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_{q}[1+(q-1)\varepsilon^{2}]+2\log_{q}[1+(q-1)\varepsilon].italic_Q ( italic_ε ) := italic_c ( italic_ε , italic_R ) = - divide start_ARG italic_q italic_ε end_ARG start_ARG 1 + ( italic_q - 1 ) italic_ε end_ARG - roman_log start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT [ 1 + ( italic_q - 1 ) italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] + 2 roman_log start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT [ 1 + ( italic_q - 1 ) italic_ε ] . | Differentiating Q(ε)𝑄𝜀Q(\varepsilon)italic_Q ( italic_ε ) with respect to ε𝜀\varepsilonitalic_ε, we have | Therefore exactly one of the two roots for Q′(ε)superscript𝑄′𝜀Q^{\prime}(\varepsilon)italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ε ) lies in the desired range 0<ε<10𝜀10<\varepsilon<10 < italic_ε < 1, and Q(ε)𝑄𝜀Q(\varepsilon)italic_Q ( italic_ε ) attains local maximum at that point. Since Q(0)=Q(1)=0𝑄0𝑄10Q(0)=Q(1)=0italic_Q ( 0 ) = italic_Q ( 1 ) = 0, we conclude that Q(ε)𝑄𝜀Q(\varepsilon)italic_Q ( italic_ε ) is positive for 0<ε<10𝜀10<\varepsilon<10 < italic_ε < 1, and therefore (44) holds for R=1−ε1+(q−1)ε𝑅1𝜀1𝑞1𝜀R=\frac{1-\varepsilon}{1+(q-1)\varepsilon}italic_R = divide start_ARG 1 - italic_ε end_ARG start_ARG 1 + ( italic_q - 1 ) italic_ε end_ARG. | Differentiating both sides with respect to κ𝜅\kappaitalic_κ and checking for 𝒢1′(κ)=0superscriptsubscript𝒢1′𝜅0\mathcal{G}_{1}^{\prime}(\kappa)=0caligraphic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_κ ) = 0, we have κ=qε21+(q−1)ε2𝜅𝑞superscript𝜀21𝑞1superscript𝜀2\kappa=\frac{q\varepsilon^{2}}{1+(q-1)\varepsilon^{2}}italic_κ = divide start_ARG italic_q italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 1 + ( italic_q - 1 ) italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. In addition, 𝒢1subscript𝒢1\mathcal{G}_{1}caligraphic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT attains local maximum at this critical point. | Differentiating c(ε,R)𝑐𝜀𝑅c(\varepsilon,R)italic_c ( italic_ε , italic_R ) twice with respect to R𝑅Ritalic_R, we have | A |
Theorem 1.11 is proved in Section 8. We conclude the paper with Section 9, which discusses and proves an auxiliary result | Ukraine, and the Institute has been severely damaged [NDGP22]. We dedicate this paper to Profs. Eremenko and Lyubich, | Jack Burkart, Alex Eremenko, Misha Lyubich, Phil Rippon, Mitsuhiro Shishikura, Dave Sixsmith and Gwyneth Stallard for interesting discussions on this topic. We are particularly grateful to Alex Eremenko for mentioning to us | Our counterexamples to Eremenko’s conjecture are constructed in Section 7. This construction is similar to | This work follows in the footsteps of two world-leading Ukrainian mathematicians, Alex Eremenko and Misha Lyubich. Their collaboration, which pioneered the use of approximation theory in complex dynamics, took place in the fall of 1983 in Kharkiv. At that time, Alex Eremenko was based at the Institute of Low Temperature Physics and Engineering, and it was there that he formulated what is now known as Eremenko’s conjecture. The city of Kharkiv has been devastated during the ongoing invasion of | B |
Let 𝒳𝒳\mathcal{X}caligraphic_X be the stack that parameterizes triples (E,∇,Ψ)𝐸∇Ψ(E,\nabla,\Psi)( italic_E , ∇ , roman_Ψ ) such that | (E,∇)𝐸∇(E,\nabla)( italic_E , ∇ ) is a tame G𝐺Gitalic_G-local systems with zero p𝑝pitalic_p-curvature, | (E,∇)𝐸∇(E,\nabla)( italic_E , ∇ ) is a tame 𝒫𝜽subscript𝒫𝜽\mathcal{P}_{\boldsymbol{\theta}}caligraphic_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT-local system with zero p𝑝pitalic_p-curvature, | (E,∇)𝐸∇(E,\nabla)( italic_E , ∇ ) is a tame G𝐺Gitalic_G-local system with zero p𝑝pitalic_p-curvature, | 𝒜𝒜\mathcal{A}caligraphic_A is the stack of tame Jpsuperscript𝐽𝑝J^{p}italic_J start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-local systems with zero p𝑝pitalic_p-curvature. | A |
Note that these two polynomials are conjugates of each other, as one can be obtained from the other by conjugating by the linear map ϕ(z)=1−zitalic-ϕ𝑧1𝑧\phi(z)=1-zitalic_ϕ ( italic_z ) = 1 - italic_z. More generally, Bd,ksubscript𝐵𝑑𝑘B_{d,k}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT is conjugate to Bd,d−k−1subscript𝐵𝑑𝑑𝑘1B_{d,d-k-1}italic_B start_POSTSUBSCRIPT italic_d , italic_d - italic_k - 1 end_POSTSUBSCRIPT via the same change of coordinates swapping 0 and 1. | Conservative dynamical Belyi polynomials can also be used to answer two questions posed by Silverman in [11] about the heights of conservative polynomials. Recall that a conservative polynomial f𝑓fitalic_f is a polynomial for which every critical point γisubscript𝛾𝑖\gamma_{i}italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT satisfies f(γi)=γi𝑓subscript𝛾𝑖subscript𝛾𝑖f(\gamma_{i})=\gamma_{i}italic_f ( italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. | We begin by showing that all conservative dynamical Belyi polynomials have persistent bad reduction at some prime. | In the sections that follow, we explore properties of these conservative dynamical Belyi maps and use them to answer questions related to post-critically finite maps and conservative polynomials in general. We refer the reader to [2] and [9, Section 3] for further background on dynamical Belyi maps. | Note that Silverman’s definition of “normalized polynomial” differs from the definition used in the previous section. Our dynamical Belyi maps Bd,ksubscript𝐵𝑑𝑘B_{d,k}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT are conservative and satisfy Bd,k(0)=0subscript𝐵𝑑𝑘00B_{d,k}(0)=0italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT ( 0 ) = 0, but they are not monic. To put them in the desired normal form, we will use scaled conjugates of these maps to provide negative answers to both parts of Question 4.3. The leading coefficient of Bd,ksubscript𝐵𝑑𝑘B_{d,k}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT is ak=(−1)k(d−1k)subscript𝑎𝑘superscript1𝑘binomial𝑑1𝑘a_{k}=(-1)^{k}\binom{d-1}{k}italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_d - 1 end_ARG start_ARG italic_k end_ARG ), so as before we choose β𝛽\betaitalic_β such that βd−1=aksuperscript𝛽𝑑1subscript𝑎𝑘\beta^{d-1}=a_{k}italic_β start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and conjugate Bd,ksubscript𝐵𝑑𝑘B_{d,k}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT by the scaling map ϕ(z)=zβitalic-ϕ𝑧𝑧𝛽\phi(z)=\frac{z}{\beta}italic_ϕ ( italic_z ) = divide start_ARG italic_z end_ARG start_ARG italic_β end_ARG to obtain the polynomial Bd,kϕsuperscriptsubscript𝐵𝑑𝑘italic-ϕB_{d,k}^{\phi}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT, just as in equation (2). The polynomial Bd,kϕsuperscriptsubscript𝐵𝑑𝑘italic-ϕB_{d,k}^{\phi}italic_B start_POSTSUBSCRIPT italic_d , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ϕ end_POSTSUPERSCRIPT is a monic conservative polynomial satisfying the hypotheses in Question 4.3. | C |
If we consider geodesic flow on an arbitrary finite polysquare translation surface, then it is typically non-integrable. | Let 𝒫𝒫\mathcal{P}caligraphic_P be a polysquare translation surface with b𝑏bitalic_b atomic squares, and let α𝛼\alphaitalic_α be a badly approximable real number. | Let 𝒫𝒫\mathcal{P}caligraphic_P be a polysquare translation surface with b𝑏bitalic_b atomic squares, and let α𝛼\alphaitalic_α be a badly approximable real number. | Let α𝛼\alphaitalic_α be a badly approximable number, and let A𝐴Aitalic_A be an upper bound on the continued fraction digits of α𝛼\alphaitalic_α. | and let I0,I1∈ℐn(𝒫;C)subscript𝐼0subscript𝐼1subscriptℐ𝑛𝒫𝐶I_{0},I_{1}\in\mathcal{I}_{n}(\mathcal{P};C)italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ caligraphic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( caligraphic_P ; italic_C ) be subintervals satisfying | A |
Last but not least, a game that characterizes Baire class 1 functions between arbitrary separable metrizable spaces | by the use of ε𝜀\varepsilonitalic_ε-δ𝛿\deltaitalic_δ characterization of Baire 1 functions. Then we modify the game defined by Kiss | In the original proof, Kiss noted that “the idea of the proof is to pick ynsubscript𝑦𝑛y_{n}italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as the image of a point | of the game, considered by Kiss, Player II was playing in a space containing the range of a function, here we let Player II play in the | Our first aim in this paper is to extend the result by Kiss. We simplify the proof of a harder implication of his result | D |
Any strategy to achieve the minimum number of pages in a matching book embedding of a graph family, such as C(n,S)𝐶𝑛𝑆C(n,S)italic_C ( italic_n , italic_S ), based on regular layout and algorithms is a kind of “polymerization process” since almost all edges are placed in a repeated pattern. | A book embedding [4] of a graph G𝐺Gitalic_G is an outerplane drawing and an edge-partition such that edges in the same part do not cross. The parts of the partition are the pages of the book embedding. The book thickness bt(G)𝑏𝑡𝐺bt(G)italic_b italic_t ( italic_G ) of G𝐺Gitalic_G is the least number of pages in any book embedding while bt(G,ω)𝑏𝑡𝐺𝜔bt(G,\omega)italic_b italic_t ( italic_G , italic_ω ) is the least number of pages for the outerplane drawing (G,ω)𝐺𝜔(G,\omega)( italic_G , italic_ω ). | To define good matching book embeddings for an infinite family of graphs, one needs to give both layout and coloring schemes: algorithms which produce the needed vertex order and edge-to-page assignment from the various integers that identify each graph in the family. Most of our families consist of circulants C(n,S)𝐶𝑛𝑆C(n,S)italic_C ( italic_n , italic_S ) with a fixed jump-length set S𝑆Sitalic_S and with the number n𝑛nitalic_n of vertices reduced modulo 2 or 4. The coloring algorithms can either be static (as in tables based on modularity) or dynamic (as in prescriptions for Hamiltonian cycles or paths). See proofs of Theorems 1 and 3, resp. | A polymer is a molecule composed of a sequence of many parts such as proteins composed of amino acids or RNA/DNA as a sequence of nucleotides. The sequence of parts may form a path or a cycle. | Periodically, 4-color the edges of length 1, and for every edge of length 3, use the same color as the unique edge of length 1 with which it is nested, thus 4-coloring all edges of length 1 or 3. | C |
{3}\text{ or }\eta M_{k}>1\}}.≤ italic_η ( ( italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT + blackboard_E [ italic_U start_POSTSUBSCRIPT 1 , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ∣ sansserif_V ] ) + 2 italic_F start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT blackboard_1 start_POSTSUBSCRIPT { italic_π start_POSTSUBSCRIPT italic_k , ∞ end_POSTSUBSCRIPT ( sansserif_V ) > ( italic_η italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT or italic_η italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > 1 } end_POSTSUBSCRIPT . | By conditioning on 𝖵𝖵\mathsf{V}sansserif_V we reduce our claim to the case of a varying environment. Hence, we may resort to [25, Lemma 2] yielding | We are ready to prove the lemma’s claim. Note that 𝖷N≥1subscript𝖷𝑁1\mathsf{X}_{N}\geq 1sansserif_X start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ≥ 1 a.s. by (23). Thus, we have | for any random variable 0≤S≤10𝑆10\leq S\leq 10 ≤ italic_S ≤ 1. This estimate follows via conditioning on Pksubscript𝑃𝑘P_{k}italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT from [25, Lemma 1], where it was derived for a varying environment. | Now, if ν∞=0subscript𝜈0\nu_{\infty}=0italic_ν start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = 0, then we may resort to Lemma 4.1 ii), yielding under the present assumption 𝔼[log𝖬N]<0𝔼delimited-[]subscript𝖬𝑁0\mathbb{E}[\log\mathsf{M}_{N}]<0blackboard_E [ roman_log sansserif_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ] < 0 for large N𝑁Nitalic_N, thus subcriticality. | A |
The corresponding Hecke algebra is isomorphic to ℂ[T]/(T−1)2ℂdelimited-[]𝑇superscript𝑇12\mathbb{C}[T]/(T-1)^{2}blackboard_C [ italic_T ] / ( italic_T - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, which is clearly not semisimple. | Theorem 1.10 then consists of controlling the d𝑑ditalic_d-torsion subfactors of 𝒢λsubscript𝒢𝜆{\euls{G}}_{\lambda}script_G start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT and ℳℳ{\euls{M}}script_M. | In this section we prove Theorem 1.10 from the introduction, which is one of the major results of the paper. This shows that the discriminant δ𝛿\deltaitalic_δ | Proposition 14.26. Moreover, for these quiver representations ℳ~=ℳ~ℳℳ{\widetilde{\euls{M}}}={\euls{M}}over~ start_ARG script_M end_ARG = script_M and so 𝒢~0=𝒢0subscript~𝒢0subscript𝒢0{\widetilde{{\euls{G}}}}_{0}={\euls{G}}_{0}over~ start_ARG script_G end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = script_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. However, 𝒢0subscript𝒢0{\euls{G}}_{0}script_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT has a complicated structure: even when n=1𝑛1n=1italic_n = 1, Corollary 14.38 shows that 𝒢0subscript𝒢0{\euls{G}}_{0}script_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT has simple socle and simple top, both isomorphic to the polynomial representation ℂ[V]ℂdelimited-[]𝑉\mathbb{C}[V]blackboard_C [ italic_V ], yet in any composition series of 𝒢0subscript𝒢0{\euls{G}}_{0}script_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT there are ℓ(2ℓ−1−1)ℓsuperscript2ℓ11\ell(2^{\ell-1}-1)roman_ℓ ( 2 start_POSTSUPERSCRIPT roman_ℓ - 1 end_POSTSUPERSCRIPT - 1 ) simple subfactors that are δ𝛿\deltaitalic_δ-torsion. | This proposition shows the remarkable nature of Theorem 1.10: despite the fact that 𝒢𝒢{\euls{G}}script_G has several δ𝛿\deltaitalic_δ-torsion subfactors, they can never appear in the socle or top of the module. | D |
Now let δi∈ℂ[𝔭i]subscript𝛿𝑖ℂdelimited-[]subscript𝔭𝑖{\delta}_{i}\in\mathbb{C}[\mathfrak{p}_{i}]italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_C [ fraktur_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] be the discriminant of the pair (Gi,𝔭i)subscript𝐺𝑖subscript𝔭𝑖(G_{i},\mathfrak{p}_{i})( italic_G start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , fraktur_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) and note that δ=δ1δ2𝛿subscript𝛿1subscript𝛿2{\delta}={\delta}_{1}{\delta}_{2}italic_δ = italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the discriminant of (G,𝔭)𝐺𝔭(G,\mathfrak{p})( italic_G , fraktur_p ). By the previous paragraph each Ai/Jisubscript𝐴𝑖subscript𝐽𝑖A_{i}/J_{i}italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT / italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is δisubscript𝛿𝑖{\delta}_{i}italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT-torsionfree. | (\mathfrak{g})script_L ( fraktur_p ) = script_K ( fraktur_p ) / script_D ( fraktur_p ) italic_τ ( fraktur_g ) | }}(\mathfrak{p}_{2})\Bigr{)}.script_K ( fraktur_p ) = ( script_K ( fraktur_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⊗ script_D ( fraktur_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) + ( script_D ( fraktur_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⊗ script_K ( fraktur_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) . | Now let D∈𝒦(𝔭)𝐷𝒦𝔭D\in{\euls{K}}(\mathfrak{p})italic_D ∈ script_K ( fraktur_p ). Then, by Lemma 9.7(2), | )={\euls{K}}(\mathfrak{p})/{\euls{D}}(\mathfrak{p})\tau(\mathfrak{g})script_K ( fraktur_p ) = { italic_d ∈ script_D ( fraktur_p ) : ∀ italic_f ∈ blackboard_C [ fraktur_p ] start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT , italic_d ( italic_f ) = 0 } , script_L ( fraktur_p ) = script_K ( fraktur_p ) / script_D ( fraktur_p ) italic_τ ( fraktur_g ) | C |
Does every complete local normal domain of mixed characteristic in dimension 3 admit a small Cohen-Macaulay algebra? | Our Theorem 1.4 is the higher dimensional extension of this example. While Theorem 1.4 can be proved by an algebraic proof as was outlined to the authors, we provide a proof for a more general statement (Proposition 3.5) with an algebraic geometric flavor which is a potential source for many other different but similar examples. Moreover, we derive Corollary 3.4 as a new result which is perhaps intrinsically interesting (this might be known to some experts and of course Example 3.1 led us to this corollary). | It was pointed out to the authors (with an outlined proof) that what follows has long been the prime example for the folklore fact that there is an example of a mixed characteristic normal local domain without any small Cohen-Macaulay algebra: | The same result concerning the non-existence of small Cohen-Macaulay algebras in positive characteristic has been proved by Bhatt [B14]; see also [SS12, Example 5.3] for an explicit graded example in prime characteristic. The proof of Theorem 1.3 is obtained by using Bhatt’s criterion via rigid cohomology ([B14]) as well as | The authors are grateful to an anonymous reader who pointed out to us Example 3.1. The authors are also grateful to Bhargav Bhatt for an enlightening comment on Remark 3.21i. | D |
Now, since α𝛼\alphaitalic_α has degree d𝑑ditalic_d, we take some N∈Nature𝑁normal-NatureN\in{\rm Nature}italic_N ∈ roman_Nature such that Φ(N)≥2dnormal-Φ𝑁2𝑑\Phi(N)\geq 2droman_Φ ( italic_N ) ≥ 2 italic_d. By testing (with the algorithms from 5 say) whether α𝛼\alphaitalic_α is a root of any Tn(x)±1plus-or-minussubscript𝑇𝑛𝑥1T_{n}(x)\pm 1italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) ± 1, for n≤N𝑛𝑁n\leq Nitalic_n ≤ italic_N we can decide whether cos−1(α)superscript1𝛼\cos^{-1}(\alpha)roman_cos start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_α ) is a rational multiple of π𝜋\piitalic_π. | The density of the positivity set of any linear recurrence sequence always exists. This fact was proved by Bell and Gerhold [BG07, Theorem 1], and is our principal starting point. With the exception of the paper above, to the best of our knowledge there is no other work that deals with the density of the positivity set. The paper [BM76] can however be interpreted as providing an algorithm to compute the density of the zero set. | As was described in the introduction, the decidability results of the preceding sections can be generalised to orbits of linear dynamical systems (lds), where the positivity set is replaced by the set of indices corresponding to the members of the orbit that belong to a given semialgebraic set. In this section we explain how to achieve this generalisation. | The negativity set is just the positivity set of the sequence ⟨−un⟩n∈Naturesubscriptdelimited-⟨⟩subscript𝑢𝑛𝑛Nature\langle-u_{n}\rangle_{n\in{\rm Nature}}⟨ - italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_n ∈ roman_Nature end_POSTSUBSCRIPT (which is plain, from (4) and the discussion above, that it can be computed). Therefore in the rest of this paper, we only deal with the density of the positivity set, which is simply referred to as the density of the sequence. | Bell and Gerhold have observed, by using an equidistribution theorem due to Weyl, a version of which can be found in Cassels’s book [Cas59], that the density is equal to the Lebesgue measure of a certain set. We proceed along the same path and go further by constructing this set, for which it is necessary to explicitly describe the multiplicative relations among the roots of the polynomial in (2). Afterwards, the problem is reduced to checking the emptiness of a semialgebraic set, which can be done using the decidability of the theory of real closed fields, i.e. Tarski’s algorithm. These tools have been successfully employed by Ouaknine, Worrell, and others, on a number of related problems, it is not surprising that they prove useful to bear on the problems of this paper as well. | B |
\right)\mapsto\dots.( 0 , 2 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ↦ ( 0 , 2 start_POSTSUPERSCRIPT 3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) ↦ ( 0 , 2 start_POSTSUPERSCRIPT 3 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) ↦ … . | (using that the composition is stable). Considering a similar calculation as above, one can see that (0,213)0superscript213(0,2^{\frac{1}{3}})( 0 , 2 start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 3 end_ARG end_POSTSUPERSCRIPT ) is indeterminate with image D𝐷Ditalic_D. Hence the total transform of E0subscript𝐸0E_{0}italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT by f2subscript𝑓2f_{2}italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is E0∪Dsubscript𝐸0𝐷E_{0}\cup Ditalic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ italic_D. This shows that | It is easy to see that the points of a minimal destabilising orbit do not contain any shorter destabilising orbits and conversely that any destabilising orbit of minimum length m𝑚mitalic_m is minimal, so these must always exist when f𝑓fitalic_f is not algebraically stable. Minimality is less natural for eventually destabilising orbits. An N𝑁Nitalic_N-eventually destabilising orbit always possesses a point that constitutes a singleton N𝑁Nitalic_N-eventually destabilising orbit, which is vacuously minimal. | If fmsubscript𝑓𝑚f_{m}italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT is algebraically stable, stop. | Hence this is not destabilising and f2subscript𝑓2f_{2}italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is algebraically stable. | D |
For part (4), we can use part (2) to conclude that |βh(x−y)|≤ℓ+2kh+1subscript𝛽ℎ𝑥𝑦ℓ2subscript𝑘ℎ1|\beta_{h}(x-y)|\leq\ell+2k_{h}+1| italic_β start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_x - italic_y ) | ≤ roman_ℓ + 2 italic_k start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + 1 whenever |h(x)−h(y)|≤ℓℎ𝑥ℎ𝑦ℓ|h(x)-h(y)|\leq\ell| italic_h ( italic_x ) - italic_h ( italic_y ) | ≤ roman_ℓ. Hence, we can take ℓ′superscriptℓ′\ell^{\prime}roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to be equal to ℓ+2kh+1|βh|ℓ2subscript𝑘ℎ1subscript𝛽ℎ\frac{\ell+2k_{h}+1}{|\beta_{h}|}divide start_ARG roman_ℓ + 2 italic_k start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + 1 end_ARG start_ARG | italic_β start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT | end_ARG. | Parts (1) and (2) in the lemma above are far from being expressible in ℒℒ\mathcal{L}caligraphic_L. However, our theory only needs their particular consequences appearing in parts (3) and (4). It is a good question to ask whether or not these properties are provable merely based on the axioms we have introduced so far. However, answering this question does not seem to be quite straightforward, hence, we choose the easier way of adding parts (3) and (4) to our axioms: | Using our observations so far and by applying our flexible notation, we are able to encode into our language some of the main features of the decimals involved in the structure 𝒵αsubscript𝒵𝛼\mathcal{Z}_{\alpha}caligraphic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT. | Although our language/theory does not literally contain the decimals, similar observations to Proposition 1.1 show that our theory is expressive enough to describe key properties of decimals by capturing their order and dense distribution in the spirit of Fact 3.1. | to be in ℒℒ\mathcal{L}caligraphic_L. In particular, all axioms and axiom schemes are ℒℒ\mathcal{L}caligraphic_L-formulas. | A |
[34, p. 582] Let ℬℬ\mathcal{B}caligraphic_B be a ∗*∗-subalgebra of a C∗-algebra 𝒜𝒜\mathcal{A}caligraphic_A. ℬℬ\mathcal{B}caligraphic_B is said to be closed under the holomorphic functional calculus of 𝒜𝒜\mathcal{A}caligraphic_A if, for every element b𝑏bitalic_b of ℬ˙˙ℬ\dot{\mathcal{B}}over˙ start_ARG caligraphic_B end_ARG and every holomorphic function f𝑓fitalic_f on an open neighborhood V⊆ℂ𝑉ℂV\subseteq\mathbb{C}italic_V ⊆ blackboard_C of σ𝒜˙(b)subscript𝜎˙𝒜𝑏\sigma_{\dot{\mathcal{A}}}(b)italic_σ start_POSTSUBSCRIPT over˙ start_ARG caligraphic_A end_ARG end_POSTSUBSCRIPT ( italic_b ), one has f(b)∈ℬ˙𝑓𝑏˙ℬf(b)\in\dot{\mathcal{B}}italic_f ( italic_b ) ∈ over˙ start_ARG caligraphic_B end_ARG. | The following theorem shows that being closed under the C∞-functional calculus of 𝒜𝒜\mathcal{A}caligraphic_A is a sufficient hypothesis on the dense ∗*∗-subalgebra ℬℬ\mathcal{B}caligraphic_B in order to guarantee uniqueness of the C∗-norm. Part of its proof adapts an argument which may be found in [5, Proposition 6.7(b)] (see also [3, Lemma 2]): | At the end of Section 3, we provide a few other applications. We begin by showing that the Fréchet ∗*∗-algebra of smooth elements for a strongly continuous Lie group representation by ∗*∗-automorphisms on a C∗-algebra admits only one C∗-norm (Theorem 3.13), illustrating this result with two algebras of pseudodifferential operators with scalar-valued symbols. Then, we prove that the “sup norm” and the “operator C∗-norm” coincide on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) when J=0𝐽0J=0italic_J = 0 (Proposition 3.15). Finally, in Theorem 3.16, we use some of our results to give very simple proofs of three propositions of Rieffel’s monograph [29]: Propositions 4.11, 5.4 and 5.6. | In Section 3, we consider the noncommutative function algebras 𝒮J𝒞(ℝn)superscriptsubscript𝒮𝐽𝒞superscriptℝ𝑛\mathcal{S}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_S start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) defined by M.A. Rieffel [29] via a deformation quantization procedure, where 𝒞𝒞\mathcal{C}caligraphic_C is a given C∗-algebra of “coefficients” and J𝐽Jitalic_J is a skew-symmetric linear transformation on ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with respect to which the usual (commutative) pointwise product is “deformed” (Definition 3.1). Using Rieffel’s deformed product ×Jsubscript𝐽\times_{J}× start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT (see Equation (3.7)), together with the pointwise involution and with the choice of standard systems of (semi)norms which are familiar from the theory of distributions (there are several variants), these are Fréchet ∗*∗-algebras. However, we will substitute the initial system of norms on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) by a more convenient one, resorting to a faithful representation of this function algebra as an algebra of bounded (pseudodifferential) operators on a Hilbert C∗-module [19]. More precisely, we first define an “operator C∗-norm” on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) (see Definition 3.1) and, under the assumption of a unital 𝒞𝒞\mathcal{C}caligraphic_C, we will define a differential norm on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ), a construction which will require several steps. In particular, we will need a version of the Calderón-Vaillancourt inequality for Hilbert C∗-modules (see Theorem 3.2 and Equation (3.23)), as well as the “symbol map” S𝑆Sitalic_S constructed in reference [23] that allows us to obtain an “inverse Calderón-Vaillancourt-type inequality” (see Equation (3.27)) which, in the scalar case (𝒞=ℂ𝒞ℂ\mathcal{C}=\mathbb{C}caligraphic_C = blackboard_C), was proved by H.O. Cordes in [9, Proposition 4.2, p. 262]. Besides showing that the natural topology of ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) is, in particular, defined by a sequence of submultiplicative ∗*∗-norms, the fact that the topology on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) is generated by a differential norm (Theorem 3.5) also implies, for a unital 𝒞𝒞\mathcal{C}caligraphic_C, that this ∗*∗-algebra is closed under the C∞-functional calculus of its C∗-completion (see Theorem 3.8). This result will put us in a position to establish the uniqueness statement for C∗-norms on ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ), by means of Theorem 2.5, for any C∗-algebra 𝒞𝒞\mathcal{C}caligraphic_C (unital, or not – see Theorem 3.9). The analogous C∗-norm uniqueness statement for 𝒮J𝒞(ℝn)superscriptsubscript𝒮𝐽𝒞superscriptℝ𝑛\mathcal{S}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_S start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) will also be obtained as a corollary, in Theorem 3.10. Moreover, due to the spectral invariance results contained in Theorems 3.8 and 3.11 (for a unital 𝒞𝒞\mathcal{C}caligraphic_C), ℬJ𝒞(ℝn)superscriptsubscriptℬ𝐽𝒞superscriptℝ𝑛\mathcal{B}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_B start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and 𝒮J𝒞(ℝn)superscriptsubscript𝒮𝐽𝒞superscriptℝ𝑛\mathcal{S}_{J}^{\mathcal{C}}(\mathbb{R}^{n})caligraphic_S start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) have the same K𝐾Kitalic_K-theory as their respective C∗-completions. | The next theorem shows some advantages of dealing with a topology which is generated by a differential (semi)norm: | D |
If (X,ω)𝑋𝜔(X,\omega)( italic_X , italic_ω ) is a Liouville domain as above, its alternative ECH capacities are a sequence of real numbers | \gamma}(H_{2}-H_{1})dt-c_{k}^{\operatorname{Alt}}(X,\omega).italic_c start_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_ϕ , italic_γ , italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≤ italic_c start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_ϕ , italic_γ , italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + ∫ start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_d italic_t - italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Alt end_POSTSUPERSCRIPT ( italic_X , italic_ω ) . | 0=c0Alt(X,ω)<c1Alt(X,ω)≤c2Alt(X,ω)≤⋯≤∞0superscriptsubscript𝑐0Alt𝑋𝜔superscriptsubscript𝑐1Alt𝑋𝜔superscriptsubscript𝑐2Alt𝑋𝜔⋯0=c_{0}^{\operatorname{Alt}}(X,\omega)<c_{1}^{\operatorname{Alt}}(X,\omega)% | limk→∞ckAlt(X)2k=4vol(X).subscript→𝑘superscriptsubscript𝑐𝑘Altsuperscript𝑋2𝑘4vol𝑋\lim_{k\to\infty}\frac{c_{k}^{\operatorname{Alt}}(X)^{2}}{k}=4\operatorname{% | \leq c_{2}^{\operatorname{Alt}}(X,\omega)\leq\cdots\leq\infty0 = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Alt end_POSTSUPERSCRIPT ( italic_X , italic_ω ) < italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Alt end_POSTSUPERSCRIPT ( italic_X , italic_ω ) ≤ italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Alt end_POSTSUPERSCRIPT ( italic_X , italic_ω ) ≤ ⋯ ≤ ∞ | B |
Acc-GT was proven to have communication complexity and gradient computation complexity O(κθ3/2log1ϵ)𝑂𝜅superscript𝜃321italic-ϵO\left(\frac{\sqrt{\kappa}}{\theta^{3/2}}\log\frac{1}{\epsilon}\right)italic_O ( divide start_ARG square-root start_ARG italic_κ end_ARG end_ARG start_ARG italic_θ start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT end_ARG roman_log divide start_ARG 1 end_ARG start_ARG italic_ϵ end_ARG ), which is optimal in the function condition number κ𝜅\kappaitalic_κ but suboptimal in 1θ1𝜃\frac{1}{\theta}divide start_ARG 1 end_ARG start_ARG italic_θ end_ARG. | Such a result implies that better answers for Question II are imperative for improving the algorithmic complexities. In particular, notice that the aforementioned methods perform one gradient computation and O(1)𝑂1O(1)italic_O ( 1 ) communication steps in one iteration. Hence they always achieve the same gradient computation complexity and communication complexity. | In this section, we show that OGT achieves the lower bounds on the gradient computation complexity and the communication complexity simultaneously. | For strongly convex and smooth objective functions, the proposed OGT method is optimal in both the gradient computation complexity and the communication complexity. | Is there a single-loop decentralized gradient-type method that achieves optimal gradient computation complexity and communication complexity simultaneously? | A |
\prime}&\end{array}start_ARRAY start_ROW start_CELL end_CELL start_CELL QET end_CELL start_CELL LPLL end_CELL end_ROW start_ROW start_CELL (S0) end_CELL start_CELL ⊢ italic_x + 0 ≖ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_x end_CELL start_CELL ⊢ italic_x + 0 ≖ italic_x end_CELL end_ROW start_ROW start_CELL (S1) end_CELL start_CELL ⊢ italic_x + italic_x ≖ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_x end_CELL start_CELL ⊢ italic_x + italic_x ≖ italic_x end_CELL end_ROW start_ROW start_CELL (S2) end_CELL start_CELL ⊢ italic_x + italic_y ≖ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_y + italic_x end_CELL start_CELL ⊢ italic_x + italic_y ≖ italic_y + italic_x end_CELL end_ROW start_ROW start_CELL (S3) end_CELL start_CELL ⊢ ( italic_x + italic_y ) + italic_z ≖ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_x + ( italic_y + italic_z ) end_CELL start_CELL ⊢ ( italic_x + italic_y ) + italic_z ≖ italic_x + ( italic_y + italic_z ) end_CELL end_ROW start_ROW start_CELL (S4) end_CELL start_CELL { italic_x ≖ start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_y ≖ start_POSTSUBSCRIPT italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } ⊢ italic_x + italic_y ≖ start_POSTSUBSCRIPT italic_ϵ ∨ italic_ϵ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY | The square on the left is Diagram 5. The right triangle is commutative. This is due to the fact that the composition of RELsubscriptREL\mathrm{R_{EL}}roman_R start_POSTSUBSCRIPT roman_EL end_POSTSUBSCRIPT with the inclusion of 𝐄𝐋𝐃Rsubscript𝐄𝐋𝐃𝑅\mathbf{ELD}_{R}bold_ELD start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT into 𝐋𝐋𝐃Rsubscript𝐋𝐋𝐃𝑅\mathbf{LLD}_{R}bold_LLD start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is the identity on 𝐄𝐋𝐃Rsubscript𝐄𝐋𝐃𝑅\mathbf{ELD}_{R}bold_ELD start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT. | from ⊢𝟏provesabsent1\vdash\mathbf{1}⊢ bold_1 we derive ⊢!r𝟏\vdash\mathsf{!}_{r}\mathbf{1}⊢ ! start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT bold_1 using [pro] and we conclude introducing 𝟏1\mathbf{1}bold_1 on the left. | ⊗tensor-product\otimes⊗ and 𝟏1\mathbf{1}bold_1 give the primary linear structure and !\mathsf{!}! the R𝑅Ritalic_R-graded modality. | This information in LPLLR is given forcing 𝟏⊢x+0≖xproves1≖𝑥0𝑥\mathbf{1}\vdash x+0\eqcirc xbold_1 ⊢ italic_x + 0 ≖ italic_x, thanks to the fact that 𝟏1\mathbf{1}bold_1 is interpreted as the function constantly equal to 00. | D |
}}\mathrm{SL}(p,\mathbb{C})})^{m}italic_μ start_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT blackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT italic_m , italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ( italic_μ start_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT blackboard_Z start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_SL ( italic_p , blackboard_C ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT, | of the trivial representation in the representation variety ℛΓG=𝖧𝗈𝗆(Γ,G)subscriptℛΓ𝐺𝖧𝗈𝗆Γ𝐺\mathcal{R}_{\Gamma}G=\mathsf{Hom}(\Gamma,G)caligraphic_R start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT italic_G = sansserif_Hom ( roman_Γ , italic_G ). | group of abelian rank r≥1𝑟1r\geq 1italic_r ≥ 1. Then, for all reductive ℂℂ\mathbb{C}blackboard_C-groups | Given two reductive groups G𝐺Gitalic_G and H𝐻Hitalic_H, both the (G×H)𝐺𝐻(G\times H)( italic_G × italic_H )-representation | with abelian rank r≥1𝑟1r\geq 1italic_r ≥ 1, and G𝐺Gitalic_G a reductive ℂℂ\mathbb{C}blackboard_C-group. | C |
In the setting of Theorem 3.17, for every Fell bundle 𝒞𝒞\mathscr{C}script_C over 𝖦𝖦{\sf G}sansserif_G , the Banach *{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT-algebra 𝔎RUCν(𝖦2|𝒞)superscriptsubscript𝔎normal-RUC𝜈conditionalsuperscript𝖦2𝒞\mathfrak{K}_{\rm RUC}^{\nu}\big{(}{\sf G}^{2}\,|\,\mathscr{C}\big{)}fraktur_K start_POSTSUBSCRIPT roman_RUC end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT ( sansserif_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | script_C ) is symmetric. | F. Flores has been partially supported by the Fondecyt Project 1171854 and the Fondecyt Project 1200884. D. Jauré acknowledges financial support from Beca de Doctorado Nacional Conicyt. M. Măntoiu has been supported by the Fondecyt Project 1200884. We are grateful to V. Nistor; due to his advice, the paper improved a lot. | Until very recently, the largest class for which general results have been found was that of crossed products attached to a global action of the group over a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. A cohomological twist has also been permitted. The simplest case of a trivial action leads to the projective tensor product between L1(𝖦)superscript𝐿1𝖦L^{1}({\sf G})italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( sansserif_G ) and a C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebra. Some references are: [23, 5, 21, 28, 27, 16, 17, 4, 14, 15, 6, 24, 12]. Groupoid algebras are much less studied in the setup of symmetry and spectral invariance; a project on this topic started in [3]. | A great deal of effort has been dedicated to Banach algebras associated to a locally compact group 𝖦𝖦{\sf G}sansserif_G . The basic example is the convolution algebra L1(𝖦)superscript𝐿1𝖦L^{1}({\sf G})italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( sansserif_G ) ; actually the interest in the symmetry property arouse around the Banach algebra interpretation and treatment of the classical result [32] of Wiener on Fourier series. But there are increasingly general other classes, as global crossed products, partial crossed products (both twisted by 2222-cocycles or not), groupoid algebras and L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-types algebras associated to Fell bundles. All of these played an important theoretical role and lead to many applications. When looking for results and examples, one aims to enlarge the collection of groups of groupoids that can be treated, as well as the class of symmetric Banach *{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT-algebras assigned to them. The present paper is concerned with these purposes, adopting the very general point of view of Fell bundles. | The space Cc(Ξ∣𝒞)subscript𝐶cconditionalΞ𝒞C_{\rm c}(\Xi\!\mid\!\mathscr{C})italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT ( roman_Ξ ∣ script_C ) of finitely-supported sections forms a dense *{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT-algebra of the Hahn algebra. The complexity of the multiplication, largely responsable for the generality of the emerging algebras, comes both from the complexity of the ’inner’ Fell multiplication ∙∙\bullet∙ and from the groupoid-type convolution inherent to the formula. | A |
\sigma_{i}\in\Sigma[x_{1},\ldots,x_{n}],\forall i=0,\ldots,m\right\}roman_QM ( italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_h start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) := { ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Σ [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] , ∀ italic_i = 0 , … , italic_m } | Let deg(p)deg𝑝{\mathrm{deg}}(p)roman_deg ( italic_p ) denote the degree of a polynomial function p𝑝pitalic_p. | Finally note that deg(ϕ(h))≤deg(h)=𝔫degitalic-ϕℎdegℎ𝔫{\mathrm{deg}}(\phi(h))\leq{\mathrm{deg}}(h)={\mathfrak{n}}roman_deg ( italic_ϕ ( italic_h ) ) ≤ roman_deg ( italic_h ) = fraktur_n. | The degree bound (1.7) is an explicit expression which only depends on the degrees of the defining polynomials deg(f),deg(g1),…,deg(gk),deg(h1),…,deg(hm)deg𝑓degsubscript𝑔1…degsubscript𝑔𝑘degsubscriptℎ1…degsubscriptℎ𝑚{\mathrm{deg}}(f),{\mathrm{deg}}(g_{1}),\ldots,{\mathrm{deg}}(g_{k}),{\mathrm{% | For any polynomial p∈ℝ[x1,…,xn]𝑝ℝsubscript𝑥1…subscript𝑥𝑛p\in{\mathbb{R}}[x_{1},\ldots,x_{n}]italic_p ∈ blackboard_R [ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ], we denote by deg(p)deg𝑝{\mathrm{deg}}(p)roman_deg ( italic_p ) the degree of p𝑝pitalic_p. | A |
Moreover, if H𝐻Hitalic_H is connected, then it is called a connected dominating subgraph of G𝐺Gitalic_G. To prove the theorem of this section, we will use the following lemma, which is a special case of [20, Proposition 2]. | Algorithm 1 Given a perfect graph G𝐺Gitalic_G on n𝑛nitalic_n vertices, an optimal colouring of G𝐺Gitalic_G and a vertex v𝑣vitalic_v, compute | Let G𝐺Gitalic_G be a graph with an optimal colouring c𝑐citalic_c of G𝐺Gitalic_G such that there exists a connected | Now, since H𝐻Hitalic_H is connected and bipartite, for any connected vertex-ordering of H𝐻Hitalic_H that starts with a minimal element of P𝑃Pitalic_P, the greedy algorithm produces a colouring that agrees with c𝑐citalic_c on H𝐻Hitalic_H. Hence, we can apply Lemma 6 to G𝐺Gitalic_G, c𝑐citalic_c and H𝐻Hitalic_H, which shows that G𝐺Gitalic_G is indeed not ugly. | A graph G=(V,E)𝐺𝑉𝐸G=(V,E)italic_G = ( italic_V , italic_E ) is a comparability graph if there exists an acyclic orientation of G𝐺Gitalic_G that is transitive. | B |
P(Zn>Aan)≤ϵ𝑃subscript𝑍𝑛𝐴subscript𝑎𝑛italic-ϵP(Z_{n}>Aa_{n})\leq\epsilonitalic_P ( italic_Z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_A italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ≤ italic_ϵ. | Wellner, 1996, Example 2.3.4) real-valued functions on a measurable space (S,𝒮)𝑆𝒮(S,\mathcal{S})( italic_S , caligraphic_S ) is said to be VC with parameters (v,A)∈(0,∞)×[1,∞)𝑣𝐴01(v,A)\in(0,\infty)\times[1,\infty)( italic_v , italic_A ) ∈ ( 0 , ∞ ) × [ 1 , ∞ ) and envelope G𝐺Gitalic_G if for any 0<ϵ<10italic-ϵ10<\epsilon<10 < italic_ϵ < 1 and any probability measure Q𝑄Qitalic_Q on (S,𝒮)𝑆𝒮(S,\mathcal{S})( italic_S , caligraphic_S ), we have | Given a probability measure Q𝑄Qitalic_Q on (S,𝒮)𝑆𝒮(S,\mathcal{S})( italic_S , caligraphic_S ), the metric space of squared-integrable functions with respect to Q𝑄Qitalic_Q is defined as | Let Q𝑄Qitalic_Q be a probability measure on S×ℝd𝑆superscriptℝ𝑑S\times\mathbb{R}^{d}italic_S × blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. Let Q1subscript𝑄1Q_{1}italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (resp. Q2subscript𝑄2Q_{2}italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) be the first (resp. second) marginal of Q𝑄Qitalic_Q and define the probability measure Q~1subscript~𝑄1\tilde{Q}_{1}over~ start_ARG italic_Q end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT on S𝑆Sitalic_S as | where, from now on, Yisubscript𝑌𝑖Y_{i}italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT will be valued in (S,𝒮)𝑆𝒮(S,\mathcal{S})( italic_S , caligraphic_S ), a general measurable space. Following the rationale of local averaging lying behind the nearest neighbor selection procedure (Györfi et al., 2006), the k𝑘kitalic_k-NN empirical measure is an estimate of μxsubscript𝜇𝑥\mu_{x}italic_μ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, the (regular) conditional measure of Y𝑌Yitalic_Y given X=x𝑋𝑥X=xitalic_X = italic_x (which further is supposed to exist). The error between two measures will be evaluated using some class 𝒢𝒢\mathcal{G}caligraphic_G of real valued measurable functions on (S,𝒮)𝑆𝒮(S,\mathcal{S})( italic_S , caligraphic_S ). | B |
}})( fraktur_d , italic_f start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT ) , ( fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_f start_POSTSUBSCRIPT fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) into (𝔡,f𝔡f𝔡′)𝔡subscript𝑓𝔡subscript𝑓superscript𝔡′(\mathfrak{d},f_{\mathfrak{d}}f_{\mathfrak{d}^{\prime}})( fraktur_d , italic_f start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) if 𝔡=𝔡′𝔡superscript𝔡′\mathfrak{d}=\mathfrak{d}^{\prime}fraktur_d = fraktur_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, the resulting 𝔇𝔇\mathfrak{D}fraktur_D is unique. | Let 𝔇𝔇\mathfrak{D}fraktur_D be a consistent scattering diagram on Mℝsubscript𝑀ℝM_{\mathbb{R}}italic_M start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT. | for every power k>0𝑘0k>0italic_k > 0, there are only a finite number of (𝔡,f𝔡)𝔡subscript𝑓𝔡(\mathfrak{d},f_{\mathfrak{d}})( fraktur_d , italic_f start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT ) with f𝔡≢1mod𝔪Rknot-equivalent-tosubscript𝑓𝔡modulo1superscriptsubscript𝔪𝑅𝑘f_{\mathfrak{d}}\not\equiv 1\mod\mathfrak{m}_{R}^{k}italic_f start_POSTSUBSCRIPT fraktur_d end_POSTSUBSCRIPT ≢ 1 roman_mod fraktur_m start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. We will also called a restriction of a scattering diagram to a convex subset of Mℝsubscript𝑀ℝM_{\mathbb{R}}italic_M start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT a scattering diagram. | The 𝒳eLsuperscriptsubscript𝒳𝑒𝐿{}^{L}\mathcal{X}_{e}start_FLOATSUPERSCRIPT italic_L end_FLOATSUPERSCRIPT caligraphic_X start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT scattering diagrams are the consistent scattering diagrams containing the initial diagram. | Such resulting diagram 𝔇𝔇\mathfrak{D}fraktur_D is called consistent. It is easy to see if a scattering diagram is consistent, then its associate asymptotic scattering diagram is also consistent. Also, a restriction of a consistent scattering diagram to a convex subset is again consistent (with respect to loops within the convex subset). | D |
A similar method that additionally exploited simultaneous decompressions of candidate compressed vectors was implemented | Table 3: Decimation class counts in [9] and solution times for simultaneous decompression-based searches in [10] | Table 3 reports the decimation class counts for the number of decimation classes of length ℓℓ\ellroman_ℓ, {0,1}01\{0,1\}{ 0 , 1 } vectors with (ℓ+1)/2ℓ12(\ell+1)/2( roman_ℓ + 1 ) / 2 ones reported in [9], and the solution times for the simultaneous decompression-based searches reported in [10]. | of 𝒃𝒃{\bm{b}}bold_italic_b’s decimation class, where the decimation class of 𝒃𝒃{\bm{b}}bold_italic_b is allowed to have multiple different decimation class representatives. | Assuming that the number of compressed pairs and the time required for decompression scales linearly on average with the number of decimation classes, we expect the Table 4 time requirements for an exhaustive search or partial search to first solution if the method | B |
We remark that the generalized persistence diagram is equivalent to the generalized rank invariant (Definitions 2.17 and 2.18). Also, the fibered barcode is equivalent to the (standard) rank invariant [16]. | Hence, in the diagram above, generalized persistence diagram and fibered barcode can be replaced by generalized rank invariant and rank invariant, respectively. | We remark that the generalized persistence diagram is equivalent to the generalized rank invariant (Definitions 2.17 and 2.18). Also, the fibered barcode is equivalent to the (standard) rank invariant [16]. | generalized persistence diagrams. In Section 3, we show that the bigraded Betti numbers can be recovered from the generalized persistence diagram. In Section 5, we discuss open questions. In the appendix, we prove that (a) the multirank invariant (introduced in [54]) also determines the bigraded Betti numbers, and that (b) in general, the generalized persistence diagram and the multirank invariant do not determine each other. | In this section we review the notions of generalized rank invariant and generalized persistence diagram [38, 51]. | A |
On [1:t:0:0]delimited-[]:1𝑡:0:0[1:t:0:0][ 1 : italic_t : 0 : 0 ] (∀t∈ℂfor-all𝑡ℂ\forall t\in{\mathbb{C}}∀ italic_t ∈ blackboard_C), this matrix turns into the shape as | [[ccccc:cccc:cccc:ccccccc]1tt2t3t4u1u1tu1t2u1t3u2u2tu2t2u2t3u12u12t⋯u13⋯u22⋯12t3t24t3u12u1t3u1t2u22u2t3u2t2u12⋯⋯⋯1tt2t32u12u1t⋯3u12⋯⋯1tt2t3⋯⋯2u2⋯].\small\setcounter{MaxMatrixCols}{20}\begin{bmatrix}[ccccc:cccc:cccc:ccccccc]1&% | [[ccccc:cccc:cccc:ccccccc]11111⋯1⋯1111⋯⋯123⋯⋯1⋯2⋯],\setcounter{MaxMatrixCols}{20}\begin{bmatrix}[ccccc:cccc:cccc:ccccccc]1&&&&&1&% | [[ccccc:cccc:cccc:ccccccc]1tt2t3t412t3t24t3𝖮1tt2t31tt2t3]\setcounter{MaxMatrixCols}{20}\begin{bmatrix}[ccccc:cccc:cccc:ccccccc]1&t&t^{2% | [vd∂vd/∂t1⋮∂vd/∂tm∂vd/∂u1⋮∂vd/∂um′](x′)matrixsubscript𝑣𝑑subscript𝑣𝑑subscript𝑡1⋮subscript𝑣𝑑subscript𝑡𝑚subscript𝑣𝑑subscript𝑢1⋮subscript𝑣𝑑subscript𝑢superscript𝑚′superscript𝑥′\displaystyle\begin{bmatrix}v_{d}\\ | C |
∏i(1−xi)λd𝐱(λ≥−12)subscriptproduct𝑖superscript1subscript𝑥𝑖𝜆𝑑𝐱𝜆12\prod_{i}(1-x_{i})^{\lambda}d\mathbf{x}~{}(\lambda\geq-\frac{1}{2})∏ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 - italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_λ end_POSTSUPERSCRIPT italic_d bold_x ( italic_λ ≥ - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ) | O(1/r2)𝑂1superscript𝑟2O(1/r^{2})italic_O ( 1 / italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | O(1/r2)𝑂1superscript𝑟2O(1/r^{2})italic_O ( 1 / italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | O(1/r2)𝑂1superscript𝑟2O(1/r^{2})italic_O ( 1 / italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | O(1/r2)𝑂1superscript𝑟2O(1/r^{2})italic_O ( 1 / italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) | A |
It is a very interesting problem to study the extent to which the best constant of Poincaré inequality is related to the geometry of the domain, especially when singularities arise in its boundary. We show that there exists a uniform constant for every definable family of domains (i.e., definable with respect to parameters) of bounded volume. | u∈W1,p(Ωk)𝑢superscript𝑊1𝑝subscriptΩ𝑘u\in W^{1,p}(\Omega_{k})italic_u ∈ italic_W start_POSTSUPERSCRIPT 1 , italic_p end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), for p∈[1,k2]𝑝1𝑘2p\in[1,\frac{k}{2}]italic_p ∈ [ 1 , divide start_ARG italic_k end_ARG start_ARG 2 end_ARG ], while 𝐭𝐫u𝐭𝐫𝑢\mathbf{tr}\,ubold_tr italic_u is not Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT on ∂reg1Ωksubscriptsuperscript1𝑟𝑒𝑔subscriptΩ𝑘\partial^{1}_{reg}\Omega_{k}∂ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r italic_e italic_g end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. The results of [15] however yield that there is p0subscript𝑝0p_{0}italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT such that for p≥p0𝑝subscript𝑝0p\geq p_{0}italic_p ≥ italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and u∈W1,p(Ω)𝑢superscript𝑊1𝑝Ωu\in W^{1,p}(\Omega)italic_u ∈ italic_W start_POSTSUPERSCRIPT 1 , italic_p end_POSTSUPERSCRIPT ( roman_Ω ), 𝐭𝐫u𝐭𝐫𝑢\mathbf{tr}\,ubold_tr italic_u is always Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT, if ΩΩ\Omegaroman_Ω is a subanalytic bounded domain (the proof actually applies to any bounded domain which is definable in a polynomially bounded o-minimal structure expanding the real field). The real number p0subscript𝑝0p_{0}italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT depends on the Lipschitz geometry of ∂ΩΩ\partial\Omega∂ roman_Ω. | We briefly recall that an o-minimal structure expanding the real field (ℝ,+,⋅)ℝ⋅(\mathbb{R},+,\cdot)( blackboard_R , + , ⋅ ) is the data for every n𝑛nitalic_n of a Boolean algebra | Some generalizations of Poincaré-Friedrichs inequality were given for example in [19, 10]. Although the boundary of a domain which is definable in an o-minimal structure may admit singularities, | Let us recall the inductive definition of cells of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Every subset of ℝ0={0}superscriptℝ00\mathbb{R}^{0}=\{0\}blackboard_R start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = { 0 } is a cell. A definable subset C𝐶Citalic_C is a cell of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT if there is a cell D𝐷Ditalic_D of ℝn−1superscriptℝ𝑛1\mathbb{R}^{n-1}blackboard_R start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT such that one of the following conditions holds: | B |
We call h∈PL(X,x)ℎPL𝑋𝑥h\in\operatorname{PL}(X,x)italic_h ∈ roman_PL ( italic_X , italic_x ) -linear if there are f1,…,fr∈𝒪X,x×subscript𝑓1…subscript𝑓𝑟superscriptsubscript𝒪𝑋𝑥f_{1},\dots,f_{r}\in{\mathscr{O}}_{X,x}^{\times}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ∈ script_O start_POSTSUBSCRIPT italic_X , italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT and ,1…,∈r{}_{1},\dots,{}_{r}\in\Lambdastart_FLOATSUBSCRIPT 1 end_FLOATSUBSCRIPT , … , start_FLOATSUBSCRIPT italic_r end_FLOATSUBSCRIPT ∈ such that h=∑j=1rlogj|fj|ℎsuperscriptsubscript𝑗1𝑟subscript𝑗subscript𝑓𝑗h=\sum_{j=1}^{r}{}_{j}\log|f_{j}|italic_h = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT roman_log | italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | in a neighbourhood of x𝑥xitalic_x. | If =𝐙absent𝐙\Lambda={\mathbf{Z}}= bold_Z or =𝐐absent𝐐\Lambda={\mathbf{Q}}= bold_Q then the minimum and maximum of two -PL functions is again -PL by [gubler-martin, Proposition 2.12(d)]. This is not true for strictly containing 𝐐𝐐{\mathbf{Q}}bold_Q. | In this section, we study positivity conditions on piecewise linear functions. We will introduce the crucial notion of a harmonic function, on which our theory of differential forms rests. We consider a good strictly K𝐾Kitalic_K-analytic space X𝑋Xitalic_X and an additive subgroup ⊂𝐑absent𝐑\Lambda\subset{\mathbf{R}}⊂ bold_R. We will always assume that =𝐙absent𝐙\Lambda={\mathbf{Z}}= bold_Z or that is divisible. | Suppose that =𝐙absent𝐙\Lambda={\mathbf{Z}}= bold_Z or that is divisible. Then there is a unique homomorphism | Suppose that =𝐙absent𝐙\Lambda={\mathbf{Z}}= bold_Z or that is divisible. A real function h:X→𝐑:ℎ→𝑋𝐑h\colon X\to{\mathbf{R}}italic_h : italic_X → bold_R is -PL if and only if for every point x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X, there exists a (para)compact strictly analytic neighbourhood U𝑈Uitalic_U of x𝑥xitalic_x, a formal K∘superscript𝐾K^{\circ}italic_K start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT-model 𝔘𝔘{\mathfrak{U}}fraktur_U of U𝑈Uitalic_U, and 𝔏∈M(𝔘)𝔏𝑀𝔘{\mathfrak{L}}\in M({\mathfrak{U}})fraktur_L ∈ italic_M ( fraktur_U ), such that h|U=h𝔏evaluated-atℎ𝑈subscriptℎ𝔏h|_{U}=h_{\mathfrak{L}}italic_h | start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT fraktur_L end_POSTSUBSCRIPT. In this case, such a neighbourhood U𝑈Uitalic_U can be chosen arbitrarily small. | C |
-graph and let B𝐵Bitalic_B be the set of sinks, that is the set of external vertices labeling vertical edges of the lattice path, let |B|=r𝐵𝑟|B|=r| italic_B | = italic_r. We define the collection ℐℐ\mathcal{I}caligraphic_I of independent sets of the positroid 𝒫(Γ)𝒫Γ\mathcal{P}(\Gamma)caligraphic_P ( roman_Γ ) to be all sets I∈𝐼absentI\initalic_I ∈ ([n]k)binomialdelimited-[]𝑛𝑘[n]\choose k( binomial start_ARG [ italic_n ] end_ARG start_ARG italic_k end_ARG ) where k≤r𝑘𝑟k\leq ritalic_k ≤ italic_r, such that there exists k𝑘kitalic_k pairwise vertex-disjoint paths from I𝐼Iitalic_I to B𝐵Bitalic_B in ΓΓ\Gammaroman_Γ. Note that if x∈I∩B𝑥𝐼𝐵x\in I\cap Bitalic_x ∈ italic_I ∩ italic_B then x𝑥xitalic_x is the trivial path from I𝐼Iitalic_I to B𝐵Bitalic_B and therefore every element of B𝐵Bitalic_B is an independent set on its own. | A matroid M=([n],ℐ)𝑀delimited-[]𝑛ℐM=([n],\mathcal{I})italic_M = ( [ italic_n ] , caligraphic_I ) is a positroid if there is a | It is easy to see that positroids are actually gammoids. For any independent set I𝐼Iitalic_I in a positroid 𝒫𝒫\mathcal{P}caligraphic_P, there is a routing between I𝐼Iitalic_I and B𝐵Bitalic_B, where B𝐵Bitalic_B is the set of all sinks in 𝒫𝒫\mathcal{P}caligraphic_P. Therefore a positroid is a gammoid with representation (Γ,B,[n])Γ𝐵delimited-[]𝑛(\Gamma,B,[n])( roman_Γ , italic_B , [ italic_n ] ) with M(Γ,B,[n])=(E,ℐ)𝑀Γ𝐵delimited-[]𝑛𝐸ℐM(\Gamma,B,[n])=(E,\mathcal{I})italic_M ( roman_Γ , italic_B , [ italic_n ] ) = ( italic_E , caligraphic_I ) where | -diagram. Equivalently, a positroid 𝒫𝒫\mathcal{P}caligraphic_P is not connected if and only if there exist two disjoint isolated blocks in its | A positroid 𝒫𝒫\mathcal{P}caligraphic_P is connected if and only if there exists no two disjoint isolated blocks in its | A |
)^{2}|\psi_{r}|^{2}}.caligraphic_I start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 99 end_POSTSUBSCRIPT = italic_r start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , caligraphic_I start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 9 , 10 end_POSTSUBSCRIPT = 0 , caligraphic_I start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 10 , 10 end_POSTSUBSCRIPT = italic_r start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | The proof of this lemma needs some involved integration. We need to expand the integrand to the third order to prove the result. Instead of diving into massive computations, we defer the proof of it to the next section and continue the main thread of our construction. Let | The calculations here and later are a little bit tedious but still manageable with bare hands. To make the life easier, we perform them with the help of Mathematica111https://www.wolframcloud.com/obj/bingomat/Published/deg2-HM-final.nb | Here the projection is with respect to the inner product of 𝒲Φsubscript𝒲Φ\mathcal{W}_{\Phi}caligraphic_W start_POSTSUBSCRIPT roman_Φ end_POSTSUBSCRIPT. Consequently | The proof of Theorem 1.5 follows the general framework as the one in our work of half-harmonic maps [9] with new essential difficulties. First, the Jacobian of the kernels is uniformly non-degenerate in [9], while it degenerates as the parameter goes to infinity in the case of harmonic maps (cf. (4.9)). One needs to expand up to the third order in the computation to observe this fact, which makes the process substantially more involved. Second, the trick of choosing a vector field corresponding to the rotation in [9] does not work here, which is the heart of the construction. Fortunately, we leverage the degenerate tendency of the Jacobian to find a new one (cf. (4.17)). We have not gotten a satisfactory explanation about why such a vector field works. | B |
One can think of Riskα(g)subscriptRisk𝛼𝑔\mathrm{Risk}_{\alpha}(g)roman_Risk start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_g ) as the risk for a “harder” version of the | σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is multiplied by a factor of 1+α1𝛼1+\alpha1 + italic_α. Later (in Proposition | σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to (1+α)σ21𝛼superscript𝜎2(1+\alpha)\sigma^{2}( 1 + italic_α ) italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. We proved that when g𝑔gitalic_g is weakly | bootstrap draws B𝐵Bitalic_B to scale with 1/α1𝛼1/\alpha1 / italic_α, for small α𝛼\alphaitalic_α, in order to | from N(0,σ2N(0,\sigma^{2}italic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT), and the error variance σ2superscript𝜎2\sigma^{2}italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is chosen to meet a | A |
\rangle^{(\infty)})\bigr{)}=m^{h+1}.roman_dim start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_k [ italic_x start_POSTSUPERSCRIPT ( ⩽ italic_h ) end_POSTSUPERSCRIPT ] / ( italic_k [ italic_x start_POSTSUPERSCRIPT ( ⩽ italic_h ) end_POSTSUPERSCRIPT ] ∩ ⟨ italic_x start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ⟩ start_POSTSUPERSCRIPT ( ∞ ) end_POSTSUPERSCRIPT ) ) = italic_m start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT . | Let I⊂k[𝐱]𝐼𝑘delimited-[]𝐱I\subset k[\mathbf{x}]italic_I ⊂ italic_k [ bold_x ] be a zero-dimensional ideal with V(I)𝑉𝐼V(I)italic_V ( italic_I ) being a single point of multiplicity m𝑚mitalic_m. | Let m𝑚mitalic_m and i𝑖iitalic_i be positive integers with 1⩽i⩽m1𝑖𝑚1\leqslant i\leqslant m1 ⩽ italic_i ⩽ italic_m. | Let m𝑚mitalic_m be a positive integer and k𝑘kitalic_k be a differential field of zero characteristic. | Let m𝑚mitalic_m and hℎhitalic_h be positive integers and k𝑘kitalic_k be a differential field of zero characteristic. Then | C |
In both cases, some extra work is needed in order to prove that the constructed Lie algebras are in fact irreducible. To this end, we prove two technical results; one concerns a special class of positively-graded Lie algebras, the other a particular type of metric Lie algebra with a distinguished nondegenerate subspace which encodes part of the Lie algebra structure in a suitable sense. | Even though all results in this paper are stated over ℝℝ\mathbb{R}blackboard_R, many of the constructions work over more general fields. | v∈ℝ4n,v∈ℝ2⊕ℝ4n.formulae-sequence𝑣superscriptℝ4𝑛𝑣direct-sumsuperscriptℝ2superscriptℝ4𝑛\displaystyle v\in\mathbb{R}^{4n},v\in\mathbb{R}^{2}\oplus\mathbb{R}^{4n}.italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 4 italic_n end_POSTSUPERSCRIPT , italic_v ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊕ blackboard_R start_POSTSUPERSCRIPT 4 italic_n end_POSTSUPERSCRIPT . | v,w∈ℝ4n,v,w∈ℝ2⊕ℝ4n;formulae-sequence𝑣𝑤superscriptℝ4𝑛𝑣𝑤direct-sumsuperscriptℝ2superscriptℝ4𝑛\displaystyle v,w\in\mathbb{R}^{4n},v,w\in\mathbb{R}^{2}\oplus\mathbb{R}^{4n};italic_v , italic_w ∈ blackboard_R start_POSTSUPERSCRIPT 4 italic_n end_POSTSUPERSCRIPT , italic_v , italic_w ∈ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊕ blackboard_R start_POSTSUPERSCRIPT 4 italic_n end_POSTSUPERSCRIPT ; | For 𝔫~~𝔫\tilde{\mathfrak{n}}over~ start_ARG fraktur_n end_ARG we follow the same proof. First note that the Nikolayevsky derivation of 𝔫~~𝔫\tilde{\mathfrak{n}}over~ start_ARG fraktur_n end_ARG vanishes. | A |
+\frac{1}{q}}.italic_δ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ≤ italic_c italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_log italic_d ) italic_d start_POSTSUPERSCRIPT 2 italic_k - divide start_ARG 1 end_ARG start_ARG italic_q end_ARG end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_q end_ARG end_POSTSUPERSCRIPT . | (b) The second way to apply the Hessian estimate is to use it within the context of the heat flow transport map of Kim and Milman [37]. This approach avoids the issues mentioned in part (a). On the other hand, the usage of this transport map is only suitable if we want to prove pointwise estimates on the Lipschitz constant of the transport map. In contrast, the Brownian transport map allows us to prove estimates on the Lipschitz constant of the transport map in expectation, which is what is needed to make the connection with the Kannan–Lovász–Simonovits conjecture; cf. Theorem 1.4. (We remark however that the heat flow map has its own advantages, as explained in [54, p.3].) | Extending the analysis of [18], we can use Lemma 4.5 to show that 1τ1𝜏\frac{1}{\tau}divide start_ARG 1 end_ARG start_ARG italic_τ end_ARG has a sub-exponential tail. | The proof of the second part of the lemma follows by noting that the solution to the ordinary differential equation \tagform@7.1, with initial condition 00 at t=0𝑡0t=0italic_t = 0, is | The statement of the lemma is essentially [18, Lemma 6], up to time-change. To make the connection with [18] we recall that, by Lemma 4.1, | D |
It is easy to see that the conclusion of (ii) fails as soon as M𝑀Mitalic_M is not weakly normal (see the proof of Corollary 3.103.103.103.10 of [26]). | We now are going to give consequences of these results in the case where the underlying manifold is an open set ΩΩ\Omegaroman_Ω. | For an open bounded subset ΩΩ{\Omega}roman_Ω of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, the set | We now wish to explain that in the case where p𝑝pitalic_p is large, the results of [26] yield that the mapping 𝐭𝐫∂Msubscript𝐭𝐫𝑀\mathbf{tr}_{\partial M}bold_tr start_POSTSUBSCRIPT ∂ italic_M end_POSTSUBSCRIPT, defined in the preceding section, is continuous in the Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT norm (see Corollary 2.7 below and Example 2.5 just above). | It is worth stressing the fact that this theory actually goes over to the more general framework of polynomially bounded o-minimal structures (expanding ℝℝ\mathbb{R}blackboard_R) [6, 10]. Indeed, the arguments being essentially local, we only need the underlying manifold M𝑀Mitalic_M to be such that every x∈M¯𝑥¯𝑀x\in{\overline{M}}italic_x ∈ over¯ start_ARG italic_M end_ARG has a neighborhood U𝑈Uitalic_U in M¯¯𝑀{\overline{M}}over¯ start_ARG italic_M end_ARG such that U∩M𝑈𝑀U\cap Mitalic_U ∩ italic_M is bi-Lipschitz homeomorphic to a set definable in some polynomially bounded o-minimal structure (that may depend on x𝑥xitalic_x), and more generally we could adapt the statements to abstract varieties, which is useful for the complex case. Moreover, since we could use “cut-off” functions at infinity, the density results actually do not require the underlying manifold to be bounded. | A |
We will see in Corollary 4.4 that, on a Riemannian manifold, a real weakly harmful structure is necessarily harmful. | Theorem 3.1 and its time-like counterpart, Theorem 3.2, show that any nondegenerate hypersurface inside an Einstein pseudo-Riemannian manifold (Z,h)𝑍ℎ(Z,h)( italic_Z , italic_h ) endowed with a Killing spinor inherits a harmful structure. If (Z,h)𝑍ℎ(Z,h)( italic_Z , italic_h ) is not assumed to be Einstein, one obtains a weakly harmful structure (see Corollary 2.2 and Corollary 2.3). | We will see in Corollary 4.4 that, on a Riemannian manifold, a real weakly harmful structure is necessarily harmful. | The main result of this paper is that any real analytic harmful structure can be extended to an Einstein manifold with a Killing spinor. The proof is akin to [1]; however, some work is needed to handle the more general signature. Indeed, the characterization of real analytic hypersurfaces inside Riemannian Einstein manifolds given in [22] extends to the pseudo-Riemannian case (Corollary 2.2, Corollary 2.3). We prove that the spinors defining the harmful structure can be extended to Killing spinors on Z𝑍Zitalic_Z by parallel translation relative to a suitable connection. This shows that any pseudo-Riemannian spin manifold with a harmful structure can be extended to an Einstein manifold with a Killing spinor (Theorem 5.4). Our result generalizes the known results quoted above in two respects: we consider pseudo-Riemannian metrics, and we allow λ≠0𝜆0\lambda\neq 0italic_λ ≠ 0 in the Killing equation. By contrast, we restrict the geometry by imposing that the metric on Z𝑍Zitalic_Z is Einstein. | A spinor ΨΨ\Psiroman_Ψ on a pseudo-Riemannian spin manifold (Z,h)𝑍ℎ(Z,h)( italic_Z , italic_h ) is Killing if it satisfies | A |
+n+k-2}{2j+k-2}\binom{2j+k-1}{j-1}.× ( 1 + italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_n - 1 ) + italic_k ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 3 italic_n ) - 3 italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_j ( 2 italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_k italic_n - italic_k - 1 ) ) ( FRACOP start_ARG italic_j + italic_n + italic_k - 2 end_ARG start_ARG 2 italic_j + italic_k - 2 end_ARG ) ( FRACOP start_ARG 2 italic_j + italic_k - 1 end_ARG start_ARG italic_j - 1 end_ARG ) . | Consider points of a form (l(k+2)−1+j,l(k+2)−1+2ln+j)𝑙𝑘21𝑗𝑙𝑘212𝑙𝑛𝑗(l(k+2)-1+j,l(k+2)-1+2ln+j)( italic_l ( italic_k + 2 ) - 1 + italic_j , italic_l ( italic_k + 2 ) - 1 + 2 italic_l italic_n + italic_j ). For such points, the proof is the same as for the Formula (10) for the hypotenuse of region II. | The proof is the same as the proof of the main theorem in [13], except that instead of Lemma 4.9 in [13], for the slightly tweaked model one should use Lemma 2.5. | The proof is the same, as for Lemma 4.1. When proving the inductive step, we still can apply Corollary 3.8 as region II is still congruent to the one, translated by T𝑇Titalic_T. | The proof of this lemma concludes the proof of the identity and finishes the proof of the initial theorem. | D |
Let W𝑊Witalic_W be a Coxeter group of type Ansubscript𝐴𝑛A_{n}italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Bnsubscript𝐵𝑛B_{n}italic_B start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, Dnsubscript𝐷𝑛D_{n}italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, or I2(n)subscript𝐼2𝑛I_{2}(n)italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_n ). | Pr[Xw⊆G/B is LJ(w)-spherical]⟶0.⟶Prdelimited-[]subscript𝑋𝑤𝐺𝐵 is LJ(w)-spherical0{\rm Pr}[X_{w}\subseteq G/B\text{\ is $L_{J(w)}$-spherical}]\longrightarrow 0.roman_Pr [ italic_X start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ⊆ italic_G / italic_B is italic_L start_POSTSUBSCRIPT italic_J ( italic_w ) end_POSTSUBSCRIPT -spherical ] ⟶ 0 . | Pr[w is proper]⟶0.⟶Prdelimited-[]𝑤 is proper0{\rm Pr}[w\text{\ is proper}]\longrightarrow 0.roman_Pr [ italic_w is proper ] ⟶ 0 . | Pr[w is J(w)-spherical]⟶0.⟶Prdelimited-[]𝑤 is J(w)-spherical0{\rm Pr}[w\text{\ is $J(w)$-spherical}]\longrightarrow 0.roman_Pr [ italic_w is italic_J ( italic_w ) -spherical ] ⟶ 0 . | Pr[Xw⊆G/B is LJ(w)-spherical]≤Pr[w is proper]Prdelimited-[]subscript𝑋𝑤𝐺𝐵 is LJ(w)-sphericalPrdelimited-[]𝑤 is proper{\rm Pr}[X_{w}\subseteq G/B\text{\ is $L_{J(w)}$-spherical}]\leq{\rm Pr}[w% | C |
Each H~keffsubscriptsuperscript~𝐻eff𝑘\widetilde{H}^{\rm eff}_{k}over~ start_ARG italic_H end_ARG start_POSTSUPERSCRIPT roman_eff end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT consists of Ltsubscript𝐿𝑡L_{t}italic_L start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT selected columns of Heffsuperscript𝐻effH^{\rm eff}italic_H start_POSTSUPERSCRIPT roman_eff end_POSTSUPERSCRIPT, associated with the selected polarization reconfigurable Tx antenna elements, and is described as | H~keffH~keff†=superscriptsubscript~𝐻𝑘effsuperscriptsubscript~𝐻𝑘eff†absent\displaystyle\widetilde{H}_{k}^{\rm eff}\widetilde{H}_{k}^{\rm eff\dagger}=over~ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_eff end_POSTSUPERSCRIPT over~ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_eff † end_POSTSUPERSCRIPT = | H~keff=superscriptsubscript~𝐻𝑘effabsent\displaystyle\widetilde{H}_{k}^{\rm eff}={{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}% | H~EW,keffH~EW,keff†=superscriptsubscript~𝐻EW𝑘effsuperscriptsubscript~𝐻EW𝑘eff†absent\displaystyle{}\widetilde{H}_{{\rm EW},k}^{\rm eff}\widetilde{H}_{{\rm EW},k}^% | H~G,keffH~G,keff†=superscriptsubscript~𝐻G𝑘effsuperscriptsubscript~𝐻G𝑘eff†absent\displaystyle\widetilde{H}_{{\rm G},k}^{\rm eff}\widetilde{H}_{{\rm G},k}^{{% | B |
In Section 5 the process for transforming the differential equation, with radicals, into a system of ADEs is summarized in a procedure. | 𝕍(𝕋1)𝕍superscript𝕋1\mathbb{V}(\mathbb{T}^{1})blackboard_V ( blackboard_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) can be parametrized, for example, as | They are polynomials in the unknowns and their derivatives, and the coefficients of the polynomial are rational in the independent variables, see e.g. [11]. | By our method, symbolic solutions of the transformed equations can be computed by well-known techniques for AODEs such as the Newton polygon method [5]. In Example 5.3, we show a differential equation where the application of the method in this paper generates more satisfactory expressions of the solutions than the direct treatment on the given equation; for this, the example has been developed with the aid of the computer algebra system Maple 2023. | For the basic notions and results on algebraic geometry, which are fundamental here, see for example [7]. | D |
\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_p ( italic_π ) = italic_p start_POSTSUBSCRIPT italic_λ ( italic_π ) end_POSTSUBSCRIPT ( bold_x ) , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_m ( italic_π ) = [ ∏ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_λ ( italic_π ) ) ! ] ⋅ italic_m start_POSTSUBSCRIPT italic_λ ( italic_π ) end_POSTSUBSCRIPT ( bold_x ) , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_e ( italic_π ) = italic_λ ( italic_π ) ! ⋅ italic_e start_POSTSUBSCRIPT italic_λ ( italic_π ) end_POSTSUBSCRIPT ( bold_x ) , end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_h ( italic_π ) = italic_λ ( italic_π ) ! ⋅ italic_h start_POSTSUBSCRIPT italic_λ ( italic_π ) end_POSTSUBSCRIPT ( bold_x ) . end_CELL start_CELL end_CELL end_ROW end_ARRAY | The following definition is an extension to [Do72, §3]. The form we adopt below is also inspired by the arguments in [BB18, §3]. | and hence indicate that Conjectures 6.1 and 6.2 are in general independent. Here we shall explain that Doubilet’s results in [Do72] give closed transformation formulas between Chern numbers and Chern character numbers of irreducible hyper-Kähler manifolds. Recall in [Do72, Thm 3] that | When taking D=ℤ>0𝐷subscriptℤabsent0D=\mathbb{Z}_{>0}italic_D = blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT and x(ai,n)=xn𝑥subscript𝑎𝑖𝑛subscript𝑥𝑛x(a_{i},n)=x_{n}italic_x ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_n ) = italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT independent of ai∈Ssubscript𝑎𝑖𝑆a_{i}\in Sitalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_S, Definition 3.1 specializes to the following example, which is exactly the case considered in [Do72, Thms 1 and 5] and justifies the notation in (3.1)-(3.4). | Since the constructions and symbols used in [Do72] are different from ours, we briefly explain that Example 3.2 is exactly the case treated in [Do72] for the reader’s convenience. Following the notation in [Do72, p.379], let | D |
𝐄(ψ∙)=𝐄(Δ[ψ∙]∙).𝐄subscript𝜓∙𝐄Δsubscriptdelimited-[]subscript𝜓∙∙\mathbf{E}(\psi_{\bullet})=\mathbf{E}(\Delta[\psi_{\bullet}]_{\bullet}).bold_E ( italic_ψ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) = bold_E ( roman_Δ [ italic_ψ start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ) . | Condition (3) follows from Theorem 5.14 and Proposition 5.8. Condition (4) is a translation of (7.1). Condition (5) is obvious.Finally, (7.9) follows from (7.1) and (1.3).∎ | We verify the conditions in Definition 7.2.Condition (1) follows from Proposition 5.8. Condition (2) follows from the fact | This follows from the corresponding result for the φjsuperscript𝜑𝑗\varphi^{j}italic_φ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT’s.∎ | This follows from the corresponding inequality of geodesic rays, which in turn follows from (biblatex) Package biblatex Error: Command ’\cite’ undefinedSee the biblatex package documentation for explanation.The citation command ’\cite’ | B |
It follows from the Sturm oscillation theorem that the number of zeros of v𝑣vitalic_v is equal to the number of bound state of −d2dx2+12ϕ′(x)superscript𝑑2𝑑superscript𝑥212superscriptitalic-ϕ′𝑥-\frac{d^{2}}{dx^{2}}+\frac{1}{2}\phi^{\prime}(x)- divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x ). Our objective here is to show that v𝑣vitalic_v has precisely one zero, an endeavor well-suited to CAP techniques. This effort aligns our research with the growing application of CAP techniques to address open problems in differential equations. See, for instance, [56, 60, 35, 18, 6, 32, 61] for further insights into CAP techniques for differential equations. | We proceed by setting a fixed value for ν𝜈\nuitalic_ν, and outline the conceptual framework for the proof of Theorem 5.1, with further details elaborated in Section 6. | If one can derive sufficiently accurate numerical estimates for the solutions of (5.2), (5.3), and (5.4), complemented by additional analytical estimates, then one can rigorously determine the number of zeros in the solution of (5.1), thereby proving Theorem 5.1. All computations utilize interval arithmetic for rigorous control over rounding errors, specifically through Rump’s INTLAB package for MATLAB [52]. | Section 5 delineates the methodology underpinning our CAP approach, setting the stage, with further details elaborated in Section 6. Notably, all computations are performed using interval arithmetic, ensuring rigorous control over rounding errors. This is facilitated by the use of Rump’s INTLAB package for MATLAB [52]. | The result of Proposition 5.2, under the assumption of convergence, facilitates rigorous numerical estimates for the solutions of the first two equations of (5.2) and the former equation of (5.3) over the interval (−∞,0]0(-\infty,0]( - ∞ , 0 ]. For the proof, the infinite series in (5.5) is divided into a finite sum plus a residual term. The finite sum is computed using interval arithmetic, specifically through Rump’s INTLAB package for MATLAB [52], for rigorous control of rounding errors. Additionally, an estimate for the residual is derived and then computed by interval arithmetic. See Section 6 for further details. | C |
The paper [2] studied the first-order SDEs with a fast-varying jump process, the aforementioned difficulty | may need to require further that the coefficients are Lipschitz continuous, which we do not assume here.] | Why do we care of the second-order stochastic systems? This is because numerous problems in physics, statistical mechanics, and engineering, etc., involve such systems. | assume any structure of ξtεsubscriptsuperscript𝜉𝜀𝑡\xi^{\varepsilon}_{t}italic_ξ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. As a result, we could not use the integration by parts (Lemma 2.1) to connect the first-order and the second-order systems. | Here, we are dealing with fully nonlinear second-order stochastic systems with multi-scales, but we do not use the weak convergence method to avoid requiring the Lipschitz continuity and other growth conditions. | D |
If we take α=β𝛼𝛽\alpha=\betaitalic_α = italic_β in Theorem 3.1, it reduces to a result of Nunokawa and Sokół [9, Theorem 2.1]. | Let μ∈[0,1)𝜇01\mu\in[0,1)italic_μ ∈ [ 0 , 1 ) and f∈𝒜.𝑓𝒜f\in\mathcal{A}.italic_f ∈ caligraphic_A . If | Let α,β∈(0,1)𝛼𝛽01\alpha,\beta\in(0,1)italic_α , italic_β ∈ ( 0 , 1 ) and 0<γ≤1.0𝛾10<\gamma\leq 1.0 < italic_γ ≤ 1 . Moreover, if f∈𝒜𝑓𝒜f\in\mathcal{A}italic_f ∈ caligraphic_A satisfies | where C(μ):=(1−μ)(1+2/(1−μ)).assign𝐶𝜇1𝜇121𝜇C(\mu):=(1-\mu)(\sqrt{1+2/(1-\mu)}).italic_C ( italic_μ ) := ( 1 - italic_μ ) ( square-root start_ARG 1 + 2 / ( 1 - italic_μ ) end_ARG ) . Then we have | Let δ>0𝛿0\delta>0italic_δ > 0, λ∈[0,π/2)𝜆0𝜋2\lambda\in[0,\pi/2)italic_λ ∈ [ 0 , italic_π / 2 ) and α∈[0,1]𝛼01\alpha\in[0,1]italic_α ∈ [ 0 , 1 ]. If f∈𝒜p𝑓subscript𝒜𝑝f\in\mathcal{A}_{p}italic_f ∈ caligraphic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT, | A |
Let G𝐺Gitalic_G be an almost-simple group of type An−1subscript𝐴𝑛1A_{n-1}italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT (n≥2𝑛2n\geq 2italic_n ≥ 2), i.e. G=SLn/μr𝐺subscriptSL𝑛subscript𝜇𝑟G=\operatorname{SL}_{n}/\mu_{r}italic_G = roman_SL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT / italic_μ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT, for some r𝑟ritalic_r dividing n𝑛nitalic_n. | Type A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The isomorphism of Theorem A gives the following equalities | Type An−1subscript𝐴𝑛1A_{n-1}italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT, n≥3𝑛3n\geq 3italic_n ≥ 3. The isomorphism of Theorem A gives the following equalities | Let G𝐺Gitalic_G be an almost-simple group of type E7subscript𝐸7E_{7}italic_E start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT, E8subscript𝐸8E_{8}italic_E start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT, F4subscript𝐹4F_{4}italic_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The isomorphism of Theorem A gives the following equalities | Let G𝐺Gitalic_G be an almost-simple group of type E6subscript𝐸6E_{6}italic_E start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT. The isomorphism of Theorem A gives the following equalities | A |
In what follows, we shall show the assertion for BSsubscript𝐵𝑆B_{S}italic_B start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT. | if dimX≥2dimension𝑋2\dim X\geq 2roman_dim italic_X ≥ 2, where the infimum is taken over all prime divisors E𝐸Eitalic_E over X𝑋Xitalic_X with center cX(E)⊂Wsubscript𝑐𝑋𝐸𝑊c_{X}(E)\subset Witalic_c start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) ⊂ italic_W. | We may assume that (X,B+M)𝑋𝐵𝑀(X,B+M)( italic_X , italic_B + italic_M ) is generalized lc at η𝜂\etaitalic_η (cf. Remark 2.4(4)). | By taking general hypersurface sections (see [BZ16, Remark 4.8]), we may assume that dimX=2dimension𝑋2\dim X=2roman_dim italic_X = 2. | We may assume that E𝐸Eitalic_E is a divisor on X′superscript𝑋′X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. | C |
But there is another expansion which is 0∈I0𝐼0\in I0 ∈ italic_I and b≠0𝑏0b\neq 0italic_b ≠ 0 because a≠0𝑎0a\neq 0italic_a ≠ 0, so we get two distinct expansions of 00 modulo I𝐼Iitalic_I, which leads to a contradiction by (4). | We say that I𝐼Iitalic_I is Lech-independent if I/I2𝐼superscript𝐼2I/I^{2}italic_I / italic_I start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is free over S/I𝑆𝐼S/Iitalic_S / italic_I. We say that I𝐼Iitalic_I is strongly Lech-independent if Ii/Ii+1superscript𝐼𝑖superscript𝐼𝑖1I^{i}/I^{i+1}italic_I start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT / italic_I start_POSTSUPERSCRIPT italic_i + 1 end_POSTSUPERSCRIPT is free over S/I𝑆𝐼S/Iitalic_S / italic_I for any i𝑖iitalic_i. We say that a sequence of elements x1,…,xrsubscript𝑥1…subscript𝑥𝑟x_{1},...,x_{r}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT is Lech-independent (resp. strongly Lech-independent), if it forms a minimal generating set of an ideal which is Lech-independent (resp. strongly Lech-independent). | Let S0subscript𝑆0S_{0}italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be an Artinian local ring which is not a field and let 𝔫0subscript𝔫0\mathfrak{n}_{0}fraktur_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT be the maximal ideal of S0subscript𝑆0S_{0}italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Let S=S0[[x]]/𝔫0x2𝑆subscript𝑆0delimited-[]delimited-[]𝑥subscript𝔫0superscript𝑥2S=S_{0}[[x]]/\mathfrak{n}_{0}x^{2}italic_S = italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT [ [ italic_x ] ] / fraktur_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and I=(x)𝐼𝑥I=(x)italic_I = ( italic_x ). Then I𝐼Iitalic_I is Lech-independent, but not strongly Lech-independent. | Let I𝐼Iitalic_I be a strongly Lech-independent ideal of a local ring (S,𝔫)𝑆𝔫(S,\mathfrak{n})( italic_S , fraktur_n ). Then dim(Γ)dimensionnormal-Γ\dim(\Gamma)roman_dim ( roman_Γ ) and e(Γ)𝑒normal-Γe(\Gamma)italic_e ( roman_Γ ) are independent of the choice of Γnormal-Γ\Gammaroman_Γ whenever I𝐼Iitalic_I is Γnormal-Γ\Gammaroman_Γ-expandable from degree i𝑖iitalic_i to j𝑗jitalic_j for any i<j𝑖𝑗i<jitalic_i < italic_j. If moreover S/I𝑆𝐼S/Iitalic_S / italic_I is Artinian, then dim(Γ)=dimSdimensionnormal-Γdimension𝑆\dim(\Gamma)=\dim Sroman_dim ( roman_Γ ) = roman_dim italic_S and e(I)=l(S/I)e(Γ)𝑒𝐼𝑙𝑆𝐼𝑒normal-Γe(I)=l(S/I)e(\Gamma)italic_e ( italic_I ) = italic_l ( italic_S / italic_I ) italic_e ( roman_Γ ). In particular, if I𝐼Iitalic_I is the maximal ideal 𝔫𝔫\mathfrak{n}fraktur_n, then e(Γ)=e(S)𝑒normal-Γ𝑒𝑆e(\Gamma)=e(S)italic_e ( roman_Γ ) = italic_e ( italic_S ). | Suppose I𝐼Iitalic_I is generated by a regular sequence, or I𝐼Iitalic_I is the maximal ideal 𝔫𝔫\mathfrak{n}fraktur_n, then I𝐼Iitalic_I is strongly Lech-independent. | D |
If we write the labels in the order of movement from p1subscript𝑝1p_{1}italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to p3subscript𝑝3p_{3}italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT along m𝑚mitalic_m, we get the permutation of M𝑀Mitalic_M. Note that the labels do not depend on the choice of γ𝛾\gammaitalic_γ, so the permutation of M𝑀Mitalic_M is well-defined. | In the case of non-singular meanders the permutation uniquely determines the meander, but in the case of meanders with non-transverse intersections this is not generally true. | If we write the labels in the order of movement from p1subscript𝑝1p_{1}italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to p3subscript𝑝3p_{3}italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT along m𝑚mitalic_m, we get the permutation of M𝑀Mitalic_M. Note that the labels do not depend on the choice of γ𝛾\gammaitalic_γ, so the permutation of M𝑀Mitalic_M is well-defined. | Consider a meander M1subscript𝑀1M_{1}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT that is a concatenation of a non-singular meander with the permutation (7,6,1,2,5,8,9,4,3)761258943(7,6,1,2,5,8,9,4,3)( 7 , 6 , 1 , 2 , 5 , 8 , 9 , 4 , 3 ) and M𝑀Mitalic_M (see the example in Figure 10(a), where M𝑀Mitalic_M is the meander with the permutation (1,2,3,4,5)12345(1,2,3,4,5)( 1 , 2 , 3 , 4 , 5 )). Next we need to “double” M1subscript𝑀1M_{1}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (as in Figure 10(b)) to obtain a meander M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Finally, we can transform M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT into an almost irreducible meander M3subscript𝑀3M_{3}italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT of total order 2(n+9)+22𝑛922(n+9)+22 ( italic_n + 9 ) + 2 by adding two more intersections between points with labels 14141414 and 15151515 (see Figure 10(c)). We can also obtain an almost irreducible meander M4subscript𝑀4M_{4}italic_M start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT of odd total order in a similar way, see Figure 10(d). | For example, the permutation of the second meander in Figure 2 is (6,5,4,3,2,1)654321(6,5,4,3,2,1)( 6 , 5 , 4 , 3 , 2 , 1 ). | D |
For all g∈ℤ≥0𝑔subscriptℤabsent0g\in{\mathbb{Z}}_{\geq 0}italic_g ∈ blackboard_Z start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT and r∈ℤ>0𝑟subscriptℤabsent0r\in{\mathbb{Z}}_{>0}italic_r ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT there is a canonical isomorphism of cohomologically graded vector spaces | 𝔤¯1,rBetti≅𝐇(ℳ1,r,1Betti,ℚ¯vir)subscriptsuperscript¯𝔤Betti1𝑟𝐇subscriptsuperscriptℳBetti1𝑟1subscript¯ℚvir\underline{{\mathfrak{g}}}^{\operatorname{\scriptscriptstyle{Betti}}}_{1,r}% | 𝔤g,rBetti≅𝐇(ℳg,r,1Betti,ℚvir)subscriptsuperscript𝔤Betti𝑔𝑟𝐇subscriptsuperscriptℳBetti𝑔𝑟1subscriptℚvir\displaystyle{\mathfrak{g}}^{\operatorname{\scriptscriptstyle{Betti}}}_{g,r}% | 𝔤g,rBetti≔𝐇(ℳg,r,0Betti,ℬ𝒫𝒮g,rBetti).≔subscriptsuperscript𝔤Betti𝑔𝑟𝐇subscriptsuperscriptℳBetti𝑔𝑟0ℬ𝒫subscriptsuperscript𝒮Betti𝑔𝑟{\mathfrak{g}}^{\operatorname{\scriptscriptstyle{Betti}}}_{g,r}\coloneqq% | 𝔤¯g,rBetti≅𝐇(ℳg,r,1Betti,ℚ¯vir).subscriptsuperscript¯𝔤Betti𝑔𝑟𝐇subscriptsuperscriptℳBetti𝑔𝑟1subscript¯ℚvir\underline{{\mathfrak{g}}}^{\operatorname{\scriptscriptstyle{Betti}}}_{g,r}% | B |
Inspired by an example in Salathé and Jones [38] about the impact of community structure on disease spread, we explore the impact of the effective community structure of a contact network on the outbreak duration, final size, and outbreak peak of a disease in simulations of a susceptible–infected–recovered (SIR) model of disease spread on the network. See [22, 30] for an introduction to disease dynamics on networks. | Salathé and Jones [38] explored the effect of differences in community structure on the outbreak duration, final size, and outbreak peak in simulations of an SIR model on networks. In our work, we consider a modified version of one of the examples in [38]. | We produce parameter configurations in NS=68subscript𝑁S68N_{\rm S}=68italic_N start_POSTSUBSCRIPT roman_S end_POSTSUBSCRIPT = 68 stages [with δ∗=0.2subscript𝛿0.2\delta_{*}=0.2italic_δ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = 0.2, δ∗∗=1subscript𝛿absent1\delta_{**}=1italic_δ start_POSTSUBSCRIPT ∗ ∗ end_POSTSUBSCRIPT = 1, β∗=0.125subscript𝛽0.125\beta_{*}=0.125italic_β start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = 0.125, α=0.1𝛼0.1\alpha=0.1italic_α = 0.1, and the corresponding value of β∗∗subscript𝛽absent\beta_{**}italic_β start_POSTSUBSCRIPT ∗ ∗ end_POSTSUBSCRIPT that we obtain using (39)]. For each parameter configuration, we run Nsim=1000subscript𝑁sim1000N_{\textnormal{sim}}=1000italic_N start_POSTSUBSCRIPT sim end_POSTSUBSCRIPT = 1000 simulations of SIR dynamics on GRLsuperscript𝐺RLG^{\mathrm{RL}}italic_G start_POSTSUPERSCRIPT roman_RL end_POSTSUPERSCRIPT (see Algorithm 3). In Figures 11(a)–(c), we show the mean outbreak duration, final size, and outbreak peak for the parameter configuration in each of the 68 stages. The qualitative behavior of these epidemiological quantities is consistent with the observations in the example of Salathé and Jones [38]. Specifically, the mean final outbreak size and the mean outbreak peak decrease monotonically as we increase the stage number [see Figures 11(b,c)]. Additionally, the mean outbreak duration peaks at intermediate stages [see Figure 11(a)]. | Inspired by an example in Salathé and Jones [38] about the impact of community structure on disease spread, we explore the impact of the effective community structure of a contact network on the outbreak duration, final size, and outbreak peak of a disease in simulations of a susceptible–infected–recovered (SIR) model of disease spread on the network. See [22, 30] for an introduction to disease dynamics on networks. | Community structure can greatly influence disease dynamics on networks [26, 30]. Salathé and Jones [38] illustrated that changes in community structure are correlated with changes in disease quantities for susceptible–infected–recovered (SIR) dynamics on a network. One of their findings is that outbreak duration can achieve a maximum at intermediate modularity values. Inspired by Salathé and Jones [38], we use our adaptation of InfoMap to study an example contagion process that illustrates how absorption in disease dynamics affects community structure, which in turn affects disease spread. We investigate the association between changes in the effective community structure that is induced by the node-absorption rates with disease quantities such as outbreak size and duration. | A |
2:The estimated sets V^1,V^2subscript^𝑉1subscript^𝑉2\widehat{V}_{1},\widehat{V}_{2}over^ start_ARG italic_V end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over^ start_ARG italic_V end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. | 3:Run Algorithm 1 Pre-processing to obtain subset ℳℳ{\mathcal{M}}caligraphic_M which achieves maximal SNRSNR\mathrm{SNR}roman_SNR. | When ℳ={2}ℳ2{\mathcal{M}}=\{2\}caligraphic_M = { 2 } and k𝑘kitalic_k is fixed, SNRSNR\mathrm{SNR}roman_SNR in 1.2 is equal to (a−b)2k[a+(k−1)b]superscript𝑎𝑏2𝑘delimited-[]𝑎𝑘1𝑏\frac{(a-b)^{2}}{k[a+(k-1)b]}divide start_ARG ( italic_a - italic_b ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_k [ italic_a + ( italic_k - 1 ) italic_b ] end_ARG, which corresponds to the SNRSNR\mathrm{SNR}roman_SNR for the undirected graph in [19], see also [1, Section 6]. | 3:Run Algorithm 1 Pre-processing to obtain subset ℳℳ{\mathcal{M}}caligraphic_M which achieves the maximal SNRSNR\mathrm{SNR}roman_SNR. | where ℳℳ{\mathcal{M}}caligraphic_M is obtained from Algorithm 1, and SNRℳ(k)subscriptSNRℳ𝑘\mathrm{SNR}_{{\mathcal{M}}}(k)roman_SNR start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT ( italic_k ) and Cℳ(k)subscript𝐶ℳ𝑘C_{{\mathcal{M}}}(k)italic_C start_POSTSUBSCRIPT caligraphic_M end_POSTSUBSCRIPT ( italic_k ) are defined in (1.2), (1.3). | A |
\varepsilon C}\right\}.italic_τ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT := roman_inf { italic_t ≥ 0 : ∀ italic_v ∈ bold_v : | italic_K start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_N start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_t ) - over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( bold_v ) | < italic_ε italic_C } . | To study the typical long-term behaviour of the population, two scaling parameters are introduced: The carrying capacity K𝐾Kitalic_K, which scales the order of the population size, and the mutation probability μKsubscript𝜇𝐾\mu_{K}italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT, which scales the frequency of mutation events. For large populations (K→∞→𝐾K\to\inftyitalic_K → ∞) and rare mutations (μK→0→subscript𝜇𝐾0\mu_{K}\to 0italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT → 0), different mechanisms that change the state of the population - like mutations introducing a new type or interactions between individuals that lead to a new equilibrium state of resident traits - act on different time scales. There are three important time scales in this setting: Ecological interactions between well-established subpopulations, like the competition for resources, can change the composition of the overall population within a short time of order 1. This is related to classical Lotka-Volterra dynamics and leads to equilibrium states between the larger traits. Short-range mutations and the initial exponential growth of small mutant populations can be witnessed on a logarithmic time scale of order lnK𝐾\ln Kroman_ln italic_K. Finally, long-range mutations - in particular those that need to traverse a large fitness valley of width L𝐿Litalic_L - are quite rare and occur on a time scale of order 1/KμKL1𝐾superscriptsubscript𝜇𝐾𝐿1/K\mu_{K}^{L}1 / italic_K italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT. The distinction between long and short-range mutations depends on the choice of the mutation probability μKsubscript𝜇𝐾\mu_{K}italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT, where long ranges L𝐿Litalic_L satisfy KμKL≪1much-less-than𝐾superscriptsubscript𝜇𝐾𝐿1K\mu_{K}^{L}\ll 1italic_K italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT ≪ 1. To obtain a non-trivial limit as K→∞→𝐾K\to\inftyitalic_K → ∞, the population size is usually rescaled by K𝐾Kitalic_K. As a result, only the established resident traits are visible. Since the ecological changes of these traits happen very fast in comparison with the other time scales, the limit of the population process yields a jump process that transitions between different equilibrium states. | The present paper finally closes the gap between the previous works and gives a full description of the jump processes resulting from long-range mutations on general finite trait graphs, thus extending the results of [8] to the more general setting of [14]. This general setting entails that, for a given equilibrium state, there might be several paths to cross the surrounding fitness valley. Concentrating on the decisive, shortest paths we calculate the rate of a transition to the next evolutionary stable condition and give the precise asymptotics in Theorem 2.7 and Corollary 2.8. The length of the shortest paths determines the time scale to cross the valley. Based on this, we introduce the notion of a stability degree L𝐿Litalic_L to classify the equilibrium states. Combining multiple of these steps gives rise to a jump chain that moves on a so called metastability graph stated in Corollary 2.10. This graph typically consists of fitness valleys of different width, which can be crossed on different time scales of the form 1/KμKL1𝐾superscriptsubscript𝜇𝐾𝐿1/K\mu_{K}^{L}1 / italic_K italic_μ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT. Depending on the choice of time scale, only some of these transitions are possible (valleys of width strictly larger than L𝐿Litalic_L cannot be crossed) or visible (transitions of valleys of width strictly smaller than L𝐿Litalic_L are immediate). This leads to different limiting jump processes in Theorem 2.11. | In this section we discuss the equilibrium population sizes of the living traits once an ESC is obtained. The results from [14] only characterize the orders of population sizes βwsubscript𝛽𝑤\beta_{w}italic_β start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT and the actual size n¯(𝐯)¯𝑛𝐯\bar{n}(\mathbf{v})over¯ start_ARG italic_n end_ARG ( bold_v ) of the resident traits associated to an ESC. In order to calculate the precise transition rates from one ESC to another, we do, however, need a better estimate for the population sizes of the non-resident traits in Vα(𝐯)subscript𝑉𝛼𝐯V_{\alpha}(\mathbf{v})italic_V start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( bold_v ). | Here C𝐶Citalic_C is a constant, depending only on the competition rates c(v,w)𝑐𝑣𝑤c(v,w)italic_c ( italic_v , italic_w ), which compensates the slight shift of the equilibrium due to small fluctuations of non-resident traits. | D |
Ptf(x)−f(x)=∫0tPsLf(x)𝑑s,x∈Rdformulae-sequencesubscript𝑃𝑡𝑓𝑥𝑓𝑥superscriptsubscript0𝑡subscript𝑃𝑠𝐿𝑓𝑥differential-d𝑠𝑥superscript𝑅𝑑P_{t}f(x)-f(x)=\int_{0}^{t}P_{s}\,Lf(x)\,ds\,,\qquad x\in{\mathbb R}^{d}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_f ( italic_x ) - italic_f ( italic_x ) = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_L italic_f ( italic_x ) italic_d italic_s , italic_x ∈ italic_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | In what follows we use properties of Qtnfsuperscriptsubscript𝑄𝑡𝑛𝑓Q_{t}^{n}fitalic_Q start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_f from Fact 3.2. | that lead to the construction and basic properties of Ptsubscript𝑃𝑡P_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT. We furthermore point out key hypotheses | a fact that fnsubscript𝑓𝑛f_{n}italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT converges to f𝑓fitalic_f | We note that the above mentioned verification requires continuity properties similar to Fact 3.4 (3). | D |
When the algorithm has finished, let T𝑇Titalic_T be the total of the values xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in the bin containing xnsubscript𝑥𝑛x_{n}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT; and let T¯=1−T¯𝑇1𝑇\bar{T}=1-Tover¯ start_ARG italic_T end_ARG = 1 - italic_T, the total in the other bin. | We shall use a tail bound for random variables with a binomial or similar distribution. We use a variant of the Chernoff bounds, which follows for example from Theorems 2.1 and 2.8 of [18] by considering S/b𝑆𝑏S/bitalic_S / italic_b. | We shall use Theorem 15.1 of [28], which says that a graph parameter f(G)𝑓𝐺f(G)italic_f ( italic_G ) is estimable if and only if the following three conditions hold. | Consider the ratios on the right sides of the inequalities (4.3) and (4.4). Choose t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT sufficiently large that for t≥t0𝑡subscript𝑡0t\geq t_{0}italic_t ≥ italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | We shall use induction on n𝑛nitalic_n. The result is trivial if n=1𝑛1n=1italic_n = 1, since both sides of (3.3) are 0. Let n≥2𝑛2n\geq 2italic_n ≥ 2 and assume that the result holds for all inputs of length n−1𝑛1n-1italic_n - 1. We shall consider two cases. | D |
Then every word in ℒ2n(X)subscriptℒ2𝑛𝑋\mathcal{L}_{2n}(X)caligraphic_L start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ( italic_X ) contains a word of the form uwv𝑢𝑤𝑣uwvitalic_u italic_w italic_v. | A word w∈ℒ(X)𝑤ℒ𝑋w\in\mathcal{L}(X)italic_w ∈ caligraphic_L ( italic_X ) is called right-special if there is | A nonempty shift space X𝑋Xitalic_X is irreducible if for every u,v∈ℒ(X)𝑢𝑣ℒ𝑋u,v\in\mathcal{L}(X)italic_u , italic_v ∈ caligraphic_L ( italic_X ) | to w𝑤witalic_w is finite for every w∈ℒ(X)𝑤ℒ𝑋w\in\mathcal{L}(X)italic_w ∈ caligraphic_L ( italic_X ). | Given a shift space X𝑋Xitalic_X and w∈ℒ(X)𝑤ℒ𝑋w\in\mathcal{L}(X)italic_w ∈ caligraphic_L ( italic_X ), a return word | D |
\ell_{1}.# italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 ) and # ( italic_s ∩ [ italic_y = roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 ] ) = roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . | then s′superscript𝑠′s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT defines a maximal exceptional sequence. We say that s′superscript𝑠′s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is | Therefore, s′superscript𝑠′s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is maximal, too, | then s′superscript𝑠′s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT defines an exceptional sequence, too. | We show that s′superscript𝑠′s^{\prime}italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is exceptional, contradicting the | B |
_{2}^{2}.italic_L start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( italic_θ , italic_z , italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ italic_y - italic_θ ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ ∥ italic_M start_POSTSUPERSCRIPT ( italic_k + 1 - italic_j ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n , italic_d end_POSTSUBSCRIPT italic_z ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG italic_ρ end_ARG start_ARG 2 end_ARG ∥ italic_z - italic_D start_POSTSUBSCRIPT italic_n , italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT italic_θ + italic_u ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG italic_ρ end_ARG start_ARG 2 end_ARG ∥ italic_u ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . | by penalizing the ℓ1subscriptℓ1\ell_{1}roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT norm its graph derivatives, which are defined via an | In summary, each iteration (cycle of updates over θ,z,u𝜃𝑧𝑢\theta,z,uitalic_θ , italic_z , italic_u) of the proposed | r}-z_{1}}.italic_f [ italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] = divide start_ARG italic_f [ italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ] - italic_f [ italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_r - 1 end_POSTSUBSCRIPT ] end_ARG start_ARG italic_z start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG . | θ,z𝜃𝑧\theta,zitalic_θ , italic_z, and then updates the dual variable u𝑢uitalic_u via gradient ascent. Namely, | D |
From this figure, it is evident that Itsubscript𝐼𝑡I_{t}italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT persists. | Numerical solutions of systems are invaluable in the study of epidemic models. This section presents the numerical results of our model, shedding light on how the parameters of the deterministic model (2) and the intensity of non-Gaussian noise in the stochastic model (4) impact the dynamics. We conduct numerical experiments to illustrate the extinction and persistence of the novel coronavirus, COVID-19, in both the deterministic model and its corresponding stochastic system for comparison. | In this study, we explore a nonlinear stochastic COVID-19 system, incorporating the influence of non-Gaussian noise. The presence of non-Gaussian noise adds a layer of complexity to the modeling framework, allowing for a more realistic representation of the uncertainties and random fluctuations inherent in the dynamics of the COVID-19 epidemic. This consideration is crucial for a comprehensive understanding of the system’s behavior and its response to unpredictable environmental factors. | The subsequent sections of the paper unfold as follows: Section 2: Model formulation- In this section, we meticulously detail the formulation of the model, providing a comprehensive overview of its deterministic aspects. Section 3: Dynamics of the deterministic model- we discuss the reproduction number and stability of the system. Section 4: Formulation and description of stochastic COVID-19 model- we explore and elucidate the dynamic properties and behaviors inherent in the stochastic aspects of the model. Section 5: Numerical experiments- this section is dedicated to presenting the numerical solutions derived for the proposed model. Through numerical simulations, we offer insights into the practical implications and outcomes of the model. Section 6: Conclusion - we present the conclusive remarks and findings of the entire paper. This section serves to summarize key results, implications, and potential avenues for future research. | This paper delves into the intricate dynamics of a COVID-19 epidemic model augmented with non-Gaussian noise. The exploration extends beyond the deterministic facet of the model, leading to a rigorous proof of the existence and uniqueness of a non-negative global solution for the stochastic system (4). These novel findings not only contribute to the theoretical foundation but also augment and refine insights garnered from preceding studies, as perceptibly illustrated in the graphical representations shown throughout this manuscript. | D |
In Lee et al. (2011, 2012), persistent homology was shown to outperform eight existing graph theory features, such as clustering coefficient, small-worldness, and modularity. Kuang et al. (2019) showed persistent homology-based measures can provide more significant group difference and better classification performance compared to standard graph-based measures that characterize small-world organization and modular structure. In Chung et al. (2017b, 2019a), persistent homology was shown to outperform various matrix norm-based network distances. In Wang et al. (2018), persistent homology was shown to outperform the power spectral density and local variance methods. In Wang et al. (2017), persistent homology was shown to outperform topographic power maps. In (Yoo et al., 2017), center persistency was shown to outperform the network-based statistic and element-wise multiple corrections. In | Chung et al. (2023b), persistent homology based clustering is shown to outperform k𝑘kitalic_k-means clustering and hierarchical clustering. | Like the majority of clustering methods such as k𝑘kitalic_k-means and hierarchical clustering that use geometric distances (Johnson, 1967; Hartigan and Wong, 1979; Lee et al., 2012), we propose to develop a topological clustering method using topological distances (Figure 5). For this purpose we use the Wasserstein distance. | Figure 9: The estimated state spaces of dynamically changing brain networks. The correlations are averaged over every time point and subject within each state for k𝑘kitalic_k-means clustering (top) and Wasserstein distance based topological clustering (bottom). In k𝑘kitalic_k-means clustering, the connectivity pattern of each state is somewhat random. | In Lee et al. (2011, 2012), persistent homology was shown to outperform eight existing graph theory features, such as clustering coefficient, small-worldness, and modularity. Kuang et al. (2019) showed persistent homology-based measures can provide more significant group difference and better classification performance compared to standard graph-based measures that characterize small-world organization and modular structure. In Chung et al. (2017b, 2019a), persistent homology was shown to outperform various matrix norm-based network distances. In Wang et al. (2018), persistent homology was shown to outperform the power spectral density and local variance methods. In Wang et al. (2017), persistent homology was shown to outperform topographic power maps. In (Yoo et al., 2017), center persistency was shown to outperform the network-based statistic and element-wise multiple corrections. In | A |
\leqslant\mathrm{e}^{b_{2}\tau(\gamma)},\quad\gamma\in\mathcal{P}.italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_e start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_τ ( italic_γ ) end_POSTSUPERSCRIPT ⩽ | roman_det ( roman_Id - italic_P start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) | ⩽ roman_e start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_τ ( italic_γ ) end_POSTSUPERSCRIPT , italic_γ ∈ caligraphic_P . | showing that ηD(s)subscript𝜂D𝑠\eta_{\mathrm{D}}(s)italic_η start_POSTSUBSCRIPT roman_D end_POSTSUBSCRIPT ( italic_s ) for Res≫1much-greater-thanRe𝑠1\operatorname{Re}s\gg 1roman_Re italic_s ≫ 1 is expressed as the difference of two Dirichlet series with positive coefficients. In particular, to show that ηD(s)subscript𝜂D𝑠\eta_{\mathrm{D}}(s)italic_η start_POSTSUBSCRIPT roman_D end_POSTSUBSCRIPT ( italic_s ) has a meromorphic extension to ℂ,ℂ\mathbb{C},blackboard_C , it is sufficient to prove that both series ηN(s)subscript𝜂N𝑠\eta_{\mathrm{N}}(s)italic_η start_POSTSUBSCRIPT roman_N end_POSTSUBSCRIPT ( italic_s ) and η2(s)subscript𝜂2𝑠\eta_{2}(s)italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_s ) have this property. | By using these estimates, we define for Re(s)≫1much-greater-thanRe𝑠1\operatorname{Re}(s)\gg 1roman_Re ( italic_s ) ≫ 1 two Dirichlet series | Here for Re(s)Re𝑠\operatorname{Re}(s)roman_Re ( italic_s ) large enough and ε>0𝜀0\varepsilon>0italic_ε > 0 small, we set | From Lemma 3.1 and (3.4), we deduce that for Re(s)≫1much-greater-thanRe𝑠1\operatorname{Re}(s)\gg 1roman_Re ( italic_s ) ≫ 1, we have | B |
Stability and safety verification in Ordinary Differential Equations (ODEs) has been explored widely and a detailed list of works can be found in survey papers [1] and [2]. Two different notions have emerged in existing literature that characterize stability and safety, respectively: | then the system (4) satisfies the two conditions of Proposition 1, and is considered to be practical Input-to-State Safe (pISSf) with respect to the unsafe set 𝒰𝒰\mathscr{U}script_U. | Input-to-state stability (ISSt) [3]: Here is the objective is to study the stability of systems in the presence of external input, and examine the boundedness of the system state trajectories only in a sense proportional to the size of the input. | In the present section, we will show that the control gain conditions in Theorem 1 simultaneously satisfy the pISSf criterion in the sense of (9) and ISSt criterion in the sense of (10). | Input-to-state safety (ISSf) [4, 5, 6]: Here the objective is to ensure that the system state trajectories stay away from a predefined unsafe region, or in other words, stay close to safe region. Specifically, trajectories moving from safe zone towards unsafe region will violate safety boundary only in a sense proportional to the size of input. On the other hand, trajectories starting in the unsafe region will be brought close of the safety boundary where closeness is proportional to the size of the input. | B |
Let ℰ0(σ,π)superscriptℰ0𝜎𝜋\mathcal{E}^{0}(\sigma,\pi)caligraphic_E start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_σ , italic_π ) (resp. ℰ1(σ,π)superscriptℰ1𝜎𝜋\mathcal{E}^{1}(\sigma,\pi)caligraphic_E start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_σ , italic_π )) be the residual representation of Gk+a(𝔸)subscript𝐺𝑘𝑎𝔸G_{k+a}(\mathbb{A})italic_G start_POSTSUBSCRIPT italic_k + italic_a end_POSTSUBSCRIPT ( blackboard_A ) generated by ℰ0(ϕ)superscriptℰ0italic-ϕ\mathcal{E}^{0}(\phi)caligraphic_E start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_ϕ ) (resp. ℰ1(ϕ)superscriptℰ1italic-ϕ\mathcal{E}^{1}(\phi)caligraphic_E start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_ϕ )) for ϕ∈𝒜Pk+a,aσ⊠π(Gk+a)italic-ϕsuperscriptsubscript𝒜subscript𝑃𝑘𝑎𝑎⊠𝜎𝜋subscript𝐺𝑘𝑎\phi\in\mathscr{A}_{P_{k+a,a}}^{\sigma\boxtimes\pi}(G_{k+a})italic_ϕ ∈ script_A start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_k + italic_a , italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ ⊠ italic_π end_POSTSUPERSCRIPT ( italic_G start_POSTSUBSCRIPT italic_k + italic_a end_POSTSUBSCRIPT ). | The proof of Proposition 3.8 closely mirrors that of Proposition 3.7; therefore, to avoid redundancy, we will highlight only the key differences. In applying the argument from Proposition 3.7 to Proposition 3.8, we encounter the absence of an analogous version of [7, Theorem 2.9] for metaplectic groups. However, one of the key ingredient of the proof of [7, Theorem 2.9] is the geometric lemma for reductive groups. Since [23, Section 3.3] supplies all the necessary components to establish the geometric lemma for metaplectic groups, the proof can be adapted almost verbatim from [7, Theorem 2.9]. | The following proposition is an analog of [20, Lemma 9.4], but its proof is much simpler due to Proposition 3.7 and Proposition 3.8. | Thanks to Proposition 4.2 and Proposition 4.3, we can define the residues of the Eisenstein series to be the limits | Thanks to Proposition 4.4 and Proposition 4.5, we can similarly define the residues of the Eisenstein series to be the limits | D |
Problems involving the optimization of pathways are critical for a number of applications across different fields. The general problem stemming from optimizing collision-free paths for robots in the realm of numerical analysis is studied in [1]. In graph theory, Dijkstra’s algorithm is used to look at the shortest paths on polyhedral surfaces, noting applications in robotics, geographic information systems, and route finding in [5]. The same type of problem relating to robotics and motion planning is explored by using sequence trees in [3]. An approach based in finding the shortest path on a polyhedral surface is used to model a network design problem with applications in telecommunications and transportation in [6]. In this paper, rather than find an algorithm to be applied, we provide our polyhedra with a coordinate system so that exact formulae can be derived. As a result, once these formulae are obtained, numerical calculations can be easily performed by anyone wishing to apply these findings. | As mentioned above, several new net substructures will be introduced in this paper. To help illustrate these concepts as they are introduced through sections 2 and 3 we will develop them for the specific case of the tetrahedron. Due to this, sections 2 and 3 will include multiple results and concepts which apply to a wide array of polyhedra, but are primarily dedicated to solving the problem of shortest paths along the surface of the tetrahedron. On the other hand, sections 4 and 5 include only one general result, but address the more complex problem of determining the collection of shortest paths along the surface of the cube. | In order to develop these formulae, we will utilize nets to calculate the shortest distance between points along the surface of convex regular polyhedra. Specifically, we will restrict our view to the paths lying along the surfaces of cubes and tetrahedra. With this goal in mind, we will define new net substructures, and using them identify where these shortest paths can lie as well as the lengths of these paths (for any two given points). | We will view the labeling of the net in Figure 1 as fixed, thus yielding a fixed labeling of our tetrahedron which we will refer to as 𝒫4subscript𝒫4\mathcal{P}_{4}caligraphic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT for the duration of the paper. We will follow the convention that any net considered will preserve the direction of rotation from one of a face’s vertices to another and be a result of cutting the relevant polyhedron along a subset of its edges and unfolding the polyhedron into a subset of ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. We now develop a coordinate system for use on the surface of any convex unit polyhedron (and in particular unit tetrahedra and unit cubes). | Since the shortest path on the surface of a convex polyhedron is a line segment contained in one of the polyhedron’s nets, to calculate the length of said paths or determine the set of points along the paths it is sufficient to do so for each subset of the nets in which these paths could be contained. We now provide for reference the definition of a dual graph, and introduce the new notion of a landscape which will be a useful tool in determining exactly which subsets of a polyhedron’s nets need to be considered for such calculations. Since the shortest path between two points on the same face of a polyhedron is simply the line segment in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT connecting the two points, landscapes need only be defined for a net’s subsets of at least two faces. | B |
≥c12𝖳1e−g(𝖳1)n.absentsubscript𝑐12subscript𝖳1superscript𝑒𝑔subscript𝖳1𝑛\displaystyle\geq\frac{c_{1}}{2\mathsf{T}_{1}}e^{-g(\mathsf{T}_{1})n}.≥ divide start_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 sansserif_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - italic_g ( sansserif_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_n end_POSTSUPERSCRIPT . | We are now ready to recall the definition of ΩnsubscriptΩ𝑛\Omega_{n}roman_Ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT: | We turn to the first part of the sum in the numerator, which corresponds to ℓ≤𝖳2ηℓsuperscriptsubscript𝖳2𝜂\ell\leq\mathsf{T}_{2}^{\eta}roman_ℓ ≤ sansserif_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT. As τ∈Bn(4)(J)𝜏subscriptsuperscript𝐵4𝑛𝐽\tau\in B^{(4)}_{n}(J)italic_τ ∈ italic_B start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_J ), one can use Lemma 4.5 so that | First, we decompose the partition function according to the last point in θ¯¯𝜃\bar{\theta}over¯ start_ARG italic_θ end_ARG before n−𝗆𝑛𝗆n-\mathsf{m}italic_n - sansserif_m. As 𝒩(n)𝒩𝑛{\mathcal{N}}(n)caligraphic_N ( italic_n ) is non-decreasing in n𝑛nitalic_n, it follows that | We now turn to the numerator. We recall the definition of 𝒩(n)𝒩𝑛{\mathcal{N}}(n)caligraphic_N ( italic_n ) in (1.19) and write | D |
\leq T_{\gamma}(K),italic_T start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_M ) = divide start_ARG ( ∫ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT italic_v italic_d italic_γ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∫ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT | ∇ italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_γ end_ARG = divide start_ARG ( ∫ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT over~ start_ARG italic_v end_ARG italic_d italic_γ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∫ start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT | ∇ over~ start_ARG italic_v end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_γ end_ARG ≤ italic_T start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_K ) , | The proposition 5.6 in Livshyts [21] was proved for convex set K𝐾Kitalic_K, but the same arguments can be applied also for all the measurable sets. We do not bring this proof here. | Just to clarify, the last identity is a tautology τ=τ𝜏𝜏\tau=\tauitalic_τ = italic_τ, and the first inequality follows from (9). In view of (19) together with Lemma 2.6 (the Gaussian analogue of Saint-Venant theorem), we conclude that | for Lipschitz continuous functions u𝑢uitalic_u, v𝑣vitalic_v, and a domain K𝐾Kitalic_K. In fact, the “integration by parts” identity also serves as the definition of L𝐿Litalic_L on Sobolev spaces. Recall also the following classical existence result, see e.g. [21]. | Let us also recall the Gaussian analogue of the Saint-Venant theorem (see e.g. Proposition 5.6 in Livshyts [21]): | D |
A module Gijsubscript𝐺𝑖𝑗{G}_{ij}italic_G start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is identifiable in ℳℳ\mathcal{M}caligraphic_M if and only if the module G~jisubscript~𝐺𝑗𝑖\tilde{G}_{ji}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT is identifiable in ℳ′superscriptℳ′\mathcal{M}^{\prime}caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT; | From Definition 1, a module Gijsubscript𝐺𝑖𝑗{G}_{ij}italic_G start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is identifiable in the model set ℳℳ\mathcal{M}caligraphic_M if and only if the following implication holds: | With the relation (8), the two rank conditions in (1) can be reinterpreted based on the underlying graph of the model set: If a network model set ℳℳ\mathcal{M}caligraphic_M in (2) is (generically) identifiable, then | The network model set ℳℳ\mathcal{M}caligraphic_M is identifiable if and only if the network model set ℳ′superscriptℳ′\mathcal{M}^{\prime}caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is identifiable. | A module Gijsubscript𝐺𝑖𝑗{G}_{ij}italic_G start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is identifiable in ℳℳ\mathcal{M}caligraphic_M if and only if the module G~jisubscript~𝐺𝑗𝑖\tilde{G}_{ji}over~ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT is identifiable in ℳ′superscriptℳ′\mathcal{M}^{\prime}caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT; | C |
Dwork et al. (2006) introduced the differential privacy for protecting privacy when releasing sensitive or confidential data. | As more and more network data are being made public, data privacy has received wide attention because of sensitive information (e.g., sex-partner relationships). | The differential privacy is a good framework for privacy protection, which reduces the leakage of privacy risks and ensures the utility of the data, and has been widely used as a privacy standard when releasing network data (e.g., Hay et al. (2009); Lu and Miklau (2014); Karwa and Slavković (2016)). | Differential privacy in network data is separated into node differential privacy [Kasiviswanathan et al. (2013)] and edge differential privacy (EDP) [Nissim et al. (2007)], | Dwork et al. (2006) introduced the differential privacy for protecting privacy when releasing sensitive or confidential data. | B |
𝒞2=g𝒞1subscript𝒞2𝑔subscript𝒞1\mathcal{C}_{2}=g\,\mathcal{C}_{1}caligraphic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_g caligraphic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. | It easy to show that the set of such elements, denoted symG(𝒞)subscriptsym𝐺𝒞\text{sym}_{G}(\mathcal{C})sym start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( caligraphic_C ), is a subgroup of G𝐺Gitalic_G, called the G𝐺Gitalic_G-symmetry group of 𝒞𝒞\mathcal{C}caligraphic_C. | Given a group G𝐺Gitalic_G acting on the plane, we say that two planar curves 𝒞1subscript𝒞1\mathcal{C}_{1}caligraphic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝒞2subscript𝒞2\mathcal{C}_{2}caligraphic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are G-congruent (𝒞1≅𝐺𝒞2subscript𝒞1𝐺subscript𝒞2\mathcal{C}_{1}\underset{G}{\cong}\mathcal{C}_{2}caligraphic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT underitalic_G start_ARG ≅ end_ARG caligraphic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) if there exists g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G, such that | Given a group G𝐺Gitalic_G acting continuously on the plane, the image of a curve 𝒞𝒞\mathcal{C}caligraphic_C parametrized by γ𝛾\gammaitalic_γ, under a transformation g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G is the curve | An element g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G is a G𝐺Gitalic_G-symmetry of 𝒞𝒞\mathcal{C}caligraphic_C if | D |
\sqrt{C_{T}}G^{2}}{2}\sqrt{T}.over¯ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ≤ ( divide start_ARG 5 italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 square-root start_ARG italic_C start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG end_ARG + italic_R square-root start_ARG italic_C start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG ) square-root start_ARG italic_T end_ARG + divide start_ARG square-root start_ARG italic_C start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG square-root start_ARG italic_T end_ARG . | By using Proposition 5.1, we can establish regret bounds for Algorithms 2 and 3 using known regret bounds for online gradient descent algorithms. Moreover, if the regret bounds for online gradient descent are sublinear in T𝑇Titalic_T and ∑t=1Tαtsuperscriptsubscript𝑡1𝑇subscript𝛼𝑡\sum_{t=1}^{T}\alpha_{t}∑ start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT is also sublinear in T𝑇Titalic_T, then the established regret bounds for Algorithms 2 and 3 will be sublinear. | The following result on static regret bounds of Algorithms 2 and 3 for strongly convex functions is a direct corollary of Proposition 5.1 and Lemma 5.4. | Using Propositions 5.1 and 5.6, we can state the following result on dynamic regret bounds of Algorithms 2 and 3. | The following result on static regret bounds of Algorithms 2 and 3 is a direct corollary of Proposition 5.1 and Lemma 5.2. | C |
×P(ra→b(t)∈dy,ra→b(s)∈dx).absent𝑃formulae-sequencesuperscript𝑟→𝑎𝑏𝑡𝑑𝑦superscript𝑟→𝑎𝑏𝑠𝑑𝑥\displaystyle\qquad\times P\left(r^{a\to b}(t)\in dy,r^{a\to b}(s)\in dx\right).× italic_P ( italic_r start_POSTSUPERSCRIPT italic_a → italic_b end_POSTSUPERSCRIPT ( italic_t ) ∈ italic_d italic_y , italic_r start_POSTSUPERSCRIPT italic_a → italic_b end_POSTSUPERSCRIPT ( italic_s ) ∈ italic_d italic_x ) . | By the Markov property of Rasuperscript𝑅𝑎R^{a}italic_R start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT, we have | Using the Markov property of Rasuperscript𝑅𝑎R^{a}italic_R start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT, for 0<s<t<u0𝑠𝑡𝑢0<s<t<u0 < italic_s < italic_t < italic_u, it holds that | Using the Markov property of Rasuperscript𝑅𝑎R^{a}italic_R start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT, for 0<t<u0𝑡𝑢0<t<u0 < italic_t < italic_u, it holds that | By the Markov property of Rasuperscript𝑅𝑎R^{a}italic_R start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT, we obtain the assertion as follows: | A |
Lemma 5.8 yields the existence of c>0𝑐0c>0italic_c > 0 such that ‖∇YuEj‖g⩽c‖u‖gersubscriptnormsubscript∇subscript𝑌𝑢subscript𝐸𝑗𝑔𝑐subscriptnorm𝑢𝑔superscript𝑒𝑟\|\nabla_{Y_{u}}E_{j}\|_{g}\leqslant c\|u\|_{g}e^{r}∥ ∇ start_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ⩽ italic_c ∥ italic_u ∥ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT. | From the normal Jacobi estimates, the uniform bound on ‖S‖gsubscriptnorm𝑆𝑔\|S\|_{g}∥ italic_S ∥ start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT, the (ALCH) and (ALS) conditions and Proposition 5.5, it finally holds that there exists c>0𝑐0c>0italic_c > 0 such that | It now follows from the uniform bound on ‖η‖g0subscriptnorm𝜂subscript𝑔0\|\eta\|_{g_{0}}∥ italic_η ∥ start_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and from equation (6.1.2) that there exists C>0𝐶0C>0italic_C > 0 such that | It now follows from the (ALCH) and (ALS) conditions together with Lemmas 4.12 and 5.5 that there exists a constant C>0𝐶0C>0italic_C > 0 such that | It follows from the triangle inequality, Cauchy-Schwarz inequality, Proposition 4.11 and Lemma 4.12, that there exists a constant C>0𝐶0C>0italic_C > 0 independent of r𝑟ritalic_r, p𝑝pitalic_p, u𝑢uitalic_u and v𝑣vitalic_v such that | C |
If one wishes to use Formula (10) to define the curvature of ℌℌ{\mathfrak{H}}fraktur_H as a tensorial operator then the second fundamental form defined in 2 must be tensorial, as must the wedge product of the second fundamental form and its Hilbert space adjoint, i.e., the object corresponding to the second quantity on the right hand side of (2.32) must be well-defined. | Ideally the curvature of ℌℌ{\mathfrak{H}}fraktur_H should be independent of the ambient BLS field 𝔏𝔏{\mathfrak{L}}fraktur_L. | Let 𝔏𝔏{\mathfrak{L}}fraktur_L be a BLS field and let ℌ⊂𝔏ℌ𝔏{\mathfrak{H}}\subset{\mathfrak{L}}fraktur_H ⊂ fraktur_L be a BLS subfield. | Let 𝔏𝔏{\mathfrak{L}}fraktur_L be a BLS field and let ℌ⊂𝔏ℌ𝔏{\mathfrak{H}}\subset{\mathfrak{L}}fraktur_H ⊂ fraktur_L be a BLS subfield. | If ℌℌ{\mathfrak{H}}fraktur_H is a BLS subfield of a BLS field 𝔏𝔏{\mathfrak{L}}fraktur_L then the tuning 𝒞∞(ℌ)superscript𝒞ℌ{\mathscr{C}}^{\infty}({\mathfrak{H}})script_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( fraktur_H ) of ℌℌ{\mathfrak{H}}fraktur_H is formal. In fact, Property i of Definition 2.36.c holds because density is part of the BLS hypothesis, and the proof of Proposition 2.27 gives Property ii. | A |
Note that the latter case is true whenever I∖i≠J∖j𝐼𝑖𝐽𝑗I\setminus i\neq J\setminus jitalic_I ∖ italic_i ≠ italic_J ∖ italic_j, otherwise we obtain [tI,J(ξ),tJ,I(±ζ)]subscript𝑡𝐼𝐽𝜉subscript𝑡𝐽𝐼plus-or-minus𝜁[t_{I,J}(\xi),t_{J,I}(\pm\zeta)][ italic_t start_POSTSUBSCRIPT italic_I , italic_J end_POSTSUBSCRIPT ( italic_ξ ) , italic_t start_POSTSUBSCRIPT italic_J , italic_I end_POSTSUBSCRIPT ( ± italic_ζ ) ]. This commutator cannot be presented in a simpler form than the very definition. | \operatorname{E}_{n}(R)start_ID ⋀ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ID roman_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_R ) in detail. The following proposition can be extracted from the very definition of ⋀2(GLn(R))⋀2subscriptGL𝑛𝑅\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!2}}$}}}\big{(}% | The attentive reader can remark these calculations to be almost completely coincide with the calculations for the orthogonal and symplectic cases [18, 19, 20]. In the special case (n,m)=(4,2)𝑛𝑚42(n,m)=(4,2)( italic_n , italic_m ) = ( 4 , 2 ) calculations are the same due to the isomorphism ⋀2E4(R)≅EO6(R)⋀2subscriptE4𝑅subscriptEO6𝑅\mathord{\raisebox{2.0pt}{\hbox{$\scriptstyle{\bigwedge^{\!2}}$}}}% | The rule for commutator calculations from the latter proposition can be translated into the language of weight diagrams: | Similarly, for the case m=3𝑚3m=3italic_m = 3 the elementary calculations can be seen directly from Figure 3.3(b)𝑏(b)( italic_b ). | C |
We always have Tn=∅subscript𝑇𝑛T_{n}=\emptysetitalic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = ∅. If X=X~𝑋~𝑋X=\widetilde{X}italic_X = over~ start_ARG italic_X end_ARG is contraction free, then for all k𝑘kitalic_k, we have TK=∅subscript𝑇𝐾T_{K}=\emptysetitalic_T start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT = ∅. | We begin by showing that when X𝑋Xitalic_X is a projective, ℚℚ{\mathbb{Q}}blackboard_Q-factorial, rational, contraction-free T𝑇Titalic_T-variety of complexity one, then the Kroenecker duality map is in fact an isomorphism after tensoring with ℚℚ{\mathbb{Q}}blackboard_Q. | which by Proposition 4.5 is an isomorphism after tensoring with ℚℚ{\mathbb{Q}}blackboard_Q if X𝑋Xitalic_X is projective, ℚℚ{\mathbb{Q}}blackboard_Q-factorial and contraction-free. | As was mentioned in the introduction, this map is in general not an isomorphism. We will see in the next section that for projective, ℚℚ{\mathbb{Q}}blackboard_Q-factorial, rational, contraction-free T-varieties of complexity one, this map is indeed an isomorphism after tensoring with ℚℚ{\mathbb{Q}}blackboard_Q (Proposition 4.5). | Let X𝑋Xitalic_X be a projective, ℚℚ{\mathbb{Q}}blackboard_Q-factorial, rational, contraction-free T𝑇Titalic_T-variety of complexity one. After tensoring with ℚℚ{\mathbb{Q}}blackboard_Q, the Kroenecker duality map | A |
(cn(λ))n≥0subscriptsubscript𝑐𝑛𝜆𝑛0(c_{n}(\lambda))_{n\geq 0}( italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_λ ) ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT and (dn(λ))n≥0subscriptsubscript𝑑𝑛𝜆𝑛0(d_{n}(\lambda))_{n\geq 0}( italic_d start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_λ ) ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT are | Hankel-totally positive in the polynomial ring ℤ[λ]ℤdelimited-[]𝜆\mathbb{Z}[\lambda]blackboard_Z [ italic_λ ] | totally positive in the ring ℤ[a,b,u,v,λ]ℤ𝑎𝑏𝑢𝑣𝜆\mathbb{Z}[a,b,u,v,\lambda]blackboard_Z [ italic_a , italic_b , italic_u , italic_v , italic_λ ] equipped | binomial-totally positive in the ring ℤ[a,b,u,v,λ]ℤ𝑎𝑏𝑢𝑣𝜆\mathbb{Z}[a,b,u,v,\lambda]blackboard_Z [ italic_a , italic_b , italic_u , italic_v , italic_λ ] | is totally positive in the ring ℤ[a,b,u,v,λ]ℤ𝑎𝑏𝑢𝑣𝜆\mathbb{Z}[a,b,u,v,\lambda]blackboard_Z [ italic_a , italic_b , italic_u , italic_v , italic_λ ] | A |
𝐆¯σsubscript¯𝐆𝜎\overline{\operatorname{\mathbf{G}}}_{\sigma}over¯ start_ARG bold_G end_ARG start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT denotes the reductive quotient of the special fiber of the Bruhat-Tits 𝒪Fsubscript𝒪𝐹{\mathcal{O}}_{F}caligraphic_O start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT-model 𝐆σsubscript𝐆𝜎\operatorname{\mathbf{G}}_{\sigma}bold_G start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT of | of 𝐆𝐆\operatorname{\mathbf{G}}bold_G whose apartment contains σ𝜎\sigmaitalic_σ, and write also 𝐒𝐒\mathbf{S}bold_S for its | 𝐆𝐆\operatorname{\mathbf{G}}bold_G attached to σ𝜎\sigmaitalic_σ. To define the map (3.5), let 𝐒𝐒\mathbf{S}bold_S be a maximal split torus | 𝐒𝐒\mathbf{S}bold_S is itself a torus, that we denote by 𝐓:=𝐙assign𝐓𝐙\mathbf{T}:=\mathbf{Z}bold_T := bold_Z. According to | Let 𝐒𝐒\mathbf{S}bold_S be a maximal split torus in 𝐆𝐆\operatorname{\mathbf{G}}bold_G and let 𝐙𝐙\mathbf{Z}bold_Z be the centralizer | B |
In this paper, we consider a Bayes optimal algorithm, which computes the consequences over all the exponentially many parallel worlds. | The multi-armed bandit problem (Robbins, 1952) involves multiple treatment arms and decisions made using samples obtained sequentially via experiments. The aim is to maximize the sum of the rewards, which boils down to cumulative regret minimization (CRM). Although this problem has attracted considerable attention from the machine learning community, this study has focused on another established branch of bandit problems, BAI, in which the goal is to identify the treatment arm with the largest mean. Finding the best treatment essentially concerns simple regret minimization (SRM). | BAI has two settings, namely, the fixed-budget setting which the number of samples is fixed, and the fixed-confidence setting in which the forecaster stops when the confidence level on the best arm reaches a predefined threshold. In this paper, we consider the former setting. | This problem associates each arm with an (unknown) parameterized distribution, which is a (noisy) signal of the quality of the treatment. | This paper addresses the problem of identifying the best arm (or treatment) among multiple options from a fixed number of samples. | D |
4 Achieving ⌊𝑨𝝀𝒌(𝒎,𝒏)⌋subscriptsuperscript𝑨𝒌𝝀𝒎𝒏\boldsymbol{\lfloor A^{k}_{\lambda}(m,n)\rfloor}bold_⌊ bold_italic_A start_POSTSUPERSCRIPT bold_italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_λ end_POSTSUBSCRIPT bold_( bold_italic_m bold_, bold_italic_n bold_) bold_⌋ | We require an analogue of Theorem 3.1 that gives decompositions of λ𝜆\lambdaitalic_λ-linear multigraphs. Furthermore, we are interested in cases where λ𝜆\lambdaitalic_λ is arbitrary, rather than being fixed as the order of the host multigraph grows. We first use Theorem 3.1 to prove an analogous result for λ𝜆\lambdaitalic_λ-linear multigraphs where λ𝜆\lambdaitalic_λ is fixed, and then in turn use this result to obtain the tool we require. In proving the intermediate result we make use of the following special case of [12, Theorem 1(ii)]. | In view of Lemma 4.1, in the remainder of this section we only need to show the existence of appropriate λ𝜆\lambdaitalic_λ-linear hypergraphs in the situations specified by Theorem 1.3. We first show that we can obtain these hypergraphs for most of the interval prescribed by Theorem 1.3. The approach we take to accomplish this has similarities to one used in [3]. | The rest of the paper is arranged as follows. In Section 2 we establish the new upper bounds and prove Theorem 1.1 and Corollary 1.2. Section 3 details some preliminary results we require for our proof of Theorem 1.3 which then takes place in Sections 4 and 5. Section 4 is devoted to the cases where ⌊Aλk(m,n)⌋subscriptsuperscript𝐴𝑘𝜆𝑚𝑛\lfloor A^{k}_{\lambda}(m,n)\rfloor⌊ italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ is achieved, while Section 5 is devoted to the cases where ⌊Bλk+1(m,n)⌋subscriptsuperscript𝐵𝑘1𝜆𝑚𝑛\lfloor B^{k+1}_{\lambda}(m,n)\rfloor⌊ italic_B start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ or ⌊Bλk(m,n)⌋subscriptsuperscript𝐵𝑘𝜆𝑚𝑛\lfloor B^{k}_{\lambda}(m,n)\rfloor⌊ italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ is achieved and concludes by completing the proofs of Theorem 1.3 and Corollary 1.4. Theorem 1.5 is proved in Section 6. We conclude with some discussion of our work and possible future directions in Section 7. | Over the course of this section and the next one, we will prove Theorem 1.3. For many of our results, we will be operating in a regime in which k𝑘kitalic_k is fixed, m𝑚mitalic_m is large (as a function of k𝑘kitalic_k) and λ𝜆\lambdaitalic_λ is arbitrary. We will employ asymptotic notation with respect to this regime. | D |
The proof of Theorem 1.1 depends on Lemmas 3.2 and 3.3, which give the existence of a non-cut edge xy𝑥𝑦xyitalic_x italic_y in lk(v,Δ)∖{σ∈lk(v,Δ):t≤σ}lk𝑣Δconditional-set𝜎lk𝑣Δ𝑡𝜎\mbox{lk}\,(v,\Delta)\setminus\{\sigma\in\mbox{lk}\,(v,\Delta):t\leq\sigma\}lk ( italic_v , roman_Δ ) ∖ { italic_σ ∈ lk ( italic_v , roman_Δ ) : italic_t ≤ italic_σ }, where v𝑣vitalic_v is a singular vertex and x,y𝑥𝑦x,yitalic_x , italic_y are non-singular vertices in ΔΔ\Deltaroman_Δ. If ΔΔ\Deltaroman_Δ has more than four singular vertices, then such a non-cut edge may not exist. Therefore, if ΔΔ\Deltaroman_Δ has more than four singular vertices, then such a characterization for ΔΔ\Deltaroman_Δ may not be possible. In fact, we have an example of a g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal normal 3-pseudomanifold ΔΔ\Deltaroman_Δ with five singular vertices, among which one has 𝕋2superscript𝕋2\mathbb{T}^{2}blackboard_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-singularity and the remaining four have ℝℙ2ℝsuperscriptℙ2\mathbb{RP}^{2}blackboard_R blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-singularities (see [1]), where ΔΔ\Deltaroman_Δ cannot be obtained from a normal 3-pseudomanifold Δ′superscriptΔ′\Delta^{\prime}roman_Δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT by an edge folding. | Kalai’s findings establish a lower bound on the value of g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for a normal d𝑑ditalic_d-pseudomanifold, where 2-dimensional links are spheres, expressed in terms of the same for a link. Subsequently, Fogelsanger’s results in [8, Chapter 8] removed the restriction on 2-dimensional links. The combined implications of Kalai and Fogelsanger’s results assert that if d≥0𝑑0d\geq 0italic_d ≥ 0 and ΔΔ\Deltaroman_Δ is a normal d𝑑ditalic_d-pseudomanifold, then for any face σ𝜎\sigmaitalic_σ of ΔΔ\Deltaroman_Δ with co-dimension 3 or more, the inequality g2(Δ)≥g2(lk(σ,Δ))subscript𝑔2Δsubscript𝑔2lk𝜎Δg_{2}(\Delta)\geq g_{2}(\mbox{lk}\,(\sigma,\Delta))italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_Δ ) ≥ italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( lk ( italic_σ , roman_Δ ) ) holds. In [12], the non-negativity of the invariant g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT has been described in a different way of looking at rigidity, and in [3], a detailed study on the lower bound theorem has been given in terms of Gromov rigidity. For the case of the normal 3-pseudomanifold ΔΔ\Deltaroman_Δ, the result turned out to be g2(Δ)≥g2(lk(v,Δ))subscript𝑔2Δsubscript𝑔2lk𝑣Δg_{2}(\Delta)\geq g_{2}(\mbox{lk}\,(v,\Delta))italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_Δ ) ≥ italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( lk ( italic_v , roman_Δ ) ) for every vertex v𝑣vitalic_v in ΔΔ\Deltaroman_Δ. If g2(Δ)=g2(lk(v,Δ))subscript𝑔2Δsubscript𝑔2lk𝑣Δg_{2}(\Delta)=g_{2}(\mbox{lk}\,(v,\Delta))italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_Δ ) = italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( lk ( italic_v , roman_Δ ) ) for some vertex v𝑣vitalic_v, then such a complex ΔΔ\Deltaroman_Δ is said to be g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal (or have relatively minimal g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [9]). Basak and Swartz [9] proved that if ΔΔ\Deltaroman_Δ is a g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal normal 3-pseudomanifold with at most two singular vertices, then ΔΔ\Deltaroman_Δ is obtained from a one-vertex suspension of a surface and some boundary complexes of 4444-simplices by a sequence of operations of the form vertex foldings and connected sums. In this article, we have worked with the g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal normal 3333-pseudomanifolds with at most four singularities. More precisely, we proved the following: | After applying vertex folding and edge folding to a simplicial complex, the vertices of two facets get identified by the admissible map ψ𝜓\psiitalic_ψ in the new complex. However, to avoid ambiguity in notations in the new complex, we denote the identified vertices with the same notation as in the original complex. | Acknowledgement: The authors would like to thank the anonymous referees for their many useful comments and suggestions. The first author is supported by the Science and Engineering Research Board (CRG/2021/000859). The second author is supported by CSIR (India). The third author is supported by the Prime Minister’s Research Fellows (PMRF) Scheme. | Further, Δ′superscriptΔ′\Delta^{\prime}roman_Δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal with respect to the singular vertex t∈Δ′𝑡superscriptΔ′t\in\Delta^{\prime}italic_t ∈ roman_Δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and the first Betti number of the link of s𝑠sitalic_s in Δ′superscriptΔ′\Delta^{\prime}roman_Δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is unchanged. Notice from the definition that an edge folding along an edge tx𝑡𝑥txitalic_t italic_x reduces the first Betti number of the corresponding vertices by 1 in the resulting complex, and it follows that b1−b2≥1subscript𝑏1subscript𝑏21b_{1}-b_{2}\geq 1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ 1. On the other hand, if ΔΔ\Deltaroman_Δ has four singular vertices, including two ℝℙ2ℝsuperscriptℙ2\mathbb{RP}^{2}blackboard_R blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-singularities, and G(Δ)=G(st(t,Δ))𝐺Δ𝐺st𝑡ΔG(\Delta)=G(\mbox{st}\,(t,\Delta))italic_G ( roman_Δ ) = italic_G ( st ( italic_t , roman_Δ ) ), then ΔΔ\Deltaroman_Δ is obtained from a g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-minimal normal pseudomanifold by two repeated applications of edge folding. Therefore, by Proposition 2.5 and Theorem 3.4, we have the following: | C |
Z=ℤ(qn)𝑍subscriptℤsubscript𝑞𝑛Z={\mathbb{Z}}_{(q_{n})}italic_Z = blackboard_Z start_POSTSUBSCRIPT ( italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT for some sequence (qn)subscript𝑞𝑛(q_{n})( italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) and +11+1+ 1 denotes the homeomorphism of adding the unit 1111. | The following arithmetic lemma will be essential when finding either measurable or topological eigenvalues of a constant-length S𝑆Sitalic_S-adic shift. | We will see in Lemma 5.1 that the sequence (𝒬n)subscript𝒬𝑛(\mathcal{Q}_{n})( caligraphic_Q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) generates in measure, and therefore we can use it when we investigate measurable eigenvalues in Section 5. | We conclude by recapping the eigenvalues, measurable and topological of this S𝑆Sitalic_S-adic shift. | There are various related notions in the measurable and topological dynamics literature that produce recognizable S𝑆Sitalic_S-adic systems. In particular, cutting-and-stacking measurable systems on a Lebesgue space (see e.g. [KS67, Fer97]) are built with recursive instructions which can be interpreted as a directive sequence, and provided the latter is recognizable, the S𝑆Sitalic_S-adic shift is a symbolic representation of the cutting-and-stacking system. Also, under the condition that the morphisms σ𝜎\sigmaitalic_σ in the directive sequence are proper, i.e., σ(a)𝜎𝑎\sigma(a)italic_σ ( italic_a ) starts with the same letter for each letter a𝑎aitalic_a, recognizable S𝑆Sitalic_S-adic systems can be directly seen as topological Bratteli-Vershik systems [DHS99]. The model | A |
The graph 𝖦2(2)¯¯subscript𝖦22\overline{\mathsf{G}_{2}(2)}over¯ start_ARG sansserif_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 2 ) end_ARG is a primitive rank 3333 graph with parameters (36,21,12,12)36211212(36,21,12,12)( 36 , 21 , 12 , 12 ) and eigenmatrices | P=(12n−1(2n−1+1)22n−2−112n−2−2n−2−11−2n−12n−1−1),Q=(122n−43(2n−1+1)(2n+1)312n−23−2n+131−2n+432n+13).formulae-sequence𝑃matrix1superscript2𝑛1superscript2𝑛11superscript22𝑛211superscript2𝑛2superscript2𝑛211superscript2𝑛1superscript2𝑛11𝑄matrix1superscript22𝑛43superscript2𝑛11superscript2𝑛131superscript2𝑛23superscript2𝑛131superscript2𝑛43superscript2𝑛13P=\begin{pmatrix}1&2^{n-1}(2^{n-1}+1)&2^{2n-2}-1\\ | P=(110570117−181−32),Q=(1211541175−2251−275225).formulae-sequence𝑃matrix11057011718132𝑄matrix12115411752251275225P=\begin{pmatrix}1&105&70\\ | P=(122n−2−12n−1(2n−1−1)12n−2−1−2n−21−2n−1−12n−1),Q=(122n−43(2n−1−1)(2n−1)312n−43−2n−131−2n+232n−13).formulae-sequence𝑃matrix1superscript22𝑛21superscript2𝑛1superscript2𝑛111superscript2𝑛21superscript2𝑛21superscript2𝑛11superscript2𝑛1𝑄matrix1superscript22𝑛43superscript2𝑛11superscript2𝑛131superscript2𝑛43superscript2𝑛131superscript2𝑛23superscript2𝑛13P=\begin{pmatrix}1&2^{2n-2}-1&2^{n-1}(2^{n-1}-1)\\ | P=(1211413−41−32),Q=(1142112−31−43).formulae-sequence𝑃matrix12114134132𝑄matrix11421123143P=\begin{pmatrix}1&21&14\\ | D |
This is in contrast to Random selection, where the covering is uniform with high probability only when there are many active users, | Furthermore, in (17) it is not necessary to sum over whole 𝔑𝔑\mathfrak{N}fraktur_N to obtain a good approximation. In practice, | Note that the SINR of the combined packet is equal to the sum of the SINRs of its constituents only when the interference in each is uncorrelated. This will not be the case if a given user collides with another in more than one slot. Whether the resulting SINR will be higher or lower than the sum will depend on whether the interference adds destructively or constructively. | However, as will become evident later, in most cases ultra-reliability cannot be achieved if the traffic intensity is too high, regardless if the access scheme is based on Steiner system or Random selection. | Consequently, our approach is suitable only for Steiner systems and will not provide a good approximation if the patterns follow a Random selection. | D |
D. Alekseevsky gratefully acknowledges support by the Grant Basis-Foundation Leader n. 22-7-1-34-1. G. Manno gratefully acknowledges support by the project “Connessioni | Let uxxuyy−uxy2≠0subscript𝑢𝑥𝑥subscript𝑢𝑦𝑦superscriptsubscript𝑢𝑥𝑦20u_{xx}u_{yy}-u_{xy}^{2}\neq 0italic_u start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT - italic_u start_POSTSUBSCRIPT italic_x italic_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≠ 0 and let F𝐹Fitalic_F be given by (19). Then F=0𝐹0F=0italic_F = 0 if and only if the Fubini–Pick invariant (4) vanishes. | D. Alekseevsky gratefully acknowledges support by the Grant Basis-Foundation Leader n. 22-7-1-34-1. G. Manno gratefully acknowledges support by the project “Connessioni | and uxxxsubscript𝑢𝑥𝑥𝑥u_{xxx}italic_u start_POSTSUBSCRIPT italic_x italic_x italic_x end_POSTSUBSCRIPT is obtained from (54). | “MIUR grant Dipartimenti di Eccellenza 2018-2022 (E11G18000350001)”, “Finanziamento alla Ricerca” 53_RBA17MANGIO and 53_RBA21MANGIO, and | D |
2−kϵ≤distH(X,X′)<21−kϵ.superscript2𝑘italic-ϵsubscriptdist𝐻𝑋superscript𝑋′superscript21𝑘italic-ϵ2^{-k}\epsilon\leq\operatorname{dist}_{H}(X,X^{\prime})<2^{1-k}\epsilon.2 start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT italic_ϵ ≤ roman_dist start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) < 2 start_POSTSUPERSCRIPT 1 - italic_k end_POSTSUPERSCRIPT italic_ϵ . | In [Sch07] Schul defined for each v∈X𝑣𝑋v\in Xitalic_v ∈ italic_X a number that measures the (normalized) thickness of the thinnest cylinder that contains the set | Next, using results in Section 4, we define for each v∈Vk𝑣subscript𝑉𝑘v\in V_{k}italic_v ∈ italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT a number α𝛼\alphaitalic_α that measures how “flat” the set Vk+1subscript𝑉𝑘1V_{k+1}italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT is around the point v𝑣vitalic_v. This notion of flatness is inspired by the Jones beta numbers in the work of Jones [Jon90] and Schul [Sch07]. | Following [BNV19, Definition 2.4], for each v∈X𝑣𝑋v\in Xitalic_v ∈ italic_X, we define the set of flat pairs 𝖥𝗅𝖺𝗍(v,X,X′)𝖥𝗅𝖺𝗍𝑣𝑋superscript𝑋′\mathsf{Flat}(v,X,X^{\prime})sansserif_Flat ( italic_v , italic_X , italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) to be the set of {v,v′}⊂X𝑣superscript𝑣′𝑋\{v,v^{\prime}\}\subset X{ italic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT } ⊂ italic_X such that | Later, Schul [Sch07] provided a modification of the algorithm so that the ratio of the length of the yielded path over the length of the optimal path is bounded by a constant C𝐶Citalic_C independent of the dimension N𝑁Nitalic_N. Variation of this algorithm also appears in [BNV19]. Here and for the rest of this paper, we refer to any of these two variations as the ATSP algorithm. The purpose of this note is to show that the ATSP algorithm, in the case that V𝑉Vitalic_V is finite, has polynomial time complexity. | A |
For U⊂ℝn,V⊂ℝmformulae-sequence𝑈superscriptℝ𝑛𝑉superscriptℝ𝑚U\subset\mathbb{R}^{n},V\subset\mathbb{R}^{m}italic_U ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_V ⊂ blackboard_R start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT open sets and a Lipschitz function f:U→V:𝑓→𝑈𝑉f:U\rightarrow Vitalic_f : italic_U → italic_V, one can define an induced mapping f#subscript𝑓#f_{\#}italic_f start_POSTSUBSCRIPT # end_POSTSUBSCRIPT on the level of flat chains, i.e. for a flat chain A𝐴Aitalic_A supported in U𝑈Uitalic_U, f#Asubscript𝑓#𝐴f_{\#}Aitalic_f start_POSTSUBSCRIPT # end_POSTSUBSCRIPT italic_A is a flat chain supported in V𝑉Vitalic_V (see [32, Sec. 5] and [31, Sec. 2 and 3]). | An element P∈𝒫k𝑃superscript𝒫𝑘P\in\mathcal{P}^{k}italic_P ∈ caligraphic_P start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT can thus be written as | A property of an object (such as a function or a set) that can be achieved by an arbitrarily small perturbation of the object is called generic. | Considering a cube K∈G𝐾𝐺K\in Gitalic_K ∈ italic_G, one can think of this construction as the boundary of the three dimensional object created by filling the space between T𝑇Titalic_T and its projection onto ∂K𝐾\partial K∂ italic_K according to Step 1 and then removing the original part T𝑇Titalic_T. Another but equivalent point of view is to take T~~𝑇\tilde{T}over~ start_ARG italic_T end_ARG as all the points along the path created by projecting T ∂K 𝑇𝐾T\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.5% | Since M𝑀Mitalic_M can be chosen arbitrarily large and ϵitalic-ϵ\epsilonitalic_ϵ arbitrarily small, we can construct a diagonal sequence and obtain the claim. | B |
\mathsf{Prin}_{G}(U)sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_W ) start_ARROW start_OVERACCENT italic_g start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_OVERACCENT → end_ARROW sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_V ) start_ARROW start_OVERACCENT italic_f start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_OVERACCENT → end_ARROW sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_U ) given by pulling back the principal bundles. However if P∈𝖯𝗋𝗂𝗇G(W)𝑃subscript𝖯𝗋𝗂𝗇𝐺𝑊P\in\mathsf{Prin}_{G}(W)italic_P ∈ sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_W ), then (gf)*P≠f*g*Psuperscript𝑔𝑓𝑃superscript𝑓superscript𝑔𝑃(gf)^{*}P\neq f^{*}g^{*}P( italic_g italic_f ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_P ≠ italic_f start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_g start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_P, but there is an isomorphism (gf)*P≅f*g*Psuperscript𝑔𝑓𝑃superscript𝑓superscript𝑔𝑃(gf)^{*}P\cong f^{*}g^{*}P( italic_g italic_f ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_P ≅ italic_f start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_g start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_P. Thus 𝖯𝗋𝗂𝗇Gsubscript𝖯𝗋𝗂𝗇𝐺\mathsf{Prin}_{G}sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is called a pseudofunctor 𝖯𝗋𝗂𝗇G:𝖢𝖺𝗋𝗍op→𝖦𝗉𝖽:subscript𝖯𝗋𝗂𝗇𝐺→superscript𝖢𝖺𝗋𝗍𝑜𝑝𝖦𝗉𝖽\mathsf{Prin}_{G}:\mathsf{Cart}^{op}\to\mathsf{Gpd}sansserif_Prin start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT : sansserif_Cart start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT → sansserif_Gpd, where 𝖦𝗉𝖽𝖦𝗉𝖽\mathsf{Gpd}sansserif_Gpd is the 2222-category of groupoids. There is an elegant theory [JY20, Chapter 10] relating pseudofunctors with categories fibered in groupoids, both of which can be used to develop the theory of stacks of groupoids [Vis07]. However, the homotopy theory of presheaves of groupoids, pseudofunctors, and categories fibered in groupoids are all equivalent in the sense of [Hol08, Corollary 4.3]. Furthermore, the notion of being a 1111-stack is independent across the three models. Thus in what follows we elect to use presheaves of groupoids, as they are the simplest to connect with the theory of simplicial presheaves by applying the nerve functor objectwise. | One of the most useful aspects of simplicial model categories is being able to define homotopically invariant mapping spaces. | is an isomorphism of sets for every U∈𝖢𝖺𝗋𝗍𝑈𝖢𝖺𝗋𝗍U\in\mathsf{Cart}italic_U ∈ sansserif_Cart and good open cover 𝒰𝒰\mathcal{U}caligraphic_U. In other words, for discrete simplicial presheaves, being an ∞\infty∞-stack is equivalent to being a sheaf. | Embedding the category of diffeological spaces into the category of simplicial presheaves puts diffeological spaces into a homotopical framework that is both easy to work with and connects powerfully with the underlying homotopy theory of simplicial sets. For instance, using this embedding, we can immediately define a notion of cohomology on diffeological spaces, which we call ∞\infty∞-stack cohomology. If X𝑋Xitalic_X is a diffeological space, and A𝐴Aitalic_A is an ∞\infty∞-stack, whose k𝑘kitalic_k-fold delooping W¯kAsuperscript¯𝑊𝑘𝐴\overline{W}^{k}Aover¯ start_ARG italic_W end_ARG start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_A exists (notions we will explain in Section 5), then the ∞\infty∞-stack cohomology of X𝑋Xitalic_X with coefficients in A𝐴Aitalic_A is given by the connected components of the derived mapping space | The paper is organized as follows. In Section 2, we will give some background information about diffeological spaces. In Section 3 we turn to diffeological principal G𝐺Gitalic_G-bundles. We define G𝐺Gitalic_G-cocycles and prove a bundle construction-type theorem, Theorem 3.15. In Section 4, we give a brisk introduction to sheaf theory, and explain the Baez-Hoffnung Theorem [BH11, Proposition 24]. In Appendix A, we compare several categories of concrete sheaves on various sites, and show that they are all equivalent, proving that the category of diffeological spaces as given in Definition 2.2 is equivalent to the usual category of diffeological spaces considered in the literature. In Section 5, we will review the Čech model structure on simplicial presheaves over cartesian spaces. Proposition 5.20 provides a cofibrant replacement of a diffeological space as the nerve of a diffeological category. We compare this diffeological category to two other diffeological categories Cˇ(X)ˇ𝐶𝑋\check{C}(X)overroman_ˇ start_ARG italic_C end_ARG ( italic_X ) and B//MB//Mitalic_B / / italic_M, which have been introduced in [KWW21] and [Igl20], respectively. From these three diffeological categories, we obtain three separate notions of Čech cohomology for diffeological spaces, and compare them in Section 5.3. In Section 6, we turn to the main result of this paper, that if G𝐺Gitalic_G is a diffeological group and X𝑋Xitalic_X is a diffeological space, then the nerve of the category of principal G𝐺Gitalic_G-bundles on X𝑋Xitalic_X is weak homotopy equivalent to the nerve of the category of G𝐺Gitalic_G-principal ∞\infty∞-bundles over X𝑋Xitalic_X. | A |
Iterating through the power set 2[l]superscript2delimited-[]𝑙2^{[l]}2 start_POSTSUPERSCRIPT [ italic_l ] end_POSTSUPERSCRIPT, the family (dimπI(V)∣I⊂[l])conditionaldimensionsubscript𝜋𝐼𝑉𝐼delimited-[]𝑙(\dim\pi_{I}(V)\mid I\subset[l])( roman_dim italic_π start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_V ) ∣ italic_I ⊂ [ italic_l ] ) can be computed in the claimed complexity. Using Theorem 4.1, we can now compute supp(V)supp𝑉\operatorname{supp}(V)roman_supp ( italic_V ) by iterating through each possible format α𝛼\alphaitalic_α with |α|=|𝒏|−r𝛼𝒏𝑟|\alpha|=|\bm{n}|-r| italic_α | = | bold_italic_n | - italic_r and decide whether for every I⊂[l]𝐼delimited-[]𝑙I\subset[l]italic_I ⊂ [ italic_l ] ∑i∈Ini−αi≤dimπI(V)subscript𝑖𝐼subscript𝑛𝑖subscript𝛼𝑖dimensionsubscript𝜋𝐼𝑉\sum_{i\in I}n_{i}-\alpha_{i}\leq\dim\pi_{I}(V)∑ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ roman_dim italic_π start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT ( italic_V ) holds. The number of possible formats is bounded by (|𝒏|−rl)binomial𝒏𝑟𝑙\binom{|\bm{n}|-r}{l}( FRACOP start_ARG | bold_italic_n | - italic_r end_ARG start_ARG italic_l end_ARG ) and the number of constraints to be checked is bounded by 2lsuperscript2𝑙2^{l}2 start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT. | In this section, we will introduce the multigraded associated varieties and provide algorithms to compute them. Moreover, we extend the results of [46] to the multigraded versions of Hurwitz forms. | In this section, we provide algorithms to compute associated varieties of multiprojective varieties. For the rest of the section, V⊂ℙ𝒏=ℙn1×ℙn2×⋯×ℙnl𝑉superscriptℙ𝒏superscriptℙsubscript𝑛1superscriptℙsubscript𝑛2⋯superscriptℙsubscript𝑛𝑙V\subset\mathbb{P}^{\bm{n}}=\mathbb{P}^{n_{1}}\times\mathbb{P}^{n_{2}}\times% | In this section, we outline an interesting connection between multiprojective varieties and the polymatroid theory. | In this section, we introduce the generalization of associated hypersurfaces to multiprojective varieties. The definitions and the results of this section follow [46]. | D |
It is supersolvable via the chain 0^⊂𝒫≤1⊂𝒫^0subscript𝒫absent1𝒫\hat{0}\subset\mathscr{P}_{\leq 1}\subset\mathscr{P}over^ start_ARG 0 end_ARG ⊂ script_P start_POSTSUBSCRIPT ≤ 1 end_POSTSUBSCRIPT ⊂ script_P. | An order ideal 𝒬𝒬\mathscr{Q}script_Q in a geometric lattice ℒℒ\mathscr{L}script_L is an M-ideal if and only if | A chain-finite lattice ℒℒ\mathscr{L}script_L is called geometric if and only if, for all x,y∈ℒ𝑥𝑦ℒx,y\in\mathscr{L}italic_x , italic_y ∈ script_L: | If ℒℒ\mathscr{L}script_L is a geometric lattice, then ℒℒ\mathscr{L}script_L satisfies Definition 2.5.1 if and only if it satisfies | Via Lemma 2.4.5, a geometric lattice ℒℒ\mathscr{L}script_L satisfies Definition 2.5.1 if and only if there is a chain | C |
L𝐆,V¯,W¯,N¯(𝐊)subscript𝐿𝐆¯𝑉¯𝑊¯𝑁𝐊{L_{\mathbf{G},\underline{V},\underline{W},\underline{N}}(\mathbf{K})}{}italic_L start_POSTSUBSCRIPT bold_G , under¯ start_ARG italic_V end_ARG , under¯ start_ARG italic_W end_ARG , under¯ start_ARG italic_N end_ARG end_POSTSUBSCRIPT ( bold_K ) is |Aut(𝐆)|Aut𝐆|\operatorname{Aut}(\mathbf{G})|| roman_Aut ( bold_G ) | when 𝐊𝒞≅𝐆superscript𝐊𝒞𝐆\mathbf{K}^{\mathcal{C}}\cong\mathbf{G}bold_K start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT ≅ bold_G and 00 otherwise. | Our definition of level-𝒞𝒞\mathcal{C}caligraphic_C groups is slightly more general than the definition used in [LWZ19, LW20]. Previously, one said the level-𝒞𝒞\mathcal{C}caligraphic_C groups are the smallest set of groups containing 𝒞𝒞\mathcal{C}caligraphic_C and closed under subgroups, quotients, and products (the variety of groups generated by 𝒞𝒞\mathcal{C}caligraphic_C). It’s possible to check that the level-𝒞𝒞\mathcal{C}caligraphic_C groups in the old sense are the level-𝒟𝒟\mathcal{D}caligraphic_D groups in the new sense, where 𝒟𝒟\mathcal{D}caligraphic_D consists of all subgroups of groups in 𝒞𝒞\mathcal{C}caligraphic_C. | Let 𝒞𝒞\mathcal{C}caligraphic_C be a set of finite groups. We say a group is level-𝒞𝒞\mathcal{C}caligraphic_C if is contained in the smallest set of finite groups containing 𝒞𝒞\mathcal{C}caligraphic_C and closed under fiber products and quotients. Then for a group H𝐻Hitalic_H, we define H𝒞superscript𝐻𝒞H^{\mathcal{C}}italic_H start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT | The key feature required for the second claim is that level-𝒞𝒞\mathcal{C}caligraphic_C groups are closed under fiber products and quotients. | For a set 𝒞𝒞\mathcal{C}caligraphic_C of groups, we have defined the set of level-𝒞𝒞\mathcal{C}caligraphic_C groups to be what is also known as the formation of groups generated by 𝒞𝒞\mathcal{C}caligraphic_C, i.e. the smallest set of isomorphism classes of groups that contains 𝒞𝒞\mathcal{C}caligraphic_C and is closed under taking quotients and fiber products. (Note fiber products are the same as subdirect products.) What we call the the level-𝒞𝒞\mathcal{C}caligraphic_C completion, G𝒞superscript𝐺𝒞G^{\mathcal{C}}italic_G start_POSTSUPERSCRIPT caligraphic_C end_POSTSUPERSCRIPT, is also known as the pro-𝒞^^𝒞\hat{\mathcal{C}}over^ start_ARG caligraphic_C end_ARG completion, where 𝒞^^𝒞\hat{\mathcal{C}}over^ start_ARG caligraphic_C end_ARG is the set of level-𝒞𝒞\mathcal{C}caligraphic_C groups. | C |
_{2}\lesssim\|F\|_{L^{2}_{s,\mathrm{sloc}}}.∥ italic_G start_POSTSUPERSCRIPT ( italic_ι ) end_POSTSUPERSCRIPT italic_ψ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Q / 2 + 1 , roman_sloc end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≲ start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT ∥ italic_G ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≲ ∥ italic_F ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s , roman_sloc end_POSTSUBSCRIPT end_POSTSUBSCRIPT . | \alpha}}\sim R^{\alpha}\lVert F^{(\iota)}\rVert_{2},\quad\alpha>1/2,∥ italic_F start_POSTSUPERSCRIPT ( italic_ι ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ≲ ∥ italic_F start_POSTSUPERSCRIPT ( italic_ι ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∼ italic_R start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ∥ italic_F start_POSTSUPERSCRIPT ( italic_ι ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_α > 1 / 2 , | On the other hand, choosing N:=α+1assign𝑁𝛼1N:=\alpha+1italic_N := italic_α + 1 in 6.6 yields in particular | \rVert_{2}\quad\text{for }\alpha^{\prime}>\alpha+d_{1}/2.≲ italic_R start_POSTSUPERSCRIPT italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∥ italic_F start_POSTSUPERSCRIPT ( italic_ι ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_α + italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 . | Since F𝐹Fitalic_F is supported in [−2,−1/2]∪[1/2,2]212122[-2,-1/2]\cup[1/2,2][ - 2 , - 1 / 2 ] ∪ [ 1 / 2 , 2 ], choosing N:=α+2assign𝑁𝛼2N:=\alpha+2italic_N := italic_α + 2 in 7.2 gives | B |
(5)5(5)( 5 ) if u1,…,upsubscript𝑢1…subscript𝑢𝑝u_{1},\ldots,u_{p}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT are plurisubharmonic functions, then u=Mη(u1,…,up)𝑢subscript𝑀𝜂subscript𝑢1…subscript𝑢𝑝u=M_{\eta}(u_{1},\ldots,u_{p})italic_u = italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) is plurisubharmonic. | Using Theorem 1.6, we obtain the following estimate for L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT integrals on the sublevel sets of ΨΨ\Psiroman_Ψ: | Following the notations and assumptions in Lemma 2.1, we present the following approximation property of φ𝜑\varphiitalic_φ and ΨΨ\Psiroman_Ψ. | Recall the definition of the minimal L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT integrals G𝐺Gitalic_G in Section 1.2. Theorem 1.6 implies the following result independent of the strong openness property. | In the present article, we establish the following concavity of G(−logr;Ψ,J,f)𝐺𝑟Ψ𝐽𝑓G(-\log r;\Psi,J,f)italic_G ( - roman_log italic_r ; roman_Ψ , italic_J , italic_f ). | B |
The matrix provided in Figure 3 describes the combinatorial and geometric structure of P15subscript𝑃15P_{15}italic_P start_POSTSUBSCRIPT 15 end_POSTSUBSCRIPT. Notice that we can apply Lemma 1.9 to facet 1111, as follows from the first three rows (or columns) of the given matrix. | (◆44uuuu⋅u⋅⋅⋅⋅u⋅⋅⋅⋅∞⋅⋅⋅⋅∞⋅∞⋅⋅⋅4◆∞4444⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅4⋅⋅⋅⋅4⋅4∞◆4444⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅4⋅⋅4⋅⋅u44◆uuu⋅u⋅∞⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅∞∞⋅u44u◆uu⋅u∞⋅⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅∞∞⋅⋅⋅⋅⋅u44uu◆u⋅u⋅⋅⋅∞u⋅⋅⋅⋅⋅∞⋅⋅⋅⋅⋅⋅⋅∞⋅u44uuu◆⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅∞⋅u∞⋅∞⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅33⋅⋅⋅⋅3⋅⋅⋅u44uuuu⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅∞⋅∞u⋅⋅⋅⋅∞⋅⋅⋅⋅⋅⋅∞⋅⋅⋅⋅◆⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅∞⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅33⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆3⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅∞⋅⋅⋅⋅⋅3◆⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅◆⋅3⋅⋅⋅⋅⋅34⋅⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3⋅⋅⋅⋅⋅uu⋅⋅⋅⋅⋅⋅⋅⋅3⋅◆⋅⋅44⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅◆⋅⋅⋅3⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3⋅⋅⋅⋅◆⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅∞4⋅⋅⋅⋅⋅⋅∞⋅⋅3⋅⋅⋅4⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅∞∞3⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3∞⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅u⋅⋅⋅⋅3⋅3⋅⋅⋅⋅⋅◆⋅4⋅⋅⋅4⋅⋅4⋅⋅∞⋅∞⋅⋅⋅⋅⋅⋅4⋅⋅3⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅∞⋅4⋅∞⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅4⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅∞⋅⋅⋅⋅3⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅∞⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅3⋅⋅4∞⋅⋅⋅⋅∞⋅⋅⋅⋅43⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅4⋅∞⋅∞⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅⋅⋅◆3⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3◆)◆44uuuu⋅u⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4◆4444⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅4⋅⋅⋅⋅4⋅4◆4444⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅4⋅⋅4⋅⋅u44◆uuu⋅u⋅⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u44u◆uu⋅u⋅⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u44uu◆u⋅u⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u44uuu◆⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅33⋅⋅⋅⋅3⋅⋅⋅u44uuuu⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅33⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆3⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3◆⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅⋅⋅⋅u⋅⋅⋅⋅⋅⋅⋅◆⋅3⋅⋅⋅⋅⋅34⋅⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3⋅⋅⋅⋅⋅uu⋅⋅⋅⋅⋅⋅⋅⋅3⋅◆⋅⋅44⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅◆⋅⋅⋅3⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3⋅⋅⋅⋅◆⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅4⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅3⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅u⋅u⋅⋅⋅⋅3⋅3⋅⋅⋅⋅⋅◆⋅4⋅⋅⋅4⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅3⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅4⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅3⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅43⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅◆⋅⋅⋅4⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅4⋅⋅⋅⋅⋅◆3⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅3⋅3◆\left(\begin{array}[]{ccccccccccccccccccccccccccccc}{\scriptscriptstyle% | Let P𝑃Pitalic_P be the 4444–dimensional non–compact hyperbolic prism described by the Coxeter–Vinberg diagram in Figure 4. | VR5V−1=(0100010000001000001000001).𝑉subscript𝑅5superscript𝑉10100010000001000001000001VR_{5}V^{-1}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\ | VR1V−1=(1000001000001009/49/43/2−1/4−3/49/49/43/2−5/41/4),VR2V−1=(1000001000000100010000−111),formulae-sequence𝑉subscript𝑅1superscript𝑉110000010000010094943214349494325414𝑉subscript𝑅2superscript𝑉11000001000000100010000111VR_{1}V^{-1}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\ | A |
We show that q1subscript𝑞1q_{1}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and q2subscript𝑞2q_{2}italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT can be represented by the same reduced quadratic form with discriminant −16p16𝑝-16p- 16 italic_p. On the contrary, if q1subscript𝑞1q_{1}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and q2subscript𝑞2q_{2}italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT can be represented by the same reduced quadratic form with discriminant −16p16𝑝-16p- 16 italic_p, then 𝒪(q1,r1)≅𝒪(q2,r2)𝒪subscript𝑞1subscript𝑟1𝒪subscript𝑞2subscript𝑟2\mathcal{O}(q_{1},r_{1})\cong\mathcal{O}(q_{2},r_{2})caligraphic_O ( italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≅ caligraphic_O ( italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) obviously. | If p≡3(mod8)𝑝annotated3𝑝𝑚𝑜𝑑8p\equiv 3\pmod{8}italic_p ≡ 3 start_MODIFIER ( roman_mod start_ARG 8 end_ARG ) end_MODIFIER, then every reduced form with discriminant −p𝑝-p- italic_p divides into six reduced forms with discriminant −16p16𝑝-16p- 16 italic_p and three of them are in the non-principal genus. If p≡7(mod8)𝑝annotated7𝑝𝑚𝑜𝑑8p\equiv 7\pmod{8}italic_p ≡ 7 start_MODIFIER ( roman_mod start_ARG 8 end_ARG ) end_MODIFIER, then every reduced form with discriminant −p𝑝-p- italic_p divides into two reduced forms with discriminant −16p16𝑝-16p- 16 italic_p and one of them is in the non-principal genus. | Assume that q1subscript𝑞1q_{1}italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and q2subscript𝑞2q_{2}italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT can be represented by a primitive form (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) with discriminant −4p4𝑝-4p- 4 italic_p. For s=2𝑠2s=2italic_s = 2, the following three forms with discriminant −16p16𝑝-16p- 16 italic_p are derived from (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ): | In this section, we will establish a correspondence between 𝔽psubscript𝔽𝑝\mathbb{F}_{p}blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-isomorphism classes of supersingular elliptic curves and reduced quadratic forms with discriminant −p𝑝-p- italic_p or −16p16𝑝-16p- 16 italic_p. | In the following of this subsection, we discuss reduced primitive quadratic forms with discriminant −16p16𝑝-16p- 16 italic_p derived from forms with discriminant −p𝑝-p- italic_p. | D |
Roughly speaking, we consider a zero-sum game between an adversary and a statistician, in which the adversary chooses a deviation and the statistician, after observing the realization s𝑠sitalic_s, has to guess the deviator if s∉D𝑠𝐷s\notin Ditalic_s ∉ italic_D. A strategy for the statistican in this game is a blame function. We use the minimax theorem to establish that the statistician has a strategy that guarantees high payoff. | However, for the minimax theorem to apply, we need to make some modifications to the game.111Blackwell [5] gives an example of a statistical game without a value. | For every n∈ℕ𝑛ℕn\in\mathbb{N}italic_n ∈ blackboard_N, the value of the game Γ(Dn)Γsubscript𝐷𝑛\Gamma(D_{n})roman_Γ ( italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is at least 1−(|I|−1)ε1𝐼1𝜀1-\sqrt{(|I|-1)\varepsilon}1 - square-root start_ARG ( | italic_I | - 1 ) italic_ε end_ARG. | Even though we use game theoretic terminology (players, actions, strategies), we emphasize that we do not define a game between the players, as there are no payoff functions. | Roughly speaking, we consider a zero-sum game between an adversary and a statistician, in which the adversary chooses a deviation and the statistician, after observing the realization s𝑠sitalic_s, has to guess the deviator if s∉D𝑠𝐷s\notin Ditalic_s ∉ italic_D. A strategy for the statistican in this game is a blame function. We use the minimax theorem to establish that the statistician has a strategy that guarantees high payoff. | A |
}\mathds{E}[(M^{\ast}_{r,m})_{[t]}]= ∑ start_POSTSUBSCRIPT italic_t = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_S ( italic_d , italic_t ) ( divide start_ARG italic_θ end_ARG start_ARG italic_θ + italic_n end_ARG ) start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT blackboard_E [ ( italic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r , italic_m end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT [ italic_t ] end_POSTSUBSCRIPT ] | =[by Equation (S3.2)]absent[by Equation (S3.2)]\displaystyle\quad=\text{[by Equation \eqref{eq_momb}]}= [by Equation ( )] | Then, by a direct combination of Equation (S4) with Equation S4.6 we can write the following identities | This completes the proof of statement i) of Proposition 1. The proof of statement ii) follows along similar lines, and by letting α→0→𝛼0\alpha\rightarrow 0italic_α → 0. By combining Equation (S3.1) with Equation (S1.7) it holds | =[by Equation (S3.5)]absent[by Equation (S3.5)]\displaystyle\quad=\text{[by Equation \eqref{eq_momm_dir}]}= [by Equation ( )] | A |
On the left, it has 6 “vertex gadgets” 1,…,61…61,\dots,61 , … , 6 that correspond to the vertices of G𝐺Gitalic_G. | For i∈{1,…,n}𝑖1…𝑛i\in\{1,\ldots,n\}italic_i ∈ { 1 , … , italic_n }, add a vertex gadget 𝒱isubscript𝒱𝑖\mathcal{V}_{i}caligraphic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT into the leftmost fragment of Zone(i)Zone𝑖\textrm{Zone}(i)Zone ( italic_i ). For each 𝒱isubscript𝒱𝑖\mathcal{V}_{i}caligraphic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT there are 3 long intervals starting from it; they terminate at a join gadget that has 3 new long intervals starting from it, and so on. This produces 3 link chains of long intervals. Each such chain eventually terminates at a corresponding edge gadget. | For each vertex gadget there are three link chains constructed by alternating long intervals and join gadgets. | For each vertex gadget, there are precisely three chains of long intervals that start from it, each chain represents an edge incident to the corresponding vertex. | If an edge was incident to a vertex, then the corresponding vertex and edge gadgets were “linked” by a pair of very long intervals starting from the vertex gadgets and terminating at the edge gadget. | C |
Part (i)𝑖(i)( italic_i ) follows from Lemma 27 and Proposition 26. For part (ii)𝑖𝑖(ii)( italic_i italic_i ), observe that by an argument similar to the proof of [16, Theorem 3.2], G𝐺Gitalic_G is residually compact in this case. Now apply part (i)𝑖(i)( italic_i ). ∎ | Let us define the above subclass more rigorously. The class of elementary groups is the smallest class ℰℰ\mathscr{E}script_E of t.d.l.c.s.c. groups which contains all s.c. profinite groups and countable discrete groups, and is closed under taking topological group extensions by s.c. profinite groups and countable discrete | The class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups are recently subject of study as an important subclass of Polish groups (separable and completely metrizable topological groups). The second countability assumption is not that essential, as every t.d.l.c. | In Section 2 we introduce box spaces and define asymptotic dimension of locally compact (second countable) groups and show that the invariant is finite for large classes of residually compact groups. In Section 3 we define the Hirsch length for topological groups and prove the Hirsch formula, which is critical for inductive arguments based on the Hirsch length. This section also deals with a natural extension of virtually nilpotent groups in the topological realm, which are shown to be residually compact and so of finite asymptotic dimension. Same is done in Section 4 for topologically elementary amenable groups. A small final epilogue in Section 5 is devoted to finiteness of the asymptotic dimension of totally disconnected, locally compact, second countable groups. | Let us compare and contrast the class [TEA] with other known classes of topological groups. Our reference for the properties of various classes discussed here is [32], from which we borrow class notations as well. [TEA] clearly contains the classes [Z] of central groups (groups whose quotient over center is compact) which is included in the larger class of extensions of abelian groups by compact groups. It does not contains the class [MAP] of maximally almost periodic groups (groups with enough finite dimensional irreducible representations to separate the points, or equivalently, enough continuous almost periodic functions to separate the points), as these are subgroup of compact groups and so contains many non amenable groups (this justifies our assumption that the class [TEA] is stable only under taking closed subgroups). [MAP] includes the class [Moore] of Moore groups (groups whose irreducible representations are finite dimensional) by Gelfand-Raikov theorem. Note that a Lie group belongs to [Moore] if and only if it is a finite extension of a central group, so [TEA] contains Lie groups in the class [Moore]. For compactly generated groups, [MAP] is included in [SIN] (small invariant neighborhood groups, so the latter is not included in [TEA], indeed [SIN] includes all discrete groups (more generally, [SIN]-groups are discrete extensions of direct products of vector groups and compact groups (c.f., [15, Theorem 2.13]). However connected [SIN] groups are in [TEA] (as they are in [Z], by Freudenthal-Weil theorem), though there are connected groups in [TEA] included neither in [MAP] nor in [SIN] (c.f., [32, Example 8]) as well as compactly generated totally disconnected groups in [TEA] included neither in [MAP] nor in [SIN] (c.f., [32, Example 9]). More generally, connected [IN]-groups are extension of compact groups by vector groups, and so are in [TEA] [15, Theorem 2.9, Corollary 2.8]. A less trivial inclusion follows by a result of Grosser and Moskowitz [15, Proposition 4.5], stating that a compactly generated [FIA]- group (topologically finite inner automorphism group, that is a group with relatively compact inner automorphisms group) is an extension of a direct product of some vector group ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with a compact group by some ℤdsuperscriptℤ𝑑\mathbb{Z}^{d}blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, showing that [TEA] contains compactly generated [FIA]- groups. Since [SIN]∩\cap∩[FD]⊆−{}^{-}\subseteqstart_FLOATSUPERSCRIPT - end_FLOATSUPERSCRIPT ⊆ [FIA]- [15, Theorem 4.6], same is true for compactly generated [SIN]-groups which are in [FD]- (topologically finite derived subgroup, that is a groups with relatively compact commutator or derived subgroup). This class is quite interesting as it contains non Type I groups, showing that [TEA] is not included in [Type I]: there is a central extension of 𝕋𝕋\mathbb{T}blackboard_T by ℤ2superscriptℤ2\mathbb{Z}^{2}blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT with derived subgroup 𝕋𝕋\mathbb{T}blackboard_T, which is a compactly generated group in [SIN]∩\cap∩[FD]-, with a non Type I, ℤ2superscriptℤ2\mathbb{Z}^{2}blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT multiplier (and so non Type I) (c.f., [32, Example 7]; also there are compactly generated solvable non Type I groups, c.f., [32, Example 17]; and connected solvable non Type I groups, c.f., [32, Example 19]). Another structural result of Grosser and Moskowitz [15, Theorem 3.16], states that for a group in [FC]- (topologically finite conjugacy class groups, that is groups relatively compact conjugacy classes) the quotient by the intersection of all compact normal subgroups is a direct product of a | B |