radar-only case 2 are also drawn. We can see that the SINR at convergence is 32.49 dB, which is about 10 dB higher than that of the LFM signals. However, since the DFRC system has to spare some transmit energy to satisfy the additional communication constraints, the proposed waveforms will suffer some performance loss (compared with the upper bound and the radar-only case, the loss of the proposed waveforms are about 1.59 dB and 1.38 dB, respectively). Next we compare the detection performance of the proposed waveforms with that of the LFM waveforms. To detect the target, we set up a hypothesis test as follows:    H 0 : y = r, H 1 : y = α 0 A(θ 0 )x + r,(50) where r = Q q=1 α q A(θ q )x + n. Assume that n ∼ CN (0, σ 2 n I) and r ∼ CN (0, R x ) (R x is defined in (17)). According to the Neyman-Pearson criterion [47], we decide H 1 if 3 Re(w † y) > T h ,(51) where w = α 0 R −1 x A(θ 0 )x, and T h is the detection threshold. The detection probability associated with this detector is given by [47] 4 P D = 1 2 erfc erfc −1 (2P FA ) − SINR r ,(52) where erfc(x) = 2 √ π ∞ x e −t 2 dt is the complementary error function, P FA is the probability of false alarm, and SINR r is given by SINR r = σ 2 0 x † A † (θ 0 )R −1 x A(θ 0 )x.(53) We can observe from (52) and (53) that the detection probability P D depends on the transmit waveforms. Therefore, it can be expected that the optimized waveforms can achieve a larger SINR and a higher detection probability. To illustrate the superiority of the proposed waveforms, we let σ 2 0 = −20 dB. It can be verified that the SINR r of the proposed waveform and the LFM signals are 12.49 dB and 2.04 dB, respectively. The associated detection probability of them is shown in Fig. 4. We can see that the detection probability of the proposed waveforms is significantly higher than that of the LFM signals. SNR m,c = E{|s m,l | 2 } σ 2 z,m ,(54) s m,l is the lth symbol of s m (l = 1, 2, · · · , L), σ 2 z,m is the noise power level of the mth communication receiver, and 2000 independent trials are conducted to obtain the SER. To achieve a given SNR m,c , we keep the amplitude of the communication signals fixed and vary the noise power. We can observe that the SER performance of the synthesized signals is also close to the desired ones. Next, we vary the transmit energy of the communication signals and analyze the SINR performance for target detection. Fig. 7 illustrates the SINR curves of the DFRC system versus the number of outer iterations for different communication energy. The associated SINR values at convergence are listed in Table III. Note that SINR decreases with the transmit energy of the communication signals. This is due to that the DFRC system has to spare more transmit energy to ensure the communication performance, resulting in a degraded target detection performance. Then, we analyze the impact of the number of communication users on the system performance. Table IV. We can find that due to the increase of the number of communication users (thus, a larger number of constraints and a smaller feasibility region), the SINR performance decreases rapidly. Moreover, the propose algorithm takes a longer time to converge. Now we study how the code length affects the DFRC system performance. We use the same parameter setting as that in Fig. 2, but vary the code length and set the maximum allowed synthesis error to be ς m = 10 −3 (m = 1, 2). Fig. 9(a) and Fig. 9(b) plot the SINR of the synthesized waveforms versus the number of outer iterations and the CPU time, respectively, for L = 10, 20, and 30. Table V presents the SINR values at convergence. Fig. 9(a) indicates that the target detection performance improves for a larger L. It is because that the degree of freedom (DOF) of the waveforms increases with the code length (which implies that the system has better interference suppression capability). However, as shown in Fig. 9(b), the increased code length will also lead to a longer time for the algorithm to reach convergence. Subsequently, we study the impact of the number of antennas on the DFRC system performance. We use the same parameter setting as that in Fig. 2, but vary the number of antennas and set the maximum March 1, 2023 DRAFT Table VI. It can be seen that increasing the number of antennas improves the detection performance, but it also lead to a heavier computation load. In Fig. 11, the impact of the maximum allowed synthesis error on the SINR performance is analyzed, where we use the same parameter setting as that in Fig. 2 smaller value of the maximum allowed synthesis error, the SINR performance only degrades slightly. However, the associated communication performance improves quickly. This implies that to guarantee the quality of service for communication without affecting the detection performance, we can control the synthesis error to a reasonably low level. Finally, we demonstrate that the proposed algorithm can be extended to deal with different constraint (including the PAPR constraint, the similarity constraint, the CM and similarity constraints). We set ρ = 2, the similarity parameters δ = 0.25e T , and δ ∞ = 1.5 √ p s 5 . The reference waveform x 0 is given by: x 0 (n) = √ p s e jπ(n−1) 2 /LNT , n = 1, 2, · · · , LN T . The other parameter setting is the same as that in Fig. 2. In Fig. 13, we can observe that increasing the PAPR results in a better detection performance. On the contrary, enforcing a similarity constraint will degrade the detection performance. V. CONCLUSION An algorithm for jointly designing the transmit waveforms and receive filters was devised to maximize the detection performance of the DFRC system and ensure communications with multiple users. Results 5 The similarity constraint is written as x − x0 2 2 ≤ δ; The constant-modulus and similarity constraints are written as |x(n)| = √ ps, n = 1, 2, · · · , LNT, x − x0 ∞ ≤ δ∞. showed that the optimized waveforms and filters could form deep nulls at the interference directions while maintaining the target response. In addition, the synthesized communication signals approximated the desired ones with the synthesis error of every user being precisely controlled. Moreover, if the transmit energy of the desired communication signals or the number of communication users was increased, the target detection performance degraded. Therefore, there is a fundamental tradeoff between the target detection and the communication performance. [48] L. E. Brennan and I. S. Reed, "Theory of adaptive radar," IEEE Transactions on Aerospace and Electronic Systems, pp. 237-252, 1973. IFig. 1 . 1The identity matrix with the size determined by the subscript(·) * , (·) , (·) †Conjugate, transpose, and conjugate transpose part of a complex-valued scalar/vector/matrix A 0 (A 0) A is positive definite (semidefinite) Illustration of a MIMO DFRC system. Remark 3 : 3Due to the constant-modulus constraint and the fractional objective function, the optimization problem in (12) is non-convex and difficult to tackle. To synthesize the constant-modulus waveforms, we derive Algorithm 2, which is a nested optimization algorithm based on the cyclic optimization, Dinkinbach's transform, and ADMM. Essentially, cyclic optimization and the method based on Dinkinbach's transform are ascent algorithms. However, the convergence property of the proposed ADMM algorithm, k+1) = x (k,l) . this section, numerical examples are provided to demonstrate the performance of the proposed algorithm. The DFRC system under consideration has N T = 16 transmit antennas and N R = 8 receive antennas. Both the transmit and the receive antenna array are uniform linear arrays, with inter-element spacing being λ/2 (λ is the wavelength). The target is at the direction of θ 0 = 20 • and has a power of 0 dB. We assume that Q = 4 interference sources are present, with the directions and power being {−40 • , −20 • , 40 • , 50 • } and σ 2 q = 30 dB (q = 1, 2, 3, 4), respectively. The noise power level in the radar receiver is 0 dB. The available transmit energy is e T = 20. The elements of the channel matrix H are independent and identically distributed, obeying a Gaussian distribution with zero mean and variance of 1 (i.e., we assume a flat fading channel). Regarding the stopping criteria of the proposed algorithm, we set primal = 10 −4 , dual = 10 −2 , and ϑ I = ϑ O = 10 −5 . Finally, all the analysis is performed on a standard laptop with CPU CoRe i7 2.8 GHz and 16 GB of RAM. Firstly, we analyze the convergence of the proposed algorithm. We assume M = 2 communication users. The desired signals of user 1 and user 2 have a quadrature phase shift keying (QPSK) and an 8-quadrature amplitude modulation (8QAM) modulation, respectively. The associated information bits are randomly generated. The transmit energies of the desired communication signals are e 1 = e 2 = 20. Fig. 2 2illustrates the convergence of the SINR of the synthesized waveforms versus the number of outer iterations, where the code length is L = 20, and the maximum allowed synthesis errors are ς 1 = 10 −3 Fig. 2 . 2SINRr versus the number of outer iterations. M = 2, e1 = e2 = 20, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . Fig. 3 Fig. 3 . 33shows the beampattern associated with the designed waveforms and filters, where the beampattern1 It can be verified that the upper bound of SINRr is NTNReT.2 The waveforms for the radar-only case are synthesized by removing the communication constraints in(12), and the associated waveform design problem can be tackled by Algorithm 3 in[46]. the beampattern has the highest gain at the target direction (θ 0 = 20 • ), and forms deep nulls (lower than −120 dB) at the interference directions (−40 • , −20 • , 40 • , and 50 • ). Therefore, the designed transmit waveforms and receive filters can suppress the interference power to a very low level, ensuring the target detection performance. Beampattern of the DFRC systems. M = 2, e1 = e2 = 20, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . 56% Fig. 4 . 56%4P d =0.15% Proposed wavefroms P d =88.Target detection probability of the DFRC systems. NT = 16, NR = 8, eT = 20, L = 20, σ 2 0 = −20 dB. Fig. 5 5shows the synthesis error of the communication signals versus the number of outer iterations. We can see that the waveforms at convergence satisfy the communication constraints, implying that the communication functionality is supported. To verify this claim, Fig. 6(a) and Fig. 6(d) compare the synthesized communication signals and the desired signals for the two users. The constellation diagrams of the synthesized communication signals are displayed in Fig. 6(b) and Fig. 6(e). It can be seen that the real and imaginary part of the synthesized signals are close to the desired ones and the constellation diagrams of these signals are nearly ideal. Fig. 6(c) and Fig. 6(f) show the symbol error rate (SER) of the synthesized signals, where we define the SNR for the mth communication user as Fig. 5 . 5The synthesis error of communication signals versus the number of outer iterations. M = 2, e1 = e2 = 20, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . Fig. 8 draws 8the SINR curves versus the number of outer iterations for different number of communication users (M = 2, 3, and 4), where the code length is L = 20, and the maximum allowed synthesis errors are ς m = 10 −3 (m = 1, · · · , M ). The SINR values at convergence for different M are shown in Fig. 6 . 6Analysis of the synthesized communication signals. (a)-(c): Comparison of the synthesized communication signals with the desired ones, the constellation diagram, and the SER of the synthesized communication signals for user 1. (d)-(f): Comparison of the synthesized communication signals with the desired ones, the constellation diagram, and the SER of the synthesized communication signals for user 2. M = 2, e1 = e2 = 20, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . Fig. 7 .Fig. 8 . 78SINRr for different em. M = 2, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . SINRr for different M . L = 20, em = 20, ςm = 10 −3 , m = 1, · · · , M . (a) SINRr versus the number of outer iterations. (b) SINRr versus the CPU time. Fig. 9 . 9SINRr for different L. M = 2, e1 = e2 = 20, ς1 = ς2 = 10 −3 . (a) SINRr versus the number of outer iterations. (b) SINRr versus the CPU time. allowed synthesis error to be ς m = 10 −3 (m = 1, 2). Fig. 10(a) and Fig. 10(b) show the convergence of the proposed algorithm versus the number of outer iterations and the CPU time, respectively, for different number of transmit and receive antennas. The SINR values at convergence are displayed in Fig. 10 . 10SINRr for different NT, NR. L = 20, M = 2, e1 = e2 = 20, ς1 = ς2 = 10 −3 . (a) SINRr versus the number of outer iterations. (b) SINRr versus the CPU time. Fig. 11 . 11SINRr for different ςm. M = 2, e1 = e2 = 20, L = 20, ς1 = ς2. Fig. 12 . 12SER for different ςm. M = 2, e1 = e2 = 20, L = 20, ς1 = ς2. (a) User 1. (b) User 2. Fig. 14 analyzes the performance of the communication signals synthesized under different constraints. We can find that since all the synthesized communication signals satisfy the communication constraint, they have similar performance. Fig. 13 . 13SINRr under different constraints. M = 2, e1 = e2 = 20, L = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . Fig. 14 . 14SER under different constraints. M = 2, e1 = e2 = 20, ς1 = 10 −3 , ς2 = 5 × 10 −3 . (a) User 1. (b) User 2. TABLE I ILIST OF NOTATIONSSymbol Meaning A Matrix a Vector a Scalar Table II , IIwhere N D and N A denote the number of iterations for Dinkelbach's transform and the proposed ADMM algorithm to reach convergence, respectively. Joint Design of CM Waveforms and Receive Filters for DFRC SystemsMarch 1, 2023 DRAFT Algorithm 2: Input: θ q , σ 2 q Q q=1 , θ 0 , α 2 0 , {s m , ς m } M m=1 , H, p s Output: x, w 1 Initialize: TABLE II COMPUTATIONAL IICOMPLEXITY ANALYSIS Computation Complexity Update w Rx O(L 2 NRNT) TABLE III IIISINRr VERSUS em e1 10 20 20 30 e2 10 20 30 30 SINRr (dB) 32.91 32.59 31.87 31.73 TABLE IV IVSINRr VERSUS M M 2 3 4 SINRr (dB) 32.74 31.22 29.57 TABLE V VSINRr VERSUS L L 10 20 30 SINRr (dB) 31.54 32.38 32.59 except for varying the maximum allowed synthesis error. The SINR values at convergence for these cases are given inTable VII. The SER performance of the synthesized communication signals is analyzed inFig. 12. We can see that for aMarch 1, 2023 DRAFT TABLE VI VISINRr VERSUS NT AND NRNT 8 8 16 16 NR 8 16 8 16 SINRr (dB) 28.68 30.53 32.55 35.28 TABLE VII SINRr VERSUS VIIVERSUSςm ςm 10 −3 10 −1 1 SINRr (dB) 32.50 32.67 33.01 March 1, 2023DRAFT This detector is also called generalized matched filter[47, pp. 478-479]. We can also use the Bayesian detector proposed in[48], i.e., we decide H1 if |w † y| > Th.4 The detection probability presented in (52) represents an upper bound for the waveform x. If the prior knowledge of the clutter or the target is imprecise, the detection performance degrades.March 1, 2023 DRAFT The challenge of spectrum engineering. H Griffiths, 2014 11th European Radar Conference. H. Griffiths, "The challenge of spectrum engineering," in 2014 11th European Radar Conference, 2014. Radar spectrum engineering and management: technical and regulatory issues. 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Li, "Quadratic optimization with similarity constraint for unimodular sequence synthesis," IEEE Transactions on Signal Processing, vol. 65, no. 18, pp. 4756-4769, 2017. Efficient algorithms for constant-modulus analog beamforming. A Arora, C G Tsinos, M R B Shankar, S Chatzinotas, B Ottersten, IEEE Transactions on Signal Processing. 70A. Arora, C. G. Tsinos, M. R. B. Shankar, S. Chatzinotas, and B. Ottersten, "Efficient algorithms for constant-modulus analog beamforming," IEEE Transactions on Signal Processing, vol. 70, pp. 756-771, 2022. Fast unit-modulus least squares with applications in beamforming. J Tranter, N D Sidiropoulos, X Fu, A Swami, IEEE Transactions on Signal Processing. 6511J. Tranter, N. D. Sidiropoulos, X. Fu, and A. Swami, "Fast unit-modulus least squares with applications in beamforming," IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2875-2887, 2017. Information-theoretic waveform design for MIMO radar detection in range-spread clutter. B Tang, P Stoica, Signal Processing. 1825107961B. Tang and P. Stoica, "Information-theoretic waveform design for MIMO radar detection in range-spread clutter," Signal Processing, vol. 182, no. 5, p. 107961, 2021. Constrained radar waveform design for range profiling. B Tang, J Liu, H Wang, Y Hu, IEEE Transactions on Signal Processing. 698B. Tang, J. Liu, H. Wang, and Y. Hu, "Constrained radar waveform design for range profiling," IEEE Transactions on Signal Processing, vol. 69, no. 8, pp. 1924-1937, 2021. Transmit waveform/receive filter design for MIMO radar with multiple waveform constraints. L Wu, P Babu, D P Palomar, IEEE Transactions on Signal Processing. 666L. Wu, P. Babu, and D. P. Palomar, "Transmit waveform/receive filter design for MIMO radar with multiple waveform constraints," IEEE Transactions on Signal Processing, vol. 66, no. 6, pp. 1526-1540, 2018. Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. M Hong, Z.-Q Luo, M Razaviyayn, SIAM Journal on Optimization. 261M. Hong, Z.-Q. Luo, and M. Razaviyayn, "Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems," SIAM Journal on Optimization, vol. 26, no. 1, pp. 337-364, 2016. Signal waveform's optimal-under-restriction design for active sensing. J Li, J R Guerci, L Xu, IEEE Signal Processing Letters. 139J. Li, J. R. Guerci, and L. Xu, "Signal waveform's optimal-under-restriction design for active sensing," IEEE Signal Processing Letters, vol. 13, no. 9, pp. 565-568, 2006. Code design to optimize radar detection performance under accuracy and similarity constraints. A D Maio, S D Nicola, Y Huang, S Zhang, A Farina, IEEE Transactions on Signal Processing. 5611A. D. Maio, S. D. Nicola, Y. Huang, S. Zhang, and A. Farina, "Code design to optimize radar detection performance under accuracy and similarity constraints," IEEE Transactions on Signal Processing, vol. 56, no. 11, pp. 5618-5629, 2008. Joint design of transmit waveforms and receive filters for MIMO radar space-time adaptive processing. B Tang, J Tang, IEEE Transactions on Signal Processing. 6418B. Tang and J. Tang, "Joint design of transmit waveforms and receive filters for MIMO radar space-time adaptive processing," IEEE Transactions on Signal Processing, vol. 64, no. 18, pp. 4707-4722, 2016. S M Kay, Detection Theory. New JerseyPrentice HallIIS. M. Kay, Fundamentals of Statistical Signal Processing-Volume II: Detection Theory. New Jersey: Prentice Hall, 1998. March 1, 2023 DRAFT

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{'abstract': "We investigate the constant-modulus (CM) waveform design for dual-function radar communication systems in the presence of clutter. To minimize the interference power and enhance the target acquisition performance, we use the signal-to-interference-plus-noise-ratio as the design metric. In addition, to ensure the quality of the service for each communication user, we enforce a constraint on the synthesis error of every communication signals. An iterative algorithm, which is based on cyclic optimization, Dinkinbach's transform, and alternating direction of method of multipliers, is proposed to tackle the encountered nonconvex optimization problem. Simulations illustrate that the CM waveforms synthesized by the proposed algorithm allow to suppress the clutter efficiently and control the synthesis error of communication signals to a low level.", 'arxivid': '2302.14263', 'author': ['Wenjun Wu ', 'Bo Tang ', 'Xuyang Wang '], 'authoraffiliation': [], 'corpusid': 255706959, 'doi': '10.1109/taes.2023.3234927', 'github_urls': [], 'n_tokens_mistral': 17033, 'n_tokens_neox': 14394, 'n_words': 8405, 'pdfsha': '084e8597633cd2143193d9e9c8f13ca308d38f5c', 'pdfurls': ['https://export.arxiv.org/pdf/2302.14263v1.pdf'], 'title': ['Constant-Modulus Waveform Design for Dual-Function Radar-Communication Systems in the Presence of Clutter', 'Constant-Modulus Waveform Design for Dual-Function Radar-Communication Systems in the Presence of Clutter'], 'venue': []}

arxiv

New conjecture on species doubling of lattice fermions 30 Jan 2023 Jun Yumoto Department of Mathematical Science Akita University 1-1 Tegata-Gakuen-machi010-8502AkitaJapan Tatsuhiro Misumi Department of Physics Kindai University 3-4-1 Kowakae, Higashi-osaka577-8502OsakaJapan Research and Education Center for Natural Sciences Keio University 4-1-1 Hitoshi223-8521YokohamaKanagawaJapan New conjecture on species doubling of lattice fermions 30 Jan 20232 CONTENTS We propose a new conjecture on the relation between the species doubling of lattice fermions and the topology of manifold on which the fermion action is defined. Our conjecture claims that the maximal number of fermion species on a finite-volume and finite-spacing lattice defined by discretizing a D-dimensional manifold is equal to the summation of the Betti numbers of the manifold. We start with reconsidering species doubling of naive fermions on the lattices whose topologies are torus (T D ), hyperball (B D ) and their direct-product spaceWe find that the maximal number of species is in exact agreement with the sum of Betti numbers D r=0 β r for these manifolds. Indeed, the 4D lattice fermion on torus has up to 16 species while the sum of Betti numbers of T 4 is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space T D × B d . We study several examples of lattice fermions defined on discretized hypersphere (S D ), and find that it has up to 2 species, which is the same number as the sum of Betti numbers of S D . From these facts, we conjecture the equivalence of the maximal number of fermion species and the summation of Betti numbers. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory. * I. INTRODUCTION Quantum field theories are non-perturbatively investigated by use of the numerical Monte-Carlo simulation based on lattice field theory [1,2]. One of the most delicate problems in the application is how to introduce fermionic degrees of freedom [3][4][5][6], where we encounter serious problems such as the reconcilement of a desirable number of flavors and chiral symmetry, the realization of a single Weyl fermion, and the sign problem of the quark determinant. The lattice fermion formulations including Wilson fermions [7], Domain-wall or overlap fermions [8][9][10][11][12], and staggered fermions [13][14][15][16][17][18][19] have been proposed to bypass the problems and have been broadly used in the lattice simulation. Apart from them, relatively new approaches have been proposed, including the generalized Wilson fermions [20][21][22][23][24][25][26][27], the staggered-Wilson fermions [17,24,[28][29][30][31][32][33][34][35][36][37][38][39], the minimally doubled fermion and the central-branch Wilson fermion [24,32,61,62]. The non-torus lattice fermion formulation is also an interesting avenue, where we consider lattices with nonzero-genus topologies. In the previous work of ours, we introduced spectral graph theory to clarify the number of zero-eigenvalues of lattice Dirac operators [63]. By use of this tool, we made progress in understanding the number of fermion species on lattices with arbitrary topologies and no discrete translational symmetry 1 . In this work we propose a new conjecture on the relation between the topology of a manifold and the number of species (doublers) of lattice fermions defined on a lattice-discretized version of the manifold. All through this work, we identify the number of "species" as the number of Dirac zero-modes of free lattice fermions. The claim of our conjecture is that the maximal number of fermion species on a finite-volume and finite-spacing lattice defined by discretizing a D-dimensional manifold M is identical to the summation of the Betti numbers of the manifold D r=0 β r (M) as long as the formulation has several basic properties, including locality, γ 5 -hermiticity and central difference. The r-th Betti number is defined as the rank of the r-th homology group H r (M). We show that our conjecture holds for lattice fermions on torus (T D ), hyperball (B D ) and their product space T D × B d . We also investigate lattice fermions defined on discretized hypersphere (S 2 ) and find a empirical fact that they have up to 2 species, while the the sum of Betti numbers for S D is 2 irrespective of dimensions. This explicit investigation indicates that the conjecture holds even for spherical and non-hypercubic lattices. In the end of this paper, we discuss the way how to prove this coincidence in terms of Hodge theory [66][67][68][69][70] and spectral graph theory [63,[71][72][73][74]. We there 1 In the other context, the Dirac-Kahler fermions on generic manifolds have been investigated, where the relation between the fermion zero-modes and the Euler characteristics of the manifold are discussed [64,65]. introduce a new viewpoint ithat the lattice Wilson term corresponds to a sort of Laplacian operator giving Betti numbers. The reason we use the terminology, "the maximal number of fermion species", is two-folded: The number of species depends on the number of lattice sites when we consider a finite-volume and finite-spacing lattice. Thus, we focus just on the case that the number of species are "maximal" in this sense given a lattice-discretized manifold. It also depends on lattice fermion formulations. Thus, we focus on the case of the "maximal" number of species given a certain lattice, which is usually realized by adopting the naive fermion formulation. This paper is constructed as follows: In Sec. II, we present empirical evidences for our conjecture. In Sec. III, we propose the main conjecture. In Sec. IV, we discuss a program for the proof of the conjecture. Sec. V is devoted to the summary and discussion. II. MAXIMAL NUMBER OF SPECIES ON DISCRETIZED MANIFOLDS In this section, we present empirical evidences that the maximal number of species on the discretized manifold is equal to the summation of the Betti numbers of the continuum manifold. We firstly focus on three types of manifolds, D-dimensional torus T D , D-dimensional hyperball B D , and (D + d)-dimensional cylinder T D × B d . We also present explicit calculations of fermion species on the discretized sphere S D . As discussed in the introduction, the definition of "the number of fermion species" is the number of Dirac zero-modes of the lattice Dirac operator (Dirac matrix) in a free theory, where we have to pay attention to the difference of sizes of γ-matrices in different dimensions. A. D-dimensional torus T D The discretized D-dimensional torus is realized as the D-dimensional square (hypercubic) lattice with the periodic boundary condition (PBC) imposed in each dimension. Hereafter, we denote the discretized D-dimensional torus as * T D in order to distinguish it from the continuum manifold T D . The discretized torus in two dimensions is schematically depicted in Fig. 1. The lattice naive-fermion action on * T D , or on D-dimensional square lattice with PBC, is S T D = n D µ=1ψ n γ µ D (PBC) µ ψ n ≡ψD (PBC) ψ ,(1) where D µ ≡ (T +µ −T −µ )/2 with T ±µ ψ n = U n,±μ ψ n±μ andμ is a unit vector. In a free theory, we just set U n,±μ = 1. Unless otherwise specified, we consider a free theory hereafter. The sum n stands for the summation over lattice sites n = (n 1 , n 2 , · · · , n D ) ∈ N D , whose intervals are 1 ≤ n µ ≤ N . The vector ψ is defined as ψ ≡ n ψ n |n with |n ≡ D µ=1 |n µ = |n 1 ⊗ |n 2 ⊗ · · · ⊗ |n D . Then, the diagonalized Dirac matrix D is obtained as U † D (PBC) U = k D µ i sin 2π N (k µ − 1) γ µ |k k| ,(2)where k = (k 1 , k 2 , · · · , k D ) ∈ [1, N ] D with k µ ∈ Z. The symbol U is the unitary matrix defined as U ≡ n,k µ exp 2πi N (n µ − 1) (k µ − 1) |n k| with |k ≡ D µ=1 |k µ . The condition that the Dirac matrix has zero-eigenvalues (zero-modes) is simply given by D µ iγ µ sin 2π N (k µ − 1) = 0 =⇒ sin 2π N (k µ − 1) = 0 ,(3) as a consequence of the linear independence of γ-matrices. It shows that we have up to 2 D fermion species on * T D as long as we take an even N . For an odd N , the number of species can decrease, but never increases. Even if we adopt another fermion formulation such as Wilson and staggered fermions on the discretized torus, but the number of species for these cases are certainly smaller than 2 D . Thus, we argue that the maximal number of species is 2 D on the discretized torus. Note that we only consider a finite-volume and finite-spacing lattice here. We never take a continuum or a thermodynamic limit in this paper. Meanwhile, the r-th Betti number β r (T D ) of the D-dimensional torus T D is obtained as β r (T D ) = D C r ,(4) where r is an integer from 1 to D. For instance, Betti numbers of four-dimensional torus T 4 are β 0 (T 4 ) = 1, β 1 (T 4 ) = 4, β 2 (T 4 ) = 6, β 3 (T 4 ) = 4, β 0 (T 4 ) = 1. Then, the summation of the Betti number in Eq. (4) over r is D r=0 β r (T D ) = D r=0 D C r = 2 D .(5) From these results, we argue that the summation of the Betti numbers for the D-dimensional torus T D is equal to the maximal number of lattice fermion species on the discretized D-dimensional torus * T D . B. D-dimensional hyperball B D The discreized D-dimensional hyperball * B D is realized as the D-dimensional square lattice with Dirichlet boundary condition (DBC) imposed in each dimension. The difference from * T D is that there is a pair of unconnected boundaries in each direction. The lattice naive-fermion action on * B D is S B D = n D µ=1ψ n γ µ D (DBC) µ ψ n ≡ψD (DBC) ψ.(6) Note that this action is different from Eq. (1) in that there are no hoppings between the boundaries. The diagonalized Dirac matrix by the unitary matrix V ≡ n,k µ i kµ sin nµkµπ N +1 |n k| is V † D (DBC) V = k D µ i cos k µ π N + 1 γ µ |k k| ,(7) where k = (k 1 , k 2 , · · · , k D ) ∈ [1, N ] D with k µ ∈ Z. The Dirac matrix has zero-eigenvalues (zeromodes) when Eq. (2) satisfies D µ iγ µ cos k µ π N + 1 = 0 =⇒ cos k µ π N + 1 = 0.(8) It means that there is one Dirac zero-mode, or equivalently equivalently a single species, if we take an odd number as N . Eq. (8) has no solution since k µ = N +1 2 is not an integer if N is even. As with the case of torus, other fermion formulations never increase the number species. Hence, we argue that the maximal number of fermion species on * B D is one. We also note that we just consider the bulk fermionic degrees of freedom, not an edge mode. On the other hand, the r-th Betti numbers for the hyperball are given as β r (B D ) = δ r0 ,(9) where δ stands for the Kronecker delta. The summation of the Betti numbers in Eq. (9) is D r=0 β r (B D ) = D r=0 δ r0 = 1.(10) From these results, we claim that the maximal number of fermion species on D-dimensional latticediscretized hyperball * B D is equal to the summation of the Betti numbers over 0 ≤ r ≤ D for the D-dimensional hyperball B D . If we take an infinite-volume limit, the number of naive-fermion species on D-dimensional discretized hyperball * B D approaches 2 D , which is the same as that on D-dimensional discretized torus. It is notable that our conjecture is applicable only to the finite-volume lattice. C. (D + d)-dimensional product manifold T D × B d The (D+d)-dimensional product manifold T D ×B d consists of two manifolds, the D-dimensional The lattice naive-fermion action on this lattice is torus T D and the d-dimensional hyperball B d . It means that the discretized (D + d)-dimensional product manifold * T D × * B d consists of the discretized D-dimensional torus * T D and the discretized d-dimensional hyperball * B D . The (D + d)-dimensional square lattice is identified as * T D × * B dS T D ×B d = nψ n   D µ=1 γ µ D (PBC) µ + d ν=D+1 γ ν D (DBC) ν   ψ n ≡ψD (P+D) ψ .(11) We note that, if we add the Wilson term to this action in (4 + 1) dimensions, it results in the action of Domain-wall fermion. For this case, the zero-modes of the Dirac matrix emerge as edge modes. The Dirac matrix D (P+D) in Eq. (11) can be diagonalized by the unitary matrix W ≡ k µ exp 2πi N (n µ − 1) (k µ − 1) ν i kν sin nν kν π N +1 |n k| with |n ≡ D+d λ=1 |n λ and |k ≡ D+d λ=1 |k λ . The diagonalized Dirac matrix is W † D (P+D) W = i    D µ=1 γ µ sin 2π N (k µ − 1) + D+d ν=D+1 γ ν cos k ν π N + 1    |k k| .(12) The condition for Eq. (12) to have zero eigenvalues (zero-modes) is D µ=1 γ µ sin 2π N (k µ − 1) + D+d ν=D+1 γ ν cos k ν π N + 1 = 0 =⇒ sin 2π N (k µ − 1) = 0, cos k ν π N + 1 = 0 .(13) It shows that the maximal number of fermion species The Betti numbers of the product manifold are on * T D × * B d is 2 D = 2 D × 1,β r (T D × B d ) =      D C r (1 ≤ r ≤ D) 0 (r > D) . The summation of the r-th Betti numbers (0 ≤ r ≤ D + d) for the (D + d)-dimensional product manifold T D × B d is D+d r=0 β r (T D × B d ) = D r=0 D C r = 2 D .(15) These results indicate that the maximum number of species on * T D × * B d is equal to the summation of the Betti numbers of T D × B d . We also note that the maximal number of naive-fermion species approaches 2 D+d in an infinitevolume limit. Our statement of the conjecture holds only for finite-volume cases. D. D-dimensional sphere S D In the continuum field theory, the fermion action on spheres gives massive fermionic degrees of freedom since the curvature works as effective mass. It is, however, not the case on the discretized sphere. In this subsection, we study the number of fermion species on the discretized sphere. We empirically show that the maximal number of species on the discretized sphere is equal to the sum of the Betti number of D-dimensional sphere S D . We begin with the two-dimensional cases. We firstly consider the following discretized spherical coordinate system for 2-sphere, labeled by two integers (M, N ): x 3 = r cos θ 2 , x 2 = r sin θ 2 cos θ 1 , x 1 = r sin θ 2 sin θ 1 ,(16) where r is a radial distance and two angles are discretized as θ 2 ≡ (N − n 2 ) π/ (N − 1) , θ 1 ≡ 2n 1 π/M with n 1 , n 2 ∈ N and n 1 ∈ [1, N ], n 2 ∈ [1, M ]. For simplicity, we fix a radial distance as r = 1. We label lattice sites as (n 1 , n 2 ). Note that there are two special points (n 1 , 1), (n 1 , N ) who ignore the hopping in n 1 -direction. We call the two points the south pole, relabeled as (0, 1), and the north pole, relabeled as (0, N ), respectively. By this definition, we consider the naive-fermion action on the discretized 2-sphere labeled by (M, N ) as S (M,N ) = n 2 µ=1ψ n σ µ D µ ψ n ≡ψD (M,N ) ψ ,(17) where ψ is a vector defined as ψ = n ψ n e n with the standard basis e n and D (M,N ) is the Dirac matrix. The sum n stands for the summation over lattice site n = (n 1 , n 2 ). Here, we specify the order of components ψ n in the vector as (0, 1) → (1, 2) → (2, 2) → · · · → (M, 2) → (1, 3) → · · · → (M, N − 1) → (0, N ). Namely, it is expressed as ψ =                         ψ (0,1) ψ (1,2) ψ (2,2) . . . ψ (M,2) ψ (1,3) . . . ψ (M,N −1) ψ (0,N )                         .(18) The Dirac matrix D (M,N ) is expressed as D (M,N ) = 1 2   I N −2 ⊗ P M O 2   ⊗ σ 1 + 1 2   D N −2 ⊗ I M V M (N −2),2 −V † M (N −2),2 O 2   ⊗ σ 2 ,(19) where σ i is the Pauli matrix and I k and O k are the k dimensional identity matrix and null matrix, respectively. P M is the M × M difference matrix with the periodic boundary condition and D N −2 is the (N − 2) × (N − 2) difference matrix with the Dirichlet boundary condition as P M =                   0 1 0 0 0 −1 −1 0 1 · · · 0 0 0 0 −1 0 0 0 0 . . . . . . . . . 0 0 0 0 1 0 0 0 0 · · · −1 0 1 1 0 0 0 −1 0                   , D N −2 =                   0 1 0 −1 0 1 0 −1 0 . . . 0 1 0 −1 0 1 0 −1 0                   .(20)V M (N −2),2 is given by V M (N −2),2 ≡            1 0 0 0 . . . 0 0 0 −1            N −2,2 ⊗            1 1 . . . 1 1            M,1 ,(21) where the subscripts j, k of the matrix ( ) j,k stand for the row and column sizes. In the example of the discretized 2-sphere (M, 3), the action is S (M,3) =ψD (M,3) ψ .(22) The discretized 2-spheres (4, 3) and (6, 3) are depicted in Fig. 3. As seen from the figures, these cases correspond to a sort of the triangulation of 2-sphere. The Dirac matirx D (M,3) is D (M,3) = 1 2   P M O 2   ⊗ σ 1 + 1 2   I M V M,2 −V † M,2 O 2   ⊗ σ 2 ,(23) where V M,2 is This matrix is diagonalized by the unitary matrix V M,2 = 1 −1 1,2 ⊗            1 1 . . . 1 1            M,1 .(24)U (M,3) ≡ U (M,3) ⊗ I 2 ,(25) where I 2 is a 2 × 2 identity matrix and U (M, 3) is U (M,3) = 1 √ M ×                         M 2 0 0 0 · · · 0 0 − √ M |χ| 2 − √ M |χ| 2 0 1 1 1 · · · 1 1 χχ 0 ξ ξ 2 ξ 3 · · · ξ M −2 ξ M −1 χχ 0 ξ 2 ξ 4 ξ 6 · · · ξ 2(M −2) ξ 2(M −1) χχ . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 ξ M −3 ξ 2(M −3) ξ 3(M −3) · · · ξ (M −3)(M −2) ξ (M −3)(M −1) χχ 0 ξ M −2 ξ 2(M −2) ξ 3(M −2) · · · ξ (M −2) 2 ξ (M −2)(M −1) χχ 0 ξ M −1 ξ 2(M −1) ξ 3(M −1) · · · ξ (M −1)(M −2) ξ (M −1) 2 χχ M 2 0 0 0 · · · 0 0 √ M |χ| 2 √ M |χ| 2                         ,(26) with ξ ≡ e 2π M (n 1 −1) and χ ≡ i/ √ 2. Then, the spectra of the Dirac matrix D (M,3) is obtained as These results indicate that there are up to two fermion species on the discretized 2-sphere (M, 3). We show that the maximal number of fermion species are two also in other cases including (M, N ) = (4, 4), (5,4), (5,5), (4,5), (6,9) by numerical calculations as shown in Appendix. A 1. The results are summarized in Table. II. U (M,3) † D (M,3) U (M,3) = Diag 0, sin 2π M σ 1 , sin 4π M σ 1 , · · · , sin 2π(M −1) M σ 1 , −i M 2 σ 2 , i M 2 σ 2 .(27) The maximal number of fermion species, "two", is equal to the sum of the Betti numbers for the 2-sphere 2 r=0 β r (S 2 ) = 2 r=0 (δ r0 + δ r2 ) = 2 ,(28) where the r-th Betti number for the 2-sphere is β r (S 2 ) = δ r0 + δ r2 . In higher dimensions, we discuss the fermion action on the cellular-decomposed sphere in a parallel manner. We show that there are up to two species on the discretized 4-sphere labeled by four integers (N 1 , N 2 , N 3 , N 4 ) by numerical calculation in Appendix. A 2. On the other hand, the r-th Betti numbers (0 ≤ r ≤ D) for D-dimensional sphere S D is given as β r (S D ) = δ r0 + δ rD .(29) Thus, the sum of the Betti numbers for the D-dimensional sphere is given as From these results, we argue that the maximal number of fermion species on the D-dimensional discretized sphere * S D is equal to the sum of the Betti numbers for the D-dimensional sphere S D . It is notable that the lattice fermion action on the spherical lattice has been studied in the literature in the different context [75][76][77]. Our result is consistent with the observations obtained in the literature. III. NEW CONJECTURE In the previous section, we have discussed that the maximal number of lattice fermion in terms of the Betti numbers of the continuum manifold. In Table. I we summarize the relation of the sum of the Betti numbers and the maximal number of fermion species. From these facts, we propose a new conjecture on the number of fermion species on the discretized manifold. The conjecture is as follows: Conjecture. 1. We firstly impose the following five conditions on the fermion action of the latticediscretized D-dimensional manifold M: Torus T D (1 + 1) D 2 D Hyperball B D 1 + 0 + 0 + · · · 1 Sphere S D 1 + 0 + 0 + · · · + 1 2 T D × B d 2 D × 1 2 D i. Central difference; anti-hermiticity of the Dirac matrix in the action holds due to this condition. ii. γ 5 hermiticity; even the action with the mass term or the Wilson term satisfies this condition. iii. Four spinors; this condition assures the linear independence of the lattice action for each direction. iv. Locality; this condition leads to finite-hopping actions although it may be unnecessary for our conjecture because non-locality usually decreases the number of species. v. Finite volume lattice; our conjecture claims that the fermion action on the finite-volume lattice picks up the topology of the continuum manifold. Our conjecture claims that, as long as these conditions hold, the maximal number of fermion species on the lattice-discretized D-dimensional manifold is equal to the summation of Betti numbers β r (M) over 0 ≤ r ≤ D for the continuum manifold M. It is expressed as max[N ( * M)] = D r=0 β r (M) ,(31) where N ( * M) is the number of fermion species on the lattice-discretized manifold * M. We consider that this conjecture holds for any orientable manifolds in any dimensions, including N -holed tori. IV. PROGRAM FOR PROOF In the previous section, we propose the conjecture on the equivalence between the maximal number of fermion species on the discretized manifold and the summation of Betti numbers of the continuum manifold, by studying the D-dimensional torus-lattice * T D , the hyperball-lattice * B D , the D × d dimensional cylinder-lattice * T D × * B d and the D-dimensional sphere lattice * S D . We now propose a program for proof of the conjecture in terms of spectral graph theory and Hodge theory. In Hodge theory [67][68][69][70], the number of zero-eigenvalues of a r-th Laplacian (∆ r = ∂ r+1 ∂ * r+1 + ∂ * r ∂ r ) defined on a complex chain coincides with the r-th Betti number, where ∂ r is a r-th boundary operator. Our program for the proof is inspired by this fact. In our program, we re-interpret lattice fermions as spectral graphs [63]. We define a Laplacian operator L of a graph G(V, E) having the set of vertices V and the set of edge E as L ij ≡              d i if i = j −1 if i = j and (i, j) are linked 0 if i = j and (i, j) are linked ,(32) where d i is the number of edges sharing the site i. Vertices in graph theory correspond to lattice sites in lattice field theory while edges correspond to links. The detailed correspondence between lattice field theory and spectral graph theory is summarized in our previous work [63]. A. One-dimensional lattices We begin with one-dimensional lattices. We consider two types of one-dimensional lattices: the one with PBC (1-dim discretized torus * T 1 ) and the other with DBC (1-dim discretized hyperball * B 1 ). For the one-dimensional lattice with N vertices with PBC, the diagonal elements of Laplacian matrix d i are d i = 2 for i = 1, 2, ..., N , thus the Laplacian in Eq. (32) is L ij = 2 N k=1 δ ik δkj − δ iN δ 1j + δ i1 δ N j + N −1 k=1 δ ik δ k+1 j + N k=2 δ ik δ k−1 j .(33) For simplicity, we take an even N for this case. It is notable that the number of zero-eigenvalues of this Laplacian is one, which can be checked by direct calculation. ψLψ = N n=1ψ n [ψ n+1 + ψ n−1 − 2ψ n ] = 2 N n=1ψ n mψ n − r 2 (ψ n+1 + ψ n−1 − 2ψ n ) m=0, r=1 ≡ S Wt | m=0, r=1 .(34) This fact means that the number of zero-eigenvalues of the matrix corresponding to the Wilson term on * T 1 is equivalent to the 0-th Betti number β 0 = 1 for the continuum torus T 1 . We will later discuss that it also holds for * B 1 and B 1 . We next consider another Laplacian operator defined as It strongly indicates that the summation of Betti numbers on an one-dimensional manifold (T 1 or B 1 ) is in exact agreement with the number of zero-eigenvalues of the naive Dirac matrix on the lattice-discretized manifold ( * T 1 or * B 1 ), which gives the maximal number of species. To complete our proof for one dimension, however, we have to prove that the number of zero-eigenvalues of L ′ is β 1 of the corresponding manifold. L ′ ij ≡              −d i if i = j −1 if i = j and (i, j) are linked 0 if i = j and (i, j) are linked ,(35) B. Higher-dimensional lattices For the higher-dimensional torus-lattice * T D , the hyperball-lattice * B D and the cylinder-lattice * T D × * B d , we can utilize Kunneth theorem claiming that the homology groups of two cellular chain complexes C, C ′ and C × C ′ have the following relation: H r (C × C ′ ) ∼ = p+q=r H p (C) ⊗ H q (C ′ ) .(36) This theorem means the homology group and its rank (Betti number) of a certain two-dimensional product manifold is obtained from those of the one-dimensional manifolds. By repeating this, we can obtain the homology groups and Betti numbers for any higher-dimensional manifolds. At least for * T D , * B D and * T D × * B d , we consider that we can extend the argument in the previous subsection to higher-dimensional cases. In this extension, the Laplacian operators giving Betti numbers β r are expressed as the sum of tensor products of the Laplacians L in Eq. (33) and L ′ in Eq. (35). We now give an example: the Laplacian giving 0-th Betti number for four-dimensional lattices is expressed as L (D=4) r=0 = L ⊗ 1 N ⊗ 1 N ⊗ 1 N + 1 N ⊗ L ⊗ 1 N ⊗ 1 N + 1 N ⊗ 1 N ⊗ L ⊗ 1 N + 1 N ⊗ 1 N ⊗ 1 N ⊗ L ⊗ 1 4 ,(37) where the lattice has N 4 vertices (lattice sites). The symbol 1 N is a N × N identity matrix. The Lagrangian constructed from this Laplacian is nothing but the four-dimensional Wilson term with m = 0, r = 1. The Laplacian giving 4-th Betti number for four-dimensional lattices is expressed as This generalization to higher-dimensions is directly applicable to the manifold obtained as a product space of T D and B D . For the manifold such as S D , we have to develop a more generic way of generalization to higher-dimensions. L (D=4) r=4 = L ′ ⊗ 1 N ⊗ 1 N ⊗ 1 N + 1 N ⊗ L ′ ⊗ 1 N ⊗ 1 N + 1 N ⊗ 1 N ⊗ L ′ ⊗ 1 N + 1 N ⊗ 1 N ⊗ 1 N ⊗ L ′ ⊗ 1 4 ,(38) In the end of this section, we comment on another avenue toward proof of the conjecture: Squaring the free naive Dirac matrix leads to another Laplacian operator. If we can prove that the number of zero-modes of this Laplacian is the sum of the Betti numbers of the continuum manifold, we can easily give a generic proof for any kind of manifolds including S D . V. SUMMARY AND DISCUSSION Our conjecture claims that the maximal number of fermion species on a finite lattice is equivalent to the summation of the Betti numbers of the D-dimensional manifold from which the lattice is constructed. It is summarized as max[N ( * M)] = D r=0 β r (M) ,(39) where N ( * M) is the number of fermion species on the lattice, which is defined as a lattice-discretized version * M of the manifold M. In a sense that this conjecture relates the number of lattice fermion species (doublers) to the topology of spacetime manifold, it is complementary to the Nielsen-Ninomiya's no-go theorem which claims the emergence of pairs of fermion species as a result of the cancellation of chiral charges on torus. We note that the term, "the number of fermion species", we used in the paper is defined as the number of Dirac zero-modes of fermions on finite lattices. For infinite-volume lattices things drastically change: For example, the number of species of naive fermion on the one-dimensional lattice hyperball * B 1 approaches two in an infinite-volume limit, which is the same number as that on * T 1 . It is of great importance that our conjecture relates the topology of a continuum manifold and the fermion species on a finite lattice defined by discretizing the manifold. We believe that the program for the proof we have discussed in this paper is applicable to lattice fermions defined on generic manifolds. Future works will be devoted to the complete proof of the theorem. The Dirac matrix on the (4, 4) sphere is D (4,4) = 1 2   I 2 ⊗ P 4 O 2   ⊗ σ 1 + 1 2   D 2 ⊗ I 4 V 8,2 −V † 8,2 O 2   ⊗ σ 2 ,(A1) with V 8,2 =   1 0 0 −1   2,2 ⊗         1 1 1 1         . (A2) The eigenvalues of this Dirac matrix are depicted in Fig. 5. There is no zero-eigenvalue in D (4,4) , which means that there is no fermion species. FIG. 5: Eigenvalue distribution of the Dirac matrix D (4,4) . There is no species, or equivalently no zero-eigenvalue. • If M is odd and N is even, there is again no species. For instance, we take the discretized 2-sphere labeled by (5,4). We numerically find that the number of zero-modes or the number of species is zero as shown in Fig. 6. -2 -1 0 1 2 -3 -2 -1 0 1 2 3 Re Im FIG. 6: Eigenvalue distribution of the Dirac matrix D (5,4) . There is no species (no zero-eigenvalue) in D (5,4) . • If both of M and N are odd, there is one species. For instance, we consider the discretized 2-sphere labeled by (5,5). Then, we numerically find that the number of species is one as shown in Fig. 7 and Fig. 8. We note that a pair of eigenvalues corresponds to a single Dirac mode since the two-dimensional γ matrices are 2 × 2 matrices. Thus, the existence of the pair of zero-eigenvalues shown in Fig. 7 and Fig. 8 means that there is a single Dirac zero-mode. • If M is even and N is odd, there are two species on the discretized 2-sphere. For instance, we take two cases, (4,5) and (6,9). The discretized 2-sphere (4, 5) is depicted in Fig. 9. The Dirac matrix on the (4, 5) sphere is D (4,5) = 1 2   I 3 ⊗ P 4 O 2   ⊗ σ 1 + 1 2   D 3 ⊗ I 4 V 12,2 −V † 12,2 O 2   ⊗ σ 2 ,(A3) with The eigenvalues of this Dirac matrix are depicted in Fig. 10. There are two species on 2-sphere (4, 5) since there are four zero-eigenvalues as seen from Fig. 11. There are four zero-eigenvalues corresponding to two species in two dimensions. V 12,2 =      1 0 0 0 0 −1      3,2 ⊗         1 1 1 1         .(A4) In the case of the (6, 9) 2-sphere, the Dirac matrix is D (6,9) = 1 2   I 7 ⊗ P 6 O 2   ⊗ σ 1 + 1 2   D 7 ⊗ I 6 V 42,2 −V † 42,2 O 2   ⊗ σ 2 ,(A5) with V 42,2 =                  1 0 0 0 0 0 0 0 0 0 0 0 0 −1                  7,2 ⊗               1 1 1 1 1 1               .(A6) The eigenvalues of this Dirac matrix are depicted in Fig. 12. There are again two species as there are four zero-eigenvalues in D (6,9) as seen from from Fig. 13. FIG. 12: Eigenvalue distribution of the Dirac matrix D (6,9) . There are two species (four zero-eigenvalues). We summarize our results in Table. II. FIG. 13: Eigenvalue distribution is depicted, where the vertical axis represents the imaginary part of the Dirac matrix D (6,9) and horizontal axis represents the serial number of eigenvalues. There are four zero-eigenvalues corresponding to two species in two dimensions. We discuss the discretized four-dimensional sphere. We first consider the discretized fourdimensional spherical-coordinate system for 4-sphere labeled by four integers (N 1 , N 2 , N 3 , N 4 ) as x 5 = r cos θ 4 , x 4 = r sin θ 4 cos θ 3 , x 3 = r sin θ 4 sin θ 3 cos θ 2 , (A7) x 2 = r sin θ 4 sin θ 3 sin θ 2 cos θ 1 , x 1 = r sin θ 4 sin θ 3 sin θ 2 sin θ 1 (A8) where r is a radial distance and the four angles are discretized as θ 1 ≡ 2π N 1 (n 1 − 1) , θ i ≡ π N i −1 (n i − 1) for i = 2, 3, 4. n 1 , n i ∈ N run as n 1 ∈ [1, N 1 ], n i ∈ [1, N i ]. For simplicity, we fix a radial distance as r = 1. In a parallel manner to the discussion for 2-sphere, we label the lattice sites as (n 1 , n 2 , n 3 , n 4 ). For instance, we take N 1 = 4 and N i = 3 for i = 2, 3, 4. A graph corresponding to the (4, 3, 3, 3) 4-sphere is depicted in Fig. 14. The Dirac matrix is given as FIG. 14: Graph corresponding to the (4, 3, 3, 3) 4-sphere is the 5-orthoplex inside Petrie polygon. D (4,3,3,3) =   D 4 O 6   ⊗ γ 1 +      O 4 V 4,2 −V † 4,2 O 2 O 4      ⊗ γ 2 +      O 6 V 6,2 −V † 6,2 O 2 O 2      ⊗ γ 3 +   O 8 V 8,2 −V † 8,2 O 2   ⊗ γ 4 .(A9) This Dirac matrix D (4,3,3,3) is 10 × 10 square matrix, apart from the γ matrix structure. and Fig. 16 shows the eigenvalue distributions of the Dirac matrix D (4,3,3,3) . For the dicretized 4-sphere (4, 3, 3, 3), we find that there are two species as there are eight zero-eigenvalues of the Dirac matrix D (4,3,3,3) . By studying other cases, we find that the number of species on the discretized 4-sphere (N 1 , N 2 , N 3 , N 4 ) is two when the number of sites N 1 is even and the number of sites N i for i from 2 to 4 are odd, as with the case on the discretized 2-sphere (M, N ). We could not find any example where the number of species goes beyond two in four dimensions too. FIG. 1 : 1Two-dimensional square lattice with opposite boundaries identified corresponds to the discretized two-dimensional torus. by imposing PBC on the D-dimensions and imposing DBC on the remaining d-dimensions. The manifold in (1 + 1)-dimensions is depicted in Fig. 2.FIG. 2: (1 + 1)-dimensional square lattice with the left and right boundaries connected is identified as the discretized hollow cylinder. which is the product of the maximum numbers of fermion species on * T D and * B D . The other fermion formulations, including the Domain-wall setup, never increase the number of species, but can decrease it. FIG. 3 : 3The discretized 2-spheres (4, 3) and (6, 3) are depicted. Red points stand for lattice sites and lines between the red points stand for links. From Eq. (27), one finds that the number of Dirac zero-modes of the Dirac matrix is algebraically two for even M since there is a certain j satisfying j = M 2 + 1 ∈ N and sin 2π(j−1) M = 0 other than j = 1. On the other hand, the number of Dirac zero-modes is one for odd M since only j = 1 satisfies sin 2π(j−1) M = 0 for this case. r0 + δ rD ) = 2 . From the generic argument of Laplacian matrices in homology theory, it is known that the number of zero-eigenvalues of L of the graph G(V, E) represents the number of connected components, or equivalently the 0-th Betti number β 0 (G) of G(V, E). Since an certain manifold in one dimension and its lattice-discretized graph share the same Betti numbers (β 0 , β 1 ), the number of zero-eigenvalues of L in Eq. (33) gives the 0-th Betti number of one-dimensional torus. Indeed, the 0-th Betti number of one-dimensional torus is one. The Lagrangian constructed from the Laplacian matrix and the field vector ψ = N i=1 ψ i e i results in the Wilson term S Wt with the mass parameter m set to zero and the Wilson-fermion parameter r set to one in a free theory, where it is expressed as with d i = 2 on * T 1 . This Laplacian corresponds to the case with the mass parameter m = −2 in Eq.(34). The number of zero-eigenvalues of this Laplacian L ′ on * T 1 is one, which is equivalent to the first Betti number of the continuum torus T 1 . From this fact, we obtain one observation that the number of zero-eigenvalues of L ′ gives the first Betti number in one dimension, which we have to prove in the future.To convince readers, we study the one-dimensional lattice withN vertices with DBC ( * B 1 ), where the diagonal elements of the Laplacian matrices d i are d 1 = 1, d N = 1, and d i = 2 for i = 2, ..., N − 1, where we take an odd N for this case. For the Laplacian L in Eq. (33), the number of zero-eigenvalues of L, which again corresponds to the Wilson term on * B 1 , gives the number of connected components, thus it is gives the 0-th Betti number β 0 = 1 of B 1 . For the Laplacian L ′ in Eq. (35), the number of its zero-eigenvalues on * B 1 is zero, which is equivalent to the first Betti number of B 1 . It is notable that our observation on the equivalence of the number of zero eigenvalues of L ′ and the first Betti number also holds for * B 1 and B 1 .So far, we study the Laplacian corresponding to the Wilson term on the one-dimensional lattice (graph) and find the following facts:• The number of zero-eigenvalues of L corresponding to the Wilson term with m = 0, r = 1 is equivalent to β 0 of the corresponding manifold. It is a mathematically rigorous fact.• The number of zero-eigenvalues of L ′ corresponding to the Wilson term with m = −2, r = 1 seems to be equivalent to β 1 of the corresponding manifold.Two zero-eigenvectors of the free naive Dirac matrix (Dirac operator) on * T 1 , which are found from the condition in Eq.(3), are identical to the one of L and the one of L ′ , respectively, while one zero-eigenvector of the naive Dirac matrix on * B 1 , which is found from the condition in Eq.(8), is identified with that of L ′ . These results originate in the fact that the naive Dirac matrix and the Laplacian operators (L, L ′ ) are simultaneously diagonalizable. which corresponds to the four-dimensional Wilson term with m = −8, r = 1. The Laplacian giving other Betti numbers are also expressed as some tensor product of L and L ′ , resulting in a certainWilson term for which one of the branches of Wilson Dirac spectrum is set to the origin. As a byproduct, this procedure of proof shows that the number of fermion species on the well-known five branches of Wilson Dirac spectrum 1, 4, 6, 4, 1 stand for the Betti numbers β r (T 4 ) for r = 0, 1, 2, 3, 4. ACKNOWLEDGMENTS T. M. and J. Y. are grateful to S. Aoki, S. Matsuura and K. Ohta for the fruitful discussion on the topics of the paper. The completion of this work is owed to the discussion in the workshop "Lattice field theory and continuum field theory" at Yukawa Institute for Theoretical Physics, Kyoto University (YITP-W-22-02). This work of T. M. is supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grant Numbers 19K03817. Appendix A: Numerical analysis for D-dimensional spheres 1. Two-dimensional sphere In this appendix, we show that there are up to two species on the discretized 2-sphere labeled by (M, N ) 2 by numerical calculations: • If both the number of sites on the longitude direction M and the number of sites on the latitude direction N are even, there is no fermion species, or no Dirac zero-modes of the Dirac matrix. For instance, we consider the discretized 2-sphere labeled by (4, 4), which is depicted in Fig. 4. FIG. 4: Discretized 2-sphere labeled by (4, 4). FIG. 7 : 7Eigenvalue distribution of the Dirac matrix D(5,5) . The pair of zero eigenvalues corresponds to a single Dirac zero-mode. FIG. 8 : 8Eigenvalue distribution is depicted, where the vertical axis represents the imaginary part of the Dirac matrix D(5,5) and the horizontal axis represents the serial number of eigenvalues. The pair of zero eigenvalues corresponds to a single Dirac zero-mode. FIG. 10 : 10Eigenvalue distribution of the Dirac matrix D(4,5) . There are four zero-eigenvalues corresponding to two species in two dimensions.FIG. 11: Eigenvalue distribution is depicted, where the vertical axis represents the imaginary part of the Dirac matrix D(4,5) and the horizontal axis represents the serial number of eigenvalues. Fig. 15 15 FIG. 15 : 15Eigenvalue distribution of the Dirac matrix D(4,3,3,3) . There are two species in four dimensions, which emerge as eight zero-eigenvalues in the figure. FIG. 16 : 16Eigenvalues of the Dirac matrix D(4,3,3,3) . There are eight zero-eigenvalues corresponding to two species in four dimensions. TABLE I : IBetti numbers and Maximal numbers of speciesmanifold M sum of β r (M ) maximal # of species 1-d torus 1 + 1 2 2-d torus 1 + 2 + 1 4 3-d torus 1 + 3 + 3 + 1 8 4-d torus 1 + 4 + 6 + 4 + 1 16 TABLE II : IIMaximal number of species on 2-sphere (M, N )the number of M the number of N maximal # of species even even 0 odd even 0 odd odd 1 even odd 2 2. 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{'abstract': 'We propose a new conjecture on the relation between the species doubling of lattice fermions and the topology of manifold on which the fermion action is defined. Our conjecture claims that the maximal number of fermion species on a finite-volume and finite-spacing lattice defined by discretizing a D-dimensional manifold is equal to the summation of the Betti numbers of the manifold. We start with reconsidering species doubling of naive fermions on the lattices whose topologies are torus (T D ), hyperball (B D ) and their direct-product spaceWe find that the maximal number of species is in exact agreement with the sum of Betti numbers D r=0 β r for these manifolds. Indeed, the 4D lattice fermion on torus has up to 16 species while the sum of Betti numbers of T 4 is 16. This coincidence holds also for the D-dimensional hyperball and their direct-product space T D × B d . We study several examples of lattice fermions defined on discretized hypersphere (S D ), and find that it has up to 2 species, which is the same number as the sum of Betti numbers of S D . From these facts, we conjecture the equivalence of the maximal number of fermion species and the summation of Betti numbers. We discuss a program for proof of the conjecture in terms of Hodge theory and spectral graph theory. *', 'arxivid': '2301.09805', 'author': ['Jun Yumoto \nDepartment of Mathematical Science\nAkita University\n1-1 Tegata-Gakuen-machi010-8502AkitaJapan\n', 'Tatsuhiro Misumi \nDepartment of Physics\nKindai University\n3-4-1 Kowakae, Higashi-osaka577-8502OsakaJapan\n\nResearch and Education Center for Natural Sciences\nKeio University\n4-1-1 Hitoshi223-8521YokohamaKanagawaJapan\n'], 'authoraffiliation': ['Department of Mathematical Science\nAkita University\n1-1 Tegata-Gakuen-machi010-8502AkitaJapan', 'Department of Physics\nKindai University\n3-4-1 Kowakae, Higashi-osaka577-8502OsakaJapan', 'Research and Education Center for Natural Sciences\nKeio University\n4-1-1 Hitoshi223-8521YokohamaKanagawaJapan'], 'corpusid': 256194256, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 25256, 'n_tokens_neox': 20147, 'n_words': 10508, 'pdfsha': '508711f17f5d7428fef38a95c2db1883ad363bc4', 'pdfurls': ['https://export.arxiv.org/pdf/2301.09805v2.pdf'], 'title': ['New conjecture on species doubling of lattice fermions', 'New conjecture on species doubling of lattice fermions'], 'venue': []}

arxiv

Rational solitons of wave resonant interaction models 28 May 2013 May 30, 2013 Antonio Degasperis antonio.degasperis@roma1.infn.it Istituto Nazionale di Fisica Nucleare Dipartimento di Fisica, "Sapienza" Department of Mathematics and Information Sciences Università di Roma Italy Sara Lombardo sara.lombardo@northumbria.ac.uk Northumbria University Newcastle upon TyneUK Rational solitons of wave resonant interaction models 28 May 2013 May 30, 20130230Ik0545Yv4265Tg Keywords: Integrable PDEsNonlinear wavesDarboux-Dressing TransformationResonant InteractionRational solitonsRouge waves Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector Nonlinear Schrödinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rationalexponential) solutions and find, for the first time to our knowledge, a broad family of such solutions of the three wave resonant interaction equations.Dedicated to the memory of our colleague and friend Sergey Valentinovich Manakov. Introduction Integrable partial differential equations which model nonlinear wave propagation in 1+1 dimension have been largely investigated because of their applicative relevance. In fact, even if approximate, some of them capture important nonlinear effects. This is because they can be derived, as amplitude modulation equations, by multiscale perturbation methods from various kind of (not necessarily integrable) wave equations with the assumption of weak dispersion and nonlinearity (see for instance [1] and references therein). The universality of these integrable models has been well recognized [2], [3]. The best known and simplest example of such models is the Nonlinear Schrödinger (NLS) equation for the evolution of the amplitude of a quasi-monochromatic wave with wave number k and frequency ω, as given by the linear dispersion function ω = ω(k). Many physical applications require however that integrable models be extended to wave coupling. One important instance regards resonance phenomena. If the dispersion relation allows for resonances, multiscale perturbation methods show that the amplitudes of two or more monochromatic waves couple to each other leading to (possibly integrable) systems of nonlinear partial differential equations. The simplest of such integrable systems is the Vector Nonlinear Schrödinger (VNLS) system of equations, given by the following two coupled equations (a subscript denotes partial differentiation) u (1) t = i u (1) xx − 2 s 1 u (1) 2 + s 2 u (2) 2 u (1) u (2) t = i u (2) xx − 2 s 1 u (1) 2 + s 2 u (2) 2 u (2)(1) where, because of the integrability condition, s 2 1 = s 2 2 = 1. This system, also known as Manakov system, follows from the weak resonant condition that two quasi-monochromatic waves, with wave-numbers k 1 and k 2 , have the same group velocity, i.e. ω ′ (k 1 ) = ω ′ (k 2 ) (ω ′ (k) = dω/dk). In (1) u (1) (x, t) , u (2) (x, t) are the amplitudes of these two resonant waves. A different type of phenomena occurs when the medium nonlinearity includes quadratic terms and the dispersion relation ω(k) allows for the two wave numbers k 1 and k 2 to satisfy the strong resonant condition ω(k 1 + k 2 ) = ω(k 1 ) + ω(k 2 ). In this case a third wave is generated with amplitude w(x, t) and the three amplitudes u (1) , u (2) and w couple to each other according to the system of equations u (1) t = −c 1 u (1) x − s 1 w * u (2) u (2) t = −c 2 u (2) x + s 2 wu (1) 0 = w x + s 1 s 2 (c 1 − c 2 ) u (1) * u (2) . (2) In this article we construct particular solutions of both the systems (1) and (2). In the construction method, the physical meaning of the wave amplitudes and of the independent variables x, t does not play any essential role. On the other hand, the results given here are likely to be of applicative relevance in a rather broad range of different physical contexts (f.i. fluid dynamics, nonlinear optics, plasma physics, Bose-Einstein condensate) so it should be kept in mind that the actual meaning of all variables may vary according to context. In particular, for the system (2), if x is the evolution (f.i. time) variable, then this system is the the well known 3 wave resonant interaction (3WRI) equation [4] where the three characteristic velocities are c 1 , c 2 , 0; otherwise, if the evolution variable is t, this system models the nonlocal interaction of two waves (NL2W) [5,6]. Here rescaling transformations have been used to give the equations (1) and (2) a neat form in terms of their coefficients. As for the solutions presented below, we observe that elementary symmetries of equations (1) and (2) (such as gauge transformations and coordinate translations) and linear transformations of the (x, t) plane can be used also to eliminate some of the free parameters which may appear in analytic expressions. Indeed these parameters will be considered in the following as unessential since they can be easily introduced through simple transformations. The kind of solutions we construct here are usually referred to as rational solitons with the following specifications: they are solitons since they are spectrally characterized by the vanishing of the continuous spectrum component, however the discrete spectrum eigenvalues are so special that their corresponding solutions have a rational dependence on the variables x, t, in contrast with the standard soliton whose expression is given in terms of exponentials. Rational solutions of multicomponent wave equations such as (1) and (2) generically have a dependence on coordinates which is richer than in the scalar case by possibly having a mixed rational and exponential expression. Despite this feature, in the following we term rational solitons all these kinds of solutions. Pole singularities in the x, t variables cannot be avoided, these being the zeros of the denominator of the rational expression. However, if these singularities occur only for complex (i.e. strictly non real) values of x and t, these solutions are bounded and gain physical relevance. Rational solutions of integrable partial differential equations attracted immediate mathematical interest in the 70's, first for the Korteweg-de Vries equation, the motion of the poles being associated with integrable many-body dynamics. Then quite a number of papers have been devoted to rational solutions of various integrable equations for one dependent variable, say Boussinesq equation [7], Hirota equation [8], Kadomtsev-Petviashvili equation [9] and NLS equation (see f.i. [10]- [13]). Recently further investigations of rational solutions were extended to integrable systems of two coupled differential equations. In this direction a number of such solutions have been found for the VNLS (1) [14,15] and for two coupled Hirota equations [16]. Similar extension has been reported also for three coupled NLS equations [17]. The starting motivation of such a surge of research work goes back to the observation by Peregrine [18] that the simplest rational solution of the focusing NLS equation may well model an ocean rogue wave (for a recent survey, see [19]). This solution describes a localized lump over a background with a peak amplitude which is three times higher than the surrounding background itself and with a finite life-time. On the physical side, these new nonlinear objects were soon recognized as ubiquitous rather than just ocean events and maritime disasters. Rogue waves have been observed not only in water tanks [20] but also in fiber optics [21] and in plasma [22]. They are predicted in the atmosphere [23], in superfluids [24], in Bose-Einstein condensates [25] and in capillary waves [26]. In this paper we systematically search for all bounded rational (mixed rational-exponential) solutions of both the VNLS equation (1) and, for the first time to our knowledge, of the 3WRI equation (2). We adopt a formalism such that these two equations are simultaneously treated by using an appropriate Lax pair. Our method of construction is based on the standard Darboux-Dressing transformation (DDT) as presented in [27,28], and briefly summarized in Section 2. Section 3 describes the algebraic algorithm we use to obtain rational solutions. In Section 4 we finally display examples of such solutions. The polynomials which appear in some of the expressions are given in Appendix A. Lax pair and Darboux-Dressing transformation Equations (1) and (2) are integrable models and as such admit a Lax representation (a Lax pair). For convenience, we introduce a Lax pair which combines both models. In subsequent sections we will describe these two dynamics separately. Let ψ x = Xψ , ψ t = T ψ ,(3) where ψ, X and T are 3 × 3 square matrices, ψ = ψ(x, t, k) being a common solution of the two linear ordinary differential matrix equations (3) while X = X(x, t, k) and T = T (x, t, k) depend on the variables x, t and the complex spectral parameter k according to the definitions X(x, t, k) = ikσ + Q(x, t) ,(4a)T (x, t, k) = α T nls (x, t, k) + β T 3w (x, t, k) (4b) where Q(x, t) contains the dynamical variables u (1) (x, t) and u (2) (x, t) and introduces two signs s 1 , s 2 , s 2 1 = s 2 2 = 1 Q =   0 s 1 u (1) * s 2 u (2) * u (1) 0 0 u (2) 0 0   (5) while σ is a constant diagonal matrix defined as σ =   1 0 0 0 −1 0 0 0 −1   .(6) The matrices T nls and T 3w are defined by T nls = 2ik 2 σ + 2kQ + iσ(Q 2 − Q x ) (7) T 3w = 2ikC − σW + σ[C, Q] ,(8) where W contains the field w(x, t) W =   0 0 0 0 0 −s 1 w * 0 s 2 w 0   .(9) C is a real diagonal matrix C =   0 0 0 0 c 1 0 0 0 c 2  (10) while α and β are real parameters such that, for α = 1, β = 0, (3) is the Lax pair corresponding to the VNLS (Manakov) equation (1), and for α = 0, β = 1 (3) is the Lax pair corresponding to the 3WRI equation (2). Indeed the compatibility conditions yield the evolution equations u (1) t = iα u (1) xx − 2 s 1 u (1) 2 + s 2 u (2) 2 u (1) + β −c 1 u (1) x − s 1 w * u (2) u (2) t = iα u (2) xx − 2 s 1 u (1) 2 + s 2 u (2) 2 u (2) + β −c 2 u(2) x + s 2 wu (1) 0 = β (w x + s 1 s 2 (c 1 − c 2 ) u (1) * u (2) ) .(11) In the search for novel rational solutions of (11) we use the Darboux-Dressing construction, as developed in [27] (where the interested reader may find additional references). For completeness, we briefly recall here the essential steps towards a new solution, starting from a known (seed) solution: given a solution u (1) 0 , u(2) 0 , w 0 of (11), let Ψ 0 be a corresponding fundamental matrix solution of (3). Then, if χ is strictly complex (χ = χ * ), Ψ(x, t, k) = 1 + χ − χ * k − χ P (x, t) Ψ 0 (x, t, k)(12) is a solution of (3) with u (1) (x, t) u (2) (x, t) = u (1) 0 (x, t) u (2) 0 (x, t) + 2i(χ − χ * )ζ * |ζ| 2 − s 1 |z 1 | 2 − s 2 |z 2 | 2 z 1 z 2 , (13a) w(x, t) = w 0 (x, t) − 2is 1 s 2 (c 1 − c 2 )(χ − χ * )z * 1 z 2 |ζ| 2 − s 1 |z 1 | 2 − s 2 |z 2 | 2 ,(13b) where the vector Z(x, t) =   ζ(x, t) z 1 (x, t) z 2 (x, t)   = Ψ 0 (x, t, χ * )Z 0(14) is a solution of (3) with k = χ * (Imχ = 0) and Z 0 is an arbitrary, constant and complex vector. Moreover in (12) the projector matrix P (x, t) is P (x, t) = ZZ † |ζ| 2 − s 1 |z 1 | 2 − s 2 |z 2 | 2   1 0 0 0 −s 1 0 0 0 −s 2   .(15) Here the condition that the parameter χ is not real is crucial. Indeed, the Darboux-Dressing transformation which adds one real pole to the solution Ψ 0 (x, t, k) in the k−plane at k = χ = χ * is given by a different formula, as detailed in [27]. However this real-pole transformation will not be used here as it yields rational (or semi-rational) solutions which are singular (i.e. unbounded). The seed solution u (1) 0 , u(2) 0 , w 0 of (11) is the plane wave   u (1) 0 (x, t) u (2) 0 (x, t)   =   a 1 e i(qx−ν 1 t) a 2 e −i(qx+ν 2 t)   ,(16a)w 0 (x, t) = is 1 s 2 (c 2 − c 1 ) a 1 a 2 2q e −i[2qx+(ν 2 −ν 1 )t] ,(16b) with ν 1 = α[q 2 + 2(s 1 a 2 1 + s 2 a 2 2 )] + β[c 1 q + s 2 a 2 2 2q (c 1 − c 2 )] , ν 2 = α[q 2 + 2(s 1 a 2 1 + s 2 a 2 2 )] + β[−c 2 q + s 1 a 2 1 2q (c 1 − c 2 )] .(17) Remark 1 With no loss of generality the amplitudes a 1 and a 2 can be taken to be real. Moreover, because of Galilei invariance, one may choose the wave numbers q and −q of these two plane-waves, see (16a), to have opposite sign. In order to construct the transformation (13) in the case where the seed solution u (1) 0 , u(2) 0 , w 0 of (11) is given by (16), we have to construct first the solution Ψ 0 of the Lax equations (3). To this aim we observe that, once (16) is fixed, the corresponding Q 0 and W 0 take the form Q 0 = G   0 s 1 a 1 s 2 a 2 a 1 0 0 a 2 0 0   G −1 ,(18)W 0 = G   0 0 0 0 0 is 2 a 1 a 2 2q (c 2 − c 1 ) 0 is 1 a 1 a 2 2q (c 2 − c 1 ) 0   G −1 ,(19) with G =   1 0 0 0 e i(qx−ν 1 t) 0 0 0 e −i(qx+ν 2 t)   .(20) It follows then that Ψ 0 (x, t, k) = G(x, t)Φ(x, t, k)(21) and the Lax pair reads Φ x = iΛ(k)Φ , Φ t = −iΩ(k)Φ ,(22) where Λ(k) =   k −is 1 a 1 −is 2 a 2 −ia 1 −k − q 0 −ia 2 0 −k + q  (23) and Ω(k) = α Ω nls (k) + β Ω 3w (k) ,(24a) with Ω nls (k) =   −2k 2 − s 1 a 2 1 − s 2 a 2 2 is 1 a 1 (2k − q) is 2 a 2 (2k + q) ia 1 (2k − q) 2k 2 − q 2 − s 1 a 2 1 − 2s 2 a 2 2 s 2 a 1 a 2 ia 2 (2k + q) s 1 a 1 a 2 2k 2 − q 2 − 2s 1 a 2 1 − s 2 a 2 2   ,(24b)Ω 3w (k) =    0 −is 1 c 1 a 1 −is 2 c 2 a 2 −ic 1 a 1 −c 1 (2k + q) − s 2 a 2 2 2q (c 1 − c 2 ) s 2 a 1 a 2 2q (c 1 − c 2 ) −ic 2 a 2 s 1 a 1 a 2 2q (c 1 − c 2 ) −c 2 (2k − q) − s 1 a 2 1 2q (c 1 − c 2 )    . (24c) Since [Λ(k) , Ω(k)] = 0 ,(25) the solution Ψ 0 has the expression Ψ 0 (x, t, k) = G(x, t)e i(Λ(k)x−Ω(k)t) .(26) Finally, the vector Z(x, t), see (14), which appears in the Darboux-Dressing transformation (13) reads Z(x, t) = G(x, t)e i(Λ(χ * )x−Ω(χ * )t) Z 0 .(27) Remark 2 The Darboux-Dressing transformation (13) may lead to a singular solution of (11) due to zeros of the denominator |ζ| 2 − s 1 |z 1 | 2 − s 2 |z 2 | 2 . The condition that the signs s 1 , s 2 are both negative (s 1 = s 2 − 1) is sufficient for this solution to be bounded (i.e. nonsingular). Nevertheless we will keep the signs s 1 , s 2 arbitrary. Remark 3 The parameter q, other than the signs s 1 , s 2 , is expected to be relevant to the stability of the plane wave solution (16). Despite the importance of this point we do not discuss it here. Rational solutions This section outlines the general scheme to construct all bounded solutions of (11) which are obtained via the Darboux-Dressing method and whose dependence on coordinates is either rational or a mixture of rational and exponential functions. Two subsections are then devoted to systematically compute the explicit expressions of all these solutions. The starting observation is that no rational dependence on x , t of the solution (13) exists if the two matrices Λ(k) and Ω(k) (for k = χ * ) are similar to a diagonal matrix. Indeed, this statement stems from the expressions (13), together with (16) and (27), which imply that, in this generic case, the explicit expression (13) of the solution contains only exponential functions of x and t. Therefore we find those particular, critical, values k c of k, such that the two matrices Λ(k c ) and Ω(k c ) are instead similar to a Jordan form. Indeed this form is generically the sum of a diagonal matrix and a nonvanishing nilpotent matrix, therefore the starting elementary observation is that, if N is a nilpotent matrix, say N m+1 = 0 and N m = 0 for an integer m, then exp(zN ) is a matrix valued polynomial of z of degree m. Moreover, in order to apply the Darboux-Dressing formula (13), the critical value k c is required to be strictly complex, namely to lie off the real axis of the complex k−plane. Therefore through our investigation we disregard all those values of k which are real even if the corresponding matrices Λ(k) and Ω(k) are similar to a Jordan form. Though the matrices Λ(k) and Ω(k) play a similar role, it is convenient to focus first on Λ(k) and its characteristic polynomial P Λ (λ) = det[λ − Λ(k)] = λ 3 + A 2 (k)λ 2 + A 1 (k)λ + A 0 (k)(28) whose coefficients take the expression (see (23)) A 2 (k) = k , A 1 (k) = −k 2 − q 2 + s 1 a 2 1 + s 2 a 2 2 , A 0 (k) = −k 3 + k(q 2 + s 1 a 2 1 + s 2 a 2 2 ) + q(s 2 a 2 2 − s 1 a 2 1 ) .(29) The following proposition holds true: Proposition 1 If λ 1 (k) , λ 2 (k) , λ 3 (k) are the three roots of the characteristic polynomial (28) then a necessary condition for Λ(k c ) to be similar to a Jordan form Λ J , Λ(k c ) = T Λ J T −1 ,(30) is that either one of them, say λ 3 , is simple and λ 1 = λ 2 is double, or λ 1 = λ 2 = λ 3 . T denotes the similarity transformation matrix. In the first case, Λ(k c ) is similar to a Jordan form Λ J if and only if λ 1 = λ 2 is geometrically simple, Λ J =   λ 1 µ 0 0 λ 1 0 0 0 λ 3   , µ = 0 ; (31) while in the second case, Λ(k c ) is similar to a Jordan form Λ J if λ 1 = λ 2 = λ 3 is geometrically simple, Λ J =   λ 1 µ 1 0 0 λ 1 µ 1 0 0 λ 1   , µ 1 = 0 .(32) Remark 4 The case in which λ 1 = λ 2 = λ 3 is geometrically double is the particular case of (31) for λ 1 = λ 3 . Remark 5 We point out for future reference that, in our notation (31 and 32), for dimensional reason we prefer to leave the entry µ in (31) and µ 1 in (32) as free nonvanishing parameters rather than giving them the unit value, µ = µ 1 = 1, as commonly in use. As for the second matrix Ω(k c ), since it commutes with Λ(k c ), see (25), it is consequently taken by the same similarity transformation Ω(k c ) = T Ω T −1 (33) into a matrix Ω which commutes with Λ J but it is not necessarily a Jordan form. Indeed, if ω 1 , ω 2 , ω 3 are the three eigenvalues of Ω(k c ), in the first case (i.e. λ 1 = λ 2 ) it necessarily follows that ω 1 = ω 2 , so that Ω =   ω 1 ρ 0 0 ω 1 0 0 0 ω 3   ,(34) which is still a Jordan form if ρ = 0, while in the second case (i.e. λ 1 = λ 2 = λ 3 ) Ω =   ω 1 ρ 1 ρ 2 0 ω 1 ρ 1 0 0 ω 1   .(35) On the other hand the values of ρ in (34) and of ρ 1 and ρ 2 in (35) have no a priori conditions. Once a critical value k c has been found, setting in (27) χ = k * c yields the expression Z(x, t) = G(x, t)V (x, t) , V (x, t) =   v(x, t) v 1 (x, t) v 2 (x, t)   = T e i(Λ J x− Ωt)   γ 1 γ 2 γ 3   ,(36) where γ 1 , γ 2 , γ 3 are arbitrary complex constants. Due to the nilpotent part of Λ J and Ω, this last expression yields a dependence of V (x, t) on x and t which is partially rational. Indeed, by inserting (31) and (34) into (36) yields the semi-rational dependence V (x, t) = T   (γ 1 + γ 2 ξ)e i(λ 1 x−ω 1 t) γ 2 e i(λ 1 x−ω 1 t) γ 3 e i(λ 3 x−ω 3 t)   , ξ = i(µx − ρt)(37) in the case λ 3 and ω 3 are (algebraically) simple. In the alternative case in which λ 1 and ω 1 are (algebraically) triple, the expression of V follows by using instead (32) and (35) and it reads V (x, t) = e i(λ 1 x−ω 1 t) T   γ 1 + γ 2 ξ 1 + γ 3 ζ γ 2 + γ 3 ξ 1 γ 3   , ξ 1 = i(µ 1 x − ρ 1 t) , ζ = 1 2 ξ 2 1 − iρ 2 t . (38) Using (36) the expression (13) of the solution u (1) , u (2) , w of (11) can be written in the more explicit form: u (1) (x, t) u (2) (x, t) = e i(qx−ν 1 t) 0 0 e −i(qx+ν 2 t) a 1 a 2 + 2i(k * c − k c )v * |v| 2 − s 1 |v 1 | 2 − s 2 |v 2 | 2 v 1 v 2 (39a) w(x, t) = is 1 s 2 (c 2 − c 1 )e −i[2qx+(ν 2 −ν 1 )t] a 1 a 2 2q + 2(k * c − k c )v * 1 v 2 |v| 2 − s 1 |v 1 | 2 − s 2 |v 2 | 2 . (39b) These last expressions (39) readily show that, if the three eigenvalues λ j are all the same, λ 1 = λ 2 = λ 3 , then the solution (39) is purely rational as its expression does not contain any exponentials (see (38)). In the alternative case, λ 1 = λ 2 = λ 3 , the expression (37) shows that the solution (39) is generically expressed in terms of both exponential and rational functions. Non generically, however, the dependence on coordinates is purely rational if γ 3 = 0 while it contains only exponentials if γ 2 = 0. We summarize the step-bystep construction of all such solutions of (11) as follows: once a critical value k c off the real axis is computed, one computes the corresponding eigenvalues λ j , ω j and the off-diagonal entries ρ or ρ 1 , ρ 2 ; the corresponding similarity matrix T is then computed and thus, using the formula (39), the final expression of the solution. The following two subsections describe the computation of the critical values k c and of the corresponding similarity transformation matrix T . The case λ 1 = λ 2 = λ 3 We start by requiring that the three roots of the characteristic polynomial (28) coincide with each other, namely P Λ (λ) = (λ − λ 1 (k)) 3 , so that λ 1 (k) = λ 2 (k) = λ 3 (k) = tr(Λ(k))/3 = −k/3 .(40)q = 0 , k c = i √ 27 2 ǫq , s 1 = s 2 = −1 , a 1 = a 2 = 2q , ǫ 2 = 1 .(41) In this case the critical value k c is imaginary and the free parameters are q (real) and the sign ǫ; hence the Darboux-Dressing transformation (13) applies and the resulting solution will be considered below. It now remains to provide the similarity transformation matrix T , as well as the two matrices Λ J and Ω, namely ω 1 and ρ 1 , ρ 2 (see (35)). T is however already given by (32) with λ 1 (k c ) = −k c /3 (the non vanishing parameter µ may be fixed according to convenience). Needless to say, the expression of the similarity matrix T is not unique and the one we give below may be changed, for instance, by a multiplication factor. In the present case in which λ 1 = λ 2 = λ 3 and Λ−λ 1 is nilpotent with (Λ−λ 1 ) 2 = 0, (Λ−λ 1 ) 3 = 0, the construction of the similarity transformation matrix T requires a tedious but straight computation and we limit ourselves to give the final formula: λ 1 = λ 2 = λ 3 = −i √ 3 2 ǫq so that Λ(k c ) = λ 1 + µ 1 N , N =   ǫ √ 3 1 1 −1 θ 0 −1 0 θ *   , µ 1 = 2iq , θ = 1 2 (−ǫ √ 3 + i) ,(42) where the dimensionless matrix N is nilpotent and θ is a phase factor, namely |θ| = 1. In this case the similarity transformation (30), with (32), is provided by the matrix T =   θ 0 −i 1 θ * iǫ √ 3 iθ * i 0  (43) whose Jordanization action is specified by the formula N = T N J T −1 , N J =   0 1 0 0 0 1 0 0 0   .(44) As for the matrix Ω, ω 1 = ω 2 = ω 3 = tr(Ω)/3 = 11 2 αq 2 + βq[c 1 − c 2 − iǫ √ 3(c 1 + c 2 )] and Ω(k c ) = ω 1 + 2αq 2   8 3ǫ √ 3 + i 3ǫ √ 3 − i −3ǫ √ 3 − i −4 −2 −3ǫ √ 3 + i −2 −4   + +βq   iǫ √ 3(c 1 + c 2 ) + c 2 − c 1 2ic 1 2ic 2 −2ic 1 iǫ √ 3(c 2 − 2c 1 ) − c 2 −2(c 1 − c 2 ) −2ic 2 −2(c 1 − c 2 ) iǫ √ 3(c 1 − 2c 2 ) + c 1   , while Ω has the expression (35), namely Ω = ω 1 + ρ 1 N J + ρ 2 N 2 J which implies Ω(k c ) = ω 1 + ρ 1 N + ρ 2 N 2 ,(45) where the matrix N has the expression (42). Comparing (45) with (42) yields ρ 1 = 4αq 2 ǫ √ 3 + 2βq(θc 1 − θ * c 2 ) , ρ 2 = 4αq 2 + 2βq(c 1 − c 2 ) .(46) We now apply the Darboux-Dressing construction formula (39) with the naked solution appropriate to this case (namely (16) with a 1 = a 2 = 2q), and the vector V (x, t) as given by (38). Thus we arrive at the following expression of the solution: u (1) (x, t) u (2) (x, t) = 2q e i(qx−ν 1 t) 0 0 e −i(qx+ν 2 t) 1 1 + 3ǫ √ 3A * |A| 2 + |A 1 | 2 + |A 2 | 2 θ * A 1 θA 2 , (47a) w(x, t) = 2iq(c 2 − c 1 )e −i[2qx+(ν 2 −ν 1 )t] [1 + 3ǫ √ 3θ * A * 1 A 2 |A| 2 + |A 1 | 2 + |A 2 | 2 ] (47b) with the notation ν = −15αq 2 − 3 2 βq(c 1 − c 2 ) , ν 1 = ν + 1 2 βq(c 1 + c 2 ) , ν 2 = ν − 1 2 βq(c 1 + c 2 ) , A = γ 1 + γ 2 ξ 1 + γ 3 (ζ − iθ * ) , A 1 = γ 1 + γ 2 (ξ 1 + θ * ) + γ 3 (ζ + θ * ξ 1 + iǫ √ 3) , A 2 = γ 1 + γ 2 (ξ 1 + θ) + γ 3 (ζ + θξ 1 ) ,(48) while ξ 1 and ζ are defined by (38) with µ 1 = 2iq (see (42)). We observe that not all the three complex parameters γ 1 , γ 2 , γ 3 , as introduced via (36), are essential as one of them can be arbitrarily fixed and two more real parameters can be absorbed as translations of x and t. The analysis of this solution is detailed in section 4. The case λ 1 = λ 2 = λ 3 Here we consider the case in which, for a critical value k = k c , one eigenvalue (say λ 1 ) of Λ(k) is algebraically double but geometrically simple, so that Λ(k c ) is similar to a Jordan form, see (30) and (31). Since finding k c generically requires computing the roots of a fourth order polynomial (see below), we postpone this computation and we construct first the similarity transformation matrix T with the assumption that k = k c is known. If λ 1 = λ 1 (k c ) and λ 3 = λ 3 (k c ) are the corresponding eigenvalues of Λ we obtain the following general expression of T T =      φ 1 φ 2 φ 3 − iφ 1 a 1 (λ 1 +k+q) − iφ 2 a 1 (λ 1 +k+q) + iµφ 1 a 1 (λ 1 +k+q) 2 − iφ 3 a 1 (λ 3 +k+q) − iφ 1 a 2 (λ 1 +k−q) − iφ 2 a 2 (λ 1 +k−q) + iµφ 1 a 2 (λ 1 +k−q) 2 − iφ 3 a 2 (λ 3 +k−q)      , k = k c ,(49) which turns out to depend on the three complex parameters φ 1 , φ 2 , φ 3 , arbitrary except for the condition that the matrix T be non singular. Since the determinant det T = 2φ 2 1 φ 3 qµa 1 a 2 (λ 1 − λ 3 ) 2 [(λ 3 + k) 2 − q 2 ][(λ 1 + k) 2 − q 2 ] 2(50) does not dependent on φ 2 , we may take φ 2 = 0 and conveniently set φ 1 = (λ 1 + k) 2 − q 2 and φ 3 = (λ 3 + k) 2 − q 2 . With this choice of the parameters the matrix T takes the expression T =   (λ 1 + k) 2 − q 2 0 (λ 3 + k) 2 − q 2 −ia 1 (λ 1 + k − q) iµa 1 (λ 1 + k − q)/(λ 1 + k + q) −ia 1 (λ 3 + k − q) −ia 2 (λ 1 + k + q) iµa 2 (λ 1 + k + q)/(λ 1 + k − q) −ia 2 (λ 3 + k + q)   , k = k c ,(51) where the condition of being invertible reads qµa 1 a 2 (λ 1 − λ 3 ) = 0. We note that the derivation of this expression requires not only that P Λ (λ 1 ) = P Λ (λ 3 ) = 0 but also that P ′ Λ (λ 1 ) = 0 where P ′ Λ (λ) = dP Λ (λ)/dλ. Since this matrix T becomes singular (i.e. non invertible) if q = 0, see (50), before proceeding further we prefer to first consider this separate case here below. The assumption q = 0 leads to consider two separate cases, namely either s 1 a 2 1 +s 2 a 2 2 = 0 or s 1 a 2 1 +s 2 a 2 2 = 0. We disregard this second case as our analysis shows that its corresponding solution becomes singular because of the vanishing of the denominator in the expression (39). Thus we treat here only the case in which q = 0 and s 1 a 2 1 + s 2 a 2 2 is strictly non vanishing. With these assumptions the explicit expression of the roots of P Λ (λ) are λ 1 = k 2 − s 1 a 2 1 − s 2 a 2 2 , λ 2 = − k 2 − s 1 a 2 1 − s 2 a 2 2 , λ 3 = −k , q = 0 .(52) The conditions that λ 1 = λ 2 and that the value of k c be not real leads to the condition s 1 a 2 1 +s 2 a 2 2 < 0. This therefore excludes the choice s 1 = s 2 = 1 and leads to the two values k = k c = ip , p = ± −s 1 a 2 1 − s 2 a 2 2 , λ 1 = λ 2 = 0 , λ 3 = −k c = −ip. We find however that the condition s 1 s 2 = 1 is necessary and sufficient for the solution (13a) to be non singular (in general singularities come from the zeros of the denominator which appears in this expression). We conclude therefore that only the (focusing) case s 1 = s 2 = −1 is worth considering. Thus in this particular (and interesting, see below) case the eigenvalues are λ 1 = λ 2 = 0 , λ 3 = −ip , p = ǫ a 2 1 + a 2 2 , ǫ 2 = 1 .(53) Thus the matrix Λ reads Λ(k c ) =   ip ia 1 ia 2 −ia 1 −ip 0 −ia 2 0 −ip  (54) and is taken into the Jordan form (here we set µ = −ip, see (31)) Λ J = −ip   0 1 0 0 0 0 0 0 1   ,(55) by the similarity transformation (30) with T =   −p p 0 a 1 0 a 2 a 2 0 −a 1   .(56) Moreover, since this case does not apply to the 3WRI equations (see (16b)), we set α = 1 and β = 0 so that the matrix Ω(k c ) has the expression Ω(k c ) =   3p 2 2pa 1 2pa 2 −2pa 1 −p 2 + a 2 2 −a 1 a 2 −2pa 2 −a 1 a 2 −p 2 + a 2 1   ,(57) which is similar to the Jordan form Ω (see (33) and (34)) with ω 1 = ω 2 = ω = p 2 , ω 3 = 0 , ρ = −2p 2 .(58) These findings, together with the explicit expression (37) and the Darboux-Dressing formula (13), yield the semi-rational solution of the VNLS equations u (1) (x, t) u (2) (x, t) = e 2iωt L B a 1 a 2 + M B a 2 −a 1 ,(59) where L = 3 2 − 8ω 2 t 2 − 2p 2 x 2 + 8iωt + |f | 2 e 2px , M = 4f (px − 2iωt − 1 2 )e (px+iωt) , B = 1 2 + 8ω 2 t 2 + 2p 2 x 2 + |f | 2 e 2px , and where f is a complex arbitrary constant. It should be remarked that the dressing construction has introduced γ 1 , γ 2 , γ 3 as arbitrary parameters, see (37). However only the complex parameter γ 3 is left essential since the other parameters can be absorbed by translations of the coordinates x, t. In fact, the expression (59) is derived by setting γ 1 = 1/2, γ 2 = 1 and γ 3 = −f . We note also that the dependence of L, M and B (see (59)) on x, t is both polynomial and exponential only through the dimensionless variables ax and ωt. Moreover the vector solution (59) turns out to be a combination of the two constant orthogonal vectors (a 1 , a 2 ) T and (a 2 , −a 1 ) T . Let us proceed further to the case in which q = 0, and let us maintain the assumption that k c is known. We first aim to computing the Jordan matrices Λ J (31) and Ω (34), which amounts to computing λ 1 , λ 3 , ω 1 , ω 3 and ρ. We start from the observation that the eigenvalue λ 1 is a zero of both the polynomial P Λ (λ) and of its derivative (see (28)) P ′ Λ (λ) = 3λ 2 + 2A 2 (k c )λ + A 1 (k c ) = 3(λ − λ + )(λ − λ − ) where λ ± = − 1 3 A 2 ± A 2 3 2 − A 1 3 .(60) Therefore this readily implies the following proposition: Proposition 2 Assume k = k c , then if P Λ (λ + ) = 0, the three roots of P Λ (λ) are λ 1 = λ 2 = λ + , λ 3 = 1 2 (3λ − − λ + ) ,(61) while if P Λ (λ − ) = 0, the three roots of P Λ (λ) are λ 1 = λ 2 = λ − , λ 3 = 1 2 (3λ + − λ − ) .(62) The proof of these formulae is elementary and consistent with the fact that the discriminant of a generic third degree polynomial, see (28), is proportional to the product [P Λ (λ + )][P Λ (λ − )]. The explicit expression of λ 1 and λ 3 finally obtains by inserting in (60) the coefficients A 2 , A 1 in terms of k via (29). As for the eigenvalues ω 1 , ω 3 and of the parameter ρ, see (34), we use the similarity property (33), the matrix transformation T being given by (51), and we obtain the expressions (with k = k c ) ω 1 = ω 2 = −α 2kλ 1 + s 1 a 2 1 + s 2 a 2 2 + q s 1 a 2 1 (λ 1 +k+q) − s 2 a 2 2 (λ 1 +k−q) + − β 2 (c 1 + c 2 )(k − λ 1 ) + (c 1 − c 2 ) s 1 a 2 1 (λ 1 +k+q) − s 2 a 2 2 (λ 1 +k−q) , ω 3 = −α 2kλ 3 + s 1 a 2 1 + s 2 a 2 2 + q s 1 a 2 1 (λ 3 +k+q) − s 2 a 2 2 (λ 3 +k−q) + − β 2 (c 1 + c 2 )(k − λ 3 ) + (c 1 − c 2 ) s 1 a 2 1 (λ 3 +k+q) − s 2 a 2 2 (λ 3 +k−q) , ρ = −αµ 2k − q s 1 a 2 1 (λ 1 +k+q) 2 − s 2 a 2 2 (λ 1 +k−q) 2 + + β 2 c 1 + c 2 + (c 1 − c 2 ) s 1 a 2 1 (λ 1 +k+q) 2 − s 2 a 2 2 (λ 1 +k−q) 2 .(63) The main task now is finding the critical values k c which are in the complex k−plane strictly off the real axis (Imk c = 0). These values are zeros of the discriminant of the polynomial P Λ (λ) (28). By taking into account the expression of the coefficients (29), this discriminant turns out to be proportional to the fourth order monodic polynomial ∆(k) = k 4 + D 3 k 3 + D 2 k 2 + D 1 k + D 0 ,(64) where the coefficients are D 3 = (s 2 a 2 2 − s 1 a 2 1 )/(2q) , D 2 = −[8q 4 − (s 1 a 2 1 + s 2 a 2 2 ) 2 + 20q 2 (s 1 a 2 1 + s 2 a 2 2 )]/(2 4 q 2 ) , D 1 = −9(s 2 a 2 2 − s 1 a 2 1 )(2q 2 + s 1 a 2 1 + s 2 a 2 2 )/(2 4 q) , D 0 = (q 2 − s 1 a 2 1 − s 2 a 2 2 ) 3 /(2 4 q 2 ) − ( 3 4 ) 3 (s 2 a 2 2 − s 1 a 2 1 ) 2 .(65) Though the generic fourth degree algebraic equation is solvable, the explicit expression of its solutions is so complicate that its use does not make their computation any easier than just computing them numerically. One exception to this wisdom is the case in which this algebraic equation reduces to a second degree equation. This is the case if we assume the condition s 1 a 2 1 = s 2 a 2 2 which implies that D 1 = D 3 = 0 with the consequence that the vanishing of the polynomial (64) reads as the second degree equation ∆(k) = R(h) = h 2 + D 2 h + D 0 = 0 (66) for the new variable h = k 2 . Here the coefficients are D 2 = −(2q 4 − a 4 1 + 10sq 2 a 2 1 )/(2 2 q 2 ) , D 0 = (q 2 − 2sa 2 1 ) 3 /(2 4 q 2 ) .(67) In this special case the reality of a 1 , a 2 implies the condition s 1 = s 2 = s and a 2 1 = a 2 2 , which has been used to pass from (65) to (67). The search for the critical values k c in the parameter space, the parameters being q , a 1 and the sign s, is now simple since the four zeros of the discriminant (64) have the explicit expression k = k(η 1 , η 2 ) = η 2 − 1 2 D 2 + η 1 1 4 D 2 2 − D 0 1/2 , η 2 1 = η 2 2 = 1 ,(68) which is the starting point of our short discussion of the corresponding family of solutions we present in the subsection 4.2.2. We note here that these expressions of k c are explicit because of the assumption s 1 a 2 1 = s 2 a 2 2 . In the generic case in which q = 0 and s 1 a 2 1 − s 2 a 2 2 = 0, we prefer to compute k c numerically as roots of the discriminant (64). Analysis of the solutions and conclusions In the previous section we have shown the way of deriving a rich family of solutions of the system (11). In fact we have constructed all the bounded (rational or semi-rational) solutions which can be obtained via the DDT method. The aim of this section is to select and detail some of such solutions. We separately treat those which are solutions of the VNLS system (1) (by setting α = 1, β = 0) and those which are solutions of the 3WRI equations (2) (by setting α = 0 , β = 1). As for the parameters which appear in the expressions of our solutions, some of them are structural coefficients which enter the partial differential equations (11), say the signs s 1 , s 2 and the characteristic velocities c 1 , c 2 ; other parameters, i.e. q, a 1 , a 2 , originate from the background (see (16)) while others, γ 1 , γ 2 , γ 3 , come from the DDT transformation. In this transformation there appears also the critical value k c of the spectral variable k, which depends only on s 1 , s 2 , q, a 1 , a 2 . Although some of the parameters are not essential as they could be eliminated by using simple symmetries, in some cases we prefer to keep them because of their physical significance. We point out also that the background parameter q plays a distinctive role in our solutions as it has no counterpart in the scalar NLS equation. 4.1 λ 1 = λ 2 = λ 3 In this case the solutions are rather peculiar as they are all purely rational. Only two critical values of k are possible, namely k c = ±iq √ 27/2 as specified by (41). These solutions exist only if s 1 = s 2 = −1, which is the focusing case of the VNLS equations, together with the condition a 1 = a 2 = 2q for the background amplitudes. The general expression of the corresponding solutions is (47). As for the three complex parameters γ 1 , γ 2 , γ 3 , we omit considering γ 2 = γ 3 = 0 since in this case the expression (47) is trivially that of a plane wave. Thus we find it convenient to illustrate the dependence of the solution on these parameters by considering separately the two cases: i) γ 3 = 0 and ii) γ 2 = 0. With no loss of generality because of translation invariance, one can set γ 2 = 1 , γ 1 = 0 in the first case and γ 3 = 1 , γ 2 = 0, while γ 1 remains arbitrary and complex, in the second case. Moreover the expression of the solution is the ratio of two polynomials of second degree in the first case i), and of two polynomials of fourth degree in the second case ii). Figures 1 to 4 illustrate these two cases separately for the VNLS and for the 3WRI equations. Solutions of the VNLS Let X = qx and T = q 2 t be rescaled variables; let u (j) (x, t) = qU (j) (X, T ), j = 1, 2. Case γ 3 = 0 , γ 2 = 1 , γ 1 = 0 U (1) = 2iθe i(X+15T ) 12X 2 + 144T 2 + (4ǫ √ 3 + 6i)X − 36iT − 1 + iǫ √ 3 12X 2 + 144T 2 + 4ǫ √ 3X + 2 .(69) Since this solution satisfies the relation u (2) (x, t) = u (1) * (x, −t) we report only the component u (1) (x, t) = qU (1) (qt, q 2 t); figure 1 displays the amplitudes |u (1) (x, t)| and |u (2) (x, t)| for a choice of parameters (see caption). Case γ 3 = 1 , γ 2 = 0 , γ 1 = 0 U (1) = 2iθe i(X+15T ) P (1) 4 P 4 , U (2) = −2iθ * e −i(X−15T ) P (2) 4 P 4(70) where the fourth degree polynomials P (1) 4 , P 4 , P 4 are given in Appendix A. Figure 2 displays the amplitudes |u (1) (x, t)| and |u (2) (x, t)| (see caption). 2 , λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, a 1 = a 2 = 2, q = 1, ǫ = 1; γ 1 = i, γ 2 = 0, γ 3 = 1. Solutions of the 3WRI Let X = qx and T = qt be rescaled variables; let u (j) (x, t) = qU (j) (X, T ), j = 1, 2, w(x, t) = qW (X, T ). Case γ 3 = 0 , γ 2 = 1 , γ 1 = 0 U (1) = 2iθe i[X+T (c 1 −2c 2 )] Q (1) 2 M 2 , W = 2θ(c 1 − c 2 )e −i[2X−T (c 1 +c 2 )] Q 2 M 2 ,(71) where the second degree polynomials Q 2 , Q 2 , M 2 are given in Appendix A. Since this solution satisfies the relation u (2) (x, t, c 1 , c 2 ) = u (1) * (x, t, c 2 , c 1 ) we report the expression of the components u (1) (x, t) = qU (1) (qx, qt), w(x, t) = qW (qx, qt) only. Figure 3 displays the amplitudes |u (1) (x, t)|, |u (2) (x, t)| and |w(x, t)| (see caption). Case γ 3 = 1 , γ 2 = 0 , γ 1 = 0 U (1) = 2iθe i[X+T (c 1 −2c 2 )] Q (1) 4 M 4 , U (2) = −2iθ * e −i[X+T (c 2 −2c 1 )] Q (2) 4 M 4 , W = 2θ(c 1 − c 2 )e −i[2X−T (c 1 +c 2 )] Q 4 M 4 ,(72) where the fourth degree polynomials Q 4 , Q(1) 4 , Q 4 , M 4 are given in Appendix A. Figure 4 displays the amplitudes |u (1) (x, t)|, |u (2) (x, t)| and |w(x, t)| (see caption). The expression (39), together with (37), shows that generically these solutions feature a dependence on coordinates which is both rational and exponential. In particular, however, if γ 3 = 0 the dependence is purely rational while if γ 2 = 0 the solution has only exponential functions. In the following we disregard this last case and consider only solutions with γ 2 = 0. Here we separately consider solutions corresponding to q = 0 and different background amplitudes, a 1 = a 2 , with q = 0 but a 1 = a 2 and, finally, with q = 0 and a 1 = a 2 . These distinctions are merely due to computational reasons. However, and interestingly enough, we numerically show below that in the last two cases (i.e. with q = 0) bounded rational solutions exist not only in the focusing case s 1 = s 2 = −1, as for the Peregrine soliton of the scalar NLS equation, but also in the defocusing case s 1 = s 2 = 1 and in the mixed case s 1 s 2 = −1. q=0 and vector Peregrine solutions In this case the solution, which is well described by its expression (59), applies only to the VNLS equation. In this respect we first notice that this expression (59), with f = 0 , and a 2 = 0, coincides with the Peregrine soliton of the scalar NLS equation. We further note that, since the two components u (1) (x, t, a 1 , a 2 ) , u (2) (x, t, a 1 , a 2 ) are related to each other by the relation u (2) (x, t, a 1 , a 2 ) = u (1) (x, t, a 2 , −a 1 ), we limit our attention only to u (1) (x, t). In the rescaled variables u (1) (x, t) = U (1) (X, T ), X = x a 2 1 + a 2 2 , T = t(a 2 1 +a 2 2 ), this solution (see (59)) may be written as U (1) = e 2iT a 1 (2+8iT )+(4X 2 +16T 2 −8iT −1) tanh(X−Z) 4X 2 +16T 2 +1 + a 2 √ 2f 4|f | 8X−16iT −1 √ 4X 2 +16T 2 +1 1 cosh(X−Z) (73) where the curve X = Z(T ) is the trajectory of the soliton as implicitly defined by the formula 2|f | 2 e 2Z = 4Z 2 + 16T 2 + 1 . As a consequence of these expressions, the large T asymptotic behavior along the curve X = Z(T ) is found to be U (1) (X, T ) → e 2iT a 1 tanh(X − Z) − ia 2 √ 2f |f | signT 1 cosh(X − Z) , T → ±∞ , (75a) Z(T ) → log |T | + 1 2 log 8 |f | 2 + O log |T | |T | , T → ±∞ .(75b) We observe that, as suggested by (73) and explicitly indicated by the asymptotic expression (75a), the amplitude a 1 multiplies a kink-type profile while the amplitude a 2 multiplies a bright-type pulse. Moreover the asymptotic motion (75b) is that of a particle which comes from x = +∞ and goes back to x = +∞ where it "stops" since its velocity asymptotically vanishes, namely dZ(T )/dT → 1/T + O(log |T |/T 2 ). Figure 5 shows an instance (see caption) of the amplitudes |u (1) (x, t)| and |u (2) (x, t)|. Further instances of this solution (73) are reported in [14,15]. 2. if q 2 < 2a 2 1 then the two zeros k(1, η 2 ) are real and the other two k(−1, η 2 ) are imaginary. Therefore in this subset of the parameter plane (a 1 , q) there are two critical values with opposite sign, i.e. k c = k(−1, η 2 ) or, explicitly, k c = k(−1, η 2 ) = iη 2 1 2 D 2 + 1 4 D 2 2 − D 0 1/2 , η 2 2 = 1 ,(76) where D 0 and D 2 are given by (67) with s = 1. Proposition 4 Assume s 1 = s 2 = −1: 1. if q 2 > 1 4 a 2 1 then the four zeros k(η 1 , η 2 ), see (68), are strictly complex (namely Im[k] = 0) and therefore there are four critical values k c = k(η 1 , η 2 ). 2. if q 2 ≤ 1 4 a 2 1 then the four zeros are imaginary and the critical values are again k c = k(η 1 , η 2 ). Once k c is computed, its corresponding solution of the equations (11) is obtained through the following chain of steps: i) use Proposition 2 to compute the eigenvalues λ 1 and λ 3 , ii) compute ω 1 , ω 3 , ρ according to (63), iii) insert the expression (51) of the similarity matrix T in (37) to compute the vector V , iv) finally apply the Darboux-Dressing formula (39). Instances of solutions of the VNLS equation are shown in Figure 6 (rational, defocusing), Figure 7 (rational, focusing), Figure 8 (rational-exponential, focusing). Instances of solutions of the 3WRI equation are shown in Figure 9 (rational, s 1 = s 2 = 1), Figure 10 (rational-exponential, s 1 = s 2 = −1). We explore this case by first computing k c numerically. Then the step-by-step method of construction of the solution, as indicated in the previous subsection, produces the plots of solutions of the VNLS as displayed in Figure 11 (rational, defocusing), Figure 12 (rational, focusing), Figure 13 (rational, s 1 = −1, s 2 = 1), Figure 14 (rational, s 1 = 1, s 2 = −1). An instance of solution of the 3WRI equation is shown in Figure 15 (rational, s 1 = s 2 = 1). Figure 11: VNLS: k c = −5.600 + 4.655i, λ 1 = λ 2 = λ 3 , s 1 = s 2 = 1, q = 1, a 1 = 2, a 2 = 5; γ 2 = 1, γ 1 = γ 3 = 0. Conclusions In this article we have devised a method of construction of solutions of two integrable systems of partial differential equations of interest in a variety of applications. These systems, the VNLS equations and the 3WRI equations, model the coupling of two waves and, respectively, of three waves. Our construction is specially tailored to yield solutions which feature a rational, or mixed rational-exponential, dependence on the independent variables. While rational solutions of integrable partial differential equations attracted mathematical interest since the 70's and consequently this type of solutions were derived for a number of integrable wave equations, it was only recently that further investigations of rational solutions were extended to integrable systems of two or three coupled differential equations. The main motivation of such a renewed interest goes back to the observation by Peregrine that the simplest rational solution of the focusing NLS equation may well model an ocean rogue wave. In a variety of physical contexts it was however soon recognized that, several waves, rather than a single one, should be considered in order to account for important resonant interaction processes. For integrable partial differential equations, according to personal taste, various, yet equivalent, approaches have been adopted: spectral transform and dressing techniques, Wronskian and Hirota methods, and Darboux transformations as considered here. These solutions are all soliton solutions since their corresponding spectral data on the continuos spectrum vanish. Moreover the strategy of computation may depend on whether the soliton is superimposed to the vacuum (i.e. the vanishing solution) or to a plane wave background. Here we deal with this second type of solitons. In most of the constructions discussed in the literature, the way to obtain polynomials out of (a linear combination of) exponentials goes through an appropriate limit process by making a number of eigenvalues of the Lax equations coalesce to get all the same value. Our approach is instead based on the exponentiation of non diagonalizable matrices. This construction naturally leads to consider those critical values k c of the spectral variable k such that the matrices which appear as exponent are similar to a Jordan form. There is therefore no need to take the limit in which different eigenvalues coalesce. We believe that our investigation is able to capture all possible solutions in this class. We are confident that the broad family of solutions presented here add a contribution to the understanding of rogue wave phenomena in novel physical situations where wave resonant interactions are relevant. A Polynomials 2 = 12X 2 + 12T 2 c 2 1 − c 1 c 2 + c 2 2 − 12XT (c 1 + c 2 ) + 2X 2 √ 3ǫ + 3i + − 2 T c 1 √ 3ǫ − 3i + c 2 √ 3ǫ + 6i + i √ 3ǫ − 1 (78a) Q 2 = 12X 2 + 12 T 2 c 2 1 − c 1 c 2 + c 2 2 − 12 XT (c 1 + c 2 ) + 4 X √ 3ǫ − 3i + − 2 T (c 1 + c 2 ) √ 3ǫ − 3i − 2i √ 3ǫ − 1 (78b) M 2 = 12X 2 + 12T 2 c 2 1 − c 1 c 2 + c 2 2 − 12XT (c 1 + c 2 ) + 4 √ 3ǫX − 2 √ 3ǫT (c 1 + c 2 ) + 2 (78c) Q(1) 4 = 12X 4 + 12T 4 c 2 1 − c 1 c 2 + c 2 2 2 + 36X 2 T 2 c 2 1 + c 2 2 − 24X 3 T (c1 + c2) − 24XT 3 c 3 1 + c 3 2 + + 4X 3 2 √ 3ǫ + 3i + − 4T 3 c 3 1 4 √ 3ǫ − 3i − 3c 2 1 c2 √ 3ǫ − 3i − 3c 1 c 2 2 √ 3ǫ + 3i + 2c 3 2 2 √ 3ǫ + 3i + − 12X 2 T √ 3ǫc 1 + c 2 √ 3ǫ + 3i + 12XT 2 3 √ 3ǫc 2 1 − 4 √ 3ǫc 1 c 2 + Figure 1 : 1VNLS: k c = i √ 27 2 , λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, a 1 = a 2 = 2, q = 1, ǫ = 1; γ 2 = 1, γ 1 = γ 3 = 0. Figure 2 : 2VNLS: k c = i √ 27 Figure 3 : 33WRI: k c = i √ 272 , λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, a 1 = a 2 = 2, q = 1, ǫ = 1; γ 2 = 1, γ 1 = γ 3 = 0. Figure 4 : 43WRI: k c = i √ 27 2 , λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, a 1 = a 2 = 2, q = 1, ǫ = 1; γ 1 = i, γ 2 = 0, γ 3 = 1. 4. 2 2The case λ 1 = λ 2 = λ 3 Figure 5 : 5VNLS: k c = i √ 5 2 , λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, q = 0, a 1 = 1, a 2 = 0.5, f = 0.1i. 4.2.2 q = 0 and a 1 = a 2 This family of solutions possesses two novel features with respect to those discussed in the previous subsections. First, the choice s 1 = s 2 = 1 is compatible with the boundedness of solutions (see below). Second, the conditions on the parameter set for the existence of a critical value k c lead to threshold phenomena for the dimensionless positive parameter m = a 2 1 /q 2 . As implied by the explicit expression (68) of the zeros of the discriminant (64), alias (66), we state the following Proposition 3 Assume s 1 = s 2 = 1:1. if q 2 ≥ 2a 2 1 then the four zeros k(η 1 , η 2 ), see (68), are real and no (complex) critical value k c exists. Figure 6 : 6VNLS: k c = i 2 −13 + 16 √ 2, λ 1 = λ 2 = λ 3 , s 1 = s 2 = 1, q = 1, a 1 = a 2 = 2; γ 2 = 1, γ 1 = γ 3 = 0. Figure 7 : 7VNLSλ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, q = 1, a 1 = a 2 = 1; γ 2 = 1, γ 1 = γ 3 = 0. Figure 8 : 8VNLSλ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, q = 1, a 1 = a 2 = 1; γ 1 = γ 2 = γ 3 = 1. Figure 9 : 93WRI: k c = i 2 −13 + 16 √ 2, λ 1 = λ 2 = λ 3 , s 1 = s 2 = 1 , q = 1, a 1 = a 2 = 2, c 1 = 1, c 2 = 2; γ 2 = 1, γ 1 = γ 3 = 0. Figure 10 : 103WRIλ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, q = 1, a 1 = a 2 = 1, c 1 = 1, c 2 = 2; γ 1 = γ 2 = γ 3 = 1.4.2.3 q= 0 and a 1 = a 2 Figure 12 : 12VNLS: k c = 4.876 + 5.343i, λ 1 = λ 2 = λ 3 , s 1 = s 2 = −1, q = 1, a 1 = 2, a 2 = 5; γ 2 = 1, γ 1 = γ 3 = 0. Figure 13 : 13VNLS: k c = −1.242 + 0.636i, λ 1 = λ 2 = λ 3 , s 1 = −1, s 2 = 1, q = 1, a 1 = 1, a 2 = 2; γ 2 = 1, γ 1 = γ 3 = 0. Figure 14 : 14VNLS: k c = 0.625 + 1.879i, λ 1 = λ 2 = λ 3 , s 1 = 1, s 2 = −1, q = 1, a 1 = 1, a 2 = 2; γ 2 = 1, γ 1 = γ 3 = 0. Figure 15 : 153WRI: k c = 1.319 + 0.256i, λ 1 = λ 2 = λ 3 , s 1 = s 2 = 1, q = 1, a 1 = 2, a 2 = 0.5, c 1 = 1, c 2 = 2; γ 2 = 1, γ 1 = γ 3 = 0. Moreover, by Cayley theorem, [Λ(k) + k/3] 3 = 0 (we omit to write the identity matrix I where no confusion can arise). Therefore the requirement that the matrix [Λ(k) + k/3] be nilpotent yields the critical values k c . We disregard the case [Λ(k) + k/3] 2 = 0 because it leads to the strong reduction a 1 a 2 = 0 and to real critical values of k. Moreover the condition [Λ(k) + k/3] 2 = 0 excludes the limiting case in which (31) holds for λ 1 = λ 3 (seeRemark 4). This way we compute all critical values k c . By disregarding those values which are real, we are left with one case only, namely = 12X 4 + 1728T 4 + 288X 2 T 2 + 4 3i + 2 √ 3ǫ X 3 − 864iT 3 − 72iX 2 T + 48 XT 2 3i − 2 = 12X 4 + 1728T 4 + 288X 2 T 2 + 4 −3i + 2 √ 3ǫ X 3 − 864iT 3 − 72iX 2 T − 48 XT 2 3i + 2 √ 3ǫ + + 3 X 2 4Re[γ 1 ] − 3i √ 3ǫ − 1 − 12 T 2 12Re[γ 1 ] − 5iP (1) 4 √ 3ǫ + + 3 X 2 4Re[γ 1 ] + 3i √ 3ǫ − 1 − 12 T 2 12Re[γ 1 ] + 5i √ 3ǫ + 9 + + 12 T X 4 √ 3ǫIm[γ 1 ] − i √ 3ǫ + 9 + 6 T 2i √ 3ǫIm[γ 1 ] + 6iRe[γ 1 ] − √ 3ǫ + 3i + + 2 X −3i √ 3ǫIm[γ 1 ] + 2 √ 3ǫRe[γ 1 ] + 3iRe[γ 1 ] − 2 √ 3ǫ − 3i + 3|γ 1 | 2 + + 1 2 1 + 5i √ 3ǫ Re[γ 1 ] + 9 2 √ 3ǫ − i Im[γ 1 ] + 5 2 1 − i √ 3ǫ (77a) P (2) 4 √ 3ǫ + 9 + + 12 T X 4 √ 3ǫIm[γ 1 ] − i √ 3ǫ + 3 + 6 T −2i √ 3ǫIm[γ 1 ] + 6iRe[γ 1 ] + √ 3ǫ − 3i + + 2 X −3i √ 3ǫIm[γ 1 ] + 2 √ 3ǫRe[γ 1 ] − 3iRe[γ 1 ] − 2 √ 3ǫ − 6i + 3|γ 1 | 2 + + 1 2 1 − 5i √ 3ǫ Re[γ 1 ] + 3 2 √ 3ǫ − 3i Im[γ 1 ] − 2 1 + i √ 3ǫ (77b) P 4 = 12X 4 + 1728T 4 + 288X 2 T 2 + 8 √ 3ǫX 3 − 96 √ 3ǫXT 2 + 6X 2 (1 + 2Re[γ 1 ])+ + 72T 2 (1 − 2Re[γ 1 ]) + 12XT (6 + 4 √ 3ǫIm[γ 1 ]) + 2X √ 3ǫ(1 + 2Re[γ 1 ])+ + 3|γ 1 | 2 − Re[γ 1 ] + 3 √ 3ǫIm[γ 1 ] + 4 (77c) Q (1) 2Im[γ 1 ] − i) − 11 + 2c 1 c 2 −4Re[γ 1 ] + i Im[γ 1 ] − 3i) − 1 + − 6XT 2c 1 √ 3ǫ(Im[γ 1 ] − i) + Re[γ 1 ] + 2 + c 2 √ 3ǫ(−2Im[γ 1 ] + 5i) + 2Re[γ 1 ] − 5 + (2Re[γ 1 ] + 1) − 6T 2 c 2 1 Re[γ 1 ] − √ 3ǫIm[γ 1 ] − 7 + 2c 1 c 2 (5 − 2Re[γ 1 ])+3c 2 2 √ 3ǫ + i + + 3X 2 4Re[γ 1 ] + 3i √ 3ǫ − 1 + − 3T 2 c 2 1 2Re[γ 1 ] − √ 3ǫ(√ 3ǫ + 7 + + 2c 2 2 Re[γ 1 ] + √ 3ǫ(+ X Re[γ 1 ] 4 √ 3ǫ + 6i − 2 √ 3ǫ(2 + 3iIm[γ 1 ]) − 6i + − 2T c 1 Re[γ 1 ] √ 3ǫ + 6i + 9Im[γ 1 ] + 5 √ 3ǫ − 6i + + c 2 Re[γ 1 ] √ 3ǫ − 3i − 3Im[γ 1 ] 3 + i √ 3ǫ − 7 √ 3ǫ + 3i + + 3|γ 1 | 2 + 9 2 √ 3ǫIm[γ 1 ] − 9 2 iIm[γ 1 ] + 5 2 i √ 3ǫRe[γ 1 ] + 1 2 Re[γ 1 ] − 5 2 i √ 3ǫ + 5 2 (79a) M 4 = 12X 4 + 12T 4 c 2 1 − c 1 c 2 + c 2 2 2 + 36T 2 X 2 c 2 1 + c 2 2 − 24X 3 T (c 1 + c 2 ) − 24XT 3 c 3 1 + c 3 2 + + 8 √ 3ǫX 3 − 4 √ 3ǫT 3 4c 3 1 − 3c 2 1 c 2 − 3c 1 c 2 2 + 4c 3 2 + − 12 √ 3ǫ(c 1 + c 2 )X 2 T + 12 √ 3ǫXT 2 3c 2 1 − 4c 1 c 2 + 3c 2 2 + + 6X 2 + c 2 2 Re[γ 1 ] + √ 3ǫIm[γ 1 ] − 4 + − 12XT c 1 Re[γ 1 ] + √ 3ǫIm[γ 1 ] + 2 + c 2 Re[γ 1 ] − √ 3ǫIm[γ 1 ] − 1 + 2 √ 3ǫX(2Re[γ 1 ] + 1)+ − 2T c 1 9Im[γ 1 ] + √ 3ǫ(Re[γ 1 ] + 5) + c 2 √ 3ǫ(Re[γ 1 ] − 4) − 9Im[γ 1 ] + + 3|γ 1 | 2 + 3 √ 3ǫIm[γ 1 ] − Re[γ 1 ] + 4 (79d) (2) 4 = 12X 4 + 12T 4 c 2 1 − c 1 c 2 + c 2 2 2 + 36T 2 X 2 c 2 1 + c 2 2 − 24T X 3 (c 1 + c 2 ) − 24T 3 X c 3 1 + c 3 2 + + 4X3 Multiscale expansion and integrability of dispersive wave equations. 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Guo B, Ling L, Liu Q P, Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions arXiv:1108.2867v2 (2011) Degenerate determinant representation of solutions of the nonlinear Schrödinger equation, higher order Peregrine breathers and multi-rogue waves. P Gaillard, J. Math. Phys. 5413504Gaillard P, Degenerate determinant representation of solutions of the nonlinear Schrödinger equation, higher order Peregrine breathers and multi-rogue waves, J. Math. Phys. 54, 013504 (2013) Breathers and Bright-Dark-Rogue Solutions for the Coupled Schrödinger Equations Chin. B Guo, L Ling, Rogue Wave, Phys. Lett. 28110202Guo B, Ling L, Rogue Wave, Breathers and Bright-Dark-Rogue Solutions for the Coupled Schrödinger Equations Chin. Phys. Lett. 28, 110202 (2011) Solutions of the Vector Nonlinear Schrödinger Equations: Evidence for Deterministic Rogue Waves. F Baronio, A Degasperis, M Conforti, S Wabnitz, Phys. Rev. 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{'abstract': 'Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector Nonlinear Schrödinger equations and the equations describing the resonant interaction of three waves. The Darboux-Dressing construction of soliton solutions is applied under the condition that the solutions have rational, or mixed rational-exponential, dependence on coordinates. Our algebraic construction relies on the use of nilpotent matrices and their Jordan form. We systematically search for all bounded rational (mixed rationalexponential) solutions and find, for the first time to our knowledge, a broad family of such solutions of the three wave resonant interaction equations.Dedicated to the memory of our colleague and friend Sergey Valentinovich Manakov.', 'arxivid': '1305.6636', 'author': ['Antonio Degasperis antonio.degasperis@roma1.infn.it \nIstituto Nazionale di Fisica Nucleare Dipartimento di Fisica, "Sapienza"\nDepartment of Mathematics and Information Sciences\nUniversità di Roma\nItaly\n', 'Sara Lombardo sara.lombardo@northumbria.ac.uk \nNorthumbria University\nNewcastle upon TyneUK\n'], 'authoraffiliation': ['Istituto Nazionale di Fisica Nucleare Dipartimento di Fisica, "Sapienza"\nDepartment of Mathematics and Information Sciences\nUniversità di Roma\nItaly', 'Northumbria University\nNewcastle upon TyneUK'], 'corpusid': 6284671, 'doi': '10.1103/physreve.88.052914', 'github_urls': [], 'n_tokens_mistral': 24429, 'n_tokens_neox': 20188, 'n_words': 12121, 'pdfsha': 'ba8ce79c80fc55a9617657a77ce49629fa755343', 'pdfurls': ['https://arxiv.org/pdf/1305.6636v1.pdf'], 'title': ['Rational solitons of wave resonant interaction models', 'Rational solitons of wave resonant interaction models'], 'venue': []}

arxiv

Neutrino-Neutrino Interactions and Flavor Mixing in Dense Matter (Dated: October 11, 2018) A B Balantekin Department of Physics University of Wisconsin Madison 53706WisconsinUSA Y Pehlivan Department of Physics University of Wisconsin Madison 53706WisconsinUSA Neutrino-Neutrino Interactions and Flavor Mixing in Dense Matter (Dated: October 11, 2018)arXiv:astro-ph/0607527v2 19 Oct 2006numbers: 1460Pq2665+t9660Jw9660Ly Keywords: Supernovaenonlinear effects in neutrino propagationneutrinos in matter An algebraic approach to the neutrino propagation in dense media is presented. The Hamiltonian describing a gas of neutrinos interacting with each other and with background fermions is written in terms of the appropriate SU(N) operators, where N is the number of neutrino flavors. The evolution of the resulting many-body problem is formulated as a coherent-state path integral. Some commonly used approximations are shown to represent the saddle-point solution of the path integral for the full many-body system. I. INTRODUCTION Neutrino propagation in dense matter [1,2] is encountered near the core of a core-collapse supernovae [3,4,5,6,7,8], in the Early Universe [9,10], and possibly in the gamma-ray bursts [11]. In particular, neutrino interactions play a crucial role in core-collapse supernovae. Neutrino oscillations in a core-collapse supernova differ from the matterenhanced neutrino oscillations in the Sun as in the former there are additional effects coming from both neutrinoneutrino scattering [3,12,13] and antineutrino flavor transformations [4]. Exact integration of the neutrino evolution equations with such terms in the supernova environment turns out to be a very difficult problem. A "mean-field" type approximation was proposed in Ref. [3] which was adopted in exploratory calculations of the conditions for r-process nucleosynthesis in supernovae [5,6]. Analytical solutions were investigated in the limiting cases where off-diagonal terms dominate [7]. In addition, various collective effects were explored [14,15,16,17,18,19,20]. The purpose of this article is to formulate the problem of neutrino propagation in dense media algebraically. An algebraic formulation of the problem should make hidden symmetries evident and also provide a framework to look for exact solutions or systematic approximations. In the next section we introduce the second-quantized formalism and the appropriate SU(2) algebra for two flavors. In this section we also show that the evolution operator for the standard MSW problem of neutrinos, mixing with each other and interacting with background electrons, can be written down exactly using an algebraic ansatz. In this and subsequent sections we utilize a path-integral approach to the underlying many-body problem, the details of which are sketched in the Appendix. In Section III, we introduce the neutrino-neutrino interaction and elucidate its algebraic nature. These arguments are expanded into the situations where antineutrinos are present in Section IV. The full problem with three flavors of both neutrinos and antineutrinos is discussed in Section V. Finally, a discussion of our results in Section VI concludes the paper. II. SECOND-QUANTIZED FORMALISM In our presentation we find it convenient to use the second-quantized form [21]. For simplicity in this section we assume that only two flavors of neutrinos mix, which we take to be the electron neutrino, ν e , and a combination of muon and tau neutrinos, which we denote by ν x . (In the limit θ 13 = 0, one particular combination of the mu and tau flavors decouple and the results in this section become exact with ν x being the combination orthogonal to the decoupled one [22]). We first consider a situation where there are no antineutrinos. (We relax both of these assumptions in the subsequent sections). The Hamiltonian describing mixing and interaction of the neutrinos with the background electrons is given by H ν = 1 2 d 3 p δm 2 2p cos 2θ − √ 2G F N e (a † x (p)a x (p) − a † e (p)a e (p)) + 1 2 d 3 p δm 2 2p sin 2θ(a † x (p)a e (p) + a † e (p)a x (p)), (1) where a † e (p) and a † x (p) are the creation operators for the left-handed ν e and ν x with momentum p, respectively, and a e (p) and a x (p) are the corresponding annihilation operators. In Eq. (1), √ 2G F N e (x) = V c is the Wolfenstein potential describing the interaction of neutrinos with electrons in neutral, unpolarized matter [23]; N e = n e − − n e + is the net electron density; θ is the vacuum mixing angle; and δm 2 = m 2 2 − m 2 1 . In writing Eq. (1) we omitted a term proportional to the identity (this includes the other Wolfenstein potential V n (x) = −(1/ √ 2)G F N n (x) describing the neutral-current interaction of neutrinos with neutrons). The presence of the Mikheev, Smirnov, Wolfenstein (MSW) resonance [23,24,25] is manifest in the first term. The inherent SU(2) symmetry of the problem can be implemented by the operators J + (p) = a † x (p)a e (p), J − (p) = a † e (p)a x (p), J 0 (p) = 1 2 a † x (p)a x (p) − a † e (p)a e (p) ,(2) which satisfy the commutation relations [J + (p), J − (q)] = 2δ 3 (p − q)J 0 (p), [J 0 (p), J ± (q)] = ±δ 3 (p − q)J ± (p).(3) These equations describe as many commuting SU(2) algebras as the number of distinct values of the neutrino momenta p permitted by the appropriate physical situation. Each J(p) is realized in the j = 1/2 representation due to the fermionic nature of neutrinos. The global SU(2) operators, J ± = d 3 pJ ± (p), J 0 = d 3 pJ 0 (p),(4) also satisfy the SU(2) commutation relations and play an important role in the problem. Representation of the algebra spanned by the operators J of Eq. (4) is obtained by adding N different copies of SU(2) each with j = 1/2 where N is the number of allowed values of neutrino momenta. In terms of the SU(2) generators of Eq.(2), we can write the neutrino Hamiltonian in Eq. (1) as follows: H ν = d 3 p δm 2 2p cos 2θJ 0 (p) + 1 2 sin 2θ (J + (p) + J − (p)) − √ 2G F d 3 pN e J 0 (p).(5) Note that in the last term of Eq. (5) we kept the electron density inside the momentum integral since it depends on the direction of the neutrino momentum (neutrino traveling in different directions will, in principle, see different electron densities), but of course not on the absolute value. In general we are interested in finding the operator U , describing the evolution of the system: i ∂U ∂t = H ν U.(6) In most cases, however, the physical interest is in finding the evolution of a particular state of the system. One may start, for example, with a state in which all permitted electron neutrino states are occupied and all ν x states are empty i.e., |φ = p∈P a † e (p)|0 ,(7) where |0 is the particle vacuum. P denotes the set of all allowed neutrino momenta. The state |φ is annihilated by the operators J − (p) for all p and also by J − . It is the lowest-weight eigenstate of the representation with j = N/2 of the global SU(2) operators J 0 and J ± . The evolution operator can be found by employing the unitary ansatz U = exp d 3 pτ (p, t)J + (p) exp d 3 p log(1 + |τ (p, t)| 2 )J 0 (p) exp − d 3 pτ * (p, t)J − (p) .(8) Here τ (p, t) is a function to be determined by substituting this ansatz into Eq. (6). One can differentiate U of Eq. (8) using the operator chain rule. Differentiation introduces the operator J 0 (p ′ ) between the first and second exponentials, and the operator J − (p ′ ) between the second and third exponentials. Since, for example, J 0 (p ′ ) does not commute with the first exponential, to write this operator before the first exponential, one needs to introduce the identity operator in the form exp d 3 pτ (p, t)J + (p) exp − d 3 pτ (p, t)J + (p) after the operator J 0 (p ′ ) and use the identity exp(O 1 )O 2 exp(−O 1 ) = O 2 + [O 1 , O 2 ] + 1 2! [O 1 , [O 1 , O 2 ]] + · · · , valid for any two arbitrary operators O 1 and O 1 . Moving all such terms to the left of the exponentials we find i ∂U ∂t = d 3 p iτ (p, t)τ * (p, t) − τ (p, t)τ * (p, t) 1 + |τ (p, t)| 2 J 0 (p) + d 3 p iτ (p, t) 1 + |τ (p, t)| 2 J + (p) − iτ * (p, t) 1 + |τ (p, t)| 2 J − (p) U,(9) where the dot denotes derivative with respect to time. Eq. (6) is satisfied if τ (p, t) obeys the equations: iτ (p, t)τ * (p, t) − τ (p, t)τ * (p, t) 1 + |τ (p, t)| 2 = δm 2 2p cos 2θ − √ 2G F N e ,(10) and iτ (p, t) 1 + |τ (p, t)| 2 = −iτ * (p, t) 1 + |τ (p, t)| 2 = 1 2 δm 2 2p sin 2θ,(11) i.e. equations for different neutrino momenta decouple as expected. In order to satisfy the condition U (t = 0) = 1, τ (p, t = 0) should be zero for all p. Using Eqs. (10) and (11), one can show that τ (p, t) is the solution of the following nonlinear (Riccati-type) first-order differential equation: iτ (p, t) = 1 2 δm 2 2p sin 2θ(1 − τ (p, t) 2 ) + δm 2 2p cos 2θ − √ 2G F N e τ (p, t).(12) τ (p, t) can be interpreted as the ratio of the one-body neutrino wave functions τ (p, t) = ψ x (p, t) ψ e (p, t) .(13) To see this, one only needs to substitute Eq. (13) into Eq. (12) together with the normalization condition |ψ e | 2 + |ψ x | 2 = 1.(14) This way, it becomes clear that Eq. (12) is equivalent to the Schrodinger equation i ∂ ∂t ψ e ψ x = 1 2 A − ∆ cos 2θ ∆ sin 2θ ∆ sin 2θ −A + ∆ cos 2θ ψ e ψ x(15) where we defined ∆ = δm 2 2p , A = √ 2G F N e .(16) For τ (t = 0) to be zero, we need the initial conditions ψ x (t = 0) = 0 and ψ e (t = 0) = 1. Eq. (15) is nothing but the standard MSW evolution equation for two flavors. Note that our ansatz for the evolution operator provides the "coset-space" formulation of this problem (cf. Eq. (12) with the formulation in the Appendix of Ref. [26]). The initial state |φ , given in Eq. (7), then evolves into the state exp − 1 2 P d 3 p log(1 + |τ (p, t)| 2 ) exp P d 3 pτ (p, t)J + (p) |φ ,(17) where τ (p, t) is obtained by solving Eq. (12). Both the initial state in Eq. (7) and evolved state in Eq. (17) are normalized to unity. The evolved state in Eq. (17) is the standard normalized SU(2) coherent state. III. NEUTRINO-NEUTRINO INTERACTIONS In some astrophysical environments, such as supernovae and the Early Universe, where neutrino density can become very large [27], the Hamiltonian in Eq. (1) is no longer sufficient. Neutrino self interactions must also be taken into account. In most cases one is mainly interested in the forward scattering of neutrinos from other neutrinos. These are described by the Hamiltonian H νν = G F √ 2V d 3 p d 3 q R pq a † e (p)a e (p)a † e (q)a e (q) + a † x (p)a x (p)a † x (q)a x (q) + a † x (p)a e (p)a † e (q)a x (q) + a † e (p)a x (p)a † x (q)a e (q)(18) where we defined R pq = (1 − cos ϑ pq ).(19) Here, ϑ pq is the angle between the momentum directions of the neutrinos with momenta p and q and we used box quantization conditions for a box with volume V . In forward scattering, neutrinos exchange their momenta by exchanging a Z boson with momentum p − q or a Z boson with zero momentum. The coefficient 1 − cos ϑ pq ensures that the neutrinos which move parallel to each other do not undergo forward scattering. Numerical studies indicate that the addition of a neutrino background results in very interesting physical effects such as coherent flavor transformation. For details see, for example, Refs. [3,4,5,6,7] and [16,17,18,19,20]. The major difficulty in studying the effects of a neutrino background on flavor evolution is the inherent nonlinearity of the problem. On the other hand, the contribution from H νν is relatively easy to study in the algebraic formalism. First, one needs to write H νν given in Eq. (18) in terms of the SU(2) generators of Eq. (2). Omitting a term proportional to the identity, we find that H νν = √ 2G F V d 3 p d 3 q R pq J(p) · J(q).(20) Here, J(p) · J(q) is defined as J(p) · J(q) = J 0 (p)J 0 (q) + 1 2 J + (p)J − (q) + 1 2 J − (p)J + (q).(21) This expression is very similar, but not identical, to the global SU(2) Casimir operator. In fact, H νν commutes with the global SU(2) operators of Eq. (4) [H νν , J i ] = 0(22) although the Hamiltonian H ν does not: [H ν , J i ] = 0.(23) On the other hand, the last term of H ν in Eq. (5) is proportional to J 0 for a constant electron density. Thus if δm 2 were zero the evolution problem for H ν + H νν could have been solved in the J -basis for a constant electron density. We see that, although H νν presents a difficulty in numerical studies, it is relatively easier to work with in this algebraic approach. The noncommutativity of the first integral in Eq. (5) with generators of the total angular momentum operators J , however, is the salient difficulty of the algebraic approach to the problem. One possible approach to the problem of finding the evolution operator with H ν + H νν is to seek a path integral representation. One can appropriately use the SU(2) coherent states |z(t) = N exp P d 3 p z(p, t)J + (p) |φ .(24) Here |φ is the state defined in Eq. (7) and the normalization constant N is given by N = exp − 1 2 P d 3 p log(1 + |z(p, t)| 2 ) .(25) Clearly when z(p, t) = τ (p, t) the state in Eq. (24) becomes the exact solution of the evolution with H ν alone. Path integral representation of the matrix element of the evolution operator calculated with H ν + H νν between two states |z(t i ) and |z ′ (t f ) is given by z ′ (t f )|U |z(t i ) = D[z, z * ] e iS[z,z * ](26) where the path integral measure is D[z, z * ] = lim N →∞ e −2 N α=1 P dp log(1+|z(p,tα)| 2 ) N α=1 p∈P 2! d 2 z(p, t α ) 2πi .(27) Here, the exponential factor arises because the SU (2) coherent states are overcomplete and therefore require a weight function in the resolution of identity. A detailed derivation can be found in the Appendix. In the above formula dz 2 refers to dzdz * . The action functional is given by 1 S[z, z * ] = t f ti dt i ∂ ∂t − H ν − H νν − i log z ′ (t f )|z(t f ) .(28) The leading contribution to this path integral comes from the stationary path |z(t) which minimizes the action functional. As in any variational problem, stationary path can be found by solving the Euler-Lagrange equations d dt ∂ ∂ż − ∂ ∂z L(z, z * ) = 0, d dt ∂ ∂ż * − ∂ ∂z * L(z, z * ) = 0,(29) where L[z, z * ] = i ∂ ∂t − H ν − H νν(30) which plays the role of the Lagrangian. To solve the Euler-Lagrange equations, however, we first need to calculate the Lagrangian as a function of z and z * . This can be done by substituting H ν from Eq. (5) and H νν from Eq. (20) into the Lagrangian of Eq. (30) and using the expectation values J + (p) = J − (p) * = z * (p, t) 1 + |z(p, t)| 2 , J 0 (p) = − 1 2 1 − |z(p, t)| 2 1 + |z(p, t)| 2 ,(31) which are valid for p ∈ P. When p ∈ P, however, these expectation values are equal to zero. When p = q coherent states also obey J a (p)J † a (q) = J a (p) J † a (q)(32) for a = 0, ±. Using above formulas and Eqs. (5), (30) and (20) together with i d dt = i dpż (p)z * (p) 1 + |z(p)| 2 ,(33) we obtain the following expression for the Lagrangian: L[z, z * ] = L ν − 1 4 √ 2G F P d 3 p B(p)(1 − |z(p)| 2 ) + 2B ex (p)z(p) + 2B xe (p)z * (p) 1 + |z(p)| 2 .(34) Here we defined L ν = i P d 3 pż (p, t)z * (p, t) 1 + |z(p, t)| 2 + 1 2 P d 3 p δm 2 2p cos 2θ − √ 2G F N e 1 − |z(p, t)| 2 1 + |z(p, t)| 2 − 1 2 P d 3 p δm 2 2p sin 2θ z(p, t) + z * (p, t) 1 + |z(p, t)| 2(35) and B(p) = √ 2G F V P d 3 qR pq 1 − |z(q, t)| 2 1 + |z(q, t)| 2 , B xe (p) = B * ex = 2 √ 2G F V P d 3 qR pq z(q, t) 1 + |z(q, t)| 2 .(36) Euler-Lagrange equations which follow from this Lagrangian are iż(p, t) = β(p, t) − α(p, t)z(p, t) − β * (p, t)z(p, t) 2(37) and its complex conjugate. The coefficients α and β are given by: α(p, t) = − δm 2 2p cos 2θ + √ 2G F N e + B(p), β(p, t) = 1 2 δm 2 2p sin 2θ + 1 2 B xe (p).(38) As in the previous section, z(p, t) can be interpreted as the ratio of the one-body neutrino wave functions ψ e , ψ x : z(p, t) = ψ x (p, t) ψ e (p, t)(39) with the normalization condition |ψ e | 2 + |ψ x | 2 = 1.(40) If we substitute Eq. (39) into Eq. (37), we see that Eq. (37) is equivalent to the Schrodinger equation i ∂ ∂t ψ e ψ x = 1 2 A + B − ∆ cos 2θ B ex + ∆ sin 2θ B xe + ∆ sin 2θ −A − B + ∆ cos 2θ ψ e ψ x(41) where ∆ and A are defined in Eq. (16). B and B ex are obtained by substituting Eq. (39) into Eq. (36): B = √ 2G F V P d 3 qR pq |ψ e (q, t)| 2 − |ψ x (q, t)| 2 ,(42)B ex = 2 √ 2G F V P d 3 qR pq ψ e (q, t)ψ * x (q, t).(43) Eq. (41) is commonly used to study neutrino propagation with neutrino-neutrino interactions (see Refs. [3] through [18]). Here we illustrated that it is not an exact result, but represents the saddle point solution to the many-body problem. There is an alternative method to obtain the approximate results outlined above. This method, typically employed in the random-phase approximation approach to many-body problems, approximates the product of two commuting arbitrary operatorsÔ 1 andÔ 2 aŝ O 1Ô2 ∼Ô 1 ξ|Ô 2 |ξ + ξ|Ô 1 |ξ Ô 2 − ξ|Ô 1 |ξ ξ|Ô 2 |ξ ,(44) provided that the condition ξ|Ô 1Ô2 |ξ = ξ|Ô 1 |ξ ξ|Ô 2 |ξ(45) is satisfied. In Eq. (44) the state |ξ must be appropriately chosen so that Eq. (45) is satisfied. Since the quantity R pq is zero when the operators do not commute (i.e., when p = q), one can apply this approximation technique to the quadratic element J(p) · J(q) which appear in the neutrino-neutrino forward scattering Hamiltonian H νν given in Eq. (20). It follows from Eq. (32) that the SU(2) coherent states satisfy the condition stated in Eq. (45). Using the symmetry R pq = R qp , one can then replace H νν by H νν ∼ 2 √ 2G F V d 3 p d 3 q R pq J 0 (p) J 0 (q) + 1 2 J + (p) J − (q) + 1 2 J − (p) J + (q) ,(46) where the expectation values are calculated using the SU(2) coherent states (the variable z(t) of these coherent states must be obtained in a self-consistent way). In writing Eq. (46) we omitted a term proportional to identity. The total Hamiltonian for the two neutrino flavors propagating in the presence of electrons and other neutrinos in the background becomes approximately H ν + H νν ∼ d 3 p δm 2 2p cos 2θ − √ 2G F N e − B(p) J 0 (p) (47) + d 3 p 1 2 sin 2θ + 1 2 B xe (p) J + (p) + d 3 p 1 2 sin 2θ + 1 2 B ex (p) J − (p). Here B(p), B ex (p) and B xe (p) are given in Eq. (36) and we used the expectation values given in Eq. (31). Since the above approximate Hamiltonian is linear in the SU(2) generators we can use the same ansatz for the evolution operator of the system as we did in section II, i.e. U = exp d 3 pz(p, t)J + (p) exp d 3 p log(1 + |z(p, t)| 2 )J 0 (p) exp − d 3 pz * (p, t)J − (p) .(48) It is straightforward to show that, substituting this ansatz into the equation i d dt U = (H ν + H νν ) U(49) yields nothing but Eq. (37) for z(p, t). It is also straightforward to generalize this linearization scheme to the situations where antineutrinos and more flavors are present. Corrections to the path integral will naturally arise as a result of the deviations from the classical (i.e., stationary) path given by Eq. (37) or Eq. (41). To calculate the effect of small deviations one can carry out a Taylor expansion of the action around the classical path. It is helpful to introduce the following nonlinear transformation in order to eliminate the exponential factor in the path integral measure in Eq. (27): β(p, t) = z(p, t) 1 + |z(p, t)| 2 β * (p, t) = z * (p, t) 1 + |z(p, t)| 2 .(50) The Jacobian resulting from this change of variables conveniently cancels out the exponential factor. It is straightforward to show that e −2 N α=1 P dp log(1+|z(p,tα)| 2 ) N α=1 p∈P 2! d 2 z(p, t α ) 2πi = N α=1 p∈P 2! d 2 β(p, t α ) 2πi(51) The path integral then takes the form z ′ (t f )|U |z(t i ) = lim N →∞ N α=1 p∈P dβ(p, t α )dβ * (p, t α ) iπ e iS[β,β * ](52) where the action functional, written in terms of the new variables is as follows: S[β, β * ] = t f ti dt dp β(p, t)β * (p, t) −β(p, t)β * (p, t) 2i + 1 2 δm 2 2p cos 2θ − √ 2G F N e 1 − 2|β(p, t)| 2 − 1 2 δm 2 2p sin 2θ β(p, t) + β * (p, t) 1 − |β(p, t)| 2(53)+ 1 2 √ 2G F V d 3 p d 3 q R pq |β(p, t)| 2 + |β(q, t)| 2 − 2|β(p, t)| 2 |β(q, t)| 2 + (1 − |β(p, t)| 2 ) (1 − |β(q, t)| 2 ) (β(p, t)β * (q, t) + β * (p, t)β(q, t)) One obtains the following equation of motion (and its complex conjugate) from the variation of this action: iβ(p, t) = δm 2 2p cos 2θ − √ 2G F N e − √ 2G F V d 3 q R pq 1 − 2|β(q, t)| 2 β(p, t)(54)+ δm 2 2p sin 2θ + √ 2G F V d 3 q R pq 1 − |β(q, t)| 2 β(q, t) 3|β(p, t)| 2 − 2 2 1 − |β(p, t)| 2 + δm 2 2p sin 2θ + √ 2G F V d 3 q R pq 1 − |β(q, t)| 2 β(q, t) β(p, t) 2 1 − |β(p, t)| 2 This classical path is the same as the one given in Eq. (37). This can be shown directly by substituting the transformation in Eq. (50) into Eq. (37). We will denote the classical path as β cl . Since the first order variations are zero on the classical path by definition, the Taylor expansion around β cl yields S[β, β * ] = S[β cl , β * cl ] + 1 2 (β − β cl ) T δ 2 S δβ δβ cl (β − β cl ) (55) + (β − β cl ) T δ 2 S δβ δβ * cl (β * − β * cl ) + 1 2 (β * − β * cl ) T δ 2 S δβ * δβ * cl (β * − β * cl ) + . . . Here (. . .) cl indicates that the derivatives are to be calculated on the classical path β cl and (β − β cl ) T δ 2 S δβ δβ cl (β − β cl ) is a short hand notation for the matrix product p,k q,m (β(p, t k ) − β cl (p, t k )) T δ 2 S δβ(p, t k ) δβ(p, t m ) cl (β(p, t m ) − β cl (p, t m )) ,(56) and similarly for the other terms. The sums over p and q run through the allowed momentum modes and the sums over k and m run from 1 to N which is the number of time intervals we introduced in path integral. N → ∞ limit should be taken as explained in the Appendix. For small deviations from the classical path, one can ignore the higher order terms in the expansion (55) and substitute it in Eq. (52): z ′ (t f )|U |z(t i ) = e iS[β cl ,β * cl ] lim N →∞ N k=1 p∈P dβ(p, t k )dβ * (p, t k ) iπ e i 1 2β T δ 2 S δβ δβ clβ +β T δ 2 S δβ δβ * clβ * + 1 2β * T δ 2 S δβ * δβ * clβ * (57) where we definedβ = β − β cl . The classical action S[β cl , β * cl ] is taken out of the integration since it does not depend onβ. The lowest order quantum corrections are captured by the Gaussian integral in Eq. (57). The result of the integration is z ′ (t f )|U |z(t i ) = lim N →∞ (iπ) N +P e iS[β cl ,β * cl ] Det (KM − L T K −1 L) .(58) Here P denote the number of allowed momentum modes. The matrices K, M and L are given as follows: K(p, k, q, m) = 1 2 δ 2 S δx(p, t k ) δx(q, t m ) cl M (p, k, q, m) = 1 2 δ 2 S δy(p, t k ) δy(q, t m ) cl (59) L(p, k, q, m) = 1 2 δ 2 S δx(p, t k ) δy(q, t m ) cl(60) where x = (β +β * )/2 and y = (β −β * )/2i. The fundamental difficulty involved in the calculation of the determinant in Eq. (58) is the existence of non-diagonal terms in the matrices K, M and L. These terms are generated by the d 3 p d 3 q R pq . . . integral in the action given in Eq. (53). A complete analysis of these determinants is beyond the scope of the present paper and will be given elsewhere. IV. NEUTRINO-ANTINEUTRINO INTERACTIONS Environments such as core-collapse supernovae and the Early Universe contain copious amounts of antineutrinos as well as neutrinos. When antineutrinos are added to the picture the effective flavor evolution Hamiltonian becomes H = H ν + Hν + H νν + Hνν + H νν .(61) Here, H ν and H νν are given in Eqs. (1) and (18). Hν and Hνν are the same as H ν and H νν , respectively except that the neutrino operators a and a † are replaced by the antineutrino operators b and b † and the sign of the N e term is reversed. The neutrino-antineutrino forward scattering Hamiltonian H νν is given as follows: H νν = − √ 2G F V d 3 pd 3 qR pq a † e (p)a e (p)b † e (q)b e (q) + a † x (p)a x (p)b † x (q)b x (q)(62)+ a † x (p)a e (p)b † x (q)b e (q) + a † e (p)a x (p)b † e (q)b x (q) . In addition to the neutrino operators defined in Eq. (2), we now define the antineutrino SU(2) operators 2 J + (p) = b † e (p)b x (p),J − (p) = b † x (p)b e (p),J 0 (p) = 1 2 b † e (p)b e (p) − b † x (p)b x (p) .(63) These operators also obey SU(2) commutation relations [J + (p),J − (q)] = 2δ(p − q)J 0 (p), [J 0 (p),J ± (q)] = ±δ(p − q)J ± (p)(64) and they commute with the neutrino operators [J i (p),J j (q)] = 0.(65) Written in terms of J i (p) andJ i (p), the Hamiltonian in Eq. (61) takes the following form: H ν + Hν + H νν + Hνν + H νν (66) = d 3 p δm 2 2p cos 2θ − √ 2G F N e J 0 (p) + 1 2 d 3 p δm 2 2p sin 2θ (J + (p) + J − (p)) + d 3 p − δm 2 2p cos 2θ − √ 2G F N e J 0 (p) + 1 2 d 3 p δm 2 2p sin 2θ J + (p) +J − (p) + √ 2G F V d 3 pd 3 qR pq J(p) · J(q) +J(p) ·J(q) − √ 2G F V d 3 pd 3 qR pq −2J 0 (p)J 0 (q) + J + (p)J − (q) + J − (p)J + (q) where a term proportional to identity is omitted. As we did in the previous section, we introduce the coherent states 3 |z(t),z(t) = NN e P d 3 pz(p,t)J+(p) e P d 3pz (p,t)J−(p) |φ .(67) Here |φ is analogous to the state defined in Eq. (7), i.e., all permitted ν e andν e states are occupied and all other neutrino flavor states are empty: |φ = p∈P a † e (p) p∈P b † e (p)|0 .(68) In the above formulas, P andP denote the set of all allowed momentum modes for neutrinos and antineutrinos, respectively. In what follows, we will drop the symbols P andP from the formulas by adopting the convention that the non-overlined quantities such as p, q will take values in P whereas the overlined quantities such asp,q take values inP. The constants N = exp − 1 2 d 3 p log(1 + |z(p, t)| 2 ) ,N = exp − 1 2 d 3p log(1 + |z(p, t)| 2 )(69) in Eq. (67) normalize the coherent states: z,z|z,z = 1.(70) A path integral representation of the evolution operator can be given in terms of these coherent states as z ′ (t f ),z ′ (t f )| U |z(t i ),z(t i ) = D[z, z * ,z,z * ]e iS[z,z * ,z,z * ](71) where the action functional S[z, z * ,z,z * ] is 4 S[z, z * ,z,z * ] = t f ti dt i ∂ ∂t − (H ν + Hν + H νν + Hνν + H νν ) − i ln z ′ (t f ),z ′ (t f )|z(t f ),z(t f ) .(72) Once more we can find the stationary path by solving the Euler-Lagrange equations for L[z, z * ,z,z * ] = i ∂ ∂t − (H ν + Hν + H νν + Hνν + H νν ) .(73) The linear terms in the Lagrangian can be calculated using J + (p) = J − (p) * = z * (p, t) 1 + |z(p, t)| 2 , J 0 (p) = − 1 2 1 − |z(p, t)| 2 1 + |z(p, t)| 2 ,(74)J + (p) = J − (p) * =z (p, t) 1 + |z(p, t)| 2 , J 0 (p) = 1 2 1 − |z(p, t)| 2 1 + |z(p, t)| 2 ,(75) which are valid for p ∈ P andp ∈P. When p ∈ P orp ∈P the expectation values are zero. To calculate the quadratic terms we use the identities J a (p)J † a (q) = J a (p) J † a (q) , J a (p)J † a (q) = J a (p) J † a (q) , J a (p)J † a (q) = J a (p) J † a (q) .(76) Here a = 0, ± and we assumed p = q andp =q. Also note that i d dt = i d 3 pż (p)z * (p) 1 + |z(p)| 2 + d 3pż (p)z * (p) 1 + |z(p)| 2 .(77) Using Eqs. (74)-(77) Lagrangian can be found as follows: L = L ν + Lν − 1 4 d 3 p B(p)(1 − |z(p)| 2 ) + 2B ex (p)z(p) + 2B xe (p)z * (p) 1 + |z(p)| 2(78)+ 1 4 d 3p B(p)(1 − |z(p)| 2 ) + 2B xe (p)z(p) + 2B ex (p)z * (p) 1 + |z(p)| 2 . Here L ν is the Lagrangian given in Eq. (35) and Lν is the same as L ν except that we substitutez(p) in place of z(p) and change the sign of N e . In the above equation we also defined B(p) = √ 2G F V d 3 qR pq 1 − |z(q, t)| 2 1 + |z(q, t)| 2 − √ 2G F V d 3q R pq 1 − |z(q, t)| 2 1 + |z(q, t)| 2(79) and B xe (p) = B * ex (p) = 2 √ 2G F V d 3 qR pq z(q, t) 1 + |z(q, t)| 2 − 2 √ 2G F V d 3q R pqz * (q, t) 1 + |z(q, t)| 2 .(80) Equations of motion which result from this Lagrangian are as follows: iż(p, t) = β(p, t) − α(p, t)z(p, t) − β * (p, t)z 2 (p, t),(81) iż(p, t) = −β * (p, t) +ᾱ(p, t)z(p, t) +β(p, t)z 2 (p, t) and the complex conjugates of these equations. The coefficients α and β are given by: α(p, t) = − δm 2 2p cos 2θ + √ 2G F N e + B(p) β(p, t) = 1 2 δm 2 2p sin 2θ + B ex (p).(83) The coefficientsᾱ andβ are the same except that the sign of the δm 2 is changed. We can again write the parameters z andz in terms of the one-body neutrino and antineutrino wave functions ψ e , ψ x ,ψ e andψ x as follows: z(p, t) = ψ x (p, t) ψ e (p, t) andz(p, t) =ψ * x (p, t) ψ * e (p, t) .(84) We also have the normalization conditions |ψ e | 2 + |ψ x | 2 = 1 and |ψ e | 2 + |ψ x | 2 = 1. Substituting Eq. (84) into Eqs. (81) and (82) we see that they are equivalent to the Schrodinger equations i ∂ ∂t ψ e (p, t) ψ x (p, t) = 1 2 A + B − ∆ cos 2θ B ex + ∆ sin 2θ B xe + ∆ sin 2θ −A − B + ∆ cos 2θ ψ e (p, t) ψ x (p, t) ,(86) and i ∂ ∂t ψ e (p, t) ψ x (p, t) = 1 2 A + B + ∆ cos 2θ B ex − ∆ sin 2θ B xe − ∆ sin 2θ −A − B − ∆ cos 2θ ψ e (p, t) ψ x (p, t) .(87) In the above formula B and B ex are obtained by substituting Eq. (85) into Eqs. (79) and (80), i.e., B = √ 2G F V d 3 qR pq |ψ e (q, t)| 2 − |ψ x (q, t)| 2 − √ 2G F V d 3q R pq |ψ e (q, t)| 2 − |ψ x (q, t)| 2 ,(88)B ex = B * xe = 2 √ 2G F V d 3 qR pq ψ e (q, t)ψ * x (q, t) − 2 √ 2G F V d 3q R pqψe (q, t)ψ * x (q, t).(89) V. THREE NEUTRINO FLAVORS In this section we generalize our formalism to three neutrino flavors, i.e. when θ 13 = 0. SU(3) symmetry of the neutrinos can be represented by the operators T ij (p) = a † i (p)a j (p) andT ij (p) = b † j (p)b i (p)(90) where i and j run over the flavor indices e, µ and τ . These operators generate orthogonal SU(3) algebras: [T ij (p), T kl (q)] = δ(p − q) (δ kj T il (p) − δ il T kj (p)) ,(91)[T ij (p),T kl (q)] = −δ(p − q) δ kjTil (p) − δ ilTkj (p) , [T ij (p),T kl (q)] = 0. The effective Hamiltonian describing the propagation of neutrinos in a background of electrons is given by [29] H = i,j d 3 p (γ ij (p) + ω ij (p)) T ij (p) + (γ ij (p) − ω ij (p))T ij (p)(92)+ G F √ 2V d 3 pd 3 qR pq i,j (T ij (p) −T ij (p))(T ji (q) −T ji (q)). In this Hamiltonian, the coefficients γ ij (p) are the elements of the symmetric, traceless matrix Γ which is given below: Γ = (γ ij ) = 1 3 Q 23 Q 13 Q 12     −    Q † 12 Q † 13 Q † 23 .(93) Here Q 12 , Q 13 and Q 23 are neutrino mixing matrices in vacuum: Q 23 Q 13 Q 12 = 1 0 0 0 C 23 S 23 0 −S 23 C 23 C 13 0 S * 13 0 1 0 −S 13 0 C 13 C 12 S 12 0 −S 12 C 12 0 0 0 1 ,(94) where C 13 , etc. is the short-hand notation for cos θ 13 , etc. Since S 13 may be multiplied by a phase, we explicitly indicated its complex conjugate. Note that individual matrices, not their matrix elements, are called Q 23 , Q 13 , and Q 12 , respectively in Eq. (94). The terms in the Hamiltonian which are proportional to γ ij represent vacuum oscillations of the neutrinos. The coefficients ω ij are real and symmetric. They are the elements of the diagonal, traceless matrix Ω given below: Ω = (ω ij ) = 1 3   2V c 0 0 0 −V c 0 0 0 −V c   .(95) Here V c is the Wolfenstein potential described earlier. A path integral formula for the evolution operator of the system can be constructed using SU(3) coherent states which are given by |z,z = NN e P dp(zµ(p)Tµe(p)+zτ (p)Tτe(p)) e P dp(zµ(p)Teµ(p)+zτ (p)Teτ (p)) |φ . These coherent states are defined with respect to the reference state |φ which is defined as in Eq. (68). The normalization constants N andN are given by N = exp − 1 2 d 3 p log(1 + |z µ (p, t)| 2 + |z τ (p, t)| 2 ) ,N = exp − 1 2 d 3p log(1 + |z µ (p, t)| 2 + |z τ (p, t)| 2 ) . (97) Neutrino SU(3) coherent states are characterized by two complex numbers that we denoted by z µ and z τ in Eq. (96). We usedz µ andz τ for the SU(3) symmetry of the antineutrinos. As a practical convention, we also define z e (p) =z e (p) = 1 in what follows 5 . Evolution operator can be given by the following path integral formula in terms of the SU(3) coherent states: z ′ (t f ),z ′ (t f )|U |z(t i ),z(t i ) = D[z,z]e iS[z,z] ,(98) where the measure is given by Eq. (127) of the Appendix. In this formula S[z,z] = t f ti dt z(t),z(t)|i d dt − H(t)|z(t),z(t) − i ln z ′ (t f ),z ′ (t f )|z(t f ),z(t f )(99) plays the role of a classical action. The derivation of these formulas and the exact expression for the integral measure can be found in the Appendix. As before, we write down the Lagrangian 6 L[z,z] = i d dt − H(100)i d dt  ψ e (p) ψ µ (p) ψ τ (p)   = (−Γ(p) + Ω(p) + Y (p))  ψ e (p) ψ µ (p) ψ τ (p)   .(114) Here, the matrix Y is the matrix formed by the elements Y ij given in Eq.(107). Note that as one goes from the neutrino to antineutrino equations only the signs of the δm 2 terms change. If we substitute Eq. (110) and (111) in Eq. (107) we see that Y ij can be written in terms of the one-body wavefunctions as Y ij (p) = √ 2G F V d 3 qR pq ψ i (q)ψ * j (q) − d 3q R pqψi (q)ψ * j (q) .(115) VI. CONCLUSIONS In this article an algebraic approach to the neutrino propagation in dense media is presented. The Hamiltonian describing a gas of neutrinos interacting with each other and background fermions is written in terms of the appropriate SU(2) (for two flavors) or SU(3) (for three flavors) operators. Neutrinos as well as antineutrinos are considered. The evolution of the resulting many-body problem is formulated as either an SU (2) or an SU(3) coherent-state path integral. The evolution operator for the entire system is calculated using two different approximations, namely the saddle-point approximation and the operator product linearization approximation. In our case these two approximations yield the same answer. This approximate solution of the neutrino evolution is the only one used so far in applications to the core-collapse supernovae and the Early Universe. It is important to stress that this solution is only an approximation to the many-body problem described by Eq. (61) and the corrections to it may play a significant role. We should also emphasize that the results for the evolution operator obtained here are applicable to any initial state. An initial state such as the one depicted in Eq. (7) represents an electron neutrino gas described by a pure state, i.e. ρ 2 = ρ. Such an initial state may be a good description of neutrinos produced in earlier stages of stellar evolution, but is inadequate for neutrinos in a thermal distribution. In such cases it would be more appropriate to describe the neutrino gas using density operators satisfying the equation iρ = [H, ρ],(116) where H is the Hamiltonian of Eq. (61). The initial density matrix ρ i , then evolves as ρ = U ρ i U † . Sometimes a polarization vector is introduced. In our notation the one-body polarization vector is defined as It is straightforward to show that linearizing the Hamiltonian of Eq. (61) using Eq. (44) and substituting this linearized Hamiltonian in Eq. (118) yields the evolution equations of the polarization vectors as stated, e.g., in Refs. [5] and [6]. One can use this resolution of identity to write down the path integral formula for the matrix element of the evolution operator U . The procedure is well established [30] and will only be outlined in what follows: We start by dividing the time interval [0, T ] into N infinitesimally small pieces: t 0 = 0, t 1 = ε, t 2 = 2ε, . . . , T = t N = N ε. (120) Ignoring the time dependence of the Hamiltonian in each infinitesimal time interval we can write z ′ (T ),z ′ (T )|U (T )|z(0),z(0) = z ′ (T ),z ′ (T )|e −iεH(tN ) e −iεH(tN−1) . . . e −iεH(t1) |z(0),z(0) . We then insert a resolution of identity to the left of each exponential. We will put an additional label to t α to the variables z i (p) andz i (p) in the resolution of identity which is inserted just next to e −iεH(tα) . This way, we obtain z ′ (T ),z ′ (T )|U (T )|z(0),z(0) = N α=1 p 3! (2πi) 2 d 2 z µ (p, t α )d 2 z τ (p, t α ) ( k |z k (p, t α )| 2 ) 3 N α=1 p 3! (2πi) 2 d 2z µ (p, t α )d 2z τ (p, t α ) ( k |z k (p, t α )| 2 ) 3 × z ′ (T ),z ′ (T )|z(t N ),z(t N ) N α=1 z(t α ),z(t α )|e −iεH(tα ) |z(t α−1 ),z(t α−1 ) . Assuming that only the continuous paths will contribute to the integral when we take N → ∞ limit, we can write |z(t α−1 ),z(t α−1 ) = 1 − ε d dt |z(t α ),z(t α ) . In this case the infinitesimal propagator z(t α ),z(t α )|e −iεH(tα ) |z(t α−1 ),z(t α−1 ) can be written as z(t α ),z(t α )|(1 − iεH(t α ))(1 − ε d dt )|z(t α ) ,z(t α ) = e iε z(tα),z(tα)|(i d dt −H(tα))|z(tα),z(tα) . Substituting this into Eq.(122) and taking the limits N → ∞ and ε → 0 such that N ε = T we arrive at the following path integral formula for the propagator: 3! (2πi) 2 d 2 z µ (p, t α )d 2 z τ (p, t α ) N α=1 p 3! (2πi) 2 d 2z µ (p, t α )d 2z τ (p, t α ).(127) flavors. For three neutrino flavors one simply calculates the trace above with the SU(3) generators, resulting in an eight-dimensional vector. The polarization vector of Eq. (117) satisfies the equation iṖ(q) = Tr([J(q), H]ρ). z ′ (T ),z ′ (T )|U (T )|z(0),z(0) = D[z,z]e iS[z,z] . − H(t)|z(t),z(t) − i log z ′ (T ),Z ′ (T )|z(T ) and solve the Euler-Lagrange equationsd dt ∂ ∂ż n − ∂ ∂z n L[z,z] = 0 and d dt ∂ ∂ż n − ∂ ∂z n L[z,z] = 0 (101) We use the shorthand notation O = z(t)|O|z(t) Time-reversal invariance requires that the order of the flavors is flipped in the definition of the antineutrino algebra as compared with the definition of the neutrino algebra in Eq. (2).3 In these formulasz is an independent complex number, not the complex conjugate of z. Complex conjugates will be denoted by a star such as z * andz * . Here the expectation values are calculated using the states in Eq. (67): O = z(t),z(t)|O|z(t),z(t). For example k |z k (p)| 2 = 1 + |zµ(p)| 2 + |zτ (p)| 2 .6 Here we use the short hand notation O = z(t),z(t)|O|z(t),z(t) , etc. for n = µ, τ and those for z * n andz * n . We use the expectation values of the SU(3) generators which are given below:for i, j = e, µ, τ (with the convention z e (p) =z e (p) = 1). Here we assumed p ∈ P andp ∈P. If p ∈ P orp ∈P then the expectation values are zero. The quadratic terms in the Lagrangian are calculated using the identitiesandwhich are valid for p = p andp =q. The expectation value of the time derivative term isUsing formulas (102)-(105) we find the Lagrangian asHere Y ij (p) is given byThe equations of motion driven from this Lagrangian are given below:where n = µ, τ . We write the parameters z i andz i in terms of the neutrino wavefunctions as z i (p) = ψ i (p)/ψ e (p) and z i (p) =ψ * i (p)/ψ * e (p), i.e.,where the one body wavefunctions are normalized as follows:If the Eqs. (110) and (111) are substituted in Eqs. (108) and (109), we find that ψ i (p) andψ i (p) obey the following Schrodinger equations:Appendix: Path Integral Representation of the Evolution OperatorIn this appendix, we drive the SU(3) path integral formula in Eq. (98). The path integral formula for SU(2) coherent states follows very similar lines.The coherent states defined in Eq. (96) admit the following resolution of identity:3 |z,z z,z|. . M Prakash, J M Lattimer, R F Sawyer, R R Volkas, arXiv:astro-ph/0103095Ann. Rev. Nucl. Part. Sci. 51M. Prakash, J. M. Lattimer, R. F. Sawyer and R. R. Volkas, Ann. Rev. Nucl. Part. Sci. 51, 295 (2001) [arXiv:astro-ph/0103095]. . A B Balantekin, arXiv:hep-ph/9808281Phys. Rept. 315A. B. Balantekin, Phys. 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{'abstract': 'An algebraic approach to the neutrino propagation in dense media is presented. The Hamiltonian describing a gas of neutrinos interacting with each other and with background fermions is written in terms of the appropriate SU(N) operators, where N is the number of neutrino flavors. The evolution of the resulting many-body problem is formulated as a coherent-state path integral. Some commonly used approximations are shown to represent the saddle-point solution of the path integral for the full many-body system.', 'arxivid': 'astro-ph/0607527', 'author': ['A B Balantekin \nDepartment of Physics\nUniversity of Wisconsin Madison\n53706WisconsinUSA\n', 'Y Pehlivan \nDepartment of Physics\nUniversity of Wisconsin Madison\n53706WisconsinUSA\n'], 'authoraffiliation': ['Department of Physics\nUniversity of Wisconsin Madison\n53706WisconsinUSA', 'Department of Physics\nUniversity of Wisconsin Madison\n53706WisconsinUSA'], 'corpusid': 118898043, 'doi': '10.1088/0954-3899/34/1/004', 'github_urls': [], 'n_tokens_mistral': 17707, 'n_tokens_neox': 15303, 'n_words': 8486, 'pdfsha': '794b02ada140e2afbbf3f6ce042c404eec0e77fe', 'pdfurls': ['https://arxiv.org/pdf/astro-ph/0607527v2.pdf'], 'title': ['Neutrino-Neutrino Interactions and Flavor Mixing in Dense Matter', 'Neutrino-Neutrino Interactions and Flavor Mixing in Dense Matter'], 'venue': []}

arxiv

Unimodular gravity theory with external sources in a Lorentz-symmetry breaking scenario 12 Jan 2019 L H C Borges UNESP -Campus de Guaratinguetá -DFQ Avenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil D Dalmazi UNESP -Campus de Guaratinguetá -DFQ Avenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil Unimodular gravity theory with external sources in a Lorentz-symmetry breaking scenario 12 Jan 2019PACS numbers: This paper is dedicated to the study of interactions between stationary field sources for the linearized unimodular gravity or WTDIFF theory in a model which exhibits Lorentz symmetry breaking due to the presence of the linearized topological Chern-Simons term in 3 + 1 dimensions, where the Lorentz symmetry breaking is caused by a single background vector v µ . Since the background vector is very tiny, we treat it perturbatively up to second order and we focus on physical phenomena which have no counterpart in standard WTDIFF theory. We consider effects related to field sources describing point-like particles and cosmic strings. We show that in a Lorentz violating scenario the interaction between external sources lead to numerically different results for linerarized Eintein-Hilbert (LEH) and WTDIFF theories, however both results are qualitatively similar and can be equalized after a rescaling of the Lorentz breaking source term which makes an experimental distinction impossible at leading order in pertubation theory as far as point particles and cosmic strings are concerned. I. INTRODUCTION Theories with Lorentz-symmetry breaking have been under intensive investigation as an attempt to find a consistent description of quantum gravity. Most of these investigations have been done in the context of the Standard Model extension (SME) [1,2] that incorporates in the Standard Model the full set of gauge-invariant, renormalizable Lorentz violation interactions. Some aspects of Lorentz violation have been investigated, for example, in Maxwell electrodynamics [3][4][5][6][7][8][9], QED [10,11], linearized gravity [12][13][14][19][20][21][22][23][24][25][26][27], short-range experiments in pure gravity [28][29][30][31][32][33]. Here we are interested in the issue of Lorentz violation in linearized gravity. There are two covariant descriptions of massless spin-2 particles in 3 + 1 dimensions via a symmetric rank-2 tensor: the linearized Einstein-Hilbert (LEH) theory [34] and the Weyl plus transverse diffeomorphism (WTDIFF) invariant model [35][36][37][38]. The WTDIFF model is the linearized truncation of unimodular gravity [39][40][41][42] which, in its turn, corresponds to the EinsteinHilbert theory with the replacement g µν →ĝ µν /(−ĝ) 1/4 . The WTDIFF model can be obtained from the usual LEH theory by the singular replacement h µν → h µν − η µν h/4, and the external source is just the traceless piece of the energy-momentum tensor T µν , i.e., T µν → T µν − η µν T /4 [17,43] . One remarkable feature of the WTDIFF theory is the fact that this theory leads to the same results obtained in LEH theory for the interactions between external sources [17,18,43], at least, as far as the Lorentz symmetry of the linearized theory is preserved. From this point of view it is natural to ask whether such equivalence remains in a Lorentz violating scenario. One might try to distinguish WTDIFF from LEH based on Lorentz violating phenomena as we pursue here. Moreover there are physical phenomena produced by the presence of external field sources, with no counterpart in standard (Lorentz preserving) WTDIFF and LEH theories. Studies of this kind have not yet been considered in the literature, to the best of the authors' knowledge and deserve investigations not only for their theoretical aspects, but also because of their possible experimental relevance in the search for Lorentz symmetry breaking. This paper is devoted to this subject, where we search for effects produced by external sources. At leading order in perturbation theory, we show that is not possible to distinguish the WTDIFF-LV and LEH-LV theories through interactions between external sources. Specifically, we consider the WTDIFF Lagrangian modified by the CPT breaking linearized topological Chern-Simons term in 3 + 1 dimensions [12], where the Lorentz symmetry breaking is due to the presence of a single background vector v µ . Since the background vector is very tiny, we treat it perturbatively up to leading order. We show that a spontaneous torque on a classical rigid halter emerges and we investigate some phenomena due to the presence of cosmic strings and show that the string can interact with a point-like particle as well as with another cosmic string in the Lorentz symmetry breaking scenario considered. The paper is structured as follows; in Sect. II we describe some general aspects of the models we deal with along the paper and present our basic formula for the computation of the energy of the system. In Sect. III we consider effects due to the presence of point-like stationary particles. Sect. IV is dedicated to the study of physical phenomena due to the presence of cosmic strings, and Sect. V is dedicated to our final remarks and conclusions. II. PROPAGATORS AND VACUUM ENERGY IN THE PRESENCE OF SOURCES First, let us consider the following Lagrangian density in 3 + 1 dimensions L = − √ −g 2k 2 R − 1 2 ǫ µνkλ v µ Γ ρ λσ ∂ ν Γ σ ρk + 2 3 Γ σ να Γ α kρ ,(1) where the first term is the well-known Einstein-Hilbert Lagrangian and the second one is the so called topological Chern-Simons term in four dimensions, g = det g µν , g µν is the space-time metric, as standard in the literature, we deffine the gravitational coupling as k 2 = 8πG, R stands for the scalar curvature, R = g µν ∂ ν Γ λ µλ − ∂ λ Γ λ µν + Γ τ µλ Γ λ τ ν − Γ τ µν Γ λ τ λ ,(2) where Γ λ µν = 1 2 g kλ (∂ µ g kν + ∂ ν g µk − ∂ k g µν ) ,(3) is the affine connection, ǫ µνkλ stands for the Levi-Civita tensor ǫ 0123 = 1 , and v µ = v 0 , v is the background vector. We will restrict our attention to a graviton propagating on a flat Minkowski geometry whose metric tensor is η µν = (+, −.−, −). We use g µν (x) = η µν + kh µν (x) ,(4) where kh µν is the symmetric spin-2 field and represents, as usually, a small pertubation around flat Minkowski space-time. The model (1) at linearized level (quadratic in h µν ) is invariant under linearized general coordinate transformations, δh µν (x) = ∂ µ ξ ν (x) + ∂ ν ξ µ (x) ,(5) where ξ ν (x) are the gauge parameters. So, in order to avoid singularities it is necessary to fix this gauge invariance, a common choice in the perturbative gravity literature is the De Donder gauge-fixing term, L gf = 1 2α F µ F µ ,(6) with F µ = ∂ ν h µν − 1 2 η µν h ,(7) and h = η µν h µν . Therefore, by using the weak field approximation (4) and taking into account (6), the quadratic Lagrangian in the spin-2 field, reads L EH−LV = − 1 2 1 2 h µν h µν − 1 2 h h + h∂ µ ∂ ν h µν − h µν ∂ µ ∂ λ h λν + 1 2α −h µν ∂ µ ∂ λ h λν + h µν ∂ µ ∂ ν h − 1 4 h h − k 2 4 ǫ µνkλ v µ (h ρ λ ∂ ν h ρk − h λρ ∂ ν ∂ ρ ∂ σ h σk ) + 1 2 k h µν T µν ,(8) where in the last line we have inserted an external source, T µν , that must be conserved ∂ µ T µν = 0 in order that the source term d 4 x h µν T µν be invariant under linearized reparametrizations (5). Now let us jump to the WTDIFF model. There are only two ways of describing massless spin-2 particles covariantly in terms of a symmetric rank-2 tensor. Besides the usual linearized Einstein-Hilbert theory L EH which is the far more popular theory, we have a model invariant under Weyl (W) transformations and transverse reparametrizations (TDIFF). There are elderly [15,16] and more recent [17,18] references on this subject . In practice the so called WTDIFF model can be obtained from the Einstein-Hilbert theory (8) in D = 3 + 1 via the singular replacement by a traceless field, i.e., h µν → h µν − η µν h/4. From (8) we obtain L W T −LV = − 1 2 1 2 h µν h µν − 3 16 h h + 1 2 h∂ µ ∂ ν h µν − h µν ∂ µ ∂ λ h λν + 1 2α −h µν ∂ µ ∂ λ h λν + h µν ∂ µ ∂ ν h − 1 4 h h − k 2 4 ǫ µνkλ v µ (h ρ λ ∂ ν h ρk − h λρ ∂ ν ∂ ρ ∂ σ h σk ) + 1 2 k h µνT µν .(9) Notice that the singular redefinition has automatically generated a redefinition of the source: T µν →T µν = T µν − η µν T 4 .(10) The new source is defined out of the old conserved source T µν but it is not conserved by itself. It satisfies instead the constraints: η µνT µν = 0 ; ∂ µT µν = −∂ ν T /4 .(11) The new constraints make the source term d 4 x h µνT µν invariant under transverse reparametrizations and Weyl transformations: δh µν = ∂ µ ξ T ν + ∂ ν ξ T µ + η µν φ ; ∂ µ ξ T µ = 0 ,(12) which are the symmetries of L W T −LV , except for the gauge fixing term of course where we have not made any field redefinition, otherwise the Weyl symmetry would not be broken by such term. We have made use of the fact that the Lorentz violating term is invariant under Weyl transformations and (arbitrary) reparametrizations. Now we are going to write down the propagators corresponding to L EH−LV and L W T −LV and the respective source-source term. The presence of the LV term makes the calculation complicate. Fortunately, it has been worked out before in [14]. Both Lagrangians (8) and (9) fit in the form L = C 1 (∂ µ h µν ) 2 + C 2 ∂ µ h∂ α h αµ + h C 3 ( )h + h µν C 4 ( )h µν + h λα θ ρα S νλ h ρν ,(13) where C 1 , C 2 are constants, C 3 ( ), C 4 ( ) are functions of and S νλ = k 2 ǫ νλβσ v β ∂ σ . If we write down (13) in the form L = h µν G µν,αβ h αβ , the differential operator G µν,αβ can be decomposed in terms of a basis of operators [14] displayed in our appendix. Suppressing indices we have G = C 4 P (2) SS + C 4 − C 1 2 P (1) SS + [C 3 + C 4 − (C 1 + C 2 ) ] P (0) W W + (3 C 3 + C 4 )P (0) SS + √ 3 C 3 − C 2 2 (P (0) SW + P (0) W S ) − 2 (S σ + S τ ) .(14) Using (74) we have the inverse operator G −1 = P (2) SS C 4 + P (1) SS C 4 − C1 2 + 3 C 3 + C 4 K (0) P (0) W W + [C 3 + C 4 − (C 1 + C 2 ) ] K (0) P (0) SS − √ 3 C 3 − C 2 2 (P (0) SW + P (0) W S ) + G −1 LV ,(15) where the spin-0 determinant K (0) and the Lorentz-violation piece are given respectively by K (0) = [C 3 + C 4 − (C 1 + C 2 ) ] (3 C 3 + C 4 ) − 3 C 3 − C 2 2 2 ,(16)G −1 LV = 4 iS τ + 2S 2 τ [1 + 4f (p)] + 4 iS σ + 2S 2 σ [1 + 16f (p)] ,(17) where f (p) is given in (71) and the operators S σ , S 2 σ , S τ and S 2 τ are also displayed in the appendix. In the cases of EH and WTDIFF we have respectively C 1 = 1 + 1 2α ; C 2 = −(1 + 1 2α ) ; C 3 = − 2 − 8α ; C 4 = 2 ; K (0) = 2 8α ,(18)C 1 = 1 + 1 2α ; C 2 = − 1 2 − 1 2α ; C 3 = − 3 16 − 8α ; C 4 = 2 ; K (0) = 2 32α .(19) Explicitly we have: G −1 EH−LV = P (2) SS /2 − P (1) SS /(4α) − 3 + 8α P (0) W W − P (0) SS − θω + ωθ + G −1 LV ,(20)G −1 W T −LV = P (2) SS /2 − P (1) SS /(4α) − 12 + 2α P (0) W W − 4 + 6α P (0) SS − 4 + 2α (θω + ωθ) + G −1 LV . (21) Notice that, since G −1 LV only depends upon the Lorentz violating term and the coefficient C 4 in the operator G, see (59) and (74), we end up with the same Lorentz violating term in G −1 for both EH and WTDIFF models. After we saturate G −1 EH−LV with conserved sources it turns out that all terms containing the derivatives ω µν drop out. We have in momentum space: T µν (−p) G −1 EH−LV µν,αβ T αβ (p) = T µν (−p) 2 P (2) SS p 2 − P (0) SS p 2 + G −1 LV µν,αβ T αβ (p) = T µν (−p) η µα η νβ + η µβ η να p 2 − 2 3 + 1 3 η µν η αβ p 2 + G −1 LV µν,αβ T αβ (p) ,(22)= T µν (−p) η µα η νβ + η µβ η να p 2 − η µν η αβ p 2 µν,αβ T αβ (p) +T µν (−p) G −1 LV µν,αβ T αβ (p) .(23) Notice the factor 1/3 in (22) which comes from P (0) SS , it is responsible for the vDVZ [46,47] mass discontinuity, since it does not appear in the propagator of the massive theory, no matter how small is the mass. Now, if we saturate G −1 W T −LV with the traceless sourcesT µν , which satisfy (11), we have once again no contribution from the spin-1 sector, though this is more subtle now: T µν (−p) P (1) SS µν,αβT αβ (p) = 2T µν (−p)θ µα ω νβT αβ (p) = 1 2T µν (−p)θ µα p ν p α T = 0 ,(24) where the last equality holds due to the transverse property of θ µα . It turns out that adding up all spin-0 contributions we derive the known result [40] that no one can tell a difference between the source-source terms in LEH and WTDIFF theories as far as Lorentz preserving terms are considered, namely from (21), T µν (−p) G −1 W T −LV µν,αβT αβ (p) =T µν (−p) 2 P (2) SS p 2 − 12 p 2 + 4 3 p 2 − 8 p 2 ωω + G −1 LV µν,αβT αβ (p) = T µν (−p) η µα η νβ + η µβ η να p 2 − η µν η αβ p 2 µν,αβ T αβ (p) +T µν (−p) G −1 LV µν,αβT αβ (p) .(25) Notice that the only difference between (23) and (25) is the replacement of T µν byT µν in the Lorentz violating term. Paving the way for the next section we present here the formulae for the vacuum energy. Since we have a quadratic Lagrangian in the field variables, the contribution due to the stationary sources to the vacuum energy of the system is given in the WTDIFF case by [48] E W T −LV = k 2 4t d 4 x d 4 yT αβ (x)D αβ,γδ (x, y)T γδ (y) ,(26) where t is a time variable and the limit t → ∞ is implicit. The integration in y 0 is from −t/2 to t/2. We have introduced the propagator in the coordinate spacê D αβ,γδ (x, y) = d 4 p (2π) 4D αβ,γδ (p) e −ip·(x−y) .(27) Splitting the momentum space propagator into Lorentz preserving and Lorentz violating parts we have from (25): E W T −LV = k 2 4t d 4 p (2π) 4 d 4 x d 4 y e −ip·(x−y) T αβ (x) D (0) αβ,γδ (p) T γδ (y) +T αβ (x) D (LV ) αβ,γδ (p)T γδ (y) ,(28) where D (0) αβ,γδ (p) = η αγ η βδ + η αδ η βγ p 2 − η αβ η γδ p 2 ; D (LV ) αβ,γδ (p) = 4 iS τ + 2S 2 τ αβ,γδ p 2 [1 + 4f (p)] + 4 iS σ + 2S 2 σ αβ,γδ p 2 [1 + 16f (p)] .(29) From (28) we can compute the interaction energy between different spin-2 field sources for the WTDIFF-LV model. The same expression (28) can be used for the EH-LV case if we replaceT µν → T µν in the Lorentz violating term. As the Lorentz symmetry breaking must be very tiny, the background vector must be small. Therefore, along the paper we will treat v µ perturbatively up to second order, which is the leading order in the background vector. III. POINT-LIKE PARTICLES In this section we study the interaction energy between two stationary point-like particles in 3 + 1 dimensions. The external source which describes this system is given by T αβ (x) I = M 1 η α0 η β0 δ 3 (x − a 1 ) + M 2 η α0 η β0 δ 3 (x − a 2 ) ,(30) T αβ (x) I = M 1 η α0 η β0 − η αβ 4 δ 3 (x − a 1 ) + M 2 η α0 η β0 − η αβ 4 δ 3 (x − a 2 ) ,(31) where the location of the particles are specified by the vectors a 1 and a 2 , and the parameters M 1 and M 2 are the particles masses. Substituting Eqs. (30) and (31) into Eq. (28) and discarding the self-interacting contributions (that is, the interactions of a given point particle with itself), we obtain E I W T −LV = k 2 M 1 M 2 2t d 4 p (2π) 4 d 4 x d 4 y δ 3 (x − a 1 ) δ 3 (y − a 2 ) e −ip·(x−y) D(E I W T −LV = − k 2 M 1 M 2 2 d 3 p (2π) 3 e ip·a p 2 − 10k 4 (v · ∇ a ) 2 d 3 p (2π) 3 e ip·a p 2 +2k 4 (v 0 ) 2 − 5v 2 d 3 p (2π) 3 e ip·a ,(33) where a = a 1 − a 2 is the distance between the two massive particles and we defined the differential operator ∇ a = ∂ ∂a 1 , ∂ ∂a 2 , ∂ ∂a 3 .(34) The last term inside the brackets of Eq. (33) is the Dirac delta function δ 3 (a) and provided that a is nonzero, this term vanishes. Using the fact that d 3 p (2π) 3 e ip·a p 2 = 1 4πa ,(35) whrere a = |a|, and performing some manipulations, we obtain E I W T −LV = − GM 1 M 2 a 1 − 10 8πG a 2 3 (v · a) 2 a 2 − v 2 .(36) Equation (36) is a perturbative result and gives the interaction energy between two point-like particles mediated by the spin-2 field with the specific Lorentz violating coupling contained in Eq. (8). The v µ dependent contribution in (36) is a correction to the usual gravitational interaction due the Lorentz symmetry breaking, leading to an anisotropic interaction between the particles. If we take v µ → 0, the expression reduces to the standard gravitational interaction. The same happens for the case where v µ = v 0 , 0 , i.e., if v = 0. However, for the particular case where the distance vector a is perpendicular to the background vector v, Eq.(36) still exhibits effects due to Lorentz symmetry breaking, E I W T −LV (v · a = 0) = − GM 1 M 2 a 1 + 10 8πG a 2 v 2 .(37) The force between the two particles can be calculated from Eq. (36), resulting in F I W T −LV = −∇ a E I W T −LV = − GM 1 M 2 a 2 1 + 30 8πG a 2 v 2 − 5 (v · a) 2 a 2 â + 60 8πG a 2 (v · a) a v ,(38) whereâ is an unit vector pointing in the direction of the vector a. The interaction force in (38) shows us more explicitly the anisotropies generated by the Lorentz symmetry breaking. We remark that the authors of [49] have considered the effect of Lorentz and CPT breaking terms of mass dimension five (three derivatives) on a couple of moving point masses. Their Lorentz violating terms include ours, see (8). In the static limit their inverse cubic force contribution F ∼ a −3 disappears in agreement with (38) where our leading Lorentz breaking contribution only appears at order 1 a −4 . An important consequence of the anisotropies in expression (36) is the emergence of a spontaneous torque on an rigid halter. In order to investigate this effect we consider a typical rigid hater composed by two particles of masses M 1 and M 2 respectively, placed at the positions a 1 = R + A 2 and a 2 = R − A 2 , where we take the distance vector A fixed (and small). From Eq. (36), we obtain E halter W T −LV = − GM 1 M 2 A 1 − 10 8πG A 2 v 2 3 cos 2 (θ) − 1 ,(39) where A =| A | and θ is the angle between the vectors A and v. Notice that 0 ≤ θ ≤ 2π. The energy (39) leads to a spontaneous torque on the rigid halter, as follows: τ halter W T −LV = − ∂E halter W T −LV ∂θ = 30 GM 1 M 2 A 8πG A 2 v 2 sin (2θ) .(40) This spontaneous torque on the rigid halter is an exclusive effect due to the Lorentz violating background. If, v µ = 0 the torque vanishes, as it should, as well as for the specific cases θ = 0, π, π/2, 2π. When θ = π/4, the torque exhibits a maximum value. For the LEH-LV theory we proceed as previously, replacingT αβ (x) → T αβ (x) andT γδ (y) → T γδ (y) in the second term between brackets on the right hand side of the Eq. (28), obtaining for the interaction energy between two point-like particles up to second order in v µ , the result E I EH−LV = k 2 M 1 M 2 2t d 4 p (2π) 4 d 4 x d 4 y δ 3 (x − a 1 ) δ 3 (y − a 2 ) e −ip·(x−y) D (0) 00,00 (p) + D (LV ) 00,00 (p) = − GM 1 M 2 a 1 − 16 8πG a 2 3 (v · a) 2 a 2 − v 2 .(41) Comparing (36) with (41) we verify that for WTDIFF-LV and LEH-LV theories, the interaction energy between two point particles is numerically different already at second order in the background vector v µ . An opposite situation occurs in standard WTDIFF and LEH theories where the energies are the same ones. The Lorentz violating terms in (36) and (41) are qualitatively similar and we could turn the overall factor 10 into 16 by scaling v µ into (4/ √ 10)v µ , at least at leading order in perturbation theory. It is not clear whether such simple rescaling will fit the remaining terms beyond the quadratic approximation in v µ . This is under investigation. IV. COSMIC STRINGS Opposite to the point-like particles of last section we now focus on long size objects, namely, cosmic strings. Their contribution to the interaction energy, at leading order in perturbation theory, can only appear due to the Lorentz violating terms. We start this section considering the interaction energy between a point-like particle and an cosmic string, both of them stationary. The cosmic string shall be taken to flow parallel to the z-axis, along the straight line located at A = (A 1 , A 2 , 0). The point-like particle is concentrated at position s. This system is described by the external source, T αβ (x) II = M η α0 η β0 δ 3 (x − s) + µ η α0 η β0 − η α3 η β3 δ 2 (x ⊥ − A) ,(42) where the first term on the right hand side of the above equation stands for the external field source produced by the point-like charge, and the second one is the source produced by the cosmic string [50]. The mass parameter M and the linear mass density µ are the coupling constants between the field and the delta functions, and x ⊥ = (x 1 , x 2 , 0), is the position vector perpendicular to the cosmic string. For the traceless source we have T αβ (x) II = M η α0 η β0 − η αβ 4 δ 3 (x − s) + µ η α0 η β0 − η α3 η β3 − η αβ 2 δ 2 (x ⊥ − A) .(43) Substituting the sources (42) and (43) in (28) and discarding the self-interacting terms, which do not contribute to the interaction force between the string and the particle (the self-interacting terms are proportional to M 2 or µ 2 ), we have 71), take into account only the contributions up to second order in v µ , and evaluating the integrals: d 2 y ⊥ , d 3 x, dy 3 , dp 3 , dx 0 , dp 0 and dy 0 , we obtain E II W T −LV = k 2 M µ 2t d 4 p (2π) 4 d 4 x d 4 y δ 3 (x − s) δ 2 (y ⊥ − A) e −ip·(x−y) D(0)E II W T −LV = 2k 6 M µ (v ⊥ · ∇ a ⊥ ) 2 d 2 p ⊥ (2π) 2 e ip ⊥ ·a ⊥ p 2 ⊥ + v 2 ⊥ + 2(v 3 ) 2 d 2 p ⊥ (2π) 2 e ip ⊥ ·a ⊥ ,(45) where v 3 is the projection of the background vector v along the string, and defined p ⊥ = (p 1 , p 2 , 0), v ⊥ = (v 1 , v 2 , 0), the distance between the particle and the cosmic string a ⊥ = (s 1 − A 1 , s 2 − A 2 , 0), and the differential operator ∇ a ⊥ = ∂ ∂a 1 , ∂ ∂a 2 , 0 .(46) Provided that a ⊥ is non-zero, the last term inside the brackets in (45) vanishes. The remaining integral is divergent, in order to circumvent this problem we proceed as in references [8,51,52], introducing a mass regulator parameter, as follows: E II W T −LV = 2k 6 M µ (v ⊥ · ∇ a ⊥ ) 2 lim m→0 d 2 p ⊥ (2π) 2 e ip ⊥ ·a ⊥ p 2 ⊥ + m 2 .(47) Using the fact that [51] d 2 q ⊥ (2π) 2 e ip ⊥ ·a ⊥ p 2 ⊥ + m 2 = 1 2π K 0 (ma ⊥ ) ,(48) and acting with the differential operator (46), we arive at E II W T −LV = k 6 M µ π lim m→0 (v ⊥ · a ⊥ ) 2 a 2 ⊥ m 2 K 2 (ma ⊥ ) − v 2 ⊥ a ⊥ mK 1 (ma ⊥ ) ,(49) where a ⊥ =| a ⊥ | and K 0 (ma ⊥ ), K 1 (ma ⊥ ), K 2 (ma ⊥ ) stand for the K-Bessel functions. Using the fact that [53] mK 1 (ma ⊥ ) m→0 −→ 1 a ⊥ , m 2 K 2 (ma ⊥ ) m→0 −→ 2 a 2 ⊥ ,(50) we obtain E II W T −LV = 512π 2 G 3 M µ a 2 ⊥ 2 (v ⊥ · a ⊥ ) 2 a 2 ⊥ − v 2 ⊥ .(51) This interaction energy is an effect due solely to the Lorentz violating background up to lowest order of the background vector, having no counterpart in standard WTDIFF theory. Clearly, if the background four-vector v µ is zero, there is no interaction energy. The same happens for the case where v ⊥ = 0 The interaction force can be obtained from Eq. (51) as follows, F II W T −LV = −∇ a ⊥ E II W T −LV = − 1024π 2 G 3 M µ a 3 ⊥ v 2 ⊥ − 4 (v ⊥ · a ⊥ ) 2 a 2 ⊥ â ⊥ + 2 (v ⊥ · a ⊥ ) a ⊥ v ⊥ ,(52) whereâ ⊥ is an unit vector pointing on the direction of vector a ⊥ . As a final comment, we point out that from Eq. (51), one can also obtain a torque on the cosmic string by fixing the point particle. Denoting by φ the angle between v ⊥ and a ⊥ , we obtain τ II W T −LV = − ∂E II W T −LV ∂φ = 1024π 2 G 3 M µ a 2 ⊥ v 2 ⊥ sin 2 (φ) .(53) We notice that the torque (53) is an effect due solely to the Lorentz violating background. It does not appear in standard WTDIFF theory. If φ = 0, π/2, π, 2π or v ⊥ = 0, the torque vanishes. For φ = π/4 the toque has a maximum intensity. For LEH-LV theory we employ the same steps as above, we obtain for the interaction energy between the cosmic string and the point-like particle the expression E II EH−LV = k 2 M µ 2t d 4 p (2π) 4 d 4 x d 4 y δ 3 (x − s) δ 2 (y ⊥ − A) e −ip·(x−y) D (0) 00,00 (p) − D (0) 00,33 (p) +D (LV ) 00,00 (p) − D (LV ) 00,33 (p) = 4E II W T −LV ,(54) showing that we have a numerically different result for both WTDIFF-LV and LEH-LV theories although they can be related via a scaling of the Lorentz breaking term once again. The next and last example is given by two parallel cosmic strings. We take a coordinate system where the first string lies along the straight line located at A 1 = (A 1 1 , A 2 1 , 0), with linear mass density µ 1 , and the second string lies along the line that crosses the xy plane at A 2 = (A 1 2 , A 2 2 , 0), with linear mass density µ 2 . The corresponding external source is given by T αβ (x) III = µ 1 η α0 η β0 − η α3 η β3 δ 2 (x ⊥ − A 1 ) + µ 2 η α0 η β0 − η α3 η β3 δ 2 (x ⊥ − A 2 ) ,(55)T αβ (x) III = µ 1 η α0 η β0 − η α3 η β3 − η αβ 2 δ 2 (x ⊥ − A 1 ) + µ 2 η α0 η β0 − η α3 η β3 − η αβ 2 δ 2 (x ⊥ − A 2 ) . (56) Substituting the sources (55) and (56) in (28), discarding the self-interacting terms, using the Eqs. (29), (65), (66), (67), (68), (69), (70), (71), proceeding as in the previous cases, and identifying the length of the cosmic string as L = dx 3 , we can show that the interaction energy between the two cosmic strings up to second order in v µ is given by E III W T −LV = 4k 6 µ 1 µ 2 L (v ⊥ · ∇ a ⊥ ) 2 lim m→0 d 2 p ⊥ (2π) 2 e ip ⊥ ·a ⊥ p 2 ⊥ + m 2 = 1024π 2 G 3 µ 1 µ 2 L a 2 ⊥ 2 (v ⊥ · a ⊥ ) 2 a 2 ⊥ − v 2 ⊥ ,(57) where we have identified the distance between the strings as a ⊥ = A 1 − A 2 , and a ⊥ =| a ⊥ | . It can be seen that the energy given above vanishes in the limit v µ = 0, where we do not have Lorentz-symmetry breaking. Similarly, the energy (57) leads to an interaction force between two cosmic strings as well as to a torque on one string when we fix the other one. For the LEH-LV theory, we obtain the expression E III EH−LV = 4E III W T −LV ,(58) where the numerical difference between the results for both theories was already expected. V. CONCLUSIONS AND PERSPECTIVES In this paper we have investigated the interactions between stationary sources for the WTDIFF and LEH theories in the presence of the Lorentz violating and CPT breaking topological Chern-Simons term in 3 + 1 dimensions. All the results have been obtained perturbatively up to second order in v µ . We have shown the emergence of an spontaneous torque on a classical rigid halter. We have also investigated interactions with one or two cosmic strings. We have shown that a cosmic string has a non trivial interaction with a point-like particle as well as with another cosmic string. All those phenomena would not appear in the absence of the Lorentz violating term and have not been explored before in the literature. We have also shown that there is a numerical difference in the interactions between stationary external sources regarding WTDIFF-LV and LEH-LV theories. At leading order in perturbation theory the differences can be extinguished after a convenient scaling of the Lorentz violating source term which makes their experimental detection impossible in the cases analyzed here. It is not clear however if such scaling will keep working beyond the leading order in perturbation theory. This is under investigation as well as the investigation of more general non minimal Lorentz violating term, see [32]. VI. APPENDIX Here we give a summary of technical details about the propagator found in [14] which include a class of Lorentz violating terms. We use a slightly different notation. The Lagrangian densities for symmetric rank-2 fields can be written as L = h αβ G αβ,µν h µν . Let us assume that the differential operator G αβ,µν , suppressing indices, can be decomposed as G = A P (2) SS + B P (1) SS + A SS P (0) SS + A W W P (0) W W + A SW (P (0) W S + P (0) SW ) + a S τ + b S σ + c S 2 τ + d S 2 σ .(59) where the spin-s operators P P (0) SW λµ αβ = 1 √ 3 θ λµ ω αβ , P (0) W S λµ αβ = 1 √ 3 ω λµ θ αβ ,(62)ω µν = ∂ µ ∂ ν , θ µν = η µν − ∂ µ ∂ ν ,(63) while in the Lorentz-violating sector we have the spin-2 operators (S τ ) αβ,γδ = 1 2 (τ αγ S βδ + τ αδ S βγ + τ βγ S αδ + τ βδ S αγ ) , S 2 τ αβ,γδ = f (p) 1 2 (θ αγ τ βδ + θ αδ τ βγ + θ βγ τ αδ + θ βδ τ αγ ) − (τ αγ τ βδ + τ αδ τ βγ ) , (S σ ) αβ,γδ = 1 2 (σ αγ S βδ + σ αδ S βγ + σ βγ S αδ + σ βδ S αγ ) , S 2 σ αβ,γδ = f (p) [σ αγ σ βδ + σ αδ σ βγ − (S αγ S βδ + S αδ S βγ )] , with τ αβ = 1 f (p) (v · p) (v α p β + v β p α ) − p 2 v α v β − (v · p) 2 p α p β p 2 ,(69)σ αβ = θ αβ − τ αβ , S αβ = k 2 ε αβρφ v ρ p φ ,(70)f (p) = k 4 (v · p) 2 − v 2 p 2 ; p µ = −ı ∂ µ .(71) From the algebra P (s) IJ P (r) JK = δ rs P (s) K I ; S σ P (s) IJ = δ s2 S σ ; S τ P (s) IJ = δ s2 S τ (72) S σ · S τ = 0 ; S 3 σ = 4 f S σ ; S 3 τ = f S τ .(73) The reader can check that the inverse of the operator (59) is [14] : G −1 = P (2) SS A + P (1) SS B + A W W P (0) SS + A SS P (0) W W − A SW (P (0) W S + P (0) SW ) K (0) + a D 1 S τ + b D 2 S σ + c(A + c f ) − a 2 A D 1 S 2 τ + d(A + 4 d f ) − b 2 A D 2 S 2 σ .(74) where f is given in (71) and D 1 = a 2 f − (A + c f ) 2 ; D 2 = 4 b 2 f − (A + 4 d f ) 2 ; K (0) = A W W A SS − A 2 W S .(75) the Eqs.(29), (65), (66), (67), (68), (69), (70), (71) up to second order in v µ , computing the integrals in the following order: d 3 x, d 3 y, dx 0 , dp 0 and dy 0 , using the Fourier representation for the Dirac delta function δ(p 0 ) = dx 0 /(2π) exp(−ipx 0 ), and identifying the time interval as t = integration limits for y 0 are as in the previous section. 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{'abstract': 'This paper is dedicated to the study of interactions between stationary field sources for the linearized unimodular gravity or WTDIFF theory in a model which exhibits Lorentz symmetry breaking due to the presence of the linearized topological Chern-Simons term in 3 + 1 dimensions, where the Lorentz symmetry breaking is caused by a single background vector v µ . Since the background vector is very tiny, we treat it perturbatively up to second order and we focus on physical phenomena which have no counterpart in standard WTDIFF theory. We consider effects related to field sources describing point-like particles and cosmic strings. We show that in a Lorentz violating scenario the interaction between external sources lead to numerically different results for linerarized Eintein-Hilbert (LEH) and WTDIFF theories, however both results are qualitatively similar and can be equalized after a rescaling of the Lorentz breaking source term which makes an experimental distinction impossible at leading order in pertubation theory as far as point particles and cosmic strings are concerned.', 'arxivid': '1901.03914', 'author': ['L H C Borges \nUNESP -Campus de Guaratinguetá -DFQ\nAvenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil\n', 'D Dalmazi \nUNESP -Campus de Guaratinguetá -DFQ\nAvenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil\n'], 'authoraffiliation': ['UNESP -Campus de Guaratinguetá -DFQ\nAvenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil', 'UNESP -Campus de Guaratinguetá -DFQ\nAvenida Dr. Ariberto Pereira da Cunha 33312516-410GuaratinguetáCEP, SPBrazil'], 'corpusid': 119244391, 'doi': '10.1103/physrevd.99.024040', 'github_urls': [], 'n_tokens_mistral': 15689, 'n_tokens_neox': 12766, 'n_words': 7489, 'pdfsha': 'bbd8b291953e82de8771b33a7d5bb2e4f3696f22', 'pdfurls': ['https://arxiv.org/pdf/1901.03914v1.pdf'], 'title': ['Unimodular gravity theory with external sources in a Lorentz-symmetry breaking scenario', 'Unimodular gravity theory with external sources in a Lorentz-symmetry breaking scenario'], 'venue': []}

arxiv

Cross-Layer Federated Learning Optimization in MIMO Networks 4 Feb 2023 Student Member, IEEESihua Wang Department of Electrical and Computer Engineering are with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence Beijing University of Posts and Telecommunications 100876BeijingChina Member, IEEEMingzhe Chen mingzhec@princeton.edu. Department of Electrical and Computer Engineering are with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence Beijing University of Posts and Telecommunications 100876BeijingChina Senior Member, IEEE, Changchuan Yin, Senior Member, IEEECong Shen Department of Electrical and Computer Engineering are with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence Beijing University of Posts and Telecommunications 100876BeijingChina Senior Member, IEEEChristopher G Brinton Department of Electrical and Computer Engineering are with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence Beijing University of Posts and Telecommunications 100876BeijingChina S Wang Brown Department of Electrical and Computer Engineering Princeton University 08544PrincetonNJUSA C Yin ccyin@ieee.org. Brown Department of Electrical and Computer Engineering Princeton University 08544PrincetonNJUSA M Chen School of Electrical and Computer Engineering University of Virginia CharlottesvilleVAUSA Christopher G Brinton Purdue University West LafayetteINUSA Cross-Layer Federated Learning Optimization in MIMO Networks 4 Feb 2023Cong Shen is with the Charles L. 2Index Terms Federated learningMIMOAirCompdigital modulation In this paper, the performance optimization of federated learning (FL), when deployed over a realistic wireless multiple-input multiple-output (MIMO) communication system with digital modulation and over-the-air computation (AirComp) is studied. In particular, an MIMO system is considered in which edge devices transmit their local FL models (trained using their locally collected data) to a parameter server (PS) using beamforming to maximize the number of devices scheduled for transmission. The PS, acting as a central controller, generates a global FL model using the received local FL models and broadcasts it back to all devices. Due to the limited bandwidth in a wireless network, AirComp is adopted to enable efficient wireless data aggregation. However, fading of wireless channels can produce aggregate distortions in an AirComp-based FL scheme. To tackle this challenge, we propose a modified federated averaging (FedAvg) algorithm that combines digital modulation with AirComp to mitigate wireless fading while ensuring the communication efficiency. This is achieved by a joint transmit and receive beamforming design, which is formulated as a optimization problem to dynamically adjust the beamforming matrices based on current FL model parameters so as to minimize the transmitting error and ensure the FL performance. To achieve this goal, we first analytically characterize how the beamforming matrices affect the performance of the FedAvg in different iterations. Based on this relationship, an artificial neural network (ANN) is used to estimate the local FL models of all devices and adjust the beamforming matrices at the PS for future model transmission. The algorithmic advantages and improved performance of the proposed methodologies are demonstrated through extensive numerical experiments. most of these existing works, such as [19] and [20], focused on the use of AirComp for analog modulation due to its simplicity for FL convergence analysis, which may not be desirable for practical wireless communication systems that almost exclusively use digital modulations. In consequence, it is necessary to study the implementation of AirComp-based FL over digital modulation-based wireless systems. A. Related Works Recent works such as [21]-[27] have studied several important problems related to the implementation of AirComp-based FL over wireless networks. The authors in [21] minimized the mean-squared error (MSE) of the FL model during AirComp transmission under transmit power constraints in a multiuser multiple-input multiple-output (MIMO) system. In [22], the 3 authors maximized the number of devices that can participate in FL training under certain MSE requirements in an AirComp-based MIMO framework. A joint machine learning rate and receiver beamforming matrix optimization method was proposed in [23] to reduce the aggregate distortion and satisfy an FL performance requirement. The authors in [24] investigated the deployment of FL over an AirComp-based wireless network to minimize the energy consumption of edge devices. In [25], the authors optimized the set of participating devices in an AirComp-assisted FL framework to speed up FL convergence. A receive beamforming scheme was designed in [26] to optimize FL performance without knowing channel state information. The authors in [27] minimized the FL model aggregation error under a channel alignment constraint in an MIMO system. However, most of these existing works [21]-[27] investigated the implementation of AirComp-based FL over analog modulation-based wireless systems, which may not be directly applied for practical digital modulation based wireless systems since these works do not consider coding and digital modulation. Recently, several works [28]-[33] have studied the implementation of AirComp FL over digital modulation based wireless systems. The authors in [28] designed one-bit quantization and modulation schemes for edge devices. One-bit gradient quantization scheme is proposed in [29] to achieve fast FL model aggregation. In [30], the authors designed a joint channel decoding and aggregation decoding schemes based on binary phase shift keying (BPSK) modulation for AirComp FL. The authors in [31] evaluated the performance of FL gradient quantization in digital AirComp. In [32], the convergence of FL implemented over an AirComp-based MIMO system is derived. The authors in [33] proposed a digital transmission protocol tailored to FL over wireless device-to-device networks. However, these prior works [28]-[33] mainly used low order digital modulation (i.e., BPSK) and hence their designed AirComp FL cannot be easily extendedto modern wireless systems that use high-order digital modulation schemes such as quadrature amplitude modulation (QAM). This is because the transmitted symbols that are processed by low order digital modulation (such as the symbols -1 and +1 in BPSK) are linearly superimposed. This linear superimposition does not exist in the high-order digital modulation schemes with complex mapping relationships between bits and symbols (such as Gray code).B. ContributionsThe main contribution of this paper is to develop a novel AirComp FL framework over highorder digital modulation-based wireless systems. Our key contributions include: Index Terms Federated learning, MIMO, AirComp, digital modulation. I. INTRODUCTION Federated learning (FL) has been extensively studied as a distributed machine learning approach with data privacy [1]- [5]. During the FL training process, edge devices are required to train a local learning model using its collected data and transmit the trained learning model to a parameter server (PS) for global model aggregation. The PS, acting as a central center, can coordinate the process across edge devices and broadcast the global model to all devices. This procedure is repeated across several rounds until achieving an acceptable accuracy of the trained model. Since the PS and edge devices must exchange their trained models iteratively over the wireless channels, FL performance can be significantly affected by imperfect and dynamic wireless transmission in both uplink and downlink. Compared to the PS broadcasting FL models to edge devices, edge devices uploading local models to the PS is more challenging due to their limited transmit power [6]- [10]. To tackle this challenge, over-the-air computation (also known as AirComp) techniques have recently been integrated into the implementation of FL [11]- [15]. Instead of decoding the individual local models of each device and then aggregating, AirComp allows edge devices to transmit their model parameters simultaneously over the same radio resources and decoding the average model (global model) directly at the PS [16]- [18]. However, • We propose a novel AirComp-based MIMO system in which distributed wireless devices modulate their trained local FL parameters into symbols and simultaneously transmit these modulated symbols to a PS that directly generates the global FL model via its received symbols. To optimize the FL training performance in the proposed system, the PS and devices must dynamically adjust the transmit and receive beamforming matrices over unreliable wireless channels. To this end, we formulate this joint transmit and receive beamforming matrix design problem as an optimization problem whose goal is to minimize the FL training loss. • To solve this problem, we first analytically characterize how the errors introduced by the proposed AirComp system affect FL training loss. Our analysis shows that the introduced errors caused by wireless transmission (i.e., fading and additive white Gaussian noise) and digital post-processing (i.e., digital demodulation) determine the gap between the optimal FL model that the FL targets to converge and the trained FL model. In particular, the errors caused by wireless transmission depends on the channel conditions and the trained FL model parameters. However, the errors caused by digital post-processing depend on the adopted modulation scheme and the number of devices participated in FL training. Hence, to minimize the errors caused by both wireless transmission and digital post-processing, the PS and the devices must dynamically adjust the transmit and receive beamforming matrices based on the adopted modulation scheme, the trained FL model parameters, and channel conditions. • To find the optimal transmit and receive beamforming matrices, we first introduce an artificial neural network (ANN)-based algorithm to predict the FL model parameters of all devices since optimizing beamforming matrices requires the information of each trained local model parameters which cannot be obtained by the PS. Then, given the predicted parameters, we derive a closed-form solution of the optimal transmit and receive beamforming matrices based on the adopted modulation scheme and channel conditions that minimizes the distance between the received signals of all devices and the predicted parameters in the decision region, which ensures the accuracy for model aggregation and FL performance. • Numerical evaluation results on real-world machine learning task datasets show that our proposed AirComp-based system can improve the identification accuracy by up to 20.5% and 10.5% compared to the AirComp-based system with analog modulation and BPSK, respectively. The rest of this paper is organized as follows. The system model and problem formulation for the AirComp-based system in FL framework are described in Section II. Section III analyzes the convergence of the designed FL framework and derives a closed-form optimal design of the transmit and receive beamforming matrices based on the analysis. In Section IV, our numerical evaluation is presented and discussed. Finally, conclusions are drawn in Section V. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider an FL system implemented over a cellular network, where K wireless edge devices train their individual machine learning models and send the machine learning parameters to a central PS through a noisy wireless MAC as shown in Fig. 1. In the considered model, the PS is equipped with N r antennas while each device k is equipped with N t antennas. Each device has N k training data samples and each training data sample n in device k consists of an input feature vector x k,n ∈ R N I ×1 and a corresponding label vector y k,n ∈ R N O ×1 where N I and N O are the dimension of input and output vectors, respectively. Table I provides a summary of the notations used throughout this paper. The objective of training is to minimize the global loss function over all data samples, which is given by F (g) = min g 1 N K k=1 N k n=1 f (g, x k,n , y k,n ) ,(1) where g ∈ R V ×1 is a vector that represents the global FL model of dimension V trained across the devices with N = K k=1 N k being the total number of training data samples of all devices. f (g, x k,n , y k,n ) is the local loss function of each device k with FL model g and data sample (x k,n , y k,n ). To minimize the global loss function in (1) in a distributed manner, each device can update its FL model using its local dataset with a backward propagation (BP) algorithm based on stochastic gradient descent (SGD), which can be expressed as w k,t = g t − λ n∈N k,t ∂f (g, x k,n , y k,n ) ∂g ,(2) where λ is the learning rate, N k,t is the subset of training data samples (i.e., minibatch) selected from device k's training dataset N k at iteration t, w k,t is the updated local FL model of device k at iteration t. Given w k,t , distributed devices must simultaneously exchange their model parameters with the PS via bandwidth-limited wireless fading channels for model aggregation. The equation of model aggregation is given by g t = K k=1 |N k | N w k,t ,(3) where |N k | represents the number of data samples in N k . To ensure all devices can participate in FL model exchanging via wireless fading channels, each device adopts digital modulation to mitigate wireless fading and the PS adopts beamforming to maximize the number of devices scheduled for FL parameter transmission. Next, we will mathematically introduce the FL training and transmission process integrated with digital modulation in the considered MIMO communication system. In particular, we first introduce our designed digital modulation process that consists of two steps: (i) digital pre-processing at the devices and (ii) digital post-processing at the PS. A. Digital pre-processing at the devices To transmit w k,t over wireless fading channels, each device k leverages digital pre-processing to represent each numerical FL parameter in w k,t using a symbol vector, which is given bŷ w k,t = l (w k,t ) ,(4) whereŵ k,t ∈ R W is a modulated symbol vector with W being the number of symbols, and l (·) denotes the digital pre-processing function that combines decimal-to-binary conversion and digital modulation where the decimal-to-binary conversion is used to represent each numerical FL parameter with a binary coded bit-interleaved vector and the digital modulation is used to modulate several binary bits as a symbol [34]. For convenience, the modulated signalŵ k,t is normalized (i.e., |ŵ k,t | = 1). We use rectangular M -quadrature-amplitude modulation (QAM) for digital modulation and it can be extended to other types of digital modulation schemes. In our model, each device sendsŵ k,t to the PS at each iteration t using fully-digital beamforming with low RF complexity. Given the transmit beamforming matrix A k,t ∈ C Nt×W and the maximal transmit power P 0 at device k, the power constraint can be expressed as E |A k,tŵk,t | 2 = |A k,t | 2 ≤ P 0 .(5) where E (x) represents the expectation of x. B. Post-processing at the PS Considering the multiple access channel property of wireless communication, the received signal at the PS is given by s t (A t ) = K k=1 H k A k,tŵk,t + n t(6) where A t = [A 1,t , · · · , A K,t ] denotes the transmit beamforming matrices of all devices, H k ∈ C Nr×Nt denotes the MIMO channel vector for the link from device k to the PS, and n t ∈ C Nr denotes additive white Gaussian noise. The entries of H k and n t are assumed to be independent and identically distributed (i.i.d.) complex Gaussian variables with zero mean. Since s t (A t ) is the weighted sum of all users' local FL models, we consider directly generating the global FL model g t+1 from s t (A t ). This is a major difference between the existing works and this work. The digital beamformer output signal can be expressed aŝ s t (B t , A t ) = B t H s t (A t ),(7) where B t ∈ C Nr×W is the digital receive beamforming matrix. Given the received symbol vectorŝ t (B t , A t ), the PS can reconstruct the numerical parameters in global FL model g t+1 , which can be expressed as g t+1 (B t , A t ) = l −1 (ŝ t (B t , A t )) ,(8) where l −1 (·) is the inverse function with respect to l (·) that combines the binary-to-decimal function and the digital demodulation function. C. Problem Formulation Next, we introduce our optimization problem. Our goal is to minimize the FL training loss by designing the transmit and receive beamforming matrices under the total transmit power constraint of each device, which is formulated as follows: min B,A F (g (B T , A T )) ,(9) s.t. |A k,t | 2 ≤ P 0 , ∀k ∈ K, ∀t ∈ T . From (9), we can see that the FL training loss F (g (B T , A T )) depends on the global FL model g (B T , A T ) that is trained iteratively. Meanwhile, as shown in (6) and (7), edge devices and the PS must dynamically adjust A t and B t based on current FL model parameters to minimize the gradient deviation caused by AirComp in the considered MIMO system with digital modulation. However, the PS does not know the gradient vector of each edge device and hence the PS cannot proactively adjust the receive beamforming matrix using traditional optimization algorithms. To tackle this challenge, we propose an ANN-based algorithm that enables the PS to predict the local FL gradient parameters of each device. Based on the predicted local FL model parameters, the PS and edge devices can cooperatively optimize the beamforming matrices to improve the performance of FL. Next, we first mathematically analyze the FL update process in the considered AirComp-based system to capture the relationship between the beamforming matrix design and the FL training loss per iteration. Based on this relationship, we then derive the closed-form solution of optimal A t and B t that depends on the predicted FL models achieved by an ANNbased algorithm. III. SOLUTION FOR PROBLEM (9) To solve (9), we first analyze the convergence of the considered FL so as to find the relationship between digital beamforming matrices A t , B t , and FL training loss in (9). The analytical result shows that the optimization of beamforming matrices A t and B t depends on the FL parameters transmitted by each device. However, the PS does not know these FL parameters since it must determine the beamforming matrices A t and B t before the FL parameter transmission. Therefore, we propose to use neural networks to predict the local FL models of each device and proactively determine the beamforming matrices using these predicted FL parameters. A. Analysis of the Convergence of the Designed FL We first analyze the convergence of the considered FL system. Since the update of the global FL model depends on the instantaneous signal-to-interference-plus-noise ratio (SINR) affected by the digital beamforming matrices A t and B t , we can analyze only the expected convergence rate of FL. To analyze the expected convergence rate of FL, we first assume that a) the loss function F (g) is L−smooth with the Lipschitz constant L > 0, b) F (g) is strongly convex with positive parameter µ, c) F (g) is twice-continuously differentiable, and d) ∇f (g t , x kn , y kn ) 2 ≤ ζ 1 +ζ 2 ∇F (g t ) 2 , as done in [35]. These assumptions can be satisfied by several widely used loss functions such as mean squared error, logistic regression, and cross entropy. Based on these assumptions, next, we first derive the upper bound of the FL training loss at one FL training step. The expected convergence rate of the designed FL algorithm can now be obtained by the following theorem. Theorem 1. Given the optimal global FL model g * , the current global FL model g t , the transmit beamforming matrix A t , and the receive beamforming matrix B t , E (F (g t+1 (A t , B t )) − F (g * )) can be upper bounded as E (F (g t+1 (A t , B t )) − F (g * )) E (F (g t ) − F (g * )) − 1 2L ∇F (g t ) 2 + 1 2L E ( e t + ê t (A t , B t ) ) 2 ,(10) where e t = K k=1 n∈N k,t ∇f (g t , x n,k , y n,k ) K k=1 |N k,t | − l −1       K k=1 l n∈N k,t ∇f (g t , x n,k , y n,k ) K k=1 |N k,t |      (11) with the first term being the gradient trained by SGD and the second term being the gradient demodulated from a sum of all selected devices' symbols (i.e., K k=1 l   n∈N k,t ∇f (gt,xn,k,yn,k)   K k=1 |N k,t | ) , and e t (A t , B t ) =l −1       K k=1 l n∈N k,t ∇f (g t , x n,k , y n,k ) K k=1 |N k,t |       − l −1       B t K k=1 H k A k,t l n∈N k,t ∇f (g t , x n,k , y n,k ) + n t K k=1 |N k,t |       .(12) Proof: See Appendix A. From Theorem 1, we see that, since e t does not depend on A t or B t , the optimization of the digital beamforming matrices cannot minimize e t . In consequence, we can only minimize ê t to decrease the gap between the FL training loss at iteration t + 1 and the optimal FL training loss (i.e., E (F (g t+1 ) − F (g * ))). Thus, problem (9) can be rewritten as min Bt,At ê t 2 (13) s.t. |A k,t | 2 ≤ P 0 , ∀k ∈ K, ∀t ∈ T . (13a) To minimize ê t in (13), the PS and edge devices must obtain the information of MIMO channel vector H k as well as the trained gradients l n∈N k,t ∇f (g t , x n,k , y n,k ) so as to adjust A t and B t . However, the trained FL gradients ∆w k,t = n∈N k,t ∇f (g t , x n,k , y n,k ) cannot be obtained by the PS before edge devices sending FL model parameters. Hence, the PS must predict ∆w k,t for optimizing A t and B t and minimizing ê t . B. Prediction of the Local FL Models Next, we explain the use of neural networks to predict the local FL model updates of all devices. Since finding a relationship among each device's local FL model updates at different iterations is a regression task and the fully-connected multilayer perceptrons (MLPs) in ANNs are good at such tasks, we propose to use MLPs instead of other neural networks such as recurrent neural networks (RNN) [36]. Next, we first explain the components of the proposed ANN-based algorithm. Then, the details to implement this algorithm for predicting each local FL model update are presented. The proposed MLP-based prediction algorithm consists of three components: a) input, b) a single hidden layer, and c) output, which are defined as follows: • Input: The input of the MLP that is implemented by the PS for predicting device k's local FL model is a vector g t−1 . As we mentioned in (6), all devices are able to connect with the PS so as to provide the input information for the MLP to predict the local FL models for next iteration. • Output: The output of the MLP is a vector ∆ w k,t that represents device k's local FL model update at current iteration t. Based on the predicted ∆ w k,t , the PS can adjust the transmit and receive beamforming matrices proactively to minimize problem (16). • A Single Hidden Layer: The hidden layer of a MLP is used to learn the nonlinear relationships between input vector g t−1 and the output vector ∆ w k,t . The weight matrix that represents the connection strength between the input vector and the neurons in the hidden layer is v in ∈ C D×V where D is the number of neurons in the single hidden layer. Meanwhile, the weight matrix that captures the strengths of the connections between the neurons in the hidden layer and the output vector is v out ∈ C V ×D . Having the components of the MLP, next, we introduce the use of the MLP to predict each device's local FL model update. The states of the neurons in the hidden layer are given by v = σ v in g k,t−1 + b v ,(14) where σ (x) = 2 1+exp(−2x) − 1 and b v ∈ C D×1 is the bias. Then, the output of the MLP can be given by ∆ w k,t = v out v + b o ,(15) where b o ∈ C V ×1 is a vector of bias. To predict each device's local FL model update, the MLP must be trained by an online gradient descent method. However, in the considered model, the PS can only obtain g t that is directly demodulated from the received signal from all devices. Hence, the PS and the devices must exchange information to train the MLP cooperatively. In particular, at each iteration, device k first generates w k,t using its local dataset and g t−1 received from the PS. Then, device k calculates the training loss of MLP and transmits it to the PS. Based on the value of the training loss, the PS and device k can update its MLP synchronously. Since each device only needs to transmit its training loss, the cost for information exchange can be ignored compared with the communicated DNN model weights. C. Optimization of the Beamforming Matrices Having the predicted local FL model updates ∆ w k,t , the PS can optimize the beamforming matrices A t and B t to solve Problem (13). Substituting ∆ w k,t , (6), and (7) into (13), we have min Bt,At l −1      K k=1 l (∆ w k,t ) K k=1 |N k,t |      − l −1      B t K k=1 H k A k,t l (∆ w k,t ) + n t K k=1 |N k,t |      2 (16) s.t. |A k,t | 2 ≤ P 0 , ∀k ∈ K, ∀t ∈ T .(16a) In (16), l −1   K k=1 l(∆ w k,t) K k=1 |N k,t |   is independent of A t and B t and can be regarded as a constant. However, the existence of the inverse function l −1 (·) defined in (8) significantly increases the complexity for solving (16). Considering l −1 (·) that is used to demodulate the symbols into numerical FL parameters, the minimization of the gap between l −1   K k=1 l(∆ w k,t) K k=1 |N k,t |   and l −1    Bt K k=1 H k A k,t l(∆ w k,t) +nt K k=1 |N k,t |    is equivalent to minimize the distance between K k=1 l(∆ w k,t) K k=1 |N k,t | and Bt K k=1 H k A k,t l(∆ w k,t) +nt K k=1 |N k,t | in the decision region of digital demodulation, as shown in Fig. 2. To this end, in this section, we first derive the position of ( ) , 1 1 , 1 | | K k t k K k t k l w l − = =                ( ) , , 1 1 , 1 | | K t k k t k t t k K k t k B H A l w n l = − =      +                 Quadrature In-phase * t a 2  ( , ) t t t e A B1 K k=1 |N k,t | K k=1 ∆ŵ I k,t,i − a I i ξ 2 , 1 K k=1 |N k,t | K k=1 ∆ŵ Q k,t,i − a Q i ξ 2 ,(17) where 2 is the minimum Euclidean distance between two constellation points. Using (17), a I i and a Q i are given by 14 and a I i , a Q i ∈ M = 1− √ M 2 ξ, 3− √ M 2 ξ, . . . ,a I t,i =          x ∈ M : − ξ 2 + K k=1 ∆ŵ I k,t,i K k=1 |N k,t | x ξ 2 + K k=1 ∆ŵ I k,t,i K k=1 |N k,t | ∩ M         (18)a Q t,i =          x ∈ M : − ξ 2 + K k=1 ∆ŵ Q k,t,i K k=1 |N k,t | x ξ 2 + K k=1 ∆ŵ Q k,t,i K k=1 |N k,t | ∩ M          .(19) Given a * t = [a I t,1 a Q t,1 , . . . , a I t,W a Q t,W ], problem (16) can be rewritten as min Bt,At a * t − B t H K k=1 H k A k,t ∆ŵ k,t − B t H n t 2 (20) s.t. A k,t 2 ≤ P 0 , ∀k ∈ K, ∀t ∈ T .(20a) where ∆ŵ k,t = l (∆ w k,t ) is a modulated symbol vector of ∆ w k,t . Problem (20) can be solved by an iterative optimization algorithm. In particular, to solve problem (20), we first fix B t , then the objective functions and constraints with respect to A t are convex and can be optimally solved by using a dual method [37]. Similarly, given A t , problem (20) is minimized as B * t =   a * t K k=1 H k A * k,t ∆ŵ k,t   H . D. Implementation and Complexity Next, we introduce the implementation and complexity of the designed FL algorithm. With regards to the implementation of the proposed algorithm, the PS must a) use MLPs to predict the devices' local FL models and b) design the transmit and receive beamforming matrices based on the predicted models. To train the MLPs that are used for the predictions of devices' local FL models, the PS will use the global FL model g t−1 that is directly reconstructed from the received symbol vectorŝ t−1 at iteration t − 1. Sinceŝ t−1 that is transmitted from all devices via AirComp is originally used for the update of the global FL model, the PS does not require any additional information for training MLPs. To design the optimal transmit and receive beamforming matrices, the PS requires the maximal transmit power P 0 and the MIMO channel vector H k of each device k. Since P 0 is a fixed scalar, the data size of which is much smaller than the data size of the local FL models that the devices must transmit to the PS during each iteration. In consequence, we can ignore the overhead of each device transmitting P 0 to the PS. With regards to H k , the PS can use channel estimation methods to learn H k over each uplink channel so as to design optimal transmit and receive beamforming matrices. Algorithm 1 Proposed FL Over AirComp-based System 1: Init: Global FL model g 0 , beamforming metrics A 0 and B 0 , MIMO channel matrix H. 2: for iterations t = 0, 1, · · · , T do 3: for k ∈ {1, 2, · · · , K} in parallel over K devices do 4: Each device calculates and returns w k,t based on local dataset and g t in (2). 5: Each device leverages digital pre-processing to modulate each model parameter into a symbol. 6: Each device sends the symbol vectorŵ n,k to the PS using the optimized transmit beamforming matrix A k,t . The PS directly demodulates the global FL model g t+1 from the received superpositioned signal using (8). 9: The PS predicts the local FL modelŵ k,t+1 of each device based on demodulated g t+1 using trained ANNs. 10: The PS proactively adjusts the transmit and receive beamforming matrices using the augmented Lagrangian method and broadcast the transmit beamforming matrix A k,t+1 to each device k. 11: end for The complexity of the proposed algorithm lies in the design of the beamforming matrices. We ignore the overhead for training MLPs since we only need to train MLP once for the entire FL training process. Meanwhile, the PS has enough computational resource for training MLPs [38]. Hence, we only analyze the complexity of a dual method that is used to optimize A t and B t which has to be done for each iteration and thus the complexity scales lies in the number of iterations required to converge. For finding optimal A t and B t , problem (20) IV. SIMULATION RESULTS AND ANALYSIS We consider a circular network area having a radius r = 1500 m with one PS at its center serving K = 20 uniformly distributed devices. In particular, the PS allocates 56 subcarriers to all devices and the bandwidth of each subcarrier is 15 kHz. The channels between the PS and devices are modeled as the independent and identically distributed Rayleigh fading channels. The other parameters used in simulations are listed in Table II. All statistical results are averaged over 5,000 independent runs. For comparison purposes, we consider three baselines: a) the proposed FL algorithm implemented over noiseless wireless channels, b) an FL algorithm that uses digital beamforming and analog modulation for FL parameter transmission [25], and c) an FL algorithm that uses digital beamforming and BPSK for FL parameter transmission [30]. To evaluate the performance of the proposed FL, MNIST dataset [39] and Fashion-MNIST dataset [40] are used. In particular, for MNIST dataset, we adopt a fully-connected neural network (FNN) that consists of two full-connection layers with 7840 (=28×28×10) model parameters. And for fashion-MNIST dataset, we adopt a FNN that consists of four full-connection layers with 83900 (= 28×28×100 + 100×50 + 50×10) model parameters. Each device collects 2000 data samples for training the adopted FNNs and the PS uses one MLP that consists of three layers to predict the FL gradient vector of each device. We assume that all local datasets are independent and identically distributed across the devices. All FL algorithms are considered to have converged when the value of the FL loss variance calculated over 20 consecutive iterations is less than 0.001. converges much faster compared to analog FL and BPSK FL. In particular, the proposed method can improve FL convergence speed by up to 75% and 85% compared to analog FL and BPSK FL. A. Convergence Speed Comparisons The 75% gain stems from the fact that, the proposed FL uses digital modulation (i.e., 64 QAM) which can combat channel impairments and misalignments thus reducing the errors incurred by model transmission. The 85% gain stems from the fact that the proposed algorithm uses high-order quantization scheme instead of using one bit to represent each FL parameter so as to reduce quantization errors. Fig. 3 also shows that the convergence speed and the identification accuracy of the proposed AirComp method are very close to the proposed FL over noiseless channels, which illustrates that our proposed method can use digital modulation to significantly reduce transmission errors caused by channel fading and noise. From Fig. 3, we can also see that without using MLP for FL gradient predictions, the proposed method cannot converge. This is because the PS cannot adjust beamforming matrices without knowing the gradient vectors of each device thus introducing demodulation errors. In Fig. 4, we show how the identification accuracy of all considered algorithms changes as the number of iterations varies for Fashion-MNIST. In this figure, we can see that, the proposed algorithm improves the identification accuracy by up to 10.5% and 20.5% compared to analog FL and BPSK FL, respectively. From Fig. 4, we can also see that analog FL converges to a bad model. This is because the noise over wireless channels introduces dynamic errors into FL parameter transmission process thus affecting FL identification accuracy. Fig. 4 also shows that the proposed method without using MLP for predicting FL gradients cannot converge, which indicates the optimal beamforming matrices design depends on the prediction of FL gradients. slowly but can improve identification accuracy by up to 17% compared to BPSK FL. This is because the proposed FL uses more bits instead of one bit in BPSK FL to represent each FL parameter thus increasing the dynamics of global FL model generation. In Fig. 6, we show how the identification accuracy of the proposed AirComp methods changes as the modulation order M varies. In this figure, we can see that as M increases, the identification accuracy of the considered algorithms increases. This is due to the fact that, as M increases, each device can use more bits to represent one FL parameter thus reducing quantization errors. However, as M continues to increase, the identification accuracy of the proposed algorithm remains a constant. This is because as M is larger than 64, quantization errors are minimized. Fig. 7 also shows that as the modulation order M increases from 64 to 256, the distribution of the FL parameters are unchanged. This is because using a modulation order M that is larger than 64 cannot further decrease quantization errors and improve identification accuracy of the trained model. B. Impacts of SNR In Fig. 8, we show how the identification accuracy changes as SNR decreases. From this figure, we can see that, the identification accuracy decreases as SNR decreases (equivalently noise power increases). This is because as SNR decreases, the probability of incurring transmission error increases, which results in additional errors in FL model training and decreases the FL identification accuracy. Fig. 8 decreases. In Fig. 9, we can see that the identification accuracy of the proposed method decreases as SNR decreases while the identification accuracy of BPSK FL remains unchanged as SNR is less than 10 dB. However, the identification accuracy of BPSK FL is lower than the proposed method at any SNR values. This is because the quantization error in BPSK significantly affects the model training process and results in a degeneration of identification accuracy. Fig. 9 also shows that the proposed method with 4 receiver antennas can achieve 8% gains in terms of identification accuracy compared to that with 2 receiver antennas when SNR is 10 dB. This implies that an increase of the number of receiver antennas can improve the identification accuracy in the proposed FL framework. This is because an increase of the number of receiver antennas enables the PS to exploit transmit diversity and reduce transmission error in the AirComp-based system. From Fig. 9, we can also see that as SNR is 5 dB, the identification accuracy of the proposed algorithm and BPSK FL decreases to 0.1. In Fig. 10, we show how the number of iterations that the considered FL algorithms require to converge changes as SNR decreases. In this figure, we can see that the number of iterations required to converge for all considered algorithms increases as SNR decreases. This is due to the fact that, as SNR decreases, the probability of introducing additional transmission errors increases thus reducing the FL convergence speed. all considered algorithms with different number of receiver antennas. The error is defined as the sum of the distances between all weights in the aggregated model and that in the perfect model g * t . In Fig. 11, we can see that the proposed method achieves a lower error rate compared to the proposed FL without using MLP for FL gradient prediction. This is because without predicting FL gradient vectors, the PS cannot proactively adjust the transmit and receive beamforming matrices to minimize transmission errors and can only use fixed beamforming design that directly aggregates all local models via linear superimposition. This linear superimposition is not available for digital modulation schemes since digital modulation may introduce complex mapping relationships between bits and symbols thus resulting in additional demodulation errors. C. Impacts of Network Size In Fig. 12, we show how the identification accuracy changes as the number of devices varies. From this figure, we can see that, as the number of devices increases, the identification accuracy increases. This is because, as the number of devices that participate in FL training at current iteration increases, the gap between the aggregated model and the perfect model g * t at each iteration decreases, thus resulting in a better identification accuracy. Fig. 12 also shows that the proposed method can achieve 3.5% and 7.5% gain in terms of the identification accuracy compared to analog FL and BPSK FL, respectively. This is because the proposed method 10 15 20 30 Number of devices enables the devices to transmit their model parameters using high-order modulation schemes, thus reducing quantization and transmission errors. Fig. 13 shows how the identification accuracy changes as the number of devices varies. In Fig. 13, we can see that the identification accuracy increases as the number of devices increases. This is because as the number of devices increases, the number of data samples used for training increases, thus resulting in an increase of identification accuracy. From Fig. 13, we see that the proposed method can improve 25% identification accuracy compared to baseline BPSK FL. This is because the proposed FL framework enables the PS and the devices to utilize highorder digital modulation to reduce quantization errors. Fig. 13 also shows that as the number of receiver antennas increases, the identification accuracy of the proposed FL remains unchanged. This implies that an increase of the number of receiver antennas may not affect the gain for identification accuracy under ideal channel conditions. In Fig. 14, we show how the number of iterations required to converge varies as the number of devices changes. From this figure, we can see that, as the number of devices increases, the number of iterations needed to converge decreases. This is because as the number of devices increases, the number of data samples used for training at each FL iteration increases. Fig. 14 also shows that the proposed method can achieve the same performance in terms of convergence speed compared to the proposed FL over noiseless channels, which illustrates that our proposed method can approach perfect model aggregation when considering fading and additive white Gaussian noise. V. CONCLUSION In this article, we have developed a novel framework that enables the implementation of FL algorithms over a digital MIMO and AirComp based system. We have formulated an optimization problem that jointly considers transmit and receive beamforming matrices for the minimization of FL training loss. To solve this problem, we have analyzed the expected improvement of FL training loss between two adjacent iterations that depends on the digital modulation mode, the number of devices, and the design of beamforming matrices. To find the tightest bound, we introduced an ANN based algorithm to estimate the local FL models of all devices and then, the optimal solution of beamforming matrices is determined based on the predicted FL model and the derived expected improvement of FL training loss. Numerical evaluation on real-world machine learning tasks demonstrated that the proposed methodology yields significant gains in classification accuracy and convergence speed compared to conventional approaches. VI. APPENDIX A. Proof of Theorem 1 To prove Theorem 1, we first rewrite F (g t+1 ) using the second-order Taylor expansion and the property of the L-smooth in Assumption a), which can be expressed as F (g t+1 ) ≤ F (g t ) + (g t+1 − g t ) ∇F (g t ) + L 2 g t+1 − g t 2 . Let g t+1 − g t = ∇F (g t ) − o t and the learning rate λ = 1 L , we have E (F (g t+1 )) − E (F (g t )) ≤ −λE (∇F (g t ) − o t ) ∇F (g t ) + Lλ 2 2 E ∇F (g t ) − o t 2 (a) = − 1 2L E ∇F (g t ) 2 + 1 2L E o t 2 , where (a) stems from the fact that Lλ 2 2 ∇F (g t ) − o t 2 = 1 2L ∇F (g t ) 2 − 1 L o T ∇F (g t ) + 1 2L E o t 2 with o t being a gradient deviation caused by the errors in local FL model transmission, which can be given as follows E o t 2 =E ∇F (g t ) − (g t+1 − g t ) 2 =E       K k=1 N k,t n=1 ∇f (g t , x n,k , y n,k ) N − l −1  ŝ t   n∈N k,t ∇f (g t , x n,k , y n,k )     2       E       K k=1 N k,t n=1 ∇f (g t , x n,k , y n,k ) N − −l −1       K k=1 l n∈N k,t ∇f (g t , x n,k , y n,k ) K k=1 |N k,t |       + l −1       K k=1 l n∈N k,t ∇f (g t , x n,k , y n,k ) K k=1 |N k,t |       − l −1  ŝ t   n∈N k,t ∇f (g t , x n,k , y n,k )     2        , where ∇f (g t , x n,k , y n,k ) is the gradient trained by (x n,k , y n,k ).ŝ t n∈N k,t ∇f (g t , x n,k , y n,k ) = is the theoretical signal that is obtained via modulation at devices and demodulation at the PS without channel impairments and misalignments. Fig. 1 . 1The structure of a FL algorithm deployed over multiple devices and one PS in a MIMO communication system. = [A 1 , . . . , A T ] and B = [B 1 , . . . , B T ] are the transmit and receive beamforming matrices for all iterations, respectively. T is a constant which is large enough to guarantee the convergence of FL. k,t | in the decision region and remove l −1 (·) from (16) for simplification. Then, we present a closed-form optimal design of the transmit and receive beamforming matrices. Fig. 2 . 2An example of 16-QAM constellation at the PS with 4 devices.Given ∆ w k,t and the digital pre-processing function l(·) defined in (4), the modulated symbol vector ∆ŵ k,t = l (∆ w k,t ) = [∆ŵ I k,t,1 ∆ŵ Q k,t,1 , . . . , ∆ŵ I k,t,L ∆ŵ Q k,t,L ] can be obtained where ∆ŵ I k,t,i and ∆ŵ Q k,t,i are the i-th in-phase and quadrature symbols modulated by ∆ w k,t , respectively. Since in-phase and quadrature-phase symbols that have vertical and horizontal decision regions are mutually independent, the value of l be obtained via individually analyzing the decision region of each in-phase and quadrature-phase symbols which are ξ are the constellation points in the decision region with M being the set of all constellation points. Fig. 3 3shows how the identification accuracy of all considered algorithms changes as the number of iterations varies.From Fig. 3, we can see that the proposed AirComp methodFig. 3. Identification accuracy vs. number of iterations on MNIST dataset. Fig. 4 . 4Identification accuracy vs. number of iterations on Fashion-MNIST dataset. Fig. 5 . 5Identification accuracy vs. convergence time on Fashion-MNIST dataset. Fig. 5 Fig. 6 . 56shows how the identification accuracy changes as the convergence time varies. Here, the convergence time consists of the model training time that of each device updating its FNN model and the model transmission time. In this figure, we can see that, the proposed FL converges Identification accuracy vs. number of iterations on MNIST dataset. Fig. 7 Fig. 7 . 77shows the distribution of the trained FL parameters obtained by the proposed methods as modulation order changes. In this figure, we use different colors to represent the percentage of the trained parameters being a certain value. In particular, as the percentage of the value in trained parameters increases, the color of that value changes from blue to pink. For example, the percentage of the pink block (i.e., +1.5 in BPSK) is 0.7 while the percentage of the cyan block (i.e., -1.5 in BPSK) is 0.3. From Fig. 7, we can see that in BPSK FL, the values of 70% FL parameters in FL model are +1.5 and the values of 30% FL parameters are -1.5. This is because in BPSK FL, each FL weight is represented by 1 bit. Hence, each weight has only two possible values (i.e., +1.5/-1.5). From Fig. 7, we can also see that in the proposed method with Distributions of weights for different modulation orders on MNIST dataset. 64 QAM, the values of 30%, 30%, and 20% FL parameters in FL model are 0, -0.375, and +0.375. This is because each FL parameter in the proposed FL with 64 QAM is represented by 6 bits and hence, each FL parameter has 64 possible values which can better approach the full precision FL parameters compared to BPSK FL. Fig. 9 9shows how the identification accuracy of considered FL algorithms changes as over noiseless channels BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4Fig. 10. Number of iterations required to converge vs. SNR on Fashion-MNIST dataset. Fig . 11 shows the cumulative distribution function (CDF) curves of the value of the errors for over noiseless channels BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4Fig. 14. Number of iterations required to converge vs. number of devices on Fashion-MNIST dataset. TABLE I ILIST OF NOTATIONS Number of training data samples on device k (x k,n , y k,n ) Training data sample n on device kNotation Description Notation Description K Number of devices M Adopted modulation order N r Number of antennas on the PS N t Number of antennas on devices N k N Number of training data samples of all devices g t Global FL model w k,t Local FL model ∆w k,t Updates of w k,t w k,t Modulated symbol vector of w k,t ∆ w k,t Prediction of ∆w k,t ∆ŵ k,t Modulated symbol vector of ∆ w k,t n t Additive white Gaussian noise A k,t Transmit beamforming matrix B t Receive beamforming matrix H k Channel vector between device k and the PS P 0 Maximal transmit power on device ξ Minimum Euclidean distance in decision region a I i , a Q i Constellation point of symbol i a * Vector of predicted constellation point M Set of all constellation points can be solved by a traditional augmented Lagrangian method that approaches the optimal solution via alternating updating A t , B t , and the Lagrangian multiplier vector. Obviously, the introduced Lagrangian multiplier vector consists of K constraints where K is the number of devices in the considered FL framework. Hence, the PS is required to sequentially update K Lagrangian multipliers, A t = [A 1,t , · · · , A K,t ], and B t at each iteration. Let L O be the number of iterations until the traditional augmented Lagrangian method converges, the complexity is O(L O K 2 ). TABLE II SIMULATION PARAMETERS IIPARAMETERSParameters Values Parameters Values Parameters Values K 20 M 64 nt -50 dBW Nr 2 Nt 2 N k 2000 P0 0.001 W T 50 W 7840 NI 28 NO 10 λ 0.01 r 1500 D 128 L0 100 also shows that the proposed AirComp method improves the identification accuracy by up to 15% compared to analog FL when SNR is 15 dB. This is due tothe fact that the proposed method can reduce transmission errors introduced by wireless channel noise via digital demodulation. Meanwhile, compared to BPSK FL, the proposed method can achieve up to 10% gain in terms of identification accuracy when SNR is 25 dB. This is because BPSK FL uses one bit to represent each FL parameter thus introducing quantization errors and 45 35 25 15 5 SNR 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Identification accuracy Proposed FL over noiseless channels Analog FL BPSK FL Proposed method with 64 QAM Fig. 8. Identification accuracy vs. SNR on MNIST dataset. 25 20 15 10 5 SNR 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Identification accuracy Proposed FL over noiseless channels BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4 Fig. 9. Identification accuracy vs. SNR on Fashion-MNIST dataset. decreasing FL performance in terms of identification accuracy. From Fig. 8, we can also see that the identification accuracy of BPSK FL remains a constant as SNR decreases to 15 dB. This is because the decision threshold of BPSK in BPSK FL is larger than that of 64 QAM in the proposed FL. However, as SNR continues to decrease, the identification accuracy of BPSK FL decreases and achieves a 15% accuracy gap compared to the proposed method. Probability Proposed method without MLP BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4 Proposed method without MLP BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4Fig. 11. Cumulative distribution function of value of error on Fashion-MNIST dataset.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.505 0.51 0.515 0.5 1 1.5 2 2.5 10 -4 Fig. 12. Identification accuracy vs. number of devices on MNIST dataset. Identification accuracy Proposed FL over noiseless channels BPSK FL with Nr=2 BPSK FL with Nr=4 Proposed method with 64 QAM and Nr=2 Proposed method with 64 QAM and Nr=4Fig. 13. 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{'abstract': 'In this paper, the performance optimization of federated learning (FL), when deployed over a realistic wireless multiple-input multiple-output (MIMO) communication system with digital modulation and over-the-air computation (AirComp) is studied. In particular, an MIMO system is considered in which edge devices transmit their local FL models (trained using their locally collected data) to a parameter server (PS) using beamforming to maximize the number of devices scheduled for transmission. The PS, acting as a central controller, generates a global FL model using the received local FL models and broadcasts it back to all devices. Due to the limited bandwidth in a wireless network, AirComp is adopted to enable efficient wireless data aggregation. However, fading of wireless channels can produce aggregate distortions in an AirComp-based FL scheme. To tackle this challenge, we propose a modified federated averaging (FedAvg) algorithm that combines digital modulation with AirComp to mitigate wireless fading while ensuring the communication efficiency. This is achieved by a joint transmit and receive beamforming design, which is formulated as a optimization problem to dynamically adjust the beamforming matrices based on current FL model parameters so as to minimize the transmitting error and ensure the FL performance. To achieve this goal, we first analytically characterize how the beamforming matrices affect the performance of the FedAvg in different iterations. Based on this relationship, an artificial neural network (ANN) is used to estimate the local FL models of all devices and adjust the beamforming matrices at the PS for future model transmission. The algorithmic advantages and improved performance of the proposed methodologies are demonstrated through extensive numerical experiments. most of these existing works, such as [19] and [20], focused on the use of AirComp for analog modulation due to its simplicity for FL convergence analysis, which may not be desirable for practical wireless communication systems that almost exclusively use digital modulations. In consequence, it is necessary to study the implementation of AirComp-based FL over digital modulation-based wireless systems. A. Related Works Recent works such as [21]-[27] have studied several important problems related to the implementation of AirComp-based FL over wireless networks. The authors in [21] minimized the mean-squared error (MSE) of the FL model during AirComp transmission under transmit power constraints in a multiuser multiple-input multiple-output (MIMO) system. In [22], the 3 authors maximized the number of devices that can participate in FL training under certain MSE requirements in an AirComp-based MIMO framework. A joint machine learning rate and receiver beamforming matrix optimization method was proposed in [23] to reduce the aggregate distortion and satisfy an FL performance requirement. The authors in [24] investigated the deployment of FL over an AirComp-based wireless network to minimize the energy consumption of edge devices. In [25], the authors optimized the set of participating devices in an AirComp-assisted FL framework to speed up FL convergence. A receive beamforming scheme was designed in [26] to optimize FL performance without knowing channel state information. The authors in [27] minimized the FL model aggregation error under a channel alignment constraint in an MIMO system. However, most of these existing works [21]-[27] investigated the implementation of AirComp-based FL over analog modulation-based wireless systems, which may not be directly applied for practical digital modulation based wireless systems since these works do not consider coding and digital modulation. Recently, several works [28]-[33] have studied the implementation of AirComp FL over digital modulation based wireless systems. The authors in [28] designed one-bit quantization and modulation schemes for edge devices. One-bit gradient quantization scheme is proposed in [29] to achieve fast FL model aggregation. In [30], the authors designed a joint channel decoding and aggregation decoding schemes based on binary phase shift keying (BPSK) modulation for AirComp FL. The authors in [31] evaluated the performance of FL gradient quantization in digital AirComp. In [32], the convergence of FL implemented over an AirComp-based MIMO system is derived. The authors in [33] proposed a digital transmission protocol tailored to FL over wireless device-to-device networks. However, these prior works [28]-[33] mainly used low order digital modulation (i.e., BPSK) and hence their designed AirComp FL cannot be easily extendedto modern wireless systems that use high-order digital modulation schemes such as quadrature amplitude modulation (QAM). This is because the transmitted symbols that are processed by low order digital modulation (such as the symbols -1 and +1 in BPSK) are linearly superimposed. This linear superimposition does not exist in the high-order digital modulation schemes with complex mapping relationships between bits and symbols (such as Gray code).B. ContributionsThe main contribution of this paper is to develop a novel AirComp FL framework over highorder digital modulation-based wireless systems. Our key contributions include:', 'arxivid': '2302.14648', 'author': ['Student Member, IEEESihua Wang \nDepartment of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n', 'Member, IEEEMingzhe Chen mingzhec@princeton.edu. \nDepartment of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n', 'Senior Member, IEEE, Changchuan Yin, Senior Member, IEEECong Shen \nDepartment of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n', 'Senior Member, IEEEChristopher G Brinton \nDepartment of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n', 'S Wang \nBrown Department of Electrical and Computer Engineering\nPrinceton University\n08544PrincetonNJUSA\n', 'C Yin ccyin@ieee.org. \nBrown Department of Electrical and Computer Engineering\nPrinceton University\n08544PrincetonNJUSA\n', 'M Chen \nSchool of Electrical and Computer Engineering\nUniversity of Virginia\nCharlottesvilleVAUSA\n', 'Christopher G Brinton \nPurdue University\nWest LafayetteINUSA\n'], 'authoraffiliation': ['Department of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina', 'Department of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina', 'Department of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina', 'Department of Electrical and Computer Engineering\nare with the Beijing Laboratory of Advanced Information Network, and the Beijing Key Laboratory of Network System Architecture and Convergence\nBeijing University of Posts and Telecommunications\n100876BeijingChina', 'Brown Department of Electrical and Computer Engineering\nPrinceton University\n08544PrincetonNJUSA', 'Brown Department of Electrical and Computer Engineering\nPrinceton University\n08544PrincetonNJUSA', 'School of Electrical and Computer Engineering\nUniversity of Virginia\nCharlottesvilleVAUSA', 'Purdue University\nWest LafayetteINUSA'], 'corpusid': 257232908, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20080, 'n_tokens_neox': 17751, 'n_words': 11168, 'pdfsha': '928fc5c0145f781cc98ada3aac0ddd12adbd0bbe', 'pdfurls': ['https://export.arxiv.org/pdf/2302.14648v1.pdf'], 'title': ['Cross-Layer Federated Learning Optimization in MIMO Networks', 'Cross-Layer Federated Learning Optimization in MIMO Networks'], 'venue': []}

arxiv

FOURIER ANALYSIS OF BIOLOGICAL EVOLUTION: CONCEPT OF SELECTION MOMENT Ashok Palaniappan FOURIER ANALYSIS OF BIOLOGICAL EVOLUTION: CONCEPT OF SELECTION MOMENT Secondary structure elements of many protein families exhibit differential conservation on their opposing faces. Amphipathic helices and -sheets by definition possess this property, and play crucial functional roles. This type of evolutionary trajectory of a protein family is usually critical to the functions of the protein family, as well as in creating functions within subfamilies. That is, differential conservation maintains properties of a protein structure related to its orientation, and that are important in packing, recognition, and catalysis. Here I define and formulate a new concept, called the selection moment, that detects this evolutionary process in protein sequences. I present a detailed account of its possible applications.A multiple alignment of the members of the protein family of interest is created.Conservation at each position of the multiple alignment is calculated in one of two ways: 1) by rigorously taking into account the phylogeny of the protein family, and rate variation among sites; 2) by using the Shannon entropy measure: ( ) ln ij ij j H i p p = − ∑ ,(1) where H(i) is the entropy at position i, p ij is the fractional occurrence of residue type j at position i, and the sum is taken over all 20 residues. The Shannon entropy is a simplified measure of conservation in this context. In this way we convert the multiple alignment into a one-dimensional function of a numerical conservation score. Now the presence of a periodicity in this one-dimensional function implies a moment in the pattern of substitution in this protein family. In particular, if a peak of periodicity of this function coincides with the period of an -helix, then there exists a strong periodicity of selection at 3.6 residues. Selection moment is then defined as the modulus of the Fourier transform of the conservation function, and the selection moment for a given periodicity is given by: 1 ( ) ( ) N i n n S H n e θ θ = = ∑ ,(2) N is the length of the sequence, and radian measure is used. 100 reflects a selection moment in the period of the --180 moment in the period of the strand/sheet. We record the periodicity for which the selection moment achieves a maximum. If we know a structure for the protein family, we could compute the selection moments at periodicities matched with the protein structure. If no representative structure is available, the selection moment is computed for all blocks of contiguous residues of a preassigned length (say 24 residues, which is the average length of a transmembrane -helix), and for each block, we derive a profile of the selection moment as a function of . The maxima of this profile are then analyzed in tandem with predictions of the protein secondary structure. To ascertain the correlation between selection moment and secondary structure, we derive selection moment profiles for different types of secondary structure elements, and note where the majority of peaks fall. We would expect the maximum selection moment to coincide with the period of the repetitive element. As an extension of the utility of the selection moment concept, we define the selection moment plot, which shows the mean selection moment as a function of mean conservation. Mean conservation is the average of the sum of the numerical conservation scores of the segment (identical to the value of the selection moment profile for an infinite period), and the mean selection moment is the corresponding ratio of the selection moment to the length of the segment. A high mean conservation and a low mean selection moment would be diagnostic of an interfacial structure that has a functional constraint to preserve both its lipid-facing and solventfacing sides. Similar insights may be drawn with respect to the location of other types of structures in this plot. Instead of the directional differential conservation described above, we may also expect to find a sequential differential conservation. The quantity of interest is then the difference in average conservation between the two halves, and larger differences point to a significance for the sequential differential conservation. It is predicted that the selection moment will be an important driving force in protein evolution, whereby it achieves a maximum trade-off between selection pressure and random variation. It is plausible that the evolution of most sequences tends to maximize their selection moment. Functional constraint is the deciding factor in the evolution of periodicity of mutability. However, unlike physical properties (like hydrophobicity) that can be studied for a single sequence, oscillations in mutability are limited by contemporaneity; it is clear they cannot be observed for a single sequence. Therefore our conclusions regard protein families. The information derived from selection moments would be indispensable in critically evaluating the oscillations in physical properties themselves. For instance, if a high protein hydrophobic moment coincided with a high family selection moment, then the functional importance of the particular repetitive structure is reinforced. The permeation domain of an ion channel is a classic example of this: transmembrane -helices flanking the pore of a K + -channel evolved large conservation moments. In particular, the part of the inner helix facing the central pore is very conserved, but the part involved in packing with the outer helix is mainly variable (Palaniappan 2005). This emerges as a design feature in the architecture of ion channels. It is clear that selection pressure operates at the level of secondary structure to conserve the helical property of orientation of physical property. Selection moments might also play a crucial role in the specificity of oligomerisation of multimeric proteins. The methodology described may be generalized with little modification to understand the directional evolutionary force acting on the surface of any given repetitive structure. Selection moment is a true essence concept, and it will be valuable in investigations of evolutionary mechanisms targeting important functions. . 2005. Theory-based investigations of the potassium-ion channel membrane protein family. PhD Dissertation (Illinois): ProQuest, ISBN 0-542-50560-6. pp. 69-71.

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{'abstract': 'Secondary structure elements of many protein families exhibit differential conservation on their opposing faces. Amphipathic helices and -sheets by definition possess this property, and play crucial functional roles. This type of evolutionary trajectory of a protein family is usually critical to the functions of the protein family, as well as in creating functions within subfamilies. That is, differential conservation maintains properties of a protein structure related to its orientation, and that are important in packing, recognition, and catalysis. Here I define and formulate a new concept, called the selection moment, that detects this evolutionary process in protein sequences. I present a detailed account of its possible applications.A multiple alignment of the members of the protein family of interest is created.Conservation at each position of the multiple alignment is calculated in one of two ways:', 'arxivid': '0704.2964', 'author': ['Ashok Palaniappan '], 'authoraffiliation': [], 'corpusid': 42482715, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 1395, 'n_tokens_neox': 1256, 'n_words': 992, 'pdfsha': 'cd426b2c2d74dfde3c9dbca27d678feb236e9e11', 'pdfurls': ['https://arxiv.org/pdf/0704.2964v1.pdf'], 'title': ['FOURIER ANALYSIS OF BIOLOGICAL EVOLUTION: CONCEPT OF SELECTION MOMENT', 'FOURIER ANALYSIS OF BIOLOGICAL EVOLUTION: CONCEPT OF SELECTION MOMENT'], 'venue': []}

arxiv

A sparse version of Reznick's Positivstellensatz 13 Feb 2020 February 14, 2020 Ngoc Hoang Anh Mai Victor Magron Jean-Bernard Lasserre A sparse version of Reznick's Positivstellensatz 13 Feb 2020 February 14, 2020Reznick's Positivstellensatzsparsity patternpositive definite formsrun- ning intersection propertysums of squaresPutinar-Vasilescu's Positivstellensatzuni- form denominatorsbasic semialgebraic set If f is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift. 220 (1995), pp. 75-97] states that there exists k ∈ N such that x 2k 2 f is a sum of squares of polynomials. Assuming that f can be written as a sum of forms p l=1 f l , where each f l depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists k ∈ N such that f = p l=1 σ l /H k l , where σ l is a sum of squares of polynomials, H l is a uniform polynomial denominator, and both polynomials σ l , H l involve the same variables as f l , for each l = 1, . . . , p. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate due to Putinar and Vasilescu. Introduction and overview Before the 1990s, representations of positive polynomials, also known as Positivstellensätze, have been discovered within a purely theoretical branch of real algebraic geometry. More recently, such Positivstellensätze have become a powerful tool in polynomial optimization and control. SOS decompositions of nonnegative polynomials have a distinguishing feature with important practical implications: Indeed they are tractable and can be determined by solving a semidefinite program 1 . Namely, writing a polynomial f ∈ R[x] 2d as an SOS boils down [19] to computing the entries of a symmetric (Gram) matrix G with only nonnegative eigenvalues (denoted by "G 0") such that f = v T d Gv d , with v d being the vector of all monomials of degree at most d. Given f, g1, . . . , gm ∈ R[x], and the basic semialgebraic set S(g) := {x ∈ R n : gj(x) ≥ 0 , j = 1, . . . , m}, with g := {g1, . . . , gm}, polynomial optimization is concerned with computing f ⋆ := inf{f (x) : x ∈ S(g)}. A basic idea is to rather consider f ⋆ = sup{λ ∈ R : f − λ > 0 on S(g)} and replace the difficult constraint "f − λ > 0 on S(g)" with a more tractable SOS-based decomposition of f − λ, thanks to various certificates of positivity on S(g). For instance, if S(g) is compact and satisfies the so-called Archimedean assumption 2 , Putinar's Positivstellensatz [21] provides the decomposition f − λ = σ0 + m j=1 σj gj, with σj ∈ Σ[x]. Then one obtains the monotone non-decreasing sequence (ρ k ) k∈N of lower bounds on f ⋆ defined by: ρ k := sup λ,σ j { λ : f − λ = σ0 + m j=1 σjgj, σj ∈ Σ[x] , deg(σjgj) ≤ 2k}. (1.1) For each fixed k, (1.1) is a semidefinite program and therefore can be solved efficiently. Moreover, by invoking Putinar's Positivstellensatz, one obtains the convergence ρ k ↑ f ⋆ as k increasees. In Table 1 are listed several useful Positivstellensätze that guarantee convergence of similar sequences (ρ k ) k∈N to f ⋆ (where now in (1.1) one uses the appropriate positivity certificate). However their associated so-called dense hierarchies of linear/SDP programs are only suitable for modest size POPs (e.g., n ≤ 10 and deg(f ), deg(gj) ≤ 10). Indeed, for instance, even though (1.1) is a semidefinite program, it involves n+2k n variables and semidefinite matrices of size up to n+k n , a clear limitation for state-of-the-art semidefinite solvers. Therefore a scientific challenge with important computational implications is to develop alternative positivity certificates that scale well in terms of computational complexity, at least in some identified class of problems. Fortunately as we next see, we can provide such alternative positivity certificates for the class of problems where some structured sparsity pattern is present in the problem description (as often the case in large-scale problems). Indeed this sparsity pattern can be exploited to yield a positivity certificate in which the sparsity pattern is reflected, thus with potential significant computational savings. 1 Semidefinite programming (SDP) is an important class of convex conic optimization problems that can be solved efficiently, up to arbitrary precision, fixed in advance; the interested read is referred to e.g. [3,Chapter 4]. 2 There are σ j ∈ Σ[x] such that S({σ 0 + m j=1 σ j g j }) is compact. Author(s) Statement Application(s) Schmüdgen [25] If f is positive on S(g) and S(g) is compact, then f = α∈{0,1} m σα m j=1 g α j j for some σα ∈ Σ[x]. [7] Putinar [21] If a polynomial f is positive on S(g) satisfying Archimedian assumption 2 , then f = σ 0 + m j=1 σ j g j for some σ j ∈ Σ[x]. [12] Reznick [23] If f is a positive definite form, then x 2k 2 f ∈ Σ[x] for some k ∈ N. [1] Polya [20] If f is a homogeneous form and f > 0 on R n + \{0}, then ( j x j ) k f has nonnegative coefficients for some k ∈ N. [5] Krivine-Stengle [11,27] If a polynomial f is positive on S(g), S(g) is compact and g j ≤ 1 on S(g), then f = α,β∈N m c α,β m j=1 (g α j j (1 − g j ) β j ) for some c α,β ≥ 0. [14] Putinar-Vasilescu [22] If a polynomial f is nonnegative on S(g), then for every ε > 0, there exists k ∈ N such that θ k (f + εθ d ) = σ 0 + m j=1 σ j g j for some σ j ∈ Σ[x], where d := 1 + ⌊deg(f )/2⌋ and θ := x 2 2 + 1. [ 16] Exploiting sparsity pattern. i ∈ T }. Also denote by R[x(T )]t (resp. Σ[x(T )]t) the restriction of R[x(T )] (resp. Σ[x(T )]) to polynomials of degree at most t (resp. 2t). For R ⊂ J, we note gR := {gj : j ∈ R}. Designing alternative hierarchies for solving f ⋆ := inf{f (x) : x ∈ S(g)}, significantly (computationally) cheaper than their dense version (1.1), while maintaining convergence to the optimal value f ⋆ is a real challenge with important implications. One first such successful contribution is due to Waki et al. [29] when the input polynomial data f, gj are sparse, where by sparse we mean the following: (ii) Structured sparsity pattern for the objective function 3 : f = p l=1 f l where f l ∈ R[x(I l )] deg(f ) , l = 1, . . . , p. (iii) Structured sparsity pattern for the constraints: J = p l=1 J l and for every j ∈ J l , gj ∈ R[x(I l )], l = 1, . . . , p. (iv) Additional redundant quadratic constraints: There exists L > 0 such that x 2 2 ≤ L for all x ∈ S(g) and L − x(I l ) 2 2 ∈ gJ l , l = 1, . . . , p. With τ (≤ n) being the maximum number of variables appearing in each index subset I l of f, gj, i.e., τ := max{n l : l = 1, . . . , p}, Table 2 displays the respective computational complexity of the sparse hierarchy of Waki et al. [29] and the dense hierarchy of Lasserre [12] for SDPs with same order k ∈ N. Obviously the sparse hierarchy provides a potentially high computational saving when compared to the dense one. In addition, convergence of the hierarchy of Waki et al. to the optimal O(τ 2k ) O(n 2k ) largest size of SDP matrix O(τ k ) O(n k ) value of the original POP was proved in [13], resulting in the following sparse version of Putinar's Positivstellensatz: ∈ Σ[x(I l )] k , σ j,l ∈ Σ[x(I l )] k−u j with uj := ⌈deg(gj)/2⌉, j ∈ J l , l = 1, . . . , p such that f = p l=1   σ 0,l + j∈J l σ j,l gj   . (1.2) Compactness of the feasible set S(g) is a crucial ingredient of the proof in [13]; shortly after, Grimm et al. [6] provided another (simpler) proof where int(S(g)) = ∅ is not needed, but where compactness of S(g) is still a crucial assumption. Motivation for sparse representations on non-compact sets. We remark that Theorem 1.1 requires the additional redundant quadratic constraints (Assumption 1.1 (iv)), which is slightly stronger than just assuming the compactness of S(g). When S(g) is compact, we can always add these constraints but we need to know the radius L > 0 of a ball centered at the origin and containing S(g). In this case, adding such constraints increases the number of positive semidefinite matrices from m to m + p in each SDP. In addition, it may be hard to verify compactness of S(g) and obtain such a radius L. To the best of our knowledge, in the non-compact case there is still no Positivstellensatz allowing one to build hierarchies for POPs satisfying : -the RIP and the structured sparsity pattern from Assumption 1.1 (i)-(iii), -and a guarantee of convergence to the global optimum. In fact we provide examples 2.2, 2.3, and 2.4, which show that in both unconstrained and constrained cases, there exist sparse nonnegative polynomials which do not have a sparse SOS-based decomposition (1.2)à la Putinar. Such examples have been our motivation to investigate existence of sparse representations in the noncompact case, as well as to construct converging SDP-hierarchies for sparse polynomial optimization in general. Dense rational SOS representations and non-compact POPs. In his famous and seminal work [8], Hilbert characterized all cases where nonnegative polynomials are SOS of polynomials. In 1927, Artin proved in [2] that every nonnegative polynomial can be decomposed as an SOS of rational functions (or rational SOS for short), thereby solving Hilbert's 17th problem. Namely, a polynomial f is nonnegative if and only if there exist σ1, σ2 ∈ Σ[x] such that f = σ1/σ2. Of course one can use Hilbert-Artin's representation to obtain a hierarchy of lower bounds for unconstrained POPs: f ⋆ := inf x∈R n f (x), by computing ρ k := sup{λ ∈ R : σ2(f − λ) = σ1, σj ∈ Σ[x] k }, for every k ∈ N, so that ρ k ≤ ρ k+1 ≤ f ⋆ for all k. However for each k the resulting optimization problem is not an SDP (and not even convex) because of the nonlinear term σ2λ. (Even with an iterative dichotomy procedure on λ, one is left with an SDP hierarchy for each fixed λ.) When f is a positive definite form Reznick proposes to select a so-called uniform denominator in the Hilbert-Artin's representation, namely to replace σ2 by some power of x 2 2 (see Table 1). As a result one obtains a decomposition in SOS of rational functions for any arbitrary small perturbation of a nonnegative polynomial f as follows: For every ε > 0, there exists k ∈ N such that θ k (f + εθ d ) = σ0 for some σ0 ∈ Σ[x], with d := ⌈deg(f )/2⌉ and θ := x 2 2 + 1. For abitrary ε > 0 fixed, we obtain an SDP-based hierarchy of bounds ρ k (ε) = sup{λ ∈ R : θ k (f − λ + εθ d ) = σ0 , σ0 ∈ Σ[x] k+d }, for every k ∈ N. If f ⋆ is attained then the sequence (ρ k (ε)) k∈N converges to a value in a neighborhood of f ⋆ . A similar idea, now based on Putinar-Vasilescu's Positivstellensatz [22], can be applied for polynomials nonnegative on non-compact basic semialgebraic sets (see Table 1). This shows that rational SOS representations with fixed forms for denominators are highly useful and applicable in non-compact POPs. Contribution. Our contribution is twofold: • We first provide a rational SOS representation for a positive definite rational form which is a sum of sparse rational functions with uniform denominators, satisfying the structured sparsity pattern and the RIP stated in Assumption 1.1 (i). This representation is provided in Theorem 2.1. As a direct consequence, we obtain a sparse version of Reznick's Positivstellensatz in Corollary 2.1. • Then, we provide two positivity certificates for arbitrary small perturbations of -globally nonnegative polynomials in Corollary 2. • First, we use an idea similar to that developed in Grimm et al. [6] (in the compact case) to prove that a sparse positive definite form can be decomposed as SOS of sparse positive definite rational forms; as expected the non-compact case is rechnically more involved. This yields a sparse version of Hilbert-Artin's representation theorem in the case of positive definite forms. • Next, we use generalizations of Schmüdgen's Positivstellensatz presented by Schweighofer [26], Berr-Wörmann [4], Jacobi [9], and Marshall [17,18], for a finitely generated R-algebra in each term of the sum, to obtain again a sparse version, this time of Reznick's Positivstellensatz for positive definite forms. • Finally we combine the homogenization/dehomogenization method that we already used in [16] together with limit tools, to provide the two sparse versions of Putinar-Vasilescu's Positivstellensatz. Main results For (i, j) ∈ N 2 , we denote the Kronecker delta function by δi,j := 1 if i = j , 0 if i = j . When Assumption 1.1 (i) holds, define: Φ l := x(Î l ) 2(1−δ l,1 ) 2 p j=l+1 x(Îj) 2δ l,s j 2 if l = 1, . . . , p − 1 , x(Î l ) 2(1−δ l,1 ) 2 if l = p . Obviously, one has Φ l ∈ R[x(I l )], for each l = 1, . . . , p. Let us state the first main result of this paper which yields a sparse version of Reznick's PositivStellensatz as a particular case. Theorem 2.1. Let Assumption 1.1 (i) holds. Let f ∈ R(x) be a positive definite rational form of degree 2d with d ∈ N >0 such that f = p l=1 p l x(I l ) 2k l 2 , where p l ∈ R[x(I l )] is homogeneous of degree 2(d + k l ) for some k l ∈ N, l = 1, . . . , p. Then there exist k ∈ N and σ l ∈ Σ[x(I l )] d+k(1+deg(Φ l )/2) , l = 1, . . . , p, such that f = p l=1 σ l x(I l ) 2k 2 Φ k l . (2. 3) The proof of Theorem 2.1 can be found in Section 4.1. As a consequence, we obtain the following sparse version of Reznick's Positivstellensatz: Corollary 2.1. Let Assumption 1.1 (i) holds. Assume that f is a positive definite form of degree 2d with d ∈ N >0 and f = p l=1 f l , where f l ∈ R[x(I l )] is homogeneous of degree 2d, l = 1, . . . , p. Then there exist k ∈ N and σ l ∈ Σ[x(I l )] d+k(1+deg(Φ l )/2) , l = 1, . . . , p, such that f = p l=1 σ l H k l , (2.4) where H l := x(I l ) 2 2 Φ l , l = 1, . . . , p. To prove Corollary 2.1, we apply Theorem 2.1 with k l = 0, l = 1, . . . , p. The representation (2.4) can still hold even when f is not a positive definite form, as illustrated in the following example: Example 2.1. Let f = f1 + f2, where f1 := x 2 4 (x 4 1 x 2 2 + x 4 2 x 2 3 + x 2 1 x 4 3 − 3x 2 1 x 2 2 x 2 3 ) + x 8 3 is the so-called Delzell's polynomial and f2 := x 2 1 x 2 2 x 2 3 x 2 5 . The polynomial f1 is nonnegative, but not SOS as shown in [15,Example 2]. Let I1 := {1, 2, 3, 4} and I2 := {1, 2, 3, 5}. Then f1 ∈ R[x(I1)] and f2 ∈ R[x(I2)] are nonnegative and homogeneous of degree 8. Since f1 is nonnegative then f is nonnegative. The following statements hold: 1. f is a nonnegative form, but is not positive definite; 2. f / ∈ Σ[x(I1)] + Σ[x(I2)], but f ∈ Σ[x(I 1 )] 6 x(I 1 ) 2 2 Φ 1 + Σ[x(I 2 )] 6 x(I 2 ) 2 2 Φ 2 . The first statement follows from the fact that f (0, 0, 0, 1, 1) = 0, ensuring that f is not a positive definite form. Proof of the second statement: Assume by contradiction that f = σ1 + σ2 for some σ l ∈ Σ[x(I l )], l = 1, 2. Evaluation at x5 = 0 yields f1 = σ1+σ2(x1, x2, x3, 0), so that f1 is an SOS, which is impossible. Thus, f / ∈ Σ[x(I1)]+Σ[x(I2)] . However, (x 2 1 +x 2 2 +x 3 3 )f1 is SOS according to [24,Example 4.4], so (x 2 1 + x 2 2 + x 2 3 )f ∈ Σ[x(I1)]5 + Σ[x(I2)]5. Note that Φ1 = Φ2 = x 2 1 + x 2 2 + x 2 3 . Therefore f ∈ Σ[x(I1)]5 Φ1 + Σ[x(I2)]5 Φ2 ⊂ Σ[x(I1)]6 H1 + Σ[x(I2)]6 H2 . When Assumption 1.1 (i) holds, define the following polynomials, for each l = 1, . . . , p: • θ l := x(I l ) 2 2 + 1 andθ l := x(Î l ) 2 2 + 1; • D l := θ 1−δ l,1 l p j=l+1θ δ l,s j j if l < p , θ 1−δ l,1 l if l = p ; • Θ l := θ l D l and ω l := deg(Θ l )/2. Note that Θ l ∈ Σ[x(I l )]ω l , for each l = 1, . . . , p. We next state the following sparse version of Putinar-Vasilescu's Positivstellensatz for polynomials nonnegative on the whole R n . Example 2.2. Let f = f1 + f2, where f1 := 8 + 1 2 x 2 1 x 4 2 + (x 2 1 − 2x 3 1 )x 3 2 + (2x1 + 10x 2 1 + 4x 3 1 + 3x 4 1 )x 2 2 + 4(x1 − 2x 2 1 )x2 is the so-called Leep-Starr's polynomial and f2 := x 2 1 x 2 3 . Let I1 := {1, 2} and I2 := {1, 3}, so that f1 ∈ R[x(I1)] and f2 ∈ R[x(I2)]. As shown in [15,Example 2], f1 is nonnegative but not an SOS. In addition, f is nonnegative. We claim that f / ∈ Σ[x(I1)] + Σ[x(I2)]. Indeed, assume by contradiction that f = σ1+σ2 for some σ l ∈ Σ[x(I l )], l = 1, 2. Evaluation at x3 = 0, yields f1 = σ1+σ2(x2, 0), so that f1 is an SOS, which is impossible. However, (x 2 1 + 1) 2 f1 is a sum of three squares of polynomials according to [15,Example 2], so ( x 2 1 + 1) 2 f ∈ Σ[x(I1)]5 + Σ[x(I2)]5. Note that D1 = D2 = x 2 1 + 1. Thus, f ∈ Σ[x(I1)]5 D 2 1 + Σ[x(I2)]5 D 2 2 ⊂ Σ[x(I1)]7 Θ 2 1 + Σ[x(I2)]7 Θ 2 2 . Example 2.3. As shown in [10,Example 5.2], the nonnegative polynomial f = x 2 1 − 2x1x2 + 3x 2 2 − 2x 2 1 x2 + 2x 2 1 x 2 2 − 2x2x3 + 6x 2 3 + 18x 2 2 x3 − 54x2x 2 3 + 142x 2 2 x 2 3 satisfies f ∈ R[x(I1)] + R[x(I2)] and f / ∈ Σ[x(I1)] + Σ[x(I2)], with I1 = {1, 2} and I2 = {2, 3}. However, f ∈ Σ[x(I 1 )] 4 Θ 1 + Σ[x(I 2 )] 4 Θ 2 , where Θ1 = (x 2 2 + 1)(x 2 1 + x 2 2 + 1) and Θ2 = (x 2 2 + 1)(x 2 2 + x 2 3 + 1). It is due to the fact that f = σ 1 D 1 + σ 2 D 2 , where D1 = D2 = x 2 2 + 1 and σ1 and σ2 are SOS polynomials given in Appendix A. We next state our second main result, namely a sparse version of Putinar-Vasilescu's Positivstellensatz for polynomials nonnegative on (possibly non-compact) basic semialgebraic sets. . We claim the following statements: 1. f / ∈ Σ[x(I1)] + g1R[x(I1)] + Σ[x(I2)] + g3Σ[x(I2)]; 2. for every ε > 0, f +ε(θ 2 1 +θ 2 2 ) ∈ Σ[x(I1)] 2k+2 + g1R[x(I1)] 4k+1 Θ k 1 + Σ[x(I2)] 2k+2 + g3Σ[x(I2)] 4k+3 Θ k 2 , for some k ∈ N depending on ε. Proof of the first statement: Assume by contradiction that there exist σ1 ∈ Σ[x(I1)], ψ1 ∈ R[x(I1)] and σ2, σ3 ∈ Σ[x(I2)] such that f = σ1 + ψ1g1 + σ2 + σ3g3. Evaluation at x1 = 1 and x3 = 0 yields x2 = σ1(1, x2) + ψ1(1, x2)x 3 2 + σ2(x2, 0) ∈ Σ[x2] + x 3 2 R[x2] , which is impossible due to [16, Lemma 3.3 (i)]. Proof of the second statement: With ε > 0 fixed, f1 + εθ 2 1 = x1x2 + ε(1 + x 2 1 + x 2 2 ) 2 = x1x2 + ε + εx 2 1 + σ4 , for some σ4 ∈ Σ[x(I1)]2. Let k ∈ N ≥2 be fixed. Then D k 1 = (1 + x 2 2 ) k = 1 + kx 2 2 + x 4 2 σ5 for some σ5 ∈ Σ[x2] k−2 , which implies D k 1 (f1 + εθ 2 1 ) = x1x2 + εx 2 1 + εkx 2 2 + σ6 + ψ2x 3 2 , for some σ6 ∈ Σ[x(I1)] k+2 and ψ2 ∈ R[x(I1)] 2k+1 . Assume that k ≥ ε −2 /4. Then D k 1 (f1 + εθ 2 1 ) = x 2 1 ε − 1 4εk + x2 √ εk + x 1 2 √ εk 2 + σ6 + ψ2x 3 2 ∈ Σ[x(I1)] k+2 + g1R[x(I1)] 2k+1 , which implies f1 + εθ 2 1 ∈ Σ[x(I 1 )] 2k+2 +g 1 R[x(I 1 )] 4k+1 Θ k 1 . We also have f2 + εθ 2 2 ∈ Σ[x(I2)] 2k+2 + g3Σ[x(I2)] 4k+3 Θ k 2 since f2 ∈ g3Σ[x(I2)]1, proving the second statement. Preliminary material Given α = (α1, . . . , αn) ∈ N n , we note |α| := α1 + · · · + αn and x α := x α 1 1 . . . x αn n . Let (x α ) α∈N n be the canonical basis of monomials for R[x] (ordered according to the graded lexicographic order) and vt(x) be the vector of monomials up to degree t, with length s(t) = n+t n . A polynomial h ∈ R[x]t is written as h(x) = |α|≤t hα x α = h T v d (x), where h = (hα) ∈ R s(t) is its vector of coefficients in the canonical basis. Denote by S n−1 := {x ∈ R n : x 2 = 1} the (n − 1)-dimensional unit sphere. A function h is homogeneous of degree t if h(λx) = λ t h(x) for all x ∈ R n and each λ ∈ R. Therefore a homogeneous polynomial can be written as h = |α|=t hαx α . A function f : R n → R is even if f (x) = f (−x) for all x. A rational function h is the ratio of two polynomials and denote by R(x) the space of all rational functions. A homogeneous rational function (also called be a rational form, or form in short) can be written as the ratio of two homogeneous polynomials. The degree-d homogenizationh of h ∈ R(x1, . . . , xn) is a homogeneous rational function in R(x1, . . . , xn+1) of degree d defined byh(x, xn+1) = x d n+1 h(x/xn+1). A rational positive definite form of degree t is a homogeneous rational function of degree t which is positive everywhere except at the origin. Equivalently, a homogeneous rational function h of degree t is a rational positive definite form of degree t if and only if there exists ε > 0 such that h ≥ ε x 2t 2 . We briefly recall some algebraic tools from generalizations of Schmüdgen's Positivstellensatz [26] which will be used in the sequel. An associative algebra A is called a finitely generated R-algebra if there exists a finite set of elements a1, . . . , an of A such that every element of A can be expressed as a polynomial in a1, . . . , an, with coefficients in R. Let A be a commutative ring. We denote by ΣA 2 the set of all SOS of elements in A. A subset T of A is called a preordering if T contains all squares and is closed under addition and multiplication. The preordering T generated by elements t1, . . . , tm (so-called smallest preordering containing t1, . . . , tm) consists of all elements of the form α∈{0,1} m (σα m j=1 t α j j ), with σα ∈ ΣA 2 . The real spectrum of a ring A with fixed preordering T , denoted by Sper T A, is defined by Sper T A := {ϕ ∈ Hom(A, R) : ϕ(T ) ⊂ R+} , where Hom(A, R) is the set of all ring homomorphisms from A to R. Let A be a ring with fixed preordering T . We denote by H(A) (resp. H ′ (A)) the ring of geometrically (resp. arithmetically) bounded elements in A, i.e., Let us note h 1 := α |hα| for a given h ∈ R[x]. We start with two preliminary results. H(A) := {h ∈ A : ∃K ∈ N : K ± h ≥ 0 on Sper T A} H ′ (A) := {h ∈ A : ∃K ∈ N : K ± h ∈ T } , where "h ≥ 0 on Sper T A" means "ϕ(h) ≥ 0 for all ϕ ∈ Sper T A". From [26, (1.1)], A = H(A) ⇒ A = H ′ (A) . Lemma 3.2. For k ∈ N and d ∈ N >0 , let q be a form of degree 2(d + k) and f = q x 2k 2 ∈ R(x). Then f is continuous and homogeneous of degree 2d. Proof. The rational function f is obviously homogeneous of degree 2d. To show that f is continuous, it is sufficient to prove that f is continuous at zero. Let y ∈ S n−1 , then one has |y α | ≤ 1, for all α such that |α| = 2(d + k). Thus, |q(y)| = | α qαy α | ≤ α |qα||y α | ≤ α |qα| = q 1 . From this, one has |f (y)| = |q(y)| ≤ q 1. Let x = 0. Since f is homogeneous of degree 2d, |f (x)| x 2d 2 = f x x ≤ q 1 . Hence for all x = 0, |f (x)| ≤ q 1 x 2d 2 , thus limx→0 f (x) = 0, yielding the conclusion. Proof. We rely on [28, Theorem 1.4 (b)] with K = S n−1 and the statement (iii) in [28,Proposition 1.2]. We first need to ensure that S n−1 is the boundary of a convex domain, which is obvious since it is the boundary of the unit ball. Then we use the fact that S n−1 is twice continuously differentiable and has Gaussian curvature 1 at every point. Lemma 3.4. Assume that I = I1 ∪ I2. Let f ∈ R(x) be a rational positive definite form of degree 2d with d ∈ N >0 such that f = f1 + f2 with f1 ∈ R(x(I1)) and f2 ∈ R(x(I2)) being continuous and homogeneous of degree 2d. Then there exists a continuous rational function ϕ ∈ R(x(I1 ∩ I2)) defined by ϕ(y) = q(y) y 2k 2 , ∀y ∈ R |I 1 ∩I 2 | , where q ∈ R[x(I1 ∩ I2) ] is a form of degree 2(d + k) for some k ∈ N (only depending on d, ε and f1) such that f = h1 + h2 , where h1 := f1 − ϕ ∈ R(x(I1)) and h2 := f2 + ϕ ∈ R(x(I2)) are continuous rational positive definite forms of degree 2d. Proof. Since f ∈ R(x) is a rational positive definite form of degree 2d, there exists ε > 0 such that f ≥ ε x 2d 2 on R n . (3.8) Let us define the function h : R |I 1 ∩I 2 | → R by h(y) := min{ψ(ξ, y) : ξ ∈ R |I 1 \I 2 | } ,(3.9) where ψ(ξ, y) := f1(ξ, y) − ε 2 (ξ, y) 2d 2 . To show that h is well-defined, it is sufficient to prove that ξ → ψ(ξ, y) is coercive on R |I 1 \I 2 | with fixed y ∈ R |I 1 ∩I 2 | . Indeed, for all ξ ∈ R |I 1 \I 2 | , by (3.8), ε 2 ξ 2d 2 ≤ ε 2 (ξ, y) 2d 2 ≤ f (ξ, y, 0) − ε 2 (ξ, y) 2d 2 = f1(ξ, y) − ε 2 (ξ, y) 2d 2 + f2(y, 0) , so ψ(ξ, y) ≥ ε 2 ξ 2d 2 − f2(y, 0). Moreover, h is homogeneous of degree 2d. Indeed, for every t ∈ R\{0}, one has h(ty) = min{f1(ξ, ty) − ε 2 (ξ, ty) 2d 2 : ξ ∈ R |I 1 \I 2 | } = t 2d min{f1(ξ/t, y) − ε 2 (ξ/t, y) 2d 2 : ξ ∈ R |I 1 \I 2 | } = t 2d min{f1(ξ, y) − ε 2 (ξ, y) 2d 2 : ξ ∈ R |I 1 \I 2 | } = t 2d h(y) . To show that h is continuous, let y1, y2 ∈ R I 1 ∩I 2 . We choose ξ1, ξ2 ∈ R |I 1 \I 2 | minimizing ξ → ψ(ξ, y1) and ξ → ψ(ξ, y2), respectively. Then ψ(ξ1, y1) − ψ(ξ1, y2) ≤ ψ(ξ1, y1) − ψ(ξ2, y2) ≤ ψ(ξ2, y1) − ψ(ξ2, y2) . From this and by (3.9), |h(y1) − h(y2)| = |ψ(ξ1, y1) − ψ(ξ2, y2)| ≤ max{|ψ(ξ1, y1) − ψ(ξ1, y2)|, |ψ(ξ2, y1) − ψ(ξ2, y2)|} . This shows that h is uniformly continuous on every compact subset of R |I 1 ∩I 2 | because ψ is uniformly continuous on every compact subset of R |I 1 | . Next, we claim that f1 − h ≥ ε 2 x(I1) 2d 2 on R |I 1 | and f2 + h ≥ ε 2 x(I2) 2d 2 on R |I 2 | . (3.10) The first claim is clear by the definition of h. To prove the second one, let (y, z) ∈ R |I 2 | = R |I 1 ∩I 2 | × R |I 2 \I 1 | , and choose ξ ∈ R |I 1 \I 2 | such that h(y) = f1(ξ, y) − ε 2 (ξ, y) 2d 2 . By (3.8), observe that f2(y, z) + h(y) = f2(y, z) + f1(ξ, y) − ε 2 (ξ, y) 2d 2 = f (ξ, y, z) − ε 2 (ξ, y) 2d 2 ≥ ε (ξ, y, z) 2d 2 − ε 2 (ξ, y) 2d 2 ≥ ε 2 (y, z) 2d 2 . Next, we will approximate h by a form of even degree on S |I 1 ∩I 2 |−1 . Note that h is continuous and even since h is homogeneous of even degree. From this and by using Lemma 3.3, there exists q ∈ R[x(I1 ∩ I2)] homogeneous of degree 2K for some K ≥ d such that |q − h| ≤ ε 4 on S |I 1 ∩I 2 |−1 . Since 1 = x(I1 ∩ I2) 2 2 on S |I 1 ∩I 2 |−1 , q x(I1 ∩ I2) 2(K−d) 2 − h ≤ ε 4 on S |I 1 ∩I 2 |−1 . From this and since q x(I 1 ∩I 2 ) 2(K−d) 2 − h is homogeneous of degree 2d, one has q x(I1 ∩ I2) 2(K−d) 2 − h ≤ ε 4 x(I1 ∩ I2) 2d 2 on R |I 1 ∩I 2 | \{0} . (3.11) By setting ϕ := q x(I 1 ∩I 2 ) 2(K−d) 2 and using Lemma 3.2, ϕ is continuous on R |I 1 ∩I 2 | . By setting h1 := f1 − ϕ ∈ R(x(I1)) and h2 := f2 + ϕ ∈ R(x(I2)), one has f = h1 + h2. Let us prove that h1 and h2 are both rational positive definite forms of degree 2d. Indeed, by (3.10) and (3.11), h1 = (f1 − h) + (h − ϕ) ≥ ε 2 x(I1) 2d 2 − ε 4 x(I1 ∩ I2) 2d 2 ≥ ε 4 x(I1) 2d 2 , h2 = (f2 + h) + (ϕ − h) ≥ ε 2 x(I2) 2d 2 − ε 4 x(I1 ∩ I2) 2d 2 ≥ ε 4 x(I2) 2d 2 . Thus, h l ≥ ε 4 x(I l ) 2d 2 on R |I l | , l = 1, 2. By setting k := K − d, the conclusion follows. Building up on Lemma 3.4, the following helpful result provides a non-compact analogue of Grimm et al. [6] and as expected, the non-compact case is much more involved. , ∀y ∈ Rn l , l = 2, . . . , p , where q l ∈ R[x(Î l )] is homogeneous of degree 2(d + k l ) for some k l ∈ N, l = 2, . . . , p, such that f = p l=1 h l , where h l := f l + ϕ l − p j=l+1 δ l,s j ϕj ∈ R(x(I l )), with ϕ1 := 0, is a continuous rational positive definite forms of degree 2d, for each l = 1, . . . , p Proof. The proof is by induction on p ∈ N ≥2 . For p = 2, the desired result follows from Lemma 3.4. Next, assume that Lemma 3.5 holds for p =p − 1 and let us prove that it is also true for p =p. By applying Lemma 3.4 with I1 = p−1 j=1 Ij, I2 = Ip and f1 = p−1 l=1 f l , f2 = fp, there exists a continuous rational function ϕp ∈ R(x(Îp)) defined by ϕp(y) := qp(y) y 2kp 2 , ∀y ∈ Rnp , where qp ∈ R[x(Îp)] is homogeneous of degree 2(d+kp) for some kp ∈ N (only depending on d, ε and f1 + · · · + fp−1) such that The following result shows that one may write a sparse rational positive definite form as a rational SOS with uniform denominator. f = h {1σ ∈ Σ[x] d+k(p+1) for some k ∈ N such that f = σ x 2k 2 p l=1 x(I l ) 2k 2 . (3.12) Proof. Denote by A the R-algebra finitely generated by polynomials xj, j = 1, . . . , n, and rational functions x(I l ) α x(I l ) 2k l 2 , α ∈ N n l such that |α| = 2(d + k l ), l = 1, . . . , p. Let C(R n ) be the space of all continuous functions on R n . By Lemma 3.2, the function x(I l ) α x(I l ) 2k l 2 is continuous for each l = 1, . . . , p, and α ∈ N n l with |α| = 2(d + k l ). Then A is a commutative ring and R[x] ⊂ A ⊂ R(x) ∩ C(R n ). Denote by T the preordering generated by ±(1 − x 2 2 ), i.e., T consists of all elements of the form σ + (1 − x 2 2 )ψ, for σ ∈ ΣA 2 and ψ ∈ A. Then A is a preordered ring with fixed preordering T . We first prove that S n−1 = {x ∈ R n : h(x) ≥ 0 , ∀h ∈ T }. Obviously S n−1 ⊆ {x ∈ R n : h(x) ≥ 0 , ∀h ∈ T }. For the other inclusion, assume by contradiction that there exists a ∈ R n \S n−1 such that h(a) ≥ 0 for all h ∈ T . Then 1 − a 2 2 = 0. By selecting h := −(1 − a 2 2 )(1 − x 2 2 ) ∈ T , one obtains the contradiction 0 ≤ h(a) = −(1 − a 2 2 ) 2 < 0. Next, notice that Sper T A is a Hausdorff space and contains all mappingsâ : A → R, h → h(a) for a ∈ S n−1 (see [18]). Hereâ is well-defined by the continuity of each element in A. In addition, since S n−1 is compact, (â) a∈S n−1 is dense in Sper T A in the topology induced by the sup-norm, i.e., for each r > 0 and for each ϕ ∈ Sper T A there exists a ∈ S n−1 such that sup h∈A |h(a) − ϕ(h)| = sup h∈A |(â − ϕ)(h)| ≤ r (see [17,Section 2] and [4, Section 2]). Let H(A) (resp. H ′ (A)) be the ring of geometrically (resp. arithmetically) bounded elements in A. Since (â) a∈S n−1 is dense in Sper T A, H(A) = {h ∈ A : h is bounded on S n−1 } = A . The latter equality is due to the compactness of S n−1 and the inclusion A ⊂ C(R n ). Combining this together with (3.7), one obtains A = H ′ (A). Next we claim that f > 0 on Sper T A. Indeed f ≥ ε x 2d 2 on R n for some ε > 0, because f is a rational positive definite form of degree 2d. Therefore f ≥ ε on S n−1 . Let ϕ ∈ Sper T A be fixed, arbitrary. By denseness of S n−1 in Sper T A, there exists a ∈ S n−1 such that |f (a) − ϕ(f )| ≤ ε 2 . Thus, ϕ(f ) = f (a) − (f (a) − ϕ(f )) ≥ ε − ε 2 = ε 2 > 0, and the result follows. Next, since f ∈ A and f > 0 on Sper T A, then by Lemma 3.1 f ∈ T . Therefore f = σ + (1 − x 2 2 )ψ for some σ ∈ ΣA 2 and ψ ∈ A. By replacing x by x x 2 and noting that f is homogeneous of degree 2d, x −2d 2 f = σ( x x 2 ). By multiplying both sides with x 2(k+d) 2 p l=1 x(I l ) 2k 2 for some large enough k, there exist r ∈ N and hj , vj ∈ R[x], j = 1, . . . , r, such that hj vj . Recall that f = p l=1 q l x(I l ) 2k l 2 . Therefore assume that k is large enough to ensure that x 2k 2 p l=1 x(I l ) 2k 2 f is a polynomial. Then r j=1 (h 2 j + v 2 j x 2 2 )+2 x 2 r j=1 hjvj must be a polynomial. However since x 2 is not a polynomial, then necessarily r j=1 hj vj = 0. Hence, x 2k 2 p l=1 x(I l ) 2k 2 f = r j=1 (h 2 j + v 2 j x 2 2 ) , which yields (3.12). Proof. One has f = p l=1 f l with f l := p l x(Ĩ l ) 2k l , l = 1, . . . , p. By Lemma 3.2, the function f l ∈ R(x(I l )) is continuous and homogeneous of degree 2d, for each l = 1, . . . , p. By applying Lemma 3.5, there exist continuous functions ϕ l ∈ R(x(Î l )), l = 2, . . . , p, defined by ϕ l (y) = q l (y) y 2k l 2 , ∀y ∈ Rn l , l = 2, . . . , p , where q l ∈ R[x(Î l )] is homogeneous of degree 2(d + k l ) for some k l ∈ N, l = 2, . . . , p, and one has f = p l=1 h l , where each h l := f l + ϕ l − p j=l+1 δ l,s j ϕj ∈ R(x(I l )), l = 1, . . . , p (with ϕ1 := 0) is a continuous rational positive definite form of degree 2d. Then, we apply Lemma 3.6 with the notation f ← h l , I ← I l and I l ← I l ∪ {Îj : sj = l, j = l + 1, . . . , p}. Therefore, there existk l ∈ N and ψ l ∈ Σ[x(I l )] d+k l (1+deg(Φ l )/2) such that Then σ l ∈ Σ[x(I l )] d+k(1+deg(Φ l )/2) and (2.3) follows, yielding the conclusion. Conclusion In this paper, we have provided: -a sparse version for both Reznick's Positivstellensatz (resp. Putinar-Vasilescu's Positivstellensatz) for positive definite forms (resp. nonnegative polynomials). -a sparse version of Putinar-Vasilescu's Positivstellensatz for polynomials that are nonnegative on a possibly non-compact basic semialgebraic set. All these certificates involve sums of squares of rational functions with uniform denominators and a topic of further research is how to exploit such positivity certificates in polynomial optimization on non-compact basic semialgebraic sets. Positivstellensätze and polynomial optimization. With x = (x1, . . . , xn), let R[x] stands for the ring of real polynomials and let Σ[x] ⊂ R[x] be the subset of sums of squares (SOS) of polynomials. Let us note R[x] d and Σ[x] d the respective restrictions of these two sets to polynomials of degree at most d and 2d. For n, m ∈ N >0 , let I := {1, . . . , n} and J := {1, . . . , m}. For T ⊂ I, denote by R[x(T )] (resp. Σ[x(T )]) the ring of polynomials (resp. the subset of SOS polynomials) in the variables x(T ) := {xi : Assumption 1 . 1 . 11The following conditions hold: (i) Running intersection property (RIP): I = p l=1 I l with p ∈ N ≥2 , I l = ∅, l = 1, . . . , p, and for every l ∈ {2, . . . , p}, there exists s l ∈ {1, . . . , l − 1}, such that I l ⊂ Is l , whereÎ l := I l ∩ l−1 j=1 Ij . W.l.o.g, set s2 := 1 andÎ1 := ∅. Denote n l := |I l | andn l := |Î l |, l = 1, . . . , p. Theorem 1. 1 . 1(Lasserre, Waki et al.) Let Assumption 1.1 holds. If a polynomial f is positive on S(g), then there exist σ 0,l 2 -and polynomials nonnegative on a (possibly non-compact) basic semialgebraic set in Corollary 2.3, when the input data satisfy a similar sparsity pattern. These two certificates are obtained via a sparse version of Putinar-Vasilescu's Positivstellensatz and do not require the additional constraints from Assumption 1.1 (iv). Illustrations of such positivity certificates for polynomials nonnegative on non-compact basic semialgebraic sets are provided in Example 2.1, 2.2, 2.3 and 2.4, for which positvity certificates (1.1) do not exist. The existence of such sparse SOS-representations is proved by combining different tools: Corollary 2. 2 . 2Let f be a nonnegative polynomial such that the conditions (i) and (ii) of Assumption 1.1 hold. Let ε > 0 and d ≥ deg(f )/2. Then there exist k ∈ N and σ l ∈ Σ[x(I l )] d+kω l , l = 1, . . . , p, such that of Corollary 2.2 is postponed to Section 4.2. The representation (2.5) can still hold even if ε = 0, as illustrated in the following examples: Corollary 2 . 3 .. 4 . 234Let f ∈ R[x] be nonnegative on S(g) such that the conditions (i), (ii) and (iii) of Assumption 1.1 hold. Let ε > 0 and d ≥ 1 + ⌊deg(f )/2⌋. Recall that uj = ⌈deg(gj)/2⌉, for all j = 1, . . . , m. Then there exist k ∈ N, σ 0,l ∈ Σ[x(I l )] d+kω l and σ j,l ∈ Σ[x(I l )] d+kω l −u j , j ∈ J l , l = 1, . . . , p, such that Let f = f1 + f2, where f1 = x1x2 and f2 = x 2 2 x3. Let g = {g1, g2, g3}, where g1 = x 3 2 , g2 = −g1 and g3 = x3. It is not hard to show that f = 0 on S(g), so that f ≥ 0 on S(g). By noting I1 := {1, 2} and I2 := {2, 3}, one has {f1, g1, g2} ⊂ R[x(I1)] and {f2, g3} ⊂ R[x(I2)] . 1 . 1If Q ⊂ A and A = H ′ (A), then for any f ∈ A, f > 0 on Sper T A ⇒ f ∈ T . Lemma 3. 3 . 3Let h : R n → R be an even function such that h is continuous on S n−1 . Then there exists a sequence (q k ) k∈N of homogeneous polynomials, with deg(q k ) = 2k for all k ∈ N, converging uniformly to h on S n−1 . Lemma 3 . 5 . 35Let Assumption 1.1 (i) holds. Let f ∈ R(x) be a rational positive definite form of degree 2d such that f = p l=1 f l with f l ∈ R(x(I l )) being continuous and homogeneous of degree 2d, l = 1, . . . , p. Then there exist continuous rational functions ϕ l ∈ R(x(Î l )), l = 2, . . . , p, defined by ϕ l (y) = q l (y) y 2k l 2 ,...,p−1} + hp , where h {1,...,p−1} := f1 + · · · + fp−1 − ϕp ∈ R x p−1 j=1 Ij and hp := fp + ϕp ∈ R(x(Ip)) are continuous rational positive definite forms of degree 2d. By the RIP, there exists sp ∈ {2, . . . ,p − 1} such thatÎp ⊂ Isp , so ϕp ∈ R(x(Isp)). Then h {1,...,p−1} = p−1 j=1 (fj − δj,sp ϕp) satisfies fj − δj,sp ϕp ∈ R(x(Ij)), j = 1, . . . ,p − 1. From this and by the induction hypothesis, there exist continuous rational functions ϕ l ∈ R(x(Î l )), l = 2, . . . ,p − 1, defined by ϕ l (y) = q l (y) y 2k l 2 , ∀y ∈ Rn l , l = 2, . . . ,p − 1 , with q l ∈ R[x(Î l )] being homogeneous of degree 2(d + k l ) for some k l ∈ N, l = 2, . . . ,p − 1, such that h {1,...,p−1} =p −1 l=1 h l , where for l = 1, . . . ,p − 1, h l := (f l − δ l,sp ϕp) + ϕ l −p −1 j=l+1 δ l,s j ϕj = f l + ϕ l −p j=l+1 δ l,s j ϕj ∈ R(x(I l )) , is a continuous rational positive definite form of degree 2d. Then f = h {1,...,p−1} +hp = p l=1 h l , yielding the conclusion. Lemma 3 . 6 ., 36Let I = p l=1 I l and d ∈ N >0 . Let f ∈ R(x) be a rational positive definite form of degree 2d such that f = where q l ∈ R[x(I l )] is homogeneous of degree 2(d + k l ) for some k l ∈ N, l = 1, . . . , p. Then there exists Let k := max{k1, . . . ,kp} and define for all l = 1, . . . , p σ l := ψ l x(I l ) Table 1 : 1Several Positivstellensätze applicable in practice. Table 2 : 2Comparing the computational complexity of the sparse and dense hierarchies.SDP of order k sparse hierarchy dense hierarchy number of variables Remark 3.1. Observe that Reznick's Positivstellensatz is a particular case of Lemma 3.6 with p = 1. Our proof is similar to the one of[24, Theorem 3.7], which addresses the case p = 1.4 Proofs 4.1 Proof of Theorem 2.1 If there are f l in the sum f such that deg(f l ) > deg(f ), we can always remove the high degree redundant term in f l which cancel with each other to make degree of f l at most deg(f ) Acknowledgements. The first author was supported by the MESRI fundingProof of Corollary 2.2l .Since f is nonnegative,f is also nonnegative. We first prove thatf + ε p l=1 x(Ī l ) 2d 2 is a positive definite form. Let y ∈ R n+1 such thatf (y) + ε p l=1 y(Ī l ) 2d 2 = 0. By the nonnegativity off and x(Ī l ) 2d 2 , l = 1, . . . , p,f (y) = y(Ī1) 2d 2 = · · · = y(Īp) 2d 2 = 0 . Hence y(Ī l ) = 0, l = 1, . . . , p, and therefore sinceĪ = p l=1Ī l , y = 0. By Theorem 2.1, there exist k ∈ N and ψ l ∈ Σ[x(Ī l )] d+kω l , l = 1, . . . , p, such thatand ω l = deg(Φ l ) + 1, l = 1, . . . , p. Letting xn+1 := 1 in (4.13) yieldswith D l =Φ l (x, 1) ∈ R[x(I l )] and σ l := ψ l (x, 1) ∈ Σ[x(I l )] d+kω l . Hence the conclusion follows since Θ l = θ l D l , l = 1, . . . , p.Proof of Corollary 2.3Proof. Recall that uj = ⌈deg(gj)/2⌉, for all j = 1, . . . , m. Define λj := ( gj 1 + 1) −1 , for all j = 1, . . . , m. We claim that λjgj θ u j l ≤ gj 1 gj 1 + 1 < 1 , j ∈ J l , l = 1, . . . , p .(4.14)For each k ∈ N introduce2k+1 (by (4.14))Thus, Q k is nonnegative on B(0, M ) c for some large enough k. (II) Then, note that lim k→∞ −(1 − a) 2k 2 a 2k+1 = 0 for all a ∈ (0, 1) and lim k→∞ (1 + a) 2k 2 a 2k+1 = ∞ for all a ∈ (0, 1). By using (4.14), each term − 1 −involved in (4.15) can be written either as −(1 − a) 2k 2 a 2k+1 when gj(x) > 0 or as (III) Let K ∈ N be fixed such that QK is nonnegative. Define rj := (2K 2 + 2K + 1)uj and w j,l := (θ u j l − λjgj) 2K 2 (λj gj) 2K+1 , so that w j,l ∈ R[x(I l )]2r j , j ∈ J l , l = 1, . . . , p, and. n+1 h(x/xn+1). Then with same notationx,Ī,Ī l andÎ l as in the proof of Corollary 2.2:(4.16)Equivalently: x(Ī l ) 2dis a positive definite form of degree 2d. By Theorem 2.1, there exist k ∈ N and ψ l ∈ Σ[x(Ī l )] d+kω l , l = 1, . . . , p, such that Hence, the conclusion follows since Θ l = θ l D l , l = 1, . . . , p. R(x) associate its degree-2d homogenizationh(x, xn+1) := x 2d. every h ∈ R(x) associate its degree-2d homogenizationh(x, xn+1) := x 2d On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity. A A Ahmadi, G Hall, arXiv:1709.09307arXiv preprintA. A. Ahmadi and G. Hall. 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Mathematische Annalen, 32(3):342-350, 1888. A representation theorem for certain partially ordered commutative rings. T Jacobi, Mathematische Zeitschrift. 2372T. Jacobi. A representation theorem for certain partially ordered commutative rings. Mathematische Zeitschrift, 237(2):259-273, 2001. I Klep, V Magron, J Povh, arXiv:1909.00569Sparse noncommutative polynomial optimization. arXiv preprintI. Klep, V. Magron, and J. Povh. Sparse noncommutative polynomial optimization. arXiv preprint arXiv:1909.00569, 2019. Anneaux préordonnés. J.-L Krivine, Journal d'analyse mathématique. 121J.-L. Krivine. Anneaux préordonnés. Journal d'analyse mathématique, 12(1):307-326, 1964. Global optimization with polynomials and the problem of moments. J B Lasserre, SIAM Journal on optimization. 113J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on optimization, 11(3):796-817, 2001. Convergent SDP-relaxations in polynomial optimization with sparsity. J B Lasserre, SIAM Journal on Optimization. 173J. B. Lasserre. Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM Journal on Optimization, 17(3):822-843, 2006. A bounded degree SOS hierarchy for polynomial optimization. J B Lasserre, K.-C Toh, S Yang, EURO Journal on Computational Optimization. 51-2J. B. Lasserre, K.-C. Toh, and S. Yang. A bounded degree SOS hierarchy for polynomial optimization. EURO Journal on Computational Optimization, 5(1-2):87-117, 2017. Polynomials in R[x,y] that are sums of squares in R (x,y). D Leep, C Starr, American Mathematical Society129Proceedings of theD. Leep and C. Starr. Polynomials in R[x,y] that are sums of squares in R (x,y). Pro- ceedings of the American Mathematical Society, 129(11):3133-3141, 2001. Positivity certificates and polynomial optimization on non-compact semialgebraic sets. N H A Mai, J.-B Lasserre, V Magron, SubmittedN. H. A. Mai, J.-B. Lasserre, and V. Magron. 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Uniform denominators in Hilbert's seventeenth problem. Mathematische Zeitschrift, 220(1):75-97, 1995. Uniform denominators in hilbert's 17th problem theorems by pólya and reznick. math.uni-konstanz. R Schabert, deR. Schabert. Uniform denominators in hilbert's 17th problem theorems by pólya and reznick. math.uni-konstanz.de, 2019. The K-moment problem for compact semi-algebraic sets. K Schmüdgen, Mathematische Annalen. 2891K. Schmüdgen. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen, 289(1):203-206, 1991. Iterated rings of bounded elements and generalizations of schmüdgen's positivstellensatz. M Schweighofer, Journal für die reine und angewandte Mathematik. 554M. Schweighofer. Iterated rings of bounded elements and generalizations of schmüdgen's positivstellensatz. Journal für die reine und angewandte Mathematik, 554:19-45, 2003. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. G Stengle, 207Mathematische AnnalenG. Stengle. 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{'abstract': "If f is a positive definite form, Reznick's Positivstellensatz [Mathematische Zeitschrift. 220 (1995), pp. 75-97] states that there exists k ∈ N such that x 2k 2 f is a sum of squares of polynomials. Assuming that f can be written as a sum of forms p l=1 f l , where each f l depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick's Positivstellensatz. Namely, there exists k ∈ N such that f = p l=1 σ l /H k l , where σ l is a sum of squares of polynomials, H l is a uniform polynomial denominator, and both polynomials σ l , H l involve the same variables as f l , for each l = 1, . . . , p. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick's certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate due to Putinar and Vasilescu.", 'arxivid': '2002.05101', 'author': ['Ngoc Hoang ', 'Anh Mai ', 'Victor Magron ', 'Jean-Bernard Lasserre '], 'authoraffiliation': [], 'corpusid': 211082770, 'doi': '10.1287/moor.2022.1284', 'github_urls': [], 'n_tokens_mistral': 18999, 'n_tokens_neox': 16693, 'n_words': 8824, 'pdfsha': 'b37249a9a8cdf106af4fd9c1af5f83e828f5301f', 'pdfurls': ['https://arxiv.org/pdf/2002.05101v2.pdf'], 'title': ["A sparse version of Reznick's Positivstellensatz", "A sparse version of Reznick's Positivstellensatz"], 'venue': []}

arxiv

Three-Dimensional Smoothed Particle Hydrodynamics Method for Simulating Free Surface Flows Rizal Dwi Prayogo rizal.dp@s.itb.ac.id Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung Jl. Ganesha 1040132BandungIndonesia Graduate School of Natural Science and Technology Kanazawa University 920-1192KakumaKanazawaJapan Christian Fredy Naa Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung Jl. Ganesha 1040132BandungIndonesia Three-Dimensional Smoothed Particle Hydrodynamics Method for Simulating Free Surface Flows Smoothed particle hydrodynamicsfree surface flowsgradient kernel renormaliza- tion In this paper, we applied an improved Smoothing Particle Hydrodynamics (SPH) method by using gradient kernel renormalization in three-dimensional cases. The purpose of gradient kernel renormalization is to improve the accuracy of numerical simulation by improving gradient kernel approximation. This method is implemented for simulating free surface flows, in particular dam break case with rigid ball structures and the propagation of waves towards a slope in a rectangular tank. Introduction Computational Fluid Dynamics (CFD) using Smoothed Particle Hydrodynamics (SPH) has a wide range of applications to solve problem in engineering and science. SPH is a mesh-free Lagrangian method and well suited to the simulation of complex and free surface flows. The SPH method was originally used to model astrophysical problems by Lucy [7] and Gingold and Monaghan [8]. Threedimensional SPH method has been studied numerically by Monaghan [6] in a field of astrophysical fluid dynamics processes. We obtain the SPH equations from the continuum equations of fluid dynamics by interpolating from set of points which may be disordered. This interpolation is based on the theory of integral interpolants using interpolation kernels which approximate the delta function. The interpolants being analytic functions can be differentiated without using grids. Monaghan [4] studied the application of the particle method SPH to free surface problems in two-dimensional cases. In this paper, we applied an improved SPH method by using gradient kernel renormalization in three-dimensional cases. The purpose of gradient kernel renormalization is to improve the accuracy of the simulations [1]. In the following, first the general concept of SPH method is given. The improved SPH method using gradient kernel renormalization is introduced and described in detail. This improved SPH method is implemented for simulating free surface flows, in particular dam break case with rigid ball structures and the propagation of waves towards a slope in a rectangular tank. Smoothed particle hydrodynamics The SPH equations are described in detail by Liu and Liu [2]. In this paper, we consider the application of three-dimensional SPH to free surface problems. The SPH method represents continuous fluid using a set of particles. Each particle i has physical quantities, such as mass m i , position r i , velocity v i , density ρ i , and pressure P i . Each particle in the SPH method is associated with a support domain. The SPH approximation, which consists of the particle approximation and the kernel approximation, is performed within the current support domain. The value of a function defining a physical quantity can be approximated by its values at a number of neighboring particles. The SPH method uses the concept of integral representation of a field function f (x) by the following identity < f (r) >= Ω f (r )W (|r − r |, h)dr ,(1) where r and r are the position vectors, W is the smoothing function or kernel function, and h is the smoothing length defining the influence radius of W . In SPH approximation, there are various kernel functions. Since it is affects the accuracy and stability of numerical results, the choice of kernel function W is important to consider. The integral representation should satisfies several conditons. That is the normalization condition We converted the continuous integral representation (1) into discretized forms as a summation over all the particles in the support domain. This process is also commonly known as particle approximation in the SPH literature [2]. Writing the particle approximation as follows < f (r) >= N j=1 m j ρ j f (r j )W (|r − r j |, h), where m j and ρ j are the mass and density of the particle j, respectively, and j = 1, 2, . . . , N , where N is the total number of neighboring particles in the influence domain Ω. In this paper, we use cubic spline kernel as follows [5] W (q, h) = 1 πh 3      1 − 3 2 q 2 + 3 4 q 3 , 0 ≤ q < 1 1 4 (2 − q) 3 , 1 ≤ q < 2 0, otherwise, where q = |r i − r j | h is the relative distance of particle i and j. 3 Numerical model The continuity equation The continuity equation is based on the conservation of mass. We write the continuity equation in the form Dρ Dt = −ρ∇v,(2) where v and ρ are velocity and density, respectively. Writing (2) in SPH discretization form as in [3], we obtain Dρ i Dt = ρ i j m j ρ j (v i − v j )∇ ri W (|r i − r j |, h),(3) where ρ k and v k are density and velocity of particle k (evaluated at k = i or k = j), respectively, m j is mass of particle j and ∇ ri W (|r i − r j |, h) = r i − r j |r i − r j | ∂W ∂r . The momentum equation The momentum equation is based on the conservation of momentum which is given by Dv Dt = − 1 ρ ∇P + F,(4) where v, ρ and P are velocity, density, and pressure, respectively. Here, F is external force, in this case gravitational acceleration. Writing (4) in SPH discretization form as in [3], we get Dv i Dt = − j m j P i + P j ρ i ρ j + Π ij ∇ ri W (|r i − r j |, h) + F,(5) where P k is pressure of particle k (evaluated at k = i or k = j). In SPH, there are various formulations for viscosity. In the momentum equation, the introduction of a viscous term is necessary not only to consider viscid fluids and no slip boundary conditions, but also to provide the stability to the system and to prevent inter-particle penetration. The artificial viscosity term Π ij is added to pressure terms within the momentum equation (5). The artificial viscosity Π ij has the form [6] Π ij =      −αcµ ij + βµ 2 ij (ρ i + ρ j )/2 , (v i − v j ).(r i − r j ) < 0 0, (v i − v j ).(r i − r j ) > 0 where µ ij = h(r i − r j ).(v i − v j ) |r i − r j | 2 + η 2 . In these expressions, c is the speed of sound, η = 0.001, α and β represent shear and bulk viscosity, respectively. For the problems described here, we choose α = 0.03 and β = 0. The equation of state The equation of state is used to relate density to pressure. In this paper, the Tait's equation of state has the form p = ρ 0 c 2 γ ρ ρ 0 γ − 1 , where c, ρ 0 , and γ are the speed of sound, density reference, and the polytropic constant, respectively. Note that γ = 7 is usually used for water simulations. The speed of sound c is approximately √ 100gH and it is chosen in respect of a low Mach number (M a < 0.1) to ensure low compressibility effects [6]. Improvement of the SPH method In this paper, we applied an improvement to the standard SPH method by using renormalization. This technique is to improve the accuracy of the method [1]. Gradient kernel renormalization The velocity gradient in (3) can be approached by using ∇v = ∇v − v∇1. We can generalize this approach for any field f by using ∇f = ∇f − f ∇1(6) and transforming (6) into its continuous convoluted form we have ∇f (r) = Ω f (r )∇W dr − f (r) Ω ∇W dr . We recall the second order Taylor expansion Ω f (r )∇W dr = f (r) Ω ∇W dr + ∂f (r) ∂r 1 Ω (r 1 − r 1 )∇W dr A + ∂f (r) ∂r 2 Ω (r 2 − r 2 )∇W dr B + ∂f (r) ∂r 3 Ω (r 3 − r 3 )∇W dr C +O(h 2 ). In order to ensure gradient interpolations of linear fields, it is necessary to ensure that the discrete approximation of A, B, and C are A =   1 0 0   B =   0 1 0   C =   0 0 1   . By the renormalization procedure [1], we modify ∇W as follows a (d, d) correction matrix, d is the dimension of the case. In this paper, we consider three-dimensional cases (d = 3) and calculate L(x) to increase the accuracy of gradient kernel approximation. The continuity equation is discretized by the following manner j m j ρ j (r j − r) L(r)∇W (|r − r j |) = 1 0 0 0 1 0 0 0 1 , where L(r) isDρ i Dt = ρ i j m j ρ j (v i − v j )L(r i )∇ ri W (|r i − r j |, h). This discretized form ensures exact interpolations for both constant and linear fields. Note that we can discretize the conservation of momentum by the following manner Dv i Dt = − j m j P i + P j ρ i ρ j + Π ij L(r i )∇ ri W (|r i − r j |, h) + F. Numerical time integration with renormalization As the other explicit hydrodynamic methods, different numerical time integrations can be applied in SPH simulation, such as Leap-Frog, predictor-corrector, Runge-Kutta, and Beeman schemes. The advantages of the Leap-Frog algorithm are its low memory usage on storage and its computational efficiency. We applied it in this paper with its improvement by using gradient kernel renormalization. Therefore, r * i = r n i + dt 2 v n i , p n i = ρ 0 c 2 0 γ ρ n i ρ 0 γ − 1 , , v n+1/2 i = v n−1/2 i − dt j m j p n i + p n j ρ n i ρ n j + Π ij L(r n+1/2 i )∇ ri W (r n+1/2 ij , h) + F n+1/2 i , ρ n+1/2 i = ρ n−1/2 i + dtρ i j m j ρ j v n+1/2 i − v n+1/2 j L(r n+1/2 i )∇ ri W (r n+1/2 ij , h), r n+1 i = r n+1/2 i + dt 2 v n+1/2 i . In the following section, the results of numerical simulations for improved SPH method are given. This method was implemented in dam-break problem with rigid ball structures and water waves generated by oblique piston type wave-maker. Dam-break and structure In this implementation, we consider a rectangular tank with three-dimensional problem, in particular on interaction between waves and structures. Here we examine the impact of a single wave with rigid ball structures over the slope by means of a three-dimensional SPH method. A rectangular tank contains fixed structures and we used 10075 particles for this simulation. The geometry is shown in Fig. 2. Oblique piston type wave-maker In this implementation, we consider a rectangular tank with three-dimensional problem, in particular the propagation of waves towards a slope. This simulation involves a wave-maker in the form of an oscillating oblique piston on the left-hand side and used 80682 particles. The geometry is shown in Fig. 4. 5 shows the simulation of free surface flows. The water waves generated by oscillating piston type wave-maker were simulated. In Fig. 5 the waves are shown propagating onto the slope. The frame at T = 0.0 s shows the initial configuration with water lying on the slope. In the next frame at T = 0.2865 s, the wave is generated by the oscillating oblique piston type wave-maker. At T = 0.823125 s, T = 0.917625 s, and T = 0.9915 s the water wave propagates onto the slope. Finally, at T = 1.18575 s the water wave reaches shallow water area and hits the right wall. Summary This paper presents the application of an improved SPH method by using gradient kernel renormalization for simulating free surface flows. We have implemented this method on three-dimensional cases, in particular on the interaction between waves and structures; and the propagation of waves towards a slope for waves generated by oblique piston type wave-maker. The three-dimensional case of the model has been shown to produce three-dimensional phenomenon, i.e., the collision of a single wave with rigid ball structures and its passing around the obstacle. In summary, an improved SPH model based on renormalization can be successfully used to simulate three-dimensional wave problems. Figure 1 : 1The influence radius of W . W (|r − r |, h) = δ(|r − r |), moreover, often the compact support condition is required W (|r − r |, h) = 0 outside of support domain. Figure 2 : 2Sketch of dam-break problem considered, side view (upper panel) and top view (lower panel). Fig. 3 3shows the motion of a single wave which moves through rigid ball structures in a rectangular tank. The frame at T = 0.0 s shows the initial configuration. In the next frame at T = 0.11175 s, the wave generated by the dam break and the initial layer of water on the bottom collides with the front of the rigid ball structures and at T = 0.1425 s the wave wraps around the rigid ball structures. At T = 0.1725 s, the waves collide from both sides of the rigid ball structures then continue moving toward the right vertical wall. The wave reflects after colliding with the opposite wall of the tank at T = 0.3255 s. The last movement, at T = 0.47625 s, the reflected wave hits the back of rigid ball structures. Figure 3 : 3(T = 0.0 s) initial configuration; (T = 0.11175 s) wave hitting the rigid ball structures; (T = 0.1425 s) wave wrapping around the rigid ball structures; (T = 0.1725 s) waves colliding after passing the rigid ball structures; (T = 0.3255 s) wave colliding with the opposite wall of the tank; (T = 0.47625 s) reflected wave hitting the back of the rigid ball structures. Figure 4 : 4Sketch of oblique piston type wave-maker, side view (upper panel) and top view (lower panel). Figure 5 : 5(T = 0.0 s) initial configuration; (T = 0.2865 s) the wave is generated by the oscillating oblique piston type wave-maker; at (T = 0.823125 s), (T = 0.917625 s), and (T = 0.9915 s) the water wave propagates onto the slope; (T = 1.18575 s) the water wave reaches shallow water area and hits the right wall. Fig. Fig. 5 shows the simulation of free surface flows. The water waves generated by oscillating piston type wave-maker were simulated. In Fig. 5 the waves are shown propagating onto the slope. The frame at T = 0.0 s shows the initial configuration with water lying on the slope. In the next frame at T = 0.2865 s, the wave is generated by the oscillating oblique piston type wave-maker. At T = 0.823125 s, T = 0.917625 s, and T = 0.9915 s the water wave propagates onto the slope. Finally, at T = 1.18575 s the water wave reaches shallow water area and hits the right wall. An improved SPH method: Towards higher order convergence. G Oger, M Doring, B Alessandrini, P Ferrant, Journal of Computational Physics. 225G. Oger, M. Doring, B. Alessandrini, P. Ferrant (2007). An improved SPH method: Towards higher order convergence. Journal of Computational Physics., 225, 1472-1492. Smoothed particle hydrodynamics: a meshfree particle method. G R Liu, M B Liu, World Scientific Publishing Co. Pte. LtdSingaporeG. R. Liu and M. B. Liu (2003). Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co. Pte. Ltd, Singapore. An Introduction to SPH. J J Monaghan, Computer Physics Communications. 48J. J. Monaghan (1988). An Introduction to SPH. Computer Physics Communications., 48, 1, 89-96. Simulating Free Surface Flows with SPH. J J Monaghan, Journal of Computational Physics. 110J. J. Monaghan (1994). Simulating Free Surface Flows with SPH. Journal of Computational Physics., 110, 399 -406. A Refined Method for Astrophysical Problems. J J Monaghan, J C Lattanzio, Astron. Astrophys. 149J. J. Monaghan and J. C. Lattanzio (1985). A Refined Method for Astrophysical Problems. Astron. Astrophys., 149, 399 -406. Smoothed Particle Hydrodynamics. J J Monaghan, Annu. Rev. Astron. Astrophys. 30J. J. Monaghan (1992). Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys., 30, 543 -574. A numerical approach to the testing of the fission hypothesis. L B Lucy, Astron. J. 82L. B. Lucy (1977). A numerical approach to the testing of the fission hypothesis. Astron. J., 82, 1013 -1024. Smoothed particle hydrodynamics: theory and application to non-spherical stars. R A Gingold, J J Monaghan, Mon. Not. R. Astr. Soc. 181R. A. Gingold and J. J. Monaghan (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astr. Soc., 181, 375 -389.

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{'abstract': 'In this paper, we applied an improved Smoothing Particle Hydrodynamics (SPH) method by using gradient kernel renormalization in three-dimensional cases. The purpose of gradient kernel renormalization is to improve the accuracy of numerical simulation by improving gradient kernel approximation. This method is implemented for simulating free surface flows, in particular dam break case with rigid ball structures and the propagation of waves towards a slope in a rectangular tank.', 'arxivid': '1309.4074', 'author': ['Rizal Dwi Prayogo rizal.dp@s.itb.ac.id \nFaculty of Mathematics and Natural Sciences\nInstitut Teknologi Bandung\nJl. Ganesha 1040132BandungIndonesia\n\nGraduate School of Natural Science and Technology\nKanazawa University\n920-1192KakumaKanazawaJapan\n', 'Christian Fredy Naa \nFaculty of Mathematics and Natural Sciences\nInstitut Teknologi Bandung\nJl. Ganesha 1040132BandungIndonesia\n'], 'authoraffiliation': ['Faculty of Mathematics and Natural Sciences\nInstitut Teknologi Bandung\nJl. Ganesha 1040132BandungIndonesia', 'Graduate School of Natural Science and Technology\nKanazawa University\n920-1192KakumaKanazawaJapan', 'Faculty of Mathematics and Natural Sciences\nInstitut Teknologi Bandung\nJl. Ganesha 1040132BandungIndonesia'], 'corpusid': 119013480, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4963, 'n_tokens_neox': 4361, 'n_words': 2783, 'pdfsha': '23725fc5a9b2aa41205ed7e10581b081603509c3', 'pdfurls': ['https://arxiv.org/pdf/1309.4074v1.pdf'], 'title': ['Three-Dimensional Smoothed Particle Hydrodynamics Method for Simulating Free Surface Flows', 'Three-Dimensional Smoothed Particle Hydrodynamics Method for Simulating Free Surface Flows'], 'venue': []}

arxiv

Theta vocabulary I 7 Apr 2015 S Kharchev Institute for Theoretical and Experimental Physics MoscowRussia A Zabrodin Institute for Theoretical and Experimental Physics Laboratory of Mathematical Physics National Research University "Higher School of Economics" 20 Myasnitskaya Ulitsa101000Moscow, MoscowRussia, Russia Theta vocabulary I 7 Apr 2015arXiv:1502.04603v2 [math.CA] This paper is an annotated list of transformation properties and identities satisfied by the four theta functions θ 1 , θ 2 , θ 3 , θ 4 of one complex variable, presented in a readyto-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters τ and 2τ . Foreword The theta functions introduced by Jacobi [J] (see also [B], [W], [WW], [M1]) are doubly (quasi)periodic analogues of the basic trigonometric functions sin(πu) and cos(πu). Let the two (quasi)periods be 1 and τ ∈ C with the condition ℑ τ > 0. The basic theta functions are θ 1 (u|τ ), θ 2 (u|τ ), θ 3 (u|τ ), θ 4 (u|τ ). The theory of theta functions is a sort of "elliptically deformed" trigonometry. In essence the functions sin and cos are the same because cos x = sin(x + π 2 ), but everybody knows that in practice it is more convenient to work with the two functions rather than one. Likewise, the four theta functions can be obtained from any one of them by simple transformations like shifts of the argument and multiplying by a common factor, but it is more convenient to deal with the set of four instead of one. The "elliptic deformation" of the trigonometric functions may go in two ways depending on which property of the former one wants to preserve or generalize. One is a deformation in the class of entire functions (the north-east arrow in the diagram below). It leads to the quasi-periodic theta functions, which are regular functions in the whole complex plane. The other one is in the class of doubly periodic functions. The (infinite) second period of the trigonometric functions becomes finite (equal to τ ) at the price of breaking the global analyticity, so the elliptic functions sn, cn and dn, which are doubly periodic analogues of trigonometric sin and cos are meromorphic functions in the complex plane. θ 1 , θ 2 , θ 3 , θ 4 ր sin, cos ց sn, cn, dn In fact the basic elliptic functions are constructed as ratios of the theta functions and in this sense the latter seem to be more fundamental. In practical calculations with trigonometric functions (and their hyperbolic cousins), one needs just a few identities for the basic functions sin and cos like the addition formula sin(x + y) = sin x cos y + sin y cos x. It is not difficult to remember them all or derive any forgotten one from scratch using the definitions sin x = −i(e ix −e −ix )/2, cos x = (e ix +e −ix )/2. For the theta functions, the situation is much more involved. They are connected by a plethora of identities most of which are not obvious, not suitable for memorizing and can not be derived from scratch in any easy way. Here is what Mumford wrote in Chapter 1 of his book "Tata lectures on Theta I" [M1] after presenting a list of ponderous identities for theta functions: "We have listed these at such length to illustrate a key point in the theory of theta functions: the symmetry of the situation generates rapidly an overwhelming number of formulae, which do not however make a completely elementary pattern. To obtain a clear picture of the algebraic implications of these formulae altogether is then not usually easy." All this is aggravated by the fact that there are several different systems of notation for theta functions in use. In the present paper we make an attempt to bring some order into this conglomeration of formulae. We show that the 3, 4 and 5-term identities of degree four (i.e. with products of four theta functions in each term), referred to as Weierstrass addition formulae, Jacobi relations, and Riemann identities, respectively, can be obtained by purely algebraic manipulations from six basic 3-term theta relations of degree two connecting theta functions with modular parameters τ and 2τ . Starting with the six "elementary bricks", it is possible to derive 52 fundamental relations of degree four containing four independent variables. Besides, we give the complete list of all important particular identities which are appropriate specifications of the basic bilinear and degree four ones. Recently, Koornwinder has proved [K] that the Weierstrass addition formulae and the Riemann identities are equivalent. We reproduce this result in a very simple way. In the future we plan to address more specific questions related to the role of theta functions in the theory of integrable systems and lattice models of statistical mechanics. Theta functions as infinite products One has the following infinite product representations: θ 1 (u|τ ) = 2q 1 4 sin πu ∞ n=1 (1 − q 2n )(1 − q 2n e 2πiu )(1 − q 2n e −2πiu ), (2.10a) θ 2 (u|τ ) = 2q 1 4 cos πu ∞ n=1 (1 − q 2n )(1 + q 2n e 2πiu )(1 + q 2n e −2πiu ), (2.10b) θ 3 (u|τ ) = ∞ n=1 (1 − q 2n )(1 + q 2n−1 e 2πiu )(1 + q 2n−1 e −2πiu ), (2.10c) θ 4 (u|τ ) = ∞ n=1 (1 − q 2n )(1 − q 2n−1 e 2πiu )(1 − q 2n−1 e −2πiu ). (2.10d) To prove (2.10d), we note that the product p(u|τ ) := ∞ n=1 (1 − q 2n−1 e 2πiu )(1 − q 2n−1 e −2πiu ) has the same zeros as θ 4 (u|τ ) and the ratio θ 4 (u|τ )/p(u|τ ) is a doubly periodic function with periods 1 and τ . Hence the ratio is constant and one has θ 4 (u|τ ) = Ap(u|τ ). To find the constant A, put u = 0 thus getting A = θ 4 (0|τ )/p(0|τ ). Finally, in accordance with the Gauss formula [An,p. 23,eq. (2.2.12)], θ 4 (0|τ ) = k∈Z (−1) k q k 2 = ∞ n=1 1 − q n 1 + q n . (2.11) Rewriting ∞ n=1 (1 − q 2n−1 ) = ∞ n=1 (1 − q n )/(1 − q 2n ), one gets A = ∞ n=1 (1 − q 2n ) and formula (2.10d) is proved. Equations (2.10a)-(2.10c) are obtained from (2.10c) by appropriate shifts of u in accordance with (2.8). As a corollary of (2.10) one has infinite product representations for the theta constants: θ ′ 1 (0) = 2πq 1 4 ∞ n=1 (1 − q 2n ) 3 , (2.12a) θ 2 (0) = 2q 1 4 ∞ n=1 (1 − q 2n )(1 + q 2n ) 2 , (2.12b) θ 3 (0) = ∞ n=1 (1 − q 2n )(1 + q 2n−1 ) 2 , (2.12c) θ 4 (0) = ∞ n=1 (1 − q 2n )(1 − q 2n−1 ) 2 . (2.12d) Since n≥1 (1 + q 2n )(1 + q 2n−1 )(1 − q 2n−1 ) = 1, this implies the famous identity for the theta constants [J, p. 517]: θ ′ 1 (0) = πθ 2 (0)θ 3 (0)θ 4 (0). (2.13) Modular transformations The transformation τ → τ + 1: θ 1 (u|τ + 1) = e πi 4 θ 1 (u|τ ), (2.14a) θ 2 (u|τ + 1) = e πi 4 θ 2 (u|τ ), (2.14b) θ 3 (u|τ + 1) = θ 4 (u|τ ), (2.14c) θ 4 (u|τ + 1) = θ 3 (u|τ ). (2.14d) Since τ → τ + 1 implies q → −q, equations (2.14) follow from (2.5) or (2.10). The transformation τ → −1/τ : θ 1 (u/τ | − 1/τ ) = −i √ −iτ e πiu 2 /τ θ 1 (u|τ ), (2.15a) θ 2 (u/τ | − 1/τ ) = √ −iτ e πiu 2 /τ θ 4 (u|τ ), (2.15b) θ 3 (u/τ | − 1/τ ) = √ −iτ e πiu 2 /τ θ 3 (u|τ ), (2.15c) θ 4 (u/τ | − 1/τ ) = √ −iτ e πiu 2 /τ θ 2 (u|τ ). (2.15d) The branch of the square root here is such that ℜ √ −iτ > 0. The proof is well known. Since the ratio e −πiu 2 /τ θ 3 (u/τ | − 1/τ ) /θ 3 (u|τ ) := C is an entire doubly periodic function of u, it is a constant (which may depend only on τ ). Shifting u by 1 2 , τ 2 , τ +1 2 , one obtains three more relations of the same kind with the same constant C. Then the substitution of these formulas to (2.13), yields C 2 = −iτ . The sign of the square root is determined by the argument that if τ ∈ iR + , then both θ 3 (0| − 1/τ ) and θ 3 (0|τ ) are real and positive. Other notation for the theta functions The notations for theta functions used in the literature are of a great variety. This can be a source of confusion. Here we briefly comment on the main systems of notation other that the one adopted in this paper. In the theory of elliptic integrals, the theta functions Θ r u|τ = θ r u 2K τ , K = π 2 θ 2 3 (0|τ ) (2.16) introduced by Riemann are commonly encountered. The number K is the full elliptic integral (of the first kind). In [A] and some other books our θ r is denoted as ϑ r while Θ r defined in (2.16) is just θ r . The antiquated Jacobi notation (still preferred by some authors) are H, H 1 , Θ 1 , Θ for Θ 1 , Θ 2 , Θ 3 , Θ 4 respectively. The "multiplicative notation" θ r (z|q) for θ r (u|τ ), where q = e πiτ , z = e 2πiu , is widely used in the modern literature on elliptic hypergeometric series and related problems. Lastly, let us mention a few of the minor differences in notation encountered in the literature. In [W], [HC] the functions Θ a,b (u) have been considered which are related with θ a,b (u) by Θ W a,b (u) = e πiab θ − a 2 , b 2 (u), Θ HC a,b (u) = e − πiab 2 θa 2 , b 2 (u). (2.17) The set of our theta functions (2.5) is related with the corresponding functions in [WW] as θ r (u|τ ) = θ WW r (πu|τ ), r = 1, 2, 3, 4. Following the original notation [J], in [A] and in some other books the notation θ 0 is used instead of θ 4 . 3 Four types of identities between theta functions Preliminaries The number of identities satisfied by the theta functions is enormous. It is still fairly big if we consider identities involving up to four independent variables. They can be split into four types: B. Three-term bilinear identities involving two independent variables. They relate products of two theta functions with modular parameter τ to linear combinations (actually, sums or differences) of similar products of theta functions with modular parameter 2τ . W. Three-term identities of degree 4 (the Weierstrass addition formulae). J. Four-term identities of degree 4 (the Jacobi formulae). R. Five-term identities of degree 4 (the Riemann identities). The identities of types W, J, R include theta functions with the same modular parameter τ and contain four independent variables. The identities of type B are the most fundamental ones: all the others are algebraic consequences of these together with the evenness properties of the theta functions θ r (−u) = (−1) δ r,1 θ r (u), r = 1, 2, 3, 4. Namely, we shall show how to derive W from B etc., according to the scheme B → W → J → R. It also turns out that the Jacobi and Riemann identities are equivalent in a very simple way. At the end of this section, we prove the arrow W ← J which implies equivalence of the Weierstrass and Jacobi identities. 3.2 Three-term bilinear identities connecting theta functions with τ and 2τ B.I. There are six basis bilinear identities: θ 1 (u|τ )θ 1 (v|τ ) = θ 3 (u + v|2τ )θ 2 (u − v|2τ ) − θ 2 (u + v|2τ )θ 3 (u − v|2τ ), (3.1a) θ 1 (u|τ )θ 2 (v|τ ) = θ 1 (u + v|2τ )θ 4 (u − v|2τ ) + θ 4 (u + v|2τ )θ 1 (u − v|2τ ), (3.1b) θ 2 (u|τ )θ 2 (v|τ ) = θ 2 (u + v|2τ )θ 3 (u − v|2τ ) + θ 3 (u + v|2τ )θ 2 (u − v|2τ ), (3.1c) θ 3 (u|τ )θ 3 (v|τ ) = θ 3 (u + v|2τ )θ 3 (u − v|2τ ) + θ 2 (u + v|2τ )θ 2 (u − v|2τ ), (3.1d) θ 3 (u|τ )θ 4 (v|τ ) = θ 4 (u + v|2τ )θ 4 (u − v|2τ ) − θ 1 (u + v|2τ )θ 1 (u − v|2τ ), (3.1e) θ 4 (u|τ )θ 4 (v|τ ) = θ 3 (u + v|2τ )θ 3 (u − v|2τ ) − θ 2 (u + v|2τ )θ 2 (u − v|2τ ). (3.1f) (See [Ig], [D], [M2] for the general case of multi-dimensional theta functions.) B.II. A system equivalent to (3.1): 2θ 1 (u + v|2τ )θ 1 (u − v|2τ ) = θ 4 (u|τ )θ 3 (v|τ ) − θ 3 (u|τ )θ 4 (v|τ ), (3.2a) 2θ 1 (u + v|2τ )θ 4 (u − v|2τ ) = θ 1 (u|τ )θ 2 (v|τ ) + θ 2 (u|τ )θ 1 (v|τ ), (3.2b) 2θ 2 (u + v|2τ )θ 2 (u − v|2τ ) = θ 3 (u|τ )θ 3 (v|τ ) − θ 4 (u|τ )θ 4 (v|τ ), (3.2c) 2θ 2 (u + v|2τ )θ 3 (u − v|2τ ) = θ 2 (u|τ )θ 2 (v|τ ) − θ 1 (u|τ )θ 1 (v|τ ), (3.2d) 2θ 3 (u + v|2τ )θ 3 (u − v|2τ ) = θ 3 (u|τ )θ 3 (v|τ ) + θ 4 (u|τ )θ 4 (v|τ ), (3.2e) 2θ 4 (u + v|2τ )θ 4 (u − v|2τ ) = θ 3 (u|τ )θ 4 (v|τ ) + θ 4 (u|τ )θ 3 (v|τ ). (3.2f) Remark 3.1 Starting with any identity in (3.1), one can derive all the other ones by appropriate shifts of the variables u, v. The proof is standard. Let us prove, for example, (3.1b). Consider the function F (v) := θ 1 (u + v|2τ )θ 4 (u − v|2τ ) + θ 4 (u + v|2τ )θ 1 (u − v|2τ ). By virtue of (2.7) and (2.8), F (v + 1) = −F (v), F (v + τ ) = e −πi(2v+τ ) F (v) and F ( 1 2 ) = 0. Hence zeros of F (v) are v n,m = n + 1 2 + mτ , n, m ∈ Z and the ratio F (v)/(θ 1 (u|τ )θ 2 (v|τ )) is an entire function doubly periodic in v with periods 1, τ . Therefore, this ratio does not depend on v: F (v)/θ 1 (u|τ )θ 2 (v|τ ) = C(u). Setting v = u, one has: C(u) = θ 1 (2u|2τ ) θ 4 (0|2τ ) θ 1 (u|τ ) θ 2 (u|τ ) . By virtue of (2.10a), (2.10b) and (2.11), θ 1 (u|τ )θ 2 (u|τ ) = θ 1 (2u|2τ )θ 4 (0|2τ ) and thus C(u) ≡ 1. Three-term Weierstrass addition identities There are twelve addition formulae (see below). We start with the identity θ 1 (u + x)θ 1 (u − x)θ 1 (v + y)θ 1 (v − y) − θ 1 (u + y)θ 1 (u − y)θ 1 (v + x)θ 1 (v − x) = θ 1 (u + v)θ 1 (u − v)θ 1 (x + y)θ 1 (x − y) (3.3) which was originally discovered and proved by Weierstrass [We,p. 155]. All the identities listed below in this section can be derived from it by appropriate shifts of the variables in accordance with relations (2.8). Our approach is different. We show that all the identities of Weierstrass' type are simple algebraic consequences of the bilinear system (3.1) together with the evenness conditions θ r (−u) = (−1) δ r,1 θ r (u), r = 1, 2, 3, 4. This argument is independent of (2.8). To prove (3.3), one should rewrite (3.1a) as θ 1 (u + x|τ )θ 1 (u − x|τ ) = θ 3 (2u|2τ )θ 2 (2x|2τ ) − θ 2 (2u|2τ )θ 3 (2x|2τ ). Multiply this by the similar expression for θ 1 (v + y|τ )θ 1 (v − y|τ ) and subtract the same with the change x ↔ y. Using (3.1a) once again, we arrive at (3.3). All the equations below in this section can be obtained from system (3.1) in a similar way. W.I. Symmetric system: θ 1 (u + x)θ 1 (u − x)θ r (v + y)θ r (v − y) − θ 1 (v + x)θ 1 (v − x)θ r (u + y)θ r (u − y) = θ 1 (u + v)θ 1 (u − v)θ r (x + y)θ r (x − y),(3.θ 2 (u + x)θ 2 (u − x)θ 3 (v + y)θ 3 (v − y) − θ 2 (v + x)θ 2 (v − x)θ 3 (u + y)θ 3 (u − y) = −θ 1 (u + v)θ 1 (u − v)θ 4 (x + y)θ 4 (x − y), (3.5a) θ 2 (u + x)θ 2 (u − x)θ 4 (v + y)θ 4 (v − y) − θ 2 (v + x)θ 2 (v − x)θ 4 (u + y)θ 4 (u − y) = −θ 1 (u + v)θ 1 (u − v)θ 3 (x + y)θ 3 (x − y), (3.5b) θ 3 (u + x)θ 3 (u − x)θ 4 (v + y)θ 4 (v − y) − θ 3 (v + x)θ 3 (v − x)θ 4 (u + y)θ 4 (u − y) = −θ 1 (u + v)θ 1 (u − v)θ 2 (x + y)θ 2 (x − y). (3.5c) W.III. Asymmetric system: θ r (u + x)θ r (u − x)θ r (v + y)θ r (v − y) − θ r (u + y)θ r (u − y)θ r (v + x)θ r (v − x) = (−1) r−1 θ 1 (u + v)θ 1 (u − v)θ 1 (x + y)θ 1 (x − y),(3.θ 3 (u + x)θ 3 (u − x)θ 3 (v + y)θ 3 (v − y) − θ 4 (v + x)θ 4 (v − x)θ 4 (u + y)θ 4 (u − y) = θ 2 (u + v)θ 2 (u − v)θ 2 (x + y)θ 2 (x − y). (3.7) W.V. Mixed identity: θ 1 (u + x)θ 2 (u − x)θ 3 (v + y)θ 4 (v − y) − θ 1 (u − y)θ 2 (u + y)θ 3 (v − x)θ 4 (v + x) = θ 1 (x + y)θ 2 (x − y)θ 3 (u + v)θ 4 (u − v). (3.8) Remark 3.2 Sometimes the Weierstrass addition formula (3.3) is referred to as Fay identity. In fact, it is a generalization of Jacobi's results (see Section 4.2 below). Four-term Jacobi identities In Sect. 3.3 we have presented twelve three-term identities of degree four depending on four variables u, v, x, y. Here we introduce another set of variables W, X, Y, Z and their "dual" counterparts [WW]: W ′ = 1 2 (−W + X + Y + Z), X ′ = 1 2 (W − X + Y + Z), Y ′ = 1 2 (W + X − Y + Z), Z ′ = 1 2 (W + X + Y − Z). (3.9) One can easily verify that W, X, Y, Z are expressed via the "dual" variables W ′ , X ′ , Y ′ , Z ′ by the same formulae, i.e., the "prime procedure" applied to (3.9) yields (W ′ ) ′ = W = 1 2 (−W ′ + X ′ + Y ′ + Z ′ ) etc. We employ the short-hand notation [pqrs] := θ p (W )θ q (X)θ r (Y )θ s (Z), [pqrs] ′ := θ p (W ′ )θ q (X ′ )θ r (Y ′ )θ s (Z ′ ) which is widely used in [WW]. If all the indices of the theta functions coincide, this is further abbreviated to [r] := θ r (W )θ r (X)θ r (Y )θ r (Z), [r] ′ := θ r (W ′ )θ r (X ′ )θ r (Y ′ )θ r (Z ′ ). Below we list all four-term basic identities of degree four which were essentially obtained by Jacobi [J, p. 507]. Here we present these in a more symmetric and comprehensive form. The simplest (and most important) ones are: [ (3.10d) The system (3.10) is a direct algebraic corollary of appropriate addition formulae given in Section 3.3. To see this, we relate the variables u, v, x, y with the variables of the present section as follows:          W = u + x, X = u − x, Y = v + y, Z = v − y. ⇐⇒          W ′ = v − x, X ′ = v + x, Y ′ = u − y, Z ′ = u + y. (3.11) Further, the products of theta functions containing "inappropriate" combinations u ± v, x ± y can be excluded from the addition formulae. Then identities (3.10a), (3.10c) emerge as particular cases of (3.6). Changing v ↔ x in (3.6) (with r = 3) and (3.7), one obtains (3.10b). Finally, (3.10d) is a "dual" version of (3.10b). Remark 3.3 Equations (3.10) are a part of the system of twelve identities written in [WW,pp. 468,488]. It is easy to see that all additional relations are appropriate linear combinations of the basic ones, (3.10). For completeness, we give here the full list: [ [1] − [2] = [4] ′ − [3] ′ , [1] − [3] = [1] ′ − [3] ′ , [1] − [4] = [2] ′ − [3] ′ , [2] − [3] = [1] ′ − [4] ′ , [2] − [4] = [2] ′ − [4] ′ , [3] − [4] = [2] ′ − [1] ′ . (3.12) Now we list symmetric self-dual identities for products of type [rrss] which can also be derived from the addition formulae by algebraic manipulations: [ where r, s ∈ (1, 2, 3, 4), r < s ands,r ∈ (1, 2, 3, 4)\(r, s),r <s. Finally, there are four "fully mixed" identities: [ [11rr] − [11rr] ′ = θ 1 (u + v)θ 1 (u − v)θ r (x + y)θ r (x − y), r = 1, 2, 3, 4. Changing here x ↔ y, one gets [rr11] ′ −[rr11] = θ 1 (u+v)θ 1 (u−v)θ r (x+y)θ r (x−y) = [11rr]−[11rr] ′ . Similarly, (3.13d)-(3.13f) follow from (3.5a)-(3.5c), respectively. To prove (3.14a), we write (3.13a) in terms of the variables u, v, x, y and exchange u ↔ x. Then [2211] − [1122] = −θ 1 (u + v)θ 1 (u − v)θ 2 (x + y)θ 2 (x − y) + θ 2 (u + v)θ 2 (u − v)θ 1 (x + y)θ 1 (x − y). Now (3.14a) holds by virtue of (3.5c). Identities (3.14b)-(3.14f) can be proved in a similar way. Finally, it is easy to see that (3.16) follows from (3.8). Indeed, subtracting (3.8) from the same identity with the exchange x ↔ y yields (3.16a). All other identities in (3.16) are proved in a similar way. Remark 3.4 Identities (3.13), (3.14) differ slightly from those written by Jacobi. For example, in [J, p. 507 We should also stress that these Jacobi identities are direct corollaries of (3.4) at r = 2 and (3.5c). This is in complete agreement with the derivation of (3.13), (3.14) from the Weierstrass addition formulae. Five term Riemann identities The Riemann identities (the term is due to Mumford [M1,page 20]) are simple corollaries of the Jacobi relations (3.10), (3.13)-(3.16). They each express a "primed" quantity as a linear combination of some appropriate four "unprimed" ones. Hence from (3.10) we have the four simplest Riemann identities: Remark 3.5 The identities presented here essentially coincide with the ones given by Mumford [M1, p. 20]. See also [WW]. Equivalence of addition formulae and Jacobi identities In section 3.4, we have obtained the Jacobi identities from the addition formulae. In its turn, one can show that the system (3.10), (3.13), (3.14) implies the addition formulae (3.5)-(3.8). The proof is similar to the one given by Koornwinder for the Riemann identities [K]. In accordance with (3.11), the relation (3.10a) acquires the form θ 1 (u + x)θ 1 (u − x)θ 1 (v + y)θ 1 (v − y) + θ 2 (u + x)θ 2 (u − x)θ 2 (v + y)θ 2 (v − y) = θ 1 (v + x)θ 1 (v − x)θ 1 (u + y)θ 1 (u − y) + θ 2 (v + x)θ 2 (v − x)θ 2 (u + y)θ 2 (u − y). (3.23) Changing here u ↔ x and v ↔ x, one obtains two additional relations: −θ 1 (u + x)θ 1 (u − x)θ 1 (v + y)θ 1 (v − y) + θ 2 (u + x)θ 2 (u − x)θ 2 (v + y)θ 2 (v − y) = −θ 1 (u + v)θ 1 (u − v)θ 1 (x + y)θ 1 (x − y) + θ 2 (u + v)θ 2 (u − v)θ 2 (x + y)θ 2 (x − y), (3.24) θ 1 (u + v)θ 1 (u − v)θ 1 (x + y)θ 1 (x − y) + θ 2 (u + v)θ 2 (u − v)θ 2 (x + y)θ 2 (x − y) = −θ 1 (v + x)θ 1 (v − x)θ 1 (u + y)θ 1 (u − y) + θ 2 (v + x)θ 2 (v − x)θ 2 (u + y)θ 2 (u − y). (3.25) Introduce the notation: A j := θ j (u + x)θ j (u − x)θ j (v + y)θ j (v − y), B j := θ j (u + y)θ j (u − y)θ j (v + x)θ j (v − x), C j := θ j (u + v)θ j (u − v)θ j (x + y)θ j (x − y). (3.26) Then relations (3.23)-(3.25) acquire the form A 1 − B 1 = B 2 − A 2 , A 1 − C 1 = A 2 − C 2 , B 1 + C 1 = B 2 −C 2 which is a system of linear equations for the unknowns A 2 , B 2 , C 2 . The system is degenerate with compatibility condition A 1 −B 1 = C 1 . In terms of the theta functions, this condition is nothing but equation (3.6) with r = 1. Equivalently, one can treat the above equations for A j , B j , C j as a linear system for the unknowns A 1 , B 1 , C 1 . Then, for example, C 1 = B 2 − A 2 which is (3.6) with r = 2. The other addition formulae can be obtained in a similar way. Particular identities 4.1 Consequences of the bilinear identities One can obtain twelve particular identities from the general system (3.2) putting v = 0 or v = ±u (actually, the restriction v = −u can be applied only for identity (3.2d) which leads to (4.3c) below): 2θ 2 1 (u|2τ ) = θ 4 (u|τ )θ 3 (0|τ ) − θ 3 (u|τ )θ 4 (0|τ ), (4.1a) 2θ 2 2 (u|2τ ) = θ 3 (u|τ )θ 3 (0|τ ) − θ 4 (u|τ )θ 4 (0|τ ), (4.1b) 2θ 2 3 (u|2τ ) = θ 3 (u|τ )θ 3 (0|τ ) + θ 4 (u|τ )θ 4 (0|τ ), (4.1c) 2θ 2 4 (u|2τ ) = θ 3 (u|τ )θ 4 (0|τ ) + θ 4 (u|τ )θ 3 (0|τ ), (4.1d) 2θ 1 (u|2τ )θ 4 (u|2τ ) = θ 1 (u|τ )θ 2 (0|τ ), (4.2a) 2θ 2 (u|2τ )θ 3 (u|2τ ) = θ 2 (u|τ )θ 2 (0|τ ), (4.2b) 2θ 2 (2u|2τ )θ 2 (0|2τ ) = θ 2 3 (u|τ ) − θ 2 4 (u|τ ), (4.3a) 2θ 2 (2u|2τ )θ 3 (0|2τ ) = θ 2 2 (u|τ ) − θ 2 1 (u|τ ), (4.3b) 2θ 3 (2u|2τ )θ 2 (0|2τ ) = θ 2 2 (u|τ ) + θ 2 1 (u|τ ), (4.3c) 2θ 3 (2u|2τ )θ 3 (0|2τ ) = θ 2 3 (u|τ ) + θ 2 4 (u|τ ), (4.3d) θ 1 (2u|2τ )θ 4 (0|2τ ) = θ 1 (u|τ )θ 2 (u|τ ), (4.4a) θ 4 (2u|2τ )θ 4 (0|2τ ) = θ 3 (u|τ )θ 4 (u|τ ). (4.4b) In [WW,Section 21.52] there are two particular equations relating theta functions with modular parameters τ and 2τ which are called the transformations of Landen's type: θ 4 (2u|2τ ) θ 4 (0|2τ ) = θ 3 (u|τ )θ 4 (u|τ ) θ 3 (0|τ )θ 4 (0|τ ) , (4.5a) θ 1 (2u|2τ ) θ 4 (0|2τ ) = θ 1 (u|τ )θ 2 (u|τ ) θ 3 (0|τ )θ 4 (0|τ ) . (4.5b) The first identity is derived from (4.4b) and from the relation θ 2 4 (0|2τ ) = θ 3 (0|τ )θ 4 (0|τ ) (4.6) One can unify the sets of equations {(4.16a), (4.17b), (4.18a)}; {(4.16b), (4.17a),(4.18b)}; {(4.16c), (4.17c),(4.18c)} and {(4.16d), (4.17d),(4.18d)} by writing down all the 12 identities (4.16)-(4.18) in the compressed form (−1) β+γ θ 2 1 (u)θ 2 α+1 (u) + θ 2 β+1 (u)θ 2 γ+1 (u) = θ β+1 (2u)θ β+1 (0)θ 2 γ+1 (0), (4.19a) (−1) β+γ θ 2 1 (u)θ 2 α+1 (u) − θ 2 β+1 (u)θ 2 γ+1 (u) = − θ γ+1 (2u)θ γ+1 (0)θ 2 β+1 (0), (4.19b) θ α+1 (2u)θ 3 α+1 (0) = θ 4 α+1 (u) + (−1) α θ 4 1 (u) , (4.19c) θ α+1 (2u)θ 3 α+1 (0) = (−1) γ+1 θ 4 β+1 (u) + (−1) β+1 θ 4 γ+1 (u). (4.19d) where in (4.19a), (4.19b), and (4.19d) the indices α, β, γ are assumed to be any cyclic permutation of {1, 2, 3} and in (4.19c) α = 1, 2, 3. Together with (4.15), the system (4.19) yields the complete set of duplication formulae. Remark 4.1 The complete list of identities (4.7)-(4.11) was originally obtained by Jacobi[J, p. 510] as a particular specification of identities (3.10), (3.13)-(3.16), see also[W, pp. 76-78],[WW,[487][488]. Mumford[M1, p. 22] has obtained a part of relations (4.7)-(4.11) as specific cases of the Riemann identities (3.17)-(3.22). Acknowledgments. We are grateful to I. Marshall and A. Morozov for reading the manuscript and valuable advices. This work was supported in part by grant NSh-1500.2014.2 for support of scientific schools. The work of S.K. was supported in part by RFBR grant 15-01-99504. The work of A.Z. was supported in part by RFBR grants 15-01-05990 and 14-which is also a corollary of (4.4b). The identity (4.5b) is a ratio of (4.4a) and (4.6). Note also that (4.5b) can be obtained from (4.5a) by the shift u → u + τ 2 .Particular addition formulaeOne can obtain important particular cases of the Weierstrass addition formulae which include two variables. Here we present the complete list of eighteen identities which easily follow from (3.4)-(3.8):(4.7a)(4.7b)(4.8b)(4.10c)As a byproduct of (4.7)-(4.10), one gets some extra identities:In particular, the following identity holds:Certainly, the last relation is a corollary of (4.3). Thus, in addition to (2.13), one gets another famous identity for theta constants:θ 4 3 (0) = θ 4 2 (0) + θ 4 4 (0). (4.14)Duplication formulaeThe duplication formulae relate the functions θ a (2u|τ ), a = 1, 2, 3, 4 with appropriate combinations of the functions θ b (u|τ ). All these identities emerge as further degenerations of addition formulae (4.7)-(4.11). Here is the complete list:θ 1 (2u)θ 2 (0)θ 3 (0)θ 4 (0) = 2 θ 1 (u)θ 2 (u)θ 3 (u)θ 4 (u), (4.15) θ 2 (2u)θ 2 (0)θ 2 3 (0) = θ 2 2 (u)θ 2 3 (u) − θ 2 1 (u)θ 2 4 (u), (4.16a) θ 2 (2u)θ 2 (0)θ 2 4 (0) = θ 2 2 (u)θ 2 4 (u) − θ 2 1 (u)θ 2 3 (u), (4.16b) θ 2 (2u)θ 3 2 (0) = θ 4 2 (u) − θ 4 1 (u), (4.16c) θ 2 (2u)θ 3 2 (0) = θ 4 3 (u) − θ 4 4 (u).(4.16d)θ 3 (2u)θ 3 3 (0) = θ 4 1 (u) + θ 4 3 (u), (4.17c) θ 3 (2u)θ 3 3 (0) = θ 4 2 (u) + θ 4 4 (u).(4.17d) θ 4 (2u)θ 4 (0)θ 2 2 (0) = θ 2 2 (u)θ 2 4 (u) + θ 2 1 (u)θ 2 3 (u), (4.18a)θ 4 (2u)θ 4 (0)θ 2 3 (0) = θ 2 1 (u)θ 2 2 (u) + θ 2 3 (u)θ 2 4 (u), (4.18b) θ 4 (2u)θ 3 4 (0) = θ 4 4 (u) − θ 4 1 (u), (4.18c) θ 4 (2u)θ 3 4 (0) = θ 4 3 (u) − θ 4 2 (u).(4.18d) Elements of the Theory of Elliptic Functions. N I Akhiezer, Translations of Mathematical Monographs. 79AMSN.I. Akhiezer, Elements of the Theory of Elliptic Functions, Translations of Mathemat- ical Monographs, 79, AMS 1990. G E Andrews, The Theory of Partitions. AddisonWesley, Reading, MAG.E. Andrews, The Theory of Partitions, AddisonWesley, Reading, MA, 1976. Higher transcendental functions. H Bateman, A Erdélyi, New York, McGraw-HillH. Bateman, A. Erdélyi, Higher transcendental functions, Vol. II New York, McGraw- Hill, 1985. Theta functions and non-linear equations. B A Dubrovin, Russian Mathematical Surveys. 3621192B.A. Dubrovin, Theta functions and non-linear equations, Russian Mathematical Sur- veys, 1981, 36:2, 1192. J Igusa, Theta functions. SpringerJ. Igusa, Theta functions, Springer, 1972. . C G J Jacobi, Gessammelte Werke, B I Berlin, 1881C.G.J. Jacobi, Gessammelte Werke, B. I, Berlin, 1881. A Hurwitz, R Courant, Vorlesungenüber Allgemeine Funktionentheorie und Elliptische Funktionen. A. Hurwitz, R. Courant, Vorlesungenüber Allgemeine Funktionentheorie und Ellip- tische Funktionen, Spinger 1929. On the equivalence of two fundamental theta identities. T H Koornwinder, arXiv:1401.5368Analysis and Applications. 12T.H. Koornwinder, On the equivalence of two fundamental theta identities, Analysis and Applications, 12 (2014), 711-725; arXiv: 1401.5368. . D Mumford, Tata Lectures on Theta I, Progress in Mathematics. 28Birkhäuser BostonD. Mumford, Tata Lectures on Theta I, Progress in Mathematics, 28 Birkhäuser Boston (1983). D Mumford, Tata Lectures on Theta III. Birkhäuser Boston97D. Mumford, Tata Lectures on Theta III, Progress in Mathematics, 97 Birkhäuser Boston (1991). Lehbruch der Algebra, B. III, Elliptische Funktionen und Algebraische Zahlen. H Weber, BraunschweigH. Weber, Lehbruch der Algebra, B. III, Elliptische Funktionen und Algebraische Zahlen, Braunschweig, 1908. . K Weierstrass, . B Mathematische Werke, Iii, &amp; Mayer, Müller, BerlinK. Weierstrass, Mathematische Werke. B. III, Mayer & Müller, Berlin, 1903. A course of modern analysis. E Whittaker, G Watson, CambridgeE. Whittaker, G. Watson, A course of modern analysis, Cambridge, 1927.

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{'abstract': 'This paper is an annotated list of transformation properties and identities satisfied by the four theta functions θ 1 , θ 2 , θ 3 , θ 4 of one complex variable, presented in a readyto-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters τ and 2τ .', 'arxivid': '1502.04603', 'author': ['S Kharchev \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'A Zabrodin \nInstitute for Theoretical and Experimental Physics\nLaboratory of Mathematical Physics\nNational Research University "Higher School of Economics"\n20 Myasnitskaya Ulitsa101000Moscow, MoscowRussia, Russia\n', 'S Kharchev \nInstitute for Theoretical and Experimental Physics\nMoscowRussia\n', 'A Zabrodin \nInstitute for Theoretical and Experimental Physics\nLaboratory of Mathematical Physics\nNational Research University "Higher School of Economics"\n20 Myasnitskaya Ulitsa101000Moscow, MoscowRussia, Russia\n'], 'authoraffiliation': ['Institute for Theoretical and Experimental Physics\nMoscowRussia', 'Institute for Theoretical and Experimental Physics\nLaboratory of Mathematical Physics\nNational Research University "Higher School of Economics"\n20 Myasnitskaya Ulitsa101000Moscow, MoscowRussia, Russia', 'Institute for Theoretical and Experimental Physics\nMoscowRussia', 'Institute for Theoretical and Experimental Physics\nLaboratory of Mathematical Physics\nNational Research University "Higher School of Economics"\n20 Myasnitskaya Ulitsa101000Moscow, MoscowRussia, Russia'], 'corpusid': 119134309, 'doi': '10.1016/j.geomphys.2015.03.010', 'github_urls': [], 'n_tokens_mistral': 12296, 'n_tokens_neox': 10574, 'n_words': 5503, 'pdfsha': 'c4bbf64e466cfa5ded6c81ac5141930b36dbd615', 'pdfurls': ['https://arxiv.org/pdf/1502.04603v2.pdf'], 'title': ['Theta vocabulary I', 'Theta vocabulary I', 'Theta vocabulary I', 'Theta vocabulary I'], 'venue': []}

arxiv

VARIATIONAL METHODS FOR THE SOLUTION OF FRACTIONAL DISCRETE/CONTINUOUS STURM-LIOUVILLE PROBLEMS 9 Apr 2016 Ricardo Almeida Agnieszka B Malinowska ANDM Luísa Morgado Tatiana Odzijewicz VARIATIONAL METHODS FOR THE SOLUTION OF FRACTIONAL DISCRETE/CONTINUOUS STURM-LIOUVILLE PROBLEMS 9 Apr 2016 The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method. Introduction Fractional calculus is a mathematical approach dealing with integral and differential terms of non-integer order. The concept of fractional calculus appeared shortly after calculus itself, but the development of practical applications proceeded very slowly. Only during the last decades, fractional problems have increasingly attracted the attention of many researchers. Applications of fractional operators include chaotic dynamics [46], material sciences [29], mechanics of fractal and complex media [12,28], quantum mechanics [19], physical kinetics [47] and many others (see e.g., [15,43]). Fractional derivatives are nonlocal operators and therefore successfully applied in the study of nonlocal or time-dependent processes [39]. The well-established application of fractional calculus in physics is in the framework of anomalous diffusion behavior [10,13,17,27,33,35]: large jumps in space are modeled by space-fractional derivatives of order between 1 and 2, while long waiting times are modeled by the time derivatives of order between 0 and 1. These partial fractional differential equations can be solved by the method of separating variables, which leads to the Sturm-Liouville and the Cauchy equations. It means that, if we are able to solve the fractional Sturm-Liouville problem and the Cauchy problem, then we can find a solution to the fractional diffusion equation. In this paper, we consider two basic approaches to the fractional Sturm-Liouville problem: discrete and continuous. In both cases, we note that the problem can be formulated as a constrained fractional variational principle. A fractional variational problem consists in finding the extremizer of a functional that depends on fractional derivatives (differences) subject to boundary conditions and possibly some extra constraints. It is worthy to point out that the fractional calculus of variations has itself remarkable applications in classical mechanics. Riewe [41,42] showed that a Lagrangian involving fractional time derivatives leads to an equation of motion with non-conservative forces such as friction. For more about the fractional calculus of variations we refer the reader to [6,22,30,31] while for various approaches to fractional Sturm-Liouville problems we refer to [3,4,23,24,25,45]. The paper is divided into two main parts dedicated, respectively, to discrete (Section 2) and continuous (Section 3) fractional problems. In the first part we give a constructive proof of the existence of orthogonal solutions to the discrete fractional Sturm-Liouville eigenvalue problem (Theorem 2.4), and show that the smallest and largest eigenvalues can be characterized as the optimal values of certain functionals (Theorem 2.5 and Theorem 2.7). Our results are illustrated by an example. In the second part we recall the fractional variational principle and the spectral theorem for the continuous fractional Sturm-Liouville problem. Since for most problems involving fractional derivatives (equations or variational problems) one cannot provide methods to compute the exact solutions analytically, numerical methods should be used for solving such problems. Discretizing both the fractional Sturm-Liouville equation and related with it isoperimetric variational problem we show, by an example, how the variational method can be used for solving the fractional Sturm-Liouville problem. Discrete Fractional Calculus In this section we explain a relationship between the fractional Sturm-Liouville difference problem and a constrained discrete fractional variational principle. Namely, it is possible to look for solutions of Sturm-Liouville fractional difference equations by solving finite dimensional constrained optimization problems. We shall start with necessary preliminaries. There are various versions of the fractional differences, we can mention here those introduced by Diaz and Osler [16], Miller and Ross [36], Atici and Eloe [7,8] or the Caputo difference [1]. In this paper, we use the notion of Grünwald-Letnikov [20,39]. Let us define the mesh points x j = a + jh, j = 0, 1, . . . , N, where h denotes the uniform space step and set D = {x 0 , . . . , x N }. In what follows α ∈ R and 0 < α ≤ 1. Moreover, we set a (α) i := 1, if i = 0 (−1) i α(α−1)···(α−i+1) i! , if i = 1, 2, . . . . (2-1) Definition 2.1. The backward fractional difference of order α, where 0 < α ≤ 1, of function f : D → R is defined by 0 ∆ α k f (x k ) := 1 h α k i=0 (−1) i α(α − 1) · · · (α − i + 1) i! f (x k−i ). (2-2) while k ∆ α N f (x k ) := 1 h α N −k i=0 (−1) i α(α − 1) · · · (α − i + 1) i! f (x k+i ) (2-3) is the forward fractional difference of function f . Fractional backward and forward differences are linear operators. Theorem 2.2. (cf. [38]) Let f, g be two real functions defined on D and β, γ ∈ R. Then 0 ∆ α k [γf (x k ) + βg(x k )] = γ 0 ∆ α k f (x k ) + β 0 ∆ α k g(x k ), k ∆ α N [γf (x k ) + βg(x k )] = γ k ∆ α N f (x k ) + β k ∆ α N g(x k ), for all k. The following formula of the summation by parts for fractional operators will be essential for proving results concerning variational problems. Lemma 2.3. (cf. [11]) Let f , g be two real functions defined on D. Then N k=0 g(x k ) 0 ∆ α k f (x k ) = N k=0 f (x k ) k ∆ α N g(x k ). If f (x 0 ) = f (x N ) = 0 or g(x 0 ) = g(x N ) = 0, then N k=1 g(x k ) 0 ∆ α k f (x k ) = N −1 k=0 f (x k ) k ∆ α N g(x k ). (2-4) 2A. The Sturm-Liouville Problem. In this subsection our topic is the Sturm-Liouville fractional difference equation: k ∆ α N (p(x k ) 0 ∆ α k y(x k )) + q(x k )y(x k ) = λr(x k )y(x k ), k = 1, . . . , N − 1, (2-5) with boundary conditions: y(x 0 ) = 0, y(x N ) = 0. (2-6) We assume that p(x i ) > 0, r(x i ) > 0, q(x i ) is defined and real valued for all x i , i = 0, . . . , N , and λ is a parameter. It is required to find the eigenfunctions and the eigenvalues of the given boundary value problem, i.e., the nontrivial solutions of (2-5)-(2-6) and the corresponding values of the parameter λ. Theorem below gives an answer to this question. r(x k )y i (x k )y j (x k ) = 0, and they span R N −1 : any vector ϕ = (ϕ(x k )) N −1 k=1 ∈ R N −1 has a unique expansion ϕ(x k ) = N −1 i=1 c i y i (x k ), 1 ≤ k ≤ N − 1. The coefficients c i are given by c i = ϕ, y i r y i , y i r . Proof. Observe that equations (2-5)-(2-6) can be considered as a system of N −1 linear equations with N − 1 real unknowns y(x 1 ), . . . , y(x N −1 ). The corresponding matrix form is as follows: Ay T = λRy T , (2-7) where the entries A ij of A are A (α) ij =    1 h 2α q(x i ) + N −i k=0 (a (α) k ) 2 p(x i+k ) , i = j 1 h 2α N −i k=0 a (α) k p(x i+k ) k+i m=0 a (α) m and k − m + i = j, i = j. and R = diag{r(x 1 ), . . . , r(x N −1 )}. Writing (2-7) as R −1 Ay T = λy T (2-8) we get an eigenvalue problem with the symmetric matrix R −1 A. Because of the equivalence of problem (2-5)-(2-6) with problem (2-8) it follows from matrix theory that the Sturm-Liouville problem (2-5)-(2-6) has N − 1 linearly pairwise orthogonal real independent eigenfunctions with all eigenvalues real. Now we would like to find constants c 1 , . . . , c N −1 such that ϕ( x k ) = N −1 i=1 c i y i (x k ), 1 ≤ k ≤ N − 1. Note that ϕ, y j r = N −1 i=1 c i y i , y j r = N −1 i=1 c i y i , y j r = c j y j , y j r because of orthogonality. Therefore c i = ϕ,y i r y i ,y i r , 1 ≤ i ≤ N − 1. 2B. Isoperimetric Variational Problems. In this section we prove two theorems connecting the Sturm-Liouville problem (2-5)-(2-6) with isoperimetric problems of discrete fractional calculus of variations. Theorem 2.5. Let y 1 denote the first eigenfunction, normalized to satisfy the isoperimetric constraint I[y] = N k=1 r(x k )(y(x k )) 2 = 1 (2-9) associated to the first eigenvalue λ 1 of problem (2-5)-(2-6). Then y 1 is a minimizer of functional J[y] = N k=1 p(x k ) ( 0 ∆ α k y(x k )) 2 + q(x k )(y(x k )) 2 (2-10) subject to boundary condition y(x 0 ) = 0, y(x N ) = 0 and isoperimetric constraint (2-9). Moreover J[y 1 ] = λ 1 . Proof. Suppose that y is a minimizer of J. Then, by Theorem 5 [32], there exists a real constant λ such that y satisfies equation k ∆ α N (p(x k ) 0 ∆ α k y(x k )) + q(x k )y(x k ) − λr(x k )y(x k ) = 0, k = 1, . . . , N − 1, (2-11) together with y(x 0 ) = 0, y(x N ) = 0 and isoperimetric constraint (2-9). Let us multiply (2-11) by y(x k ) and sum up from k = 1 to N − 1, then N −1 k=1 y(x k ) k ∆ α N (p(x k ) 0 ∆ α k y(x k )) + q(x k )(y(x k )) 2 = N −1 k=1 λr(x k )(y(x k )) 2 By summation by parts (2-4) N −1 k=1 y(x k ) k ∆ α N (p(x k ) 0 ∆ α k y(x k )) = N k=1 p(x k ) ( 0 ∆ α k y(x k )) 2 . As (2-9) holds and y(x N ) = 0 we obtain J[y] = λ. Any solution to problem (2-9)-(2-10) that satisfies equation (2-11) must be nontrivial since (2)(3)(4)(5)(6)(7)(8)(9) holds, so λ must be an eigenvalue. According to Theorem 2.4 there is the least element in the spectrum being eigenvalue λ 1 , and the corresponding eigenfunction y (1) normalized to meet the isoperimetric condition. Therefore J[y (1) ] = λ 1 . Definition 2.6. We will call functional R defined by R[y] = J[y] I[y] , where J[y] is given by (2-10) and I[y] by (2-9), the Rayleigh quotient for the fractional discrete Sturm-Liouville problem (2-5)-(2-6). Theorem 2.7. Assume that y satisfies boundary conditions y(x 0 ) = y(x N ) = 0 and is nontrivial. (i) If y is a minimizer of Rayleigh quotient R for the Sturm-Liouville problem (2-5)-(2-6), then value of R in y is equal to the smallest eigenvalue λ 1 , i.e., R[y] = λ 1 . (ii) If y is a maximizer of Rayleigh quotient R for the Sturm-Liouville problem (2-5)-(2-6), then value of R in y is equal to the largest eigenvalue λ N −1 , i.e., R[y] = λ N −1 . Proof. We give the proof only for the case (i) as the second case can be proved similarly. Suppose that y satisfying boundary conditions y(x 0 ) = y(x N ) = 0 and being nontrivial, is a minimizer of Rayleigh quotient R and that value of R in y is equal to λ. Consider the following functions φ : [−ε, ε] −→ R h −→ I[y + hη] = N k=1 r(x k )(y(x k ) + hη(x k )) 2 ψ : [−ε, ε] −→ R h −→ J[y + hη] = N k=1 p(x k ) ( 0 ∆ α k (y(x k ) + hη(x k ))) 2 + q(x k )(y(x k ) + hη(x k )) 2 and ζ : [−ε, ε] −→ R h −→ R[y + hη] = J[y+hη] I[y+hη] , where η : D → R, η(x 0 ) = η(x N ) = 0, η = 0. Since ζ is of class C 1 on [−ε, ε] and ζ(0) ≤ ζ(h), |h| ≤ ε, we deduce that ζ ′ (0) = d dh R[y + hη] h=0 = 0. Moreover, notice that ζ ′ (h) = 1 φ(h) ψ ′ (h) − ψ(h) φ(h) φ ′ (h) and ψ ′ (0) = d dh J[y + hη] h=0 = 2 N k=1 [p(x k ) 0 ∆ α k y(x k ) 0 ∆ α k η(x k ) + q(x k )y(x k )η(x k )] , φ ′ (0) = d dh I[y + hη] h=0 = 2 N k=1 r(x k )y(x k )η(x k ). Therefore ζ ′ (0) = d dh R[y + hη] h=0 = 2 I[y] N k=1 [p(x k ) 0 ∆ α k y(x k ) 0 ∆ α k η(x k ) + q(x k )y(x k )η(x k )] − J[y] I[y] N k=1 r(x k )y(x k )η(x k ) = 0. Having in mind that J[y] I[y] = λ, η(x 0 ) = η(x N ) = 0 and using the summation by parts formula (2-4) we obtain N −1 k=1 [ k ∆ α N (p(x k ) 0 ∆ α k y(x k )) + q(x k )y(x k ) − λr(x k )y(x k )] η(x k ) = 0. Since η is arbitrary, we have k ∆ α N (p(x k ) 0 ∆ α k y(x k )) + q(x k )y(x k ) − λr(x k )y(x k ) = 0, k = 1, . . . , N − 1. (2-12) As y = 0 we have that λ is an eigenvalue of (2-12). On the other hand, let λ i be an eigenvalue and y i the corresponding eigenfunction, then k ∆ α N (p(x k ) 0 ∆ α k y i (x k )) + q(x k )y i (x k ) = λ i r(x k )y i (x k ). (2-13) Similarly to the proof of Theorem 2.5, we can obtain N k=1 p(x k ) 0 ∆ α k y i (x k ) 2 + q(x k )(y i (x k )) 2 N k=1 r(x k )(y i (x k )) 2 = λ i , for any 1 ≤ i ≤ N − 1. That is R[y i ] = J[y i ] I[y i ] = λ i . Finally, since the minimum value of R at y is equal to λ, i.e., λ ≤ R[y i ] = λ i ∀i ∈ {1, . . . , N − 1} we have λ = λ 1 . Example 2.8. Let us consider the following problem: minimize J[y] = N k=1 ( 0 ∆ α k y(x k )) 2 (2-14) subject to I[y] = N k=1 (y(x k )) 2 = 1 (2-15) and y(x 0 ) = y(x N ) = 0, where N is fixed. In this case the Euler-Lagrange equation takes the form Table 1. Those results are obtained by solving the matrix eigenvalue problem of the form (2-8). Observe that problem (2-14)-(2-15) can be treated as a finite dimensional constrained optimization problem. Namely, the problem is to minimize function J of N − 1 variables: k ∆ α N 0 ∆ α k y(x k ) = λy(x k ), k = 1, . . . , N − 1.y 1 = y(x 1 ), . . . , y N −1 = y(x N −1 ) on the N − 1 dimensional sphere with equation N −1 k=1 y 2 k = 1. Table 2 and Figure 1 present the solution to problem (2-14)- (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) for N = 4, h = 1 and different values of α's. By Theorem 2.5 the first eigenvalue λ 1 of (2-16) is the minimum value of J on N 1 k=1 y 2 k = 1 and the first eigenfunction of (2-16) is the minimizer of this problem. Other eigenfunctions and eigenvalues of (2-16) we can found by using the first order necessary optimality conditions (Karush-Kuhn-Tucker conditions), that is, by solving the following system of equations: ∂J ∂y k = λ ∂I ∂y k , k = 1, . . . , N − 1, N −1 k=1 y 2 k = 1. (2-17) α y(x 1 ) y(x 2 ) y(x 3 ) Continuous Fractional Calculus This section is devoted to the continuous fractional Sturm-Liouville problem and its formulation as a constrained fractional variational principle. Namely, we shall show that this formulation can be used to approximate the solutions. As in the discrete case there are several different definitions for fractional derivatives [21], the most well known are the Grünwald-Letnikov, the Riemann-Liouville and the Caputo fractional derivatives. a D α x f (x) := 1 Γ(1 − α) d dx x a (x − t) −α f (t)dt, and x D α b f (x) := −1 Γ(1 − α) d dx b x (t − x) −α f (t)dt, respectively; (2) the left and right Caputo fractional derivatives of order α by C a D α x f (x) := 1 Γ(1 − α) x a (x − t) −α f ′ (t)dt, and C x D α b f (x) := −1 Γ(1 − α) b x (t − x) −α f ′ (t)dt, respectively. The Caputo derivative seems more suitable in applications. Let us recall that the Caputo derivative of a constant is zero, whereas for the Riemann-Liouville is not. Moreover, the Laplace transform, which is used for solving fractional differential equations, of the Riemann-Liouville derivative contains the limit values of the Riemann-Liouville fractional derivatives (of order α − 1) at the lower terminal x = a. Mathematically such problems can be solved, but there is no physical interpretation for such type of conditions. On the other hand the Laplace transform of the Caputo derivative imposes boundary conditions involving the value of function at the lower point x = a which usually are acceptable physical conditions. The Grünwald-Letnikov definition is a generalization of the ordinary discretization formulas for integer order derivatives. a D α x f (x) := lim h→0 + 1 h α ∞ k=0 (−1) k α k f (x − kh), and GL x D α b f (x) := lim h→0 + 1 h α ∞ k=0 (−1) k α k f (x + kh), respectively. Here α k stands for the generalization of binomial coefficients to real numbers (see ). However in this section, for historical reasons, we denote (w α k ) := (−1) k α k rather than a (α) i . Relations between those three types of derivatives are given below and can be found respectively in [39,21]. a D α x f (x) = a D α x f (x) − f (a) Γ(1 − α) (x − a) −α (3-1) and C x D α b f (x) = x D α b f (x) − f (b) Γ(1 − α) (b − x) −α . If f (a) = 0 or f (b) = 0, then C a D α x f (x) = a D α x f (x) or C x D α b f (x) = x D α b f (x) , respectively. It is well known that we can approximate the Riemann-Liouville fractional derivative using the Grünwald-Letnikov fractional derivative. Given the interval [a, b] and a partition of the interval x j = a + jh, for j = 0, 1, ..., N and some h > 0 such that x N = b, we have a D α x j f (x j ) = 1 h α j k=0 (w α k )f (x j−k ) + O(h), x j D α b f (x j ) = 1 h α N −j k=0 (w α k )f (x j+k ) + O(h), that is, the truncated Grünwald-Letnikov fractional derivatives are first-order approximations of the Riemann-Liouville fractional derivatives. Using the relation (3-1), we deduce a decomposition sum for the Caputo fractional derivatives: C a D α x j f (x j ) ≈ 1 h α j k=0 (w α k )f (x j−k ) − f (a) Γ(1 − α) (x j − a) −α =: C aD α x j f (x j ), (3-2) C x j D α b f (x j ) ≈ 1 h α N −j k=0 (w α k )f (x j+k ) − f (b) Γ(1 − α) (b − x j ) −α =: C x jD α b f (x j ). (3-3) 3A. Variational Problem. Consider the following variational problem: to minimize the functional I[y] = b a L(x, y(x), C a D α x y(x)) dx,(3-4) subject to the boundary conditions y(a) = y a and y(b) = y b , y a , y b ∈ R, (3)(4)(5) where 0 < α < 1 and the Lagrange function L : [a, b] × R 2 → R is differentiable with respect to the second and third arguments. Theorem 3.5. ([2]) If y is a solution to (3-4)-(3-5) , then y satisfies the following fractional differential equation ∂L ∂y (x, y(x), C a D α x y(x)) + x D α b ∂L ∂ C a D α x y (x, y(x), C a D α x y(x)) = 0, t ∈ [a, b]. (3-6) Relations like (3)(4)(5)(6) are known in the literature as the Euler-Lagrange equation, and provide a necessary condition that every solution of the variational problem must verify. Adding to problem (3-4)- (3)(4)(5) an integral constraint b a g(x, y(x), C a D α x y(x)) dx = K, (3-7) where K is a fixed constant and g : [a, b] × R 2 → R is a differentiable function with respect to the second and third arguments, we get an isoperimetric variational problem. In order to obtain a necessary condition for a minimizer we define the new function F := λ 0 L(x, y(x), C a D α x y(x)) − λg(x, y(x), C a D α x y(x)), (3-8) where λ 0 , λ are Lagrange multipliers. Then every solution y of the fractional isoperimetric problem given by (3-4)- (3)(4)(5) and (3-7) is also a solution to the fractional differential equation (c.f. [5]) ∂F ∂y (x, y(x), C a D α x y(x)) + x D α b ∂F ∂ C a D α x y (x, y(x), C a D α x y(x)) = 0, t ∈ [a, b]. (3-9) Moreover, if y is not a solution to ∂g ∂y (x, y(x), C a D α x y(x)) + x D α b ∂g ∂ C a D α x y (x, y(x), C a D α x y(x)) = 0, t ∈ [a, b], (3-10) then we can put λ 0 = 1 in (3-8). 3A.1. Discretization Method 1. Using the approximation formula for the Caputo fractional derivative given by , we can discretize functional (3)(4) in the following way. Let N ∈ N, h = (b − a)/N and the grid x j = a + jh, j = 0, 1, . . . , N . Then I[y] = N k=1 x k x k−1 L(x, y(x), C a D α x y(x)) dx ≈ N k=1 hL(x k , y(x k ), C a D α x k y(x k )) ≈ N k=1 hL(x k , y(x k ), C aD α x k y(x k )). (3)(4)(5)(6)(7)(8)(9)(10)(11) This is the direct way to solve the problem, using discretization techniques. hL(x k , y(x k ), C aD α x k y(x k )) → min, subject to y 0 = y a and y N = y b where y k := y(x k ). Using the first order necessary optimality conditions given by the following system of N − 1 equations: ∂Φ ∂y j = 0, ∀j = 1, . . . , N − 1, we get ∂L ∂y (x j , y(x j ), C aD α x j y(x j )) + N −j k=0 (w α k ) h α ∂L ∂ C a D α x y (x j+k , y(x j+k ), C aD α x j+k y(x j+k )) = 0 (3-12) j = 1, . . . , N − 1. As N → ∞, that is, as h → 0, the solutions of system (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) converge to the solutions of the fractional Euler-Lagrange equation associated to the variational problem (see [40,Theorem 4.1]). Constrained variational problem given by (3-4)- (3)(4)(5) and (3-7) can be solved similarly. More precisely, in this case we have to replace the Lagrange function L by the augmented function F = λ 0 L − λg, and proceed with similar calculations. 3B. Sturm-Liouville problem. Consider the fractional differential equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) subject to the boundary conditions y(a) = y(b) = 0. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) Equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) together with condition (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) is called the fractional Sturm-Liouville problem. As in the discrete case, it is required to find the eigenfunctions and the eigenvalues of the given boundary value problem, i.e., the nontrivial solutions of (3-13)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and the corresponding values of the parameter λ. C D α b p(x) C D α a + q(x) y(x) = λr α (x)y(x), In what follows we assume: (A): Let 1 2 < α < 1 and p, q, r α be given functions such that: p ∈ C 1 [a, b] and p(x) > 0 for all x ∈ [a, b]; q, r α ∈ C[a, b], r α (x) > 0 for all x ∈ [a, b] and ( √ r α ) ′ is Hölderian, of order (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) has an infinite increasing sequence of eigenvalues λ 1 , λ 2 , ..., and to each eigenvalue λ k there is a corresponding continuous eigenfunction y k which is unique up to a constant factor. β ≤ α − 1 2 , on [a, b]. The fractional Sturm-Liouville problem can be remodeled as a fractional isoperimetric variational problem. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) associated to the first eigenvalue λ 1 of problem (3-13)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and assume that function D α b (p C D α a y 1 ) is continuous. Then, y 1 is a minimizer of the following variational functional: I[y] = b a r α (x)y 2 (x) dx = 1,J[y] = b a p(x)( C D α a y(x)) 2 + q(x)y 2 (x) dx, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) in the class of C[a, b] functions with C D α a y and D α b (p C D α a y) continuous in [a, b], subject to the boundary conditions y(a) = y(b) = 0 (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and isoperimetric constraint (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15). Moreover, J[y 1 ] = λ 1 . 3B.1. Discretization Method 3. Using the approximation formula for the Caputo fractional derivatives given by (3-2)-(3-3), we can discretize equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) in the following way. Let N ∈ N, h = (b − a)/N and the grid x j = a + jh, j = 0, 1, . . . , N . Then at x = x i , (3-13) may be discretized as: h −2α r α (x i ) N −i k=0 (w α k ) p(x i+k ) i+k l=0 (w α l ) y i+k−l + q(x i ) r α (x i ) y i = λy i , i = 1, . . . , N − 1, which in the matrix form, may be written as AY = λY, (3-18) where Y = [y 1 y 2 . . . y N −1 ], y i = y(x i ), and A = (c ik ), i = 1, 2, . . . , N − 1, k = 1, 2, . . . , N − 1, with c ik =          h −2α rα(x i ) N −i j=0 w α j 2 p(x j+i ) + q(x i ) rα(x i ) , i = k h −2α rα(x i ) N −i j=0 w α j w α j+i−k p(x j+i ), i > k h −2α rα(x i ) N −i j=k−i w α j w α j+i−k p(x j+i ), i < k , reducing in this way the Sturm-Liouville problem to an algebraic eigenvalue problem. Example 3.8. Let us consider the following problem: minimize the functional 1 0 ( C 0 D α x y(x)) 2 dx, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) under the restrictions 1 0 y 2 (x) dx = 1 and y(0) = y(1) = 0, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) where α = 3/4. Since y(0) = 0, we have C 0 D α x y(x) = 0 D α x y(x). Discretizing the problem, as explained in Section 3A.1, we obtain a finite dimensional constrained optimization problem: (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) subject to Using the Maple package Optimization, we get approximations of the optimal solutions to (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) for different values of N . Table 3 shows values of λ 1 for N = 5, 10, 15. Note that λ 1 is the value of (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21), where y = [0, y 1 , . . . , y N −1 , 0] is the optimal solution to (3-21)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22). N k=1 N 2α−1 k i=0 (w α i )y k−i 2 → min, In other words, λ 1 is an approximation of the minimum value of functional (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) and the first eigenvalue of the Sturm-Liouville (which is the Euler-Lagrange equation for considered variational problem). Figure 2 presents minimizers y for N = 5, 10, 15. Table 3. Values of λ 1 for N = 5, 10, 15. Observe that the unique solution to the Euler-Lagrange equation (cf. (3)(4)(5)(6)(7)(8)(9)(10)) associated to the integral constraint is y(x) = 0. As y(x) = 0 is not a solution to (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)-(3-20) (condition 1 0 y 2 (x) dx = 1 fails), we can consider λ 0 = 1 in (3)(4)(5)(6)(7)(8). Therefore the auxiliary function is F := ( C 0 D α x y(x)) 2 − λy 2 (x). Thus, Φ(y 1 , . . . , y N −1 ) := N k=1 h ( C 0 D α x k y k ) 2 − λy 2 k , and the computation of ∂Φ/∂y j leads to − λy j + N 2α N −j k=0 (w α k ) j+k l=0 (w α l )y j+k−l = 0, j = 1, . . . , N − 1. (3-23) Solving system of equations (3-23) together with (3-22) we obtain not only an approximation of the optimal solution to problem (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20), but also other solutions to the Euler-Lagrange equation (3)(4)(5)(6)(7)(8)(9) as N → ∞. In other words, we get some approximations of the eigenvalues and eigenfunctions of the Sturm-Liouville problem. Table 4 presents approximations of the eigenvalues obtained by this procedure for N = 5, 10, 15. In Figure 3 we present the optimal solution for this procedure, that corresponds to the eigenvector associated with the eigenvalue λ 1 . In Conclusions Since 1986, when the seminal works were published [37,44], fractional differential equations have become a popular way to model anomalous diffusion. As it is stated in [33] this type of approach is the most reasonable: the fractional derivative in space codes large particle jumps (that lead to anomalous super-diffusion) while the time-fractional derivative models time delays between particle motion. Fractional diffusion equations have been used, e.g., to model pollution in ground water [9] and flow in porous media [18]. Many other examples can be found in [33,34]. Table 5. Approximation of the eigenvalues using method 3B.1. It was proved in [26] that, under appropriate assumptions, the following space-time fractional diffusion equation where, 0 < β < 1, 1 2 < α < 1, and C D β 0+,t , C D α b−,x , C D α a+,x are partial fractional derivatives, with the boundary and initial conditions: y k , f E β (−λ k t β )y k (x). In (4-4): f, g := b a r α (x)f (x)g(x) dx, E β the one-parameter Mittag-Leffler function, y k and λ k (k = 1, 2, . . .) are the eigenfunctions and the eigenvalues of the fractional Sturm-Liouville problem (3-13)- (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). Thus numerical methods, presented in this paper, for finding the eigenvalues and eigenfunctions of fractional Sturm-Liouville problems can be also used to approximate solution to fractional diffusion problems of the form (4-1)-(4-3). We have presented a link between fractional Sturm-Liouville and fractional isoperimetric variational problems that provides a possible method for solution of those firstly mentioned problems. Discrete problems with the Grünwald-Letnikov difference were analyzed: we proved the existence of orthogonal solutions to the discrete fractional Sturm-Liouville eigenvalue problem and showed that its eigenvalues can be characterized as values of certain functionals. For continuous problems with the Caputo fractional derivatives, in order to examine the performance of the proposed method, the approximation based on the shifted Grünwald-Letnikov definition were used. This type of discretization is most popular in practical application, when solving numerically fractional diffusion equations, due to the fact that codes are mass-preserving [14]. Theorem 2 . 4 . 24The Sturm-Liouville problem (2-5)-(2-6) has N − 1 real eigenvalues, which we denote by λ 1 ≤ λ 2 ≤ · · · ≤ λ N −1 . The corresponding eigenfunctions,y 1 , y 2 , . . . , y N −1 : {x 1 , . . . , x N −1 } → R, are mutually orthogonal: if i = j, ( 2 - 16 ) 216Together with boundary condition y(x 0 ) = y(x N ) = 0 it is the Sturm-Liouville eigenvalue problem where p(x i ) = 1, r(x i ) = 1 and q(x i ) = 0 for k = 1, . . . , N − 1. Let us choose N = 4 and h = 1. Eigenvalues of (2-16) for different values of α's are presented in Figure 1 . 1The solution to problem (2-14)-(2-15) for different values of α's: Table 2 . 2The solution to problem (2-14)-(2-15) for different values of α's: 1/4, 1/2, 3/4, 1. Definition 3. 1 . 1Let f : [a, b] → R be a function and α a positive real number such that 0 < α < 1. We define (1) the left and right Riemann-Liouville fractional derivatives of order α by Definition 3 . 2 . 32Let 0 < α < 1 be a real. The left and right Grünwald-Letnikov fractional derivative of a function f , of order α, is defined as GL Proposition 3 . 3 . 33Let us assume that the function f is integrable in [a, b]. Then, the Riemann-Liouville fractional derivatives exist and coincide with Grünwald-Letnikov fractional derivatives. Proposition 3 . 4 . 34Let us assume that f is a function for which the Caputo fractional derivatives exist together with the Riemann-Liouville fractional derivatives in [a, b]. Then C 3A. 2 . 2Discretization Method 2. By the previous discussion, the initial problem of minimization of the functional (3-4), subject to boundary conditions (3-5), can be numerically replaced by the finite dimensional optimization problem Φ(y 1 , . . . , y N −1 ) := N k=1 Theorem 3.6. ([23]) Under assumption (A), the fractional Sturm-Liouville Problem (3-13)- Theorem 3 . 7 . 37([23]) Let assumption (A) holds and y 1 be the eigenfunction, normalized to satisfy the isoperimetric constraint Figure 2 . 2Approximation of solutions to problem (3-19)-(3-20) (Method 1): ⋄(N = 5); •(N = 10); +(N = 15). Figures 4, 5 and 6 we compare the approximation of the optimal solutions to (3-19)-(3-20), obtained by solving (3-21)-(3-22) (Method 1) and (3-23)-(3-22) (Method 2), for N = 5, 10, 15. N Figure 3 . 3Approximation of solutions to problem (3-19)-(3-20) (Method 2): ⋄(N = 5); •(N = 10); +(N = 15).Now let us consider the Sturmthe boundary conditions y(0) = y(1) = 0. (3-25) Note that, under conditions of Theorem 3.7, equation (3-24) is the Euler-Lagrange equation for isoperimetric problem (3-19)-(3-20).Table 3.8 presents approximations of the eigenvalues of (3-24) obtained by discretization method 3B.1 for N = 5, 10, 20, 40, 80, 160 (for N = Figure 4 . 4N = 5: •(Method 1); +(Method 2). Figure 5 . 5N = 10: •(Method 1); +(Method 2). 20, 40, 80, 160 only the first 14 eigenvalues are listed). In Figure 3.8 we present normalized eigenfunctions, obtained for N = 100, corresponding to the eigenvalues λ 1 , λ 2 , λ 3 and λ 4 . Figure 6 . 6N = 15: •(Method 1); +(Method 2). N x p(x) C D α a+,x + q(x) u(t, x), for all (t, x) ∈ (0, ∞)× [a, b], Figure 7 . 7u(t, a) = u(t, b) = 0, t ∈ (0, ∞), (4-2) u(0, x) = f (x), x ∈ [a, b], Normalized eigenfunctions obtained with N = 100, corresponding to the eigenvalues λ 1 , λ 2 , λ 3 and λ 4 (top left, top right, bottom left and bottom right, respectively). has a continuous solution u : [0, ∞) × [a, b] → R given by the series u(t, x) = ∞ k=1 Table 1. Eigenvalues of (2-16) for different values of α's: 1/4, 1/2, 3/4, 1.α λ 1 λ 2 λ 3 0.25 0.7102065750 1.148567387 1.349294886 0.50 0.6004483933 1.353660384 1.831047473 0.75 0.5779798778 1.632135974 2.496488153 1 0.5857864376 2.0 3.414213562 Table 4. Values of λ i for N = 5, 10, 15.5 10 15 λ 1 4.603751969 4.491185168 4.426964909 λ 2 13.67144835 14.31569449 14.33350940 λ 3 22.69092491 26.35335634 26.90113751 λ 4 29.24531071 39.48118456 41.37391615 λ 5 - 52.54234156 56.93534748 λ 6 - 64.64953668 73.03700902 λ 7 - 74.96494602 89.07875858 λ 8 - 82.83813371 104.5749014 λ 9 - 87.76536891 119.0339408 λ 10 - - 132.0436041 λ 11 - - 143.2212682 λ 12 - - 152.2566950 λ 13 - - 158.8942685 λ 14 - - 162.9518168 4.603751972 4.491185175 4.387575384 4.314056432 4.264767769 4.231946921 λ 2 13.67144835 14.31569450 14.29943076 14.18275912 14.08194289 14.01015799 λ 3 22.69092491 26.35335634 27.02784640 27.01132309 26.88877847 26.78184511 λ 4 29.24531071 39.48118456 41.95747334 42.33045300 42.25841874 42.134291285 10 20 40 80 160 λ 1 λ 5 − 52.54234157 58.40981791 59.60122278 59.68496380 59.56753673 λ 6 − 64.64953668 75.99486098 78.61012095 79.00578911 78.93437596 λ 7 − 74.96494602 94.25189512 99.05856280 99.96479334 99.98764503 λ 8 − 82.83813372 112.8375161 120.7904806 122.4632696 122.6454529 λ 9 − 87.76536891 131.3694072 143.5891552 146.3337902 146.7516690 λ 10 − − 149.5318910 167.3194776 171.5033012 172.2513810 λ 11 − − 166.9946039 191.8029693 197.8472756 199.0332700 λ 12 − − 183.4744810 216.9142113 225.3053712 227.0563318 λ 13 − − 198.6917619 242.4962397 253.7773704 256.2351380 λ 14 − − 212.4063351 268.4292940 283.2100111 286.5368179 RICARDO ALMEIDA, AGNIESZKA B. 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{'abstract': 'The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method.', 'arxivid': '1604.04272', 'author': ['Ricardo Almeida ', 'Agnieszka B Malinowska ', 'ANDM Luísa Morgado ', 'Tatiana Odzijewicz '], 'authoraffiliation': [], 'corpusid': 119680624, 'doi': '10.2140/jomms.2017.12.3', 'github_urls': [], 'n_tokens_mistral': 19027, 'n_tokens_neox': 15829, 'n_words': 7900, 'pdfsha': '178ec14c95059b888ec049e9558f6c8a7107d6de', 'pdfurls': ['https://arxiv.org/pdf/1604.04272v1.pdf'], 'title': ['VARIATIONAL METHODS FOR THE SOLUTION OF FRACTIONAL DISCRETE/CONTINUOUS STURM-LIOUVILLE PROBLEMS', 'VARIATIONAL METHODS FOR THE SOLUTION OF FRACTIONAL DISCRETE/CONTINUOUS STURM-LIOUVILLE PROBLEMS'], 'venue': []}

arxiv

NONMONOTONE BARZILAI-BORWEIN GRADIENT ALGORITHM FOR ℓ 1 -REGULARIZED NONSMOOTH MINIMIZATION IN COMPRESSIVE SENSING 19 Jul 2012 Yunhai Xiao Soon-Yi Wu ANDLiqun Qi NONMONOTONE BARZILAI-BORWEIN GRADIENT ALGORITHM FOR ℓ 1 -REGULARIZED NONSMOOTH MINIMIZATION IN COMPRESSIVE SENSING 19 Jul 2012nonsmooth optimizationnonconvex optimizationBarzilai-Borwein gradient algorithmnonmonotone line searchℓ 1 regularizationcompressive sensing AMS subject classifications 65L0965K0590C3090C25 This paper is devoted to minimizing the sum of a smooth function and a nonsmooth ℓ 1 -regularized term. This problem as a special cases includes the ℓ 1 -regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the ℓ 1 -norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the ℓ 1 -norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive ℓ 1 -regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for ℓ 1 -regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.AMS subject classifications. 65L09, 65K05, 90C30, 90C25 research activities over the past years[11,12,10,13,20]. Compressive sensing is based on the fact that if the original signal is sparse or approximately sparse in some orthogonal basis, an exact restoration can be produced via solving problem (1.2). 1. Introduction. The focus of this paper is on the following structured minimization min x∈R n F (x) = f (x) + µ x 1 ,(1.1) where f : R n → R is a continuously differentiable (may be nonconvex) function that is bounded below; · 1 denotes the ℓ 1 -norm of a vector; parameter µ > 0 is used to trade off both terms for minimization. Due to its structure, problem (1.1) covers a wide range of apparently related formulations in different scientific fields including linear inverse problem, signal/image processing, compressive sensing, and machine learning. Problem formulations. A popular special case of model (1.1) is the ℓ 1 -norm regularized least square problem min x∈R n 1 2 Ax − b 2 2 + µ x 1 ,(1.2) where A ∈ R m×n (m ≪ n) is a linear operator, and b ∈ R m is an observation. Model (1.2) mainly appears in compressive sensing -an emerging methodology in digital signal processing, and has attracted intensive Another prevalent case of (1.1) that has been achieved much interest in machine learning is the linear and logistic regression. Given the training date A = [a 1 , · · · , a m ] ⊤ ∈ R m×n and class labels y ∈ {−1, +1} m . A linear classifier is a hyperplane {w i : x ⊤ a i + b = 0}, where x ∈ R n is a set of weights and b ∈ R is the intercept. A frequently used model is the ℓ 2 -loss support vector machine min x∈R m ,b∈R m i=1 max{0, 1 − y i (x ⊤ a i + b)} 2 + µ x 1 ,(1.3) Because of the "max" operation, the ℓ 2 -loos function is continuous, but not differentiable. Based on the conditional probability, another popular model is the logistic regression min x∈R n ,b∈R m i=1 log 1 + e −(x ⊤ ai+b)yi + µ x 1 . (1.4) Obviously, the logistic loss function is twice differentiable. Although the models of these problems have similar structures, they may be very different from real-data point of view. For example, in compressive sensing, the length of measurement m is much smaller than the length of original signal (m ≪ n) and the encoding matrix A is dense. However, in machine learning, the numbers of instance m and features n are both large and the data A is very sparse. Existing algorithms. Since the ℓ 1 -regularized term is non-differentiable when x contains values of zero, the use of the standard unconstrained smooth optimization tools are generally precluded. In the past decades, a wide variety of approaches has been proposed, analyzed, and implemented in compressive sensing and machine learning literatures. This includes a variety of algorithms for special cases where f (x) has a specific functional form such as the least square (1.2), the square loss (1.3) and the logistic loss (1.4). In the following, we briefly review some of them in each literature. The first popular approach falls into the coordinate descent method. At the current iterate x k , the simple coordinate descent method updates one component at a time to generate x j k , j = 1, . . . , n + 1, such that x 1 k = x k , x n+1 k = x k+1 , and solves a one-dimensional subproblem min z F (x j k + ze j ) − F (x j k ), (1.5) where e j is defined as the j-th column of an identity matrix. Clearly, the objective function has one variable, and one non-differentiable point at z = −e j . To solve the logistic regression model (1.4), BBR [23] solves the sub-problem approximately by the use of trust region method with Newton step; CDN [14] improves BBR's performance by applying a one-dimensional Newton method and a line search technique. Instead of cyclically updating one component at each time, the stochastic coordinate descent method [35] randomly selects the working components to attain better performance; the block coordinate gradient descent algorithm -CGD [38,43] is based on the approximated convex quadratic model for f , and selects the working variables with some rules. The second type of approach is to transform model (1.1) into an equivalent box-constrained optimization problem by variable splitting. Let x = u − v with u i = max{0, x i } and v i = max{0, −x i }. Then, model (1.1) can be reformulated equivalently as min u,v f (u − v) + µ n i=1 (u i + v i ), s.t. u ≥ 0, v ≥ 0. (1.6) The objective function and constraints are smooth, and therefore, it can be solved by any standard boxconstrained optimization technique. However, an obvious drawback of this approach is that it doubles the number of variables. GPSR [22] solves (1.6), and subsequently solves (1.2), by using Barzilai-Borwein gradient method [2] with an efficient nonmonotone line search [24]. It is actually an application of the wellknown spectral projection gradient [8] in compressive sensing. Trust region Newton algorithm -TRON [28,42] The third type of method is to approximate the ℓ 1 -regularized term with a differentiable function. The simple approach replaces the ℓ 1 -norm with a sum of multi-quadric functions l(x) n i x 2 i + ǫ, where ǫ is a small positive scalar. This function is twice-differentiable and lim ǫ→0 + l(x) = x 1 . Subsequently, several smooth unconstrained optimization approaches can be applied, based on this approximation. However, the performance of these algorithms is much influenced by the parameter values, and the condition number of the corresponding Hessian matrix becomes larger as ǫ decreases. The Nesterov's smoothing technique [29] is to construct smooth functions to approximate any general convex nonsmooth function. Based on this technique, NESTA [4] solves problem (1.2) by using first-order gradient information. The fourth type of approach falls into the subgradient-based Newton-type algorithm. The important attempt in this class is from Andrew and Gao [1], who extend the well-known limited memory BFGS method [31] to solve ℓ 1 -regularized logistic regression model (1.4), and propose an orthant-wise limited memory quasi-Newton method -OWL-QN. At each iteration, this method computes a search direction over an orthant containing the previous point. The subspace BFGS method -subBFGS [41] involves an inner iteration approach to find the descent quasi-Newton direction and a subgradient Wolfe-condtions to determine the stepsize which ensures that the objective functions are decreasing. This method enjoys global convergence and is capable of solving general nonsmooth convex minimization problems. Finally, to solve model (1.2), besides GPSR and NESTA, there are other numerous specially designed solvers. By an operator splitting technique, Hale, Yin and Zhang derive the iterative shrinkage/thresholding fixed-point continuation algorithm (FPC) [25]. By combining the interior-point algorithm in [26], FPC is also extended to solve large-scale ℓ 1 -regularized logistic regression in [36]. TwIST [7] and FISTA [3] speed up the performance of IST and have virtually the same complexity but with better convergence properties. Another closely related method is the sparse reconstruction algorithm SpaRSA [39], which is to minimize non-smooth convex problem with separable structures. SPGL1 [5] solves the lasso model (1.2) by the spectral gradient projection method with an efficient Euclidean projection on ℓ 1 -norm ball. The alternating directions method -YALL1 [40], investigates ℓ 1 -norm problems from either the primal or the dual forms and solves ℓ 1 -regularized problems with different types. All the reviewed algorithms differ in various aspects such as the convergence speed, ease of implementation, and practical applicability. Moreover, there is no enough evidence to verify that which algorithm outperforms the others under all scenarios. Contributions and organization. Although much progress has been achieved in solving the problem (1.1), these algorithms mainly deal with the case where f is a convex function even a least square. In this paper, unlike all the reviewed algorithms, we propose a Barzilai-Borwein gradient algorithm for solving ℓ 1 -regularized nonsmooth minimization problems. At each iteration, we approximate f locally by a convex quadratic model, where the Hessian is replaced by the multiplies of a spectral coefficient with an identity matrix. The search direction is determined by minimizing the quadratic model and taking full use of the ℓ 1 -norm structure. We show that the generated direction is descent which guarantees that there exists a positive stepsize along the direction. In our algorithm, we adopt the nonmonotone line search of Grippo, Lampariello, and Lucidi [24], which allows the function values to increase occasionally in some iteration but decrease in the whole iterative process. The attractive property of the nonmonotone line search is that it saves much number of function evaluations which should be the main computational burden in large dataset. The method is easily performed, where only the value of objective function and the gradient of the smooth term are needed at each iteration. We show that each cluster of the iterates generated by this algorithm is a stationary point of F . In this paper, although we mainly consider the ℓ 1 -regularizer, the ℓ 2 -norm regularization problem and the matrix trace norm problems can also be readily included in our framework. Thus, this broaden the capability of the algorithm. We implement the algorithm to solve problem (1.1) where f is a nonconvex smooth function from CUTEr library to show its efficiency. Moreover, we also run the algorithm to solve ℓ 1 -regularized least square, and do performance comparisons with the state-of-the-art algorithms NESTA, CGD, TwIST, FPC and GPSR. The comparisons results show that the proposed algorithm is effective, comparable, and promising. We organize the rest of this paper as follows. In Section 2, we briefly recall some preliminary results in optimization literature to motivate our work, construct the search direction, and present the steps of our algorithm along with some remarks. In Section 3, we establish the global convergence theorem under some mild conditions. In Section 4, we show that how to extend the algorithm to solve ℓ 2 -norm and matrix trace norm minimization problems. In Section 5, we present experiments to show the efficiency of the algorithm in solving the ℓ 1 -regularized nonconvex problem and least square problem. Finally, we conclude our paper in Section 6. Algorithm. 2.1. Preliminary results. First, consider the minimization of the smooth function without the ℓ 1norm regularization min x∈R f (x). (2.1) The basic idea of Newton's method for this problem is to iteratively use the quadratic approximation q k to the objective function f (x) at the current iterate x k and to minimize the approximation q k . Let f : R n → R be twice continuously differentiable, and its Hessian G k = ∇ 2 f (x k ) be positive definite. Function f at the current x k is modeled by the quadratic approximation q k , f (x k + s) ≈ q k (s) = f (x k ) + ∇f (x k ) ⊤ s + 1 2 s ⊤ G k s, where s = x − x k . Minimizing q k (s) yields x k+1 = x k − G −1 k ∇f (x k ), which is Newton's formula and s k = x k+1 − x k = −G −1 k ∇f (x k ) is the so-called Newton's direction. For the positive definite quadratic function, Newton's method can reach the minimizer with one iteration. However, when the starting point is far away from the solution, it is not sure that G k is positive definite and Newton's direction d k is a descent direction. Let the quadratic model of f at x k+1 be f (x) ≈ f (x k+1 ) + ∇f (x k+1 ) ⊤ (x − x k+1 ) + 1 2 (x − x k+1 ) ⊤ G k+1 (x − x k+1 ). Finding the derivative yields ∇f (x) ≈ ∇f (x k+1 ) + G k+1 (x − x k+1 ). Setting x = x k , s k = x k+1 − x k , and y k = ∇f (x k+1 ) − ∇f (x k ) we get G k+1 s k ≈ y k . (2.2) For various practical problems, the computing efforts of the Hessian matrices are very expensive, or the evaluation of the Hessian is difficult; even the Hessian is not available analytically. These lead to the quasi-Newton method which generates a series of Hessian approximations by the use of the gradient, and at the same time maintains a fast rate of convergence. Instead of computing the Hessian G k , quasi-Newton method constructs the Hessian approximation B k , where the sequence {B k } possesses positive definiteness and satisfies B k+1 s k = y k . (2.3) In general, such B k+1 will be produced by updating B k with some typical and popular formulae such as BFGS, DFP, and SR1. Unfortunately, the standard quasi-Newton algorithm, or even its limited memory versions, doesn't scale well enough to train very large-scale models involving millions of variables and training instances, which are commonly encountered, for example, in natural language processing. The main computational burden of Newton-type algorithm is the storage of a large matrix at per-iteration, which may be out of the memory capability for a PC. It should be develop a matrix-free algorithm to deal with large-scale problems but also belongs to the quasi-Newton framework. For this purpose, it would like to furthermore simplify the approximation Hessian B k as a diagonal matrix with positive components, i.e., B k = λ k I with an identity matrix I and λ k > 0. Then, the quasi-Newton condition changes to the form λ k+1 Is k = y k . Multiplying both sides by s ⊤ k , gives λ (1) k+1 = s ⊤ k y k s k 2 2 . (2.4) Similarly, multiplying both sides by y ⊤ k , yields λ (2) k+1 = y k 2 2 s ⊤ k y k . (2.5) Observing both formulae, it indicates that if s ⊤ k y k > 0, the matrix λ k+1 I is positive definite, which ensures that the search direction −λ −1 k ∇f (x k ) is descent at current point. The formulae (2.4) and (2.5) were firstly developed by Barzilai and Borwein [2] for the quadratic case of f . This method essentially consists the steepest descent method, and adopts the choice of (2.4) or (2.4) as the stepsize along a negative gradient direction. Barzilai and Borwein [2] showed that the corresponding iterative algorithm is R-superlinearly convergent for the quadratic case. Raydan [33] presented a globalization strategy based on nonmonotone line search [24] for the general non-quadratic case. Other developments in Barzilai 2.2. Algorithm. Due to its simplicity and numerical efficiency, the Barzilai-Borwein gradient method is very effective to deal with large-scale smooth unconstrained minimization problems. However, the application of the Barilai-Borwein gradient algorithm to ℓ 1 -regularized nonsmooth optimization is problematic since the regularization is non-differentiable. In this subsection, we construct an iterative algorithm to solve the ℓ 1regularized structured nonconvex optimization problem. The algorithm can be described as the iterative form x k+1 = x k + α k d k , where α k is a stepsize, and d k is a search direction defined by minimizing a quadratic approximated model of F . Now, we turn to our attention to consider the original problem with ℓ 1 -regularizer. Since ℓ 1 -term is not differentiable, hence, at current x k , objective function F is approximated by the quadratic approximation Q k , F (x k + d) =f (x k + d) + µ x k + d 1 ≈f (x k ) + ∇f (x k ) ⊤ d + λ k 2 d 2 2 + µ x k 1 + x k + hd 1 − x k 1 h Q k (d), (2.6) where h is a small positive number. The term in [·] can be considered as an approximate Taylor expansion of x k + d 1 with a small h, and the case h = 1 reduces the equivalent form x k + d 1 . Minimizing (2.6) yields min d∈R n Q k (d) ⇔ min d∈R n ∇f (x k ) ⊤ d + λ k 2 d 2 2 + µ h x k + hd 1 ⇔ min d∈R n h 2 λ k ∇f (x k ) ⊤ d + λ k 2 d 2 2 + µ h x k + hd 1 ⇔ min d∈R n 1 2 x k + hd − x k − h λ k ∇f (x k ) 2 2 + µh λ k x k + hd 1 ⇔ min d∈R n n i=1 1 2 x i k + hd i − x i k − h λ k ∇f i (x k ) 2 + µh λ k |x i k + hd i | (2.7) where x i k , d i , and ∇f i (x k ) denote the i-th component of x k , d, and ∇f (x k ) respectively. The favorable structure of (2.7) admits the explicit solution x i k + hd i k = max x i k − h λ k ∇f i (x i k ) − µh λ k , 0 x i k − h λ k ∇f i (x k ) |x i k − h λ k ∇f i (x k )| . Hence, the search direction at current point is d k = − 1 h x k − max x k − h λ k ∇f (x k ) − µh λ k , 0 x k − h λ k ∇f (x k ) |x k − h λ k ∇f (x k )| . (2.8) where | · | and " max " are interpreted as componentwise and the convention 0 · 0/0 = 0 is followed. When µ = 0, (2.8) reduces to d k = −λ −1 k ∇f (x k ), i.e., the traditional Barizilai-Borwein gradient algorithm in smooth optimization. The key motivation for this formulation is that the optimization problem in Eq. (2.7) can be easily solved by exploiting the structure of the ℓ 1 -norm. Lemma 2.1. For any real vectors a ∈ R n and b ∈ R n , the following function L(x) is non-decreasing L(x) = a + bx 1 − a 1 x , x ∈ (0, ∞). (2.9) Proof. Note that L(x) = a + bx 1 − a 1 x = n i |a i + b i x| − |a i | x n i l i (x), hence, it reduces to prove that l i (x) is non-decreasing for each i. (a). When a i ≥ 0 and a i x + b i ≥ 0. It is clear that l i (x) = b i . (b). When a i ≥ 0 and a i x + b i ≤ 0, we have l i (x) = −2a i − b i x x = −2a i x − b i . (c). When a i ≤ 0 and a i x + b i ≥ 0, we have l i (x) = 2a i + b i x x = 2a i x + b i . (d). When a i ≤ 0 and a i x + b i ≥ 0, we have l i (x) = −b i . It is not difficult to see that l i (x) is non-decreasing at each case. Hence, L(x) is non-decreasing. The following lemma shows that the direction defined by (2.8) is descent if d k = 0. Lemma 2.2. Suppose that λ k > 0 and d k is determined by (2.8). Then F (x k + θd k ) ≤ F (x k ) + θ ∇f (x k ) ⊤ d k + µ x k + hd k 1 − µ x k 1 h + o(θ) θ ∈ (0, h],(2. 10) and ∇f (x k ) ⊤ d k + µ x k + hd k 1 − µ x k 1 h ≤ − λ k 2 d k 2 2 . (2.11) Proof. By the differentiability of f and the convexity of x 1 , we have that for any θ ∈ (0, h] (θ/h ∈ (0, 1]), F (x k + θd k ) − F (x k ) = f (x k + θd k ) − f (x k ) + µ x k + θd k 1 − µ x k 1 = f (x k + θd k ) − f (x k ) + µ θ h (x k + hd k ) + (1 − θ h )x k 1 − µ x k 1 ≤ f (x k + θd k ) − f (x k ) + θµ h x k + hd k 1 + (1 − θ h )µ x k 1 − µ x k 1 = θ∇f (x k ) ⊤ d k + o(θ) + θ µ h x k + hd k 1 − µ h x k 1 , which is exactly (2.10). Noting that d k is the minimizer of (2.6) and θ ∈ (0, h], from (2.6) and the convexity of x 1 , we have ∇f (x k ) ⊤ d k + λ k 2 d k 2 2 + µ x k + hd k 1 − µ x k 1 h ≤θ∇f (x k ) ⊤ d k + λ k 2 θd k 2 2 + µ h x k + θhd k 1 − µ h x k 1 ≤θ∇f (x k ) ⊤ d k + λ k θ 2 2 d k 2 2 + θµ h 2 x k + h 2 d k 1 + µ h (1 − θ h ) x k 1 − µ h x k 1 . Hence, (1 − θ)∇f (x k ) ⊤ d k + µ h x k + hd k 1 − θµ h 2 x k + h 2 d k 1 − µ h (1 − θ h ) x k 1 ≤ − λ k 2 (1 − θ 2 ) d k 2 2 . (2.12) The last three terms of the left side in (2.12) can be re-organized as µ h x k + hd k 1 − θ h x k + h 2 d k 1 − (1 − θ h ) x k 1 = µ h x k + hd k 1 − x k 1 − θ x k + h 2 d k 1 − x k 1 h = µ h x k + hd k 1 − x k 1 − θ h · x k + h 2 d k 1 − x k 1 h 2 ≥ µ h x k + hd k 1 − x k 1 − θ h · x k + hd k 1 − x k 1 h = µ h (1 − θ){ x k + hd k 1 − x k 1 }, (2.13) where the inequality is from Lemma 2.1. Combining (2.12) with (2.13), it produces (1 − θ)∇f (x k ) ⊤ d k + (1 − θ) µ x k + hd k 1 − µ x k 1 h ≤ − λ k 2 (1 − θ 2 ) d k 2 2 . (2.14) Dividing both sides of (2.14) by (1 − θ) and noting θ ∈ (0, h], we get the desirable result (2.11). When the search direction is determined, a suitable stepsize along this direction should be found to determine the next iterative point. In this paper, unlike the traditional Armijo line search or the Wolfe-Powell line search, we pay particular attention to a nonmonotone line search strategy. The traditional Armijo line search requires the function value to decrease monotonically at each iteration. As a result, it may cause the sequence of iterations following the bottom of a curved narrow valley, which commonly occurs in difficult nonlinear problems. To overcome this difficultly, a credible alternative is to allow an occasional increase in the objective function at each iteration. To easy comprehension of the proposed algorithm, we briefly recall the earliest nonmonotone line search technique by Grippo, Lampariello, and Lucidi [24]. Let δ k ∈ (0, 1), ρ ∈ (0, 1) andm be a positive integer. The nonmonotone line search is to choose the smallest nonnegative integer j k such as the stepsize α k =αρ j k satisfing f (x k + α k d k ) ≤ max 0≤j≤m(k) f (x k−j ) + δα k ∇f (x k ) ⊤ d k ,(2.15) where m(0) = 0 and 0 ≤ m(k) ≤ min{m(k − 1) + 1,m}. If m(k) = 0, the above nonmonotone line search reduces to the standard Armijo line search. For the ℓ 1 -regularized nonsmooth problem (1.1), based on Lemma 2.2, the inequality (2.15) should be modified as F (x k + α k d k ) ≤ max 0≤j≤m(k) F (x k−j ) + δα k ∆ k ,(2.16) where In light of all derivations above, we now describe the nonmonotone Barzilai-Borwein gradient algorithm (abbreviated as NBBL1) as follows. ∆ k = ∇f (x k ) ⊤ d k + µ x k + hd k 1 − µ x k 1 h .(2. Algorithm 1. (NBBL1) Initialization: Choose x 0 and constants µ > 0. Constantsα > 0, ρ ∈ (0, 1), δ ∈ (0, 1), h ∈ (0, 1] and positive integerm. Set k = 0. Step 1. Stop if d k 2 = 0. Otherwise, continue. Step 2. Compute d k via (2.8). Step 3. Compute α k via (2.16). Step 4. Let x k+1 = x k + α k d k . Step 5. Let k = k + 1. Go to Step 1. Remark 1. We have shown that if λ k > 0, then the generated direction is descent. However, in this case, the condition λ k > 0 may fail to be fulfilled and the hereditary descent property is not guaranteed any more. To cope with this defect, we should keep the sequence {λ k } uniformly bounded; that is, for sufficiently small λ (min) > 0 and sufficiently large λ (max) > 0, the λ k is forced as λ k = min{λ (max) , max{λ k , λ (min) }}. This approach ensures that λ k is bounded from zero and subsequently ensures that d k is descent at periteration. Remark 2. From Lemma 2.2, it is clear that there exists a constant θ ∈ (0, h] such that x k + θd k is a descent point in sense of (2.10). Hence, in practical computation, it is suggested to choose the initial stepsize asα = h. 3. Convergence analysis. This section is devoted to presenting some favorable properties of the generated direction and establishing the global convergence of Algorithm 1 subsequently. Our convergence result utilizes the following assumptions. Proof. If d k = 0, then Lemma 2.2 shows that d k is descent direction at x k , which implies that x k is not a stationary point of F . On the other hand, if d k = 0 is the solution of (2.7), for any αd ∈ R n with α > 0 we have α∇f (x k ) ⊤ d + λ k α 2 2 d 2 2 + µ h x k + αhd 1 ≥ µ h x k 1 . (3.1) Since f (x k + αd) − f (x k ) = α∇f (x k ) ⊤ d + o(α), this together with (3.1) yields F ′ (x k ; d) = lim α↓0 f (x k + αd) − f (x k ) + µ x k + αd 1 − µ x k 1 α = lim α↓0 α∇f (x k ) ⊤ d + o(α) + µ x k + αd 1 − µ x k 1 α ≥ lim α↓0 − λ k α 2 2 d 2 2 + o(α) α + µ x k + αd 1 − µ x k 1 − µ h x k + αhd 1 − µ h x k 1 α . ≥ lim α↓0 − λ k α 2 2 d 2 2 + o(α) α =0, where the second inequality is from Lemma 2.1. Hence, x k is a stationary point of F . The proof of the following lemma is similar with the Theorem in [24]. Lemma 3.2. Let l(k) be an integer such that k − m(k) ≤ l(k) ≤ k and F (x l(k) ) = max 0≤j≤m(k) F (x k−j ). Then the sequence {F (x l(k) )} is nonincreasing and the search direction d l(k) satisfies lim k→∞ α l(k) d l(k) 2 = 0. (3.2) Proof. From the definition of m(k), we have m(k + 1) ≤ m(k) + 1. Hence F (x l(k+1) ) = max 0≤j≤m(k+1) F (x k+1−j ) ≤ max 0≤j≤m(k)+1 F (x k+1−j ) = max{F (x l(k) ), F (x k+1 )} =F (x l(k) ). Moreover, by (2.16), we have for all k >m, F (x l(k) ) = F (x l(k)−1 + α l(k)−1 d l(k)−1 ) ≤ max 0≤j≤m(l(k)−1) F (x l(k)−1−j ) + δα l(k)−1 ∆ l(k)−1 = F (x l(l(k)−1) ) + δα l(k)−1 ∆ l(k)−1 . By assumption 1, the sequence {F (x l(k) )} admits a limit for k → ∞. Hence, it follows that lim k→∞ α l(k) ∆ l(k) = 0. In this case, we assume that there exists a constant ǫ > 0 such that d k 2 ≥ ǫ, ∀ k ∈ K. (3.7) Since α k is the first value for satisfying (2.16), it follows from Step 3 in Algorithm 1 that there exists an indexk such that, for all k ≥k and k ∈ K, F (x k + α k ρ d k ) > max 0≤j≤m(k) F (x k−j ) + δ α k ρ ∆ k ≥ F (x k ) + δ α k ρ ∆ k . (3.8) Since f is continuous differentiable, by the mean-value theorem on f , we can find there exists a constant θ k ∈ (0, 1), such that f (x k + α k ρ d k ) − f (x k ) = α k ρ ∇f (x k + θ k α k ρ d k ) ⊤ d k . By combining with (3.8), we have ∇f (x k + θ k α k ρ d k ) ⊤ d k + µ x k + α k ρ d k 1 − µ x k 1 α k /ρ > δ∆ k . (3.9) Sinceα = h and α k → 0 in (3.6), we have α k < ρh as k → ∞. It follows from Lemma 2.1 that µ x k + α k ρ d k 1 − µ x k 1 α k /ρ − µ x k + hd k 1 − µ x k 1 h ≤ 0. Subtracting both sides of (3.9) by ∆ k and noting the definition of ∆ k , it is clear that ∇f (x k + θ k α k ρ d k ) ⊤ d k − ∇f (x k ) ⊤ d k ≥∇f (x k + θ k α k ρ d k ) ⊤ d k − ∇f (x k ) ⊤ d k + µ x k + α k ρ d k 1 − µ x k 1 α k /ρ − µ x k + hd k 1 − µ x k 1 h > − (1 − δ)∆ k ≥(1 − δ) λ (min) 2 d k 2 2 . (3.10) Taking the limit as k ∈ K, k → ∞ in the both sides of (3.10) and using the smoothness of f , we obtain 0 = ∇f (x) ⊤d − ∇f (x) ⊤d ≥ (1 − δ) λ (min) 2 d 2 2 , which implies d k 2 → 0 as k ∈ K, k → ∞. This yields a contradiction because (3.7) indicates that d k 2 is bounded. 4. Some extensions. In this section, we show that our algorithm can be readily extended to solve ℓ 2 -norm and matrix trace norm minimization problems in machine learning; thus, broaden the applicable range of our approach significantly. Firstly, we consider the ℓ 2 -regularization problem min x∈R n F (x) = f (x) + µ x 2 . It is not difficult to deduce that, the search direction d k is determined by minimizing min d∈R n 1 2 x k + hd − x k − h λ k ∇f (x k ) 2 2 + µh λ k x k + hd 2 . From [21], the explicit solution is x k + hd k = max x k − h λ k ∇f (x k ) 2 − µh λ k , 0 x k − h λ k ∇f (x k ) x k − h λ k ∇f (x k ) 2 , i.e., d k = − 1 h x k − max x k − h λ k ∇f (x k ) 2 − µh λ k , 0 x k − h λ k ∇f (x k ) x k − h λ k ∇f (x k ) 2 . Now, we consider the matrix trace norm minimization problem min X∈R m×n F (X) = f (X) + µ X * ,(4.1) where the functional X * is the trace norm of matrix X, which is defined as the sum of its singular values. That is, assume that X has r positive singular values of σ 1 ≥ σ 2 ≥ . . . ≥ σ r ≥ 0, then X * = r i=1 σ i . The matrix trace norm is alternatively known as the Schatten ℓ 1 -norm, Ky Fan norm, and nuclear norm [34]. Such problem has been received much attention because it is closely related to the affine rank minimization problem, which has appeared in many control applications including controller design, realization theory and model reduction. As it has been done in the previous sections, we can readily reformulate (2.6) as the following quadratic model to determine the search direction, min D∈R m×n 1 2 X k + hD − X k − h λ k ∇f (X k ) 2 2 + µh λ k X k + hD * . (4.2) To get the exact solution of (4.2), we now consider the singular value decomposition (SVD) of a matrix Y ∈ R m×n with rank r, Y = U ΣV ⊤ , Σ = diag({σ i } 1≤i≤r ), where U and V are m × r and r × n matrices respectively with orthonormal columns, and the singular value σ i is positive. For each τ > 0, we let D τ (Y ) = U D τ (Σ)V ⊤ , D τ (Σ) = diag([σ i − τ ] + ), where [·] + = max{0, ·}. It is shown that D τ (Y ) obeys the following nuclear norm minimization problem [9], i.e., D τ (Y ) = arg min X τ X * + 1 2 X − Y 2 F . (4.3) Comparing (4.2) to (4.3), we deduce that X k + hD k = U D µh/λ k (Σ)V ⊤ and D µh/λ k (Σ) = diag [σ i − µh λ k ] + , or, equivalently, D k = − 1 h X k − U D µh/λ k (Σ)V ⊤ . Subsequently, it is easily to derive the nonmonotone Barzilai and Borwein gradient algorithmic framework for solving ℓ 2 -norm and matrix trace norm regularization problems. Numerical experiments. In this section, we present numerical results to illustrate the feasibility and efficiency of NBBL1. We partition our experiments into three classes based on different types of f . In the first class, we perform our algorithm to solve ℓ 1 -regularized nonconvex problem. In the second class, we test our algorithm to solve ℓ 1 -regularized least squares which mainly appear in compressive sensing. In the third class, we compare some state-of-the-art algorithms in compressive sensing to show the efficiency of our algorithm. All experiments are performed under Windows XP and Matlab 7.8 (2009a) running on a Lenovo laptop with an Intel Atom CPU at 1.6 GHz and 1 GB of memory. 5.1. Test on ℓ 1 -regularized nonconvex problem. Our first test is performed on a set of the nonconvex unconstrained problems from the CUTEr [16] library. The second-order derivatives of all the selected problems are available. Since we are interested in large problems, we only consider the problems with size at least 100. For these problems, we use the dimensions that is admissible of the "double large" installation of CUTEr. The algorithm stops if the norm of the search direction is small enough; that is, d k 2 ≤ tol 1 . (5.1) The iterative process is also stopped if the number of iterations exceeds 10000 without achieving convergence. In this experiment, we take tol 1 = 10 −8 , h = 1, λ (min) = 10 −20 , λ (max) = 10 20 . In the line search, we chooseα 0 = 1, ρ = 0.35, δ = 10 −4 andm = 5. We test NBBL1 with different parameter values µ = {0, 1/4, 1/2, 2}. The numerical results are presented in Table 5 From Table 5.1, we see that NBBL1 works successfully for all the test problems in each case. Particularly, NBBL1 always produces great accuracy solutions within little consuming time. The proposed algorithm requires large number of iterations for some special problems, such as problems FLETCHER, NONCVXU2, BROYDN7D with parameter µ = 0, problems FLETCHER and BROYDN7D with µ = 0.25, problem FLETCHER with µ = 0.5, and VARDIM with µ = 2. However, if lower precision is permitted, the number can be decreased dramatically. The first part of Table 5.1 presents the numerical results of NBBL1 for solving a smooth nonconvex minimization problem without any regularization. From the last second column in this part, we observe that the norm of the final gradient is sufficiently small. The important observation verifies that the proposed algorithm is very efficient to solve unconstrained smooth minimization problems. It is not a pleasant supervise, because our algorithm reduces to the well-known nonmonotone Barzailai-Borwein gradient of Raydan [33] in this case. 5.2. Test on ℓ 1 -regularized least square. Letx be a sparse or a nearly sparse original signal, A ∈ R m×n (m ≪ n) be a linear operator, ω ∈ R m be a zero-mean Gaussian white noise, and b ∈ R m be an observation which satisfies the relationship b = Ax + ω. Recent compressive sensing results show that, under some technical conditions, the desirable signal can be reconstructed almost exactly by solving the ℓ 1 -regularized least square (1.2). In this subsection, we perform two classes of numerical experiments for solving (1.2) by using the Gaussian matrices as the encoder. In the first class, we show that our algorithm performs well to decode a sparse signal, while in the second class we do a series of experiments with different h to choose the best one. We measure the quality of restoration x * by means of the relative error to the original signalx; that is RelErr = x * −x 2 x 2 . (5.2) In the first test, we use a random matrix A with independent identically distributions Gaussian entries. The ω is the additive Gaussian noise of zero mean and standard deviation σ. Due to the storage limitations of PC, we test a small size signal with n = 2 11 , m = 2 9 . The original contains randomly p = 2 6 non-zero elements. Besides, we also choose the noise level σ = 10 −3 . The proposed algorithm starts at a zero point and terminates when the relative change of two successive points are sufficient small, i.e., Comparing the left plot to the right one in Figure 5.1, we clearly see that the original sparse signal is restored almost exactly. We see that all the blue peaks are circled by the red circles, which illustrates that the original signal has been found almost exactly. All together, this simple experiment shows that our algorithm performs quite well, and provides an efficient approach to recover large sparse non-negative signal. x k − x k−1 2 x k−1 2 < tol 2 . We have clearly known that the last term in the approximate quadratic model (2.6) is equivalent to x k + d 1 exactly when h = 1. Next, we provide evidence to show that other values can be potentially and dramatically better than h = 1. We conduct a series of experiments and compare the performance at each case. In our experiments, we set all the parameters values as the previous test except for n = 2 10 . We present, in Figure 5 In Figure 5.2, the number of iterations, the computing time and the quality of restorations are greatly influenced by the h values. Generally, as h increases, NBBL1 always has good performance. The right plot clearly demonstrates that the relative error decreases dramatically at the very beginning and then becomes slightly after 0.2. However, the quality of restoration can not be improved any more after 0.7. On the other hand, the left and the middle plots show that the number of iterations and the computing time slightly increase after h = 0.8. Taking three plots together, these plots verify that the performance of NBBL1 is sensitive to the h values, and the value h ∈ [0.7, 0.8] may be the better choice. 5.3. Comparisons with NESTA-Ct, GPSR-BB, CGD, TwIST and FPC-BB. The third class of the experiment is to test against several state-of-the-art algorithms which are specifically designed in recent years to solve ℓ 1 -regularized problems in compressive sensing or linear inverse problems. It is difficult to compare each algorithm in a very fair way, because each algorithm is compiled with different parameter settings, such as the termination criterions, the staring points, or the continuation techniques. Hence, as usual, in our performance comparisons, we run each code from the same initial point, use all the default parameter values, and only observe the convergence behavior of each algorithm to attain a similar accuracy solution. NESTA 1 uses Nesterov's smoothing technique [37] and gradient method [30] to solve basis pursuit denoising problem. The current version is capable of solving ℓ 1 -norm regularization problems with different types including (1.2). In this experiment, we test NESTA with continuation (named NESTA-Ct) for comparison, where this algorithm solves a sequence of problems (1.2) by using a decreasing sequence of values of µ. Additionally, NESTA-Ct uses the intermediated solution as a warm start for the next problem. In running NESTA, all the parameters are taken as default except TolVar is set to be 1.e − 5 to obtain similar quality solutions with others. GPSR-BB 2 (Gradient Projections for Sparse Reconstruction) [22] reformulates the original problem (1.2) as a box-constrained quadric programming problem (1.6) by splitting x = u − v. Figueiredo, Nowak and Wright use a gradient projection method with Barziali-Borwein steplength [2] for its solution. Moreover, the nonmonotone line search [24] is also used to improve its performance. For the comparison with GPSR-BB, we use its continuation variant and set all parameters as default. The well-known CGD 3 uses gradient algorithm to solve (1.5) in order to obtain the search direction d i k = ze i in i ∈ J , where J is a nonempty subset of {1, ..., n}, and choose the index subset J a Gausssouthwell rule. The iterative process x k+1 = x k + α k d k continues until some termination critera are met, where d i k = 0 with i / ∈ J and the stepsize α k by using a Armijo rule. In running CGD, we use the code CGD in its Matlab package, and set all the parameter as default except for init=2 to start the iterative process at x 0 = A ⊤ b. TwIST 4 is a two-step IST algorithm for solving a class of linear inverse problems. Specifically, TwIST is designed to solve min u J (u) + µ 2 Au − f 2 2 , (5.4) where A is a linear operator, and J (·) is a general regularizer, which can be either the ℓ 1 -norm or the TV. The iteration framework of TwIST is u k+1 = (1 − α)u k−1 + (α − δ)u k + δΦ µ (ξ k ), where α, δ > 0 are parameters, ξ k = u k + A ⊤ (f − Au k ) and Φ µ (ξ k ) = arg min u J (u) + µ 2 u − ξ k 2 2 . (5.5) We use the default parameters in TwIST and terminate the iteration process when the relative variation of function value falls below 10 −4 . FPC 5 is the fixed-point continuation algorithm to solve the general ℓ 1 -regularized minimization problem (1.1), where f is a continuous differentiable convex function. At current x k and any scalar τ > 0, the next iteration is produced by the so-called fixed point iteration x k+1 = sgn(x k − τ ∇f (x k )) max |x k − τ ∇f (x k )| − µτ, 0 , where "sgn" is a componentwise sign function. In order to obtain a good practical performance, a continuation approach is also augmented in FPC. Moreover, the FPC is further modified by using Barzilai we set xtol = 1e-5 to stop the algorithm when the relative change between successive points is below xtol. In this test, A is a partial discrete cosine coefficients matrix (DCT), whose m rows are chosen randomly from the n × n DCT matrix. Such encoding matrix A does not require storage and enables fast matrixvector multiplications involving A and A ⊤ . Therefore, it is able to be used to test much larger size problems than using Gaussian matrices. In NBBL1, we take tol 2 = 10 −4 , h = 0.8, λ (min) = 10 −30 , λ (max) = 10 30 . In the line search, we chooseα 0 = 0.8, ρ = 0.35, δ = 10 −5 andm = 5. In this comparison, we let n = 2 12 , m = floor(n/4). The original signalx contains p = floor(m/6) number of nonzero components, where floor is a Matlab command used to round an element to the nearest integers towards minus infinity. Moreover, the observation b is contaminated by Gaussian noise with level σ = 1e − 3. The goal is to use each algorithm to reconstructx from the observation b by solving (1.2) with µ = 2 −8 . All the tested algorithms start at x 0 = A ⊤ b and terminate with different stopping criterions to produce similar quality resolutions. To specifically illustrate the performance of each algorithm, we draw four figures to show their convergence behavior from the point of objective function values and relative error as the iteration numbers and computing time increase, which given in Figure 5.3. From the top plots in Figure 5.3, NBBL1 usually decreases relative errors faster than NESTA-Ct, CGD and GPSR-BB throughout the entire iteration process, and meanwhile requires less number of iterations. The top right plot shows that TwIST needs less steps than NBBL1 to obtain similar level of relative error. However, TwIST is much slower because it has to solve a de-noising subproblem (5.5) at each iteration. Unfortunately, NBBL1 needs further improvement to challenge the well-known code FPC-BB. We now turn our attention to observe the function values behavior of each algorithm. Similarly, NBBL1 is superior to NEST-Ct, CGD, GPSR-BB and TwIST from the computing time points of view. FPC-BB reaches the lowest function values at the very beginning, and then starts to increase it to meet nearly equal final values at the end. In this test, CGD appears to be much slower than the others, because it is sensitive to the choice of starting points. If CGD starts at x 0 = 0 with all the other settings unchanged, its performance should be significantly improved [40]. Taking everything together, from the limited numerical experiment, we conclude that NBBL1 provides an efficient approach for solving ℓ 1 -regularized nonsmooth problem and is competitive with or performs better than NESTA-Ct, GPSR-BB, CGD, TwIST and FPC-BB. 6. Conclusions. In this paper, we proposed, analyzed, and tested a new practical algorithm to solve the separable nonsmooth minimization problem consisting of a ℓ 1 -norm regularized term and a continuously differentiable term. The type of the problem mainly appears in signal/image processing, compressive sensing, machine learning, and linear inverse problems. However, the problem is challenging due to the non-smoothness of the regularization term. Our approach minimizes an approximal local quadratic model to determine a search direction at each iteration. The search direction reduces to the classic Barzilai-Borwein gradient method in the case of µ = 0. We show that the objective function is descent along this direction providing that the initial stepsize is less than h. We also establish the algorithm's global convergence theorem by incorporating a nonmonotone line search technique and assuming that f is bounded below. Extensive experimental results show that the proposed algorithm is an effective toll to solve ℓ 1 -regularized nonconvex problems from CUTEr library. Moreover, we also run our algorithm to recover a large sparse signal from its noisy measurement, and numerical comparisons illustrate that our algorithm outperforms or is competitive with several state-of-the-art solvers which specifically designed to solve ℓ 1 -regularized compressive sensing problems. Unlike all the existing algorithms in this literature, our approach uses an linear model to approximate x k + d 1 for computing the search direction with a small scalar h; that is x k + d 1 ≈ x k 1 + x k + hd 1 − x k 1 h . Although the equations may hold exactly in the case of h = 1, a series of numerical experiments show that h ∈ [0.7, 0.8] may produce better performance with suitable experiment settings. This approach is distinctive and novel; therefore, it is one of the important contributions of this paper. As we all know, the nonmonotone Barzilai-Borwein gradient algorithm of Raydan [33] is very effective for smooth unconstrained minimization, and its remarkable effectiveness in signal reconstruction problems involving ℓ 1 -regularized problems has not been clearly explored. Hence, our approach can be considered as a modification or extension, to broaden the university of [33]. Moreover, the numerical experiments illustrate that our approach performs comparable to or even better than several state-of-the-art algorithms. Surely, this is the numerical contribution of our paper. Although the proposed algorithm needs further improvement to challenge the well-known code FPC-BB, the enhancement of it to deal with non-convex problems is noticeable. Our algorithm is readily to solve the ℓ 1 -regularized logistic regression, the ℓ 2 -norm and matrix trace norm minimization problems in machine learning. However, we do not test them in this paper. This should be interesting for further investigations. Assumption 1 . 1The level set Ω = {x : f (x) ≤ f (x 0 )} is bounded. Lemma 3. 1 . 1Suppose that λ k > 0 and d k is defined by (2.8) with h ∈ (0, 1]. Then x k is a stationary point of problem (1.1) if and only if d k = 0. other hand, from the definition of ∆ k in (2.17) and the inequality (2.11), it is not difficult to Theorem 3. 3 . 3Let the sequence {x k } and {d k } generated by Algorithm 1. Then, there exists a subsequence K such that lim k→∞,k∈K d k 2 = 0. (3.4) Proof. From [24], it is clear that (3.2) also implies lim k→∞ α k d k 2 = 0. (3.5) Now, letx be a limit point of {x k }, and {x k } K1 be a subsequence of {x k } converging tox. Then by ( .1, which contains the name of the problem (Problem), the dimensions of the problem (Dim), the number of iterations (Iter), the number of function evaluations (Nf), the CPU time required in seconds (Time), the final objective function values (Fun), the norm of the final gradient of f (Normg), and the norm of final direction (Normd). ( 5 . 3 ) 53In this experiment, we take tol = 10 −4 , h = 10 −2 , λ (min) = 10 −30 , λ (max) = 10 30 . In the line search, we chooseα 0 = 10 −2 , ρ = 0.35, δ = 10 −4 andm = 5. The original signal, the limited measurement, and the reconstructed signal are given inFigure 5.1. Fig. 5 . 1 . 51Left: original signal with length 4096 and 64 positive non-zero elements; Middle: the noisy measurement with length 512; Right: recovered signal by NBBL1 (red circle) versus original signal (blue peaks). .2, the impact of the parameter h values on the total number of iterations, the computing time, and the quality of the recovered signal. In each plot, the level axis denotes the values of h from 0.01 to 1 in a log scale. Fig. 5. 2 . 2Performance of NBBL1: number of iterations (left), computing time (middle) and final relative error (right). In each plot, the horizontal axis represents the value of h in log scale. Fig. 5 . 3 . 53Comparison result of NBBL1, NESTA-Ct, CGD, TwIST, GPSR-BB, and FPC-BB. The x-axes represent the CPU time in seconds (left column) and the number of iterations (right column). The y-axes represent the relative error (top row) and the function values (bottom row). and Borwein gradient algorithm can be found in[6,15,17,18,19,32,44]. Table 5 .1 5Test result for NBBL1Problem Dim µ Iter Nf Time Fun Normg Normd VARDIM 1000 0.0 49 94 0.48 3.2506e-26 3.6059e-13 2.5893e-09 FLETCHER 100 0.0 1217 1983 3.75 3.0113e-10 2.2576e-05 9.9505e-09 COSINE 10000 0.0 51 350 23.41 -9.9990e+03 2.5387e-03 4.4188e-09 SINQUAD 1000 0.0 180 908 10.22 6.4479e-05 4.9743e-05 6.8482e-09 GENROSE 200 0.0 323 646 0.80 1.0000e+00 1.3870e-05 9.9488e-09 WOODS 1000 0.0 322 579 3.00 9.9104e-13 7.2693e-06 7.5155e-09 NONCVXU2 200 0.0 4987 8476 21.17 4.6373e+02 2.0726e-07 7.0300e-09 BROYDN7D 500 0.0 1305 2402 10.38 3.8234e+00 9.3966e-07 9.2037e-09 CHAIWOO 1000 0.0 757 1335 9.80 1.0000e+00 8.0006e-06 4.4747e-09 VARDIM 1000 0.25 49 94 0.63 2.5000e+02 6.8487e+00 6.4503e-09 FLETCHER 100 0.25 5042 8657 15.83 2.4497e+01 2.5000e+00 9.7495e-09 COSINE 10000 0.25 47 108 20.33 -1.6829e+03 2.4999e+01 4.8382e-09 SINQUAD 1000 0.25 46 92 1.95 2.8084e-01 3.5357e-01 2.2567e-13 GENROSE 200 0.25 9 57 0.06 1.9846e+02 3.5267e+00 3.7742e-09 WOODS 1000 0.25 645 1527 6.69 2.4911e+02 7.9057e+00 7.0300e-09 NONCVXU2 200 0.25 998 1957 4.78 5.6230e+02 3.3990e+00 8.2357e-09 BROYDN7D 500 0.25 1314 2366 10.95 8.9609e+01 5.5902e+00 8.1753e-09 CHAIWOO 1000 0.25 435 746 6.11 2.5055e+02 7.9057e+00 6.6783e-09 VARDIM 1000 0.5 448 1540 4.80 4.8920e+02 1.4141e+01 6.6304e-09 FLETCHER 100 0.5 2654 4513 8.00 4.8953e+01 5.0000e+00 8.4947e-09 COSINE 10000 0.5 21 66 8.45 1.0042e+02 4.9995e+01 9.2370e-09 SINQUAD 1000 0.5 36 67 1.16 4.2508e-01 7.0724e-01 7.0416e-09 GENROSE 200 0.5 9 60 0.08 1.9887e+02 7.0534e+00 5.5793e-09 WOODS 1000 0.5 643 1321 6.03 4.9644e+02 1.5811e+01 2.9571e-09 NONCVXU2 200 0.5 595 906 2.59 5.7366e+02 6.7493e+00 1.5667e-09 BROYDN7D 500 0.5 1020 2096 8.30 1.7207e+02 1.1180e+01 8.8339e-09 CHAIWOO 1000 0.5 1265 2109 17.00 5.0253e+02 1.5811e+01 5.3247e-09 VARDIM 1000 2.0 1506 4816 14.81 1.7511e+03 6.2435e+01 2.4783e-09 FLETCHER 100 2.0 996 1721 3.13 2.4344e+02 2.0000e+01 2.8322e-09 COSINE 10000 2.0 2 3 1.00 9.9990e+03 0.0000e+00 0.0000e+00 SINQUAD 1000 2.0 67 110 2.30 8.6555e-01 2.8291e+00 5.0413e-11 GENROSE 200 2.0 2 3 0.03 2.0000e+02 2.8213e+01 0.0000e+00 WOODS 1000 2.0 102 283 1.05 3.3572e+03 6.3246e+01 3.8964e-09 NONCVXU2 200 2.0 251 825 1.50 7.3779e+02 1.7429e+01 9.7233e-09 BROYDN7D 500 2.0 152 455 1.09 5.0216e+02 6.7458e+00 5.0308e-09 CHAIWOO 1000 2.0 927 1634 13.45 1.9739e+03 6.3246e+01 9.5918e-09 -Borwein stepsize (code FPC-BB in Matlab package FPC v2). 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{'abstract': 'This paper is devoted to minimizing the sum of a smooth function and a nonsmooth ℓ 1 -regularized term. This problem as a special cases includes the ℓ 1 -regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the ℓ 1 -norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the ℓ 1 -norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive ℓ 1 -regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for ℓ 1 -regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.AMS subject classifications. 65L09, 65K05, 90C30, 90C25 research activities over the past years[11,12,10,13,20]. Compressive sensing is based on the fact that if the original signal is sparse or approximately sparse in some orthogonal basis, an exact restoration can be produced via solving problem (1.2).', 'arxivid': '1207.4538', 'author': ['Yunhai Xiao ', 'Soon-Yi Wu ', 'ANDLiqun Qi '], 'authoraffiliation': [], 'corpusid': 26500440, 'doi': '10.1007/s10915-013-9815-8', 'github_urls': [], 'n_tokens_mistral': 22101, 'n_tokens_neox': 19119, 'n_words': 11060, 'pdfsha': '654d75f013043c0f169c966d3bec1952ee211c5f', 'pdfurls': ['https://arxiv.org/pdf/1207.4538v1.pdf'], 'title': ['NONMONOTONE BARZILAI-BORWEIN GRADIENT ALGORITHM FOR ℓ 1 -REGULARIZED NONSMOOTH MINIMIZATION IN COMPRESSIVE SENSING', 'NONMONOTONE BARZILAI-BORWEIN GRADIENT ALGORITHM FOR ℓ 1 -REGULARIZED NONSMOOTH MINIMIZATION IN COMPRESSIVE SENSING'], 'venue': []}

arxiv

arXiv:physics/0003084v1 [physics.bio-ph] Spatial-temporal correlations in the process to self-organized criticality 28 Mar 2000 C B Yang Institute of Particle Physics Hua-Zhong Normal University 430079WuhanChina X Cai Institute of Particle Physics Hua-Zhong Normal University 430079WuhanChina Z M Zhou Physics Department Hua-Zhong University of Science and Technology 430074WuhanChina arXiv:physics/0003084v1 [physics.bio-ph] Spatial-temporal correlations in the process to self-organized criticality 28 Mar 2000(March 31, 2022) A new type of spatial-temporal correlation in the process approaching to the self-organized criticality is investigated for the two simple models for biological evolution. The change behaviors of the position with minimum barrier are shown to be quantitatively different in the two models. Different results of the correlation are given for the two models. We argue that the correlation can be used, together with the power-law distributions, as criteria for self-organized criticality.The phenomenon of "self-organized criticality" (SOC), with potential applications ranging from the behavior of sandpile and the description of the growth of surfaces to generic description of biological evolution, has become as a topic of considerable interest[1][2][3][4][5][6][7][8]. It is observed that the dynamics of complex systems in nature does not follow a smooth, gradual path, instead it often occurs in terms of punctuations, or "avalanches" in other word. The appearance of the spatial-temporal complexity in nature, containing information over a wide range of length and time scale, presents a fascinating but longstanding puzzle. Such complexity also shows up in simple mathematical models for biological evolution and growth phenomena far from equilibrium. In former studies, power-law distributions for the spatial size and lifetime of the "avalanches" have been observed in various complex systems and are regarded as "fingerprints" for SOC. It seems that there is no general agreement on a suitable definition of SOC [9,10], although a minimal definition was given in[11]. Because there is no universally accepted "black-box" tests for the presence or absence of SOC based solely on observables, systems with a wide range of characteristics have all been designated as "self-organized critical".While numerous numerical studies have claimed SOC to occur in specific models, and although the transition to the SOC state was studied in[12][13][14], a question has never been answered: How is the process approaching to the final dynamical SOC attractor characterized? One may even ask whether the phenomenon SOC can be adequately characterized by such power-law distributions. The answer to the latter question seems to be negative, as concluded in[15]. In Ref.[15] were pointed out "some striking observable differences between two 'self-organized critical' models which have a remarkable structural similarity". The two models, as called the Bak-Sneppen (B-S) models, are introduced in [16-18] and are used to mimic biological evolution. The models involve a one-dimensional random array on L sites. Each site represents a species in the "food-chain". The random number (or barrier) assigned to each site is a measure of the "survivability" of the species. Initially, the random number for each species is drawn uniformly from the interval (0, 1). In each update, the least survivable species (the update center) and some others undergo mutations and obtain new random numbers which are also drawn uniformly from (0, 1). In the first version of the model (the local or nearest-neighbor model), only the update center and its two nearest neighbors participate the mutations. In the second version, K − 1 other sites chosen randomly besides the update center are involved in the update and assigned new random survivabilities (so this version is called random neighbor model). Periodic boundary conditions are adopted in the first model. As shown in[18][19][20], the second version is analytically solvable. Investigation in[15]shows that some behaviors of the local and random neighbor models are qualitatively identical. They both have a nontrivial distribution of barrier heights of minimum barriers, and each has a power-law avalanche distribution. But the spatial and temporal correlations between the minimum barriers show different behaviors in the two models and thus can be used to distinguish them.In all the studies mentioned above, spatial and/or temporal distributions of the "avalanches" and correlations between positions with minimum of barriers are investigated separately. As shown in many studies, however, spatial and/or temporal distribution of the "avalanches" alone cannot be used as a criterion for SOC, nor can the spatial or temporal correlation do. In this paper, it is attempted to study a new kind of correlation between minimum barriers in the process of the updating in the two models for biological evolution. The correlation between the positions with minimum barriers at time (or update) s and s + 1 is investigated. Since the new correlation involves two sites at different times, it is of spatial-temporal type. Thus it may be suitable for the study of spatial-temporal complexity.Consider the update process of the local neighbor model. Initially, each site is assigned a random number. All the random numbers are drawn uniformly from interval (0,1). Denote X(s) the site number with minimum barrier after s updates. The sites can be numbered such that 1 ≤ X(s) ≤ L. To see how X(s) changes in updating process in the model X(s) is shown inFig. 1as a function of s for an arbitrary update process for lattice size L=200 with s from 1 to 2000. The lower part ofFig. 1is a zoomed part of the upper one for small s. It is clear that X(s) seems to be random when s is small. With the going-on of updating, X(s) becomes more and more likely to be in the neighborhood of last update center, X(s − 1). So there appear some plateau like parts in Fig. 1. In other word, there appears some correlation between X(s) when the system is self-organized to approach the critical state. So, it may be fruitful to study the self-correlation of X(s) in searching quantities characterizing the process to SOC. For this purpose, one can define a quantity C(s) = X(s)X(s + 1) − X(s) X(s + 1) , with average over different events of updating. Obviously, if there is no correlation between the sites with minimum barrier at time s and s + 1, or X(s)X(s + 1) = X(s) X(s + 1) , C(s) will be zero. Thus, C(s) can show whether there is correlation between X(s) and also give a measure of the strength of the correlation. Because of the randomness of the survivability at each site, X(s) can be 1, 2, · · ·, L with equal probability, 1/L. Thus, X(s) = (L + 1)/2 for every time s. It should be pointed out that X(s) = (L + 1)/2 does not mean any privilege of sites with numbering about (L+1)/2. In fact, all sites can be the update center with equal chance at time s if the update process is repeated many times from the initial state. Due to the randomness of the updated survivability X(s + 1) can also take any integer from 1 to L. However, the distribution of X(s + 1) is peaked at X(s) when s is large, see [13] for detail. With the update going on, the width of the distribution becomes more and more narrower. When the width becomes narrow enough, X(s)X(s + 1) will turn out to be X 2 (s) = (2L 2 + 3L + 1)/6. So, C(s) will approach (L 2 − 1)/12 for large s. In above definition for C(s), however, the neighboring relation between X(s) and X(s + 1) cannot be realized once the numbering for the sites is given. Due to the periodic boundary conditions adopted in the model, one of the nearest neighbors of the site with numbering 1 is the one numbered L. To overcome this shortcoming, one can introduce an orientational shorter distance ∆(s) between X(s) and X(s + 1). Imagine the L sites with numbering 1, 2, · · · , L are placed on a circle in clockwise order. Then |∆(s)| is the shorter distance between the two sites on the circle. If X(s + 1) is reached along the shorter curve from X(s) in clockwise direction, ∆(s) is positive. Otherwise ∆(s) is negative. For definiteness, one can assume −L/2 ≤ ∆(s) < L/2. With ∆(s), one can use X ′ (s + 1) = X(s) + ∆(s)(2) in place of X(s + 1) in the definition of C(s). Since X ′ (s) can cross the (non-existing) boundary between 1 and L and reflect the neighboring relation with X(s), the effect of periodic boundary conditions on the correlation can be taken into account. (In the simulation of the B-S model numbering the L sites with integer numbers 1, 2, · · · , L is necessary, but the start position can be arbitrary. Different numbering scheme will give the same results for C(s), as physically demanded. This in return is also an indication of the equivalence of all sites in the presence of periodic boundary conditions.) To normalize the dependence of C(s) on the size of the one-dimensional array, we can renormalize C(s) by (L 2 − 1)/12. In the following, we use a normalized definition of C(s) as C(s) = X(s)X ′ (s + 1) − X(s) X ′ (s + 1) (L 2 − 1)/12 .(3) In current study X(s) and ∆(s) are determined from Monte Carlo simulations, and 500,000 simulation events are used to determine the averages involved. For each event, 2000 updates are performed from an initial state with random barriers on the sites uniformly distributed in (0, 1). The normalized correlation function C(s) is shown as a function of s in Fig. 2 for L = 50, 100, and 200. One can see that C(s) is a monotonously increasing function of time s. As in our naive consideration, C(s) is very small in the early stage of updates and becomes larger and larger for larger s, indicating the increase of the strength of correlation between the sites with minimum barrier at different times. The behavior of C(s) with s exhibits different characteristics for small and large s. C(s) increases with s very quickly for small s, but the rate becomes quite slow after a knee point. The knee point appears earlier for smaller L, showing the existence of a finite-size effect. Also, the seemly saturating value of C(s) depends on the size L of the lattice, or more clearly, it increases with the lattice size L. Since only 500,000 simulation events are used in current study, there shows the effect of fluctuations in the figure. The correlation between X(s) can be investigated for the random neighbor model for biological evolution in the same way. For simplicity only the case with K = 3 is taken into account. The generalization to other cases is straight forward. First, one can have a look on how X(s) changes with update. X(s) is shown as a function of s in the upper part of Fig. 3. This plot may look as a random scatter of points at first sight. But it is not. A close look reveals correlations: X(s) often has almost same value for several consecutive or almost consecutive s values. However, no obvious plateau like part can be seen in the figure, showing the difference between the two versions of B-S model. C(s) is also studied and shown in the lower part of Fig. 3 as a function of s for the lattice size L = 200. In the random neighbor version of the B-S model, sites numbered with 1 and L are no longer neighbors. So, in the calculation of C(s) from Eq. (3), X(s + 1) is used instead of X ′ (s + 1). The counterpart for the nearest neighbor model is also drawn in the figure for comparison. One can see that the saturating value is much smaller than in the case of the local neighbor version of the model. From the discussions above one can see that the correlation between the sites with minimum barrier may play an important role in investigating SOC. The power-law distributions for the size and lifetime of the "avalanches" together with the new kind of correlation may be used as criteria for SOC. This work was supported in part by the NNSF in China and NSF in Hubei, China. One of the authors (C.B.Yang) would like to thank Alexander von Humboldt Foundation of Germany for the Research Fellowship granted to him. Figure Captions Fig. 1 1The change of site X(s) with time s for an arbitrary event in the nearest neighbor version of the B-S model for biological evolution. Fig. 2 2The correlation function C(s) as a function of s for lattice size L=50, 100, and 200 for the same model as inFig. 1. Fig. 3 3Upper part: The change of site X(s) with s for the random neighbor version of the B-S model for biological evolution; Lower part: The correlation function C(s) for the two versions as functions of s for L=200. . P Bak, C Tang, K Wiesenfeld, Phys. Rev. Lett. 59381Phys. Rev. A38P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A38, (1987) 364; Phys. Rev. Lett. 59, (1987) 381. . K Chen, P Bak, S P Obukhov, Phys. Rev. A43. 625K. Chen, P. Bak, and S.P. Obukhov, Phys. Rev. A43, (1991) 625. . P Bak, K Chen, M Creuts, Nature. 342780P. Bak, K. Chen, and M. Creuts, Nature 342, (1989) 780. . K Sneppen, Phys. Rev. Lett. 693539K. Sneppen, Phys. Rev. Lett. 69, (1992) 3539; . K Sneppen, M H Jensen, ibid. 7071101K. Sneppen and M.H. Jensen, ibid. 70, (1993) 3833; 71, (1993) 101. . P Bak, K Chen, Sci. Am. 264146P. Bak and K. Chen, Sci. Am. 264 (1), (1991) 46. . K Chen, P Bak, Phys. Lett. 14046K. Chen and P. Bak, Phys. Lett. A140, (1989) 46. . P Bak, K Chen, C Tang, Phys. Lett. 147297P. Bak, K. Chen, and C. Tang, Phys. Lett. A147, (1990) 297. . A Sornette, D Sornette, Europhys. Lett. 9197A. Sornette and D. Sornette, Europhys. Lett. 9, (1989) 197. . D Sornette, Phys. Rev. Lett. 722306D. Sornette, Phys. Rev. Lett. 72, (1994) 2306. . G Canelli, R Cantelli, F Cordero, Phys. Rev. Lett. 722307G. Canelli, R. Cantelli, and F. Cordero, Phys. Rev. Lett. 72, (1994) 2307. . H Flyvbjerg, Phys. Rev. Lett. 76940H. Flyvbjerg, Phys. Rev. Lett. 76, (1996) 940. . M Paczuski, S Maslov, P Bak, Europhys. Lett. 2797M. Paczuski, S. Maslov and P. Bak, Europhys. Lett. 27, (1994) 97. . M Paczuski, S Maslov, P Bak, Phys. Rev. E53. 414M. Paczuski, S. Maslov and P. Bak, Phys. Rev. E53, (1996) 414. . A Corral, M Paczuski, Phys. Rev. Lett. 83572A. Corral and M. Paczuski, Phys. Rev. Lett. 83, (1999) 572. . J Boer, A D Jackson, Tilo Wetig, Phys. Rev. E51. 1059J. de Boer, A.D. Jackson, and Tilo Wetig, Phys. Rev. E51, (1995) 1059. . P Bak, K Sneppen, Phys. Rev. Lett. 714083P. Bak and K. Sneppen, Phys. Rev. Lett. 71, (1993) 4083. . H Flyvbjerg, P Bak, K Sneppen, Phys. Rev. Lett. 714087H. Flyvbjerg, P. Bak, and K. Sneppen, Phys. Rev. Lett. 71, (1993) 4087. . J Boer, B Derrida, H Flyvbjerg, A D Jackson, T Wettig, Phys. Rev. Lett. 73906J. de Boer, B. Derrida, H. Flyvbjerg, A.D. Jackson, and T. Wettig, Phys. Rev. Lett. 73, (1994) 906. . Yu M Pis&apos;mak, J. Phys. A: Math. Gen. 283109Yu.M. Pis'mak, J. Phys. A: Math. Gen. 28, (1995) 3109. . Yu M Pis&apos;mak, Phys. Rev. E56. 1326Yu.M. Pis'mak, Phys. Rev. E56, (1997) R1326.

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{'abstract': 'A new type of spatial-temporal correlation in the process approaching to the self-organized criticality is investigated for the two simple models for biological evolution. The change behaviors of the position with minimum barrier are shown to be quantitatively different in the two models. Different results of the correlation are given for the two models. We argue that the correlation can be used, together with the power-law distributions, as criteria for self-organized criticality.The phenomenon of "self-organized criticality" (SOC), with potential applications ranging from the behavior of sandpile and the description of the growth of surfaces to generic description of biological evolution, has become as a topic of considerable interest[1][2][3][4][5][6][7][8]. It is observed that the dynamics of complex systems in nature does not follow a smooth, gradual path, instead it often occurs in terms of punctuations, or "avalanches" in other word. The appearance of the spatial-temporal complexity in nature, containing information over a wide range of length and time scale, presents a fascinating but longstanding puzzle. Such complexity also shows up in simple mathematical models for biological evolution and growth phenomena far from equilibrium. In former studies, power-law distributions for the spatial size and lifetime of the "avalanches" have been observed in various complex systems and are regarded as "fingerprints" for SOC. It seems that there is no general agreement on a suitable definition of SOC [9,10], although a minimal definition was given in[11]. Because there is no universally accepted "black-box" tests for the presence or absence of SOC based solely on observables, systems with a wide range of characteristics have all been designated as "self-organized critical".While numerous numerical studies have claimed SOC to occur in specific models, and although the transition to the SOC state was studied in[12][13][14], a question has never been answered: How is the process approaching to the final dynamical SOC attractor characterized? One may even ask whether the phenomenon SOC can be adequately characterized by such power-law distributions. The answer to the latter question seems to be negative, as concluded in[15]. In Ref.[15] were pointed out "some striking observable differences between two \'self-organized critical\' models which have a remarkable structural similarity". The two models, as called the Bak-Sneppen (B-S) models, are introduced in [16-18] and are used to mimic biological evolution. The models involve a one-dimensional random array on L sites. Each site represents a species in the "food-chain". The random number (or barrier) assigned to each site is a measure of the "survivability" of the species. Initially, the random number for each species is drawn uniformly from the interval (0, 1). In each update, the least survivable species (the update center) and some others undergo mutations and obtain new random numbers which are also drawn uniformly from (0, 1). In the first version of the model (the local or nearest-neighbor model), only the update center and its two nearest neighbors participate the mutations. In the second version, K − 1 other sites chosen randomly besides the update center are involved in the update and assigned new random survivabilities (so this version is called random neighbor model). Periodic boundary conditions are adopted in the first model. As shown in[18][19][20], the second version is analytically solvable. Investigation in[15]shows that some behaviors of the local and random neighbor models are qualitatively identical. They both have a nontrivial distribution of barrier heights of minimum barriers, and each has a power-law avalanche distribution. But the spatial and temporal correlations between the minimum barriers show different behaviors in the two models and thus can be used to distinguish them.In all the studies mentioned above, spatial and/or temporal distributions of the "avalanches" and correlations between positions with minimum of barriers are investigated separately. As shown in many studies, however, spatial and/or temporal distribution of the "avalanches" alone cannot be used as a criterion for SOC, nor can the spatial or temporal correlation do. In this paper, it is attempted to study a new kind of correlation between minimum barriers in the process of the updating in the two models for biological evolution. The correlation between the positions with minimum barriers at time (or update) s and s + 1 is investigated. Since the new correlation involves two sites at different times, it is of spatial-temporal type. Thus it may be suitable for the study of spatial-temporal complexity.Consider the update process of the local neighbor model. Initially, each site is assigned a random number. All the random numbers are drawn uniformly from interval (0,1). Denote X(s) the site number with minimum barrier after s updates. The sites can be numbered such that 1 ≤ X(s) ≤ L. To see how X(s) changes in updating process in the model X(s) is shown inFig. 1as a function of s for an arbitrary update process for lattice size L=200 with s from 1 to 2000. The lower part ofFig. 1is a zoomed part of the upper one for small s. It is clear that X(s) seems', 'arxivid': 'physics/0003084', 'author': ['C B Yang \nInstitute of Particle Physics\nHua-Zhong Normal University\n430079WuhanChina\n', 'X Cai \nInstitute of Particle Physics\nHua-Zhong Normal University\n430079WuhanChina\n', 'Z M Zhou \nPhysics Department\nHua-Zhong University of Science and Technology\n430074WuhanChina\n'], 'authoraffiliation': ['Institute of Particle Physics\nHua-Zhong Normal University\n430079WuhanChina', 'Institute of Particle Physics\nHua-Zhong Normal University\n430079WuhanChina', 'Physics Department\nHua-Zhong University of Science and Technology\n430074WuhanChina'], 'corpusid': 2630664, 'doi': '10.1103/physreve.61.7243', 'github_urls': [], 'n_tokens_mistral': 4590, 'n_tokens_neox': 4025, 'n_words': 2503, 'pdfsha': '7575ef4207155a77bec42eeeb28d3013d6f46b90', 'pdfurls': ['https://export.arxiv.org/pdf/physics/0003084v1.pdf'], 'title': ['arXiv:physics/0003084v1 [physics.bio-ph] Spatial-temporal correlations in the process to self-organized criticality', 'arXiv:physics/0003084v1 [physics.bio-ph] Spatial-temporal correlations in the process to self-organized criticality'], 'venue': []}

arxiv

Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes 20 Apr 2019 Eloy Ayón-Beato Departamento de Física CINVESTAV-IPN Apdo. Postal 14-74007000 CDMX México Moisés Bravo-Gaete Facultad de Ciencias Básicas Universidad Católica del Maule Casilla 617TalcaChile Francisco Correa Instituto de Ciencias Físicas y Matemáticas Universidad Austral de Chile Casilla 567ValdiviaChile Mokhtar Hassaïne Instituto de Matemática y Física Universidad de Talca Casilla 747TalcaChile María Montserrat Juárez-Aubry Arkansas State University Carretera estatal #100 km. 17.576270Municipio Colón, Querétaro, QuerétaroMéxico Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes 20 Apr 2019arXiv:1904.09391v1 [hep-th] In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higherdimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves to calculate the correct expressions for the masses of the higher-dimensional Lifshitz black hole as well as their corresponding soliton. The robustness of this Cardy-like formula is checked by showing that the microscopic entropy is in perfect agreement with the gravitational Wald entropy. Consequently, the calculated global charges are compatible with the first law of thermodynamics. We also verify that all the configurations satisfy an anisotropic higher-dimensional version of the Smarr I. INTRODUCTION Gauge/gravity duality can be extended to nonrelativistic systems by using anisotropic spacetimes. In this context, the archetypal example is the Lifshitz spacetime [1] ds 2 = − r 2z l 2z dt 2 + l 2 r 2 dr 2 + r 2 l 2 D−2 i=1 dx 2 i ,(1) whose main feature is the isometry allowing time and space scale with different exponents. Here, z is the dynamical critical exponent responsible of the anisotropic scaling characterizing nonrelativistic systems. As it was preliminarily emphasized in [1], standard vacuum Einstein gravity cannot allows Lifshitz spacetimes, except in the isotropic case z = 1 where they turn out to be antide Sitter (AdS) spaces. Nevertheless, this problem can be circumvented by considering instead higher-order corrections to gravity theories or by introducing specific matter sources. It has then becomes important to find specific gravity models that can accommodate the Lifshitz spacetimes together with their black hole extensions recovering the anisotropic scaling asymptotically. These so-called Lifshitz black holes are supposed to holographically capture the finite-temperature behavior of their strongly correlated nonrelativistic dual systems. New Massive Gravity [2] was one of the first gravity models that was shown to admit an analytic Lifshitz black hole as part of its vacua [3]. A property that later resulted to be generic for higher-order pure gravity theories in higher dimensions [4]. In presence of specific matter sources, Lifshitz solutions have also been investigated, see e.g. [5][6][7][8]. Also charged Lifshitz solutions can be engineering through a Maxwell-Proca model [9] or its nonlinear generalization [10] or in the presence of dilaton scalar fields [11]. The relevance of Lifshitz black holes lies in the hope that strongly coupled condensed matter systems can be better understood at finite temperature from an holographic point of view. But because of their unconventional asymptotic behavior, these black holes present interesting features which deserve more profound investigations. For example, their thermodynamic properties are usually quite different from those of the isotropic AdS black holes and in particular if the solutions are charged, see e.g. [12,13]. On the other hand, threedimensional configurations are usually excellent laboratories to investigate important conceptual questions about the gauge/gravity duality. For example, it has been shown that the semiclassical entropy of three-dimensional black holes with Lifshitz asymptotics can be recovered through a Cardy-like formula where the mass of their corresponding Lifshitz solitons explicitly appears, giving a prominent role to these regular configurations [14]. The solitons are obtained from the black holes by means of a double Wick rotation that involves inverting the dynamical critical exponent, as result they enjoy the same sort of uniqueness than the black holes [15]. The robustness of this formula has been successfully tested in a system exhibiting a wide spectrum of Lifshitz configurations as is the case of self-interacting scalar fields nonminimally coupled to New Massive Gravity [16]. Recently, this Cardylike formula has been extended to higher-dimensional anisotropic black holes [17]. In the present work, we pretend to test the validity of this higher-dimensional Cardy-like formula by considering again self-interacting scalar fields, but nonminimally coupled now to the most general quadratic curvature corrections of Einstein gravity in higher dimensions. We hope this study will contribute to highlight the importance of the role played by the soliton in the description of the thermal properties of black holes, and particulary for those with unconventional asymptotic behaviors. The paper is organized as follows. In the next section, we will present the theory, the field equations as well as a specific ansatz which permits to obtain particular Lifshitz black hole solutions. Using this ansatz unspecifically, the thermodynamics quantities of interest as the entropy, the temperature and the mass of the black holes will be generically computed together with the mass of the soliton. In section IV, we will explicitly present four concrete classes of Lifshitz black holes fitting our ansatz. For these solutions, we will check that their gravitational entropy, calculated with the standard Wald formula, can be correctly reproduced by means of the Cardy-like formula. In the last section, we show the existence of a fifth class of Lifshitz black hole that is slightly different of our working ansatz. For this specific solution, we also test the robustness of the Cardy-like formula. In all cases we verify the fulfillment of the first law of black hole thermodynamics, which point the correctness of the quasilocal off-shell extension of the ADT formalism that we use to compute global charges. Finally, we provide an Appendix for reporting some of the involved expressions for the (coupling) constants of the specifics solutions. II. SET-UP OF THE PROBLEM In arbitrary dimension D, we consider a gravity action given by the most general quadratic-curvature corrections of the Einstein-Hilbert action sourced by a selfinteracting nonminimally coupled scalar field S[gµν , Φ] = d D x √ −g(Lg + Ls),(2) with Lg = 1 2κ R − 2λ + β1R 2 + β2R αβ R αβ + β3R αβµν R αβµν , Ls = − 1 2 ∇µΦ∇ µ Φ − 1 2 ξRΦ 2 − U (Φ). The cosmological constant λ, the coupling constants β i , the self-interacting potential U (Φ) and eventually the nonminimal coupling parameter ξ will depend explicitly on the concrete solutions presented in the sections that follow. The field equations obtained by varying the action with respect to the metric and the scalar field read Gµν + λgµν + Kµν = κTµν ,(3a)Φ − ξRΦ = dU (Φ) dΦ ,(3b) where we have defined Kµν = (β2 + 4β3) Rµν + 1 2 (4β1 + β2)gµν R − (2β1 + β2 + 2β3)∇µ∇ν R + 2β3R µγαβ R γαβ ν + 2(β2 + 2β3)R µανβ R αβ − 4β3RµαR α ν + 2β1RRµν − 1 2 β1R 2 + β2R αβ R αβ + β3R αβγδ R αβγδ gµν , (3c) and the energy-momentum tensor is given by Tµν = ∇µΦ∇ν Φ − gµν 1 2 ∇σΦ∇ σ Φ + U (Φ) + ξ (gµν − ∇µ∇ν + Gµν ) Φ 2 . (3d) In order to look for Lifshitz black holes, we will opt for the following ansatz ds 2 = − r 2z l 2z f (r)dt 2 + l 2 r 2 dr 2 f (r) + r 2 l 2 D−2 i=1 dx 2 i , Φ = Φ(r),(4) where the structural metric function satisfies that lim r→∞ f (r) = 1, condition ensuring the metric to reproduce the Lifshitz asymptotic (1). As shown below, four of the five classes of Lifshitz solutions that will be presented can be generically parameterized as f (r) = 1 − r h r χ , Φ(r) = Φ 0 r h r χ 2 ,(5) where χ is a nonnegative decay exponent modulating the Lifshitz asymptotics, r h denotes the location of the horizon and Φ 0 characterizes the positive strength of the field. The positivity of the scalar field can be explained from the fact that for these four classes of solutions, the discrete transformation Φ → −Φ will be a symmetry of the problem. The ansatz (4)(5) is also motivated by the fact that for Φ 0 = 0 most of the vacuum Lifshitz black hole solutions known for the theory have precisely this form [4], which also occurs for their charged extensions [12]. The remaining solution belongs to a different class where the structural metric function involves two different radial powers and will be presented in Sec. V. One the main aims of this work is to confirm the importance of the role played by the gravitational soliton for the thermal properties of the Lifshitz black holes. In order to achieve this task correctly, we will need the Lifshitz soliton counterparts of the black holes (4)(5). The solitons will be generically described by the following metric ds 2 = −r 2 l 2 dt 2 + l 2 r 2 dr 2 f (r) +r 2z l 2z f (r)dx 2 1 +r 2 l 2 D−2 i=2 dx 2 i , (6) with f (r) = 1 − 2 χ 1/z l r χ , Φ(r) = Φ 0 2 χ 1/z l r χ 2 .(7) The solitons are obtained from the black holes by means of a double Wick rotationt = −ix 1 andx 1 = −it supplemented by a re-definition of the horizon location r h = l 2 χ 1 z , which ensures the correct identification. The configurations that will be described below are fully determined in term of three parameters, namely the decay exponent χ, the strength Φ 0 and the dynamical exponent z. In order to simplify the discussion, we start by evaluating first the formulas of interest as the entropy, temperature and mass of the black hole configurations as well as the mass of the solitons for generic values of these constants, i.e. not for those that actually satisfy all the system constraints. The precise thermodynamic quantities are given later for each genuine solution with the help of these formulae. III. PRELIMINARY THERMODYNAMIC QUANTITIES First of all, the Wald entropy formula [18] for black holes that eventually extreme action (2) by fitting ansatz (4-5) generically reads S W ≡ − 2π r h l D−2 Ω D−2 P abcd ε ab ε cd r=r h = 2π κ 1 − κξΦ 2 0 + χ l 2 2(χ − 3z − 2D + 4)β 1 + (χ − 3z − D + 2)β 2 + 2(χ − 3z)β 3 × r h l D−2 Ω D−2 ,(8) where P abcd ≡ ∂(L g + L s )/∂R abcd and Ω D−2 represents the finite volume of the (D − 2)-dimensional planar base manifold. On the other hand, their temperature is given by T = 1 4π r z+1 h l z+1 f ′ (r h ) = χ 4πl r h l z .(9) In order to compute the masses of the black hole and soliton configurations defined in Eqs. (4-7), we will opt for the quasilocal formalism as defined in [19,20]. Notice that this formalism has proved to be well suited for correctly computing the masses of black holes of higherorder gravity theories with rather unconventional asymptotic, see e.g. [16]. The quasilocal formalism is based on an off-shell prescription [19] for the ADT potential [21] which allows the following concise expression for the conserved charge associated to a Killing vector field k Q(k) = B d D−2 x µν ∆N µν (k) − 2k [µ 1 0 ds Θ ν] (k|s) ,(10) where s is a parameter interpolating between the solution of interest at s = 1 and the asymptotic one at s = 0, the difference between their off-shell Noether potentials is denoted by ∆N µν (k) ≡ N µν s=1 (k) − N µν s=0 (k) and Θ ν is the surface term arising after varying the action. In the present case, these tensors are given by Θ µ = 2 √ −g P µ(αβ)γ ∇ γ δg αβ − δg αβ ∇ γ P µ(αβ)γ + 1 2 ∂L s ∂ (∂ µ Φ) δΦ , N µν = 2 √ −g (P µνρσ ∇ ρ k σ − 2k σ ∇ ρ P µνρσ ) . For a timelike Killing vector field, ∂ t = k µ ∂ µ , the evaluation of the mass formula for action (2) in the black hole ansatz (4)(5) gives rise to the expression M bh (k) = 4 Ψ 1 + κ Φ 2 0 [2 (2 χ + 2 z − D + 2) ξ − χ] × r h l 2χ l r 2χ−z−D+2 Ω D−2 8 κ l + 2 Ψ 2 − κ Φ 2 0 [4(χ + z)ξ − χ] × r h l χ l r χ−z−D+2 Ω D−2 4 κ l ,(11) where Ψ 1 and Ψ 2 are two dimensionless linear combinations of the squared corrections coupling constants reported in App. A. For the soliton ansatz (4-7) with timelike Killing vector field ∂t = k µ ∂ µ the mass formula reads M sol (k) = 4 Ξ 1 + κ Φ 2 0 [2 (4χ − 2 z − D + 6) ξ − χ] × 2 χ 2χ/z l r 2χ−z−D+2 Ω D−2 8 κ l + 2 Ξ 2 − κ Φ 2 0 [4(χ + 1)ξ − χ] × 2 χ χ/z l r χ−z−D+2 Ω D−2 4 κ l ,(12) where the dimensionless coupling constants combinations Ξ 1 and Ξ 2 are also defined in App. A. For actual solutions the mass expressions (11) and (12) cannot depend on the radial coordinates r andr, respectively. Interestingly, this imposes constraints on the constants z, χ and Φ 0 giving indications on the possible solutions within the ansatz; concretely, only two families of exponents χ are possible since they are the only giving rise to a nontrivial global charge mass. In what follows, we will report four different classes of Lifshitz black hole solutions fitting our ansatz (4)(5). For each solution, we will check that its gravitational Wald entropy (8) is correctly reproduced by means of a higherdimensional anisotropic Cardy-like formula [17] given by SC = 2πl(z + D − 2) D − 2 −(D − 2)M sol z z z+D−2 M bh D−2 z+D−2 .(13) This expression is the higher-dimensional extension of the one obtained for two-dimensional Lifshitz field theory [14]. Here we have used the notation S C for the microscopic entropy in order to reflect that the anisotropic Cardy-like expression is a priori different from the gravitational Wald entropy (8). Nevertheless, as shown below, both entropies will coincide for the different classes of solutions reported. For completeness, we will also verify that the first law of black hole thermodynamics dM bh = T dS W ,(14) consistently holds for each Lifshitz black hole solution. IV. FOUR CLASSES OF LIFSHITZ BLACK HOLES Here, we will present the four classes of solutions that fit within ansatz (4)(5) and compute their definitive thermodynamic quantities through the preliminary formulas derived in the previous section. For each solution, we will corroborate that their Wald entropy can be reproduced from the anisotropic Cardy-like formula (13). A. Class with arbitrary dynamical exponent and arbitrary nonminimal coupling parameter The first family of solutions is obtained for the standard potential where a mass term is supplemented by a quartic interaction and exists for arbitrary values of the dynamical exponent z and of the nonminimal coupling parameter ξ. Because of cumbersome formulae, the concrete form of the potential, and the parameterizations obeying the different coupling constants as well as the cosmological constant are reported in App. B. Its line element and the nontrivial scalar field are given by ds 2 = − r l 2z 1 − r h r (z+D−2)/2 dt 2 + l 2 r 2 1 − r h r (z+D−2)/2 −1 dr 2 + r 2 l 2 D−2 i=1 dx 2 i , Φ(r) = [3z 2 + (D − 2)(D + 2)]P 3 (z)l 2 − 2(D − 3)(D − 4)(z + D − 2)P 4 (z)β 3 κ l 2 P 5 (z; ξ) r h r (z+D−2)/4 ,(15) where the polynomials P n and the remaining details of the solution are defined in App. B. This solution is obtained from the proposed ansatz (5) by using one of the only two decay exponents allowing a well-defined Lifshitz mass, namely χ = (z + D − 2)/2. The result is the higher-dimensional lifting from D = 3 of the black hole family with Lifshitz decay (z + 1)/2 originally derived in Ref. [8] for New Massive Gravity, whose thermodynamics was studied in Ref. [16]. It is interesting to notice that the above higher-dimensional line element has been previously obtained also as a vacuum solution in [4], but for a more restrictive election of the coupling constant. The vacuum limit of [4] is easily recovered by fixing the coupling constant β 3 in solution (15) in order to obtain a vanishing scalar strength. In this sense the present solution is a generalization of the one of [4] allowing the same black hole to be dressed by a self-interacting nonminimally coupled scalar field. The thermodynamic properties of the lifted configuration follow from the following expressions, first, the preliminary formula (8) gives the Wald entropy S W = 2πΥ 1 κ r h l D−2 Ω D−2 ,(16) where the dimensionless coefficient Υ 1 , depending on the free coupling constants, is mutual to all the extensive thermodynamic quantities associated with the solution and is defined in App. D. It also measures the depar-ture of the theory from the behavior of standard gravity, whose areal interpretation of black holes entropy forces Υ 1 = 1. The Hawking temperature (9) in this case reads T = z + D − 2 8πl r h l z .(17) The formulae (11) and (12) lead to the Lifshitz black hole and soliton masses, respectively M bh = (D − 2)Υ 1 4κ r h l z+D−2 Ω D−2 l ,(18)M sol = − zΥ 1 4κ 4 z + D − 2 z+D−2 z Ω D−2 l .(19) Now, it is straightforward to verify that the Cardy-like formula (13) for the entropy correctly reproduces the gravitational Wald entropy, that is S W = S C . Another interesting feature of these thermodynamic quantities is that they obey the following anisotropic higher-dimensional version of the Smarr formula [24] M bh = D − 2 z + D − 2 T S W ,(20) which in fact is not unexpected since they consequently respect the first law (14). For the other decay exponent compatible with a welldefined mass, χ = z + D − 2, it happens that one obtains a vanishing mass, as will be exhibited in Subsec. IV C. However, there is an exception for the critical exponent z = D, this is the solution we present below. B. Solution with a fixed value of the dynamical exponent z = D The second family of Lifshitz black hole solutions exists for a dynamical exponent z = D and for a nonminimal coupling parameter ξ < (D − 1)/(5D − 2), ds 2 = − r 2D l 2D 1 − r h r 2(D−1) dt 2 + l 2 r 2 1 − r h r 2(D−1) −1 dr 2 + r 2 l 2 D−2 i=1 dx 2 i , Φ(r) = 1 √ κ (D − 2) D − 1 − (5D − 2)ξ r h r D−1 .(21) In this case, the self-interacting potential is also given by a mass term plus a Φ 4 −interaction and the coupling constants β 1 and β 3 are arbitrary U (Φ) = − (D − 1)[D − 1 − (5D − 2)ξ] 4(D − 2)l 2 × 2(D − 2)Φ 2 + [D − 1 − (D − 2)ξ]κΦ 4 , λ = (D − 1)[8D(D − 3)(D − 4)β 3 − (5D − 2)l 2 ] 4l 4 , β 2 = − 2(5D − 2)(D − 1)β 1 + 4(2D + 1)β 3 − l 2 2(D + 2)(D − 1) . This solution consistently fits the working ansatz (5) within the other admisible family of decay exponents χ = z +D−2 = 2(D−1) when the critical exponent takes the value z = D. Its Hawking temperature becomes T = (D − 1) 2πl r h l D , while the entropy together with the masses of the black hole and its soliton counterpart are given by S W = 2πΥ 2 κ r h l D−2 Ω D−2 , M bh = (D − 2)Υ 2 2κ r h l 2(D−1) Ω D−2 l , M sol = − DΥ 2 2κ 1 D − 1 2(D−1) D Ω D−2 l , where the mutual extensive coefficient is again defined in App. D. As before, one can check the validity of the first law and the Cardy-like formula (13), as well as of the Smarr formula (20). C. Lifshitz black holes with vanishing mass There exist two other families fitting our ansatz (5) which are defined for an arbitrary value of the dynamical exponent. These solutions present the peculiarity of having a vanishing mass and a zero Wald entropy, and hence the first law of thermodynamics is trivially satisfied. These zero mass solutions are also an interesting ground to test the validity of the Cardy-like formula since their solitons counterpart must also have a vanishing mass which in turn trivially implies that S W = 0 = S C . Here, we just report for completeness these Lifshitz black hole configurations of vanishing mass. The first solution is obtained by choosing the other decay exponent compatible with a well-defined Lifshitz mass χ = z + D − 2, this gives ds 2 = − r 2z l 2z 1 − r h r z+D−2 dt 2 + l 2 r 2 1 − r h r z+D−2 −1 dr 2 + r 2 l 2 D−2 i=1 dx 2 i , Φ(r) = 2 (D − 2)(z − 1) κP 2 (z; ξ) r h r z+D−2 2 ,(22) where the second grade polynomial in the critical exponent at the denominator of the scalar strength is defined as P 2 (z; ξ) ≡ (z + D − 2) 2 − 4[2z 2 + (D − 2)(2z + D − 1)]ξ. This solution is the higher-dimensional lifting of the family with Lifshitz decay z + 1 obtained in Ref. [8]. The self-interaction potential supporting the solution and the values of the coupling constants are extended as U (Φ) = − P 2 (z; ξ) 16l 2 2Φ 2 + (z + D − 2) 2 4(D − 2)(z − 1) − ξ κΦ 4 , β 1 = (D − 2)z 2 + 2(D − 3)z − (D − 2) 2 (D − 2)(z − 1)(z + D − 2) β 3 + (1 − 4ξ)l 2 2P 2 (z; ξ) , β 2 = − (D − 2)(4z 2 − D 2 + 3D − 4) + 2D(D − 3)z (D − 2)(z − 1)(z + D − 2) β 3 − l 2 2P 2 (z; ξ) , while the cosmological constant λ takes the same expression given in App. B. It is straightforward to check by means of formulas (11)(12) that the masses of the black hole and its soliton counterpart are zero. A property also shared by the Wald entropy (8). The other zero-mass solution is obtained for the decay exponent χ = 2(z − 1) which does not gives in general a global charge. However, this exponent works just because it exactly cancels the coefficients in front of the decaying powers preventing the mass formula to become a conserved charge, causing at the same time its vanishing. The resulting solution is ds 2 = − r l 2z 1 − r h r 2(z−1) dt 2 + l 2 r 2 1 − r h r 2(z−1) −1 dr 2 + r 2 l 2 D−2 i=1 dx i 2 , Φ(r) = 1 √ κl 4(D − 3)(D − 4)z(z − D)β 3 − (D − 1)(2z − D − 2)l 2 P 2 (z; ξ) r h r z−1 ,(23) where the polynomial in the critical exponent is given bỹ P 2 (z; ξ) ≡ [2z 2 + (D − 2)(2z + D − 1)]ξ − (D − 1)(z − 1), and the specific parameterizations of the coupling constants together with the cosmological one are presented in App. C. It is interesting to emphasize that this line element corresponds to other of the vacuum solutions previously obtained in [4] for the same theory, but with a more restrictive choice of the coupling constants. Notice that if we fix the coupling constant β 3 by demanding the vanishing of the scalar strength we recover the black hole of [4] without scalar field. In other words, this solution generalize the other vacuum example of higher-dimensional Lifshitz black hole by dressing it with a self-interacting nonminimally coupled scalar field. V. LAST CLASS OF LIFSHITZ BLACK HOLE There exists a fifth class of Lifshitz black holes that does not fit within our ansatz (5). This solution has a fixed value of the dynamical exponent z = D and is valid also for a precise value of the nonminimal coupling parameter ξ = (2D − 1)(D − 1) 2(3D 2 − 2D + 4)(D + 1) . The configuration in question reads ds 2 = − r 2D l 2D 1 − M l r 2(D−1) − α √ M l r D−1 dt 2 + l 2 r 2 1 − M l r 2(D−1) − α √ M l r D−1 −1 dr 2 + r 2 l 2 D−2 i=1 dx 2 i , Φ(r) = 1 √ κ 2 M (3 D 2 − 2 D + 4) (D + 1) (D − 1) P 2 (D) l r D−1 ,(24) where α is a coupling constant appearing in the potential and P 2 (D) = 6D 2 − 16D − 1. Indeed, this solution exists provided that the potential and the coupling constants are given by U (Φ) = − (D − 1) 3 P 2 (D)(D 3 − 4D 2 + 19D + 2)κ 16(D + 1) 2 (3D 2 − 2D + 4) 2 l 2 Φ 4 − α(2D − 1) 2l 2 (D − 1) 5 P 2 (D)κ 2(D + 1)(3D 2 − 2D + 4) 3 Φ 3 − (D − 2)(D − 1) 2 P 2 (D) 4(D + 1)(3D 2 − 2D + 4)l 2 Φ 2 , β 1 = (2D 3 − 6D 2 + 25D + 3)l 2 2(D − 4)(D − 1) 2 P 2 (D) , β 2 = − 2(D + 1)(D 2 + 3D − 1)l 2 (D − 4)(D − 1) 2 P 2 (D) , β 3 = 3(D + 1)l 2 2(D − 4)P 2 (D) , λ = − (D − 1)(D − 2)(18D 2 − 32D − 1) 4P 2 (D)l 2 . The event horizon of this black hole is located at the radius r D−1 h = l D−1 √ M 2 α + α 2 + 4 , and in terms of this radius the quantities of interest to corroborate the first law (14), the validity of the Cardylike (13) and Smarr (20) formulas are given by S W = 2πΥ 3 κ r h l D−2 Ω D−2 , T = (D − 1) √ α 2 + 4 2πl(α + √ α 2 + 4) r h l D , M bh = (D − 2) √ α 2 + 4Υ 3 2κ(α + √ α 2 + 4) r h l 2(D−1) Ω D−2 l , M sol = −D √ α 2 + 4Υ 3 2κ(α + √ α 2 + 4) α + √ α 2 + 4 (D − 1) √ α 2 + 4 2(D−1) D Ω D−2 l , where the extensive coefficient is expressed as all the previous ones in App. D. VI. CONCLUSION Here, we have extended the work done in three dimensions in the case of a scalar field nonminimally coupled to New Massive Gravity [16]. Indeed, we have considered a gravity theory given by the most general quadratic corrections to Einstein gravity supplemented by a source action describing a self-interacting nonminimally coupled scalar field. For this theory, we have presented five different classes of Lifshitz black hole solutions. Each solution is specified with a particular self-interacting potential and for a certain parametrization of the coupling constants β i accompanying the different gravity invariants. Interestingly, some of the obtained solutions describe Lifshitz black hole that were known previously as part of the vacuum of the studied theories [4], but for more restrictive elections of the coupling constants. Hence, they constitute generalizations of these vacuum higher-dimensional Lifshitz black holes that turn to be dressed by self-interacting nonminimally coupled scalar fields. It must be emphasized that, in contrast with the three-dimensional case, none of the different choices of the coupling constants β i corresponds to the recently discussed critical gravity points [22,23]. We would like to stress that our work constitute a new example putting in light the importance played by the gravitational solitons in order to describe the thermal properties of black holes with (un)usual asymptotics. In this spirit, it will be desirable to keep exploring this issue from the holographic point of view. In particular, a promising work to be done will consist in identifying or interpreting the role of the soliton in the field theory side. Another interesting aspect that has to do with these solutions concerns the Smarr formula [24]. Indeed, since all the solutions reported here verify the higher-dimensional anisotropic Cardy-like formula (13) as well as the first law of thermodynamics, they will also satisfy an anisotropic higherdimensional version of the Smarr formula (20). This last formula is in perfect accordance with the one obtained in Ref. [25] for different theories admitting Lifshitz black holes. It is evident that the emergence of the solutions presented here is essentially due to the higher-order nature of the gravity theory together with the nonminimal coupling of the scalar field to these gravity through the term RΦ 2 . One eventually can pursue the exploration on this issue by studying other Lifshitz black hole solutions that may arise from other nonminimal couplings as those recently put in spotlight through the Horndeski Lagrangian [26]. The dimensionless combinations of coupling constants appearing in the black hole mass formula (11) are defined by l 2 Ψ 1 = − (2 β 1 + β 2 + 2 β 3 ) χ 3 + 2 (2 β 1 + β 2 + 2 β 3 ) z + (D − 2) (5 β 1 + 2 β 2 + 3 β 3 ) χ 2 + (2 β 1 + β 2 + 2 β 3 ) z 2 − 3 (D − 2) (β 1 + β 2 + 3 β 3 ) z − (D − 2) [2 (2 D − 3) β 1 + (D − 2) β 2 − 2 β 3 ] χ − 2 (2 β 1 + β 2 + 2 β 3 ) z 3 − (D − 2) (2 β 1 − β 2 − 6 β 3 ) z 2 − 2 (D − 2) (β 1 + β 2 + 3 β 3 ) z + (D − 2) (D − 2) [(D − 1) β 1 + β 2 ] + 2 β 3 , l 2 Ψ 2 = 2 (2 β 1 + β 2 + 2 β 3 ) χ 3 − 4 (2 β 1 + β 2 + 2 β 3 ) z + (D − 2) (8 β 1 + 3 β 2 + 4 β 3 ) χ 2 − 2 (2 β 1 + β 2 + 2 β 3 ) z 2 − 3 (D − 2) (β 2 + 4 β 3 ) z − (D − 2) [4 (D − 1) β 1 + (D − 2) β 2 − 4 β 3 ] χ + 4 (2 β 1 + β 2 + 2 β 3 ) z 3 + 2 (D − 2) (2 β 1 − β 2 − 6 β 3 ) z 2 + 4 (D − 2) (β 1 + β 2 + 3 β 3 ) z − (D − 2) 2 (D − 2) [(D − 1) β 1 + β 2 ] + 4 β 3 − l 2 . The corresponding dimensionless combinations ap-pearing in the soliton mass formula (12) are l 2 Ξ 1 = − (4 β 1 + β 2 ) χ 3 + [(9 β 1 − β 3 + 2 β 2 ) D − 4 β 2 + 6 β 3 − 22 β 1 + (14 β 1 + 4 β 2 + 2 β 3 ) z] χ 2 + − (6 β 3 + 5 β 2 + 14 β 1 ) z 2 + ((3 β 3 − 15 β 1 − 3 β 2 ) D − 2 β 3 + 10 β 2 + 42 β 1 ) z − (6 β 1 + β 2 ) D 2 + (28 β 1 − 6 β 3 + 2 β 2 ) D − 32 β 1 + 8 β 3 ] χ + (4 β 3 + 4 β 1 + 2 β 2 ) z 3 + [−20 β 1 + (6 β 1 + β 2 − 2 β 3 ) D − 6 β 2 − 4 β 3 ] z 2 + 4 β 3 + 28 β 1 + 4 β 1 D 2 − (22 β 1 − 2 β 2 − 2 β 3 ) D − 4 β 2 z + (D − 6) β 1 D 2 − (3 β 1 − β 2 ) D + 2 β 3 + 2 β 1 − 2 β 2 , l 2 Ξ 2 = (β 2 + 4 β 1 ) χ 3 + [(−12 β 1 − 3 β 2 ) z + (−2 β 2 − 8 β 1 ) D − 4 β 3 + 4 β 2 + 20 β 1 ] χ 2 + {[(12 β 1 + 3 β 2 ) D − 36 β 1 − 4 β 3 − 10 β 2 ] z + (12 β 1 + 4 β 2 + 4 β 3 ) z 2 + (6 β 1 + β 2 ) D 2 + (−26 β 1 + 4 β 3 − 2 β 2 ) D − 4 β 3 + 28 β 1 − l 2 χ + (−8 β 3 − 8 β 1 − 4 β 2 ) z 3 + 8 β 3 + (−2 β 2 − 12 β 1 + 4 β 3 ) D + 40 β 1 + 12 β 2 z 2 + 2 l 2 − 56 β 1 − 8 β 1 D 2 + 4 (11 β 1 − β 2 − β 3 ) D − 8 β 3 + 8 β 2 ] z − 24 β 2 − 2 β 1 D 3 + 2 (9 β 1 − β 2 ) D 2 + −40 β 1 + l 2 + 16 β 2 − 4 β 3 D − 4 l 2 + 24 β 3 + 24 β 1 . Appendix B: Parameters associated to the first class of solutions (15) The potential associated to the first class of solutions of Subsec. IV A reads where for simplicity we have defined where the polinomial P 5 (z; ξ) was previously defined in App. B. The dimensionless extensive coefficient related to the second class of solutions of Subsec. IV B is written as l 2 Υ 2 = 8(D − 2)(D − 1) 2 D + 2 β 1 − 4(D 3 − D 2 − 2D − 4) D + 2 β 3 − (D − 1)(D − 2)(4ξ − 1)l 2 (D + 2) (5D − 2)ξ − D + 1 . The two classes of solution with vanishing mass of Subsec. IV C consequently have vanishing extensive coefficients. Finally, the dimensionless extensive coefficient of the last class in Sec. V is l 2 Υ 3 = − 2(D + 1)(2D − 1) √ α 2 + 4 (6D 2 − 16D − 1)(α + √ α 2 + 4) . ACKNOWLEDGMENTS This work has been partially funded by Conacyt grant A1-S-11548. M.B. is supported by grant Conicyt/Programa Fondecyt de Iniciación en Investigación No. 11170037. F.C. is supported by the Alexander von Humboldt Foundation and Fondecyt grant 1171475. F.C. is grateful for the warm hospitality at Leibniz Universität Hannover. U (Φ) = κ 4 3z 2 + (D − 2)(D + 2) ξ − (z + D − 2) 2 P 5 (z; ξ) 64 2(D − 4)(z + D − 2)(D − 3)β 3 P 4 (z) − 3z 2 + (D − 2)(D + 2) l 2 P 3 (z) Φ 4 + 16 2z 2 + 2(D − 2)z + (D − 1)(D − 2) ξ − 3(z + D − 2) 2 32l 2 Φ 2 ,while the coupling constants and the cosmological one are tied as followsβ 1 = 4P 4 (z) (z + D − 2) 2 β 3 + l 2 (z + D − 2)P 3 (z) − P 6 (z)β 3 − 4 l 2 (z + D − 2) P 3 (z) ξ 2 (z + D − 2) P 5 (z; + D − 2) P 5 (z; ξ) (z − 1) (z + D − 2) D 2 + 3 z 2 − 4 l 2 + β 3P4 (z) (z + D − 2) 2 − 4 β 3 2 z 2 + (2 D − 4) z + (D − 1) (D − 2) P 4 (z) ξ , λ = − 1 4l 2 2z 2 + (D − 2)(2z + D − 1) − 4(D − 3)(D − 4)z(z + D − 2)β 3 l 2 , − (z) ≡ − 15 z 3 + (19 D − 22) z 2 + 3 (D − 2) (D − 18) z + (D − 2) D 2 − 4 D + 36 , P 4 (z) ≡ − 3 z 4 + (7 D − 10) z 3 + 3 D 2 − 2 D − 25 z 2 + 4 − 14 D 2 + 30 D + D 3 z + (2 D + 7) (D − 2) 2 , P 4 (z) ≡ 9 z 4 − (18 D − 24) z 3 − 8 D 2 − 80 z 2 − 8 D 2 − 2 D 3 − 64 D + 160 z − (D − 2) D 3 + 2 D 2 − 12 D + 24 , P 4 (z) ≡ 6 z 4 − 2 (7 D − 10) z 3 − 13 D 2 + 34 − 53 D z 2 − 32 − 2 D − 2 D 3 + 2 D 2 z − (D − 2) D 3 + D 2 − 10 D + 20 , P 6 (z) ≡ − 144 z 6 + (144 D − 96) z 5 + 32 D 2 + 1440 D − 4400 z 4 + 352 D 3 − 1472 D 2 − 544 D + 5504 z 3 + 16 (D − 2) 7 D 3 − 62 D 2 + 212 D − 256 z 2 + 16 (D − 2) (D − 4) D 3 − 8 D 2 + 34 D − 76 z (5z + 3D − 6)(z + D − 2) 2 β 3 − (z + D − 2) 3z 2 + (D − 2)(D + 2) l 2 4P 5 (z; ξ) , Appendix A: Expressions for the coefficients of the mass formulae(11)and(12)Appendix C: Parameters associated to the third class of solutions(23)The parameters which ensure the existence of the zeromass solution (23) are given byAppendix D: Extensive coefficientsThe extensive thermodynamic quantities of each presented solution are proportional to a mutual coefficient encompassing the details to which the theory is thermodynamically sensitive, i.e. the values of this coefficient determine all the points in the parameter space of the theory probed by the solution which corresponds to the same thermodynamic behavior. 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{'abstract': 'In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higherdimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves to calculate the correct expressions for the masses of the higher-dimensional Lifshitz black hole as well as their corresponding soliton. The robustness of this Cardy-like formula is checked by showing that the microscopic entropy is in perfect agreement with the gravitational Wald entropy. Consequently, the calculated global charges are compatible with the first law of thermodynamics. We also verify that all the configurations satisfy an anisotropic higher-dimensional version of the Smarr', 'arxivid': '1904.09391', 'author': ['Eloy Ayón-Beato \nDepartamento de Física\nCINVESTAV-IPN\nApdo. Postal 14-74007000\n\nCDMX\nMéxico\n', 'Moisés Bravo-Gaete \nFacultad de Ciencias Básicas\nUniversidad Católica del Maule\nCasilla 617TalcaChile\n', 'Francisco Correa \nInstituto de Ciencias Físicas y Matemáticas\nUniversidad Austral de Chile\nCasilla 567ValdiviaChile\n', 'Mokhtar Hassaïne \nInstituto de Matemática y Física\nUniversidad de Talca\nCasilla 747TalcaChile\n', 'María Montserrat Juárez-Aubry \nArkansas State University\nCarretera estatal #100 km. 17.576270Municipio Colón, Querétaro, QuerétaroMéxico\n'], 'authoraffiliation': ['Departamento de Física\nCINVESTAV-IPN\nApdo. Postal 14-74007000', 'CDMX\nMéxico', 'Facultad de Ciencias Básicas\nUniversidad Católica del Maule\nCasilla 617TalcaChile', 'Instituto de Ciencias Físicas y Matemáticas\nUniversidad Austral de Chile\nCasilla 567ValdiviaChile', 'Instituto de Matemática y Física\nUniversidad de Talca\nCasilla 747TalcaChile', 'Arkansas State University\nCarretera estatal #100 km. 17.576270Municipio Colón, Querétaro, QuerétaroMéxico'], 'corpusid': 128305482, 'doi': '10.1103/physrevd.100.044024', 'github_urls': [], 'n_tokens_mistral': 14796, 'n_tokens_neox': 11893, 'n_words': 7147, 'pdfsha': '329a236cdb288d7da2dfa5f7b857d77febd9fa92', 'pdfurls': ['https://arxiv.org/pdf/1904.09391v1.pdf'], 'title': ['Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes', 'Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes'], 'venue': []}

arxiv

Robust covariance estimation with missing values and cell-wise contamination Karim Lounici karim.lounici@polytechnique.edu CMAP Ecole Polytechnique Palaiseau CMAP Ecole Polytechnique Palaiseau France, France Gregoire Pacreau gregoire.pacreau@polytechnique.edu CMAP Ecole Polytechnique Palaiseau CMAP Ecole Polytechnique Palaiseau France, France Robust covariance estimation with missing values and cell-wise contamination Large datasets are often affected by cell-wise outliers in the form of missing or erroneous data. However, discarding any samples containing outliers may result in a dataset that is too small to accurately estimate the covariance matrix. Moreover, most robust procedures designed to address this problem are not effective on highdimensional data as they rely crucially on invertibility of the covariance operator. In this paper, we propose an unbiased estimator for the covariance in the presence of missing values that does not require any imputation step and still achieves minimax statistical accuracy with the operator norm. We also advocate for its use in combination with cell-wise outlier detection methods to tackle cell-wise contamination in a high-dimensional and low-rank setting, where state-of-the-art methods may suffer from numerical instability and long computation times. To complement our theoretical findings, we conducted an experimental study which demonstrates the superiority of our approach over the state of the art both in low and high dimension settings.Preprint. Under review. Introduction Outliers are a common occurrence in datasets, and they can significantly affect the accuracy of data analysis. While research on outlier detection and treatment has been ongoing since the 1960s, much of it has focused on cases where entire samples are outliers, as demonstrated by Huber's work [7,30,9]. While sample-wise contamination is a common issue in many datasets, modern data analysis often involves combining data from multiple sources. For example, data may be collected from an array of sensors, each with an independent probability of failure, or financial data may come from multiple companies, where reporting errors from one source do not necessarily impact the validity of the information from the other sources. Discarding an entire sample as an outlier when only a few features are contaminated can result in the loss of valuable information, especially in high-dimensional datasets where samples are already scarce. It is important to identify and address the specific contaminated features, rather than simply treating the entire sample as an outlier. In fact, if each dimension of a sample has a contamination probability of ϵ, then the probability of that sample containing at least one outlier is given by 1 − (1 − ϵ) p , where p is the dimensionality of the sample. In high dimension, this probability can quickly exceed 50%, surpassing the breakdown point of many Right: For each method, mean computation time (in seconds) over 20 repetitions and whether it uses matrix inversion. For p = 100, we had to raise r (Σ) to 10 otherwise both DI and TSGS would fail due to numerical instability. robust estimators designed for the Huber sample-wise contamination setting. Hence, it is crucial to develop robust methods that can handle cell-wise contaminations and still provide accurate results. The issue of cell-wise contamination, where individual cells in a dataset may be contaminated, was first introduced in [2]. However, the issue of missing data due to outliers was studied much earlier, dating back to the work of [24]. Although missing values in a dataset are much easier to detect than outliers, they can still have a significant impact on the accuracy of statistical analysis and supervised learning tasks. Specifically, missing data can lead to errors in estimating the location and scale of the underlying distribution [16] and can negatively affect the performance of supervised learning algorithms [11]. Several robust estimation methods have been proposed to handle missing data, including Expectation Maximization (EM)-based algorithms [5], maximum likelihood estimation [10] and Multiple Imputation [16], among which we can find k-nearest neighbor imputation [28] and iterative imputation [31]. For the covariance matrix estimation in the Missing Completely At Random (MCAR) framework of [24], [17] provides suboptimal theoretical guarantees and a debiasing scheme for the estimation of the covariance. In comparison to data missingness or its sample-wise counterpart, the cell-wise contamination problem is less studied. The Detection Imputation (DI) algorithm of [21] is an EM type procedure combining a robust covariance estimation method with an outlier detection method to iteratively update the covariance estimation. Other methods include adapting methodology created for Huber contamination for the cell-wise problem, such as in [4] or [1]. In high dimensional statistics, however, most of these methods fail due to high computation time and numerical instability. They are simply not designed to work in this regime since they are based on the Mahalanobis distance, which requires an inversion of the estimated covariance matrix. This is a major issue since classical covariance matrix estimators have many eigenvalues close to zero or even exactly equal to zero in high-dimension. Furthermore, to the best of our knowledge, no theoretical result exists concerning the statistical accuracy of these methods in the cell-wise contamination setting contrarily to the extensive literature on Huber's contamination. Contributions. In this paper, we address the problem of high-dimensional covariance estimation in the presence of missing observations and cell-wise contamination. To formalize this problem, we adopt and generalize the setting introduced in [6]. We propose a computationally efficient and numerically stable procedure that avoids matrix inversion, making it well-suited for high-dimensional data. We derive non-asymptotic estimation bounds of the covariance with the operator norm and matching minimax lower bounds (up to log), which clarify the impact of the missing value rate and outlier contamination rate. Our theoretical results also provide a significant improvement over [17] in the MCAR and no contamination setting. Next, we conduct an experimental study on synthetic data, comparing our proposed method to the state-of-the-art (SOTA) methods. Our results demonstrate that SOTA methods fail in the high-dimensional regime due to matrix inversions, while our proposed method performs well in this regime, highlighting its effectiveness. Then we demonstrate the practical utility of our approach by applying it to real-life datasets, which highlights that the use of existing estimation methods significantly alters the spectral properties of the estimated covariance matrices. This implies that cell-wise contamination can significantly impact the results of dimension reduction techniques like principal component analysis (PCA) by completely altering the computed principal directions. Our experiments demonstrate that our method is more robust to cell-wise contamination than SOTA methods and produces reliable estimates of the covariance. 2 Missing values and cell-wise contamination setting Let X 1 , . . . , X n be n i.i.d. copies of zero mean vector random vector X admitting unknown covariance operator Σ = E [X ⊗ X], where ⊗ is the outer product. Denote by X (j) i the jth component of vector X i for any j ∈ [p]. All our results are non-asymptotic and cover all configurations of n, p including the high-dimensional setting p ≫ n. In this paper, we consider the following two realistic scenarios where the measurements are potentially corrupted. Missing values. We assume that each component X (j) i is observed independently from the others with probability δ ∈ (0, 1]. Formally, we observe the random vector Y ∈ R p defined as follows: Y (j) i = d i,j X (j) i , 1 ≤ i ≤ n, 1 ≤ j ≤ p(1) where d ij are independent realisations of a bernoulli random variable of parameter δ. This corresponds the Missing Completely at Random (MCAR) setting of [24]. Cell-wise contamination. Here we assume that some missing components X (j) i can be replaced with probability ε by some independent noise variables , representing either a poisoning of the data or random mistakes in measurements. The observation vector Y then satisfies: Y (j) i = d i,j X (j) i + (1 − d i,j )e i,j ξ (j) i , 1 ≤ i ≤ n, 1 ≤ j ≤ p(2) where ξ 1 , . . . ξ n are i.i.d. erroneous measurements and e i,j are i.i.d. bernoulli random variables with parameter ε. We also assume that all the variables X i , ξ i , d i,j , e i,j are mutually independent. In this scenario, a component X (j) i is either perfectly observed with probability δ, replaced by a random noise with probability ε ′ = ε(1 − δ) or missing with probability (1 − δ)(1 − ε). Cell-wise contamination as introduced in [2] corresponds to the case where ε = 1, and thus ε ′ = 1 − δ. If we consider the mean estimation problem, then the cell-wise contamination problem is indistinguishable from the classical Huber contamination problem since estimation of a mean vector is equivalent to the estimation of each marginal mean independently from the others. Since cell-wise contamination is equivalent to contamination of each marginal independently from the other marginals following Huber's paradigm, we can use for instance Tuker's median as a robust estimator of the mean. However, as argued in [2], the situation is quite different for covariance estimation. Our proposal is based on a correction of the classical covariance estimator on Y 1 , . . . , Y n first introduced in [17] for the missing values scenario. The procedure is based on the following observation, with Σ Y the covariance of the data with missing values and Σ the true covariance: Σ = δ −1 − δ −2 diag(Σ Y ) + δ −2 Σ Y(3) Note that this formula assumes the knowledge of δ. In the missing values scenario, δ can be efficiently estimated by a simple count of the the values exactly set to 0 or equal to NaN (not a number). However, in presence of contamination as in (2), one does not know the exact location and number of outliers. In our experiments, we will estimate δ by the proportion of data remaining after a filtering procedure. Notations. We denote by ⊙ the Hadamard (or term by term) product of two matrices and by ⊗ the outer product of vectors, i.e. ∀x, y ∈ R d , x ⊗ y = xy ⊤ . We denote by ∥.∥ and ∥.∥ F the operator and Frobenius norms of a matrix respectively. Optimal estimation of covariance matrices with missing values We consider the scenario outlined in (1) where the matrix Σ is of approximately low rank. To quantify this, we use the concept of effective rank, which provides a useful measure of the inherent complexity of a matrix. Specifically, the effective rank of Σ is defined as follows r(Σ) := E ∥X∥ 2 ∥Σ∥ = tr (Σ) ∥Σ∥(4) We note that 0 ≤ r(Σ) ≤ rank(Σ). Furthermore, for approximately low rank matrices with rapidly decaying eigenvalues, we have r(Σ) ≪ rank(Σ). This section presents a novel analysis of the estimator defined in equation (3), which yields a non-asymptotic minimax optimal estimation bound in the operator norm. Our findings represent a substantial enhancement over the suboptimal guarantees reported in [17]. Non-asymptotic upper-bound in the operator norm. We provide an upper bound of the estimation error in operator norm. We write Y i = d i ⊙ X i . Let Σ Y = n i=1 Y i ⊗ Y i be the classical covariance estimator of the covariance of Y . When the dataset contains missing values and corruptions, Σ Y is a biased estimator of Σ. Exploiting (3), [17] proposed the following unbiased estimator of the covariance matrix Σ: Σ = δ −2 Σ Y + (δ −1 − δ −2 )diag Σ Y .(5)Σ − Σ ≤ C ∥Σ∥ δ r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n(6) This bound improves upon [17,Proposition 3] which proved with probability at least 1 − e −t : Σ − Σ ≤ C ∥Σ∥ r(Σ)(t + log(2p)) δ 2 n ∨ r(Σ)(t + log(2p)) δ 2 n (δ + t + log n) Contrarily to the previous display, the bound in (6) admits an improved dependence on the parameter δ as we replaced δ 2 by δ in the denominator. Actually, this bound is minimax optimal (up to log) and even sharp minimax when r(Σ) ≥ log p as we will prove it in Theorem 2. The complete proof argument for Theorem 1 is provided in appendix E.2. It relies on a recent generic chaining result for quadratic processes. Comparatively, the bound in [17,Proposition 3] was based on non-commutative Bernstein inequality and is never minimax-optimal in any settings of δ, n, p, r(Σ). Sketch of proof. We note that Σ − Σ ≤ δ −2 Σ Y − E[ Σ Y ] + δ −1 diag Σ Y − E[Σ Y ] . We bound Σ Y − E[Σ Y ] using a generic chaining argument and diag Σ Y − E[Σ Y ] via Bernstein's inequality. Minimax lower-bound. We now provide a minimax lower bound for the covariance estimation with missing values problem. Let S p the set of p × p symmetric semi-positive matrices. Then, define C r = {S ∈ S p : r(S) ≤ r} the set of matrices of S p with effective rank at most r. Theorem 2. Let p, n, r be strictly positive integers such that p ≥ max{n, 2r}. Let X 1 , . . . , X n be i.i.d. random vectors in R p with covariance matrix Σ ∈ C r . Let (d i,j ) 1≤i≤n,1≤j≤p be an i.i.d. sequence of Bernoulli random variables with probability of success δ ∈ (0, 1], independent from the X 1 , . . . , X n . We observe n i.i.d. vectors Y 1 , . . . , Y n ∈ R p such that Y (j) i = d i,j X (j) i , i ∈ [n], j ∈ [p] . Then there exists two absolute constants C > 0 and β ∈ (0, 1) such that: inf Σ max Σ∈Cr P Σ Σ − Σ ≥ C ∥Σ∥ δ r(Σ) n ≥ β(7) where infΣ represents the infimum over all estimatorsΣ of matrix Σ based on Y 1 , . . . , Y n . This lower bound improves upon [17,Theorem 2] as it relaxes the hypotheses on n and r. More specifically, the lower bound in [17] requires n ≥ 2r 2 /δ 2 while we only need the mild assumption p ≥ max{n, 2r}. Furthermore, the above lower bound matches the upper bound of Theorem 1 up to a logarithmic factor in the high-dimensional regime p ≥ max{n, 2r} and n ≥ r(Σ), hence clarifying the impact of missing data on the estimation rate via the parameter δ. Furthermore, if we have in addition r(Σ) ≥ log p, then the lower bound matches perfectly the upper-bound. Our proof argument leverages the properties of the Grassmann manifold, which has been previously utilized in different settings such as sparse PCA without missing values or contamination [33] and lowrank covariance estimation without missing values or contamination [14]. However, tackling missing values in the Grassmann approach is the main technical challenge as it modifies the distribution of observations and requires several additional nontrivial arguments to control the distribution divergences, which is a crucial step in deriving the minimax lower bound for our problem. Sketch of proof. We first build a sufficiently large test set of hard-to-learn covariance operators exploiting entropy properties of the Grassmann manifold such that the distance between any two distinct covariance operator is at least of the order ∥Σ∥ δ r(Σ) n . Next, in order to control the Kullback-Leibler divergence of the observations with missing values, we exploit in particular interlacing properties of eigenvalues of the perturbed covariance operators [26]. Heterogeneous missingness. In the MCAR scenario, we assume now that each feature has a different missing value rate. We denote by δ j ∈ (0, 1] the probability to observe feature X (j) , 1 ≤ j ≤ p. We define nextδ = max j {δ j } and δ = min j {δ j } the largest and smallest probabilities to observe a feature. By a straightforward modification of Σ and the proof of Theorem 1, under the same assumptions on X, we get, for any t > 0, with probability at least 1 − e −t Σ − Σ ≤ Cδ ∥Σ∥ δ 2 r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n .(8) Similarly we also obtain the following lower bound. For δ ∈ [1/2, 1], let p, n, r be strictly positive integers such that n ≥ 2r/δ 2 and p ≥ 2r. Then we have inf Σ max Σ∈Cr P Σ Σ − Σ ≥ C ∥Σ∥ r(Σ) δ 2 n ≥ β.(9) Ifδ ≍ δ, then the rates in (8) and (9) are matching and the minimax optimality result remains valid. Optimal estimation of covariance matrices with cell-wise contamination We consider the contamination scenario described in (2). We further assume that the ξ 1 , . . . ξ n are subgaussian r.v. and that Λ : = E[ξ 1 ⊗ ξ 1 ] is diagonal. In the presence of cell-wise contaminations, the operator Σ Y = E (Y ⊗ Y ) satisfies Σ Y = δ 2 Σ + (δ − δ 2 )diag(Σ) + ε(1 − δ)Λ. Note that the additional term ε(1 − δ)Λ in the cell-wise contamination setting becomes negligible when δ ≈ 1 or ε ≈ 0. Using the DDC detection procedure procedure of [21], we can detect the contaminations and accurately estimate both δ, ϵ and the diagonal operator Λ. We will not develop this aspect further and simply assume that these are known in the following result. Hence we propose the following unbiased estimator of Σ. Let Σ Y = n −1 n i=1 Y i ⊗ Y i and Σ = (δ −1 − δ −2 )diag Σ Y + δ −2 Σ Y − ε(1 − δ) δ Λ.(10) Non-asymptotic upper-bound in the operator norm. We prove the following result. Theorem 3. Let the assumptions of Theorem 1 be satisfied. We assume in addition that the observations Y 1 , . . . , Y n satisfy (2) with ε ∈ [0, 1) and δ ∈ (0, 1]. Then, for any t > 0, with probability at least 1 − e −t : Σ − Σ ≲ ∥Σ∥ δ r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n + ε(1 − δ) ∥Λ∥ δ r(Λ) n ∨ r(Λ) n ∨ log p n ∨ log p n ∨ t n ∨ t n + 1 δ ε(1 − δ) + ε(1 − δ) √ δ δ 2 ∥Λ∥ ∥Σ∥ r(Λ) ∨ r(Σ) t + log(r(Λ) ∨ r(Σ)) n + δε(1 − δ) δ 2 ∥Λ∥ ∥Σ∥ r(Λ) ∨ r(Σ) (t + log(r(Λ) ∨ r(Σ))) log n n .(11) Sketch of proof. We first note that Σ − Σ ≤ δ −2 Σ Y − Σ Y + δ −1 diag Σ Y − Σ Y + δ −2 Σ X,ξ,δ,ε .(12) The triangular inequality gives Σ Y − Σ Y = Σ δ − Σ δ + Λ ε − EΛ ε + Σ X,ξ,δ,ε ≤ Σ δ − Σ δ + Λ ε − E Λ ε + Σ X,ξ,δ,ε , where the three empirical matrices are 1. Σ δ = n −1 n i=1 (d i ⊗ d i ) ⊙ (X i ⊗ X i ), the empirical covariance matrix of the d i ⊙ X i ; 2. Λ ε = n −1 n i=1 ([(1 − d i ) ⊙ e i ] ⊗ [(1 − d i ) ⊙ e i ]) ⊙ (ξ i ⊗ ξ i ), the empirical covariance of the (1 − d i ) ⊙ e i ⊙ ξ i is such that E Λ ε = ε(1−δ) δ Λ; 3. Σ X,ξ,δ,ε = n −1 n i=1 (d i ⊗ [(1 − d i ) ⊙ e i ]) ⊙ (X i ⊗ ξ i ) + ([(1 − d i ) ⊙ e i ] ⊗ d i ) ⊙ (ξ i ⊗ X i ) is the empirical covariance between the d i ⊙ X i and the (1 − d i ) ⊙ e i ⊙ ξ i . We tackle Σ δ − Σ δ and Λ ε − E Λ ε similarly as in the proof of Thm 1. We tackle Σ X,ξ,δ,ε via a dimension-free non-commutative Bernstein inequality [18,27]. See App E.3 for the full details. As emphasized in [12], the effective rank r(Σ) provides a measure of the statistical complexity of the covariance learning problem in the absence of any contamination. However, when cell-wise contamination is present, the statistical complexity of the problem may increase if r(Λ) ≥ r(Σ). Fortunately, if the filtering process reduces the proportion of cell-wise contamination from ϵ to ϵ ′ such that ϵ ′ tr(Λ) ≤ tr(Σ) and ϵ ′ ∥Λ∥ ≤ ∥Σ∥, then we can effectively mitigate the impact of cell-wise contamination, as highlighted in the following result. Corollary 1. Let the assumptions of Theorem 3 be satisfied. Assume in addition that ϵ tr(Λ) ≤ tr(Σ) and ϵ ∥Λ∥ ≤ ∥Σ∥. Then, for any t > 0, with probability at least 1 − e −t Σ − Σ ≲ ∥Σ∥ δ r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n + ∥Σ∥ ε(1 − δ) δ 2 + ∥Σ∥ δ r(Σ) n (1 − δ)(t + log p) + ∥Σ∥ δ r(Σ) n √ 1 − δ [t + log p] log n √ δ n .(13) Proof. This is a straightforward consequence of Theorem 3. 940 ± 350 2, 800 ± 900 1.7 × 10 6 ± 3.8 × 10 5 Minimax lower-bound. The lower bound for missing values still applies to the contaminated case as missing values are a particular case of contamination. However replacing missing values by adversarial contaminations and using the proof argument of [3] for Huber's contamination, we obtain in the cell-wise setting the following minimax lower bound. Theorem 4. Let p, n, r be strictly positive integers such that p ≥ max{n, 2r}. Let X 1 , . . . , X n be i.i.d. random vectors in R p with covariance matrix Σ ∈ C r . Let (d i,j ) 1≤i≤n,1≤j≤p be i.i.d. sequence of bernoulli random variables of probability of success δ ∈ (0, 1], independent to the X 1 , . . . , X n . We observe n i.i.d. vectors Y 1 , . . . , Y n ∈ R p satisfying (2) where ξ i are i.i.d. of arbitrary distribution Q. Then there exists two absolute constants C > 0 and β ∈ (0, 1) such that: inf Σ max Σ∈Cr max Q P Σ,Q Σ − Σ ≥ C ∥Σ∥ δ r(Σ) n ε(1 − δ) ≥ β(14) where infΣ represents the infimum over all estimators of matrix Σ and max Q is the maximum over all contamination Q. This lower bound combined with the upper bound of Corollary (1) clarifies the impact of the cell-wise contamination parameter ϵ. The proof can be found in Appendix F.3. Experiments In our experiments, MV refers to the debiased covariance estimator (5). The synthetic data generation is described in Appendix A. We also performed experiments on real life datasets described in App. B. All experiments were conducted on a 2020 MacBook Air with a M1 processor (8 cores, 3.4 GHz). 1 Missing Values sklearn [20] provides two popular imputation methods: KNNImputer, which imputes the missing values based on the k-nearest neighbours [28], and IterativeImputer, which is inspired by the R package MICE [31]. In Figures 2, 3 and Table 1, we compare our estimator MV defined in (5) to these two imputation methods combined with the usual covariance estimator on synthetic data (see appendix A for details of data generation) in terms of statistical accuracy and execution time. We show that MV achieved a statistical accuracy similar to that of the SOTA IterativeImputer but is significantly faster even on moderately high dimensional data (less than 10 milliseconds for MV against about 28 minutes for IterativeImputer). MV also significantly beats KNNImputer both in term of statistical accuracy and computation time. We also see that trivial marginal imputation simply does not work. Based on these results, we can also argue that imputation of missing values is not mandatory for accurate estimation of the covariance operator : another viable option is to apply a debiasing correction to the empirical covariance computed on the original data containing missing values. The advantage of this approach is its low computational cost. Cell-wise contamination Methods tested. Our baselines are the classical covariance estimator applied without care for contamination and an oracle which knows the position of every outlier, deletes them and then computes the MV bias correction procedure (5). In view of Theorems 1 and 2, this oracle procedure is the best possible in the setting of cell-wise contamination. Hence, we have a practical framework to assess the performance of any procedure designed to handle cell-wise contamination. Spectral error Classical Mean Imputation Median Imputation II KNN MV (ours) Figure 3: Estimation error on a synthetic dataset with p = 500, n = 300, r (Σ) = 5. The SOTA methods in the cell-wise contamination setting are the DI (Detection-Inputation) method [21] and the TSGS method (Two Step Generalised S-estimator) [1]. Both these methods were designed to work in the standard setting n > p but cannot handle the high-dimensional setting as we already mentioned. Nevertheless, we included comparisons of our methods to them in the standard setting n > p. The code for DI and TSGS are from the R packages cellwise and GSE respectively. Our estimators are referenced as DDCMV (short for Detecting Deviating Cells Missing Values), which uses the DDC detection procedure of [23] to first remove outliers and then compute the debiaised covariance of (5) on the filtered data, and tailMV, which detects outliers through thresholding and then uses again (5). We also combined the filtering step with KNNimpute and IterativeImputer to define two additional novel robust procedures which we call DDCKNN and DDCII. To the best of our knowledge, this second alternative approach combining filtering with missing values imputation has never been tested to deal with cell-wise contamination. A detailed description of each method is provided in appendix C. Outlier detection and estimation error under cell-wise contamination on synthetic data. We showed that the error of a covariance estimator under cell-wise contamination depends on the proportion of remaining outliers after a filtration. In table 2 we investigate the filtering power of the Tail Cut and DDC methods in presence of Dirac contamination. We consider the cell-wise contamination setting (2) in the most difficult case ϵ = 1 which means that an entry is either correctly observed or replaced by an outlier (in other words, the dataset does not contain any missing value). For each values of δ in a grid, the quantitiesδ andε are the proportions of true entries and remaining contaminations after filtering averaged over 20 repetitions. The DDC based methods are particularly efficient since the proportion of Dirac contamination drops from 1 − δ to virtually 0 for any δ ≥ 0.74. In Fig. 4, we see that the performance of our method is virtually the same as the oracle OracleMV as long as the filtering procedure correctly eliminates the Dirac contaminations. As soon as the filtering procedure fails, the statistical accuracy brutally collapses and our DDC based estimators no longer do better than the usual empirical covariance. In Table 6 in App. H and Fig. 5, we repeated the same experiment but with a centered Gaussian contamination. Contrarily to the Dirac contamination scenario, we see in Fig. 5 that the statistical accuracy of our DDC based methods slowly degrades as the contamination rate increases but their performance remains significantly better than that of the usual empirical covariance. The effect of Cell-wise contamination on real-life datasets We tested the methods on 8 datasets from sklearn and Woolridge's book on econometrics [34]. These are low dimensional datasets (less than 20 features) representing various medical, social and economic phenomena. We also included 2 high-dimensional datasets. See App. B for the list of the datasets. One interesting observation is that the instability of Mahalanobis distance-based algorithms is not limited to high-dimensional datasets. Even datasets with a relatively small number of features can exhibit instability. This can be seen in the performance of DI on the Attend dataset, as depicted in On the Abalone dataset, once we have removed 4 obvious outliers (which are detected by both DDC and the tail procedure), all estimators reached a consensus with the non-robust classical estimator, meaning that this dataset provides a ground truth against which we can evaluate and compare the performance of robust procedures in our study. To this end, we artificially contaminate 5% of the cells at random in the dataset with a Dirac contamination and compare the spectral error of the different robust estimators. As expected, TSGS and all our new procedures succeed at correcting the error, however DI becomes unstable (see Table 3). On Breast Cancer, DI also disagrees with every other procedures (see Figure 6), casting some doubt on the reliability of its estimate. We also performed experiments on 2 high-dimensional datasets, where our methods return stable estimates of the covariance (DDCMV99 and DDCMV95 are within ≈ 3% of each other) and farther away from the classical estimator (See Figures 13 and 14 in App. H). Note also that DDCII's computation time explodes and even returns out-of-memory errors due to the high computation cost of IterativeImputer that we already highlighted in Table 1. Conclusion and future work In this paper, we have derived sharp theoretical upper bounds on the spectral error of our covariance estimator robust to missing data with matching minimax lower bounds in the missing value setting. We have also derived the first theoretical guarantees in the cell-wise contamination setting. We highlighted in our numerical experimental study that in the missing value setting, our debiased estimator designed to tackle missing values without imputation offers statistical accuracy similar to the SOTA IterativeImputer for a dramatic computational gain. We also found that SOTA algorithms in the cell-wise contamination setting often fail in the standard setting p < n for dataset with fast decreasing eigenvalues (resulting in approximately low rank covariance), a setting which is commonly encountered in many real life applications. This is due to the fact that these methods use matrix inversion which is unstable to small eigenvalues in the covariance structure and can even fail Probability that a cell be observed correctly ϵ Probability that an unobserved cell be contaminated Σ True covariance matrix of X Σ Y True covariance matrix of Y r(Σ) Effective rank of Σ Σ Unbiased estimator of the covariance of X Σ Y Empirical covariance of Y Λ Noise covariance matrix ∥X∥ Operator norm of X ∥X∥ F Forbenius norm of X ∥X∥ ψ1 , ∥X∥ ψ2 Orlicz norms of X ⊙ Hadamard or term by term product of matrices ⊗ Outer product of vectors Appendix A presents the synthetic data generation procedure used throughout our experiments. Appendix B and in particular Table 5 list the real life datasets presented in the paper. The cell-wise contamination correction methods are shown in Appendix C, with the DDC algorithm of [23] further detailed in Appendix D for convenience. The upper bound proofs can be found in Appendix E and the lower bound proofs in Appendix F, so that similar proof techniques be grouped together for clarity. Additional technical elements of these proofs were grouped in Appendix G when we felt that they impacted the latter's readability. Finally, we show the full results of our experiments in Appendix H. A Synthetic data generation We generate synthetic datasets of n realisations of a multivariate centered normal distribution. Its covariance matrix is defined as follows. We first set the eigenvalues as exp (−j/r) for j ∈ {1, p}, where r is the requested effective rank of the matrix. This approximation guaranties that the true effective rank is below r +1 for r << p. Then, using the ortho-group tool from scipy.stats, we create a random orthonormal matrix H and set Σ = Hdiag(g(Λ))H ⊤ , which is symmetric and of low effective rank at most r + 1. Finally, we divide Σ by its largest diagonal term so that the variances of the marginals be closer to 1. We contaminate our synthetic datasets using a binary mask obtained by computing the realisation of n × p i.i.d. bernoulli random variables. We fill the resulting missing data with either n samples of a isotropic gaussian of covariance σI p , where σ is the strength of the contamination (which we call the Gaussian contamination) or a n × p array of value ±σ (which we call the Dirac contamination). Let ξ be a random vector following one of those two contaminations, the data we feed all algorithms is then Y = mask ⊙ X + (1 − mask) ⊙ ξ. B Real life data set For our real data experiments, we removed any categorical variable from the datasets as well as variables that appeared to be mixtures of two distribution with different means. We also applied a log transform to skewed variables. The list of datasets can be found in Table 5. Finally, the Abalone dataset contains four obvious outliers that we removed in our experiments although they were easily detected by both DDC and the thresholding procedure. oracleMV is an oracle that knows which cells are contaminated. This method shows the performance of our corrected estimator in the case of a perfect outlier detection algorithm, hence providing an idea of the optimal precision attainable with regard to the available information. C.2 Our methods tailMV or tail Missing Values, is an estimator built by deleting extreme values in the dataset. It is actually one of the intermediary steps of DDC and we wanted to test how efficient it was on its own. We use the robust Huber estimator of the python package Statsmodel.robust [8] to compute the standard deviation of each marginal and eliminate any cell with value above 3 times these estimates. DDCMV short for Detecting Deviating Cells Missing Values, is an estimator built using the DDC detection procedure of [21], where detected outliers are removed and considered as missing values. A detailed description of DDC is provided in appendix D. We then apply our corrected covariance estimator. We will add to the name of the method the quantile at which we consider a data as an outlier (DDCMV99 uses the 99-percentile of χ 2 1 for instance). When nothing is mentioned, assume that DDCMV99 is used. In our experiments, we use the R implementation found in the package cellWise, whose results are then sent to a python script for formatting. DDCKNN detects outliers with the DDC procedure, removes them and imputes the missing values using the k-nearest neighbour procedure of [28] as implemented in sklearn under the name KNNImputer. DDCII also detects and removes outliers with the DDC procedure, then imputes the missing values using sklearn's Iterative Imputer class. C.3 SOTA methods for cell-wise contamination DI or Detection Imputation [22] Is an iterative algorithm made of two alternating steps inspired by the Expectation Maximisation (EM) algorithm. The first detects outliers with regard to a previously estimated covariance matrix, then the second computes a new covariance matrix having removed the previously detected outliers using the M step of EM, but with bias correction. This new matrix is then the basis for the next detection step and so on. The authors found their algorithm to have a O(T np 3 ) complexity, with T the number of iterations, and make the assumption that the covariance matrix is of full rank to perform matrix inversion, both facts that make it difficult to use in high dimensions. TSGS or Two Steps Generalised S-estimator [1] and [15] is also based on a two step process of detection then correction. Detection is based on the same DDC procedure while the estimation phase is based on the Generalised S-estimator of [4]. S-estimators are based on the Mahalonobis distance and thus require the true covariance matrix to be invertible. This may lead to numerically instability in our approximately low rank setting. However, if the matrix is of full rank, the generalised version of these estimators are proven to be consistent in the Missing Completely At Random setting. D The Detecting Deviating Cells algorithm This section is entirely based on [23], whose algorithm we describe here for convenience. DDC (Detecting Deviating Cells) is a 7 steps algorithm. In the following, let (X j i ) i∈[n],j∈[p] be our dataset of n samples from data with dimension p. Step 1: standardisation We start by assuming that the X i follow a normal distribution and we set Z j i = X j i − µ j X σ j X with µ j X being the empirical mean of marginal j, and σ j X its standard deviation. Step 2: cutoff DDC sets to NA all values of Z j i if |Z j i | ≥ χ 2 1,p with χ 2 1,p the p th centile of a χ 2 1 distribution, where p = 99% by default. Step 3: bivariate relationship The algorithm then computes the correlation between each couple of marginals. If |ρ i (Z j , Z k )| ≤ 0.5; set b jk = 0. Otherwise, b jk = slope(Z j |Z k ) with slope(x|y) the robust slope in the linear regression of x using y. Step 4: comparison Then DDC tries to predict the expected values of each Z j i according to a weighted mean of the values of the other marginals, using the previously computed correlations as weights.Ẑ j i = G {b jk Z k i , k ∈ [p] , k ̸ = j} with G the weighted mean using ρ(Z j , Z h ) as weights. Step 5: deshrinkage DDC adjusts the mean to account for shrinkage. a j = slope(Z j i |Ẑ j i ) Z j⋆ i = a jẐ j i Step 6: residual computation Then, one can take the residuals: r j i = Z j i −Ẑ j i µ Z j −Ẑ j Step 7: destandardisation Finally, DDC returns the data to its actual location and scale. The residuals can then be tested using a χ 2 1 law to determine whether or not they are outliers. E Proof of upper bounds E.1 Tools and definitions We recall the definition and some basic properties of sub-exponential random vectors. Definition 1. For any α ≥ 1, the ψ α -norms of a real-valued random variable V are defined as: ∥V ∥ ψα = inf{u > 0, E exp (|V | α /u α ) ≤ 2} We say that a random variable V with values in R is sub-exponential if ∥V ∥ ψα < ∞ for some α ≥ 1. If α = 2, we say that V is sub-Gaussian. If a real-valued random variable V is sub-Gaussian, then V 2 is sub-exponential. Indeed, we have: V 2 ψ1 ≤ 2 ∥V ∥ 2 ψ2 Definition 2. A random vector X ∈ R p is sub-exponential if ⟨X, x⟩ are sub-exponential random variables for x ∈ R p . The ψ α -norms of a random vector X are defined as: ∥X∥ ψα = sup x∈R p ,|x|2=1 ∥⟨X, x⟩∥ ψα , α ≥ 1 Bernstein's inequality can be adapted to the matrix setup as follows (see corollary 5.17 in [32]): Proposition 1. Let Z 1 , . . . Z n be independent sub-exponential zero mean real-valued random variables. Set K = max i ∥Z i ∥ ψ1 . Then, for t > 0, with probability at least 1 − e −t : n −1 n i=1 Z i ≤ CK t n ∨ t n (15) where C is an absolute constant. E.2 Proof of theorem 1 Let X 1 , . . . , X n be i.i.d. copies of random vector X satisfying X = p k=1 λ k z k θ k ,(16) where λ 1 ≥ λ 2 ≥ · · · ≥ λ p ≥ 0, {θ k } p k=1 is an orthonormal basis of R p and {z k } p k=1 is an i.i.d. sequence of subgaussian random variables. Without loss of generality, we assume that ∥z k ∥ ψ2 = 1. In this framework, X admits covariance operator: Σ = p k=1 λ k θ k ⊗ θ k . Let for 1 ≤ i ≤ n and 1 ≤ j ≤ p, d ij follows an Bernoulli lax B(δ), with δ ∈ [0, 1], such that d ij is independent both from X (j) i , that is the jth component of X i , and of any other Bernoulli random variable. Let finally Y (h) i = d ij X (j) i the observed random variable with missing values. We will denote by ≲ the fact that the left side term is dominated by the right side term. This proof is the consequence of two lemmas: Lemma 1. Under the same assumptions, let Σ Y = EY ⊗ Y and Σ Y = n −1 n i=1 Y i ⊗ Y i . There exist an absolute constant c 1 such that, for t > 0, with probability at least 1 − e −t : Σ Y − Σ Y ≤ c 1 δ ∥Σ∥ r(Σ) n ∨ r(Σ) n ∨ t n ∨ t n(17) and Lemma 2. Under the same assumptions and notations, there exist an absolute constant c 2 such that, for t > 0, with probability at least 1 − e −t : diag Σ Y − Σ Y ≤ c 2 max j {Σ jj } t n ∨ log p n ∨ t n ∨ log p n .(18) E.2.1 Proof of lemma 1 By definition of the operator norm, we can express this error in terms of Rayleigh's quotient: n −1 i Y i ⊗ Y i − E [Y ⊗ Y ] = max ∥u∥=1 u ⊤ n −1 i Y i ⊗ Y i − E [Y ⊗ Y ] u = max ∥u∥=1 n −1 i u ⊤ (Y i ⊗ Y i )u ⊤ − uE [Y ⊗ Y ] u = max ∥u∥=1 n −1 i ⟨d i ⊙ X i , u⟩ 2 − u ⊤ E [Y ⊗ Y ] u.(19) Let X andδ be two random variables of same distribution to, respectively, the X i and d i . We can rewrite the expectation as: u ⊤ E [Y ⊗ Y ] u = Eu ⊤ (δ ⊙ X) ⊗ (δ ⊙ X)u = E⟨δ ⊙ X, u⟩ 2(20) Let F = {⟨·, u⟩, ∥u∥ ≤ 1}. Since X is subgaussian,δ ⊙ X is too. This means that the ψ 1 and ψ 2 norms of linear functionals ⟨δ ⊙ X, u⟩ are both equivalent to the L 2 -norm. Thus: sup f ∈F ∥f ∥ ψ1 ≲ sup ∥u∥≤1 E 1 /2 ⟨δ ⊙ X, u⟩ 2 ≤ E 1 /2 δ ⊙ X 2 = E 1 /2 p i=1δ 2 i X 2 i (21) Sinceδ is a Boolean vector, ∀i,δ 2 i =δ i . Thus, by the tower property: sup f ∈F ∥f ∥ ψ1 ≲ E 1 /2 Eδ p i=1δ i X 2 i = E 1 /2 δ ∥X∥ 2 ≤ δ ∥Σ∥.(22) Now let us focus on γ 2 (F, ψ 2 ). The norm equivalence and Talagrand's theorem [25] gives γ 2 (F, ψ 2 ) ≲ γ 2 (F, L 2 ) ≲ E sup ∥u∥≤1 ⟨δ ⊙ X, u⟩ ≤ √ δE ∥X∥ .(23) Thus, under theorem 3 of [13], there exist an absolute constant c 1 such that, for t > 0, with probability at least 1 − e −t : E Σ Y − Σ Y ≲ max √ δ ∥Σ∥ 1 /2 √ δE ∥X∥ √ n , δE ∥X∥ 2 n , δ ∥Σ∥ t n , δ ∥Σ∥ t n = δ ∥Σ∥ r(Σ) n ∨ r(Σ) n ∨ t n ∨ t n .(24) E.2.2 Proof of lemma 2 Since taking the operator norm of a diagonal matrix is the same as taking the operator norm of the vector containing the diagonal values, we get: diag n −1 n 1 Y i ⊗ Y i − diag (EY ⊗ Y ) = max j={1,...p} n −1 n i=1 d (j) i X (j) i 2 − δΣ jj (25) Following a similar convexity argument as that of Appendix G.1, we obtain that d (j) i X (j) i 2 ψ1 ≤ 2 d (j) i X (j) i 2 ψ2 ≤ 2δ X (j) i 2 ψ2 .(26) Next, using the model assumption (16), we get, for any given j ∈ {1, . . . , p}, X (j) i 2 ψ2 ≤ c 2 Σ jj ≤ c 2 ∥Σ∥ ,(27) where c 2 is an absolute constant. Combining the last two displays with Proposition 1 with an union bound, we get for t > 0, with probability at least 1 − e −t , diag Σ Y − Σ Y ≤ c ′ 2 δ ∥Σ∥ t + log p n ∨ t + log p n ,(28) where c ′ 2 > 0 is an absolute constant. E.2.3 Proof of theorem 1 Now that we have proven lemmas 1 and 2, we can combine them to obtain the final upper bound. We are looking for an upper bound on: Σ − Σ = (δ −1 − δ −2 )diag Σ Y − Σ Y + δ −2 Σ Y − Σ Y ≤ (δ −1 − δ −2 ) diag Σ Y − Σ Y + δ −2 Σ Y − Σ Y ≤ δ −1 diag Σ Y − Σ Y + δ −2 Σ Y − Σ Y .(29) Combining lemmas 1 and 2 with a union bound argument, and by reajusting the constants, we get that, for t > 0, with probability at least 1 − e −t : Σ − Σ ≤ C ∥Σ∥ δ r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n ,(30) with C > (c 1 ∨ c 2 ) an absolute constant. E.3 Proof of the upper bound in the contaminated case E.3.1 Bounding the error on the full matrix Using the previous result, we know that, with probability at least 1 − e −t and for an absolute constant C: Σ δ − Σ δ ≤ C ∥Σ∥ δ r(Σ) n ∨ r(Σ) n ∨ log p n ∨ log p n ∨ t n ∨ t n ,(31) and Λ ε − E Λ ε ≤ C(1 − δ)ε ∥Λ∥ r(Λ) n ∨ r(Λ) n ∨ log p n ∨ log p n ∨ t n ∨ t n .(32) Now we need to control the norm of Σ X,ξ,δ,ε = n −1 n i=1 (d i ⊗ [(1 − d i ) ⊙ e i ]) ⊙ (X i ⊗ ξ i ) + ([(1 − d i ) ⊙ e i ] ⊗ d i ) ⊙ (ξ i ⊗ X i ). To this end, we apply Theorem 7.3.1 in [27] in combination with a truncation argument as in [17] to obtain with probability at least 1 − e −t Σ X,ξ,δ,ε ≲ δ ε(1 − δ) + ε(1 − δ) √ δ ∥Λ∥ ∥Σ∥ r(Λ) ∨ r(Σ) t + log(r(Λ) ∨ r(Σ)) n + δε(1 − δ) ∥Λ∥ ∥Σ∥ r(Λ) ∨ r(Σ) (t + log(r(Λ) ∨ r(Σ))) log n n .(33) Notice next that Σ X,ξ,δ,ε has all its entries on the diagonal equal to zero. Hence applying the correction to obtain Σ, we get a control on Σ X,ξ,δ,ε /δ 2 . Finally, we need to bound diag( Λ ε − E[ Λ ε ]) . We apply a similar reasoning as that used to derive (28). We obtain, for any t > 0 with probability at least 1 − e −t diag( Λ ε − E[ Λ ε ]) ≲ ε(1 − δ) ∥Λ∥ t + log p n ∨ t + log p n . Combining the last two displays with (12), (31) and (32) gives the result. E.4 Adapting the proof to the heterogeneous missingness Let δ = (δ 1 , . . . , δ p ) andδ = max j δ j and δ = min j δ j . It is quite obvious to see that equation 21 adapts as: sup f ∈F ∥f ∥ ψ1 ≲ δ ∥Σ∥(34) Similarly, equations 23 becomes: γ 2 (F, ψ 2 ) ≲ δ ∥Σ∥(35) and finally, we have that ∆ ⊙ Σ Y − Σ Y ≤ δ −2 Σ Y − Σ Y(36) F Proof of lower bounds The first two subsections deal with the lower bound of theorem 2, the third extends it to the contaminated case. F.1 Hypothesis construction in a Grassmannian manifold Let p ≥ 2 be the dimension of our observations and let 1 ≤ r ≤ p be the intrinsic dimension of Σ. Although the problem at hand is p-dimensional, we are most interested in correctly estimating the r eigenspaces related to the r largest eigenvalues. We will thus look at p dimensional matrices that are projection in R p of r dimensional kernels. Let H be a p × r matrix with orthonormal rows. Each matrix H describes a subspace U H of R p , where dim(U H ) = r and H ⊤ H is its projector in R p . The set of all U H is the Grassmannian manifold G r (R p ), which is the set of all r-dimensional subspaces of R p . The Grassmannian manifold is a smooth manifold of dimension d = r(p − r), where one can define a metric for all subspaces U,Ū ∈ G r (R p ): d(U,Ū ) = ∥P U − PŪ ∥ F = H ⊤ H −H ⊤H(37) where P U and PŪ are the projectors to the subspaces U andŪ respectively and H andH are the r × p matrix with orthonormal rows associated with U andŪ respectively. In the remainder of the proof, we will identify the projectors to the subspaces. A result on the entropy of Grassmanian manifolds [19] shows that: Proposition 2. For all ε > 0, there exists a family of orthonormal projectors U ⊂ G r (R p ) such that: |U| ≥ c ε d(38) and, ∀P, Q ∈ G r (R p ), P ̸ = Q,c ε √ r ≤ ∥P − Q∥ F ≤ ε √ r c(39) for some small enough absolute constantc, where |U| is the cardinal of set U. Without loss of generality, we assume that the block matrix P 1 = I r 0 0 0 belongs to the set U. Indeed, the Frobenius norm is invariant through a change of basis. Let us then build such a set U of hypotheses. Let γ = a p /δ 2 n where a > 0 is an absolute constant We set N = |U| and U = {P 1 , . . . , P N } where P 1 was introduced above. Let us define the family of p × p symmetric matrices Σ 1 , . . . , Σ N , ∀j ∈ {1, N } as follows : Σ j = I p + γP j , where I p is the p × p identity matrix. These covariance matrices belongs to the class of spiked covariance matrices. Then, we can see that, for i, j ∈ {1, . . . N }, by setting ε = 1 /2: ∥Σ i − Σ j ∥ 2 F = γ 2 ∥P i − P j ∥ 2 F > a 2c2 pr 2δ 2 n(40) F.2 KL-divergence of hypotheses Now that we have our candidate covariances Σ 1 , . . . , Σ N , let us define the associated distributions. For j ∈ {1, N }, let X 1 , . . . X n be i.i.d. random variables following a gaussian N (0, Σ j ) law. Let d 1 , . . . d n be each vectors of p i.i.d bernoulli random variables of probability of success δ > 0, and let Y 1 , . . . Y n be random variables such that, ∀i ∈ {1, n}, Y i = d i ⊙ X i , with ⊙ the Hadamard or term-by-term product. Let us also define as P j the distribution of Y 1 , . . . Y n and P (δ) j the conditional distribution of the Y 1 , . . . Y n knowing d 1 , . . . d n . Finally, let E j be the expectation given the distribution associated with the j-th projector and E δ the expectation given d 1 , . . . d n . For j ∈ {2, . . . , N }, let us compute the Kullback-Leibler divergence from P 1 to P j . KL(P 1 , P j ) = E 1 log dP 1 dP j = E 1 log dP δ ⊗ P (δ) 1 dP δ ⊗ P (δ) j = E δ KL(P (δ) 1 , P (δ) j ) = n i=1 E δ KL(P (di) 1 , P (di) j ) (41) Since ∀i ∈ {1, . . . , n}, Y i |d i ∼ N (0, (d i ⊗ d i ) ⊙ Σ) , for all j ∈ {1, . . . N } and for each realisation δ(ω) ∈ {0, 1} p , P j ≫ P 1 , thus KL(P 1 , P j ) < ∞. Define J i = {j : d i,j = 1, 1 ≤ j ≤ p} the set of indices kept by vector d i and p i = p j=1 d i,j ∼ B(p, δ). Then, define the mapping Q i : R p → R di such that Q i (x) = x Ji , such that x Ji is a p i dimensional vector containing the components of x whose index are in J i . Let Q * i : R di → R p the right inverse of Q i . Note that ∀j ∈ {1, N − 1}, Σ j = (1 + γ)P j + P ⊥ j , with P ⊥ j the projector to the subspace of R p orthogonal to the one described by P j . Let us define Σ (di) j = Q i Σ j Q * i . Then, observe that Σ (di) 1 is invertible, with inverse Q i 1 γ+1 P 1 + P ⊥ 1 Q * i since P 1 and P ⊥ 1 are diagonal matrices. We thus get, for i ∈ {1, . . . n}: KL(P (di) 1 , P (di) j ) = 1 2 tr Σ (di) −1 1 Σ (di) j − p i − log(det(Σ (di) −1 1 Σ (di) j ))(42) First, using a result of linear algebra described in section G.3, we show that: −E δ log(det(Σ (di) −1 1 Σ (di) j )) ≤ ar p/n.(43) In the high-dimensional regime p ≥ n, we obtain −n E δ log(det(Σ (di) −1 1 Σ (di) j )) ≤ ar √ n p ≤ a r p.(44) Next, let us focus on bounding 1 2 tr Σ (di) −1 1 (Σ (di) j − Σ (di) 1 ) . Remember that Σ 1 is diagonal. Using the fact that Σ −1 1 = 1 1+γ P 1 + P ⊥ 1 , we get: tr Σ (di) −1 1 (Σ (di) j − Σ (di) 1 ) = γ 1 + γ tr (Q i P 1 (P j − P 1 )Q * i ) + γtr Q i P ⊥ 1 (P j − P 1 )Q * i = γ 1 + γ (tr (Q i P 1 P j Q * i ) − tr (Q i P 1 Q * i )) + γtr (Q i (I p − P 1 ) P j Q * i ) = γ 1 + γ − γ (tr (Q i P 1 P j Q * i ) − p i ) = γ 2 2(1 + γ) ∥Q i (P j − P 1 )Q * i ∥ 2 F(45) Finally, using the fact demonstrated in appendix G.5 and the upper bound of proposition 2, we get that: KL(P 1 , P j ) ≤ n i=1 E δ γ 2 2(1 + γ) ∥Q i (P j − P 1 )Q * i ∥ 2 F ≤ n i=1 γ 2 δ 2(1 + γ) ∥P j − P 1 ∥ 2 F ≤ n i=1 γδr 8c 2 ≤ a 8c 2 r √ p n ≤ a 2 4c 2 r p.(46) Thus, since N ≥ ⌊2c⌋ r(p−r) , and since we assumed that p > 2r: KL(P 1 , P j ) ≤ α log(N )(47) for α = a 2 /8c 2 . According to theorem 2.5 of [29], the previous display combined with (40) gives inf Σ sup PΣ P Σ Σ − Σ 2 F ≥ C r δ 2 n p ≥ β(48) where C > 0 and β > 0 are two absolute constants. This fact, in turn, implies the lower bound of theorem 2, since, for all Σ 1 , Σ 2 matrices of our hypothesis set: ∥Σ 1 − Σ 2 ∥ 2 ≥ C r δ 2 n(49) Indeed, otherwise, we would get ∥Σ 1 − Σ 2 ∥ 2 F < p ∥Σ 1 − Σ 2 ∥ 2 < C r δ 2 n p(50) which contradicts equation 40. F.3 Lower bound in the contaminated case The bound of theorem 4 is made of two terms. The left term is the missing values lower bound, since missingness is a particular case of contamination. The second term is a result from the Huber contamination analysis of [3], which we develop here. Let us define the Huber contamination with missing values. Let P be a distribution and Q an arbitrary contamination distribtuion. The Huber contaminated distributionP is defined as: P = δP + ϵ(1 − δ)Q where δ, ϵ ∈ [0, 1]. Let P be a parametric distribution P θ , whose parameter θ ∈ Θ we want to estimate. Theorem 5.1 in [3] states that for two parametric distributions P θ1 and P θ2 with parameters θ 1 , θ 2 ∈ Θ respectively, there exist two contaminations Q 1 and Q 2 such thatP 1 andP 2 are not identifiable, as long as the parameter distance L(θ 1 , θ 2 ) is below the modulus of continuity: ω (ϵ(1 − δ), Θ) = sup L (θ 1 , θ 2 ) : TV (P θ1 , P θ2 ) ≤ ϵ(1 − δ) δ , θ 1 , θ 2 ∈ Θ This formulation is slightly different than the one in [3] to account for missingness without contamination. In appendix E, [3] gives a lower bound to this value in the case where θ is a covariance matrix. Let Σ 1 = I p and Σ 2 = I p + ϵ(1 − δ)E 1,1 , where E 1,1 is the matrix with zeros except in the (1, 1) entry, which is equal to 1. Then, set P 1 = N (0, Σ 1 ) and P 2 = N (0, Σ 2 ). Observe that: TV (P 1 , P 2 ) ≤ 1 2 KL (P 1 , P 2 ) ≤ 1 8 ∥P 1 − P 2 ∥ 2 F = (ϵ(1 − δ)) 2 8 and L(Σ 1 , Σ 2 ) = ∥Σ 1 − Σ 2 ∥ 2 = (ϵ(1 − δ)) Thus ω(ϵ(1 − δ), Θ) > (ϵ(1 − δ)) 2 . They conclude using Le Cam's two point method (see e.g. chapter 2.3 of [29]) that there exist two absolute constants C, c > 0 such that: inf Σ sup Σ sup QP Σ − Σ ≥ Cϵ(1 − δ) > c This is for Huber contamination. Let us now examine what happens for cell-wise contamination using the same hypotheses P 1 and P 2 . NoteP 1 andP 2 the distributions of P 1 and P 2 cell-wise contaminated by two distributions Q 1 and Q 2 that we will both set with independent components. Since P 1 and P 2 are isotropic Gaussians and the contamination is completely at random, we can decompose the distributions as follows: P 1 = p i=1 δP 1,i + ϵ(1 − δ)Q 1,i andP 2 = p i=1 δP 2,i + ϵ(1 − δ)Q 2,i Notice that taken separately, the components can be considered to be univariate Gaussian distributions under a Huber contamination. We can now try to build Q 1 and Q 2 so thatP 1 andP 2 are equal in distribution. Let us first set Q 1,i = Q 2,i = N (0, 1) for i ̸ = 1, since the components are equal in distribution for i ̸ = 1 the contamination we choose here doesn't matter much. As ω(ϵ(1 − δ), Θ) > (ϵ(1 − δ)) 2 , using the construction of theorem 5.1 in [3], we can build two distributions q 1 and q 2 such that: δP 1,1 + ϵ(1 − δ)q 1 = δP 2,1 + ϵ(1 − δ)q 2 Let us thus set Q 1,1 = q 1 and Q 2,1 = q 2 . Under this contamination, we haveP 1 =P 2 . By applying Le Cam's two point argument, we find here again that there exist two absolute constants C, c > 0 such that: inf Let X and ξ be two one-dimensional random variables following a sub-gaussian distribution, and let d, d ′ be independent bernoulli random variable of mean δ. The Orlicz ψ 1 norm of d(1 − d ′ )Xξ is: ∥d(1 − d ′ )Xξ∥ ψ1 = inf{u > 0, E exp (|d(1 − d ′ )Xξ|/u) ≤ 2} = inf{u > 0, E exp (d(1 − d ′ )|Xξ|/u) ≤ 2}(51) Since the bernoulli variables are binary. By the tower property and Jensen's inequality, we have that, ∀u such that the expectation is well defined : E exp (d(1 − d ′ )|Xξ|/u) = EE d,d ′ exp (|d(1 − d ′ )Xξ|/u) ≥ E exp (δ(1 − δ)|Xξ|/u)(52) which implies that {u > 0, E exp (δ(1 − δ)|Xξ|/u) ≤ 2} ⊂ {u > 0, E exp (d(1 − d ′ )|Xξ|/u) ≤ 2}(53) With a simple change of variable, one can see that: inf{u > 0, E exp (δ(1 − δ)|Xξ|/u) ≤ 2} = δ(1 − δ) inf{u > 0, E exp (|Xξ|/u) ≤ 2} = δ(1 − δ) ∥Xξ∥ ψ1 ≤ δ(1 − δ) ∥X∥ ψ2 ∥ξ∥ ψ2(54) Hence inf{u > 0, E exp (d(1 − d ′ )|Xξ|/u) ≤ 2} ≤ δ(1 − δ) ∥X∥ ψ2 ∥ξ∥ ψ2 .(55) G.2 Proof of the correction formula of equation 10 Let X be a zero mean random vector of R p admitting covariance matrix Σ. Let ξ be a zero mean random vector, independent from X, with diagonal covariance matrix Λ. Let (d j ) 1≤j≤p and (e j ) 1≤j≤p sequences of Bernoulli random variables of probability respectively δ and ε(1 − δ), independent from both X and ξ and such that 1 ≤ j ≤ p, d j e j = 0. Then, let Y (j) i = d j ⊙ X (j) + e j ⊙ ξ (j) . We have that: (Y ⊗ Y ) jk = d j X (j) 2 + e j ξ (j) 2 if j = k d j d k X (j) X (k) + d j e k X (j) ξ (k) + e j d k ξ (j) X (k) + e j e k ξ (j) ξ (k) otherwise(56) This means that we have, by independence of the X (j) and the ξ (j) , and by independence of the ξ (j) with each other: Σ Y jk = E (Y ⊗ Y ) jk = δΣ jj + ε(1 − δ)Λ jj if j = k δ 2 Σ jk otherwise(57) Thus: Σ jk = δ −1 Σ Y jj − ε(1 − δ)Λ jj if j = k δ −2 Σ Y jk otherwise(58) Which in turn means that: Σ = (δ −1 − δ −2 )diag(Σ Y ) + δ −2 Σ Y + ε(1 − δ) δ Λ(59) This gives the general correction formula with independent contamination. For the missing values correction, simply set Λ = 0 the p × p zero matrix. G.3 Bounds on the determinant of in equation 44 Theorem 13 of [26] states that, for any matrix A of size p with eigenvalues λ 1 , . . . λ s , each with multiplicity µ 1 , . . . µ s such that s i=1 µ i = p, then any principal submatrix A(j|j), that is, a matrix created by removing line j and column j from A, has eigenvalues λ i with multiplicity max(0, µ i − 1). The remaining eigenvalues have values between min i λ i and max i λ i . In our case, the matrix Σ j has only two eigenvalues: 1 + γ and 1, with multiplicity r and p − r respectively. One will easily find by recurrence on the number of deleted dimensions, which is p − p i with p i = p j=1 d i,j , that: det Σ (di) j = (1 + γ) max(0,r−p+pi) p−pi k=1 λ k(60) where ∀k ∈ {1, p i }, 1 ≤ λ k ≤ 1 + γ. This means, in particular, that: (1 + γ) max(0,r−p+pi) ≤ det Σ (pi) j ≤ (1 + γ) min(r,pi)(61) Now, let us demonstrate the statement in equation 44. We have Σ 1 and Σ j having the same eigenvalues 1+γ and 1 with multiplicity respectively r and p−r. Let p i = p k=1 d i,k be the number of remaining components after applying the boolean filter d i (thus there are p − p i deleted components). Since Σ 1 is diagonal, we know that Σ (di) 1 will also have eigenvalues 1 + γ and 1, with multiplicity a i and b i respectively, where a i ∼ B(r, δ) and b i ∼ B(p − r, δ) where B is the binomial distribution. Then, using the lower bound we just demonstrated, we get that: −E δ log det Σ (di)−1 1 Σ (di) j = E δ a i log(1 + γ) + b i log(1) − log det Σ (di) j ≤ E δ a i log(1 + γ) − max(0, r − p + p i ) log(1 + γ) ≤ (rδ + min(0, p − p i − r)) log(1 + γ) ≤ rδ log(1 + γ)(62) In particular, we know that γ > 0, so log(1 + γ) ≤ γ and −E δ log det Σ (di)−1 1 Σ (di) j ≤ rδγ ≤ a r p/n.(63) G.4 Behaviour of the Q i with regard to matrix multiplication We know that Q i Q * i = I di . Furthermore, Q * i Q i = I (Ji) p , where I (ji) p is the diagonal matrix where the jth diagonal term is 1 if only if j ∈ J i , and 0 otherwise. Finally, notice that in the general case, Q i AQ * i Q i BQ * i ̸ = Q i ABQ * i , except when either A or B is diagonal. Indeed, for k, l ∈ {1, p}: (Q i AQ * i Q i BQ * i ) kl = p m=1 A km B ml I k∈Ji I l∈Ji I m∈Ji(64) Which, if A is diagonal, simply gives: (Q i AQ * i Q i BQ * i ) kl = A kk B kl I k∈Ji I l∈Ji = (Q i ABQ * i ) kl(65) G.5 Proof of the upper bound of the frobenius norm with missing values Let P ∈ R p×p be any matrix, then, using the fact that the d i are boolean vectors: E δ ∥(d i ⊗ d i ) ⊙ P ∥ 2 F = E δ tr ((d i ⊗ d i ) ⊙ P ) ⊤ ((d i ⊗ d i ) ⊙ P ) = E δ p k=1 p l=1 d k i d l i P 2 kl = p k=1   δPkk + p l=1 l̸ =k δ 2 P 2 kl    ≤ δ ∥P ∥ 2 F(66) H Tables Table 6: We consider the cell-wise contamination model ( (2)) with a Gaussian contamination of high intensity, ε = 1 and for several values of δ in a grid. For each δ, we average the proportion of real dataδ and contaminated dataε after filtering over 20 repetitions. Values are displayed in percentages (δ must be high,ε low, both are expressed in percentages). Figure 9: DI fails on ATTEND since the covariance matrix is approximately low rank. The dataset has only 8 features and the effective rank of its covariance matrix is below 2. Figure 11: Woolridge's CEOSAL dataset fails both TSGS and DI with its dimension of 13 and effective rank of around 2.5. Figure 1 : 1Left: Estimation error of the covariance matrix for n = 100, p = 50, r(Σ) = 2 under a Dirac contamination (tailMV and DDCMV are our methods). Here ϵ = 1 and δ varies in (0, 1). Figure 2 : 2Estimation error on a synthetic dataset with p = 50, n = 300, r (Σ) Figure 4 :Figure 5 : 45Estimation error as a function of the contamination rate for n = 500, p = 400, r(Σ) = 5 and Dirac contamination . Estimation error as a function of the contamination rate for n = 500, p = 400, r(Σ) = 5 and Gaussian contamination . Figure 9 , 9where it fails to provide accurate results. Similarly, both TSGS and DI fail to perform well on the CEOSAL2 dataset, as shown inFigure 11, despite both datasets having fewer than 15 features. 1-norm of the components of Σ X,ξ,δ Figure 14 : 14Relative spectral difference (in percentages) between estimated covariance matrices of NASDAQ stock returns over 2021 and 2022. Here, DDCII fails due to out-of-memory errors. Theorem 1. Let X 1 , . . . , X n be i.i.d. subgaussian random variables in R p , with covariance matrix Σ, and let d ij , i ∈ [1, n], j ∈ [1, p] be i.i.d bernoulli random variables with probability of success δ > 0. Then there exists an absolute constant C such that, for t > 0, with probability at least 1 − e −t : Table 1 : 1Execution time of the covariance estimation procedures (in milliseconds) with n = 300 averaged over all values of the contamination rate δ and 20 repetitions.method p = 50 p = 100 p = 500 MV (ours) 0.29 ± 0.03 0.49 ± 0.08 9.7 ± 4.5 KNNImputer (KNN) 26 ± 9.8 45 ± 17 470 ± 190 IterativeImputer (II) Table 2 : 2We consider contaminated data following model (2) contaminated with a Dirac contamination of high intensity with ϵ = 1 and for several values of δ in a grid. For each δ, we average the proportion of real dataδ and contaminated dataε after filtering over 20 repetitions. Values are displayed in percentages (δ must be high,ε low)).CONTAMINATION TAIL CUT DDC 99% DDC 90% RATE (1 − δ)δ STDε STDδ STDε STDδ STDε STD 0.1 % 99.6 0.023 0.000 0.000 99.1 0.029 0.000 0.000 94.8 0.054 0.00 0.00 1% 98.8 0.027 0.000 0.000 98.2 0.037 0.000 0.00 94.3 0.102 0.00 0.00 5% 94.9 0.013 0.000 0.000 94.6 0.018 0.000 0.000 91.8 0.060 0.00 0.000 10% 90.0 0.004 0.000 0.000 89.9 0.016 0.00 0.000 88.2 0.109 0.000 0.000 20% 80.0 0.000 20.0 0.000 80.0 0.003 0.017 0.035 79.4 0.035 0.009 0.022 30% 70.0 0.000 30.0 0.000 70.0 0.001 3.48 2.19 69.9 0.015 2.930 2.31 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Dataset contamination rate (1 − δ) Table 3 : 3Relative spectral difference (in percentages) between estimated covariance matrices on the Abalone data set with 5% synthetic contamination (δ = 0.95, ϵ = 1). On the cleaned dataset, all the robust estimators are very close to the empirical covariance (relative differences < 5%), so we consider its empirical covariance matrix as the truth. Notice that the DI procedure fails, probably due to numerical errors.relative Classical DDCMV99 DDCMV95 DDC KNN DDC II TSGS DI error to estimator Truth 18.8 3.99 7.47 2.24 1.93 3.23 35.1 Classical - 16.3 15.6 11.8 12.0 18.6 18.6 DDCMV99 - - 3.79 7.05 7.25 5.75 34.1 DDCMV95 - - - 2.57 2.93 9.40 34.4 DDC KNN - - - - 0.40 4.06 34.2 DDC II - - - - - 3.82 34.3 TSGS - - - - - - 34.3 to return any estimate. In contrast, we showed that our strategy combining filtering with estimation procedures designed to tackle missing values produce far more stable and reliable results. In future work, we plan to improve our theoretical upper and lower bounds in the cell-wise contamination setting to fully clarify the impact of this type of contamination in covariance estimation. Table 4 : 4NotationsSymbol Description X The random variable of interest Y The observed contaminated random variable p Dimension of the random variable n Number of samples δ Table 5 : 5Datasets used in our real-life experiments.Name Source p n r (Σ) Description Abalone sklearn 7 4173 1.0 Caracteristics of abalone specimens Breast Cancer sklearn 13 178 2.3 Data on cell nuclei Wine sklearn 30 69 2.8 Chemical data on wine varieties Cameras R 11 1038 2.7 Camera caracteristics over different models Attend [34] 8 680 2.0 Class attendance Barium [34] 11 131 2.4 Barium exports CEOSAL2 [34] 13 177 2.5 Firm accountancy data INTDEF [34] 12 49 2.2 USA deficit SP 500 yfinance 496 502 2.7 Returns of SP 500 companies in 2021/2022 NASDAQ yfinance 1442 502 4.0 Returns of NASDAQ companies in 2021/2022 C Methods compared in the cell-wise contamination setting C.1 Baseline methods Classical denotes the classical covariance estimator applied without care for contamination. We expect all other methods to perform better than it. Table 7 : 7Same table on the Abalone dataset, contaminated with a Dirac contamination.Figure 7: Relative spectral difference (in percentages) between estimated covariance matrices of the 13 features of sklearn's Wine dataset.Figure 8: Relative spectral difference (in percentages) between estimated covariance matrices of the 11 features of the R camera dataset.CONTAMINATION TAIL CUT DDC 99% DDC 90% Figure 10: Relative spectral difference (in percentages) between estimated covariance matrices of the 11 features of the Woolridge Barium dataset.DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS DI classical DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS 9.6 15 7.9 3.8 25 25 6.3 3.7 7.1 25 22 9.1 13 29 20 4.9 23 22 23 23 35 5 10 15 20 25 30 35 DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS DI classical DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS 26 27 40 68 4.2 15 66 16 66 66 10 20 30 40 50 60 Table 8 : 8Same table on the Abalone dataset, contaminatedwith a Gauss contamination.Figure 12: Relative spectral difference (in percentages) between estimated covariance matrices of the 13 features of Woolridge's INTDEF dataset.Figure 13: Relative spectral difference (in percentages) between estimated covariance matrices of SP500 stock returns over 2021 and 2022.On high-dimensional data, DDCII looses accucary as compared to our other procedures DDC-MV and DDC KNN, mainly because IterativeImputer does not scale well with dimension.CONTAMINATION TAIL CUT DDC 99% DDC 90% RATEδ STDε STDδ STDε STDδ STDε STD 0.1% 69.5 0.001 0.016 0.010 98.0 0.013 0.059 0.009 93.2 0.019 0.056 0.009 1% 68.9 0.004 0.162 0.029 97.7 0.044 0.570 0.040 92.6 0.075 0.545 0.042 5% 66.2 0.028 0.852 0.055 93.5 0.058 2.86 0.045 89.8 0.119 2.73 0.050 10% 62.8 0.012 1.80 0.072 88.8 0.047 5.84 0.089 85.9 0.111 5.56 0.100 20% 55.9 0.008 3.95 0.088 79.6 0.044 12.5 0.098 77.7 0.123 11.6 0.103 30% 49.0 0.003 6.62 0.093 68.0 0.553 21.3 0.892 66.8 0.746 19.5 0.662 DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS DI classical DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS 27 28 24 15 43 50 5.5 5.4 16 37 47 7.1 18 36 46 17 36 45 39 46 21 10 15 20 25 30 35 40 45 DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS DI classical DDC MV 99 DDC MV 95 DDC KNN 99 DDC II 99 TSGS 5.9 6.1 4.8 3.4 2.2 2.3 3.6 3 4 1.9 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Code at https://github.com/GregoirePacreau/Missing-Values-Experiments.git Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination. 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{'abstract': 'Large datasets are often affected by cell-wise outliers in the form of missing or erroneous data. However, discarding any samples containing outliers may result in a dataset that is too small to accurately estimate the covariance matrix. Moreover, most robust procedures designed to address this problem are not effective on highdimensional data as they rely crucially on invertibility of the covariance operator. In this paper, we propose an unbiased estimator for the covariance in the presence of missing values that does not require any imputation step and still achieves minimax statistical accuracy with the operator norm. We also advocate for its use in combination with cell-wise outlier detection methods to tackle cell-wise contamination in a high-dimensional and low-rank setting, where state-of-the-art methods may suffer from numerical instability and long computation times. To complement our theoretical findings, we conducted an experimental study which demonstrates the superiority of our approach over the state of the art both in low and high dimension settings.Preprint. Under review.', 'arxivid': '2306.00752', 'author': ['Karim Lounici karim.lounici@polytechnique.edu \nCMAP Ecole Polytechnique Palaiseau\nCMAP Ecole Polytechnique Palaiseau\nFrance, France\n', 'Gregoire Pacreau gregoire.pacreau@polytechnique.edu \nCMAP Ecole Polytechnique Palaiseau\nCMAP Ecole Polytechnique Palaiseau\nFrance, France\n'], 'authoraffiliation': ['CMAP Ecole Polytechnique Palaiseau\nCMAP Ecole Polytechnique Palaiseau\nFrance, France', 'CMAP Ecole Polytechnique Palaiseau\nCMAP Ecole Polytechnique Palaiseau\nFrance, France'], 'corpusid': 258999401, 'doi': None, 'github_urls': ['https://github.com/GregoirePacreau/Missing-Values-Experiments.git'], 'n_tokens_mistral': 26405, 'n_tokens_neox': 22430, 'n_words': 13457, 'pdfsha': '6ce5cd37f5083edbf97bcf183156ed1cbf7fccc2', 'pdfurls': ['https://export.arxiv.org/pdf/2306.00752v1.pdf'], 'title': ['Robust covariance estimation with missing values and cell-wise contamination', 'Robust covariance estimation with missing values and cell-wise contamination'], 'venue': []}

arxiv

Simultaneous Inference of Covariances 2 Sep 2011 Han Xiao hxiao@stat.rutgers.edu Department of Statistics 501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL Wei Biao Wu wbwu@galton.uchicago.edu Department of Statistics 501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL Simultaneous Inference of Covariances 2 Sep 2011AMS 2000 subject classifications: Primary 62H1562H10; secondary 62E20 Keywords and phrases: Covariance matrixhigh dimensional analysismaximal deviationtaperingtest for bandednesstest for covariance struc- turetest for stationarity We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances. Introduction Let X n = (X ij ) 1≤i≤n,1≤j≤m be a data matrix whose n rows form independent samples from some population distribution with mean vector µ n and covariance matrix Σ n . High dimensional data increasingly occur in modern statistical applications in biology, finance and wireless communication, where the dimension m may be comparable to the number of observations n, or even much larger than n. Therefore, it is necessary to study the asymptotic behavior of statistics of X n under the setting that m = m n grows to infinity as n goes to infinity. In many empirical examples, it is often assumed that Σ n = I m , where I m is the m × m identity matrix, so it is important to perform the test H 0 : Σ n = I m(1) before carrying out further estimation or inference procedures. Due to high dimensionality, conventional tests often do not work well or cannot be implemented. For example, when m > n, the likelihood ratio test (LRT) cannot be used because the sample covariance matrix is singular; and even when m < n, the LRT is drifted to infinity and lead to many false rejections if m is also large (Bai et al., 2009). Ledoit and Wolf (2002) found that the empirical distance test (Nagao, 1973) is not consistent when both m and n are large. The problem has been studied by several authors under the "large n, large m" paradigm. Bai et al. (2009) and Ledoit and Wolf (2002) proposed corrections to the LRT and the empirical distance test respectively. Assuming that the population distribution is Gaussian with µ n = 0, Johnstone (2001) used the largest eigenvalue of the sample covariance matrix X ⊤ n X n as the test statistic, and proved that its limiting distribution follows the Tracy-Widom law (Tracy and Widom, 1994). Here we use the superscript ⊤ to denote the transpose of a matrix or a vector. His work was extended to the non-Gaussian case by Soshnikov (2002) and Péché (2009), where they assumed the entries of X n are independent and identically distributed (i.i.d.) with sub-Gaussian tails. Let x 1 , x 2 , . . . , x m be the m columns of X n . In practice, the entries of the mean vector µ n are often unknown, and are estimated byx i = (1/n) n k=1 X ki . Write x i −x i for the vector x i −x i 1 n , where 1 n is the n-dimensional vector with all entries being one. Let σ ij = Cov(X 1i , X 1j ), 1 ≤ i, j ≤ m, be the covariance function, namely, the (i, j)th entry of Σ n . The sample covariance between columns x i and x j is defined aŝ σ ij = 1 n (x i −x i ) ⊤ (x j −x j ). In high-dimensional covariance inference, a fundamental problem is to establish an asymptotic distributional theory for the maximum deviation M n = max 1≤i<j≤m |σ ij − σ ij |. With such a distributional theory, one can perform statistical inference for structures of covariance matrices. For example, one can use M n to test the null hypothesis H 0 : Σ n = Σ (0) , where Σ (0) is a pre-specified matrix. Here the null hypothesis can be that the population distribution is a stationary process so that Σ n is Toeplitz, or that Σ n has a banded structure. It is very challenging to derive an asymptotic theory for M n if we allow dependence among X 11 , . . . , X 1m . Many of the earlier results assume that the entries of the data matrix X n are i.i.d.. In this case σ ij = 0 if i = j. Jiang (2004) derived the asymptotic distribution of L n = max 1≤i<j≤m |σ ij |. Theorem 1 (Jiang, 2004). Suppose X i,j , i, j = 1, 2, . . . are independent and identically distributed as ξ which has variance one. Suppose E|ξ| 30−ǫ < ∞ for any ǫ > 0. If n/m → c ∈ (0, ∞), then for any y ∈ R, lim n→∞ P nL 2 n − 4 log m + log(log m) + log(8π) ≤ y = exp −e −y/2 . Jiang's work has attracted considerable attention, and been followed by , Liu et al. (2008), Zhou (2007) and Li and Rosalsky (2006). Under the same setup that X n consists of i.i.d. entries, these works focus on three directions (i) reduce the moment condition; (ii) allow a wider range of p; and (iii) show that some moment condition is necessary. In a recent article, Cai and Jiang (2011) extended those results in two ways: (i) the dimension p could grow exponentially as the sample size n provided exponential moment conditions; and (ii) they showed that the test statistic max |i−j|>sn |σ ij | also converges to the Gumbel distribution if each row of X n is Gaussian and is s n -dependent. The latter generalization is important since it is one of the very few results that allow dependent entries. In this paper we shall show that a self-normalized version of M n converges to the Gumbel distribution under mild dependence conditions on the vector (X 11 , . . . , X 1m ). Thus our result provides a theoretical foundation for highdimensional simultaneous inference of covariances. The rest of this article is organized as follows. We present the main result in Section 2. In Section 3, we use two examples on linear processes and nonlinear processes to demonstrate that the technical conditions are easily satisfied. We discuss three tests for the covariance structure using our main result in Section 4. The proof is given in Section 5, and some auxiliary results are collected in Section 6. Main result We consider a slightly more general situation where population distribution can depend on n. Let X n = (X n,k,i ) 1≤k≤n,1≤i≤m be a data matrix whose n rows are i.i.d. m-dimensional random vectors with mean µ n = (µ n,i ) 1≤i≤m and covariance matrix Σ n = (σ n,i,j ) 1≤i,j≤m . Let x 1 , x 2 , . . . , x m be the m columns of X n . Letx i = (1/n) n k=1 X n,k,i , and write x i −x i for the vector x i −x i 1 n . The sample covariance between x i and x j is defined aŝ σ n,i,j = 1 n (x i −x i ) ⊤ (x j −x j ). It is unnatural to study the maximum of a collection of random variables which are on different scales, so we consider the normalized version |σ n,i,j − σ n,i,j |/τ n,i,j , where τ n,i,j = Var [(X n,1,i − µ n,i )(X n,1,j − µ n,j )] . In practice, τ n,i,j are usually unknown, and can be estimated bŷ τ n,i,j = 1 n |(x i −x i ) • (x j −x j ) −σ n,i,j · 1 n | 2 . where • denotes the Hadamard product defined as A • B := (a ij b ij ) for two matrices A = (a ij ) and B = (b ij ) with the same dimensions. We thus consider M n = max 1≤i<j≤m |σ n,i,j − σ n,i,j | τ n,i,j .(2) Due to the normalization procedure, we can assume without loss of generality that σ n,i,i = 1 and µ n,i = 0 for each 1 ≤ i ≤ m. Define the index set I n = {(i, j) : 1 ≤ i < j ≤ m}, and for α = (i, j) ∈ I n , let X n,α := X n,1,i X n,1,j . Define K n (t, p) = sup 1≤i≤m E exp (t|X n,1,i | p ) , M n (p) = sup 1≤i≤m E(|X n,1,i | p ), τ n = inf 1≤i<j≤m τ n,i,j , γ n = sup α,β∈In and α =β |Cor(X n,α , X n,β )| , γ n (b) = sup α∈In sup A⊂In,|A|=b inf β∈A |Cor(X n,α , X n,β )| . We need the following technical conditions. (A1). lim inf n→∞ τ n > 0. (A2). lim sup n γ n < 1. (A3). γ n (b n ) · (log b n ) = o(1) for any sequence (b n ) such that b n → ∞. (A3 ′ ). γ n (b n ) = o(1) for any sequence (b n ) such that b n → ∞, and α,β∈In [Cov(X n,α , X n,β )] 2 = O(m 4−ǫ )for some constant ǫ > 0. (A4). log m = o n p/(4+2p) and lim sup n→∞ K n (t, p) < ∞ for some constants t > 0 and 0 < p ≤ 4. (A4 ′ ). m = O(n q ) and lim sup n→∞ M n (4q + 4 + δ) < ∞ for some constants q > 0 and δ > 0. The two conditions (A3) and (A3 ′ ) require that the dependence among X n,α , α ∈ I n , are not too strong. They are translations of (B1) and (B2) in Section 6.1 (see Remark 2 for some equivalent versions), and either of them will make our results valid. We use (A2) to get rid of the case where they may be lots of pairs (α, β) ∈ I n such that X n,α and X n,β are perfectly correlated. Assumptions (A4) and (A4 ′ ) connect the growth speed of m relative to n and the moment conditions. They are typical in the context of high dimensional covariance matrix estimation. Condition (A1) excludes the case that X n,α is a constant. Theorem 2. Suppose that X n = (X n,k,i ) 1≤k≤n,1≤i≤m is a data matrix whose n rows are i.i.d. m-dimensional random vectors, and whose entries have mean zero and variance one. Assume (A1), (A2), either of (A3) and (A3 ′ ), and either of (A4) and (A4 ′ ), then for any y ∈ R, lim n→∞ P nM 2 n − 4 log m + log(log m) + log(8π) ≤ y = exp −e −y/2 . Examples Except for (A4) and (A4 ′ ), which put conditions on every single entry of the random vector (X n,1,i ) 1≤i≤m , all the other conditions of Theorem 2 are related to the dependence among these entries, which can be arbitrarily complicated. In this section we shall provide examples which satisfy the four conditions (A1), (A2), (A3) and (A3 ′ ). Observe that if each row of X n is a random vector with uncorrelated entries (specifically, the entries are independent), then all these conditions are automatically satisfied. They are also satisfied if the number of non-zero covariances is bounded. Stationary Processes Suppose (X n,k,i ) = (X k,i ), and each row of (X k,i ) 1≤i≤m is distributed as a stationary process (X i ) 1≤i≤m of the form X i = g(ǫ i , ǫ i−1 , . . .) where ǫ i 's are i.i.d. random variables, and g is a measurable function such that X i is well-defined. Let (ǫ ′ i ) i∈Z be an i.i.d. copy of (ǫ i ) i∈Z , and X ′ i = g(ǫ i , . . . , ǫ 1 , ǫ ′ 0 , ǫ −1 , ǫ −2 , . . .). Following Wu (2005), define the physical dependence measure of order p by δ p (i) = X i − X ′ i p . Define the squared tail sum Ψ p (k) =   ∞ j=k (δ p (i)) 2   1/2 , and use Ψ p as a shorthand for Ψ p (0). We give sufficient conditions for (A1), (A2), (A3) and (A3 ′ ) in the following lemma and leave its proof to the supplementary file. Lemma 3. (i) If 0 < Ψ 4 < ∞ and Var(X i X j ) > 0 for all i, j ∈ Z, then (A1) holds. (ii) If in addition, | Cor(X i X j , X k X l )| < 1 for all i, j, k, l such that they are not all the same, then (A2) holds. (iii) Assume that the conditions of (i) and (ii) hold. If Ψ p (k) = o(1/ log k) as k → ∞, then (A3) holds. If m j=0 (Ψ 4 (j)) 2 = O(m 1−δ ) for some δ > 0, then (A3 ′ ) holds. Remark 1. Let g be a linear function with g(ǫ i , ǫ i−1 , . . .) = ∞ j=0 a j ǫ i−j , where ǫ j are i.i.d. with mean 0 and E(|ǫ j | p ) < ∞ and a j are real coefficients with ∞ j=0 a 2 j < ∞. Then the physical dependence measure δ p (i) = |a i | ǫ 0 − ǫ ′ 0 p . If a i = i −β ℓ(i), where 1/2 < β < 1 and ℓ is a slowly varying function, then (X i ) is a long memory process. Smaller β indicates stronger dependence. Condition (iii) holds for all β ∈ (1/2, 1). Moreover, if a i = i −1/2 (log(i)) −2 , i ≥ 2, which corresponds to the extremal case with very strong dependence β = 1/2, we also have Ψ p (k) = O((log k) −3/2 ) = o(1/ log k). So our dependence conditions are actually quite mild. If (X i ) is a linear process which is not identically zero, then the following regularity conditions are automatically satisfied: Ψ 4 > 0, Var(X i X j ) > 0 for all i, j ∈ Z, and | Cor(X i X j , X k X l )| < 1 for all i, j, k, l such that they are not all the same. Non-stationary Linear Processes Assume that each row of (X n,k,i ) is distributed as (X n,i ) 1≤i≤m , which is of the form X n,i = t∈Z f n,i,t ǫ i−t , where ǫ i , i ∈ Z are i.i.d. random variables with mean zero, variance one and finite fourth moment, and the sequence (f n,i,t ) satisfies t∈Z f 2 n,i,t = 1. Denote by κ 4 the fourth cumulant of ǫ 0 . For 1 ≤ i, j, k, l ≤ m, we have σ n,i,j = t∈Z f n,i,i−t f n,j,j−t , Cov(X n,i X n,j , X n,k X n,l ) = Cum(X n,i , X n,j , X n,k , X n,l ) + σ n,i,k σ n,j,l + σ n,i,l σ n,j,k , where Cum(X n,i , X n,j , X n,k , X n,l ) is the fourth order joint cumulant of the random vector (X n,i , X n,j , X n,k , X n,l ) ⊤ , which can be expressed as Cum(X n,i , X n,j , X n,k , X n,l ) = t∈Z f n,i,i−t f n,j,j−t f n,k,k−t f n,l,l−t κ 4 , by the multilinearity of cumulants. In particular, we have Var(X i X j ) = 1 + σ 2 n,i,j + κ 4 · t∈Z f 2 n,i,t f 2 n,j,t . Since κ 4 = Var(ǫ 2 0 ) − 2 Eǫ 2 0 2 ≥ −2, the condition κ 4 > −2 (3) guarantees (A1) in view of Var(X i X j ) ≥ (1 + σ 2 n,i,j )(1 + min{κ/2, 0}) ≥ min{1, 1 + κ/2} > 0. To ensure the validity of (A2), it is natural to assume that no pairs X n,i and X n,j are strongly correlated, i.e. lim sup n→∞ sup 1≤i<j≤m t∈Z f n,i,i−t f n,j,j−t < 1.(4) We need the following lemma, whose proof is elementary and will be given in the supplementary file. Lemma 4. The condition (4) suffices for (A2) if ǫ i 's are i.i.d. N (0, 1). As an immediate consequence, when ǫ i 's are i.i.d. N (0, 1), we have ℓ := lim sup n→∞ inf * inf ρ∈R Var (X n,i X n,j − ρX n,k X n,l ) > 0, where inf * is taken over all 1 ≤ i, j, k, l ≤ m such that i < j, k < l and (i, j) = (k, l). Observe that when ǫ i 's are i.i.d. N (0, 1), Var (X n,i X n,j − ρX n,k X n,l ) = 2 · t∈Z (f n,i,i−t f n,j,j−t − ρf n,k,k−t f n,l,l−t ) 2 (5) + s<t (f n,i,i−t f n,j,j−s + f n,i,i−s f n,j,j−t −ρf n,k,k−t f n,l,l−s − ρf n,k,k−s f n,l,l−t ) 2 ; and when ǫ i 's are arbitrary variables, the variance is given by the same formula with the number 2 in (5) being replaced by 2 + κ 4 . Therefore, if (3) holds, then lim sup n→∞ inf * inf ρ∈R Var (X n,i X n,j − ρX n,k X n,l ) ≥ min{1, 1 + κ 4 /2} · ℓ > 0, which implies (A2) holds. To summarize, we have shown that (3) and (4) suffice for (A2). Now we turn to Conditions (A3) and (A3 ′ ). Set h n (k) = sup 1≤i≤m   ∞ |t|=⌊k/2⌋ f 2 n,i,t   1/2 , where ⌊x⌋ = max{y ∈ Z : y ≤ x} for any x ∈ E, then we have |σ n,i,j | ≤ 2h n (0)h n (|i − j|) = 2h n (|i − j|).| Cov(X n,i X n,j , X n,k X n,l )| ≤ (4 + 2κ 4 )h n (b), and hence (A3) holds if we assume h n (k n ) log k n = o(1) for any positive sequence (k n ) such that k n → ∞. (A3 ′ ) holds if we assume m k=1 [h n (k)] 2 = O m 1−δ . for some δ > 0, because |Cov(X n,i X n,j , X n,k X n,l )| ≤ 2κ 4 h n (|i − j|) + 2h n (|i − k|) + 2h n (|i − l|). Testing for covariance structures The asymptotic distribution given in Theorem 2 has several statistical applications. One of them is in high dimensional covariance matrix regularization, because Theorem 2 implies a uniform convergence rate for all sample covariances. Recently, Cai and Liu (2011) explored this direction, and proposed a thresholding procedure for sparse covariance matrix estimation, which is adaptive to the variability of each individual entry. Their method is superior to the uniform thresholding approach studied by Bickel and Levina (2008b). Testing structures of covariance matrices is also a very important statistical problem. As mentioned in the introduction, when the data dimension is high, conventional tests often cannot be implemented or do not work well. Let Σ n and R n be the covariance matrix and correlation matrix of the random vector (X n,1,i ) 1≤i≤m respectively. Two types of tests have been studied under the large n, large m paradigm. Chen et al. (2010), Bai et al. (2009), Ledoit and Wolf (2002) and Johnstone (2001) considered the test H 0 : Σ n = I m ;(6) and Liu et al. (2008), Schott (2005), Srivastava (2005) and Jiang (2004) studied the problem of testing for complete independence H 0 : R n = I m .(7) Their testing procedures are all based on the critical assumption that the entries of the data matrix X n are i.i.d., while the hypotheses themselves only require the entries of (X n,1,i ) 1≤i≤m to be uncorrelated. Evidently, we can use M n in (2) to test (7), and we only require the uncorrelatedness for the validity of the limiting distribution established in Theorem 2, as long as the mild conditions of the theorem are satisfied. On the other hand, we can also take the sample variances into consideration, and use the following test statistic M ′ n = max 1≤i≤j≤m |σ n,i,j − σ n,i,j | τ n,i,j . to test the identity hypothesis (6), where σ n,i,j = I{i = j}. It is not difficult to verify that M ′ n has the same asymptotic distribution as M n under the same conditions with the only difference being that we now have to take sample variances into account as well, namely, the index set I n in Section 2 is redefined as I n = {(i, j) : 1 ≤ i ≤ j ≤ m}. Clearly, we can also use M ′ n to test H 0 : Σ n = Σ 0 for some known covariance matrix Σ 0 . By checking the proof of Theorem 2, it can be seen that if instead of taking the maximum over the set I n = {(i, j) : 1 ≤ i < j ≤ m}, we only take the maximum over some subset A n ⊂ I n whose cardinality |A n | converges to infinity, then the maximum also has the Gumbel type convergence with normalization constants which are functions of the cardinality of the set A n . Based on this observation, we are able to consider three more testing problems. Test for stationarity Suppose we want to test whether the population is a stationary time series. Under the null hypothesis, each row of the data matrix X n is distributed as a stationary process (X i ) 1≤i≤m . Let γ l = Cov(X 0 , X l ) be the autocovariance at lag l. In principle, we can use the following test statistic T n = max 1≤i≤j≤m |σ n,i,j − γ i−j | τ n,i,j . The problem is that γ l are unknown. Fortunately, they can not only be estimated, but also be estimated with higher accuracŷ γ n,l = 1 nm n k=1 n i=|l|+1 (X n,k,i−|l| −μ n )(X n,k,i −μ n ), whereμ n = (1/nm) n k=1 m i=1 X n,k,i , and we are lead to the test statistic T n = max 1≤i≤j≤m |σ n,i,j −γ i−j | τ n,i,j . Using similar arguments of Theorem 2 of , under suitable conditions, we have max 0≤l≤m−1 |γ n,l − γ l | = O P ( log m/nm). Therefore, the limiting distribution for M n in Theorem 2 also holds for T n . Test for bandedness In time series and longitudinal data analysis, it can be of interest to test whether Σ m has the banded structure. The hypothesis to be tested is H 0 : σ n,i,j = 0 if |i − j| > B,(8) where B = B n may depend on n. Cai and Jiang (2011) studied this problem under the assumption that each row of the data matrix X n is a Gaussian random vector. They proposed to use the maximum sample correlation outside the band T n = max |i−j|>Bσ n,i,j σ n,i,iσn,j,j as the test statistic, and proved that T n also has the Gumbel type convergence provided that B n = o(m) and several other technical conditions hold. Apparently, our Theorem 2 can be employed to test (8). If all the conditions of the theorem are satisfied, the test statistic T n = max |i−j|>Bn |σ n,i,j | τ n,i,j . has the same asymptotic distribution as M n as long as B n = o(m). Our theory does not need the normality assumption. Assess the tapering procedure Banding and tapering are commonly used regularization procedures in high dimensional covariance matrix estimation. Convergence rates were first obtained by Bickel and Levina (2008a), and later on improved by Cai et al. (2010). Let us introduce a weaker version of the latter result. Suppose each row of X n is distributed as the random vector X = (X i ) 1≤i≤m with mean µ and covariance matrix Σ = (σ ij ). Let K 0 , K and t be positive constants, and C η (K 0 , K, t) be the class of m-dimensional distributions which satisfy the following conditions max |i−j|=k |σ ij | ≤ Kk −(1+η) for all k; (9) λ max (Σ) ≤ K 0 ; P |v ⊤ (X − µ)| > x ≤ e −tx 2 /2 for all x > 0 and v = 1; where λ max (Σ) is the largest eigenvalue of Σ. For a given even integer 1 ≤ B ≤ m, define the tapered estimate of the covariance matrix Σ Σ n,Bn = (w ijσn,i,j ) , where the weights correspond to a flat top kernel and are given by w ij =    1, when |i − j| ≤ B n /2, 2 − 2|i − j|/B n , when B n /2 < |i − j| ≤ B n , 0, otherwise. Theorem 5 (Cai et al., 2010). If m ≥ n 1/(2η+1) , log m = o(n) and B n = n 1/(2η+1) , then there exists a constant C > 0 such that sup Cη E λ(Σ n,Bn − Σ) 2 ≤ Cn −2η/(2η+1) + C log m n . We see that it is the parameter η that decides the convergence rate under the operator norm. After such a tapering procedure has been applied, it is important to ask whether it is appropriate, and in particular, whether (9) is satisfied. We propose to use T n = max |i−j|>Bn |σ n,i,j | τ n,i,j as the test statistic. According to the observation made at the beginning of Section 4, if the conditions of Theorem 2 are satisfied, then T ′ n = max |i−j|>Bn |σ n,i,j − σ i,j | τ n,i,j has the same limiting law as M n . On the other hand, (9) implies that max |i−j|>Bn |σ i,j | = O n −(1+η)/(2η+1) , so T n has the same limiting distribution as T ′ n if we further assume log m = o n 2/(4η+2) . Proof The proofs of Theorem 2 under (A4) and (A4 ′ ) are very similar, and they share a common Poisson approximation step, which we will formulate in Section 5.1 under a more general context, where the limiting distribution of the maximum of sample means is obtained. Since the proof under (A4 ′ ) is more involved, we provide the detailed proof under this assumption in Section 5.2, and point out in Section 5.3 how it can be adapted to give a proof under (A4). Maximum of Sample Means: An Intermediate Step In this section we provide a general result on the maximum of sample means. Let Y n = (Y n,k,i ) 1≤k≤n, i∈In be a data matrix whose n rows are independent and identically distributed, and whose entries have mean zero and variance one, where I n is an index set with cardinality |I n | = s n . For each i ∈ I n , let y i be the i-th column of Y n ,ȳ i = (1/n) n k=1 Y n,k,i . Define W n = max i∈In |ȳ i |. Let Σ n be the covariance matrix of the s n -dimensional random vector (Y n,1,i ) i∈In . Lemma 6. Assume Σ n satisfies either (B1) or (B2) of Section 6.1 and log s n = o(n 1/3 ). Suppose there is a constant C > 0 such that Y n,k,i ∈ B(1, Ct n ) for each 1 ≤ k ≤ n, i ∈ I n , with t n = √ nδ n (log s n ) 3/2 , where (δ n ) is a sequence of positive numbers such that δ n = o(1) and (log s n ) 3 /n = o(δ n ), and the definition of the collection B(d, τ ) is given in (27). Then lim n→∞ P nW 2 n − 2 log s n + log(log s n ) + log π ≤ z = exp −e −z/2 . Proof. For each z ∈ R, let z n = a 2sn z/2 + b 2sn . Let (Z n,i ) i∈In be a mean zero normal random vector with covariance matrix Σ n . For any subset A = {i 1 , i 2 , . . . , i d } ⊂ I n , let y A = √ n(ȳ i1 ,ȳ i2 , . . . ,ȳ i d ) ⊤ and Z A = (Z i1 , Z i2 , . . . , Z i d ). By Lemma 8, we have for θ n = δ 1/2 n / √ log s n that P (|y A | • > z n ) ≤ P (|Z A | • > z n − θ n ) + C d exp − θ n C d δ n (log s n ) −3/2 ≤ P (|Z A | • > z n − θ n ) + C d exp −(log s n )δ −1/2 n Therefore, A⊂In,|A|=d P (|y A | • > z n ) ≤ A⊂In,|A|=d P (|Z A | • > z n − θ n ) + C d s d n exp −(log s n )δ −1/2 n . Similarly, we have A⊂In,|A|=d P (|y A | • > z n ) ≥ A⊂In,|A|=d P (|Z A | • > z n + θ n ) − C d s d n exp −(log s n )δ −1/2 n . Since (z n ± θ n ) 2 = 2 log s n − log(log s n ) − log π + z + o(1), by Lemma 7, we know lim n→∞ A⊂In,|A|=d P (|Z A | • > z n ± θ n ) = e −dz/2 d ! , and hence lim n→∞ A⊂In,|A|=d P (|y A | • > z n ) = e −dz/2 d ! . The proof is complete in view of Lemma 9. Proof under (A4 ′ ) We divide the proof into three steps. The first one is a truncation step, which will make the Gaussian approximation result Lemma 8 and the Bernstein inequality applicable, so that we can prove Theorem 2 under the assumption that all the involved mean and variance parameters are known. In the next two steps we show that plugging in estimated mean and variance parameters does not change the limiting distribution. Step 1: Truncation For notational simplicity we let q = p/(4 + 2p). Definẽ X n,k,i = X n,k,i I |X n,k,i | ≤ n 1/(4+2p) , and defineM n similarly as M n with X n,k,i being replaced by its truncated versionX n,k,i . Since log m = o(n q ), we have P M n = M n ≤ n k=1 m i=1 P |X n,k,i | > n 1/(4+2p) ≤ nmK n (t, p) exp −tn p/(4+2p) = K n (t, p) exp {−tn q + log m + log n} = o(1). Therefore, in the rest of the proof, it suffices to considerX n,k,i . For notational simplicity, we still useX n,k,i to denote its centered version with mean zero. Defineσ n,i,j = E X n,1,iXn,1,j , andτ n,i,j = Var X n,1,iXn,1,j . Set Cov(X n,α ,X n,β ) − Cov(X n,α , X n,β ) ≤ C exp {−tn q /2} . By (14), we know the covariance matrix of (X n,α ) α∈In satisfies either (B1) E exp{C p t|X n,α | p/2 } < ∞. It follows that when 0 < p < 2, for each integer r ≥ 3 E|X n,α | r ≤ E|X n,α | rp/2 · 4n 2/(4+2p) r(1−p/2) ≤ 4n 2/(4+2p) r(1−p/2) r!(C p t) −r E exp{C p t|X n,α | p/2 }. Therefore, E 0Xn,α ∈ B 1, C √ n n 2p/(4+2p) . When 2 ≤ p ≤ 4, it is easily seen that E 0Xn,α ∈ B(1, C). Since log m = o(n q ), we know all the conditions of Lemma 6 are satisfied, and hence lim n→∞ P nM 2 n,1 − 4 log m + log(log m) + log(8π) ≤ y = exp −e −y/2 . (15) Combining (13) and (14), we know the preceding equation (15) also holds with M n,1 being replaced by M n,2 . Step 2: Effect of Estimated Means SetX n,i = (1/n) n k=1X n,k,i . Define (X n,k,i −X n,i )(X n,k,j −X n,j ) − σ n,i,j . In this step we show that (15) also holds for M n,3 . Observe that |M n,3 − M n,2 | ≤ max 1≤i<j≤m |X n,iXn,j | τ n,i,j ≤ max 1≤i≤m |X n,i | 2 · min 1≤i<j≤mτ n,i,j −1/2 . Since each X n,k,i is bounded by 2n 1/(4+2p) , by Bernstein's inequality we have for any constant K > 0, max 1≤i≤m P |X n,i | > 2K log m n ≤ C exp − 2K 2 n log m Cn + 2K √ n log m · 2n 1/(4+2p) ≤ Cm −K 2 /C , and hence max 1≤i≤m |X n,i | = O P log m n ,(16) which together with (14) implies that |M n,3 − M n,2 | = O P log m n = o P 1 n log m . Therefore, (15) also holds for M n,3 . Step 3: Effect of Estimated Variances Denote byσ n,i,j the estimate ofσ n,i,ǰ σ n,i,j = 1 n n k=1 (X n,k,i −X n,i )(X n,k,j −X n,j ). In the definition ofM n ,τ n,i,j is unknown, and is estimated by τ n,i,j = 1 n n k=1 (X n,k,i −X n,i )(X n,k,j −X n,j ) −σ n,i,j 2 In this step we show that (15) holds forM n . Since n M 2 n,3 −M 2 n ≤ nM 2 n,3 · max 1≤i<j≤m |1 −τ n,i,j /τ n,i,j |, it suffices to show that max 1≤i<j≤m |τ n,i,j −τ n,i,j | = o P (1/ log m).(17) Setτ n,i,j,1 = 1 n n k=1 (X n,k,i −X n,i )(X n,k,j −X n,j ) −σ n,i,j 2 τ n,i,j,2 = 1 n n k=1 X n,k,iXn,k,j −σ n,i,j 2 . Observe thatτ n,i,j,1 −τ n,i,j = (σ n,i,j −σ n,i,j ) 2 which in together with (15) implies that max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j | = O P (log m/n) . Note thatX n,k,i,j are uniformly bounded according to the truncation (12), so X n,k,iXn,k,j −σ n,i,j 2 ≤ 64n 4/(4+2p) . By Bernstein's inequality, we have max 1≤i<j≤m P |τ n,i,j,2 −τ n,i,j | ≥ 2n −q ≤ exp − 2n 2(1−q) Cn + 2n 1−q · 128n 4/(4+2p) /3 ≤ exp (−n q /100) , and it follows that max 1≤i<j≤m |τ n,i,j,2 −τ n,i,j | = O P (n −q ).(19) In view of (18), (19), and the assumption log m = o(n q ), we know to show (17), it remains to prove max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j,2 | = o P (1/ log m). Elementary calculations show that max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j,2 | ≤ 4h 2 n,1 h n,2 + 3h 4 n,1 + 4h By (16), we know h n,1 = O P ( log m/n). By (19) we have h n,4 = O P (1). Combining (12) and the Bernstein's inequality, we can show that h n,3 = O P log m/n . As an immediate consequence, we know h n,2 = O P (1). Therefore, max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j,2 | = O P log m/n , and (20) holds by using the assumption log m = o(n q ) = o(n 1/3 ). The proof of Theorem 2 under (A4 ′ ) is now complete. Proof under (A4) We follow the proof in Section 5.2, and point out necessary modifications to make it work under (A4). If not specified, all the notations have the same definitions as in Section 5.2. For notational simplicity, we let p = 4(1 + q) + δ. Step 1: Truncation We truncate X n,k,i bỹ X n,k,i = X n,k,i I |X n,k,i | ≤ n 1/4 / log n , then P M n = M n ≤ nmM n (p)n −p/4 (log n) p ≤ CM n (p)n −δ/4 (log n) p = o(1). Therefore, in the rest of the proof, it suffices to considerX n,k,i . For notational simplicity, we still useX n,k,i to denote its centered version with mean zero. Elementary calculations show that max 1≤i≤j≤m |σ n,i,j − σ n,i,j | ≤ Cn −(p−2)/4 (log n) p−2 , and max α,β∈In Cov(X n,α ,X n,β ) − Cov(X n,α , X n,β ) ≤ Cn −(p−4)/4 (log n) p−4 . By (21), we know the covariance matrix of (X n,α ) α∈In satisfies either (B1) or (B2) if Σ n satisfies (B1) or (B2) correspondingly. Since E 0Xn,α ∈ B 1, 8 √ n/(log n) 2 , we know all the conditions of Lemma 6 are satisfied, and hence (15) holds for M n,1 . Combining (21) and (22), we know (15) also holds with if we replace M n,1 by M n,2 . Step 2: Effect of Estimated Means Using Bernstein's inequality, we can show max 1≤i≤m |X n,i | = O P log n n , which implies that |M n,3 − M n,2 | = O P log n n and hence (15) also holds for M n,3 . Step 3: Effect of Estimated Variances It suffices to show that max 1≤i<j≤m |τ n,i,j −τ n,i,j | = o P (1/ log n). Using (15), we know max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j | = O P (log n/n) . Since X n,k,iXn,k,j −σ n,i,j 2 ≤ 64n/(log n) 4 . By Corollary 1.6 of Nagaev (1979) (with x = n/(log n) 2 and y = n/[2(log n) 3 ] in their inequality (1.22)), we have max 1≤i<j≤m P |τ n,i,j,2 −τ n,i,j | ≥ (log n) −2 ≤ Cn n(log n) −2 · [n(log n) −3 /2] q∧1 log n ≤ C(log n) 5 n q∧1 log n , and it follows that max 1≤i<j≤m |τ n,i,j,2 −τ n,i,j | = O P (log n) −2 .(25) In view of (24), (25), we know to show (23), it remains to prove max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j,2 | = o P (1/ log n). We know h n,1 = O P ( log n/n) and h n,4 = O P (1). Using the Bernstein's inequality, we can show that h n,3 = O P log n/n , and it follows that h n,2 = O P (1). Therefore, max 1≤i<j≤m |τ n,i,j,1 −τ n,i,j,2 | = O P log n/n , and (26) holds. The proof of Theorem 2 under (A4) is now complete. Some auxiliary results In this section we provide a normal comparison principle and a Gaussian approximation result, and a Poisson convergence theorem. A normal comparison principle Suppose for each n ≥ 1, (X n,i ) i∈In is a Gaussian random vector whose entries have mean zero and variance one, where I n is an index set with cardinality |I n | = s n . Let Σ n = (r n,i,j ) i,j∈In be the covariance matrix of (X n,i ) i∈In . Assume that s n → ∞ as n → ∞. We impose either of the following two conditions. If z n satisfies that z 2 n = 2 log s n − log log s n − log π + 2z + o(1), then for all d ≥ 1. lim n→∞ Q ′ n,d = e −dz d ! , Lemma 7 is a refined version of Lemma 20 in , so we omit the proof and put the details in a supplementary file. Remark 2. The conditions imposed on γ(n, b n ) seem a little involved. We have the following equivalent versions. Define G n (t) = max i∈In j∈In I{|r n,i,j | > t}. Then (i) γ(n, b n ) = o(1) for any sequence b n → ∞ if and only if the sequence [G n (t)] n≥1 is bounded for all t > 0; and (ii) γ(n, b n )(log b n ) = o(1) for any sequence b n → ∞ if and only if G n (t n ) = exp{o(1/t n )} for any positive sequence (t n ) converging to zero. A Gaussian approximation result For a positive integer d, let B d be the Borel σ-field on the Euclidean space R d . For two probability measures P and Q on R d , B d and λ > 0, define the quantity π(P, Q; λ) = sup A∈B d max P (A) − Q A λ , Q(A) − P A λ , where A λ is the λ-neighborhood of A A λ := x ∈ R d : inf y∈A |x − y| < λ . For τ > 0, let B(d, τ ) be the collection of d-dimensional random variables which satisfy the multivariate analogue of the Bernstein's condition. Denote by (x, y) the inner product of two vectors x and y. B(d, τ ) = ξ is a random variable : Eξ = 0, and E (ξ, t) 2 (ξ, u) m−2 ≤ 1 2 m!τ m−2 u m−2 E (ξ, t) 2 for every m = 3, 4, . . . and for all t, u ∈ R d . The following Lemma on the Gaussian approximation is taken from Zaȋtsev (1987). Lemma 8. Let τ > 0, and ξ 1 , ξ 2 , . . . , ξ n ∈ R d be independent random vectors such that ξ i ∈ B(d, τ ) for i = 1, 2, . . . , n. Let S = ξ 1 + ξ 2 + . . . + ξ n , and L (S) be the induced distribution on R d . Let Φ be the Gaussian distribution with the zero mean and the same covariance matrix as that of S. Then for all λ > 0 π[L (S), Φ; λ] ≤ c 1,d exp − λ c 2,d τ , where the constants c j,d , j = 1, 2 may be taken in the form c j,d = c j d 5/2 . Poisson approximation: moment method Lemma 9. Suppose for each n ≥ 1, (A n,i ) i∈In is a finite collection of events. Let I An,i be the indicator function of A n,i , and W n = i∈I I An,i . For each d ≥ 1, define Q n,d = A⊂In,|A|=d P i∈A A n,i . Suppose there exists a λ > 0 such that lim n→∞ Q n,d = λ d /d ! for each d ≥ 1. Then lim n→∞ P (W n = k) = λ k e −λ /k ! for each k ≥ 0. Observe that for each d ≥ 1, the d-th factorial moment of W n is given by E [W n (W n − 1) · · · (W n − d + 1)] = d ! · Q n,d , so Lemma 9 is essentially the moment method. The proof is elementary, and we omit details. In this document we give the proofs of Lemma 3, Lemma 4 and Lemma 7 of the main article. Proof of Lemma 3. Assume X i has mean zero and variance one. Let γ k = E(X 0 X k ) be the autocovariance of lag k. Then by Proposition 8, Eq. (34) of , we know |γ k | ≤ Ψ 2 · Ψ 2 (|k|). (S.1) (i) Since Ψ 4 < ∞, we know for any η > 0, there exists a N 1 > 0 such that |γ k | < η when k ≥ N 1 . For j ≤ k, defineX k,j = g(ǫ k , . . . , ǫ j+1 , ǫ ′ j , ǫ ′ j−1 , . . .), where (ǫ ′ i ) i∈Z is an i.i.d. copy of (ǫ i ) i∈Z . By Eq. (38) of , we know there exists a N 2 > 0 such that when k ≥ N 2 , X k −X k 4 ≤ η. Set N = max{N 1 , N 2 }, when k ≥ N , we have Var(X 0 X k ) = E(X 2 0 X 2 k ) − γ 2 k = E X 2 k X 2 k,j + E X 2 0 (X 2 k − X 2 k,j ) − γ 2 k ≥ 1 − η 2 − 2 X 0 3 4 · η. Therefore, (A1) holds because η can be arbitrarily small. (ii) We need to show that sup j≥0, 0≤k≤l, (0,j) =(k,l) Cor(X 0 X j , X k X l ) < 1. It suffices to show that for some N > 0 sup j≥0, 0≤k≤l, (0,j) =(k,l), j+k+l≥N Cor(X 0 X j , X k X l ) < 1. If j + k + l ≥ N , then the set {0, j, k, l} can be partitioned into two non-empty subsets B 1 and B 2 whose distance is no less than N/6. We only consider this type of partitions. If there is a partition such that 1 one of B 1 and B 2 has cardinality one, then similarly as (i), we know for any η > 0, when N is large enough, | Cov(X 0 X j , X k X l )| = |E(X 0 X j X k X l ) − γ j γ l−k | ≤ η. If for any partition both B 1 and B 2 has cardinality two, there are two sub-cases. (a) j < k ≤ l and k − j ≥ N/6. For any η > 0, when N is large enough, we have | Cov(X 0 X j , X k X l )| = |E [X 0 X j (X k X l − X k,j X l,j )]| ≤ η. (b) min{j, l} − k ≥ N/6. As in (i), for any η > 0, when N is large enough, we have Var(X 0 X j ) ≥ 1 − η, Var(X k X l ) ≥ 1 − η, and |γ j γ l−k | < η. On the other hand, the condition Ψ 4 > 0 guarantees that the process is non-deterministic, and hence γ := sup t≥1 |γ t | < 1. It follows that when N is large enough |E(X 0 X j X k X l )| = |E(X 0 X j,k X k X l,k ) + E[X 0 X k (X j X l − X j,k X l,k )]| ≤ γ + η. Therefore, | Cor(X 0 X j , X k X l )| ≤ (γ + 2η)/(1 − η) < 1 when η is small enough. The proof of (ii) is now complete. (iii) We first consider (A3). Note that Cov(X i X j , X k X l ) = Cum(X i , X j , X k , X l ) + γ i−k γ j−l + γ i−l γ j−k , where Cum(X i , X j , X k , X l ) is the fourth order joint cumulant of the random vector (X i , X j , X k , X l ) ⊤ . On the other hand, using similar arguments as Theorem 21 of , we can show that |Cum(X i , X j , X k , X l )| ≤ CΨ 4 (⌊b/2⌋). Therefore, if Ψ 4 (k) = o(1/ log k) as k → ∞, then (A3) holds. Now we turn to (A3 ′ ). Write Cov(X i X j , X k X l ) = E(X i X j X k X l ) − γ i−j γ k−l . By (S.1), it is easily seen that 1≤i,j,k.l≤m γ 2 i−j γ 2 k−l = O(m 4−2δ ). Lemma S.2. Assume either (B1) or (B2). For a positive real number z n , define the event A n,i and Q n,d as A n,i = {X n,i > z n } and Q n,d = A⊂In,|A|=d P i∈A A n,i . If z n satisfies that z 2 n = 2 log s n − log log s n − log(4π) + 2z + o(1), then for all d ≥ 1 lim n→∞ Q n,d = e −dz d ! . Proof. The following facts about normal tail probabilities are well-known: When the X n,i 's are dependent, the result is still trivially true when d = 1. Now we deal with the d ≥ 2 P (X 1 ≥ x) ≤ 1 √ 2πx e − case. Suppose (b n ) is a sequence of positive numbers which converges to infinity. For each subset J of I n with cardinality |J| = d, we define an undirected graph G (J) by identifying each i ∈ J with a node and saying i and j are adjacent if |r n,i,j | > γ(n, b n ). Suppose the graph G (J) has d − s connected components B 1 , . . . , B d−s . If s ≥ 1, assume w.l.o.g. that |B 1 | ≥ 2. Pick k 0 , k 1 ∈ B 1 , and k p ∈ B p for 2 ≤ p ≤ d − s, and set K = {k 0 , k 1 , k 2 , . . . , k d−s }. Define Q J = P (∩ k∈J A k ) and Q K similarly, then Q J ≤ Q K . By (S.3) of Lemma S.1, there exists a number M > 1 depending on d and the sequences (γ n ) and (b n ), such that when n ≥ M , Q K ≤ C d−s exp − (1 − γ n ) 2 + d − s 2 − C d−s γ(n, b n ) z 2 n ≤ C d−s exp − d − s 2 + (1 − γ n ) 2 3 z 2 n . Note that z 2 n = 2 log s n − log log s n + O(1). Pick b n = ⌊s α n ⌋ for some α < (1 − γ n ) 2 /3d. For any 1 ≤ a ≤ d − 1, since there are at most O b a n s d−a n subsets J ⊂ I n such that |J| = d and the graph G (L) has d − a connected components, we know the sum of Q J over these J is dominated by C d−a exp log s n (d − a) + 2(d − 1)(1 − γ n ) 2 3d − (d − a) − 2(1 − γ n ) 2 3 when n is large enough, which converges to zero. Therefore, it remains to consider all the subsets J ⊂ I n such that the graph G (J) has no edges Let J ⊂ I n be a subset such that |J| = d, and |r n,i,j | < γ(n, b n ) for all pairs i, j such that i, j ∈ J and i = j, and J (d, b n ) be the collection of all such subsets. Let (r ij ) i,j∈J be the d-dimensional covariance matrix of X J := (X n,i ) i∈J . There exists a matrix R J = θ(r ij ) i,j∈J + (1 − θ)I d for some 0 < θ < 1 such that Q J − Q d (I d ,z a subset {i, j}, for any integer b > 0, there are at most 8b 2 subsets {k, l} such that {k, l} ⊂ B(i; b)∪B(j; b), where B(x; r) is the open ball {y : |x−y| < r}. For all other subsets {k, l}, we have n,i,j − σ n,i,j | ≤ C exp {−tn q /2} , ( B1 )Lemma 7 . B17For any sequence (b n ) such that b n → ∞, γ(n, b n ) = o (1/log b n ) ; and lim sup n→∞ γ n < 1.(B2) For any sequence (b n ) such that b n → ∞, γ(n, b n ) = o(Assume either (B1) or (B2). For a positive real number z n , define A ′ n,i = {|X n,i | > z n } and Q ′ Fix a subset {i, j}, for any integer b > 0, there are at most 8b 2 subsets {k, l} such that {k.l} ⊂ B(i; b) ∪ B(j; b), where B(x; r) is the open ball {y : |x − y| < r}. For all other subsets {k, l}, by (S.1), we have |γ i−k γ j−l + γ i−l γ j−k | ≤ CΨ 4 (b). assumption on z n , if for each n, X n,i , i ∈ I n are i.i.d., then by (S. Supplementary file of Simultaneous Inference of CovariancesHan Xiao and Wei Biao Wu501 Hill Center 110 Frelinghuysen Road Piscataway, NJ 08854 e-mail: xiao@stat.rutgers.edu Department of Statistics 5734 S. University Ave Chicago, IL 60637 e-mail: wbwu@galton.uchicago.edu n ) = h,l∈J,h<l ∂Q d ∂r hl [R J ; z n ]r hl . [E(X i X j X k X l )] 2 = O(m 4−δ ), which is true because by Eq. (38) of [E(X i X j X k X l )] 2 = [E(X i X j X k (X l − X l,k ))] 2 ≤ 12 X 0 6 4 [Ψ 4 (l − k)] 2 .The proof of Lemma 3 is now complete.We now give the proof of Lemma 4.Proof of Lemma 4. Suppose (Y 1 , Y 2 , Y 3 , Y 4 ) has a joint normal distribution. We can write Y i = α ⊤ i Z, where Z is a four dimensional standard Gaussian random vector. For any 0 < ν < 1, define the subset of R 16 ,| is a continuous function on D ν , and D ν is compact, the maximum correlation is attained at some point in D ν .On the other hand, elementary calculation shows thatall perfectly correlated. The proof is now complete.The proof of Lemma 7 is a refined version of that of Lemma 20 in . We need the following bounds on normal tail probabilities, which are taken from Lemma 19 of .Denote by ϕ d ((r ij ); x 1 , . . . , x d ) the density of a d-dimensional multivariate normal random vector X = (X 1 , . . . , X d ) ⊤ with mean zero and covariance matrix (r ij ), where we always assume r ii = 1 for 1 ≤ i ≤ dLemma S.1. For every z > 0, 0 < s < 1, d ≥ 1 and ǫ > 0, there exists positive constants C d and ǫ d suchWe first give a one-sided version of Lemma 7 and its proof, then we show how it implies Lemma 7.It follows thatwhere the sum * i,j∈In is over all the pair (i, j) such that |r n,i,j | ≤ γ(n, b n ). Under the assumption (B1), we haveSince lim n→∞ γ(n, b n ) log b n = 0, it also holds that lim n→∞ γ(n, b n ) log s n = 0. Note that lim n→∞ (log s n ) 1/2 /z n = 2 −1/2 , it follows that lim n→∞ f d−2 (γ(n, b n ), 1/z n )(log s n ) d/2−1 = 2 −d/2+1 . Therefore, the term in (S.6) converges to zero, and the theorem holds under (B1).Alternatively, if (B2) is true, from (S.5) we have Proof of Lemma 7. In the proof of Theorem S.2, the upper bounds on Q J and |Q J − Q(I d ; z n )| are expressed through the absolute values of the covariances, so we can obtain the same bounds for probabilities of the form P (∩ 1≤i≤d {(−1) ai X ti ≥ z n }) for any (a 1 , . . . , a d ) ∈ {0, 1} d . 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Craig A Tracy, Harold Widom, Comm. Math. Phys. 1591Craig A. Tracy and Harold Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys., 159(1):151-174, 1994. Nonlinear system theory: another look at dependence. Wei Biao Wu, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA102Wei Biao Wu. Nonlinear system theory: another look at dependence. Proc. Natl. Acad. Sci. USA, 102(40):14150-14154 (electronic), 2005. Asymptotic inference of autocovariances of stationary processes. Han Xiao, Wei Biao Wu, preprintHan Xiao and Wei Biao Wu. Asymptotic inference of autocovariances of stationary processes. preprint, available at http://arxiv.org/abs/1105.3423, 2011. On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein's inequality conditions. A Yu, Zaȋtsev, Probab. Theory Related Fields. 744A. Yu. Zaȋtsev. On the Gaussian approximation of convolutions under multidimen- sional analogues of S. N. Bernstein's inequality conditions. Probab. Theory Related Fields, 74(4):535-566, 1987. Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Wang Zhou, Trans. Amer. Math. Soc. 35911Wang Zhou. Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc., 359(11):5345-5363, 2007. Asymptotic inference of autocovariances of stationary processes. Han Xiao, Wei Biao Wu, preprintHan Xiao and Wei Biao Wu. Asymptotic inference of autocovariances of stationary processes. preprint, available at http://arxiv.org/abs/1105.3423, 2011.

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{'abstract': 'We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances.', 'arxivid': '1109.0524', 'author': ['Han Xiao hxiao@stat.rutgers.edu \nDepartment of Statistics\n501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL\n', 'Wei Biao Wu wbwu@galton.uchicago.edu \nDepartment of Statistics\n501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL\n'], 'authoraffiliation': ['Department of Statistics\n501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL', 'Department of Statistics\n501 Hill Center 110 Frelinghuysen Road Piscataway, 5734 S. University Ave Chicago08854, 60637NJ, IL'], 'corpusid': 88519509, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19198, 'n_tokens_neox': 17202, 'n_words': 9572, 'pdfsha': '1c4bb218fa60edb34d728b5b5fb4a7e3fa583c2e', 'pdfurls': ['https://arxiv.org/pdf/1109.0524v1.pdf'], 'title': ['Simultaneous Inference of Covariances', 'Simultaneous Inference of Covariances'], 'venue': []}

arxiv

Characterization of spin wave propagation in (111) YIG thin films with large anisotropy A Krysztofik Institute of Molecular Physics Polish Academy of Sciences M. Smoluchowskiego 17PL-60-179PoznańPoland H Głowiński Institute of Molecular Physics Polish Academy of Sciences M. Smoluchowskiego 17PL-60-179PoznańPoland P Kuświk Institute of Molecular Physics Polish Academy of Sciences M. Smoluchowskiego 17PL-60-179PoznańPoland Centre of Advanced Technology Adam Mickiewicz University Umultowska 89cPL-61-614PoznańPoland S Ziętek Department of Electronics AGH University of Science and Technology Al. Mickiewicza 40PL-30-059KrakówPoland L E Coy NanoBioMedical Centre Adam Mickiewicz University Umultowska 85PL-61-614PoznańPoland J N Rychły Faculty of Physics Adam Mickiewicz University Umultowska 85PL-61-614PoznańPoland S Jurga NanoBioMedical Centre Adam Mickiewicz University Umultowska 85PL-61-614PoznańPoland T W Stobiecki Department of Electronics AGH University of Science and Technology Al. Mickiewicza 40PL-30-059KrakówPoland J Dubowik Institute of Molecular Physics Polish Academy of Sciences M. Smoluchowskiego 17PL-60-179PoznańPoland Faculty of Physics and Applied Computer Science AGH University of Science and Technology Al. Mickiewicza 30PL-30-059KrakówPoland Characterization of spin wave propagation in (111) YIG thin films with large anisotropy 1 We report on long-range spin wave (SW) propagation in nanometer-thick Yttrium Iron Garnet (YIG) film with an ultralow Gilbert damping. The knowledge of a wavenumber value | ⃗ | is essential for designing SW devices. Although determining the wavenumber | ⃗ | in experiments like Brillouin light scattering spectroscopy is straightforward, quantifying the wavenumber in all-electrical experiments has not been widely commented so far.We analyze magnetostatic spin wave (SW) propagation in YIG films in order to determine SW wavenumber | ⃗ | excited by the coplanar waveguide. We show that it is crucial to consider influence of magnetic anisotropy fields present in YIG thin films for precise determination of SW wavenumber. With the proposed methods we find that experimentally derived values of | ⃗ | are in perfect agreement with that obtained from electromagnetic simulation only if anisotropy fields are included. 2 Spin wave (SW) propagation in magnetic thin film structures has become intensively investigated topic in recent years due to promising applications in modern electronics [ 1,2,3,4 ]. The wavenumber (or equivalentlythe wavelength = 2 /| ⃗ |) is an important parameter to account for propagation characteristics. For example, it is essential to choose SW wavenumber and correlate it to certain device dimension in order to ensure observation of expected phenomena in SW devices e.g. in magnonic crystals [ 5,6 ] or devices based on wave interference such as SW transistor [ 2 ], SW logic gates [ 2 ], Mach-Zender type interferometers [ 7 ]. The knowledge of SW wavenumber is also very important in the assessment of the effective magnitude of Dzaloshinskii-Moriya interaction using collective spin-wave dynamics [ 8 ]. In propagating SW spectroscopy experiments two shorted coplanar waveguides (CPWs) are commonly used as a transmitter and a receiver [ 9 ]. Each CPW, integrated within the film, consists of a signal line and two ground lines connected at one end. When a rf-current flows through the transmitter it induces an oscillating magnetic field around the lines that exerts a torque and causes spin precession in the magnetic material beneath. The inverse effect is then used for SW detection by the receiver. Since the generated magnetic field is not homogenous with reference to the film plane and solely depends on CPW geometry, it determines the distribution of SW wavenumber that can be excited. It is assumed that the transmitter excites a broad spectrum of SW wavevectors of wavenumber extending to ≈ / ( is a width of CPW line) with a maximum of excitation amplitude approximately around ≈ /2 [ 10 ]. The question now is: what is the actual wavenumber of the SW with the largest amplitude detected by the receiver situated at a certain distance from the transmitter. It appears that while in Brillouin light scattering spectroscopy is easily accessible, in all electrical spin wave spectroscopic experiments the determination of SW wavenumber is rather challenging [ 11 ]. We aim to answer this question by analyzing our experimental results of SW propagation in yttrium iron garnet (Y 3 Fe 5 O 12 , YIG) thin films. YIG films are known as possessing the lowest Gilbert damping parameter enabling the SW transmission over the distances of several hundred micrometers [ 2,12 ]. However, YIG films synthesized by pulsed laser deposition (PLD) exhibit substantially disparate values of anisotropy fields and saturation magnetization, depending on the growth process parameters and, consequently, stoichiometry of the obtained film [ 13,14,15 ]. It has already been theoretically predicted 3 that anisotropy may significantly affect SW propagation and the transmission characteristics [ 16,17 ]. Therefore, for such YIG films, SW spectra analysis requires careful consideration of anisotropic properties of a given film. Here, we compare two methods of experimental determination of the SW wavenumber which include anisotropy fields. The experimental results are then compared with electromagnetic simulations. ). After the growth, the sample was additionally annealed ex situ at 800℃ for 5 . X-ray diffraction and reflection measurements showed that the YIG film was single-phase, epitaxial with the GGG substrate with the thickness of 82 and RMS roughness of 0.8 . XRD θ-2θ scan, presented in Fig. 1, clearly shows the high crystallinity of the YIG film, displaying well defined Laue oscillations, typical for highly epitaxial films, which clearly point to the high quality and well textured YIG (111) film [ 18 ]. Subsequently, a system of two CPWs made of 100 thick aluminum was integrated onto YIG film (Fig.2) using a maskless photolithography technique. The width of signal and ground lines was equal to 9.8 and the gaps between them were 4 wide. The distance between the centers of signal lines was 150 . To investigate SW propagation we followed approach presented in Ref. [ 9 ] and [ 12 ]. Using a Vector Network Analyzer transmission signal S 21 was measured for Damon-Eshbach surface modes with wavevector ⃗ perpendicular to the magnetization for magnetic fields ranging from −310 to +310 ( Fig. 3(a)). Exemplary S 21 signals (imaginary part), which are shown in Figs 3(b) and (c), reveal a series of oscillations as a function of frequency with a Gaussian-like envelope corresponding to the excited SW wavenumber distribution. Figure 3(c) shows that frequency separation ∆ between two oscillation maxima differs noticeably in value depending on the magnetic field. The decrease in signal amplitude is also observed since SW decay length is inversely proportional to the frequency, so that the low-frequency SWs propagate further away [ 12,19 ]. For the frequencies of the highest signal amplitude, the wavenumber can be determined according to the dispersion relation derived for (111) crystalline orientation of the YIG film [ 16,17 ]: (1) where is the microwave frequency, the Bohr magneton constant, ℏthe reduced Planck constant,the spectroscopic splitting factor,the external magnetic field, the saturation magnetization,the film thickness,the wavenumber, the cubic anisotropy field and the out-of-plane uniaxial anisotropy field. As can be seen from Eq. 1, in order to determine wavenumber one needs to evaluate many material constants, namely , , , , in the first instance. = 2 ℏ √( + 2 ) ( − 1 2 − + 4 − 2 ) − 1 2 ( sin (3 )) 2 , This problem can be partially solved with a broadband ferromagnetic resonance measurement of the film. For = 0 Eq.1 simplifies to the formula, which allows for the determination of the spectroscopic factor and the effective magnetization 4 * = − 1 2 − + 4 : =0 = 2 ℏ √ ( + 4 * ).(2) Therefore, within this approach, the film thickness and the saturation magnetization should be determined using other experimental methods. To investigate ferromagnetic resonance of the YIG film, the reflection signal S 11 was measured. In order to avoid extrinsic contribution to the resonance linewidth caused by nonmonochromatic excitation of the CPW (2 ∆ = ∆ ) [ 21 ] and, consequently, possible ambiguities in the interpretation of resonance peak position, it is recommended to perform this measurement with the use of a wide CPW. Note that the full width at half maximum of a CPW excitation spectra ∆ ≈ [ 21 ]. In our study we used a CPW with signal and ground lines of the width equal to 450 and with the 20 wide gaps between them. For such a CPW, the simulated value of is equal to 49 −1 and, therefore, yields negligible broadening that is of the order of a few MHz. The measured S 11 signal (imaginary part) is depicted in Fig. 3(a) [ 14 ]. From the analysis of resonance linewidth vs. frequency [ 23 ] we additionally extracted Gilbert damping parameter of the YIG film, which equals to = (5.5 ± 0.6) × 10 −4 and implies low damping of magnetization precession. Typical values of cubic magnetocrystalline anisotropy field range from −18 to −64 for PLD grown YIG films [ 14,15,22 ], what indicates that resonance measurements as well as spin wave propagation are governed by the out-of-plane uniaxial anisotropy. For the film employed in our study, the value is of about −600 in agreement with previous reports [ 14,15,22 ]. For any more complex architecture of magnonic waveguides and circuits it is likewise imperative to investigate the in-plane anisotropy properties [ 24 ]. As can be seen from Eq. 1 one would expect a six-fold anisotropy in the plane of (111)-oriented single crystals, that is common among rare-earth substituted YIG garnets and LPE-YIG films [ 18,25,26,27 ]. To examine this issue, we performed VSM and angular resolved ferromagnetic resonance measurements. Hysteresis loops for all measured in-plane directions exhibit no substantial differences regarding coercive field (≈ 1.2 ), saturation field and saturation magnetization ( Fig. 4(a)). The angular resolved resonance measurements confirm this result and show that the (111) YIG film is isotropic in the film plane ( Fig. 4(b)). The main reason for this behavior is the low value of cubic anisotropy field which causes the resonance frequency modulation by a value 8 of the fraction of MHz. Such small differences do not surpass the experimental error, nor would they significantly affect the coherent SW propagation. It is expected that the SW propagation characteristics, measured for any other crystallographic orientation, would therefore remain unaltered. Another method of extracting SW wavenumber involves the analysis of the SW group velocity . Following Ref. [ 21 ], can be determined from frequency difference ∆ between two oscillation maxima in S 21 signal according to the relation: = ∆ ,(3) where is the distance between two CPWs. To determine ∆ we chose two neighboring oscillation maxima of the highest S 21 signal amplitude as it is shown in Fig. 3(b) and (c). In Fig. 5 the derived values of group velocity are shown as a function of magnetic field. It is found that reaches the value of 7.6 / for the field of 1.3 (preferable in magnonic information processing devices of high efficiency) and 1.4 / for the field of 285 . It should be highlighted that such big differences in values can be further utilized to design tunable, impulse-response delay lines as changes up to five times with the magnetic field. At a distance of 150 between CPWs it would allow to achieve 20 to 110 delay times of an impulse. With the red line in Fig. 5 a fitting is depicted according to: = 2 = ℏ 2 (− 1 2 − +4 −4 ) 2√( +2 )( − 1 2 − +4 −2 ) . The main advantage of extracting SW wavenumber from ( ) dependence is that it does not require additional measurement of which is often notably influenced by an error in the estimated film volume. Since the saturation magnetization can be treated as a fitting As can be seen from Figure 5, SW group velocity attains the maximum value as the magnetic field approaches = 0. The maximum value of is given by: ( =0) ≅ ℏ √ 2 (− 1 2 − + 4 [1 − ]).(5) The zero-field region may therefore become the subject of interest for magnonic applications. Moreover, Eq. 5 shows that the maximum value of depends on the anisotropy fields. PLDgrown YIG films possessing a high anisotropy would allow faster information processing in SW circuits than LPE films for which the value of − 1 2 − is smaller (as it was pointed out above). To confront our experimental results with the expected, theoretical value of , we performed electromagnetic simulations in Comsol Multiphysics. Here, CPW was modeled according to the geometry of the performed CPW (Fig. 2), assuming lossless conductor metallization, relative permittivity of the substrate = 12 and 50 port impedance. From the simulated in-plane distribution of the dynamic magnetic field ℎ (inset of Fig. 6) an excitation spectra of CPW was obtained using discrete Fourier transformation of ℎ ( ). The highest excitation strength is observed for = 1838 −1 , which corresponds well to the experimentally obtained values within 7% accuracy. The second observed maxima is at 2 = 6770 −1 . However, as its amplitude is 20 times lower with respect to the amplitude of it is not observed in the measured S 21 signal. To extend our study, we performed a series of further simulations for the CPW dimensions, which are achievable with electron-and photolithography. We assumed equal widths of signal and ground lines ( ) as well as equal widths of gaps between them ( ). The results are presented in Fig. 7. It is found that for the widths ranging from 300 to 40 , the wavenumber vary between 70000 −1 and 250 −1 , respectively, revealing the CPW wavenumber probing limits. We also note that the gap width significantly affects . In order to accurately extrapolate its contribution to , we developed empirical formula which incorporates width : = 2.27 +0.6 .(6) The fittings, according to the Eq. 6, are depicted in Fig. 7 with solid lines. We found that Eq. 6 is valid for gap width 0.1 < < 2 . For = 0.74 this formula is equivalent to the one previously proposed in Ref. [ To conclude, we reported on long-range spin wave propagation in the 82 thick YIG film over the distance as large as 150 . In order to precisely determine excited wavenumber by the coplanar antenna, it is essential to take into account anisotropy fields present in YIG films. We showed that anisotropy significantly affects SW propagation characteristics, namely it causes an increase in SW frequency as well as in SW group velocity. The main contribution comes from the out-of-plane uniaxial anisotropy field. The cubic anisotropy field is negligibly small in the YIG (111) film and it does not affect magnetization dynamics in the film plane. We explained that the wavenumber determination from group velocity vs. magnetic field dependence requires only two types of measurement, that is broadband SW spectroscopy and the measurement of film thickness. Fig. 1 . 1A θ-2θ XRD scan of epitaxial YIG film on GGG (111) substrate near the GGG (444) reflection. YIG film was grown on a monocrystalline, 111-oriented Gadolinium Gallium Garnet substrate (Gd 3 Ga 5 O 12 , GGG) by means of PLD technique. Substrate temperature was set to 650℃ and under the 1.2 × 10 −4 oxygen pressure (8 × 10 −8 base pressure) thin film was deposited at the 0.8 / growth rate using third harmonic of Nd:YAG Laser ( = 355 Fig. 2 . 2SEM image of the integrated CPWs on the YIG film. The distance between the transmitter and the receiver is equal to 150. The depicted Cartesian and crystallographic coordinate system is used throughout this paper. The width of signal and ground lines is marked with . denotes the gap width between the lines. 5 Fig. 3 . 53(a) Color-coded SW propagation data S 21 measured at different magnetic fields. With a red line ( ) dependence of the uniform excitation ( = 0) is depicted. The red line corresponds to the maximum in S 11 signal in (b). The blue dashed line represents a dispersion relation with = = 0. (b) Reflection (S 11 , 11= 0) and transmission (S 21 , ≠ 0) signals. The plot illustrates a magnified cross-section of (a) at = −67.5 . (c) SW spectra measured at different magnetic fields. Color-coding in (b) and (c) corresponds to the one defined in (a). equivalent to the one originally obtained by Damon and Eshbach[ 20 ]. The azimuthal angle defines the in-plane orientation of magnetization direction with respect to the (112 ̅ ) axis of YIG film. In our study the term is parallel to (112 ̅ ) axis and = 0°. Substitution of the , * , and values into Eq. 1 enabled the determination of wavenumber = 1980 ± 102 −1 . It should be noted that if anisotropy fields were neglected in the Eq.1 ( = = 0), yet only saturation magnetization was taken into account, a fitting to the experimental data would not converge. The calculated dispersion relation with the derived value of , assuming = = 0 is depicted with blue dashed line inFig. 3 (a). Omission of anisotropy fields in magnetization dynamic measurements may therefore lead to the significant misinterpretation of experimental results for YIG thin films. Fig. 4 . 4(a) VSM hysteresis loops measured in the film plane for three different crystallographic directions. The magnetization is normalized to the saturation magnetization = 120 ± 19 / 3 . A paramagnetic contribution of the GGG substrate was subtracted for each loop. (b) Resonance frequency as a function of azimuthal angle taken at = 150 . The red line depicts the calculated values of resonance frequency according to Eq. Fig. 5 . 5Spin wave group velocity as a function of the external magnetic field. The red line represents a fit according to Eq. 4. Fig. 6 . 6Excitation spectrum of the CPW with 9.8 wide signal lines and 4 gaps. The inset shows in-plane component of the dynamic magnetic field excited by the CPW. Fig. 7. Wavenumber of the highest amplitude as a function of CPW signal line width. The solid lines represent a fit according to Eq. 6.10 ] ( ≈ /2 ). 0 4000 8000 12000 0.0 0.2 0.4 0.6 0.8 1.0 -50 -25 0 25 50 -25 0 25 50 h x [Oe] x [m] -50 -25 0 25 50 -25 0 25 50 h x [Oe] x [m] k 2 = 6770 cm -1 Amplitude [a.u.] k [cm -1 ] -50 -25 0 25 50 -25 0 25 50 h x [Oe] x [m] k maxAmp = 1838 cm -1 Acknowledgements Spin wave nonreciprocity for logic device applications. M Jamali, J H Kwon, S-M Seo, K-J Lee, Yang H , Sci. Rep. 33160Jamali M, Kwon J H, Seo S-M, Lee K-J and Yang H 2016 Spin wave nonreciprocity for logic device applications. Sci. Rep. 3, 3160 Magnon transistor for all-magnon data processing. A V Chumak, A Serga, B Hillebrands, Nat. Comun. 54700Chumak A V, Serga A A and Hillebrands B 2014 Magnon transistor for all-magnon data processing. Nat. Comun. 5, 4700 Realization of a spin-wave multiplexer. K Vogt, F Y Fradin, J E Pearson, T Sebastian, S D Bader, B Hillebrands, A Hoffmann, H Schultheiss, Nat. 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B. 91104421Bessonov V D, Mruczkiewicz M, Gieniusz G, Guzowska U, Maziewski A, Stognij A I, Krawczyk M 2015 Magnonic band gaps in YIG-based one-dimensional magnonic crystals: An array of grooves versus an array of metallic stripes Phys. Rev. B 91, 104421 Spin-wave logical gates. M P Kostylev, A A Serga, T Schneider, Leven, B Hillebrands, Appl. Phys. Lett. 87153501Kostylev M P, Serga A A, Schneider T, Leven, B and Hillebrands B 2005 Spin-wave logical gates Appl. Phys. Lett. 87, 153501 All-Electrical Measurement of Interfacial Dzyaloshinskii-Moriya Interaction Using Collective Spin-Wave. J M Lee, C Jang, B C Min, S W Lee, K J Lee, J Chang, Dynamics Nano Lett. 16Lee J M, Jang C, Min B C, Lee S W, Lee K J, Chang J 2016 All-Electrical Measurement of Interfacial Dzyaloshinskii-Moriya Interaction Using Collective Spin- Wave Dynamics Nano Lett. 16, 62-67 Propagating spin wave spectroscopy in a permalloy film: A quantitative analysis. M Bailleul, D Olligs, C Fermon, Appl. Phys. Lett. 83972Bailleul M, Olligs D and Fermon C 2003 Propagating spin wave spectroscopy in a permalloy film: A quantitative analysis. Appl. Phys. Lett. 83, 972 Anisotropic Propagation and Damping of SpinWaves in a Nanopatterned Antidot Lattice. S Neusser, G Duerr, H G Bauer, S Tacchi, M Madami, G Woltersdorf, G Gubbiotti, C H Back, D Grundler, Phys. Rev. Lett. 10567208Neusser S, Duerr G, Bauer H G, Tacchi S, Madami M, Woltersdorf G, Gubbiotti G, Back C H, Grundler D 2010 Anisotropic Propagation and Damping of SpinWaves in a Nanopatterned Antidot Lattice. Phys. Rev. Lett. 105, 067208 Characterization of magnetostatic surface spin waves in magnetic thin films: evaluation for microelectronic applications. J H Kwon, S S Mukherjee, P Deorani, M Hayashi, H Yang, Appl. Phys. A. 111Kwon J H, Mukherjee S S, Deorani P, Hayashi M and Yang H, Characterization of magnetostatic surface spin waves in magnetic thin films: evaluation for microelectronic applications. Appl. Phys. A 111, 369-378 Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics. H Yu, O Helly, V Cros, R Bernard, P Bartolotti, A Anane, F Brandl, R Huber, I Stasinopoulos, D Grundler, Sci. Rep. 46848Yu H, Helly O, Cros V, Bernard R, Bartolotti P, Anane A, Brandl F, Huber R, Stasinopoulos I, Grundler D 2014 Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics. Sci. Rep. 4, 6848 Large spin pumping from epitaxial Y3Fe5O12 thin films to Pt and W layers. H L Wang, C H Du, Y Pu, R Adur, P C Hammel, F Yang, Phys. Rev. B. 88100406Wang H L, Du C H, Pu Y, Adur R, Hammel P C, Yang F Y 2013 Large spin pumping from epitaxial Y3Fe5O12 thin films to Pt and W layers. Phys. Rev. B 88, 100406 Pulsed laser deposited Y3Fe5O12 films: Nature of magnetic anisotropy. S A Manuilov, S Khartsev, A M Grishin, J. Appl. Phys. 106123917Manuilov S A, Khartsev S I and Grishin A M 2009 Pulsed laser deposited Y3Fe5O12 films: Nature of magnetic anisotropy J. Appl. Phys. 106, 123917 W-C Chiang, M Y Chern, J G Lin, C Y Huang, FMR studies of. Chiang W-C, Chern M Y, Lin J G and Huang C Y 2002 FMR studies of YIG/GGG) superlattices and YIG thin films. Y3fe5o12/Gd3ga5o12, J. Magn. Magn. Mater. 239332Y3Fe5O12/Gd3Ga5O12 (YIG/GGG) superlattices and YIG thin films J. Magn. Magn. Mater. 239, 332 The dipole-exchange spin wave spectrum for anisotropic ferromagnetic films with mixed exchange boundary conditions. B A Kalinikos, M P Kostylev, N Kozhus, A N Slavin, J. Phys.: Condens. Matter. 2Kalinikos B A, Kostylev M P, Kozhus N V and Slavin A N 1990 The dipole-exchange spin wave spectrum for anisotropic ferromagnetic films with mixed exchange boundary conditions J. Phys.: Condens. Matter 2, 9861-9877 Spin waves and magnetic anisotropy in ultrathin (111)-oriented cubic films. G Gubbiotti, G Carlotti, B Hillebrands, J. Phys.: Condens. Matter. 102171Gubbiotti G, Carlotti G and Hillebrands B 1998 Spin waves and magnetic anisotropy in ultrathin (111)-oriented cubic films J. Phys.: Condens. Matter 10, 2171 Enhanced Magneto-optic Kerr Effect and Magnetic Properties of CeY2Fe5O12 Epitaxial Thin Films. A Kehlberger, K Richter, M C Onbasli, Jakob G Kim, D H Goto, T Ross, C A Götz, G Reiss, G Kushel, T Kläui, M , Phys. Rev. Appl. 414008Kehlberger A, Richter K, Onbasli M C, Jakob G, Kim D H, Goto T, Ross C A, Götz G, Reiss G, Kushel T, Kläui M 2015 Enhanced Magneto-optic Kerr Effect and Magnetic Properties of CeY2Fe5O12 Epitaxial Thin Films Phys. Rev. Appl. 4 014008 Spin-wave transduction at the submicrometer scale: Experiment and modeling. V Vlaminck, M Bailleul, Phys. Rev. B. 8114425Vlaminck V and Bailleul M 2010 Spin-wave transduction at the submicrometer scale: Experiment and modeling Phys. Rev. B 81, 014425 Surface Magnetostatic Modes and Surface Spin Waves. J Eshbach, R W Damon, Phys. Rev. B. 1181208Eshbach J R and Damon R W 1960 Surface Magnetostatic Modes and Surface Spin Waves. Phys. Rev. B 118, 1208 Spin Waves in 2D and 3D Magnonic Crystals: From Nanostructured Ferromagnetic Materials to Chiral Helimagnets. T Schwarze, PhD ThesisSchwarze T 2013 Spin Waves in 2D and 3D Magnonic Crystals: From Nanostructured Ferromagnetic Materials to Chiral Helimagnets. PhD Thesis Pulsed laser deposited Y3Fe5O12 films: Nature of magnetic anisotropy II. S A Manuilov, A M Grishin, J. Appl. Phys. 10813902Manuilov S A and Grishin A M 2010 Pulsed laser deposited Y3Fe5O12 films: Nature of magnetic anisotropy II. J. Appl. Phys. 108, 013902 Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert damping and bulk-like magnetization. M C Onbasli, A Kehlberger, D H Kim, Jakob G Kläui, M Chumak, A V Hillebrands, B Ross, C , APL Mater. 2106102Onbasli M C, Kehlberger A, Kim D H, Jakob G, Kläui M, Chumak A V, Hillebrands B, Ross C A 2014 Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert damping and bulk-like magnetization. APL Mater. 2, 106102 Spin-wave propagation in ultra-thin YIG based waveguides. M Collet, O Gladii, M Evelt, V Bessonov, L Soumah, S O Demokritov, Y Henry, Y Cros, M Bailleul, V Demidov, A Anane, Appl. Phys. Lett. 11092408Collet M, Gladii O, Evelt M, Bessonov V, Soumah L, Demokritov S O, Henry Y, Cros Y, Bailleul M, Demidov V E and Anane A 2017 Spin-wave propagation in ultra-thin YIG based waveguides Appl. Phys. Lett. 110 092408 Optical and magneto-optical behavior of Cerium Yttrium Iron Garnet thin films at wavelengths of 200-1770 nm. M C Onbasli, L Beran, M Zahradník, M Kučera, R Antoš, J Mistrík, G F Dionne, M Veis, C A Ross, Sci. Rep. 615Onbasli M C, Beran L, Zahradník M, Kučera M, Antoš R, Mistrík J, Dionne G F, Veis M and Ross C A 2016 Optical and magneto-optical behavior of Cerium Yttrium Iron Garnet thin films at wavelengths of 200-1770 nm Sci. Rep. 6 23640 15 Ultrafast Magneto-Optical and Magnetization-Induced Second Harmonic Generation Techniques for Studies of Magnetic Nanostructures. A Bonda, S Uba, L Uba, Acta Phys. Polon. A. 1211225Bonda A, Uba S and Uba L 2012 Ultrafast Magneto-Optical and Magnetization- Induced Second Harmonic Generation Techniques for Studies of Magnetic Nanostructures. Acta Phys. Polon. A 121, 1225 C Dubs, O Surzhenko, R Linke, A Danilewsky, U Bruckner, J Dellith, arXiv:1608.08043v1Submicrometer yttrium iron garnet LPE resonance losses. Dubs C, Surzhenko O, Linke R, Danilewsky A, Bruckner U, Dellith J 2016 Sub- micrometer yttrium iron garnet LPE resonance losses. arXiv:1608.08043v1 Observation of the propagation and interference of spin waves in ferromagnetic thin films. K Perzlmaier, G Woltersdorf, C H Back, Phys. Rev. B. 7754425Perzlmaier K, Woltersdorf G and Back C H 2008 Observation of the propagation and interference of spin waves in ferromagnetic thin films Phys. Rev. B 77, 054425

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{'abstract': 'We report on long-range spin wave (SW) propagation in nanometer-thick Yttrium Iron Garnet (YIG) film with an ultralow Gilbert damping. The knowledge of a wavenumber value | ⃗ | is essential for designing SW devices. Although determining the wavenumber | ⃗ | in experiments like Brillouin light scattering spectroscopy is straightforward, quantifying the wavenumber in all-electrical experiments has not been widely commented so far.We analyze magnetostatic spin wave (SW) propagation in YIG films in order to determine SW wavenumber | ⃗ | excited by the coplanar waveguide. We show that it is crucial to consider influence of magnetic anisotropy fields present in YIG thin films for precise determination of SW wavenumber. With the proposed methods we find that experimentally derived values of | ⃗ | are in perfect agreement with that obtained from electromagnetic simulation only if anisotropy fields are included.', 'arxivid': '1902.04608', 'author': ['A Krysztofik \nInstitute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland\n', 'H Głowiński \nInstitute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland\n', 'P Kuświk \nInstitute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland\n\nCentre of Advanced Technology\nAdam Mickiewicz University\nUmultowska 89cPL-61-614PoznańPoland\n', 'S Ziętek \nDepartment of Electronics\nAGH University of Science and Technology\nAl. Mickiewicza 40PL-30-059KrakówPoland\n', 'L E Coy \nNanoBioMedical Centre\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland\n', 'J N Rychły \nFaculty of Physics\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland\n', 'S Jurga \nNanoBioMedical Centre\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland\n', 'T W Stobiecki \nDepartment of Electronics\nAGH University of Science and Technology\nAl. Mickiewicza 40PL-30-059KrakówPoland\n', 'J Dubowik \nInstitute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland\n', '\nFaculty of Physics and Applied Computer Science\nAGH University of Science and Technology\nAl. Mickiewicza 30PL-30-059KrakówPoland\n'], 'authoraffiliation': ['Institute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland', 'Institute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland', 'Institute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland', 'Centre of Advanced Technology\nAdam Mickiewicz University\nUmultowska 89cPL-61-614PoznańPoland', 'Department of Electronics\nAGH University of Science and Technology\nAl. Mickiewicza 40PL-30-059KrakówPoland', 'NanoBioMedical Centre\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland', 'Faculty of Physics\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland', 'NanoBioMedical Centre\nAdam Mickiewicz University\nUmultowska 85PL-61-614PoznańPoland', 'Department of Electronics\nAGH University of Science and Technology\nAl. Mickiewicza 40PL-30-059KrakówPoland', 'Institute of Molecular Physics\nPolish Academy of Sciences\nM. Smoluchowskiego 17PL-60-179PoznańPoland', 'Faculty of Physics and Applied Computer Science\nAGH University of Science and Technology\nAl. Mickiewicza 30PL-30-059KrakówPoland'], 'corpusid': 119329965, 'doi': '10.1088/1361-6463/aa6df0', 'github_urls': [], 'n_tokens_mistral': 9117, 'n_tokens_neox': 7555, 'n_words': 4438, 'pdfsha': '8445cb8b433ac0aa9816f4dc6c15658ce08d5105', 'pdfurls': ['https://arxiv.org/pdf/1902.04608v1.pdf'], 'title': ['Characterization of spin wave propagation in (111) YIG thin films with large anisotropy', 'Characterization of spin wave propagation in (111) YIG thin films with large anisotropy'], 'venue': []}

arxiv

A Frequency Domain Approach to Predict Power System Transients Wenqi Cui Weiwei Yang Baosen Zhang A Frequency Domain Approach to Predict Power System Transients The dynamics of power grids are governed by a large number of nonlinear differential and algebraic equations (DAEs). To safely operate the system, operators need to check that the states described by these DAEs stay within prescribed limits after various potential faults. However, current numerical solvers of DAEs are often too slow for real-time system operations. In addition, detailed system parameters are often not exactly known. Machine learning approaches have been proposed to reduce the computational efforts, but existing methods generally suffer from overfitting and failures to predict unstable behaviors.This paper proposes a novel framework to predict power system transients by learning in the frequency domain. The intuition is that although the system behavior is complex in the time domain, there are relatively few dominate modes in the frequency domain. Therefore, we learn to predict by constructing neural networks with Fourier transform and filtering layers. System topology and fault information are encoded by taking a multi-dimensional Fourier transform, allowing us to leverage the fact that the trajectories are sparse both in time and spatial frequencies. We show that the proposed approach does not need detailed system parameters, greatly speeds up prediction computations and is highly accurate for different fault types. arXiv:2111.01103v3 [eess.SY] 31 Jan 2023 I. INTRODUCTION Increasing the amount of renewable resources integrated in the electric grid is fundamental to reducing carbon emissions and mitigating climate change. Many governments and companies have set ambitious goals to generate their electricity with close to 100% renewables by 2050 [1]. So far, much of the attention has been paid to increasing the aggregate generation capacities. However, increased renewable generation capacities also lead to challenging problems in dynamic stability of the grid [2]. A grid can be thought as a large interconnected system of generators, loads and power electronic components, governed by nonlinear differential and algebraic equations (DAEs) [3]. This system also undergos constant disturbances, from load changes to line outages [4]. The predominant goal of power system operations is to make sure that the system stays within acceptable limits under these disturbances. The system states are governed by different DAEs during the transient across pre-fault, fault-on and post-fault stages [5]. Fig. 1 shows a tripped line in Florida that led to rolling blackouts that impacted the lower two-thirds of state [6]. Ideally, a system should withstand these types of single events without performance degradation (N − 1 security) [4], but this was not the case in Fig. 1. A fundamental reason is that solving the governing DAEs is extremely computationally challenging and not all contingencies can be checked. These DAEs are W. Cui highly nonlinear and numerical methods (e.g., implicit and explicit integration) are used to solve them [7]. To study transient stability, numerical algorithms need to use fairly fine discretizations. Even for a moderately sized system, existing numerical solvers may take minutes to simulate only seconds of system trajectories [8]. Consequently, only a limited number of scenarios are studied offline and operators tend to restrict the system to operate close to these scenarios. As shown in Fig. 2, for certain given conditions of power generation, demand and topology, system operators simulate the system trajectories after distrubances through solving the shifted DAEs. By studying scenarios of system conditions offline, system operators will have a checklist about what action needs to be taken in real time to ensure that the systems' states are within the permissible range after critical contingencies. Since renewables have much larger uncertainties than conventional resources, operators often curtail them to artificially limit their generation to avoid operating at "unknown" regions [3]. For example, some European grids are not allowed to operate at above 40% wind, no matter how much wind is actually blowing [9]. Therefore, fast and accurate dynamic simulations would greatly increase the actual utilization of renewables in the system. Recently, machine learning (ML) approaches have been proposed for dynamical simulation instead of DAE solvers and can reduce the computation time by orders of magnitude. Given the fault information and present measurement of states, ML methods learn feed-forward neural networks to predict the future trajectories. Most works focus on binary classification to identify whether a system is stable. Long short-term memory based recurrent neural networks are commonly utilized to process sequential data [11], [12]. A shaplet learning approach is proposed to extract spatial-temporal correlations [13]. Convolutional neural networks are adopted in [14], [15] to process data from multiple sources. However, a binary prediction Fig. 2. Power system operators conduct dynamic power system transient prediction by solving DAEs [5], [10], where x and y represent the set of state variables and algebraic variables, respectively. The dynamics are described by differential equations f (·) and algebraic equations h(·), with subscript F and P F indicates the fault-on and post-fault period divided by the fault time t f and the fault-clear time t cf , respectively. Under certain conditions of power generation, demand and topology, the system trajectory after distrubances is simulated through solving the shifted DAEs. may not provide sufficient information for real-time decision making. For example, to prevent cascading outages, operators need to know the magnitude of the states and the actual trajectories [5]. And binary predictions are too coarse for making these types of assessments. To alleviate the limitation of binary prediction, some recent works provide finer trajectory prediction by learning the time domain solutions to the governing DAEs. Polynomial basis are used in [16] to approximate the solutions, but the number of basis functions grows exponentially with the system size. Extreme learning machine is utilized in [17] for online voltage stability margin prediction, but the traces of the system variables are not provided. Deep neural networks are used in [18], [19] to directly learn to predict the future trajectory from past and current measurements. However, since power systems are large and sampled sparsely in time, direct regression on time-domain data does not perform very well. A rolling prediction is used in [18] to improve accuracy, but with a high computational cost. Instead of the timedomain approach, eigenstates of linear swing equations after eigenspace transformation are utilized in [20] to infer system dynamics. However, the assumptions on the linearized system model and uniform damping ratios may not hold for realistic and large-scale systems. In addition, since the majority of trajectories used in training is stable, the learned networks fail predict unstable behaviors. Physics informed neural networks that directly attempt to solve the DAEs have been proposed as an alternative [21], but it does not currently scale beyond small networks. In this paper, we propose a novel framework for predicting power system transients by learning and making predictions in the frequency domain, which provides a computation speed up of more than 400 times compared to existing power system tools. This approach follows the intuition that the system tend to undergo oscillations that have a few dominate temporalspatial modes. We adopt and extend the structure of Fourier Neural Operator in [22] to learn in the frequency domain and recover the time domain trajectories through the inverse Fourier transform. Specifically, we design the dataframe to encode the power system topology and fault information, which lead to a 3D Fourier transform. This method is able to make smooth and accurate predictions, capturing both stable and unstable behaviors without the need to manually tune the training data. It improves the MSE prediction error by more than 70% compared with state-of-the-art AI methods, and vastly improves the detection of unstable behavior. Code and data are available at https://github.com/Wenqi-Cui/Predict-Power-System-Dynamics-Frequency-Domain. In summary, the main contributions of the paper are: 1) We propose a novel machine learning approach to predict transient dynamics in the frequency domain, which can accurately predict state trajectories based on a few measurements. 2) We develop a dataframe that encodes spatial-temporal information about the system topology, which greatly reduces the computational complexity in multi-dimensional Fourier transforms. 3) The time-varying active/reactive power injection and fault-on/clear actions are incorporated in the proposed framework, enabling the prediction of the transients subject to different net power injections and actions. The remaining of this paper is organized as follows. Section II introduces the problem formulation for predicting power system dynamics and transients. Section III provides the setup and intuition of learning in the frequency domain. Section IV shows the proposed framework for dynamic transient prediction and Section V illustrates the construction of dataframe to encode spatial-temporal relationships. Section VI shows the simulation results. Section VII concludes the paper. II. MODEL AND PROBLEM FORMULATION A. Power System Swing Equations The dynamics of power systems depend on the interactions of a myriad of components including governors, exciters, stabilizers, etc., as illustrated in Fig. 3 [4]. Let x ∈ R n , y ∈ R m , a ∈ R d be all the state variables, algebraic variables and external input variables, respectively. 1 The complete power system model for calculating system dynamic response relative to a disturbance can be described by a set of DAEs as follows [5], [16]: ẋ = f (x, y, a) 0 = h(x, y, a)(1) where the differential equation f : R n × R m × R d → R n typically describes the internal dynamics of devices such as the speed and angle of generator rotors, the response of generator control systems (e.g., excitation systems, turbines, governors), the dynamics of equipment including DC lines, dynamically modeled loads and their control systems. Correspondingly, x ∈ R n is the state variables such as generator rotor angles, generator velocity deviations (speeds), electromagnetic flux, various control system internal variables, etc. The set of algebraic equations h : R n × R m × R d → R m describes the electrical transmission system and interface equations. Correspondingly, y ∈ R m is the algebraic variable such as voltage magnitude and angles. The external input variables a ∈ R d acting on the equations are power injection from generators, automatic generation control systems, fault-response actions and so on [5], [23]. In this paper, we mainly consider the uncertainties in power generation and demand, as well as the fault-response actions u. Let the active and reactive net power injection be p and q, then the external input variables is sometimes written as the tuple a = (p, q, u). Under changing generator and load conditions, power systems are operated to withstand the occurrence of certain contingencies. To ensure that cascading outages will not occur for the set of critical disturbances, the state variables need to stay within permissible ranges during the transient process of power system after disturbances [5], [24]. For even a moderate power system with tens or hundreds of buses, it may be governed by hundreds or thousands of DAEs. B. Transient Dynamics Disturbances lead to deviations in the states x and variables y through a step change of parameters in (1). For example, a short circuit on a transmission line will results in a sudden change of the susceptance and conductance in set of algebraic equations, depending on the specific fault types (e.g., single-phase-to-ground, two-phase-to-ground, lineto-line, three-phase-to-ground, etc.). Suppose a fault happens at the time t f and is cleared at the time t cl . 2 The pre-fault stage is defined as the period before the fault happens at t f . The system evolves from the initial state x(t 0 ) as: ẋ = f (x, y, a; x(t 0 )) 0 = h(x, y, a) , t 0 ≤ t < t f(2) The sudden parameter changes after disturbances will lead to a shift of the swing equations. The fault-on system evolves with k subsequent actions from system relays and circuit breaks. Suppose the j-th action is taken at t F,j , the fault-on system is described by several set of equations [5] 2 The disturbance such as a short circuit on a transmission line is automatically cleared by protective relay operation after a certain amount of time. ẋ = f F,1 (x, y, a; x(t f )) 0 = h F,1 (x, y, a) , t f ≤ t < t F,1 . . . ẋ = f F,k (x, y, a; x(t F,k )) 0 = h F,k (x, y, a) , t F,k−1 ≤ t < t cl(3) The post-fault stage refers to the system after the fault is cleared. The post-fault system evolves with the differential equation starting from the post-fault initial state x(t cl ), written as [5] ẋ = f P F (x, y, a; x(t cl )) 0 = h P F (x, y, a) , t ≥ t cl .(4) Despite of large numbers of variables x and y, not all of them are observable to system operators. δ := (δ i , i ∈ [N ]) ∈ R N , ω := (ω i , i ∈ [N ]) ∈ R N , v := (v i , i ∈ [N ]) ∈ R N . These variables of interest are described by a three-tuple, denoted by s = (δ, ω, v) ∈ R 3N . Note that other variables can also be included in s if they are observable. The key to safe dynamic operation of power system is to predict the future of the system trajectories, given the fault information, some observations s and the expected clearing actions u. Based on these trajectories, interim actions like load shedding or emergency generation can be taken to reduce the impact of the faults [4], [7]. C. Current Approaches and Challenges Current approaches in power system dynamic prediction is based on solving (2)-(4), which are highly nonlinear equations. System operators typically rely on numerical integration, such as Runge-Kutta (RK) methods or trapezoidal rule, to iteratively approximate the solution of (2)-(4) in small time intervals [7]. However, because of the highly nonlinear nature of the DAEs, very fine discretization steps are required for these numerical methods. As a result, these approaches may be too slow for real-time decision-making. Some solvers use reduced-order model and convert DAEs to ordinary differential equations (ODEs) to simulate the dynamic response of generators [25]. For a moderately sized system, existing numerical solvers take minutes to simulate only seconds of system trajectories [8]. As an alternative, system operators also use manual heuristics to take actions, but this strategy is becoming less tenable as renewables introduce distinctly different operating scenarios. III. LEARNING FOR DYNAMIC TRANSIENT PREDICTIONS A. Problem Setup Learning-based approaches try to find a mapping from present measurements to future trajectories. The predictions are then obtained through function evaluations, which significantly reduces the computational time compared to the conventional numerical approaches. The problem we are interested in is to predict the trajectory of the states s starting at the time stamp t on for τ out number of time steps with the sampling interval ∆t, as illustrated in Fig. 4. The input are τ in observations of the states from s(t on −τ in ) to s(t on − 1), the external inputs from a(t on − τ in ) to a(t on − 1), and the expected fault-clear actions from u(t on ) to u(t on + τ out − 1). We write s in = (s(t on − τ in ), · · · , s(t on − 1)), a = (a(t on − τ in ), · · · , a(t on − 1)), u out = (u(t on ), · · · , u(t on + τ out − 1)) and s out = (s(t on ), · · · , s(t on + τ out − 1)). Our goal is to find a mapping G from the space of input (s in , a, u out ) to output trajectories s out . In this paper we consider the mapping realized through (deep) neural networks with parameters Φ. The prediction is then given bŷ s out = G Φ (s in , a, u out ).(5) The exact form of Φ depends on the structure of neural network used. In this paper, we adopt the Fourier neural operator to learn in the frequency domain, and details will be specified later in Section IV. Let H be the batch size andŝ i out be the prediction of the i-th sample for i = 1, · · · , H. The weights of neural network Φ are updated by back-propagation to minimize loss function L(Φ) defined by the mean absolute percentage error (MAPE) between predicted trajectory and the actual trajectory L(Φ) = 1 H H i=1 ||ŝ i out −s i out ||1 ||s i out ||1 [22]. B. Current ML Approaches and Limitations Learning the power system transient dynamics is not trivial because states undergo nonlinear oscillations. Predictions with three existing approaches are illustrated in Fig. 5. The blue line is the trajectory of the frequency deviation on a bus before and after a fault. The blue squares are true trajectory sampled at discrete times and the grey area is the prediction horizon. A standard approach is to use a neural network to learn the time-domain mapping from the input to output. As illustrated in Fig. 5(a) and Fig. 5(b), purely learning in the time domain will easily overfit and cannot learn a smooth curve like the true trajectories. More importantly, generic machine learning approaches prone to false negative errors. Since the vast majority of historical trajectories are stable, a ML method tends to not predict unstable trajectories. This would lead to catastrophic consequences if the system operator does not take actions to mitigate instabilities. Similarly, fitting the nonlinear dynamics with polynomial basis will also easily lead to over-fitting, as illustrated in Fig. 5(c). Recently, Physics-Informed Neural Networks have been proposed in [26] to learn solutions that satisfy equations from implicit Runge-Kutta (RK) integration. This approach has been applied to power system swing dynamics in [21]. Since RK method is the weighted sum of ODE solutions in discretized intervals, its accuracy decreases sharply when predicting trajectories with large oscillations for a longer horizon (e.g., larger than 1 second), as illustrated in Fig. 5(d) (the prediction errors are larger than then limit of the y-axis). δ i = ω i (6a) M iωi = p i − D i ω i − N j=1,j =i B ij (δ i − δ j ) (6b) where i ∈ [N ] := {1, . . . , N } is the index of buses, M := diag(M i , i ∈ [N ]) ∈ R N ×N are the generator inertia constants, D := diag(D i , i ∈ [N ]) ∈ R N ×N are damping coefficients, p := (p i , i ∈ [N ]) ∈ R N are the net power injections, B := [B ij ] ∈ R N ×N is the susceptance matrix. With the assumption that the inertia and damping of the buses are proportional to its power ratings (i.e, D i /M i = γ for all i ∈ [N ]), (6) can be explictly solved [27]. Let C be the incidence matrix. The (scaled) graph Laplacian matrix is Γ = M −1/2 CBC M −1/2 , where 0 = λ 1 < λ 2 ≤ · · · ≤ λ n are the eigenvalues with corresponding orthonormal eigenvectors r 1 , r 2 , . . . , r n . Suppose there is a step change ∆p in the net power injection, and its decomposition along the eigenvectors is ∆p = ip i M 1/2 r i . Then, the solution of equations (6) is [27] ω(t) = n i=1p i γ 2 − 4λ i e φi,+t − e φi,−t M −1/2 r i (7) where φ i,+ := −γ+ √ γ 2 −4λi 2 φ i,− := −γ− √ γ 2 −4λi 2 . Note that φ i,+ are complex numbers with non-zero imaginary part if γ 2 − 4λ i < 0, which results in sinusoidal oscillations. Thus, the sinusoidal basis in Fourier transform (and inverse Fourier transform) is a natural fit for power system transient dynamics. Because of the finite set of eigenvalues, there is a finite number of modes in the frequency domain. As a result, the trajectories are sparse in the frequency domain, making it easier to learn after Fourier transform. However, the analysis based on the linear model (6) cannot be applied to more realistic systems as illustrated in Fig. 3, where high-order nonlinear differential equations are involved. This is the reason why we need learning to predict the transient dynamics. In the next sections, we will show the framework of learning in the frequency domain and conduct numerical verification using full-order model for system dynamics. IV. LEARNING IN THE FREQUENCY DOMAIN A. Structure of the Neural Network We construct the structure of neural network shown in Fig. 6, which consists of several Fourier layers for learning in the frequency domain. The input trajectory is first passed through an encoder to integrate the spatial-temporal relationships and the fault information. The encoded data is then passed through l level of Fourier Layers [22], where the input of the j-th layer is g j and the output is g j+1 for j = 1, · · · , l. Each Fourier layer consists of one path with trainable weights θ j that learns periodic components in the frequency domain, and another path with trainable weights W j that directly operate on the time-domain data. Intuitively, the second path (often called the pass-through layer in Machine Learning literature) with weights W j helps to keep the track of aperiodic and high-frequency component. In the following, we illustrate the structure of each Fourier Layer. The dimension for each tensor will be specified later after we elaborate the encoder in Section V. Fig. 6. The structure of neural networks for power system transient prediction using frequency-domain information. The input trajectory is first encoded using the framework introduced later in Section V. The encoded data is then passed through several Fourier Layers [22], where the input and output of the j-th layer is g j and g j+1 , respectively. The j-th layer consists of trainable weights θ j for learning in the Fourier domain, and trainable weights W j to keep the track of aperiodic and distorted waveform. B. Fourier Layer For the input of each layer, we conduct discrete Fourier transform F to convert the input trajectory into the frequency domain [28]. Inspired by the work in [22], we use neural networks parameterized by θ j to learn in the frequency in each layer j, and then recover the time-domain sequences by inverse Fourier transform F −1 . This process is defined as Fourier neural operator K θj (·) represented by K θj (g j ) = F −1 (θ j · ψ (Fg j )) ,(8) where the function ψ(·) is a low-pass filter that truncates the Fourier series at a maximum number of modes k max for efficient computation [22]. Then θ j is the weight tensor that conducts linear combination of the modes in the frequency domain. The output of the j-th layer adds up Fourier neural operator with the initial time-domain sequence weighted by W j to recover aperiodic and high-frequency components g j+1 = σ W j g j + K θj (g j ) ,(9) where σ is a nonlinear activation function whose action is defined component-wise. We use ReLU in this paper. Even though we limit at most k max Fourier modes after the low-pass filter ψ after the Fourier transform, the linear transform W j maintains high-frequency modes. The cut-off frequency k max of the low-pass filter is a tradeoff between the number of frequency component that are kept versus the computational complexity. If too few frequencies are selected, there is not enough information in the frequency domain to learn well. If too many frequencies are selected, then we need to learn a high dimensional set of weights, negating the benefit of learning in the frequency domain. The trade-off we adopted is to keep a small number of modes in the frequency domain and pass them through nonlinear layers, while still using a direct path as shown in Fig. 6. Intuitively, this means that the low frequency modes should be learned in frequency domain, while the higher frequency modes can be directly handled using the time domain signal. The exact value of k max involves some trial and error. We will show the numerical study about the effect of k max in Section VI. C. Multi-Dimensional Fourier Transforms The above approach of learning the weights in the frequency domain and recovering the trajectory with inverse Fourier transform provides the advantage in fitting oscillatory functions, by learning smooth curvatures and avoiding over-fitting. For a system with N buses, there are 3N state variables we are interested: the voltage, angle and frequency at each bus. However, conducting Fourier transform with 3N dimensions is time consuming even for moderatedly sized power systems. Another choice is to neglect the dependence and purely conduct 1D Fourier transform on the time dimension. However, this will degrade the prediction performance since the networked structure is an important cause of the oscillations. Moreover, the time varying parameters and the fault-clear actions should be considered as well. To overcome these challenges, we design a novel dataframe that encodes time-varying parameters, fault information and spatial-temporal relationships in transient dynamics in the next section. V. ENCODING SPATIAL-TEMPORAL RELATIONSHIPS A. Spatial-Temporal Relationship in Transient Dynamics We construct 3D tensors to encode the input trajectories such that the spatial-temporal relationships in the power system can be included. In addition, computation complexities of Fourier transforms are also reduced. The proposed framework is shown in Fig. 7 (a). Each data point is indexed by three dimensions and written as (g 0 ) x,y,zin . The x axis indexes the buses from 0 to N − 1. The y axis indexes different state variables for each bus with y = 0, 1, 2 stands for δ x , ω x , V x , respectively. The z in axis is for input time interval and z in = t on − τ in , · · · , t on − 1. For example, (g 0 ) 5,1,10 is the frequency at bus 5 at the time step 10, and (g 0 ) 6,2,10 is the voltage at bus 6 at the time step 10. We would like to highlight that the spatial information in this manuscript can be understood as the inherent spatial similarity of signals. For example, it is well known that the angle speed of generators tend to appear as groups that swing in similar patterns [29]. This indicates the correlation along the axis of bus indexes. Moreover, the angle, angle speed and voltage also typically oscillate at similar frequency. This indicates the correlation along the axis of the type of signals. The Fourier transform is most commonly defined along the axis of time, namely, the 1D Fourier transform along the time series. In this paper, we conduct 3D Fourier transform along the dimension of x, y and the time horizon. This way, the correlations along different buses, along type of signals and along the time horizon are extracted. B. Encoding On-Fault and Fault-Clear Information Importantly, we aim to predict the trajectories under changing net injections and faults. This is different from most previous works that learn a static mapping from input to output time sequences for fixed parameters. Therefore, we encode parameters and fault-clear actions explicitly in the input tensor as shown in Fig. 7 (a). Note that the fault-clear action may not known in advance, and we set up an expected relay time to predict the dynamic behaviors. The aim is to use the learningbased method to reproduce the results from solvers. That is, given the relay actions after the fault happens, the solvers can compute the trajectories after the fault. Likewise, we envision the relay actions as an extra input encoded in the data frame. The fault information is encoded in u 1 (t) and u 2 (t) , which are variables that contain the location of fault and the type of fault at the time step t, respectively. For example, suppose the fault at the time step z out + t on includes the trip of line 100 and the fault type is line-to-line fault. We encode the fault location as 100 and this fault type as 20. Then, u 1 (z out + t on ) = 100 and u 2 (z out + t on ) = 20, respectively. For the time step z out + t on between the line tripping and line relay, u 1 (z out + t on ) = 0 and u 2 (z out + t on ) = 0. If there is no fault happening at the time step z out + t on , then u 2 (z out + t on ) = 0 and u 2 (z out +t on ) = 0. Hence, the information of line tripping and relay is inherently included when we add u 1 (t) and u 2 (t) to the data frame for t = t on − τ in , · · · , t on + τ out − 1. Next, we show how to attach u and net injections to the data-frame. C. Encoding Time-Varying Parameters and Actions Time-varying parameters include the net active power injection p(t) = (p 1 (t), · · · , p N (t)) and reactive power q(t) = (q 1 (t), · · · , q N (t)). We stack them on the y axis on y = 3 and y = 4 as shown in blue part of Fig. 7(a). The benefit of this design is that the variance of p(t) and q(t) through time is naturally incorporated in the z in axis. The fault information u 1 (t) and u 2 (t) for t = t on − τ in , · · · , t on − 1 are stacked to the y-axis as y = 5 and y = 6, shown in red part of Fig. 7(a). Fault-clear actions may happen in the predicted time horizon [t on , t on + τ out − 1]. To incorporate future actions and temporal dependence in the prediction time steps, we expand the 3D input tensor in Fig. 7(b) along the output time sequence, with the new axis z out = 0, · · · , τ out − 1 correspond to the time stamp from t on to t on + τ out − 1. This is visualized in the green part of the Tensor in Fig. 7(b), where the axis of z in is attached with z out + t on , u 1 (z out + t on ) and u 2 (z out + t on ), respectively. Each data point in the 4D tensor is written as (g 1 ) zout,x,y,zin , where x indexes bus, y indexes the type of signals (e.g., y = 0, 1, 2 corresponds to the angle, angle speed, and voltage, respectively), and z out indexes the output time steps. The index z in = 0, · · · , t on − 1 corresponds to the time stamp of the input trajectory. The index z in = t on stands for the location of the fault, z in = t on + 1 stands for the type of the fault, and z in = t on + 2 stands for the time stamp, respectively. For example, suppose the fault at the time stamp z out + t on includes the trip of line 100 and the fault type is line-to-line fault. We encode this fault type as number 20 and the fault location as number 100. Then (g 1 ) zout,x,y,ton = 100, (g 1 ) zout,x,y,ton+1 = 20 and (g 1 ) zout,x,y,ton+2 = (z out + t on )∆t for all x and y. Namely, the fault type as number 20, the fault location as number 100, and the time (z out +t on )∆t are duplicated along the dimension of x and y. This way, fault-clearing actions and the output time stamps are encoded in the dataframe without adding extra complexity for batch operation. After such an encoder, the data frame is converted from a 3D tensor with dimension R N ×Y ×τin to a 4D tensor with dimension R τout×N ×Y ×(τin+3) . We train the neural network on a batch of trajectories and the number of trajectories is B. Hence, the full size of input data is with the dimension R B×τout×N ×Y ×(τin+3) . We use the Great Britain transmission network with 2224 nodes that will appear later in the case study to give an impression of the size of the data. For the time steps τ in = 20 and τ out = 150, the memory space for one input trajectory (B=1) in the Great Britain transmission network is 0.016GB, and the memory space for 200 input trajectory (B=200) in training is 3.2GB. The memory space for other sizes of systems will scale linearly with the number of buses. The faults of interest in the paper are mainly transmission line faults, and therefore we envision that the p and q during the prediction horizon of less then 10s will not deviate too much from the values in the input trajectory. Hence, the encoding of p and q in horizon of the input trajectory already provides enough information to guide a good prediction. The goal is to use neural networks to map the the input tensor to output tensor with dimension R τout×N ×3 for the predicted dynamics of δ, ω and V along τ out time steps for N buses. D. 3D Fourier Transform After the encoder, the Fourier transform in (8) is reduced to 3D Fourier transform along the axis of x, y and z out computed as (Fg j ) ξ1,ξ2,ξ3,zin = τout−1 zout=0 N −1 x=0 Y −1 y=0 e −2πi( z out ξ 1 τ out + xξ 2 N + yξ 3 Y ) · (g j ) zout,x,y,zin ,(10) where ξ 1 , ξ 2 and ξ 3 are modes in the frequency domain in the three dimensions after the discrete Fourier transform. Importantly, 3D FFT has been supported by most machine learning frameworks (e.g., Pytorch), which is computational efficient for both backward propagation in training and forward propagation in prediction. The structure of Fourier layer with the 3D Fourier Transform is visualized in Fig. 7(c). After truncating the Fourier series at a maximum number of modes k max,i for the ith dimension, an equivalent convolution in the frequency domain is conducted using dot-product with weights θ j ∈ R kmax,1×kmax,2×kmax,3×(τin+3)×(τin+3) defined by (θ j · ψ (Fg j )) ξ1,ξ2,ξ3,zin = τin+2 v=0 (θ j ) ξ1,ξ2,ξ3,zin,v (Fg j ) ξ1,ξ2,ξ3,v (11) for ξ 1 = 0, . . . , k max,1 − 1, ξ 2 = 0, . . . , k max,2 − 1, ξ 3 = 0, . . . , k max,3 − 1, z in = 0, . . . , τ in + 2 and j = 1, . . . , l. The time domain signal is recovered by inverse Fourier transform as follows: K θj (g j ) zout,x,y,zin = F −1 (θ j · ψ (Fg j )) zout,x,y,zin = kmax,1−1 ξ1=0 kmax,2−1 ξ2=0 kmax,3−1 ξ3=0 e 2πi( z out ξ 1 τ out + xξ 2 N + yξ 3 Y ) · (θ j · ψ (Fg j )) ξ1,ξ2,ξ3,zin ,(12) for z out = 0, . . . , τ out −1, x = 0, . . . , N −1, y = 0, . . . , Y −1, z in = 0, . . . , τ in + 2 and j = 1, . . . , l. Plugging (12) into (9) gives the output of the j-th layer g j+1 ∈ R τout×N ×Y ×(τin+3) . The predicted trajectoryŝ out ∈ R τout×N ×3 are obtained from the output of the last layer after a dense combination weight W l+1 ∈ R (τin+3)×1 . (ŝ out ) zout,x,y = τin+2 zin=0 (g l+1 ) zout,x,y,zin (W l+1 ) zin(13) for z out = 0, . . . , τ out − 1, x = 0, . . . , N − 1, y = 0, 1, 2. The index y = 0, 1, 2 corresponds to the prediction of angle, angle speed, and voltage, respectively. Typically, increased layer l enable the structure to learn more complex dynamic patterns. In simulation, we found that four layers are sufficient. In practice, the encoder may also conduct a linear combination on the 4D tensor along the dimension of z in to increase the representation capability of the neural networks [22]. In that case, the dimension of W l+1 needs to be adjusted accordingly and all the other computations still remain the same. E. Algorithm The pseudo-code for our proposed method is given in Algorithm 1. The variables to be trained are weights Φ = {θ, W } shown in Fig. 6. Adam algorithm is adopted to update weights in each episode. The main practical benefit is that the learned neural networks are feedforward functions, which can be evaluated orders-of-magnitude faster than conventional iterative solvers. To be clear, these neural networks are not replacement for conventional solvers. Rather, they can be used by system operators to study a much larger set of scenarios of how the system would behave under various types of disturbances. They would be valuable tools that would enable better characterization of the dynamic behavior of the systems, and complement existing tools such as high fidelity simulators. Algorithm 1 Training and Predicting Power System Transients Training: Learning rate α, batch size H, number of episodes I, dataset for training Initialisation :Initial weights Φ of the neural network 1: for episode = 1 to I do 2: Encode the input trajectory s i in and the output trajectory s i out for i = 1, · · · , H with fault information in the dataset 3: Using the current weights Φ of the neural network, compute the predicted trajectoryŝ i out for i = 1, · · · , H 4: Calculate total loss of all the batches Loss = 1 H H i=1 ||ŝ i out −s i out ||1 ||s i out ||1 . 5: Update weights in the neural network by passing Loss to Adam optimizer: Φ ← Φ − αAdam(Loss) 6: end for Predicting: Pre-trained weights Φ of the neural network 7: Encode the input trajectory s in with the setup of faultclear actions 8: Using the pre-trained weights Φ of the neural network, compute the predicted trajectoryŝ out VI. CASE STUDY In this section, we conduct several case studies to illustrate the effectiveness of the proposed method. We validate the performance of the proposed approach on a realistic power grid by the case studies with the Northeastern Power Coordinating Council (NPCC) 48-machine, 140-bus test system as shown in Fig. 8 [25], [30]. Last, we verify the performance on largescale power systems using the Great Britain (GB) transmission Fig. 8. Topology of Northeastern Power Coordinating Council (NPCC) 48machine, 140-bus power system [25], [30] network, which consists of 2224 nodes, 3207 branches and 394 generators [31]. In the appendix-A, a simple single-machine infinite bus system is used to show the benefit of learning in the frequency domain compared to the time domain. A. Simulation and Hyper-Parameter Setup We construct the network in Fig. 6 with four Fourier layers. We normalize the data of different physical meanings and thus eliminate the effect of the magnitude of features brought by different unit. The maximum number of modes in the frequency domain is set to be k max,1 = 6, k max,2 = 3 and k max,3 = 3. The episode number and the batch size are 4000 and 800, respectively. Weights of neural networks are updated using Adam with learning rate initializes at 0.02 and decays every 100 steps with a base of 0.85. We use Pytorch and a single Nvidia Tesla P100 GPU with 16GB memory. We use generic deep neural network (DNN) as a benchmark to compare the performance. The DNN has a dense structure and seven layers with ReLU activation, where the width of each layer is 20. The hyper-parameters of DNN are also tuned to achieve their best performances for different tasks. The episode number and the batch size are set the same as FNO. B. Performance on NPCC test system The performance of the proposed method on a practical power system is verified by simulations on Northeastern Power Coordinating Council (NPCC) test system, which represents the power grid of the northeastern United States and Canada and was involved in the 2003 blackout event [30]. The power system toolbox in MATLAB is used to generate dataset of power system transient dynamics with the full 6-order generator model, turbine-governing system and exciters [25]. The power system toolbox utilizes kron-reduced admittance matrix and simulate dynamics with equivalent ordinary differential equations [25]. The trajectories are generated considering the actions of protective relays in 4-20 cycles [32]. We create cases with stressed conditions (stable and unstable) by increasing the level of loads until the system is unstable. The cases with stressed conditions account for 15% in the training set. The input trajectories evolves τ in = 20 time steps, with time interval ∆t between neighbouring time step to be 0.03 (i.e., approximate two cycles that can be attained by most phasor measurement unit (PMU)). We predict the subsequent trajectories of the length τ out = 150 for total duration of 4.5s. The training time of 4000 episodes is 4424.33s. We quantify the prediction accuracy through the relative mean squared error (RMSE) defined as ||s out −ŝ out || 2 2 /||s out || 2 2 . Fig. 9 shows true (i.e., simulated) and predicted trajectories of the system after a three-phase line fault between bus 54 and bus 103 cleared at the time of 0.3s. Let the time of fault happens as the time t = 0. The prediction starts at t on = 0.06s, which means one time step in the input trajectories corresponds to the fault-on system. The grey area is the envelope of trajectories in all generator buses and the lines are the trajectories in ten generator buses. For all the three states variables (i.e., δ, ω and V ), the predicted trajectories in Fig. 9(b) has similar envelope as the accurate trajectories in Fig. 9(a). The RMSE for the prediction in Fig. 9 is 0.0041. The convergence of the envelope in frequency deviation ω to zeros indicates that the system is stable after the fault and its clear action. Moreover, both the magnitude and the periodic oscillations in the ten generator buses are all captured by the prediction with FNO for the on-fault and post-fault period. As illustrated in Fig.7, the type of fault and the fault-clear action at t = 0.3s is encoded in the input tensor of FNO. Correspondingly, Fig. 9(b) predicts a step increase of voltage at t = 0.3s, which is the same as Fig. 9(a). Notably, the magnitude of voltage at the bus 54 and bus 101 is below 0.8 p.u. before t = 1s, exceeding the permissible ranges of 5% from nominal. This may cause lowvoltage curtailment of the generators and warrant attention from system operators. Therefore, the proposed prediction can provide sufficient information for identifying how danger the system is. To illustrate the performance of the proposed method in predicting an unstable system, Fig. 10 shows true (i.e., simulated) and predicted trajectories of the system after a lineto-line fault between bus 75 and bus 124 and recovered at the time of 0.3s. Especially, it is a stressed unstable test case by gradually increasing the level of loads until the system is unstable. Although the magnitude of the trajectories are still bounded, the drifted angle and the collapse of voltage have reflected the unstable behaviors. The proposed method capture both the trend and oscillations of the unstable behaviors. The RMSE for the prediction in Fig. 10 is 0.1198. Moreover, the accuracy in terms of predicting unstable cases is to identify the unstable behaviors. In the next subsection, we verify in the test set that the proposed method can predict all the unstable system accurately shortly after a fault happens. The trajectory in the frequency domain (computed by Fast Fourier Transform) for the stable case in Fig. 9 and the unstable case in Fig. 10 is given in Fig. 11 and Fig. 12, respectively. The proposed method also achieves high accuracy in the frequency domain. Moreover, the high-frequency component in both Fig. 11 and Fig. 12 is almost zero. This provides the intuition why the low-pass filter can reduce computational complexity without affecting the prediction performances. Numerical studies about the effect of k max on the lowpass filter can be found in Appendix-B. The visualization of the predictions with different influence factors including fault-on/clear actions, fault type and fault location is shown in Appendix-C. C. Quantifying the Performance on NPCC As shown in Fig. 9 and Fig. 10, the dynamics of the power system transient states differ greatly with different fault types and system parameters. To quantify the performance of the proposed method in stochastic scenarios, we calculate the mean prediction error in the test set with 100 cases where initial states, location of fault, type of fault and fault-clearing time are randomly generated. Three metrics are included: 1) Relative mean squared error (RMSE) 2) Type1-error: Percentage of unstable cases predicted to be stable. This is the more severe type of error and may cause blackouts of power systems (instability is declared when the average value of ω from t = 4s to t = 4.5s exceed 0.5Hz). 3) Type2-error: Percentage of stable cases predicted to be unstable. Notably, we fix the length of the input trajectories s in to be τ in = 20. The input trajectories contain data before and after the fault (similar to a rolling window). As illustrated in Fig. 4, more data point after fault will be observed if the prediction starting point t on is larger (if we wait longer after the fault to do a prediction). The more steps after the fault in s in , the better the prediction performance. Table I summarize the metrics for the prediction error corresponding to different number of on-fault cycles (one cycle is 1/60=0.017s) involved in the input trajectories s in . From Table I, the RMSE of FNO is much lower than the case in DNN, respectively. Interestingly, DNN has the Type2error to be approximate zero while extremely high Type1error. The reason is that DNN will easily overfit since the majority (93%) of training samples. By contrast, FNO brings zero Type1-error and Type2-error once there is an on-fault data point entered in the input trajectories. Therefore, the proposed prediction with FNO will capture all the dangerous unstable case. The low RMSE indicates that the proposed method can also simulate the dynamics of trajectories accurately. Considering that unstable behaviors may not be observed in real measurements, we also investigate the performance of the proposed method where unstable behaviors are not present in the training set. Table II shows the comparison of RMSE on the dataset without unstable cases. The RMSE decreases greatly after eliminating the unstable cases. Of course, this is an easier problem, and the RMSE decreases greatly after eliminating the unstable cases. Lastly, we compare the average computational time in the test set for FNO and Power System Toolbox in MATLAB as shown in Table III. The execution time for 3D Fourier Transform and 3D inverse Fourier Transform in one layer is 7.79×10 −5 s and 1.21×10 −4 s, respectively. For the prediction time horizon ranges from 3s, 4.5, and 6s, the computational time of FNO is 0.0036s, 0.0037s, 0.0039s, respectively. By constrast, the computational time of MATLAB toolbox are 469, 605 and 867 times slower than FNO. Therefore, the proposed approach will significantly speed up the simulation for power system transient dynamics. D. Case study on the Great Britain transmission network We conduct case study on GB system to test the performance of the proposed method on large power networks. We use ANDES (an open source package for power system dynamic simulation) to generate dataset of power system transient dynamics with the full 6-order generator model, turbinegoverning system and exciters [31]. Differential-algebraic equations are solved for dynamic simulation [31]. The input trajectories evolves τ in = 20 time steps, with a sampling period of 1/30s (i.e., two cycles). We predict the subsequent trajectories of the length τ out = 150 steps, for a total duration of 5s. The number of trajectories we used to train the neural network is 200. The training time of 4000 episodes is 9256.96s. We believe that increasing the number of trajectories can further improve the performance of the learned neural networks, but our computing resources (a single Nvidia Tesla P100 GPU with 16GB memory) limit us to 200 trajectories. Despite of this, the following simulation result on the test set shows that the performance of the learned neural network is actually good enough. Let the fault happens as the time t = 0. The prediction starts at t on = 2/30s, which means one time step in the input trajectories corresponds to the on-fault system. Fig. 13 shows the prediction on a line fault (a three-phase-to-ground fault between bus 56 and bus 637 and recovered at the time of 0.13s) that is not covered in the training set. The grey area is the envelope of trajectories in all generator buses and the lines are the trajectories in ten generator buses. For all the three states variables (i.e., δ, ω and V ), the predicted trajectories in Fig. 13(b) have similar envelope as the simulated (i.e., groundtruth) trajectories in Fig. 13 (a). Moreover, both the magnitude and the periodic oscillations in the ten generator buses are all captured by the prediction with FNO for the on-fault and post-fault period. In 100 test cases where initial states, location of fault, and fault-clearing time are randomly generated, the mean value of RMSE for DNN is 0.0034. By contrast, the mean value of RMSE for FNO is 0.0001, which is 97% smaller than DNN. Hence, the proposed method achieves much higher accuracy compared with generic deep neural networks. The above experiments show that the neural network is not simply memorizing but generalizing as well [33], [34]. Since the power system is a synchronized and connected network, transients from different faults could be related. We conjecture that there should be some sparse pattern behind the transient dynamics of the system, and this is the reason why we can learn well with a moderate amount of data. These relationships may be hard to visualize or analytically characterize, which makes machine learning useful. Theoretical analysis of the phenomenon is an important future direction for us. This paper proposes a frequency domain approach for predicting power system transient dynamics. Inspired by the intuition that there are relatively few dominate modes in the frequency domain, we construct neural networks with Fourier transform and filtering layers. We design the dataframe to encode the power system topology and fault-on/clear information in transient dynamics, allowing the extraction of spatial-temporal relationship through 3D Fourier transform. Simulation results show that the proposed approach speeds up prediction computations by orders of magnitude and is highly accurate for different fault types. Compared with state-of-the art AI methods, the proposed method reduce MSE prediction error by more than 50% and vastly improves the detection of unstable dynamics. The simulation results point to an interesting observation that there are sparse patterns behind the transient dynamics of the system, and it is this sparsity that allows the neural networks to learn and predict. Making the theory rigorous is an important future direction for us. The RAM space of GPU resources constrains the amount of data that can be processed to train the neural networks. Investigating the parallel training on multiple GPU resources and the better usage of RAM space are also important future directions for the proposed method to be utilized in larger systems. A. A Single-Machine Infinite Bus System Example To visualize and compare the performance of different prediction approaches, we show an illustrative example on a generator connected to an infinite bus, modeling a connection to a large grid that appears as a voltage source [35]. The proposed method using Fourier Neural Operator (FNO) is compared with Physics-Informed Neural Networks (PINN) and DNN. The parameters for PINN is the same as [21] where the case study is also a single-machine infinite bus system. For the prediction with 4.5 seconds, the average relative mse on the test set for FNO, DNN, PINN are 0.0098, 0.1811, and 15.53, respectively. The extreme large mse for PINN reflects that it fails when a long prediction horizon is needed. Fig. 15 shows the dynamics of frequency deviation ω and angle δ in a prediction of 4.5 seconds for FNO and DNN. The ground truth is the trajectory found by numerical simulation. FNO fits the ground truth almost perfectly. By contrast, the learned dynamics from DNN are not smooth and show much larger deviations from the ground truth compared with FNO. This verifies the analysis illustrated in Fig. 5 that purely timedomain tends to overfit easily and have difficulties learning smooth oscillations. B. The influence of the low-pass filter To show the effect of the low pass filter, we fix k max,2 = 3, k max,3 = 3 and compare the performances of the trained neural networks by varying k max,1 to be 4, 6, 8 and 16. Note that the largest possible frequency is the Nyquist frequency where k max,1 = τ /2 for the trajectory of lentgh τ , so the cut-off frequency is much lower than the highest frequency component (where k max,1 = 75 for τ = 150 in this case study). The relative mean squared error (mse) on the test set, the average prediction time and the training time is shown in Table IV. Increasing k max,1 reduces relative mse slightly but greatly increases the prediction time and the training time. Especially, the case without the low-pass filter has the relative mse slightly lower than the case k max,1 = 6, but the prediction and training time is 1.35 and 2.12 times as long as the value of the case k max,1 = 6, respectively. The prediction of the angle speed ω for stressed stable and stressed unstable cases corresponding to the different setups of the low-pass filters are shown in Fig. 16. For all the setup of k max , the shape of aperiodic and distorted waveform are all captured. Since the improvement brought by increasing k max,1 is not large after k max,1 > 6 , we select k max,1 = 6 to report the results. C. Influence factors including fault-on/clear actions, fault type and fault location This subsection visualizes the predictions for the NPCC system with different influence factors. The base case is shown in Fig. 9, where a three-phase line fault between bus 54 and bus 103 is cleared at the time of 0.3s. We then alter faulton/clear actions, fault type and fault location. The detailed simulation results are given below. 1) Altering fault-on/ clear actions In this case, we alter the fault-clear action from the time of 0.3s to 0.15s after the fault. The performance is shown in Fig. 17, where the largest frequency deviation and the duration of voltage drop are reduced compared with Fig. 9. Hence, the proposed method captures the difference brought by different fault-clear time. 2) Altering fault type In this case, we alter the fault type from a three-phase line fault to a two-phase line fault. The performance is shown in Fig. 18, where the major difference compared with Fig. 9 is the lower magnitude of angles because of the slightly different rate-of-change of the angle speed compared with Fig. 9. Hence, the proposed method captures the difference brought by different types of faults. 3) Altering the fault location In this case, we alter the fault location to the line between bus 75 and bus 124. The performance is shown in Fig. 19, where the dynamics are very different compared with Fig. 9. The proposed method predicts the different shapes and magnitudes of oscillations. Hence, the proposed method captures the difference brought by different location of faults. Therefore, the proposed architecture handles all the factors well and predicts both the magnitude and oscillations accurately. Fig. 1 . 1Frequency oscillations during the Florida blackout resulted from a tripped line event in 2008[6]. Fig. 3 . 3Different components contributing to power system transient dynamics[4]. For a system with N buses, typically the main variables of interest are the angle δ i , rotor angle speeds (frequency) deviation ω i and voltage v i at each bus i. Denote [N ] := {1, . . . , N }, Fig. 4 . 4Illustration of the trajectory prediction starting at the time stamp ton for τout number of time steps, using τ in observations. Fig. 5 . 5Illustration of power system transient prediction with existing machine learning approaches. (a) Purely learning in the time domain tend to overfit. (b) Since the vast majority of historical data are stable, a ML method tends to not predict unstable trajectories. (c) Fitting the nonlinear dynamics with polynomial basis also easily lead to over-fitting. (d) The accuracy of PINN decreases sharply and fails to provide meaningful results for longer horizons. C. The Proposed Approach: System Dynamics in the Frequency Domain Because of the above challenges when learning in the time domain, we propose a new approach for learning power system transient dynamics in the frequency domain. Here we use a simple swing equation model for transient dynamics to illustrate the intuition for this approach [7], [27]: Fig. 7 . 7Structure of power system transient prediction using Fourier Neural Operator (a) The data frame that incorporate time-varying parameters and fault-on/clear actions within input time slots. (b) Incorporation of future fault-clear actions in the output time slots. (c) Fourier layer for Learning in the the frequency domain[22]. trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s (b) Predicted trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s Fig. 9. Stable dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately. (a) True trajectories (lines) and envelope (grey area) after a line-to-line fault between bus 75 and bus 124 and recovered at the time of 0.3s (b) Predicted trajectories (lines) and envelope (grey area) after a line-to-line fault between bus 75 and bus 124 and recovered at the time of 0.3s Fig. 10. Unstable dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately. of true trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s (b) FFT of predicted trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s Fig. 11. Frequency-domain trajectories of angle δ (left), angle speed deviation ω (middle) and voltage V (right) for stable dynamics in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. (a) FFT of true trajectories (lines) and envelope (grey area) after a line-to-line fault between bus 75 and bus 124 and recovered at the time of 0.3s (b) FFT of predicted trajectories (lines) and envelope (grey area) after a line-to-line fault between bus 75 and bus 124 and recovered at the time of 0.3s Fig. 12. Frequency-domain trajectories of angle δ (left), angle speed deviation ω (middle) and voltage V (right) for unstable dynamics in NPCC corresponding to (a) true trajectories the ground truth produced by a solver. (b) prediction of FNO. Fig. 14 shows the prediction on another fault (a three-phaseto-ground fault between bus 482 and bus 944 ) that is far-away from the fault shown in Fig. 13. The dynamics of Fig. 13 and Fig. 14 look to be very different, so the transient under one fault appears to have no relationship with another far-away fault. Despite of this, Fig. 13 and Fig. 14 show that the neural network can capture both patterns of the dynamics. Fig. 13 . 13Dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in GB corresponding to (a) the ground truth produced by a solver (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately. Fig. 15 . 15Example trajectories on a single-machine infinite bus system. FNO almost exactly matches the true one, while DNN shows much larger errors. w.o. low-pass filter (c) k max,1 = 4 (d) k max,1 = 6 (e) k max,1 = 8 (f) k max,1 = 16 Fig. 16. Trajectories (lines) and envelope (grey area) of the angle speed ω after a line-to-line fault between bus 75 and bus 124 and recovered at the time of 0.3s. We generate the stressed stable (the first row) and unstable case (the second row) by varying load levels. For all the three setup of kmax, The proposed method captures the dynamics for both stable and unstable cases. ( a ) aTrue trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.15s (b) Predicted trajectories (lines) and envelope (grey area) after a three-phase line fault between bus 54 and bus 103 and recovered at the time of 0.15sFig. 17. Stable dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately.(a) True trajectories (lines) and envelope (grey area) after a two-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s (b) Predicted trajectories (lines) and envelope (grey area) after a two-phase line fault between bus 54 and bus 103 and recovered at the time of 0.3s Fig. 18. Stable dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately.(a) True trajectories (lines) and envelope (grey area) after a two-phase line fault between bus 75 and bus 124 and recovered at the time of 0.3s (b) Predicted trajectories (lines) and envelope (grey area) after a two-phase line fault between bus 75 and bus 124 and recovered at the time of 0.3s Fig. 19. Stable dynamics of angle δ (left), frequency deviation ω (middle) and voltage V (right) in NPCC corresponding to (a) the ground truth produced by a solver. (b) prediction of FNO. The grey area shows the envelope of the trajectories for all generator buses. Lines with different colors shows the trajectories in selected generator buses. The proposed method predict both the magnitude and oscillations accurately. and B. Zhang are with the Department of Electrical and Computer Engineering, University of Washington Seattle, WA 98195, USA {wenqicui, zhangbao}@uw.edu W. Yang is with Microsoft Research, Redmond, WA 98052, USA wei-wya@microsoft.com. The authors are supported in part by the Climate Change AI. TABLE I IPERFORMANCE -ON FAULT True trajectories (lines) and envelope (grey area) after a three-phase-to-ground fault between bus 56 and bus 637 and recovered at the time of 0.13s (b) Predicted trajectories (lines) and envelope (grey area) after a three-phase-to-ground fault between bus 56 and bus 637 and recovered at the time of 0.13sMetric Relative mse Error-Type1 Error-Type2 Cycle after fault 0 2 4 10 20 0 2 4 10 20 0 2 4 10 20 FNO 0.0144 0.0063 0.0055 0.0053 0.0051 0 0 0 0 0 0.011 0 0 0 0 DNN 0.0778 0.0712 0.0655 0.0654 0.0656 1 1 1 0.667 0.167 0 0 0 0.011 0 (a) TABLE II RMSE IIFOR THE DATASET WITHOUT STRESSED UNSTABLE CASESCycle after fault 0 2 4 10 20 FNO 0.0077 0.0011 0.0009 0.0008 0.0007 DNN 0.0154 0.0146 0.0141 0.0125 0.0104 TABLE III AVERAGE COMPUTATIONAL TIME Methods FNO Matlab toolbox Speed up Prediction horizon = 3s 0.0036 1.69 469x Prediction horizon = 4.5s 0.0037 2.24 605x Prediction horizon = 6s 0.0039 3.38 867x VII. 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{'abstract': 'The dynamics of power grids are governed by a large number of nonlinear differential and algebraic equations (DAEs). To safely operate the system, operators need to check that the states described by these DAEs stay within prescribed limits after various potential faults. However, current numerical solvers of DAEs are often too slow for real-time system operations. In addition, detailed system parameters are often not exactly known. Machine learning approaches have been proposed to reduce the computational efforts, but existing methods generally suffer from overfitting and failures to predict unstable behaviors.This paper proposes a novel framework to predict power system transients by learning in the frequency domain. The intuition is that although the system behavior is complex in the time domain, there are relatively few dominate modes in the frequency domain. Therefore, we learn to predict by constructing neural networks with Fourier transform and filtering layers. System topology and fault information are encoded by taking a multi-dimensional Fourier transform, allowing us to leverage the fact that the trajectories are sparse both in time and spatial frequencies. We show that the proposed approach does not need detailed system parameters, greatly speeds up prediction computations and is highly accurate for different fault types. arXiv:2111.01103v3 [eess.SY] 31 Jan 2023', 'arxivid': '2111.01103', 'author': ['Wenqi Cui ', 'Weiwei Yang ', 'Baosen Zhang '], 'authoraffiliation': [], 'corpusid': 256416539, 'doi': '10.1109/tpwrs.2023.3259960', 'github_urls': ['https://github.com/Wenqi-Cui/Predict-Power-System-Dynamics-Frequency-Domain.'], 'n_tokens_mistral': 21359, 'n_tokens_neox': 18759, 'n_words': 12705, 'pdfsha': 'e03c7ad0d99958b6a163432154b9b2d5f1a1ba37', 'pdfurls': ['https://export.arxiv.org/pdf/2111.01103v3.pdf'], 'title': ['A Frequency Domain Approach to Predict Power System Transients', 'A Frequency Domain Approach to Predict Power System Transients'], 'venue': []}

arxiv

H 2 optimal model reduction on general domains 2 May 2023 Alessandro Borghi Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 13610623Berlin, BerlinGermany † Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 13610623Berlin, BerlinGermany Tobias Breiten tobias.breiten@tu-berlin.de Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 13610623Berlin, BerlinGermany H 2 optimal model reduction on general domains 2 May 2023Springer Nature 2021 L A T E X template † These authors contributed equally to this work.rational interpolationmodel reductionconformal mapsHardy spaces Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space H2, a new H2 like optimal model reduction problem is introduced and first order optimality conditions are derived. As in the classical H2 case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method. Introduction We consider large-scale single input single output (SISO) linear time invariant (LTI) dynamical systems of the form ẋ(t) = Ax(t) + bu(t), x(0) = 0, y(t) = c * x(t),(1) where A ∈ C n×n and c, b ∈ C n . Here, for fixed time t, x(t) ∈ C n , u(t) ∈ C, and y(t) ∈ C are the state, input, and output of the system, respectively. Let us emphasize that the results in this article similarly hold true for systems with multiple inputs and outputs and the restriction to the SISO case is made for the ease of presentation. Accompanied by the time domain description (1) we have the (frequency domain) transfer function H(s) = c * (sI − A) −1 b. If (1) is minimal, then H is a rational function of degree n. As an accurate modeling of such systems in both time and frequency domain can be computationally expensive for large values of n, we are interested in the construction of a reduced order surrogate model of the form ˙ x r (t) = A r x r (t) + b r u(t), x r = 0, y r (t) = c * r x r (t), with transfer function H(s) = c * r (sI − A r ) −1 b r , where A r ∈ C r×r and c r , b r ∈ C r . Here, the goal for constructing a "good" reduced order model is twofold: on the one hand, we are interested in (2) being efficiently solvable such that we demand r ≪ n; on the other hand, the reduced model is expected to yield an accurate approximation such that the outputs of both full and reduced system are close to each other, i.e., y r (t) ≈ y(t) for t ≥ 0. The latter condition requires a more precise notion of similarity. For example, it is well known, see, e.g., [1] that sup t≥0 |y(t) − y r (t)|≤ H − H r H2 u L2 which has led to the study of H 2 optimal model reduction problems, see [2,3]. More generally, model reduction techniques for linear systems of the form (1) have been addressed from a multitude of different areas such as system theory [3][4][5], reduced basis methods [6,7], rational interpolation [8][9][10], proper orthogonal decomposition [11,12] and, more recently, data driven techniques [13][14][15]. While a complete overview of the existing literature is out of the scope of this article, let us refer to the monographs [1,[16][17][18] and the references therein. The results discussed throughout this article are related to the error bound in (3) and the corresponding H 2 approximation problem of the underlying transfer functions. First order optimality conditions for H 2 model reduction of linear systems have already been derived in [3,19]. Later on, in [2] the iterative rational Krylov algorithm (IRKA) has been proposed to numerically compute H 2 optimal reduced order models, see also [20,21]. A similar method, called MIRIAm, has been discussed in a discrete time setting in [22]. Several extensions for frequency-weighted [23,24], frequency-limited [25], structurepreserving [26], parametric [27] or data-driven [28] H 2 problems have been developed over the last years. One of the essential theoretical assumptions made in these works is that the full order model (1) is (asymptotically) stable. For unstable systems, available methods are rather scarce. Some notable exceptions are the approaches discussed in [29,30]. Moreover, let us particularly mention the h 2,α model reduction technique proposed in [31] which allows to treat discrete time systems with system poles outside of the unit circle by extending the classical (discrete time) Hardy space to a circle of radius α. The strategy we follow here is based on similar ideas and also relates to the recent more general L 2 optimal model reduction framework from [32,33]. We build upon the existing theory of optimal H 2 model reduction and extend it by an appropriate use of specific conformal maps. The main contributions are the following: (i) We consider rational transfer functions with poles in specific domains in C, thereby covering typical cases such as the open left half plane and the open unit disk. For this purpose, we define the H 2 (Ā c ) space in Definition 1. (ii) We study the resulting H 2 (Ā c ) optimal model reduction problem and derive structured first order necessary interpolation optimality conditions in Theorem 3. (iii) Under additional assumptions, we show in Corollary 3 that more explicit interpolation conditions can be obtained for which we propose a numerical algorithm by a modification of IRKA, see Algorithm 1. (iv) The proposed algorithm is shown to be applicable to the Schrödinger and the undamped wave equation where the system poles are positioned along the imaginary axis. The rest of the paper is organized as follows. In Section 2 we briefly recall the concept of interpolatory model order reduction and existing interpolationbased H 2 optimality conditions. In Section 3 we introduce a Hardy space for functions with poles in general domains. We refer to this space as H 2 (Ā c ). We discuss its connection to the classical H 2 space and characterize the arising inner products. In Section 4 we introduce an H 2 (Ā c ) optimal model reduction problem for which we derive first order necessary interpolation conditions for local optimality. In addition, under specific assumptions, we propose a method based on IRKA for computing a solution to the model reduction problem. In Section 5 we demonstrate the effectiveness of our approach through numerical experiments for two spatially discretized partial differential equations with spectra residing along the imaginary axis. Notation By C, C + , C − and D we denote the complex plane, the open right half and open left half complex plane, and the open unit disk, respectively. For an open subset X ⊂ C, we denote δX to be its boundary, X c its complement,X its closure, andX c = C\ {X ∪ δX} its exterior. We denote the complex conjugation of a scalar and the Hermitian of a matrix by (·) * . We denote the Euclidean norm by · 2 and the absolute value over the complex numbers by |x|= √ xx * where x ∈ C. The symbol i denotes the imaginary unit. Let X ∈ C n×n be a matrix, then Ran(X) is its range. The first and second derivative of the function f at a point x, i.e., d ds f (s) s=x and d 2 ds 2 f (s) s=x , are denoted by f ′ (x) and f ′′ (x), respectively. Being f and g two complex valued functions we denote their composition by f •g and its evaluation at x by f (g(x)). The inverse of a function f is denoted by f −1 . Let f be analytic with a Taylor series around x 0 equal to f (x) = ∞ j=0 a j (x − x 0 ) j . Then f (x) = ∞ j=0 a * j (x − x * 0 ) j = f (x * ) * . In other words, f is f but with its coefficients replaced by their complex conjugates. For a rational function F with poles {λ j } n j=1 ∈ X, we denote the residue of F in λ by res[F (s), λ]. If λ is a simple pole, then res[F (s), λ] = lim s→λ (s − λ)F (s), if λ is a double pole, then res[F (s), λ] = lim s→λ d dx (x − λ) 2 F (x) x=s . Interpolatory model order reduction In this section, we briefly recall the concept of interpolatory model reduction. In particular, we summarize the problem of H 2 optimal model reduction and how it relates to rational (Hermite) interpolation as it lays the foundations for our main results in Section 3. For the construction of the reduced model in (2), we consider a Petrov-Galerkin projection. In other words, given two matrices V r , W r ∈ C n×r , we consider an approximation of the form x(t) ≈ V r x r (t) such that the residual for (1) satisfies the following orthogonality condition Ran(W r ) ⊥ V r˙ x r (t) − AV r x r (t) − bu(t) . If W * r V r is invertible, this leads to the reduced order system matrices A r = (W * r V r ) −1 W * r AV r , b r = (W * r V r ) −1 W * r b, and c * r = c * V r .(4) It is well-known, see, e.g., [2,8,34] that by choosing V r and W r as rational Krylov subspaces characterized by the resolvent operator (σI − A) −1 , the reduced order model (2) satisfies the following Hermite type interpolation conditions. such that (σ j I − A) and (σ j I − Ar) are both nonsingular. Let the two projection matrices Vr and Wr be chosen such that Ran(Vr) = span (σ 1 I − A) −1 b, . . . , (σrI − A) −1 b , Ran(Wr) = span (σ * 1 I − A * ) −1 c, . . . , (σ * r I − A * ) −1 c .(5) Then the reduced transfer function H with matrices as in (4) satisfies H(σ j ) = H(σ j ) and H ′ (σ j ) = H ′ (σ j ) for j = 1, . . . , r. Since the choice of interpolation points has a significant influence on the quality of the reduced model, different selection strategies for σ j have been proposed. For our purposes, so-called H 2 optimal interpolation points, see [2,22], will be of particular relevance. Optimal H 2 model reduction Recall that for functions G, H that are analytic in the open right half plane, the Hardy space H 2 (C + ) ([1, Section 5.1.3]) is defined as H 2 (C + ) := H : C + → C analytic sup x>0 ∞ −∞ |H(x + iω)| 2 dω < ∞ . Moreover, H 2 (C + ) becomes a Hilbert space when endowed with the inner product H, G H2(C+) := 1 2π ∞ −∞ H(iω) * G(iω) dω,(6) and induced norm H H2(C+) := 1 2π ∞ −∞ |H(iω)| 2 dω 1 2 .H −H H2(C+) .(7) In [2,3] it was proved that if H, with poles { λ j } r j=1 , is a local minimizer of (7), then H(− λ * j ) = H(− λ * j ) and H ′ (− λ * j ) = H ′ (− λ * j ) for j = 1, . . . , r.(8) These interpolation conditions are usually referred to as Meier-Luenberger conditions, see [2]. A similar result was proved for discrete time systems. In this case, the transfer functions H andH are analytic inD c (see [2,22]). The optimal H 2 framework minimizes the error norm H −H H2(D c ) := 1 2π 2π 0 H(e iϑ ) −H(e iϑ ) 2 dϑ 1 2 , and for H being a local minimizer we have that H 1/ λ * j = H 1/ λ * j and H ′ 1/ λ * j = H ′ 1/ λ * j for j = 1, . . . , r. (9) Throughout this manuscript, we refer to (8) and (9) as the H 2 (C + ) and H 2 (D c ) optimality conditions thereby emphasizing in which set the functions are analytic. This will turn out useful for the upcoming results. Note that as long as the systems are assumed to be asymptotically stable, considering Hardy spaces on the specific sets C + andD c is sufficient. For cases in which the poles reside in different sets or when a more granular view of the spectrum is desired, the conditions (8) and (9) may not be an ideal choice. In this context, [31] discussed generalizations of (8) and (9) to the shifted right half plane and disks with arbitrary radii, thereby allowing for unstable models. Albeit these findings can be applied to a wider set of functions, they still restrict to a very specific shape of the considered domains. In the next section we show that it is possible to generalize these frameworks to rational transfer functions with poles located in domains, i.e., non-empty connected open sets. 3 The H 2 (Ā c ) space Consider A ⊂ C to be a non-empty connected open set in the complex plane. We define F and G as the rational functions F (s) = n i=1 φ i s − λ i , G(s) = q j=1 ν j s − µ j(10) where φ i = res[F (s), λ i ] and ν j = res[G(s), µ j ]. In particular, we assume both F and G to have only simple poles. Throughout this section we make extensive use of conformal maps and therefore recall the conformal mapping theorem. Similar statements on the properties of conformal maps can be found in [36,Theorem I.5.15]. From [37, Section 2.6] and [38, Section 1.1] we also have the following properties: (i) Since ψ is analytic in X and ψ ′ does not vanish, ψ is injective in X. (ii) If ψ : X → Y is conformal and bijective, then also its inverse ψ −1 : Y → X is conformal and bijective. (iii) If ψ : X → Y and ξ : Y → E are conformal mappings, then ψ • ξ is conformal. For ψ being analytic in X, it holds that ψ(s 0 ) * = ψ(s * 0 ) for s 0 ∈ X, cf. also the notation in Section 1.1. Before we introduce the space H 2 (Ā c ), which we subsequently use to define our optimal model reduction framework, we make the following assumption. Assumption 1 Let the meromorphic function ψ : C → C be given. We assume ψ : X → A, with X ⊆ C − , to be bijective conformal. LetX ⊆X c be an open subset such that {s ∈ C − s * ∈ X} ⊆X. We assume ψ to mapX intoĀ c . In addition, ψ ′ is zero in a finite amount of points inX c . In summary, we assume ψ to fulfill ψ : X → A is bijective conformal, ψ :X →Ā c is meromorphic. From now on, we assume that Assumption 1 is satisfied. In Figure 1 we depict the domains X, A,X and their exteriors, along with the mapping ψ. Consider F as in (10) with its poles λ i ∈ A, i = 1, . . . , n, such that it is analytic inĀ c . It is easy to prove that if ψ : C + →Ā c is analytic, then F (ψ(·)) and F (ψ(· * )) * are analytic in C + (see also the proof of [35, Theorem 6.6.2]). In addition, F (ψ(−·)) and F (ψ(−·)) = F (ψ(−· * )) * are analytic in C − . Similar arguments apply in the case that ψ is analytic in X ⊂ C − . This would then result in F (ψ(−· * )) * being analytic in the set {s ∈ C − s * ∈ X}. We now define the H 2 space for functions analytic inĀ c . This definition relies on the concepts introduced by Duren in [39,Chapter 10]. The main difference is the set in which ψ is conformal. Definition 1 (H 2 (Ā c ) space) Let f :Ā c → C and g :Ā c → C be analytic. Denote by H f (s) = f (ψ(s))ψ ′ (s) 1 2 ,(11) then the H 2 (Ā c ) inner product is defined as f, g H2(Ā c ) := H f , Hg H2(C+) with the induced H 2 (Ā c )-norm f H2(Ā c ) := H f H2(C+) = H f , Hg H2(C+) 1 2 . The space H 2 (Ā c ) is defined as H 2 (Ā c ) := f :Ā c → C analytic f H2(Ā c ) < ∞ . It follows from Definition 1 that if f ∈ H 2 (Ā c ) then H f ∈ H 2 (C + ). Note that the norm strongly depends on the set A as well as the chosen map ψ. Obviously, for ψ(s) = s we have H 2 (Ā c ) ≡ H 2 (C + ) leading to the results mentioned in Section 2.1. Instead of computing the H 2 (Ā c ) inner product through the integral in (6), we now derive a pole residue expression similar to [2, Lemma 2.4]. Lemma 1 Consider the functions f, g ∈ H 2 (Ā c ). Let H f and Hg have finitely many poles {l i } n i=1 ∈ C − and {m j } q j=1 ∈ C − . Then f, g H2(Ā c ) = q j=1 res H f (−s)Hg(s), m j = n i=1 res H g (−s)H f (s), l i * , where H f (−s) = f (ψ(−s))ψ ′ (−s) 1 2 . Proof The proof directly follows from [2, Lemma 2.4] applied to H f and Hg, respectively. Lemma 1 allows to compute the H 2 (Ā c )-norm as follows f 2 H2(Ā c ) = n i=1 H f (−l i )res [H f (s), l i ] . We now consider the specific case where f is a rational function as in (10) and therefore adopt the notation F that we previously introduced for functions that have a rational structure. We begin by examining the case where ψ and ψ ′ (·) 1 2 share the same poles in the complex plane which happens to be the case if ψ is a rational function with simple poles. If F is rational as in (10), then this leads to H F having the same poles as F • ψ. In other words, if we denote by {λ i } n i=1 the poles of F and by {l i } n i=1 the poles of H F , then {ψ −1 (λ i )} n i=1 = {l i } n i=1 . Lemma 2 Let F (·) = n i=1 φi ·−λi , with λ i ∈ A for i = 1, . . . , n. Let ψ and ψ ′ (·) 1 2 have the same poles. Then res H F (s), ψ −1 (λ i ) = φ i ψ ′ (ψ −1 (λ i )) − 1 2 ,(12) for i = 1, . . . , n. Proof Note that since F is a rational function with poles in A, we have that F ∈ H 2 (Ā c ). By expanding the left hand side of (12) and using (10), we get res H F (s), ψ −1 (λ i ) = res F (ψ(s))ψ ′ (s) 1 2 , ψ −1 (λ i ) = lim s→ψ −1 (λi) (s − ψ −1 (λ i )) n j=1 φ j ψ(s) − λ j ψ ′ (s) 1 2 = φ i ψ ′ (ψ −1 (λ i )) 1 2 lim s→ψ −1 (λi) s − ψ −1 (λ i ) ψ(s) − λ i . Since both the numerator and denominator approach zero, we apply L'Hôpital's rule [40,Chapter 9] resulting in lim s→ψ −1 (µ) s − ψ −1 (µ) ψ(s) − µ = lim s→ψ −1 (µ) 1 ψ ′ (s) = 1 ψ ′ (ψ −1 (µ)) . This leads to res H F (s), ψ −1 (λ i ) = φ i ψ ′ (ψ −1 (λ i )) 1 2 ψ ′ (ψ −1 (λ i )) = φ i ψ ′ (ψ −1 (λ i )) − 1 2 . Combining the previous result with Lemma 1 provides a simpler characterization of the H 2 (Ā c ) inner product. Corollary 1 Let F (·) = n i=1 φi ·−λi and G(·) = q j=1 νj ·−µj have simple poles {λ i } n i=1 ∈ A and {µ j } q j=1 ∈ A, respectively. Assume that H F and F • ψ as well as H G and G • ψ have the same poles. Then F, G H2(Ā c ) = q j=1 H F (−ψ −1 (µ j )) ν j ψ ′ (ψ −1 (µ j )) − 1 2 . For F ∈ H 2 (Ā c ) and ψ ′ (·) 1 2 with simple poles {γ ℓ } m ℓ=1 ∈ C − , we define F F (s) := H F (−ψ −1 (s))ψ ′ (ψ −1 (s)) − 1 2 + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ ψ(γ ℓ ) − s .(14) A similar function has been introduced in [23] for a frequency-weighted H 2 model reduction problem. In particular, as in [23,Corollary 4], the evaluation of F F and its derivative in µ can be expressed as a specific H 2 (Ā c ) inner product involving rational functions of degree one and two, respectively. Lemma 3 Let F ∈ H 2 (Ā c ) be as in Lemma 2 and µ ∈ A. Consider ψ and ψ ′ (·) 1 2 with different poles. Let {γ ℓ } m ℓ=1 ∈ C − with γ ℓ = ψ −1 (µ), ℓ = 1, . . . , m, be the poles of ψ ′ (·) 1 2 . Then F, 1 · − µ H2(Ā c ) = F F (µ), and F, 1 (· − µ) 2 H2(Ā c ) = F ′ F (µ).(15) Proof Once again, recall that since µ ∈ A, it holds that 1 ·−µ ∈ H 2 (Ā c ) and 1 (·−µ) 2 ∈ H 2 (Ā c ). Using Lemma 1, for the first inner product we obtain F, 1 · − µ H2(Ā c ) = res H F (−s) ψ ′ (s) 1 2 ψ(s) − µ , ψ −1 (µ) + m ℓ=1 res H F (−s) ψ ′ (s) 1 2 ψ(s) − µ , γ ℓ . Since 1 ·−µ is a rational function, we can apply Lemma 2 resulting in res 1 s − µ , ψ −1 (µ) = ψ ′ (ψ −1 (µ)) − 1 2 .(16) It is worth noting that F ∈ H 2 (Ā c ) implies that H F (−·) is analytic in C − . Using (16), we now obtain the first assertion in (15) since F, 1 · − µ H2(Ā c ) = H F (−ψ −1 (µ))res ψ ′ (s) 1 2 ψ(s) − µ , ψ −1 (µ) + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ ψ(γ ℓ ) − s = H F (−ψ −1 (µ))ψ ′ (ψ −1 (µ)) − 1 2 + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ ψ(γ ℓ ) − s = F F (µ). From ψ being bijective conformal in X, with the implicit function theorem (see [36, Theorem I.5.7]), we conclude that d ds ψ(ψ −1 (s)) = ψ ′ (ψ −1 (s)) ψ −1 ′ (s) = 1, leading to ψ −1 ′ (s) = 1 ψ ′ (ψ −1 (s)) . Moreover, we also know that ψ −1 is analytic on the image of ψ. For the second inner product we have that the poles of ψ ′ (·) 1 2 differ from ψ −1 (µ). In addition, because H F (−·) is analytic in C − , the term H F (−·)ψ ′ (·) 1 2 (ψ(·) − µ) −2 has only a double pole in ψ −1 (µ). Hence, for the inner product it follows F, 1 (· − µ) 2 H2(Ā c ) = res H F (−s) (ψ(s) − µ) 2 ψ ′ (s) 1 2 , ψ −1 (µ) + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 = lim s0→ψ −1 (µ) d ds (s − ψ −1 (µ)) 2 H F (−s) (ψ(s) − µ) 2 ψ ′ (s) 1 2 s=s0(18)+ m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 = lim s0→ψ −1 (µ) −H ′ F (−s) s − ψ −1 (µ) ψ(s) − µ 2 ψ ′ (s) 1 2 + H F (−s) s − ψ −1 (µ) ψ(s) − µ 2 ψ ′ (s) − 1 2 2 ψ ′′ (s) + H F (−s) 2 s − ψ −1 (µ) ψ(s) − µ ψ −1 (µ) − s (ψ(s) − µ) 2 ψ ′ (s) + 1 ψ(s) − µ ψ ′ (s) 1 2 s=s0 + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 . We first focus on the terms lim s→ψ −1 (µ) s − ψ −1 (µ) ψ(s) − µ ,(19) and lim s→ψ −1 (µ) 2 s − ψ −1 (µ) ψ(s) − µ ψ −1 (µ) − s (ψ(s) − µ) 2 ψ ′ (s) + 1 ψ(s) − µ .(20) Again, as in (13) and (17) with L'Hôpital's rule we arrive at lim s→ψ −1 (µ) s − ψ −1 (µ) ψ(s) − µ = ψ −1 ′ (µ).(21) Similarly, for (20) we find ψ −1 ′ (µ) lim s→ψ −1 (µ) 2 ψ −1 (µ) − s (ψ(s) − µ) 2 ψ ′ (s) + 1 ψ(s) − µ = ψ −1 ′ (µ) lim s→ψ −1 (µ) ψ ′′ (s) ψ ′ (s) ψ −1 (µ) − s ψ(s) − µ = − ψ −1 ′ (µ) 2 ψ ′′ (ψ −1 (µ)) ψ ′ (ψ −1 (µ)) Using these equalities in (18) gives F, 1 (· − µ) 2 H2(Ā c ) = −H ′ F (−ψ −1 (µ)) ψ −1 ′ (µ) 2 ψ ′ (ψ −1 (µ)) 1 2 + H F (−ψ −1 (µ)) ψ −1 ′ (µ) 2 ψ ′ (ψ −1 (µ)) − 1 2 2 ψ ′′ (ψ −1 (µ)) − H F (−ψ −1 (µ)) ψ −1 ′ (µ) 2 ψ ′′ (ψ −1 (µ)) ψ ′ (ψ −1 (µ)) ψ ′ (ψ −1 (µ)) 1 2 + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 = −H ′ F (−ψ −1 (µ)) ψ −1 ′ (µ) 2 ψ ′ (ψ −1 (µ)) 1 2 − 1 2 H F (−ψ −1 (µ)) ψ −1 ′ (µ) 2 ψ ′ (ψ −1 (µ)) − 1 2 ψ ′′ (ψ −1 (µ)) + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 . Using (21) eventually leads to the second equality of (15) F, 1 (· − µ) 2 H2(Ā c ) = −H ′ F (−ψ −1 (µ)) ψ −1 ′ (µ)ψ ′ (ψ −1 (µ)) − 1 2 − 1 2 H F (−ψ −1 (µ))ψ ′ (ψ −1 (µ)) − 3 2 ψ ′′ (ψ −1 (µ)) ψ −1 ′ (µ) + m ℓ=1 H F (−γ ℓ ) res ψ ′ (s) 1 2 , γ ℓ (ψ(γ ℓ ) − µ) 2 = F ′ F (µ). It is worth noting that if {γ ℓ } m ℓ=1 ∈X\C c + and ψ ′ (·) 1 2 analytic in X, then F F is analytic in A. This is due to both summands in (14) being analytic in A. Let us now consider the case where H F has the same poles as F • ψ as in Corollary 1. Then we have a simplification of Lemma 3. As a matter of fact, F F (s) becomes F F (s) = H F (−ψ −1 (s))ψ ′ (ψ −1 (s)) − 1 2 , leading to the following corollary. Corollary 2 Consider F ∈ H 2 (Ā c ) and µ ∈ A. If ψ and ψ ′ (·) 1 2 have the same poles, then F, 1 · − µ H2(Ā c ) = H F (−ψ −1 (µ))ψ ′ (ψ −1 (µ)) − 1 2 , and F, 1 (· − µ) 2 H2(Ā c ) = d ds H F (−ψ −1 (s))ψ ′ (ψ −1 (s)) − 1 2 s=µ . 4 Optimal H 2 (Ā c ) model reduction The H 2 optimal model order reduction framework seeks a solution for the optimization problem in (7). We now introduce a similar approach based on the H 2 (Ā c ) space described in Definition 1. The objective is to find an optimal reduced order model (2) that minimizes the error norm miñ H H −H H2(Ā c ) .(22) By definition we can recast this optimization problem as miñ H H H − HH H2(C+) .(23) However, note that it is not possible to perform classical H 2 (C + ) optimal model reduction for H H , the reason being that the reduced model HH would have to possess a very particular structure. In fact, (23) presents a structurepreserving H 2 model reduction problem that similarly arises for the frequencyweighted case, see, once again, [23]. Contrary to the latter work, under certain assumptions on ψ, ψ ′ and ψ −1 , we will be able to derive optimality conditions which allow for a numerical approach based on a slight modification of IRKA. Of course, the nonconvexity of (22) makes the computation of a global minimizer very challenging if not impossible. For this reason, one seeks local minimizers, making the task more feasible. Consequently, let us consider the transfer function H of the reduced model in (2) to be a local minimizer of (22). The following theorem states the interpolation conditions that H needs to fulfill. The proof follows a perturbation argument that has previously been used in, e.g., [2,23,41]. In particular, consider a transfer function H (ε) with H − H (ε) H2(Ā c ) = O(ε) so that the local minimality of H implies H − H H2(Ā c ) ≤ H − H (ε) H2(Ā c ) .(24) Theorem 3 Let A be a non-empty connected open set and H ∈ H 2 (Ā c ). Consider ψ as in Assumption 1 and let ψ ′ (·) 1 2 be analytic in X and have the left-half plane poles {γ ℓ } m ℓ=1 ∈X\C c + . Let H ∈ H 2 (Ā c ) be a local minimizer of (22) with poles { λ j } r j=1 ∈ A. Let {ψ −1 ( λ j )} r j=1 be different from the poles of ψ ′ (·) 1 2 and ψ. Then the following interpolation conditions hold for j = 1, . . . , r F H ( λ j ) = F H ( λ j ) and F ′ H ( λ j ) = F ′ H ( λ j ).(25) Proof For the first condition, we consider a perturbation of the local optimum in the p-th residue such that H (ε) (s) = φp + εe iθ1 s − λp + j =p φ j s − λ j , with ε > 0 small and θ 1 arbitrary. We then get from (24) H − H 2 H2(Ā c ) ≤ H − H (ε) 2 H2(Ā c ) = H − H + H − H (ε) 2 H2(Ā c ) = H − H 2 H2(Ā c ) + 2Re H − H, H − H (ε) H2(Ā c ) + H − H (ε) 2 H2(Ā c ) which leads to 0 ≤ 2Re H − H, H − H (ε) H2(Ā c ) + H − H (ε) 2 H2(Ā c ) .(26) We use Lemma 3 and (14) to evaluate the two terms in (26) H − H, H − H (ε) H2(Ā c ) = H − H, −εe iθ1 s − λp H2(Ā c ) = −εe iθ1 F H ( λp) − F H ( λp) ,(27) and H − H (ε) 2 H2(A) = −εe iθ1 s − λp 2 H2(Ā c ) = O(ε 2 ), for ε → 0. Consider θ 1 chosen such that e iθ1 F H ( λp) − F H ( λp) is positive and real valued, i.e. θ 1 = −arg F H ( λp) − F H ( λp) . This implies that (24) becomes 0 ≤ 2 F H ( λp) − F H ( λp) ≤ O(ε). As this holds for arbitrary ε > 0, in the limit we obtain F H ( λp) = F H ( λp),(28) resulting in the first part of (25). The second interpolation condition is proven as above but with a perturbation of the p-th pole, i.e., H (ε) (s) = φp s − ( λp + εe iθ2 ) + j =p φ j s − λ j . Let us first expand the inner product H − H, H − H (ε) H2(Ā c ) = H − H, φp s − λp − φp s − ( λp + εe iθ2 ) H2(Ā c ) = φp F H ( λp) − F H ( λp) − φp F H ( λp + εe iθ2 ) − F H ( λp + εe iθ2 ) = − φp F H ( λp + εe iθ2 ) − F H ( λp + εe iθ2 ) where in the last equality we have used the first optimality condition (28). Because {γ ℓ } m ℓ=1 ∈X\C c + and ψ ′ (·) 1 2 is analytic in X, we have that both F H and F H are analytic in A. For ε → 0, we consider the Taylor expansion of the two functions around λp (see also [41,Theorem 5 .1.1]). This leads to H − H, H − H (ε) H2(Ā c ) = −εe iθ2 φp F ′ H ( λp) − F ′ H ( λp) . For ε → 0 we also have that H − H (ε) 2 H2(A) = O(ε 2 ). Inserting these equalities in (26) results in 0 ≤ −2εRe e iθ2 φp F ′ H ( λp) − F ′ H ( λp) + O(ε 2 ).(29) Choosing θ 2 such that e iθ2 φp(F ′ H ( λp) − F ′ H ( λp)) is positive and real valued gives us F ′ H ( λp) = F ′ H ( λp) , obtaining the second equality of (25). This is then repeated for p = 1, . . . , r. Theorem 3 gives us necessary optimality conditions that must hold for H ∈ H 2 (Ā c ) to solve (22). Due to the particular structure of F it becomes practically very challenging to compute a reduced modelH such that Hermite interpolation between FH and F H is fulfilled. Similar challenges were discussed for the weighted H 2 model reduction problem in [23,24]. Corollary 3 Let the assumptions in Theorem 3 hold. Let ψ and ψ ′ (·) 1 2 have the same poles. Assume that in neighborhoods of the points −ψ −1 ( λ j ) * the derivative ψ ′ exists and is not zero for j = 1, . . . , r. Then the optimality conditions in (25) become H(ϕ( λ j )) = H(ϕ( λ j )) and H ′ (ϕ( λ j )) = H ′ (ϕ( λ j )),(30) where ϕ(s) = ψ(−ψ −1 (s)) * . Proof The proof directly follows using Corollary 2 in Theorem 3. Since ψ and ψ ′ (·) 1 2 share the same poles with Corollary 2, we simplify (25) according to H H (−ψ −1 ( λp)) = H H (−ψ −1 ( λp)) and H ′ H (−ψ −1 ( λp)) = H ′ H (−ψ −1 ( λp) ). (31) Using (11) in the first equality of (31) results in H(ϕ( λp)) * ψ ′ (−ψ −1 ( λp)) 1 2 = H(ϕ( λp)) * ψ ′ (−ψ −1 ( λp)) 1 2 ,(32) where ϕ(s) = ψ(−ψ −1 (s)) * . The assumption made on ψ ′ states that ψ ′ (−s * ) * = ψ ′ (−s) is non zero in s = ψ −1 ( λp). This allows to simplify (32) such that H(ϕ( λp)) = H(ϕ( λp)). Before continuing with the second equality we calculate the following derivative for s = ψ −1 ( λp) d ds H H (−s) = d ds H(ψ(−s))ψ ′ (−s) 1 2 = d ds H(ψ(−s) * ) * ψ ′ (−s) 1 2 = d ds H(ψ(−s * )) * ψ ′ (−s) 1 2 = −H ′ (ψ(−s * )) * ψ ′ (−s * ) * ψ ′ (−s) 1 2 + H(ψ(−s * )) * d ds ψ ′ (−s) 1 2 .(33) Using (33) in the second equality of (31) and utilizing the first interpolation conditions in (30) leads to H ′ (ϕ( λp)) = H ′ (ϕ( λp)), concluding the proof. We will adopt the above simplified optimality conditions to design a generalized version of IRKA in Section 4.2. Analysis of specific conformal mappings In what follows we discuss the validity of Assumption 1 for some particular conformal maps. In addition, we show that these functions also meet the assumptions made in Corollary 3. These functions will subsequently be used in the numerical examples of Section 5. Obtaining the H 2 (D c ) optimality conditions We first recover the H 2 (D c ) optimality conditions (9) as a specific case of the H 2 (Ā c ) framework. The meromorphic function ψ that conformally maps C − into D is the following Möbius transformation [35] ψ(s) = s + 1 s − 1 ,(34) with inverse ψ −1 (s) = s+1 s−1 . Let us note that ψ also conformally maps C + \{1} intoD c . Being the derivative of (34) ψ ′ (s) = −2 (s − 1) 2 ,(35) we can see that it is not zero in the complex plane, excluding the double pole in 1. If we consider only C − \{−1} as domain of ψ, instead of the entire left half plane, then the mapping (34) meets Assumption 1. Let us note that, because of the previews consideration, we now have that ψ : C − \{−1} → D\{0} is bijective and conformal. Let us now calculate the square root of (35) ψ ′ (s) 1 2 = i √ 2 s − 1 . We have that ψ ′ (·) 1 2 has the same poles as ψ and is non-zero in C\{1}. Furthermore, for the full order transfer function H(s) = n j=1 φ j s − λ j ,(36) we have that H H (s) = H(ψ(s))ψ ′ (s) 1 2 = n j=1 i √ 2φ j s + 1 − λ j (s − 1) . Hence, the poles of H H (s) are given by ψ −1 (λ j ) = λj +1 λj −1 . For λ j ∈ D\{0} and ψ : C − \{−1} → D\{0} being bijective and conformal, we then have that ψ −1 (λ j ) ∈ C − \{−1} for j = 1, . . . , n. In addition, the poles of H H (s) are the same as the ones of H(ψ(s)) = n j=1 φ j (s − 1) s + 1 − λ j (s − 1) . For this reason we can apply Corollary 3 to retrieve the optimal interpolation conditions. Knowing that ψ(−s) = −s+1 −s−1 , we first expand the function ϕ(s) = ψ −ψ −1 (s) * = − s+1 s−1 + 1 − s+1 s−1 − 1 * = 1 s * .(37) From Corollary 3 we have that H(ϕ( λ p )) = H(ϕ( λ p )) and H ′ (ϕ( λ p )) = H ′ (ϕ( λ p )). Plugging in (37) gives us the optimal interpolation conditions in (9) H 1/ λ * p = H 1/ λ * p and H ′ 1/ λ * p = H ′ 1/ λ * p . It is worth mentioning that there exists a slight discrepancy between the above framework and the original H 2 (D c ) formulation for discrete time systems. This is due to the restriction of the poles to be in D\{0} such that we avoid the singularity of ψ in (34). Optimality conditions for the upper half complex plane We now study the case where the full order transfer function in (36) has poles on the upper half complex plane C ↑ = z ∈ C Im(z) > 0 . In this case, we choose the conformal map ψ(s) = −is with inverse ψ −1 (s) = is. The function is analytic in the entire complex plane and conformally maps C − into C ↑ and C + into C ↓ = z ∈ C Im(z) < 0 . For this reason, Assumption 1 is met. In addition, neither ψ nor ψ ′ have poles, they are both analytic in C, and ψ ′ is non-zero everywhere. Thanks to these properties we meet the assumptions of Corollary 3. By expanding the function ϕ(s) = ψ(−ψ −1 (s)) * = ψ(−is) * = s * ,(38) we then get the interpolation conditions H λ * p = H λ * p and H ′ λ * p = H ′ λ * p . In other words, the interpolation points mirror the poles of H with respect to the real axis. This is to be expected as we are rotating the framework of IRKA by π 2 . Optimality conditions for an ellipse In this last example, we consider the Bernstein ellipse δB, i.e., an ellipse with foci at 1 and −1 (see [42,Chapter 8]). We refer to the interior of δB as B. Letting [−1, 1] be a slit in the real axis, we assume the full order model to have poles in B\[−1, 1]. In this case, the conformal map is designed following the process depicted in Figure 2. From the w plane to the s plane we use conformal mappings in the following order: Möbius transformation, scaling by R > 1, and the Joukowski transform [37,Section 6]. This leads to ψ(w) = 1 2 R w + 1 w − 1 + R −1 w − 1 w + 1 , ψ −1 (s) = s + (s 2 − 1) 1 2 R −1 + 1 s + (s 2 − 1) 1 2 R −1 + 1 . (39) Let us focus on the Joukowski transform and its inverse J(z) = 1 2 z + 1 z , J −1 (s) = s ± (s 2 − 1) 1 2 .(40) This function maps bothD c and D\{0} into C\[−1, 1]. In particular, it maps circles of radius R and 1/R, with R > 1, into Bernstein ellipses with major axis (R + R −1 )/2 and minor axis (R − R −1 )/2. Since we have J(z) = J(1/z), the Joukowski transform is obviously not bijective and, as a remedy, we choose the positive root of J −1 and the exterior of D as domain of J. In the z plane of Figure 2, the domain of J is represented as the blue torus and the red plane outside the disk of radius R. In particular, the blue torus is mapped into B\[−1, 1] and the red plane intoB c . We now analyze the conformal map (39) in more detail. This meromorphic function in C has two poles in 1 and −1. Its derivative ψ ′ (w) = 1 2 −R (w − 1) 2 + R −1 (w + 1) 2 = 1 2 R −1 (w − 1) 2 − R(w + 1) 2 (w − 1) 2 (w + 1) 2 , exists everywhere in C\{−1, 1} and has two double poles in 1 and −1. It presents two zeroes in (±R −1 + 1)/(±R −1 − 1) and a double one at infinity. Since we choose R > 1, the two zeroes are in the left half w plane. In particular, these are marked with a cross in Figure 2 along the boundary ofX c . Function ψ conformally and bijectively maps X into B\[−1, 1], andX = C + \{1} intoB c . We do not consider the disk in the w plane given byX c \X otherwise we would lose bijectivity. With these considerations we meet Assumption 1. In addition, we have that ψ ′ (·) 1 2 and ψ share the same poles, and ψ ′ is zero in a finite amount of points in C\{−1, 1}. As a consequence, (39) meets the assumptions of Corollary 3. The optimal interpolation points are then given by the function ϕ(s) = ψ(−ψ −1 (s)) * = ψ   − s + (s 2 − 1) 1 2 R −1 + 1 s + (s 2 − 1) 1 2 R −1 + 1   * . Because ψ(·) = ψ(·) we have that ϕ(s) = ψ   − s + (s 2 − 1) 1 2 R −1 + 1 s + (s 2 − 1) 1 2 R −1 + 1   * = 1 2 R 2 s + (s 2 − 1) 1 2 + s + (s 2 − 1) 1 2 R 2 * .(41) Using (41) in (30) we obtain optimality conditions for transfer functions with poles in B\[−1, 1]. In Section 5, we will consider the conformal map (39) with two minor modifications: 1) a translation of the ellipse by c ∈ C and 2) a scaling and rotation by M ∈ C. The resulting conformal map becomes ψ(w) = c + M 2 R w + 1 w − 1 + R −1 w − 1 w + 1 ,(42) with inverse ψ −1 (s) = (s − c)M −1 + (s − c)M −1 2 − 1 1 2 R −1 + 1 (s − c)M −1 + ((s − c)M −1 ) 2 − 1 1 2 R −1 − 1 . The interpolation points are then computed by ϕ(s) = c + M 2    R 2 (s − c)M −1 + ((s − c)M −1 ) 2 − 1 1 2 + (s − c)M −1 + (s − c)M −1 2 − 1 1 2 R 2    * .(43) IRKA with conformal maps In view of the interpolation conditions from Corollary 3, it now seems natural to consider an iterative algorithm to solve (22) by modification of IRKA. In particular, instead of updating the interpolation points according to a reflection along the imaginary axis via − λ * j , we use ϕ( λ j ) for j = 1, . . . , r. This modified version of IRKA allows to reduce transfer functions with poles in general domains that are characterized by a specific set of conformal maps (see Corollary 2 and 3). In Algorithm 1, we provide an appropriate pseudocode. Algorithm 1 Iterative Rational Krylov Algorithm with Conformal Maps Require: The full order system matrices (A, b, c), the conformal map ψ, and the reduced order r < n. 1: Make initial guess of interpolation points σ 0 2: Construct the projection matrices V r and W r as in (5) 3: while σ i+1 − σ i / σ i > tol do 4: Assign A r = (W * r V r ) −1 W * r AV r 5: Solve the eigenvalue problem A r v j = λ j v j and set σ (j) i+1 = ϕ( λ j ) for j = 1, . . . , r 6: Update the projection matrices such that Ran(V r ) = span (σ (1) i+1 I − A) −1 b, . . . , (σ (r) i+1 I − A) −1 b , Ran(W r ) = span ((σ (1) i+1 ) * I − A * ) −1 c, . . . , ((σ (r) i+1 ) * I − A * ) −1 c 7: end while 8: Construct the reduced order model matrices A r , b r , c r using (4). Let us emphasize that the optimal H 2 (Ā c ) model reduction problem aims at minimizing the error H H −HH H2(C+) with respect toH. In particular, the conformal map ψ might cause H H to have poles very close to the imaginary axis resulting in a potentially poor convergence behavior. An appropriate choice of ψ might therefore require an individual analysis of the problem at hand. In the next section, we will report on such issues and also demonstrate when and how the use of Algorithm 1 is beneficial compared to the classical version of IRKA. Numerical experiments In this section, we test our theoretical results with two numerical examples. More in detail, we show the effectiveness of Algorithm 1 applied to systems that are not asymptotically stable. The two considered cases are the Schrödinger and the undamped wave equations. Our main purpose is to show that Algorithm 1 is able to effectively reduce systems with poles along the imaginary axis, a case where the H 2 (C + ) framework would fail. In the first example, we compare the results of Algorithm 1 against IRKA. In the second experiment, we apply a more complex conformal map, developed in Section 4.1.3, and show the performance of the resulting reduced model. All simulations were generated with MATLAB ® 2021b on a laptop computer equipped with an Intel ® i5-1135G7 processor with 8 cores and 16GB of RAM. The H 2 (Ā c ) error norms are computed through the integral command while the trajectories are a result of the ode23 routine. Both these MAT-LAB ® functions were used with default relative and absolute tolerances. The implementation is also publicly available 1 . Schrödinger equation In the first example, we consider the following boundary controlled Schrödinger equation (see, e.g., [43, Example 6.7.3, Section 11.6.1]) ∂w(x, t) ∂t = −i ∂ 2 w(x, t) ∂x 2 , on (0, 1) × (0, T ), w(0, t) = 0, w(1, t) = u(t), on (0, T ),y(t) = 1 0 w(x, t) dx, on (0, T ), w(x, 0) = 0, in (0, 1), where u and y are the (scalar) input and output of the system. We use a spatial semi discretization by centered finite differences resulting in a full order system of dimension n = 1000. As this system has its poles on the upper part of the imaginary axis, we apply the following conformal map from Section 4.1. This leads to the function in (38) for computing the interpolation points. The initial shifts for Algorithm 1 are chosen with fixed imaginary part at −100i and a normally distributed random choice of the real part. In Figure 3, we depict the relative H 2 (Ā c ) error defined as H − H H2(Ā c ) H H2(Ā c ) , withĀ c = C ↓ , see Section 4.1.1, and reduced orders varying from r = 5 to r = 25. Here, as expected, Algorithm 1 clearly outperforms IRKA with regard to the H 2 (Ā c ) error. Let us emphasize that Algorithm 1 first of all tries to ensure that the resulting reduced model has its poles also in the open upper half plane. However, the reduced poles generally cannot be expected to remain exactly on the imaginary axis, therefore resulting in unstable systems. Nevertheless, in Figure 4 the trajectories of the full order model (FOM) and the resulting reduced order model (ROM) with r = 15 are considered for a sinusoidal input of 1 Hz. We see that the output of the reduced model y r almost exactly replicates y with a relatively low absolute error. Figure 4 shows how the trajectories of a ROM computed by IRKA, with r = 15, behaves under the same sinusoidal input as above. Here the initial shifts were chosen specifically so that IRKA would converge. In Figure 4 we can see that the absolute error given by the ROM of Algorithm 1 outperforms the one computed by IRKA. This is due to the poles of the system being on the upper part of the imaginary axis, making the H 2 (C + ) framework, and consequently IRKA, unfeasible for this problem. Similar conclusions can be drawn for the step response in Figure 5. Wave equation As a second example, we consider the linear undamped wave equation subject to distributed control and observation given by ∂ 2 w(x, t) ∂t = ∂ 2 w(x, t) ∂x 2 + χ [ where χ [0.6,0.7] denotes the indicator function on the interval I = [0.6, 0.7]. Again, we employ a finite difference discretization with 5000 inner grid points leading to a first order ODE system of dimension n = 10000. Here, the poles are located on the imaginary axis but they are now symmetrically distributed according to the real axis. For this example, we choose the conformal map described in Section 4.1.3. We utilize (42) where we include a translation by c ∈ C, and scaling and rotation by M ∈ C. To include all the FOM poles we choose c = −1 × 10 −3 and M = 1.5 × 10 4 i. To restrict the poles of the reduced model on the imaginary axis we choose R = 1 + 1 × 10 −6 . This makes the minor axis (R − R −1 )/2 approach 0 and so constraining Algorithm 1 to position the poles on the imaginary axis. However, because the poles of the FOM transfer function H will be close to the boundary of A, i.e., the ellipse, the poles of H H will get closer to the imaginary axis. This can lead to some numerical issues in the construction of the reduced model and the computation of the H 2 (Ā c ) norm. The interpolation points are chosen according to (43) in each iteration of Algorithm 1. Here, the initial shifts are taken with fixed real part at 0.1 and a normally distributed random choice of the imaginary part. It is worth mentioning that the choice of parameters in (42), and eventually in (43), strictly depends on the position of the system poles. This requires the user to have some knowledge regarding the location of the spectrum for the correct use of the conformal map. Figure 6 shows the impulse response of the FOM and a reduced model of order r = 20. We see that the two trajectories almost match with low absolute error. Even if in this example Algorithm 1 shows potentially good performance for low frequencies, it must be pointed out that this is dependent on the choice of the initial shifts. As mentioned above, the boundary of the ellipse is mapped into the imaginary axis by ψ −1 . Having the poles of the FOM very close to the boundary of the ellipse makes the computation of the reduced system more sensible to the choice of the initial shifts. Nevertheless, with this approach, we can restrict the poles of the ROM to be almost on the imaginary axis. Conclusions In this paper, we introduced a novel H 2 optimal model reduction framework that can treat transfer functions with poles in general domains. For this purpose, we used conformal maps to define the H 2 (Ā c ) space and derived first order optimality conditions. With some additional assumptions, we retrieved simplified optimal interpolation conditions which we used to develop a modified version of IRKA. Through numerical experiments, we demonstrated how our method overcomes the challenges faced by IRKA in the case of non-asymptotically stable systems. Several ideas for future research can be further investigated. One direction is to develop a fully data-driven version of Algorithm 1, see also [28]. Future research should also include the extension of our theoretical results to the H ∞ model reduction setting. Additionally, a connection between the H 2 (Ā c ) error and the L ∞ error (in time domain) of the system output could lead to interesting insights into the theory of this paper. Moreover, other conformal maps could be analyzed and tested with the proposed framework. Theorem 2 ([ 35 , 235Theorem 6.1.2]) Let ψ : X → Y, with X, Y ⊂ C open, be Fréchet differentiable as a function of two real variables. The mapping ψ is conformal in X if and only if it is analytic in X and ψ ′ (s 0 ) = 0 for every s 0 ∈ X. Fig. 1 1Depiction of the sets introduced in Assumption 1 and the mapping ψ along with its inverse ψ −1 . Fig. 2 2Mapping from the left half plane into the interior of a Bernstein ellipse with major and minor axis (R + R −1 )/2 and (R − R −1 )/2 respectively. Fig. 3 3The H 2 (Ā c ) relative error of Algorithm 1 and IRKA for different reduced orders r and n = 1000.which rotates (clockwise) the left half plane by π 2 ψ(s) = −is. Fig. 4 ( 4Top) the real and imaginary output responses of the FOM (y) and ROM ( yr) to a sinusoidal input of 1 Hz. Here the ROM system matrices are computed with Algorithm 1. (Bottom) the absolute error of the reduced model computed with Algorithm 1 and IRKA. Fig. 5 ( 5Top) the real and imaginary output step responses of the FOM (y) and ROM ( yr). Here the ROM system matrices are computed with Algorithm 1. (Bottom) the absolute error of the reduced model computed with Algorithm 1 and IRKA. Fig. 6 ( 6Top) the output impulse response of the FOM (y) and ROM ( yr). 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{'abstract': 'Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space H2, a new H2 like optimal model reduction problem is introduced and first order optimality conditions are derived. As in the classical H2 case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.', 'arxivid': '2305.01511', 'author': ['Alessandro Borghi \nInstitut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany\n', '† \nInstitut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany\n', 'Tobias Breiten tobias.breiten@tu-berlin.de \nInstitut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany\n'], 'authoraffiliation': ['Institut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany', 'Institut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany', 'Institut für Mathematik\nTechnische Universität Berlin\nStraße des 17. Juni 13610623Berlin, BerlinGermany'], 'corpusid': 258437230, 'doi': '10.48550/arxiv.2305.01511', 'github_urls': ['https://github.com/aaborghi/H2-arbitrary-domains'], 'n_tokens_mistral': 22574, 'n_tokens_neox': 19358, 'n_words': 10971, 'pdfsha': 'a99d4da08a9be60cac57a21924b13c58dcdff74c', 'pdfurls': ['https://export.arxiv.org/pdf/2305.01511v1.pdf'], 'title': ['H 2 optimal model reduction on general domains', 'H 2 optimal model reduction on general domains'], 'venue': []}

arxiv

Light, matter, and quantum randomness generation: A relativistic quantum information perspective Richard Lopp Department of Applied Mathematics University of Waterloo N2L 3G1WaterlooOntarioCanada Institute for Quantum Computing University of Waterloo N2L 3G1WaterlooOntarioCanada Eduardo Martín-Martínez Department of Applied Mathematics University of Waterloo N2L 3G1WaterlooOntarioCanada Institute for Quantum Computing University of Waterloo N2L 3G1WaterlooOntarioCanada Perimeter Institute for Theoretical Physics 31 Caroline St NN2L 2Y5WaterlooOntarioCanada Light, matter, and quantum randomness generation: A relativistic quantum information perspective We study how quantum randomness generation based on unbiased measurements on a hydrogenlike atom can get compromised by the unavoidable coupling of the atom with the electromagnetic field. We improve on previous literature by analyzing the light-atom interaction in 3+1 dimensions with no single-mode or rotating-wave approximations and taking into account the non-pointlike nature of the atom, its orbital structure, and the exchanges of angular momentum between atom and field. We show that preparing the atom in the ground state in the presence of no field excitations is not universally the safest state to generate randomness. arXiv:1710.06875v2 [quant-ph] I. INTRODUCTION Randomness is in itself a valuable resource for vastly different fields of science spanning game theory, chaos theory and cryptography. However, classical sources cannot generate true randomness since they might depend on prior information [1]. After all, classical mechanics is a deterministic theory and thus predictable. On the other hand, quantum theory provides a fundamental source of randomness. For example, the outcome of an unbiased measurement of an observable of a quantum system in a basis complementary to the basis in which it was prepared is a priori unpredictable. In that sense, one can think of a plethora of quantum systems that one could use for extracting randomness. The majority of current quantum random number generators is of optical nature, e.g. photon counting or phase noise of lasers. Another major branch consists of electronic setups, e.g. noise generation in Zener diodes or electronic shot noise [2]. Both branches share the feature that the system used to generate quantum randomness is fundamentally and intrinsically coupled to the electromagnetic field. As such, the quantum system can become correlated with the electromagnetic field. In principle, those correlations can be exploited by adversaries to remotely make an educated guess on the outcome of the measurement without having physical access to the quantum system. In order to understand how randomness extraction can get compromised by the coupling between the quantum system used for randomness generation and the electromagnetic field, we want to study a very simple example from atomic physics: preparing a hydrogen-like atom in a given state and measuring in a complementary basis. One may think that if the preparation and the measurement are done fast enough, no information about the outcome of the measurement can possibly be leaked to the electromagnetic field; given that the atom is in its ground state (so as to minimize spontaneous emission) and placed in the vacuum of the electromagnetic field in absence of charges or currents. In fact, this first intuition happens to be confirmed under the common approximations in Quantum Optics (QO), namely the rotating-wave approximation (RWA) and single-mode approximation (SMA). A quick quantum-optical calculation shows that an adversary cannot increase their chances of guessing the outcome of the measurement correctly if the joint state of atom and field is in its ground state. However, this intuition, and the calculation that backs it up, are not revealing the full story: If the coupling strength between the atom and the field is strong enough (strong-coupling in QO [3][4][5] or ultra-strong coupling in superconducting circuits [6]), or if the time between preparation and measurement is short enough, the most common approximations in QO (that happen to violate the local covariance of the interaction and thus render QO non-relativistic [7]), such as the single mode approximation, break down. An idealized model on a fully relativistic footing has been analyzed already for the case of a massless scalar field in 1+1 dimensions [8]. There, a simplified atom is modeled as an Unruh-DeWitt detector [9,10]. It was found that information is always leaked to the quantum field due to interactions of the atom-field system which entangle the atom and the field even when they start in their respective ground states, and even if the time between preparation and measurement is small. Moreover it turned out that a superposition of ground and excited state is the optimal state in maximizing the randomness extracted. It is important to clarify how our work is distinguished from previous work on the same issue [8]. Namely, in this study we go beyond a simple scalar 1+1 dimensional field-atom toy model employed in past literature, and we consider a fully-featured hydrogen-like atom interacting with a quantum electromagnetic field via a dipole coupling in 3+1 dimensions, with the appropriate physical values for all the fundamental constants in the problem. This allows us to capture in the model the inherently anisotropic nature of the atomic transitions, as well as the exchange of angular momentum between the atom and the field, both argued as weak points of previous literature studies. We also consider a wider variety of time dependence of the interaction strength with time to model adiabatic versus sudden effects. These are the main novel points of this work. The structure is as follows: In Section II we will provide the physical background and the setup. We will also give a measure of randomness. The results for the amount of randomness generated are presented in Section III for the different considered time-dependent couplings between atom and quantum field as well as the comparison to the scalar field models of previous studies. Finally, in Section IV we will summarize and discuss the results. II. SETUP In the following, we will use natural units (c = = 1) and the Minkowski metric η = diag(−, +, +, +). We will consider a fully featured hydrogen-like atom coupled dipolarly to the electromagnetic field. We will not choose any of the usual approximations. Concretely we will not use the rotating-wave approximation [11] (whose limitations were pointed out in [7]) and we will not consider the atom to be point-like, and instead take into account its fully-featured orbital structure. We will focus on the randomness that can be extracted from a general electric dipolar transition between two levels of the atom: a ground state |g a and an excited state |e a . For the state of the field, an intuitive approach would be to consider the case where no field excitations are present near the atom if we wanted the atom to not be correlated with the field. If there were 'field quanta' around the atom, surely the probability of finding the atom in one or another state would be biased towards the excited state (through photon absorption with the field) and an adversary could use that to predict the outcome of a measurement on the atom with more than 50% accuracy, thus compromising the extraction of randomness from the atom. We would expect then that preparing the field in the vacuum state would be the best way to circumvent the bias of the probability to find the atom in the ground or excited state. For these reasons, in the same spirit as in [8], we will consider that the electromagnetic field is in the vacuum state. However, even in the vacuum we expect atom-field interaction to create correlations between atom and field, which in turn can reduce the extracted randomness. Previous studies of randomness generation in atomic systems coupled to quantum fields used a simplified model of light-matter interaction (the Unruh-DeWitt model) and generic spherically symmetric smearing functions. These approaches did not account for anisotropies in the spatial distribution of the atomic wave functions of the orbitals considered for the transition, and, furthermore, they did not take into account effects coming from the exchange of orbital angular momentum between the quantum field and the atom. We will include these aspects by considering the full features of the atom. The dipole coupling as leading multipole term between an atom and the electromagnetic field is given by the following interaction Hamiltonian H I = ex ·Ê(x, t). (1) wherex is the position operator of the electron with charge e in the hydrogen-like atom, andÊ is the electric field. If we express this interaction Hamiltonian in the position representation (described in full detail in section II of [12]), we obtain that the Hamiltonian, in the interaction picture, can be written aŝ H I (t) = e a e|x ·Ê(x, t) |g a e iΩt |e a g| + H.c. = R 3 dx F (x) ·Ê(x, t)e iΩt |e a g| + H.c. = R 3 d 3 xd(x, t) ·Ê(x, t).(2) Here, Ω is the energy gap between the energy eigenstates corresponding to the two orbitals considered, and the second equality holds by insertion of unity resolved in the position eigenbasis. The dipole momentd (restricted to the two relevant orbitals between which it mediates the particular transition we study) takes the form d(x, t) = e F (x)e iΩtσ+ + F * (x)e −iΩtσ− ,(3) whereσ + = |e a g|,σ − = |g a e| are SU(2) ladder operators. Finally, the spatial smearing vector F (x) is fixed by the wave functions of the orbitals involved in the atomic transition. Explicitly, as shown in [12], the smearing vector takes the form F (x) = Ψ * e (x)xΨ g (x),(4) where Ψ g (x) and Ψ e (x) are, respectively, the orbital wave functions of the ground and excited states of the atomic transition considered. To study interactions at finite times, we will add a switching function χ(t) controlling the coupling strength as a function of time such that H I (t) = χ(t) R 3 d 3 xd(x, t) ·Ê(x, t).(5) From a practical perspective, the switching function enables us to let the boundaries of the integration of the time evolution to ±∞. We also assume that in the asymptotic past and future atom and field are uncoupled, i.e. the switching function falls off rapidly enough or has compact support. Moreover, from a physical side it can be thought of as a way to account for the finite time between preparation and measurement: a compactly supported switching function sets a clear time stamp of the preparation time (the initial interaction time after preparation) and the measurement time (the amount of time from preparation to measurement). In addition, a switching function allows us to model more accurately experimental setups. For instance, we could initially place the atom inside a small enough cavity such that the lowest energy mode of the cavity determined by the IR cut-off lies well above the energy gap of the atom. In that case, the atom being placed inside a Faraday cage effectively does not interact with the cavity field nor with the field outside. When we subsequently remove the cavity, we create a coupling between atom and field. The interaction time is finite if we bring back the cavity. This could correspond to a sudden top-hat switching function given the cavity is removed and brought back quick enough. Another example modeled by a Gaussian switching function could correspond to an atom moving transversely through a cavity since the ground transversal mode of a cavity has a Gaussian-shaped amplitude profile. More generally, even though we study the behavior of atomfield interaction in free space in the following, one can model the evolution in a highly controlled light-matter interaction setup. It is possible to temporally vary the coupling strength between a superconducting qubit and the electromagnetic field inside a microwave cavity. In that way one can design a range of switching functions [8,13]. For our purposes, we expand the electric field operator into plane-wave modes of momentum k and polarization s, with their respective creation and annihilation operatorsâ † k,s andâ k,s , satisfying the canonical equal time commutation relations. In this form, the field operator can be written aŝ E(x, t) = 2 i=1 d 3 k (2π) 3/2 |k| 2 −iâ k,si (k, s i )e ik·x + H.c. ,(6) where k and x are 4-vectors and we denoted as (k, s 1 ) and (k, s 2 ) an arbitrary set of two independent transverse polarization vectors (k · (k, s i ) = 0). The time evolution of the coupled system of atom and quantum field is captured by the unitary operatorÛ acting on the initial joint state of the systemρ i such that after the interaction the joint system is in the statê ρ af =|Ψ af˜ Ψ| =Ûρ iÛ † ,(7)whereÛ = T exp −i ∞ −∞ dtĤ I (t) .(8) T denotes the time-ordering operation. We will assume that initially field and detector are uncorrelated and hence in a product state of the form ρ i = |Ψ a Ψ| ⊗ |0 f 0| ,(9) where |Ψ a is some arbitrary superposition of the energy eigenstates of the atom, and, as noted before, the field is in the vacuum state. Generating randomness from an atomic probe (i.e., two energy levels) is conceptually easy: one prepares an initial state of the atom, and then performs a von-Neumann measurement in a complementary basis. However, even theoretically, this protocol for extracting randomness was too naive: atoms are always intrinsically coupled to the electromagnetic field. In between the preparation of the atom and the projective measurement, the atom interacts with the electromagnetic field which will generally correlate both, giving an adversary with access to the field means to make an educated guess on the result of the measurement. Contrary to intuition, the acquisition of correlations between the field and the state of the atom can happen even if the time between preparation and projection is small, and even if both atom and field start in the ground state [7,8]. These correlations serve as a bias which can be exploited by an adversary who has access to the field to infer the measurement outcome better than just by chance. In order to prevent this, two options are at disposal. First, one can try to change the initial state of the atom to minimize these correlations, and secondly a different measurement basis might allow to re-establish an unbiased situation. Let us formalize the problem: The joint system (atomfield) is prepared in its initial state in some basis at some time. Following preparation, atom and field interact with each other, and after some time T the von-Neumann measurement {P x } will be performed in some other arbitrary basis on the atom with the objective of generating randomness. From this measurement one obtains the result x = {0, 1}, eigenvalues of some observableX. This yields the new total stateρ x xf = |x a x| ⊗τ x f . The state of the field after the projection (τ x f ) can be obtained by tracing out the atom and it is given bŷ τ x f = tr a P xρaf tr P xρaf .(10) This state can possibly be accessed by an adversary in order to infer the measurement result x. The conditional min-entropy [8,14,15] will be used to quantify a lower bound on the extracted randomness by an adversary with access to the quantum field after the initial measurement and is defined as H min (X|F )ρ xf = − log 2 [P g (X|F )ρ xf ] ,(11) where P g (X|F )ρ xf denotes the probability of guessing correctly the outcome of a measurement on the random variable X associated to the observableX given access to the partial state of the field F , and wherê ρ xf = x p X (x)ρ x xf is the statistical ensemble of the possible measurement outcomes. The choice of the min-entropy as a figure of merit to quantify randomness is justified by the following rationale: Since the min-entropy takes the value k if all outcomes of a distribution occur at most with probability 2 −k , we have a necessary condition to generate k random bits from the distribution. More generally, the distribution only has to be -close to a distribution that has min-entropy k [16]. The min-entropy also constitutes a much better estimator of randomness than the Shannon entropy, which coin-cides with the min-entropy for homogeneous (flat) distributions. The reason is that the Shannon entropy yields the gain of information about a distribution obtained per individual sampling after taking the average over (asymptotically infinitely) many independent samples, whereas the min-entropy quantifies the gain of information when taking only one sample in the 'worst-case' scenario [17]. Due to this averaging, we cannot conclude that having access to a random variable with a high Shannon entropy we are in possession of a good randomness source. Therefore, the min-entropy functions as a more conservative estimator of randomness. Indeed, the min-entropy is always bounded from above by the Shannon entropy. Accordingly, it is known that often the Shannon entropy significantly overestimates the amount of randomness obtainable from a random variable [16]. Another point to take into account is the fact that the quantum field is infinite-dimensional. From the point of view of randomness extraction, the issue of the infinitedimensionality of the field can be reduced to a problem of finite number of degrees of freedom since, by construction, the atom possesses a finite number of energy eigenstates. For example, in this paper we will consider the conservative case where we quantify the randomness that can be extracted from only two levels connected by an electric dipole transition, such that the field can excite the ground state of the atom only to one higher energy state (in the same fashion as it was done in [8] for a scalar field). We can write the final pure state of the joint system after interaction via Schmidt decomposition as |Ψ af = λ 0 |0 a ⊗ |f 0 f + λ 1 |1 a ⊗ |f 1 f ,(12) where {|i a } are the eigenstates of the observableX (not necessarily the energy eigenstates of the atom), and {|f i f } are two orthonormal basis states out of the field's infinite-dimensional Hilbert space. A priori these basis states of the field are not known, and their precise form is not even needed to arrive at an analytic expression for the amount of generated randomness. If the adversary wants to implement a protocol to optimize the guessing probability for the measurement outcome, then they would indeed need to construct {|f i f } by a Schmidt decomposition algorithm, and may involve many (possibly infinite) field modes. However, we do not concern ourselves with finding that specific decomposition as doing so is the adversary's task. Rather, our objective is to reduce their ability to make educated guesses on the randomly generated data by probing the field. Thus, we should keep the most conservative assumptions on the adversary's ability. Considering that the min-entropy is invariant under local isometries [18], we can devise a unitary operation that transfers the information from the field to an ancillary qubit E in possession of the adversary, e.g. swapping entanglement between field and E. Therefore, the new final joint state reads |Ψ ae = λ 0 |0 a ⊗ |0 e + λ 1 |1 a ⊗ |1 e .(13) Accordingly, after the von-Neumann measurement on atom A, the ensemble corresponding to the different outcomes isρ x ae = x |x a x| ⊗τ x e ,(14) with the qubit E being, for the outcome x, in the statê τ x e = tr a P xρae tr P xρae .(15) The probability of guessing correctly the outcome is equivalent to the optimal success probability of the adversary to distinguish the states of the qubitτ x e : P g X|E ρae = max E x p X (x) e x|Ê(τ x e ) |x e = max Πx x p X (x) tr Π xτ x e ,(16) where we optimize over CPTP mapsÊ or equivalently over POVMs {Π x =Ê † (|x e x|)}. It is assumed that the adversary knows the measurement basis given byX, rendering the adversary more powerful. By the Helstrom bound [19] for the minimum-error probability of distinguishing two states by optimizing over POVMs we find P g X|E ρae = 1 2 1 + p X (0)τ 0 e − p X (1)τ 1 e 1 ,(17) where Ô 1 = tr Ô †Ô is the Schatten 1-norm. Counteracting the adversary to yield the maximum (denoted by superscript * ) amount of randomness H * min which can be extracted from the atom, we have to optimize over all von-Neumann measurements on atom A. Any arbitrary complex two-dimensional projector decomposition can be written as a linear combination of projectors of the form P i = |m i a m i | with |m 0 a = cos θ |0 a + e iφ sin θ |1 a , |m 1 a = sin θ |0 a − e iφ cos θ |1 a .(18) Then we find that p X (x)τ x e = |n x e n x |, where |n 0 e = λ 0 a m 0 |0 a |0 e + λ 1 a m 0 |1 a |1 e ,|n 1 e = λ 0 a m 1 |0 a |0 e + λ 1 a m 1 |1 a |1 e .(19) This allows us to write the optimized guessing probability, by using Eq. (17), as P * g X|E ρae = min {|mi a } 1 2 1 + 1 − 4 | e n 0 |n 1 e | 2 = 1 2 1 + 1 − (λ 0 − λ 1 ) 2 = 1 2 + 1 4 − 1 2 tr (ρ 2 a ) − 1 4 ,(20) whereρ a is the reduced density matrix of the atom after its interaction with the field from preparation to measurement. Finally we find the expression for the optimized min-entropy: H * min = − log 2 1 2 + 1 − tr (ρ 2 a ) 2 .(21) Thus, it is sufficient to know the state of the atom after the interaction to fully quantify the extractable randomness. This measure of randomness generation can also be viewed from the point of view of device-independent random number generation and quantum key distribution -see, for instance, [20][21][22][23]. In that context the minentropy gives an quantitative estimate allowing to certify whether the output is truly random whilst treating the random number generator as a black box. III. RESULTS A. Final atomic state The time-evolved state will be calculated by a perturbative Dyson expansion of (8), granted the relevant parameters are small enough: U = 1 1 −i ∞ −∞ dtĤ I (t) Û (1) − ∞ −∞ dt t −∞ dt Ĥ I (t)Ĥ I (t ) Û (2) + . . .(22) Thus, to second order in the coupling constant e the evolved state takes the form ρ af =ρ i +Û (1)ρ i +ρ iÛ (1) † +Û (2)ρ i +ρ iÛ (2) † +Û (1)ρ iÛ (1) † + O(e 3 ).(23) We assume that the initial state of the field is its ground state |0 f and the atom is in some superposition of its energy eigenstates |Ψ a = a |g a + √ 1 − a 2 |e a , where we restrict a to be real and a = 1 (a = 0) corresponds to the ground state (excited state). Hence the initial state of the atom readŝ ρ a,i = a 2 a √ 1 − a 2 a √ 1 − a 2 1 − a 2(24) in the {|g a , |e a } basis. Then after interaction between atom and field, and before subsequent measurement on the atom, the final atomic state reads to second order ρ a =ρ a,i + tr f Û (1)ρ iÛ (1) † + tr f Û (2)ρ i + H.c. ∆ρ . (25) Therefore we call ∆ρ the correction to the initial state carrying the time evolution of the atom to leading order. Note that in (25) there are no first order terms. This is because for the vacuum state tr f Û (1)ρ i = 0. In Appendix A the derivation is explicitly shown in general form for arbitrary atomic transitions and switching functions. In particular, the final results for the exemplary 1s → 2p z atomic transition are given for the following switching functions (see Appendix A 1): 1) Gaussian switching χ g (t) = e −t 2 /σ 2 , 2) sudden Heaviside tophat switching χ s (t) = Θ(t)Θ(−t + σ) and 3) Dirac delta switching χ d (t) = Cδ(t), where σ is the interaction time scale and the constant C is needed for correct dimensionality. This yields for the change in the atomic state to second order in perturbation theory respectively ∆ρ g = 24576(a 0 eσ) 2 π ∞ 0 d|k| |k| 3 e − 1 2 σ 2 (|k|+Ω) 2 (4a 2 0 |k| 2 + 9) 6 2 (1 − a 2 )e 2|k|σ 2 Ω − a 2 1 0 0 −1 + a 1 − a 2 e 2|k|σ 2 Ω erf iσ(|k| − Ω) √ 2 − erf iσ(|k| + Ω) √ 2 − 1 − e |k|σ 2 Ω 2 0 0 1 0 + H.c. ,(26)∆ρ s = 49152(a 0 e) 2 π 2 ∞ 0 d|k| |k| 3 (4a 2 0 |k| 2 + 9) 6 (|k| 2 − Ω 2 ) 2 2a 2 (|k| − Ω) 2 cos(σ(|k| + Ω)) +2 1 − 2a 2 |k| 2 + Ω 2 + 4|k|Ω + 2(a 2 − 1 cos(σ(|k| − Ω))(|k| + Ω) 2 1 0 0 −1 + e 2iσΩ (|k| 2 − Ω 2 ) + |k| 2 (2iσΩ − 1) + 4Ωe iσΩ (Ω cos (|k|σ) − i|k| sin (|k|σ)) − Ω 2 (3 + 2iσΩ) ×a 1 − a 2 0 0 1 0 + H.c. ,(27)∆ρ h Ω=0 = 512e 2 295245π 2 1 − 2a 2 24 − √ πG 2,1 1,3 0 0,5, 1 2 9σ 2 16a 2 0 1 0 0 −1 ,(28)∆ρ d = 128C 2 e 2 10935π 2 a 2 0 (1 − 2a 2 ) 1 0 0 −1 ,(29) where G m,n p,q a1,...,ap b1,...,bq z is the Meijer G-function, erf(z) is the error function and a 0 is the generalized Bohr radius. Eq. (28) is obtained from (27) for degenerate atomic transitions (Ω = 0), also called gapless sudden switching. B. Extracted randomness In the previous section we have obtained the time evolved density matrix of the atom from the time of preparation to the time when the measurement is performed for the different switching functions considered, namely Gaussian (26), sudden (27)-(28), and delta (29). With this information at hand, we can now calculate the number of bits of randomness that can be generated with each measurement. We will present the results for Gaussian switching, gapless sudden switching and delta switching separately. The first step is to choose physically meaningful values for the parameters of the problem. As a baseline, we start with the parameters a 0 ≈ 2.68 · 10 −4 eV −1 , e ≈ 137 −1/2 ≈ 8.54 · 10 −2 , Ω ≈ 3.73 eV. These have been chosen such that the atomic radius corresponds to the Bohr radius, e to the standard electric charge in vacuum (the square root of the fine structure constant in natural units) and a 0 Ω ≈ 0.001 is of the same order of a typical transition from the ground state to the first excited state in a hydrogen-like atom [24]. By varying a 0 , e, Ω we will study how the generated randomness is dependent on these parameters. B1. Gaussian switching: χ g (t) = e −t 2 /σ 2 For a Gaussian switching function, the amount of randomness that can be generated as a function of the interaction time σ and initial superposition parameter a is shown in Fig. 1. As a general feature we note that for shorter interaction times the amount of randomness is compromised more severely. In fact, we see that for the regular free-space coupling of Fig. 1a, Gaussian switching provides a good source of randomness for interaction times above ≈ 10 −2 eV −1 , which in principle tells us that an adiabatic switching (smooth switching that depends only on one timescale, such as Gaussian) prevents the generation of atom-field correlations well enough to guarantee a reliable extraction of randomness. However, this is not true for regimes of strong coupling: as we will comment on below, the amount of randomness extracted decays fast with the interaction strength and becomes relevant for strong coupling strengths. Remarkably, and contrary to intuition the ground state is not the most secure choice of initial atom preparation for short interaction times. It turns out that an equal superposition of ground and excited state is most resilient and in fact yields min-entropy very close to 1 bit. Moreover, the initial guess that the excited state of the atom may be the worst preparation (because of its probability of spontaneously decay) is not the complete picture. Surprisingly, the ground state is almost as bad a choice as the excited state in terms of generation of randomness. This stresses our claim: for fast randomness generation the equal superposition state provides the best possible initialization of the system. Nonetheless, as we would expect, for the late interaction time regime we recover that the ground state yields maximum randomness generation whereas all other state preparations, including excited state and equal superposition, experience a decrease in randomness for longer interaction times (see Fig. 2). In Fig. 3 we show the dependence of the extracted randomness on the parameters e, a 0 and Ω. For the chosen interaction time of σ = 2.5 × 10 −3 , we are in a regime where the rotating wave approximation is not valid, i.e. σΩ 1. The stronger the coupling e between atom and field the less randomness will be generated overall since it results in the enhancements of acquired atom-field correlations. The extracted randomness falls off more quickly for states which are closer to being either of the two energy eigenstates. This is particularly relevant as it is shown in Fig. 1b: in regimes of strong coupling the loss of randomness at short times can still be relatively significant for timescales of 10 −1 eV −1 . One concern may be that the perturbative expansion is not valid for regimes of very low min-entropy as shown in Fig. 1b. For instance, H min = 0.4 bit means that the final atomic state has a purity of, using (21), tr ρ 2 a = 0.87. In the Appendix A 1 b we will present numerical values for the change of the atomic state (for all switching functions), showing that perturbation theory holds for low min-entropy in our analysis up to some value of the coupling strength (depending on the switching function) for which we provide a lower bound in the appendix. For the dependence on the atomic radius we find that for large values of a 0 the generated randomness asymptotically approaches a constant value after passing through a minimum. The depth of the minimum is larger for states that are closer to either of the energy eigenstates. Hence the equal superposition of them shows to be very close to constant. The extracted randomness decreases with larger values for the energy gap Ω for atomic states with parameter a ≤ 1/ √ 2 and increases for the remaining states. Hence the ground state or in general states with the major probability of being in the ground state after preparation become more secure when the gap between the energy eigenstates increases. This is consistent with the intu- ition that a larger gap makes it more difficult for the ground state to get excited through a counter-rotating process (emitting excitations that could be captured by an adversary). At the same time, increasing the gap increases the probability that the excited states decayed emitting light, which in turn can be captured to infer the measurement outcome. An equal superposition state is overall most resistant to variations in these parameters and, moreover, is close to being constant in all three parameter cases. 1 • • • • ▲ ▲ ▲ ▲ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ○ ○ ○ ○ ★ ★ ★ ★ B2. Sudden switching: χ s (t) = Θ(t)Θ(−t + σ) We consider here the case of an infinitely fast switching on and off, modelled by a square function. For the sudden top-hat switching we will study the case of degenerate atomic transitions (Ω = 0) due to numerical simplicity. The min-entropy portraits a different picture (see Fig. 4a) than for the Gaussian switching function. It is still true that an equal superposition of ground and excited state is the most secure state to generate randomness for general interaction times. However, short interaction times between field and atom yield a larger min-entropy than longer interaction times. On the other hand, for later times the min-entropy varies very little with the interaction time for fixed a. It suggests that the amount of randomness that can be extracted takes an asymptotic value for fixed a. In this case, we observe that it is preferable to perform the measurement very fast in order to avoid the loss of randomness coming from the regime of long interaction times. This stands in contrast to the Gaussian switching where it is better to choose a longer interaction time between atom and electromagnetic field. From (28) it is obvious that to second order in perturbation theory the equal superposition provides us with a state that yields H min = 1 bit since that state is a fixed point in time evolution (does not vary in time) for the degenerate transition case. We want to highlight that the symmetry around a = 1/ √ 2 is exact in Fig. 4a. Details are given in the Appendix A 1 c. There we will also comment on the exact symmetry in the case of Dirac switching. In Fig. 4b and Fig. 4c the dependence of the min-entropy on its parameters e and a 0 is shown for fixed times σ. As in the case of Gaussian switching, a stronger coupling implies a decrease of extracted randomness. It also holds that states prepared close to being in an equal superposition show a slower decrease in the min-entropy than for states which are prepared close to being in an energy eigenstate. Moreover, for small atomic radii a 0 the extracted randomness shows a minimum and increases then asymptotically to a constant value of the min-entropy, depending on a. In summary, the equal superposition provides the optimal state to extract randomness as it is in fact independent of the parameters to leading order in perturbation theory. It should be noted that because the eigenstates of the atom are degenerate (Ω = 0) there is no regime where the RWA condition is satisfied. B3. Delta switching: χ d (t) = Cδ(t) We consider here the effect of a fast kick of the system, modelled by a delta coupling. This can be seen as the limit of a succession of thinner Gaussian (or top-hat) functions of equal area (recall that this way of interpreting the delta as a limit is important for the results at hand, as discussed in detail in [25], and in Appendix A). Due to the pointlike-in-time nature of the interaction, we are explicitly in a regime where the rotating wave approximation does not hold. Studying the delta switching, we take C = σ = 2.5 × 10 −3 eV −1 (reading (29) we note that C acts in the same way as the coupling constant). The particular choice for C means that the time-integrated switching function is proportional to σ, as was in the case for Gaussian and sudden switching. Eq. (29) shows that once again the equal superposition yields perfect randomness extraction H min = 1 bit. This can be seen in Fig. 5. Fig. 5a shows that the min-entropy peaks at equal superposition of the atom's initial state and quickly decreases at either side, resulting in a much larger loss of randomness than for any of the previously studied switching functions. Moreover the peak becomes narrower the stronger the coupling between atom and field is, spoiling quickly any randomness extraction if it is not in an equal superposition state. From Fig. 5b and Fig. 5c we see that, as expected, a stronger coupling between atom and electromagnetic field causes larger correlations and reduces the min-entropy. In addition, the dependence on the atomic radius displays an increase to an asymptotic value of the min-entropy. In contrast to the two previous switching functions, the delta switching shows much larger variations in the minentropy. C. Comparison with scalar field models Let us now compare our results to earlier studies where the atom was modeled as an Unruh-DeWitt (UDW) detector coupled to a scalar field φ(x, t). The UDW model has been shown to capture the fundamental features to leading order of light-matter interactions as long as there is no exchange of orbital angular momentum [12,26,27]. We will consider two different kinds of UDW detectors, namely the original UDW model [9,10] and the derivative coupling (that we will denote as UDW d ) [28]. The respective Hamiltonians are whereμ(t) is the monopole moment capturing the internal degrees of freedom of the detector, and F (x) is the ad hoc included spatial smearing function of the detector. In particular, (31) has been used in previous literature to analyze the loss of randomness due to coupling to relativistic fields [8], so it makes sense to compare the results of the simplified scalar model with the realistic hydrogenlike model employed here. H UDW = eχ(t) dx 3 F (x)μ(t)φ(x, t),(30)H UDW d = eχ(t) dx 3 F (x)μ(t)∂ tφ (x, t),(31) The difference between the EM coupling and these two models has been analyzed in the past in the context of entanglement harvesting [12]. The UDW d model can be thought of as a scalar analogue of the dipole coupling by noting that in the Coulomb gauge E = −∂ t A and one may perhaps expect that it should resemble the dipole interaction to some extent (as discussed in [8]). Both scalar models do not allow transitions where there is exchange of angular momentum. In particular, the 1s → 2p z transition is not permitted. Same as in [12] we will consider the closest scalar analogue to that transition to compare to the EM case; that is 1s → 2s for the scalar models, but keeping the 1s → 2p z transition for the EM model. The change in the density matrix of the atomic state after an interaction of time σ takes for the scalar couplings the form ∆ρ UDW = − 32768 π a 2 0 eσ 2 ∞ 0 d|k| |k| 5 e − 1 2 σ 2 (|k|+Ω) 2 (4a 2 0 |k| 2 + 9) 6σ z ,(32)∆ρ UDW d = − 32768 π a 2 0 eσ 2 ∞ 0 d|k| |k| 7 e − 1 2 σ 2 (|k|+Ω) 2 (4a 2 0 |k| 2 + 9) 6σ z ,(33) withσ z being the Pauli Z matrix. The scalar models were derived by assuming a Gaussian switching function and the initial ground state of the detector (a = 1). In addition the smearing function was chosen as the scalar version of the smearing vector: F (x) = ψ e (x)ψ g (x). Consequently, we have to analyze the electric dipole model in the respective configuration slice. It should be noted that the coupling constants e of the different couplings do not all have the same dimensionality. In particular, for the dipole and direct scalar interaction we find [e] = 0, whilst for the derivative coupling [e] = −1 (in mass dimensions). We choose the parameters a 0 = 2.68·10 −4 eV −1 , e = 10 −3 , Ω = 3.73 eV, taking into account that for stronger couplings the perturbative expansion of the UDW d model breaks down by virtue of the additional |k| 2 dependence in (33). In Fig. 6 one finds that the derivative model vastly underestimates the extracted randomness for early times and is off by up to over 40 %. On the other hand the UDW model slightly overestimates it for short interaction times by the order of 10 −2 %. For long interaction times both scalar models approach the realistic dipole model. Since σΩ 1 in the plots, we again go beyond the validity of the RWA. IV. CONCLUSIONS We quantified a lower bound for the randomness that can be extracted from a hydrogen-like atom coupled to the electromagnetic field. This work is an advancement of the previous work [8] by having considered a fully-featured hydrogen-like atom coupling to an electromagnetic field in 3+1 dimensions (instead of a monopole detector coupling to a scalar field in 1+1D). In doing so we have tackled previous criticisms to former literature in the choice of the smearing function (here derived from the atomic orbital wavefunctions from first principles), the anisotropic nature of atomic transitions and exchange of angular momentum between atom and quantum field, as well as the choice of the value of the physical parameters of the problem (here coming from first principles). Lastly, we compared adiabatic and sudden time-dependencies of the interaction strength. We emphasize that, same as in the simpler models employed in [8], we also did not make any use of the usual simplifications of the interaction in the context of quantum optics. Namely, we did not assume the rotating wave approximation or the single mode approximation. We analyzed how much information an adversary with access to the EM field but not the atom can obtain about a supposedly random measurement outcome. We found (consistently with studies that considered simplified scalar field interaction models [8]) that generally the ground state of the atom and the vacuum state of the field is not the optimal state to generate randomness out of a succession of preparation and measurement in unbiased bases for the atomic state basis. We have analyzed a variety of switching regimes and found that for the switching function as well as the duration of the interaction between atom and electromagnetic field there are two possibilities for choosing the optimal state in terms of randomness generation: For short time between preparation and measurement in the unbiased basis, the equal superposition between ground and excited atomic states yields the optimal randomness. For coupling and decoupling times much shorter than the inverse of the frequency of the atomic transition (that could be thought of as preparation-to-measurement times), the equal superposition between ground and excited yields the best results even for long times between preparation and measurement. In contrast, for adiabatic switching and long times, the ground state of the atom yields the optimal randomness generation. Furthermore, we also showed that in the cases where the equal superposition is optimal, the ground state is one of the two worst choices (together with the excited atomic state) in order to generate randomness, something that contradicts the intuition coming from the rotating wave approximation that basically would suggest that 'if everything is in the ground state the field and the atom will remain uncrorrelated'. Finally, we compared the realistic model of the electromagnetic field coupled to the atom via a dipole moment to simplified scalar models used in previous studies. We found that both the Unruh-Dewitt coupling [10] and the derivative coupling [28] provide a good approximation for the full electromagnetic model for long enough interaction times. For short interaction times, the Unruh-Dewitt model is a better approximation than the derivative coupling, which significantly deviates from the full electromagnetic calculation. This information is useful when considering scalar approximations to the lightmatter interaction We would like to emphasize that the scope of this paper is not as much to describe a particular experimental setup, but rather study how the flow of quantum information in special relativistic quantum regimes deviates from non-relativistic scenarios (which make use rotating wave and single mode approximations) by virtue of the entanglement between the atom and the electromagnetic field. In this context we aimed to analyze how this flow of information impacts the ability of obtaining certified randomness, even under ideal assumptions regarding preparation and measurement procedures, as a matter of first principles. In this section we will derive in detail the change of the reduced density matrix of the atom after interaction with the electromagnetic field. Starting from Eq. (25) by recalling Eq. (22) we find tr f Û (2)ρ i = tr f   − R dt t −∞ dt χ(t)χ(t ) R 3 dx R 3 dx 3 i,j=1d i (x, t)Ê i (x, t)d j (x , t )Ê j (x , t )ρ i   = − R dt t −∞ dt χ(t)χ(t ) R 3 dx R 3 dx 3 i,j=1d i (x, t)d j (x , t ) |Ψ a Ψ| e 0|Ê i (x, t)Ê j (x , t ) |0 e . (A1) Since the atomic state outer product |Ψ a Ψ| is on the right-hand side of the product of the dipole operators, only terms with one raising and one lowering operator survive. Similarly, tr f Û (1)ρ iÛ (1) † = R dt R dt χ(t)χ(t ) R 3 dx R 3 dx 3 i,j=1d i (x, t) |Ψ a Ψ|d j (x , t ) tr f Ê i (x, t) |0 e 0|Ê j (x , t ) ,(A2) where we used thatd i andÊ i are Hermitian. This yields then tr f Û (2)ρ i = −e 2 R dt t −∞ dt χ(t)χ(t ) R 3 dx R 3 dx 3 i,j=1 1 − a 2 |e a e| + a 1 − a 2 |e a g| F T i (x)W ij (x, x ; t, t )F * j (x )e iΩ(t−t ) + a 2 |g a g| + a 1 − a 2 |g a e| F * T i (x)W ij (x, x ; t, t )F j (x )e −iΩ(t−t ) ,(A3)tr f Û (1)ρ iÛ (1) † =e 2 R dt R dt χ(t)χ(t ) R 3 dx R 3 dx 3 i,j=1 a 2 |e a e| F * T i (x )W ij (x , x; t , t)F j (x)e iΩ(t−t ) + 1 − a 2 |g a g| F T i (x )W ij (x , x; t , t)F * j (x)e −iΩ(t−t ) + a 1 − a 2 |g a e| F * T i (x )W ij (x , x; t , t)F * j (x)e −iΩ(t+t ) + a 1 − a 2 |e a g| F T i (x )W ij (x , x; t , t)F j (x)e iΩ(t+t ) ,(A4) where we have defined the Wightman 2-tensor for the electric field W ij (x 2 , x 1 ; t 2 , t 1 ) = e 0|Ê i (x 2 , t 2 )Ê j (x 1 , t 1 ) |0 e = R 3 d 3 k (2π) 3 |k| 2 e −i|k|(t2−t1) e ik·(x2−x1) δ i,j − k i k j |k| 2 .(A5) To arrive at that expression the completeness relation of the polarization vectors (k, s i ) was used: 2 i=1 (k, s i ) ⊗ (k, s i ) = 1 1 − k ⊗ k |k| 2 .(A6) In the following we will drop the subscripts of the outer products belonging to the Hilbert space of atom A. We will separate the terms in the Wightman tensor according to the identity 1 1 and the dyadic product k ⊗ k (such that their sum corresponds to the complete expression), denoted by the corresponding subscripts. We wish to integrate over spherical coordinates, naturally suggested by the wave function Ψ nlm (x) = R nl (|x|)Y lm (x), where Y lm (x) are the spherical harmonics withx = (θ x , φ x ) as the angular components of the unit radial vector, and R nl (|x|) are the radial wave functions of a hydrogenoid atom [29]. The following two decompositions are helpful: e ix·y = ∞ l=0 l m=−l 4πi l j l (|x||y|)Y lm (x)Y * lm (ŷ) = ∞ l=0 l m=−l 4πi l j l (|x||y|)Y * lm (x)Y lm (ŷ), (A7) x · y = 4π 3 |x||y| [Y 10 (x)Y 10 (ŷ) − Y 11 (x)Y 1−1 (ŷ) − Y 1−1 (x)Y 11 (ŷ)] ,(A8) with the spherical Bessel functions j l (x). The first contribution to the time evolved density matrix then reads tr f Û (2)ρ i 1 1 = −e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 l m=−l 4πi l ∞ l =0 l m =−l 4πi l (−1) l 4π 3 R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) × ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) × dΩ k Y lm (k)Y l m (k) dΩ x dΩ x Y * lm (x)Y * l m (x ) Y 10 (x)Y 10 (x ) − Y 11 (x)Y 1−1 (x ) − Y 1−1 (x)Y 11 (x ) × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) Y * leme (x)Y lgmg (x)Y leme (x )Y * lgmg (x ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) Y leme (x)Y * lgmg (x)Y * leme (x )Y lgmg (x ) ,(A9) where we have used the identity Y lm (−x) = (−1) l Y lm (x) and that R nl (|x|) is real. Also dΩ = d(cos Θ)dφ is the standard solid angle differential. The integral over dΩ k reads dΩ k Y lm (k)Y l m (k) = (−1) m δ l,l δ m,−m by using Y * lm (x) = (−1) m Y l−m (x) . This simplifies the integrals over the other two solid angles drastically such that we can use the following identity of spherical harmonics integrated over the unit sphere S 2 dΩ Y * l1,m1 (x)Y * l3,m3 (x)Y l2,m2 (x)Y l4,m4 (x) = ∞ λ=0 λ µ=−λ 2λ + 1 4π (2l 1 + 1)(2l 2 + 1)(2l 3 + 1)(2l 4 + 1) l 1 l 3 λ 0 0 0 l 2 l 4 λ 0 0 0 l 1 l 3 λ −m 1 −m 3 −µ l 2 l 4 λ m 2 m 4 µ ,(A10) with l 1 l 2 l 3 m 1 m 2 m 3 as the Wigner 3j-symbols (see, for instance, section 34.2 of [30]). With this formula, the sums over l , m, m and the integrals over the all solid angles can be executed. Let us concentrate first on the second term of the sum in the curly brackets of (A9), coming from F * T i W ij F j , which yields ∞ l =0 l m=−l l m =−l i l+l (−1) l j l (|k||x |) dΩ k Y lm (k)Y l m (k) dΩ x Y leme (x)Y * lgmg (x)Y * lm (x) × dΩ x Y * leme (x )Y lgmg (x )Y * l m (x ) Y 10 (x)Y 10 (x ) − Y 11 (x)Y 1−1 (x ) − Y 1−1 (x)Y 11 (x ) = 3(−1) mg−me i 2l (−1) l (4π) 2 (2l + 1)(2l e + 1)(2l g + 1) ∞ λ,λ =0 (2λ + 1)(2λ + 1)j l (|k||x |l g 1 λ m g −1 1 − m g ,(A11) where the sums over µ and µ can be executed by using that 3j-symbols are zero unless the sum over the entries of the lower row is zero. The first term from (A9) can be obtained from (A11) by noting that effectively l and l , thus also m and m , are interchanged and hence it requires to take (−1) mg−me → (−1) −mg+me . Since this is equivalent, the first term can also be described by (A11). Therefore, in all generality (A9) reads tr f Û (2)ρ i 1 1 = −e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 (4π) 2 4π 3 R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) × ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) 3(−1) mg−me (4π) 2 (2l + 1)(2l e + 1)(2l g + 1)l g 1 λ m g −1 1 − m g × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) .(A12) Before specifying atomic transition or the switching function of the coupling to the electric field, we will derive the general expressions of the remaining terms, having derived terms containing F * T i W ij F j and F T i W ij F * j of the 1 1 part. Secondly we look at the remaining 1 1 contribution: tr f Û (1)ρ iÛ (1) † 1 1 = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 l m=−l 4πi l ∞ l =0 l m =−l 4πi l (−1) l 4π 3 R dt R dt χ(t)χ(t )e −i|k|(t −t) dΩ k Y lm (k)Y l m (k) × ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) dΩ x dΩ x Y * lm (x)Y * l m (x ) × Y 10 (x)Y 10 (x )−Y 11 (x)Y 1−1 (x )−Y 1−1 (x)Y 11 (x ) 1 − a 2 |g g|e −iΩ(t−t ) Y * leme (x )Y lgmg (x )Y leme (x)Y * lgmg (x) + a 2 |e e| e iΩ(t−t ) Y leme (x )Y * lgmg (x )Y * leme (x)Y lgmg (x) + a 1 − a 2 |g e| e −iΩ(t+t ) Y leme (x )Y * lgmg (x )Y leme (x) ×Y * lgmg (x) + a 1 − a 2 |e g| e iΩ(t+t ) Y * leme (x )Y lgmg (x )Y * leme (x)Y lgmg (x) .(A13) From (A11) we already know how to compute the first two terms in the curly brackets. The other two follow immediately by noting that they can be obtained from the known terms by including or removing the conjugate of one of the smearing functions. Either way, effectively l e ↔ l g and m e ↔ m g change in the corresponding term. As we recall from (A11), we had the requirement that m = −m and hence all contributions disappear except when m e = m g since the Wigner 3-j symbols vanish in case the sum of the lower components does not equal zero. Thus we find tr f Û (1)ρ iÛ (1) † 1 1 = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 (4π) 2 4π 3 R dt R dt χ(t)χ(t )e −i|k|(t −t) ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) × ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) 3(−1) mg−me (4π) 2 (2l + 1)(2l e + 1)(2l g + 1) = 9(2l + 1)(2l + 1)(2l g + 1)(2l e + 1) × l g 1 λ m g 1 −1 − m g l l g λ m g − m e − 1 −m g m e + 1 l e 1 λ m e 1 −1 − m e .(A17) This yields hence tr f Û (2)ρ i k⊗k = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 l m=−l 4πi l ∞ l =0 l m =−l 4πi l (−1) l 4π 3 2 R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) × ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) × dΩ k Y lm (k)Y l m (k) dΩ x dΩ x Y * lm (x)Y * l m (x ) Y 10 (x)Y 10 (k) − Y 11 (x)Y 1−1 (k) − Y 1−1 (x)Y 11 (k) × Y 10 (k)Y 10 (x ) − Y 11 (k)Y 1−1 (x ) − Y 1−1 (k)Y 11 (x ) × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) Y * leme (x)Y lgmg (x)Y leme (x )Y * lgmg (x ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) Y leme (x)Y * lgmg (x)Y * leme (x )Y lgmg (x ) = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 4π R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) ∞ l,l =0 i l+l (2l + 1)(2l + 1)(2l g + 1)(2l e + 1) × ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |) × ∞ λ ,λ =0 (2λ + 1)(2λ + 1) l l g λ 0 0 0 l e 1 λ 0 0 0 l l e λ 0 0 0 l g 1 λ 0 0 0 (A + B) × (−1) l j l (|k||x |)j l (|k||x|) 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) +(−1) l j l (|k||x|)j l (|k||x |) a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) ,(A18) where we redefined l ↔ l for the first term in the curly brackets to derive the last formula. The last contribution we have to calculate is analogously tr f Û (1)ρ iÛ (1) † k⊗k = −e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 ∞ l=0 l m=−l 4πi l ∞ l =0 l m =−l 4πi l (−1) l 4π 3 2 R dt R dt χ(t)χ(t )e −i|k|(t −t) × ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|)j l (|k||x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |)j l (|k||x |) × dΩ k Y lm (k)Y l m (k) dΩ x dΩ x Y * lm (x)Y * l m (x ) Y 10 (x)Y 10 (k) − Y 11 (x)Y 1−1 (k) − Y 1−1 (x)Y 11 (k) × Y 10 (k)Y 10 (x )−Y 11 (k)Y 1−1 (x )−Y 1−1 (k)Y 11 (x ) 1 − a 2 |g g| e −iΩ(t−t ) Y * leme (x )Y lgmg (x )Y leme (x)Y * lgmg (x) + a 2 |e e| e iΩ(t−t ) Y leme (x )Y * lgmg (x )Y * leme (x)Y lgmg (x) + a 1 − a 2 |g e| e −iΩ(t+t ) Y leme (x )Y * lgmg (x )Y leme (x)Y * lgmg (x) +a 1 − a 2 |e g| e iΩ(t+t ) Y * leme (x )Y lgmg (x )Y * leme (x)Y lgmg (x) = −e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 4π R dt R dt χ(t)χ(t )e −i|k|(t −t) ∞ 0 d|x||x| 3 R nele (|x|)R nglg (|x|) ∞ 0 d|x ||x | 3 R nele (|x |)R nglg (|x |) × ∞ l,l =0 i l+l (2l + 1)(2l + 1)(2l g + 1)(2l e + 1) ∞ λ ,λ =0 (2λ + 1)(2λ + 1) l l g λ 0 0 0 l e 1 λ 0 0 0 l l e λ 0 0 0 l g 1 λ 0 0 0 = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 4π R dt R dt χ(t)χ(t )e −i|k|(t −t) ∞ 0 d|x||x| 3 R 2,1 (|x|)R 1,0 (|x|)j l (|k||x|) × ∞ 0 d|x ||x | 3 R 2,1 (|x |)R 1,0 (|x |)j l (|k||x |) 1 3 [j 0 (|k||x|)j 0 (|k||x |) + 2j 2 (|k||x|)j 2 (|k||x |)] × 1 − a 2 |g g| e −iΩ(t−t ) + a 2 |e e| e iΩ(t−t ) + a 1 − a 2 |e g| e iΩ(t+t ) + a 1 − a 2 |g e| e −iΩ(t+t ) , (A24) where we used for last term in the curly brackets that λ = λ = l and λ = 0, 1, 2. For the penultimate term in the curly brackets one finds λ = λ = 1 and l = 0, 1, 2. By virtue of (A22) we arrive at tr f Û (1)ρ iÛ (1) † 1 1 = e 2 a 2 0 663552 π 2 ∞ 0 d|k||k| 3 R dt R dt χ(t)χ(t )e −i|k|(t −t) 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 × 1 − a 2 |g g| e −iΩ(t−t ) + a 2 |e e| e iΩ(t−t ) + a 1 − a 2 |e g| e iΩ(t+t ) + a 1 − a 2 |g e| e −iΩ(t+t ) . (A25) The same will be shown now for the dyadic contributions. Starting from (A18) it yields tr f Û (2)ρ i k⊗k = e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 4π R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) ∞ 0 d|x||x| 3 R 2,1 (|x|)R 1,0 (|x|) × ∞ 0 d|x ||x | 3 R 2,1 (|x |)R 1,0 (|x |) 1 9 [j 0 (|k||x|) − 2j 2 (|k||x|)] [j 0 (|k||x |) − 2j 2 (|k||x |)] × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) ,(A26) by noting that λ = λ = 1 and l, l , λ = 0, 1, 2. Solving again the spatial integrals we obtain the form tr f Û (2)ρ i k⊗k = e 2 a 2 0 24576 π 2 ∞ 0 d|k||k| 3 R dt t −∞ dt χ(t)χ(t )e −i|k|(t−t ) 9 − 20a 2 0 |k| 2 2 (4a 2 0 |k| 2 + 9) 8 × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) . (A27) Finally, (A19) gives tr f Û (1)ρ iÛ (1) † k⊗k = −e 2 ∞ 0 d|k| (2π) 3 |k| 3 2 4π R dt R dt χ(t)χ(t )e −i|k|(t −t) ∞ 0 d|x||x| 3 R 2,1 (|x|)R 1,0 (|x|) ∞ 0 d|x ||x | 3 R 2,1 (|x |)R 1,0 (|x |) × 1 9 [j 0 (|k||x|) − 2j 2 (|k||x|)] [j 0 (|k||x |) − 2j 2 (|k||x |)] × 1 − a 2 |g g| e −iΩ(t−t ) + a 2 |e e| e iΩ(t−t ) + a 1 − a 2 |g e| e −iΩ(t+t ) + a 1 − a 2 |e g| e iΩ(t+t ) = −e 2 a 2 0 24576 π 2 ∞ 0 d|k||k| 3 R dt R dt χ(t)χ(t )e −i|k|(t −t) 9 − 20a 2 0 |k| 2 2 (4a 2 0 |k| 2 + 9) 8 × 1 − a 2 |g g| e −iΩ(t−t ) + a 2 |e e| e iΩ(t−t ) + a 1 − a 2 |g e| e −iΩ(t+t ) + a 1 − a 2 |e g| e iΩ(t+t ) .(A28) Now that we have found the analytic expressions evaluated except for the wave vector and time integrals, we can particularize to the desired switching functions and execute the remaining integrals. a. Gaussian, sudden, and delta switching Here we present the results for three different switching functions: (i) χ g (t) = e −t 2 /σ 2 tr f Û (1)ρ iÛ (1) † 1 1 = 663552 π (ea 0 σ) 2 ∞ 0 d|k||k| 3 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 (1 − a 2 )e − 1 2 σ 2 (|k|−Ω) 2 a √ 1 − a 2 e − 1 2 σ 2 (|k| 2 +Ω 2 ) a √ 1 − a 2 e − 1 2 σ 2 (|k| 2 +Ω 2 ) a 2 e − 1 2 σ 2 (|k|+Ω) 2 ,(A29)tr f Û (1)ρ iÛ (1) † k⊗k = − 24576 π (ea 0 σ) 2 ∞ 0 d|k||k| 3 (9 − 20a 2 0 |k| 2 ) 2 (4a 2 0 |k| 2 + 9) 8 (1 − a 2 )e − 1 2 σ 2 (|k|−Ω) 2 a √ 1 − a 2 e − 1 2 σ 2 (|k| 2 +Ω 2 ) a √ 1 − a 2 e − 1 2 σ 2 (|k| 2 +Ω 2 ) a 2 e − 1 2 σ 2 (|k|+Ω) 2 ,(A30)tr f Û (2)ρ i 1 1 = − 331776 π (ea 0 σ) 2 ∞ 0 d|k||k| 3 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 erfc iσ(|k| + Ω) √ 2 e − 1 2 σ 2 (|k|+Ω) 2 a 2 a √ 1 − a 2 0 0 +erfc iσ(|k| − Ω) √ 2 e − 1 2 σ 2 (|k|−Ω) 2 0 0 a √ 1 − a 2 1 − a 2 ,(A31)tr f Û (2)ρ i k⊗k = 12288 π (ea 0 σ) 2 ∞ 0 d|k||k| 3 (9 − 20a 2 0 |k| 2 ) 2 (4a 2 0 |k| 2 + 9) 8 erfc iσ(|k| + Ω) √ 2 e − 1 2 σ 2 (|k|+Ω) 2 a 2 a √ 1 − a 2 0 0 +erfc iσ(|k| − Ω) √ 2 e − 1 2 σ 2 (|k|−Ω) 2 0 0 a √ 1 − a 2 1 − a 2 ,(A32) where we have used the following for the nested time integrals: ∞ −∞ dt t −∞ dt e ±iΩ(t−t ) e − t 2 σ 2 e − t 2 σ 2 e −i|k|(t−t ) = √ π σ 2 ∞ −∞ dte − 1 4 (|k|∓Ω) 2 σ 2 e i(±Ω−|k|)t e − t 2 σ 2 = πσ 2 2 e − 1 2 (Ω 2 +|k| 2 )σ 2 e ±|k|Ωσ 2 I σ 2 (|k| ∓ Ω), σ(|k| ∓ Ω) = πσ 2 2 erfc iσ(|k| ∓ Ω) √ 2 e 1 2 (−|k| 2 σ 2 ±|k|Ωσ 2 −Ω 2 σ 2 ) = πσ 2 2 erfc iσ(|k| ∓ Ω) √ 2 e 1 2 σ 2 (|k|∓Ω) 2 ,(A33) with I(a, b) = ∞ −∞ dxe −a 2 −ibx−x 2 erf(x − ia) = −i √ πe −a 2 − b 2 4 erf a + b 2 √ 2 . (A34) Then the whole change in the atomic state is ∆ρ g = 24576(a 0 eσ) 2 π ∞ 0 d|k| |k| 3 e − 1 2 σ 2 (|k|+Ω) 2 (4a 2 0 |k| 2 + 9) 6 2 (1 − a 2 )e 2|k|σ 2 Ω − a 2 1 0 0 −1 + a 1 − a 2 e 2|k|σ 2 Ω erf iσ(|k| − Ω) √ 2 − erf iσ(|k| + Ω) √ 2 − 1 − e |k|σ 2 Ω 2 0 0 1 0 + H.c. . (A35) For the sudden top-hat switching (ii) χ s (t) = Θ(t)Θ(−t + σ), with the duration of interaction σ, Hence we find the total change of the atomic state to be ∆ρ s = 49152(a 0 e) 2 π 2 ∞ 0 d|k| |k| 3 (4a 2 0 |k| 2 + 9) 6 (|k| 2 − Ω 2 ) 2 2a 2 (|k| − Ω) 2 cos(σ(|k| + Ω)) +2 1 − 2a 2 |k| 2 + Ω 2 + 4|k|Ω + 2(a 2 − 1 cos(σ(|k| − Ω))(|k| + Ω) 2 1 0 0 −1 + e 2iσΩ (|k| 2 − Ω 2 ) + |k| 2 (2iσΩ − 1) + 4Ωe iσΩ (Ω cos (|k|σ) − i|k| sin (|k|σ)) − Ω 2 (3 + 2iσΩ) tr f Û (1)ρ iÛ (1) † 1 1 = 1327104 π 2 (ea 0 ) 2 ∞ 0 d|k||k| 3 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 1 − cos (σ(|k| − Ω)) (|k| − Ω) 2 1 − a 2 0 0 0 + 1 − cos (σ(|k| + Ω)) (|k| + Ω) 2 0 0 0 a 2 − a 1 − a 2 cos (σ|k|) − cos (σΩ) |k| 2 + Ω 2 0 e −iΩσ e iΩσ 0 ,(A36)tr f Û (1)ρ iÛ (1) † k⊗k = − 49152 π 2 (ea 0 ) 2 ∞ 0 d|k||k| 3 (9 − 20a 2 0 |k| 2 ) 2 (4a 2 0 |k| 2 + 9) 8 1 − cos (σ(|k| − Ω)) (|k| − Ω) 2 1 − tr f Û (2)ρ i 1 1 = − 331776C 2 π 2 (ea 0 ) 2 ∞ 0 d|k||k| 3 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 a 2 a √ 1 − a 2 a √ 1 − a 2 1 − a 2 ,(A44)tr f Û (2)ρ i k⊗k = 12288C 2 π 2 (ea 0 ) 2 ∞ 0 d|k||k| 3 (9 − 20a 2 0 |k| 2 ) 2 (4a 2 0 |k| 2 + 9) 8 a 2 a √ 1 − a 2 a √ 1 − a 2 1 − a 2 .(A45) The nested time integrals over the two Dirac distributions are mathematically ambiguous (see [25]) and require us to understand them as some sort of limit of a sequence of functions. If the delta distribution is understood as the limit of a sequence of symmetric peaked functions of smaller and smaller width and constant area (e.g., the Dirac distribution is the short width limit -symmetrically taken-of a sudden top-hat or Gaussian function), as it is shown in the appendix of [25], one can show that In an equal superposition (a = 1 √ 2 ) the atomic state does not get perturbed to second order in perturbation theory for the delta and gapless sudden switching. Thus the purity is preserved which yields H min = 1 bit. As can be easily checked, for all cases the perturbation is traceless and Hermitian. b. Assessing the validity of the perturbative approach In the following we will present numerical results for the change in the atomic state after interaction for the different switching functions. We will focus on regimes of low min-entropy, establishing that perturbation theory indeed holds in these cases below a certain threshold of the coupling strength e depending on the switching function. Since the excited state always led to the least generated H min , we will particularize to a = 0 that will yield the worst-case-scenario for perturbation theory to hold. Assuming this, the initial state readŝ ρ 0 = 0 0 0 1 .(A48) We saw earlier (Fig. 3a, 4b, and 5a) that increasing the coupling strength can drastically reduce the generated randomness. We choose in the following a 0 = 2.68 · 10 −4 eV −1 , Ω = 3.73 eV, and σ = 2.5 × 10 −3 . Then we find that the coupling strength needs to be below the following values such that the magnitude of the perturbation to the initial state is at most 0.1 times the original state for Gaussian and gapless sudden switching: with both having a purity of tr ρ 2 a = 0.82. We declare that the end-limit of the safe applicability of the perturbative analysis. On the other end, for the largest perturbation shown in the plots in the case of Dirac switching (Fig. 5a) ∆ρ d (e = 4 × 8.54 × 10 −2 ) = 0.012 0. 0. −0.012 , tr ρ 2 a = 0.98, so we stay well within the regime of perturbation theory for all regimes studied of delta switching. However, for Gaussian and sudden switching in the plots Fig. 3a and Fig. 4b we have gone slightly above those 'safe' numbers. The changes in the density matrices in the worst-case scenario for the plotted figures are Therefore they are still under control, even if they are outside of the perturbative regime (higher order corrections will still be smaller). c. Symmetry property of the min-entropy In order to see why to leading order in perturbation theory H min is symmetric around a = 1 √ 2 for Dirac and gapless (Ω = 0) sudden switching, let us focus on the term 1 − 2a 2 1 0 0 −1 ,(A54) which is present in (28) and (29). Under the transformation a 2 → 1 − a 2 , (A54) yields an additional factor of (−1). Since the min-entropy (see (21)) depends on the purity of the state after the interaction, tr ρ 2 a = tr (ρ a,i + ∆ρ) 2 = tr ρ 2 a,i + 2ρ a,i ∆ρ + O(e 4 ), Clearly (A56) is then invariant under a 2 → 1 − a 2 . Therefore, we expect the min-entropy to be symmetric around a = 1 FIG. 1 .FIG. 2 . 12Min-entropy Hmin plotted against duration of interaction σ and a 2 (proportional to the z component of the initial Bloch vector) for Gaussian switching with the parameters a0 = 2.68 · 10 −4 eV −1 , Ω = 3.73 eV, and (a) free-space coupling e = 8.54 · 10 −2 or (b) strong coupling e = 5. Recall that a = 0 corresponds to |e a and a = 1 to |g a . Hmin = 1 bit coincides with maximal randomness and is in fact never absolutely reached. The highest amount of randomness can be found for an equal superposition a = 1/ √ 2 (red dashed line). The excited state and, surprisingly, the ground state are the least favored preparations for short interactions. The lack of symmetry is related to the non-homogeneous nature of the switching and it is explained in detail in Appendix A 1 c. Min-entropy Hmin for longer interaction times σ with parameters a0 = 2.68 · 10 −4 eV −1 , Ω = 3.73 eV, e = 8.54 · 10 −2 for the ground (a = 1) and excited state (a = 0), and equal superposition (a = 1/ √ 2) in the case of Gaussian switching. The ground state recovers Hmin = 1 bit, the other initial atomic preparations witness a fall-off of the extracted randomness. FIG. 3 . 3Extracted min-entropy as a function of (a) electric charge e, (b) atomic radius a0, and (c) energy gap Ω for a fixed interaction time σ = 2.5 · 10 −3 eV −1 and different atomic state parameters a in the case of Gaussian switching. For (a) a0 = 2.68 · 10 −4 eV −1 , Ω = 3.73 eV, in (b) e = 8.54 · 10 −2 , Ω = 3.73 eV, and in (c) a0 = 2.68 · 10 −4 eV −1 , e = 8.54 · 10 −2 . In (a) and (b) σΩ ≈ 0.01, and in (c) σΩ < 0.045, and hence all plots are beyond the validity of the RWA. For (a), as explained in Appendix A 1 b, perturbation theory holds for e < 12.8 (see Appendix A 1 b to find a discussion about the whole plot domain). FIG. 4 .FIG 4(a) Min-entropy Hmin plotted against duration of interaction σ and a 2 (proportional to the z component of the Bloch vector) with the parameters a0 = 2.68·10 −4 eV −1 , e = 8.54·10 −2 for gapless sudden switching. The red dashed line corresponds to equal superposition and yields a maximum min-entropy of Hmin = 1 bit. Ground and excited state witness the least amount of randomness that can be extracted. (b, c) Extracted min-entropy as a function of the parameters (b) electric charge e and (c) atomic radius a0 for a fixed interaction time σ = 2.5 · 10 −3 eV −1 and different atomic state parameters a in the case of sudden switching. In (b) we keep a0 = 2.68 · 10 −4 eV −1 fixed and for (c) we have e = 8.54 · 10 −2 . According to the discussion in Appendix A 1 b, in (b) perturbation theory holds safely for e < 4.7. (see Appendix A 1 b to find a discussion about the whole plot domain). . 5. (a) Min-entropy Hmin plotted against a 2 (proportional to the z component of the Bloch vector) with the atomic radius a0 = 2.68 · 10 −4 eV −1 for delta switching and different values of the coupling strength e, taking C = σ = 2.5 × 10 −3 eV −1 . At equal superposition a = 1/ √ 2 the extracted randomness has its maximum with Hmin = 1 bit. (b, c) Extracted min-entropy as a function of the parameters (b) electric charge e and (c) atomic radius a0 for different atomic state parameters a in the case of delta switching. In (b) a0 is taken to be 2.68 · 10 −4 eV −1 , and in (c) the coupling constant is 8.54 · 10 −2 . FIG. 6 . 6(a) Min-entropy Hmin plotted against duration of interaction σ with parameters a0 = 2.68 · 10 −4 eV −1 , e = 10 −3 , Ω = 3.73 eV in the ground state a = 1 for Gaussian switching of the electric dipole model (EM) for 1s → 2pz, scalar coupling (UDW) and coupling to the time derivative of the scalar field (UDW d ). Both scalar models assume a 1s → 2s transition. (b) Corresponds to a zoomed-in region of (a) marked by the black box. The considered interactions are ultra-short, i.e. σΩ < 0.03, hence beyond any RWA. the funding of the NSERC Discovery program and the Ontario Early Researcher Award. Appendix A: Deriving the time-evolved density matrix m e − m g −m e m g − 1 m e − m g −m e m g − 1 cos (σ(|k| + Ω)) (|k| + Ω) 2 0 0 0 a 2 − a 1 − a 2 cos (σ|k|) − cos (σΩ)|k| 2 + Ω 2 0 e −iΩσ e iΩσ 0 ,(A37)tr f Û (2)ρ i 1 1 = 663552 π 2 (ea 0 ) 2 ∞ 0 d|k||k| 3 iσ(|k| + Ω) + e −iσ(|k|+Ω) − 1 (|k| + Ω) particularize to the gapless case (Ω = 0), allowing then to perform the last remaining integral such ...,bq z is the Meijer G-function. Finally for (iii) χ d (t) = Cδ(t), where C is some constant with mass dimension [C] = can be integrated analytically and hence the change in the atomic state due to the delta switching the object of interest is, using(24),tr (ρ a,i ∆ρ) ∝ a 2 − (1 − a 2 ) (1 − 2a 2 ) = −(1 − 2a 2 ) 2 . × Y 10 (x)Y 10 (k) − Y 11 (x)Y 1−1 (k) − Y 1−1 (x)Y 11 (k) Y 10 (k)Y 10 (x ) − Y 11 (k)Y 1−1 (x ) − Y 1−1 (k)Y 11 (x ) √ 2 in the case for gapless sudden and Dirac switching, as is shown inFig. 4a and 5a. ∞ λ,λ =0 (2λ + 1)(2λ + 1)where we explicitly used the Kronecker delta to indicate the dependence on m e and m g . One can derive the second from the first term in the curly brackets by exchanging l e ↔ l g in terms associated to λ, and derive the third from the first term by exchanging l e ↔ l g in terms associated to λ . Now we can determine the formulae for the dyadic part k ⊗ k. We start fromwithx indicating l e ↔ l g (m e ↔ m g ) in 3j-symbols involving λ (therefore also for the m-component of l ) andx that l e ↔ l g (m e ↔ m g ) in 3j-symbols involving λ (also for the m-component of l). For instance,This implies that the coefficients contain terms of the formTherefore, unless m e = m g , the third and last term of the curly brackets vanish in (A19). The Kronecker delta has been explicitly added to stress this fact. Now all expressions are in generality and cannot be simplified more without specifying the atomic transition and the switching function.Transition from ground to first excited stateIn the following we will derive the time evolved density matrix by studying the 1s → 2p z transition (l g = 0, m g = 0, l e = 1, m e = 1). Then (A9) can be simplified by using the properties of the Wigner 3j-symbols. In particular the first 3j-symbol forces l = λ, moreover we see that λ = 1 and λ = 0, 1, 2. Thus it yieldsWe solve the integral over |x| and |x | by using the following identitywith 2F1 (a, b; c; z) := 2 F 1 (a, b; c; z)/Γ(z) as the regularized hypergeometric function. Therefore we finddt χ(t)χ(t )e −i|k|(t−t ) 16a 4 0 |k| 4 − 8a 2 0 |k| 2 + 9 (4a 2 0 |k| 2 + 9) 8 × 1 − a 2 |e e| + a 1 − a 2 |e g| e iΩ(t−t ) + a 2 |g g| + a 1 − a 2 |g e| e −iΩ(t−t ) . (A23)Before specifying the switching function χ(t) to integrate over the time integrals, we will simplify the other contributions to the time evolved density matrix. In particular we find analogously for (A13) Starting from eq. (26), one sees that the off-diagonal elements are invariant under a 2 → 1 − a 2 , the diagonal elements however would yield an additional factor of (−1) (as needed for symmetry of the min-entropy under the transformation) if it were not for the factor exp 2|k|σ 2 Ω . It is therefore the case that if we use a non-homogeneous switching function (there is real time variation of the coupling) for atoms with non-zero internal dynamics. On the other hand, for Gaussian switching there is none such invariance of the min-entropy. Ω > 0) the free time evolution distorts the symmetry of the plotsOn the other hand, for Gaussian switching there is none such invariance of the min-entropy. Starting from eq. (26), one sees that the off-diagonal elements are invariant under a 2 → 1 − a 2 , the diagonal elements however would yield an additional factor of (−1) (as needed for symmetry of the min-entropy under the transformation) if it were not for the factor exp 2|k|σ 2 Ω . It is therefore the case that if we use a non-homogeneous switching function (there is real time variation of the coupling) for atoms with non-zero internal dynamics (Ω > 0) the free time evolution distorts the symmetry of the plots. . 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{'abstract': 'We study how quantum randomness generation based on unbiased measurements on a hydrogenlike atom can get compromised by the unavoidable coupling of the atom with the electromagnetic field. We improve on previous literature by analyzing the light-atom interaction in 3+1 dimensions with no single-mode or rotating-wave approximations and taking into account the non-pointlike nature of the atom, its orbital structure, and the exchanges of angular momentum between atom and field. We show that preparing the atom in the ground state in the presence of no field excitations is not universally the safest state to generate randomness. arXiv:1710.06875v2 [quant-ph]', 'arxivid': '1710.06875', 'author': ['Richard Lopp \nDepartment of Applied Mathematics\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n\nInstitute for Quantum Computing\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n', 'Eduardo Martín-Martínez \nDepartment of Applied Mathematics\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n\nInstitute for Quantum Computing\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada\n\nPerimeter Institute for Theoretical Physics\n31 Caroline St NN2L 2Y5WaterlooOntarioCanada\n'], 'authoraffiliation': ['Department of Applied Mathematics\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Institute for Quantum Computing\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Department of Applied Mathematics\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Institute for Quantum Computing\nUniversity of Waterloo\nN2L 3G1WaterlooOntarioCanada', 'Perimeter Institute for Theoretical Physics\n31 Caroline St NN2L 2Y5WaterlooOntarioCanada'], 'corpusid': 55418815, 'doi': '10.1016/j.optcom.2018.03.056', 'github_urls': [], 'n_tokens_mistral': 27217, 'n_tokens_neox': 23507, 'n_words': 14342, 'pdfsha': 'cd2c5f0874fc0f0db4c5c96ad59e0389038c7786', 'pdfurls': ['https://arxiv.org/pdf/1710.06875v2.pdf'], 'title': ['Light, matter, and quantum randomness generation: A relativistic quantum information perspective', 'Light, matter, and quantum randomness generation: A relativistic quantum information perspective'], 'venue': []}