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𝒮×𝒢=(𝕊×[𝕊],⊕,⊗,(−∞,∅),(0,[])),𝒮𝒢𝕊delimited-[]𝕊direct-sumtensor-product0\mathcal{S}\times\mathcal{G}=\left(\mathbb{S}\times\left[\mathbb{S}\right],%

\mathbb{S}\right)\to\mathbb{S}.italic_f : ( blackboard_S → blackboard_S → blackboard_S ) → ( blackboard_S → blackboard_S → blackboard_S ) → blackboard_S → blackboard_S → ( blackboard_X → blackboard_S ) → blackboard_S .

\mathbb{S}\right)\to\mathbb{S},italic_f : ( blackboard_S → blackboard_S → blackboard_S ) → ( blackboard_S → blackboard_S → blackboard_S ) → blackboard_S → blackboard_S → ( blackboard_X → blackboard_S ) → blackboard_S ,

\left[\,\right]\right\}\right)\right),caligraphic_S × caligraphic_G = ( blackboard_S × { [ blackboard_S ] } , ⊕ , ⊗ , ( - ∞ , ∅ ) , ( 0 , { [ ] } ) ) ,

\right),caligraphic_S × caligraphic_G = ( blackboard_S × [ blackboard_S ] , ⊕ , ⊗ , ( - ∞ , ∅ ) , ( 0 , [ ] ) ) ,

D

As a trivial corollary, we deduce that the half-infinite 1111-direction geodesic spends asymptotically the same amount of time

As in Theorem 1, any polygon of the 2222-colorings can be replaced by a circle, an ellipse, or any other piecewise smooth closed curve.

We remark that any polygon in the given 2222-coloring can be replaced by a circle, an ellipse, or any other piecewise smooth closed curve.

Here we investigate how any polygon of the 2222-colorings can be replaced by a circle, an ellipse, or any other piecewise smooth closed curve.

with edges that are piecewise smooth closed curves, satisfying some mild technical requirements as stated in Lemma 6.1,

B

Snell envelope theory [20] and special utility-based arguments for certain applications [28] (see also subgame-perfect equilibrium methods [50] and ε𝜀\varepsilonitalic_ε–equilibria [41, 32]).

When λ𝜆\lambdaitalic_λ is very large, the exercise opportunities of player P𝑃Pitalic_P are almost as frequent as those of player C𝐶Citalic_C.

Two retailers (players C𝐶Citalic_C and P𝑃Pitalic_P) are competing for purchasing a limited supply/inventory of a specific product from a common supplier.

Beyond existence results, examples where such equilibria have been constructed are very limited and are always under a setting of

(a) our already obtained results on Nash equilibria in threshold strategies (Propositions 4.12 and 4.13);

C

\mathbb{R}.over^ start_ARG caligraphic_M end_ARG start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ ) caligraphic_M start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ψ ) / blackboard_R .

Based on the above construction, Juhász [30] defined a differential on S⁢F⁢C⁢(ℋ)𝑆𝐹𝐶ℋSFC(\mathcal{H})italic_S italic_F italic_C ( caligraphic_H ) by

Suppose (M,γ)𝑀𝛾(M,\gamma)( italic_M , italic_γ ) is a balanced sutured manifold. Then there is an admissible balanced diagram ℋℋ\mathcal{H}caligraphic_H compatible with (M,γ)𝑀𝛾(M,\gamma)( italic_M , italic_γ ). The vector spaces H⁢(S⁢F⁢C⁢(ℋ),∂Js)𝐻𝑆𝐹𝐶ℋsubscriptsubscript𝐽𝑠H(SFC(\mathcal{H}),\partial_{J_{s}})italic_H ( italic_S italic_F italic_C ( caligraphic_H ) , ∂ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) for different choices of ℋℋ\mathcal{H}caligraphic_H and Jssubscript𝐽𝑠J_{s}italic_J start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT, together with some canonical maps, form a transitive system S⁢F⁢H⁢(M,γ)𝑆𝐹𝐻𝑀𝛾SFH(M,\gamma)italic_S italic_F italic_H ( italic_M , italic_γ ) over 𝔽2subscript𝔽2\mathbb{F}_{2}blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

We prove this equation by introducing a decomposition 𝒮⁢ℱ⁢ℋT⁢(−M,−γ)𝒮ℱsubscriptℋ𝑇𝑀𝛾\mathcal{SFH}_{T}(-M,-\gamma)caligraphic_S caligraphic_F caligraphic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( - italic_M , - italic_γ ) of S⁢F⁢H⁢(−M,−γ)𝑆𝐹𝐻𝑀𝛾SFH(-M,-\gamma)italic_S italic_F italic_H ( - italic_M , - italic_γ ) similar to 𝒮⁢ℋ⁢𝒢T⁢(−M,−γ)𝒮ℋsubscript𝒢𝑇𝑀𝛾\mathcal{SHG}_{T}(-M,-\gamma)caligraphic_S caligraphic_H caligraphic_G start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( - italic_M , - italic_γ ). However, the construction of S⁢F⁢H𝑆𝐹𝐻SFHitalic_S italic_F italic_H is based on balanced diagrams of balanced sutured manifolds, which is different from the construction of SHG¯¯SHG\underline{\rm SHG}under¯ start_ARG roman_SHG end_ARG by closures. So we have to show that S⁢F⁢H𝑆𝐹𝐻SFHitalic_S italic_F italic_H satisfies the similar setups of SHG¯¯SHG\underline{\rm SHG}under¯ start_ARG roman_SHG end_ARG to construct 𝒮⁢ℱ⁢ℋT⁢(−M,−γ)𝒮ℱsubscriptℋ𝑇𝑀𝛾\mathcal{SFH}_{T}(-M,-\gamma)caligraphic_S caligraphic_F caligraphic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( - italic_M , - italic_γ ). This is the main goal of Section 4, where we collect results for S⁢F⁢H𝑆𝐹𝐻SFHitalic_S italic_F italic_H parallel to SHG¯¯SHG\underline{\rm SHG}under¯ start_ARG roman_SHG end_ARG, including gradings associated to admissible surfaces, the surgery exact triangle, the bypass exact triangle, and contact gluing maps.

Based on the ℤnsuperscriptℤ𝑛\mathbb{Z}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT grading from Theorem 2.8, we can define the graded Euler characteristic.

A

If the data is split in training, validation, and test set, the local R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

Figure 4: R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT scores comparison between local versions varying the number of

the local R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT score is not met. In general, the local R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT score is evaluated

Figure 6: R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT scores comparison between global AS and local versions varying

Remark 3 (Estimator based on local R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT scores).

D

𝚺¯=1M⁢∑j=1M𝚺⁢(Tj)=[1M−1M⁢ρ2…M−1M⁢ρ2ρM−1M⁢ρ21…M−1M⁢ρ2ρ⋮⋮⋱⋮⋮M−1M⁢ρ2M−1M⁢ρ2…1ρ\hdashline⁢[2⁢p⁢t/2⁢p⁢t]⁢ρρ…ρ

𝚺¯=1p−2⁢∑j=1p−2𝚺⁢(Tj)=[10…0ρρ/(p-2)01…0ρρ/(p-2)⋮⋮⋱⋮⋮⋮00…1ρρ/(p-2)\hdashline⁢[2⁢p⁢t/2⁢p⁢t]⁢ρρ…ρ1ρ^2ρ/(p−2)ρ/(p−2)…ρ/(p−2)ρ^21].¯𝚺1𝑝2superscriptsubscript𝑗1𝑝2𝚺subscript𝑇𝑗delimited-[]10…0ρρ/(p-2)missing-subexpressionmissing-subexpression01…0ρρ/(p-2)missing-subexpressionmissing-subexpression⋮⋮⋱⋮⋮⋮missing-subexpressionmissing-subexpression00…1ρρ/(p-2)missing-subexpressionmissing-subexpression\hdashlinedelimited-[]2𝑝𝑡2𝑝𝑡𝜌𝜌…𝜌1ρ^2missing-subexpressionmissing-subexpression𝜌𝑝2𝜌𝑝2…𝜌𝑝2ρ^21missing-subexpressionmissing-subexpression\overline{\bm{\Sigma}}=\frac{1}{p-2}\sum_{j=1}^{p-2}\bm{\Sigma}(T_{j})=\left[%

1]≔[𝑨𝑩𝑩⊤𝑫].¯𝚺1𝑀superscriptsubscript𝑗1𝑀𝚺subscript𝑇𝑗delimited-[]1𝑀1𝑀superscript𝜌2…𝑀1𝑀superscript𝜌2ρmissing-subexpressionmissing-subexpression𝑀1𝑀superscript𝜌21…𝑀1𝑀superscript𝜌2ρmissing-subexpressionmissing-subexpression⋮⋮⋱⋮⋮missing-subexpressionmissing-subexpression𝑀1𝑀superscript𝜌2𝑀1𝑀superscript𝜌2…1ρmissing-subexpressionmissing-subexpression\hdashlinedelimited-[]2𝑝𝑡2𝑝𝑡𝜌𝜌…𝜌

𝚺⁢(T1)=[10…0ρρ01…0ρ0⋮⋮⋱⋮⋮⋮00…1ρ0\hdashline⁢[2⁢p⁢t/2⁢p⁢t]⁢ρρ…ρ1ρ^2ρ0…0ρ^21]≔[𝑨𝑩𝑩⊤𝑫]𝚺subscript𝑇1delimited-[]10…0ρρmissing-subexpressionmissing-subexpression01…0ρ0missing-subexpressionmissing-subexpression⋮⋮⋱⋮⋮⋮missing-subexpressionmissing-subexpression00…1ρ0missing-subexpressionmissing-subexpression\hdashlinedelimited-[]2𝑝𝑡2𝑝𝑡𝜌𝜌…𝜌1ρ^2missing-subexpressionmissing-subexpression𝜌0…0ρ^21missing-subexpressionmissing-subexpression≔matrix𝑨𝑩superscript𝑩top𝑫\bm{\Sigma}(T_{1})=\left[\begin{array}[]{cccc;{2pt/2pt}cc}1&0&\ldots&0&\rho&%

ρ…ρ1]𝚺subscript𝑇1delimited-[]10ρ^2…superscript𝜌2𝜌missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression01ρ^2…superscript𝜌2𝜌missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression\hdashlinedelimited-[]2𝑝𝑡2𝑝𝑡superscript𝜌2superscript𝜌21…superscript𝜌2𝜌missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression⋮⋮⋮⋱⋮⋮missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionsuperscript𝜌2superscript𝜌2ρ^2…1𝜌missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression\hdashlinedelimited-[]2𝑝𝑡2𝑝𝑡𝜌𝜌

B

Step 3 Conclusion. We inject the bounds (7.16), (7.8), (7.9) and (7.10) in the decomposition (7.6), this establishes the desired formula (1.10) upon choosing K′superscript𝐾′K^{\prime}italic_K start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT large enough.

u^{n}(t=0)=0,\end{array},\quad n\geq 1\right.{ start_ARRAY start_ROW start_CELL italic_i ∂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_ω ( italic_D ) italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_λ ∑ start_POSTSUBSCRIPT italic_j + italic_k = italic_n - 1 end_POSTSUBSCRIPT italic_M ( italic_M italic_u start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT + italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG ) ( italic_M italic_u start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT + italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_ARG ) end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_t = 0 ) = 0 , end_CELL end_ROW end_ARRAY , italic_n ≥ 1

The formula corresponding to (6.6) for the expectation of the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT norm of unsuperscript𝑢𝑛u^{n}italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is for t≥0𝑡0t\geq 0italic_t ≥ 0

The iterates unsuperscript𝑢𝑛u^{n}italic_u start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, defined through (1.5), satisfy the bounds

{u^{0}}Mu^{0}+(M\overline{u^{0}})^{2}( italic_M italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( italic_M italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_M italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG + italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG italic_M italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT + ( italic_M over¯ start_ARG italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, are represented by the graphs below:

C

Let M∈ℤn×n𝑀superscriptℤ𝑛𝑛M\in\mathbb{Z}^{n\times n}italic_M ∈ blackboard_Z start_POSTSUPERSCRIPT italic_n × italic_n end_POSTSUPERSCRIPT be an integer matrix which is expanding in the sense that each of its eigenvalues has modulus strictly greater than one. Moreover, let 𝒟⊂ℤn𝒟superscriptℤ𝑛\mathcal{D}\subset\mathbb{Z}^{n}caligraphic_D ⊂ blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be a digit set with |detM|𝑀|\det M|| roman_det italic_M | elements.

Then it follows from the theory of iterated function systems (see e.g. Hutchinson [17]) that there is a unique nonempty compact set T=T⁢(M,𝒟)𝑇𝑇𝑀𝒟T=T(M,\mathcal{D})italic_T = italic_T ( italic_M , caligraphic_D ) such that

Before we state our main results, we introduce some notation. Let M𝑀Mitalic_M be an expanding 3×3333\times 33 × 3 integer matrix and let 𝒟⊂ℤ3𝒟superscriptℤ3\mathcal{D}\subset\mathbb{Z}^{3}caligraphic_D ⊂ blackboard_Z start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT be a digit set such that the unique nonempty compact set T=T⁢(M,𝒟)𝑇𝑇𝑀𝒟T=T(M,\mathcal{D})italic_T = italic_T ( italic_M , caligraphic_D ) defined by the set equation

If T𝑇Titalic_T has nonempty interior then it is called an integral self-affine tile, or just a self-affine tile for short. If 𝒟𝒟\mathcal{D}caligraphic_D is a complete set of coset representatives of the residue class ring ℤn/M⁢ℤnsuperscriptℤ𝑛𝑀superscriptℤ𝑛\mathbb{Z}^{n}/M\mathbb{Z}^{n}blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT / italic_M blackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, it is called a standard digit set. For standard digit sets it is known that the nonempty compact set T𝑇Titalic_T defined by (1.1) always has nonempty interior (see Bandt [2]).

Let T=T⁢(M,𝒟)𝑇𝑇𝑀𝒟T=T(M,\mathcal{D})italic_T = italic_T ( italic_M , caligraphic_D ) be an A⁢B⁢C𝐴𝐵𝐶ABCitalic_A italic_B italic_C-tile.

A

In [1, Theorem 2.1], Bamerni, Kadets and Kiliçman established that if A𝐴Aitalic_A is a dense subset of a Banach space X𝑋Xitalic_X, then there is a nontrivial closed subspace M𝑀Mitalic_M of X𝑋Xitalic_X such that A∩M𝐴𝑀A\cap Mitalic_A ∩ italic_M is dense in M𝑀Mitalic_M.

In [1, Theorem 2.1], Bamerni, Kadets and Kiliçman established that if A𝐴Aitalic_A is a dense subset of a Banach space X𝑋Xitalic_X, then there is a nontrivial closed subspace M𝑀Mitalic_M of X𝑋Xitalic_X such that A∩M𝐴𝑀A\cap Mitalic_A ∩ italic_M is dense in M𝑀Mitalic_M.

Proof of Theorem 2.1. First Aαsubscript𝐴𝛼A_{\alpha}italic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT is dense in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT by Lemma 2.2. Now suppose that there is a nontrivial subspace M𝑀Mitalic_M of ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT of dimension r𝑟ritalic_r (1≤r≤n−11𝑟𝑛11\leq r\leq n-11 ≤ italic_r ≤ italic_n - 1) such that Aα∩Msubscript𝐴𝛼𝑀A_{\alpha}\cap Mitalic_A start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∩ italic_M is dense in M𝑀Mitalic_M. One can choose a basis {v1,v2,…,vr}subscript𝑣1subscript𝑣2…subscript𝑣𝑟\{v_{1},v_{2},\dots,v_{r}\}{ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT } of M𝑀Mitalic_M so that

By a nontrivial subspace M𝑀Mitalic_M of X𝑋Xitalic_X, we mean that M𝑀Mitalic_M is non-zero and distinct from X𝑋Xitalic_X. We acknowledge here that the authors of [1]

We point out that when X𝑋Xitalic_X is of finite dimension, Lemma 2.3 of [1] used in the proof of Theorem [1, Theorem 2.1] is not true.

C

𝖼𝗈𝖠𝗅𝗀U⁡(ℳ)subscript𝖼𝗈𝖠𝗅𝗀Uℳ{\mathsf{coAlg}_{{\operatorname{U}(\mathcal{M})}}}sansserif_coAlg start_POSTSUBSCRIPT roman_U ( caligraphic_M ) end_POSTSUBSCRIPT𝖼𝖲𝖾𝗍𝖼𝖲𝖾𝗍{\mathsf{cSet}}sansserif_cSet𝖼𝗈𝖠𝗅𝗀.𝖼𝗈𝖠𝗅𝗀{\mathsf{coAlg}.}sansserif_coAlg .NU⁡(ℳ)□subscriptsuperscriptN□Uℳ\scriptstyle{\operatorname{N^{\square}_{{\operatorname{U}(\mathcal{M})}}}}roman_N start_POSTSUPERSCRIPT □ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_U ( caligraphic_M ) end_POSTSUBSCRIPTN𝒜⁢𝗌□subscriptsuperscriptN□𝒜𝗌\scriptstyle{\operatorname{N^{\square}_{\!{\mathcal{A}\mathsf{s}}}}}roman_N start_POSTSUPERSCRIPT □ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_A sansserif_s end_POSTSUBSCRIPT

The product defined above in Equation 7 differs from the one defined in [medina2022cube_einfty] by the sign (−1)xsuperscript1𝑥(-1)^{x}( - 1 ) start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT.

In the diagram of the above theorem, the arrow from 𝗌𝖲𝖾𝗍0superscript𝗌𝖲𝖾𝗍0\mathsf{sSet}^{0}sansserif_sSet start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT to 𝖬𝗈𝗇𝖢𝗁subscript𝖬𝗈𝗇𝖢𝗁\mathsf{Mon}_{\mathsf{Ch}}sansserif_Mon start_POSTSUBSCRIPT sansserif_Ch end_POSTSUBSCRIPT is Adams’ cobar construction, the one from 𝗌𝖲𝖾𝗍0superscript𝗌𝖲𝖾𝗍0\mathsf{sSet}^{0}sansserif_sSet start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT to 𝖬𝗈𝗇𝖼𝗈𝖠𝗅𝗀subscript𝖬𝗈𝗇𝖼𝗈𝖠𝗅𝗀\mathsf{Mon}_{\mathsf{coAlg}}sansserif_Mon start_POSTSUBSCRIPT sansserif_coAlg end_POSTSUBSCRIPT is Baues’ enhancement, and the one from 𝗌𝖲𝖾𝗍0superscript𝗌𝖲𝖾𝗍0\mathsf{sSet}^{0}sansserif_sSet start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT to 𝖬𝗈𝗇𝖼𝗈𝖠𝗅𝗀U⁡(ℳ)subscript𝖬𝗈𝗇subscript𝖼𝗈𝖠𝗅𝗀Uℳ\mathsf{Mon}_{\mathsf{coAlg}_{{\operatorname{U}(\mathcal{M})}}}sansserif_Mon start_POSTSUBSCRIPT sansserif_coAlg start_POSTSUBSCRIPT roman_U ( caligraphic_M ) end_POSTSUBSCRIPT end_POSTSUBSCRIPT is our lift.

More specifically, we verify that the monoid structure induced on S□⁡(Ωx⁢𝔛)superscriptS□subscriptΩ𝑥𝔛\operatorname{S^{\square}}(\Omega_{x}\mathfrak{X})start_OPFUNCTION roman_S start_POSTSUPERSCRIPT □ end_POSTSUPERSCRIPT end_OPFUNCTION ( roman_Ω start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT fraktur_X ) by the concatenation of loops is compatible with a natural E∞subscript𝐸E_{\infty}italic_E start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-coalgebra structure on cubical singular chains, similar to the one defined in [medina2022cube_einfty].

}\to\mathsf{coAlg}_{\operatorname{U}(\mathcal{M})}start_OPFUNCTION roman_N start_POSTSUPERSCRIPT □ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_U ( caligraphic_M ) end_POSTSUBSCRIPT end_OPFUNCTION : sansserif_cSet → sansserif_coAlg start_POSTSUBSCRIPT roman_U ( caligraphic_M ) end_POSTSUBSCRIPT – defined in [medina2022cube_einfty] with a different sign convention – is monoidal.

A

{0.4286,0.5000,0.5397,0.5143,0.5306,1.0000,0.6190}0.42860.50000.53970.51430.53061.00000.6190\{0.4286,0.5000,0.5397,0.5143,0.5306,1.0000,0.6190\}{ 0.4286 , 0.5000 , 0.5397 , 0.5143 , 0.5306 , 1.0000 , 0.6190 }

where V1,T,β,m⁢(ω;u=1)subscript𝑉1𝑇𝛽𝑚𝜔𝑢1V_{1,T,\beta,m}(\omega;u=1)italic_V start_POSTSUBSCRIPT 1 , italic_T , italic_β , italic_m end_POSTSUBSCRIPT ( italic_ω ; italic_u = 1 ) and V1,T,β,m⁢(ω;u=0)subscript𝑉1𝑇𝛽𝑚𝜔𝑢0V_{1,T,\beta,m}(\omega;u=0)italic_V start_POSTSUBSCRIPT 1 , italic_T , italic_β , italic_m end_POSTSUBSCRIPT ( italic_ω ; italic_u = 0 ) are respectively given by (25) and (26) in terms of V1,T−1,β,m⁢(⋅)subscript𝑉1𝑇1𝛽𝑚⋅V_{1,T-1,\beta,m}(\cdot)italic_V start_POSTSUBSCRIPT 1 , italic_T - 1 , italic_β , italic_m end_POSTSUBSCRIPT ( ⋅ ). Clearly, the number of observed belief states grows exponentially with both the number of arms (as arms are not decoupled by any relaxation) and the time horizon T𝑇Titalic_T due to the tree-expansion type of the belief update given in (4). In contrast, our approximate Whittle index has a linear complexity in both T𝑇Titalic_T and the number of arms. In Fig. 6 and Fig. 6, we compare the real running times between the optimal policy and our algorithm to illustrate the efficiency of the latter. Note that the optimal policy for any finite time horizon T𝑇Titalic_T provides an upper bound on the total discounted reward over T𝑇Titalic_T achieved by the infinite-horizon optimal policy. This is because one can definitely apply the infinite-horizon optimal policy to the finite-horizon problem up to time T𝑇Titalic_T. Henceforth, the near-optimality of our algorithm is well demonstrated by comparing to the finite-horizon optimal policy over the first T𝑇Titalic_T steps as shown in Figures 8-22 (see Table 1 and Table 2 for system parameters). Furthermore, all numerical experiments with randomly generated system parameters showed that setting the iteration number k=4𝑘4k=4italic_k = 4 is sufficient for the Whittle indices of all arms to converge such that their rank remains the same as k𝑘kitalic_k increases at each time step t𝑡titalic_t. In other words, setting k=4𝑘4k=4italic_k = 4 makes the approximated Whittle index have the same action path as the exact Whittle index, leading to the same performance. When k𝑘kitalic_k becomes smaller, the approximation error will be larger and cause more performance loss as shown in Fig. 23.

Algorithm 1 outperforms the myopic policy. Interestingly, the Whittle index policy may have some performance loss in the middle but eventually catches up with the optimal policy as time goes. This is consistent with the conjecture that Whittle index policy is asymptotically optimal as time goes to infinity as the Lagrangian relaxation should not fundamentally alter the state and action paths of the optimal policy for the original problem from the perspective of large deviation theory (Weber and Weiss, 1990). On the contrary, the myopic policy is unable to follow the optimal action path and never catches up! Since the myopic policy only cares about maximizing the immediate reward, its performance for T=1𝑇1T=1italic_T = 1 is optimal (thus better than any other policy) because the state transitions do not matter in this case. Definitely, the myopic policy has the lowest complexity but this advantage is negligible given its significant performance loss and the efficiency of our algorithm compared to the optimal policy as shown in Fig. 6 and Fig. 6.

Rahul et al. (2018), Varun et al. (2018) and Kesav et al. (2019) considered similar models (except for some nuances) and established indexability under much stricter conditions on the system parameters. For example, all the three papers require that the Markov transition probabilities of each arm have differences bounded by 1/3131/31 / 3 while we do not need any restriction on the transition probabilities. Furthermore, the three papers require that the discount factor β𝛽\betaitalic_β is less than 1/3131/31 / 3 while our condition on β𝛽\betaitalic_β is a more relaxed closed-form expression of the system parameters. In terms of the computation of the Whittle index, their algorithm is a direct application of general reinforcement learning while our algorithmic framework is based on the detailed analysis of the value functions with a quick convergence to the exact Whittle index function. In this paper, we also plot the performance of the optimal policy to demonstrate the near-optimality of our algorithm in addition to the comparison with the myopic policy. For homogeneous systems (stochastically identical arms), we show that our algorithm is equivalent to the myopic policy and theoretically prove its optimality under certain conditions. Wang et al. (2018) also considered our model and assumed the optimality of the threshold policy for a single arm while using a very coarse linear approximation to compute the Whittle index function (the key step (a) for the second equality in the proof of Lemma 6 in Wang et al. (2018) is incorrect). In this paper, we rigorously prove the optimality of the threshold policy for a single arm and establish the indexability under certain conditions and subsequently construct an efficient algorithm for computing the Whittle index function with arbitrary precision with its optimality numerically verified in general and formally proved for a class of homogeneous systems.

Authors’ contributions Keqin Liu constructed the proof sketch for each theorem and the main algorithm and contributed to the writing of the paper. Richard Weber outlined the proof strategy for the optimality of the myopic policy in homogeneous systems and contributed to the verification and writing of the paper. Chengzhong Zhang filled out the details of the proofs and conducted the numerical simulations.

D

We cast the proof in the context of Weyl modules for the general linear group G⁢Ln⁢(K)𝐺subscript𝐿𝑛𝐾GL_{n}(K)italic_G italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K ), see Theorem 2.1, and use a classic result of Carter and Lusztig [3] to descend to Specht modules, see Remark 2.2. Our proof is elementary, as it is based on computations with tableaux, and yields an explicit isomorphism on the level of homomorphism spaces of Weyl modules.

Let ν∈∧(n,r)𝜈𝑛𝑟\nu\in\wedge(n,r)italic_ν ∈ ∧ ( italic_n , italic_r ) and μ=(μ1,…,μn)∈∧+(n,r)𝜇subscript𝜇1…subscript𝜇𝑛superscript𝑛𝑟\mu=(\mu_{1},...,\mu_{n})\in\wedge^{+}(n,r)italic_μ = ( italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ ∧ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_n , italic_r ). According to [1], equation (11) of Section 2, a basis of the K𝐾Kitalic_K-vector space

We cast the proof in the context of Weyl modules for the general linear group G⁢Ln⁢(K)𝐺subscript𝐿𝑛𝐾GL_{n}(K)italic_G italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_K ), see Theorem 2.1, and use a classic result of Carter and Lusztig [3] to descend to Specht modules, see Remark 2.2. Our proof is elementary, as it is based on computations with tableaux, and yields an explicit isomorphism on the level of homomorphism spaces of Weyl modules.

In Section 2 we state the main result and in Section 3 we gather the necessary recollections for its proof which is given in Section 4.

Now we apply Lemma 4.6 (2) to each term in the right hand side of (4.5) and (4.6). Substituting in (4.2) and (4.3) we obtain

C

In particular, we see that under quite general conditions (not implying ‘smoothness’, see for example [13]), the asymptotic joint distribution of certain components in the fragment of Rnsubscript𝑅𝑛R_{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is a Boltzmann Poisson distribution, and this may allow us to find the limiting probability that Rnsubscript𝑅𝑛R_{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is connected.

The plan of the rest of the paper is that the results on pendant appearances are proved in Section 2;

1.1 Pendant appearances in the random graph Rnsubscript𝑅𝑛R_{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT

For example, in the special case when 𝒜𝒜\mathcal{A}caligraphic_A is the class ℰSsuperscriptℰ𝑆\mathcal{E}^{S}caligraphic_E start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT of graphs embeddable in a given surface S𝑆Sitalic_S and H𝐻Hitalic_H is a connected planar graph, it is known further that pend⁢(Rn,H)pendsubscript𝑅𝑛𝐻{\rm pend}(R_{n},H)roman_pend ( italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_H ) is asymptotically normally distributed [8, 4]. See Section 6.7.2 of [27] for related results on ‘block-stable’ classes of graphs with weights on the blocks.

Here we give results on pendant appearances in Rnsubscript𝑅𝑛R_{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT which are both more precise and more general than earlier results, see Section 1.1; and we use these results to deduce properties of the fragment of Rnsubscript𝑅𝑛R_{n}italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT (the part of the graph not in the giant component), see Section 1.2.

A

Recently, some spectral methods for PDEs have been extended to neural networks (Fan et al., 2019a, c; Kashinath et al., 2020). In Section 4.4, we build on these works by proposing the Fourier neural operator architecture defined directly in Fourier space with quasi-linear time complexity and state-of-the-art approximation capabilities.

however, it is not as efficient both in terms of computational complexity and the number of parameters required.

However we do not study these other two sources of error in this work. Furthermore we do not study how the number of parameters in our approximation grows as the error tolerance is refined. Generally, this growth may be super-exponential as shown in (Kovachki et al., 2021). However, for certain classes of operators and related approximation methods, it is possible to beat the curse of dimensionality;

The approximation theory of operators is much more complex and challenging compared to that of functions over Euclidean spaces. It is important to study the class of neural operators with respect to their architecture – what spaces the true solution operators lie in, and which classes of PDEs the neural operator approximate efficiently. We leave these as exciting, but

We note that, while the application of our methodology is based on having point-wise evaluations of the function, it is not limited by it. One may, for example, represent a function numerically as a finite set of truncated basis coefficients. Invariance of the representation would then be with respect to the size of this set. Our methodology can, in principle, be modified to accommodate this scenario through a suitably chosen architecture. We do not pursue this direction in the current work. From the construction of neural operators, when the input and output functions are evaluated on fixed grids, the architecture of neural operators on these fixed grids coincide with the class of neural networks.

B

Typically, the distributed control protocol that is designed for a platoon must guarantee internal stability and string stability of the platoon system, e.g., see [3, 4, 5, 6]. Internal stability refers to a vehicle matching its speed to the speed of the vehicle in front of it, or the platoon leader, while keeping a desired inter-vehicle distance [7]. String stability enables vehicle platoons to attenuate the error signals as they propagate down the vehicle string [8]. To be more precise, if the system is string stable, then: (i) vehicles can attain and keep the desired configuration; (ii) the effects of disturbances are attenuated along the string [9]. Historically, the work on string stability can be traced back to [10] and to the California PATH program [11].

The CTHS policy, on the other hand, is another way to ensure string stability under the PF topology. In this policy, each vehicle regulates its desired distance from its predecessor by using a linear function of speed (leader/predecessor speed or its own speed) with a constant time headway as the proportional gain [14]. Apart from the advantages of the CTHS policy, it may compromise the transport throughput, since by using the time headway, the inter-vehicle distance increases as the speed grows. It is therefore desirable to reduce the time headway while guaranteeing string stability [15]. The work in [16] demonstrates that by increasing the number of connected predecessors, as in the multiple-predecessor-following (MPF) topology (an example is shown in Fig. 1), the minimum employable time headway will be decreased.

we consider the amplification of spacing errors since spacing errors directly affect the platoon safety. Define the predecessor-follower spacing error as

String stability is known to be dramatically influenced by the spacing policy, which is one of the main components of a platoon that determines the desired inter-vehicle distance. There are two main spacing policies: constant distance spacing policy (CDS) and constant time headway spacing (CTHS) policy. Information flow topology (IFT) is another important component that may impact string stability.

Typically, the distributed control protocol that is designed for a platoon must guarantee internal stability and string stability of the platoon system, e.g., see [3, 4, 5, 6]. Internal stability refers to a vehicle matching its speed to the speed of the vehicle in front of it, or the platoon leader, while keeping a desired inter-vehicle distance [7]. String stability enables vehicle platoons to attenuate the error signals as they propagate down the vehicle string [8]. To be more precise, if the system is string stable, then: (i) vehicles can attain and keep the desired configuration; (ii) the effects of disturbances are attenuated along the string [9]. Historically, the work on string stability can be traced back to [10] and to the California PATH program [11].

C

In 1988 J. Franks defined what seems to be a deeper form of linking, for which one requires from a periodic orbit to have a non-zero rotation number around a fixed point. Franks asked whether every periodic orbit of an orientation preserving homeomorphism is linked in that sense to a fixed point [3]. This difficult open problem was resolved in the affirmative by P. Le Calvez in 2006 [17]. It seems that Le Calvez’s result cannot be extended to bounded orbits. A result that seems related to both forms of linking is proven in [12], where a way of locating fixed points in proximity of recurrent orbits is given, by the means of topological hulls of unions of arcs, connecting a finite number of points of an ϵitalic-ϵ\epsilonitalic_ϵ-periodic orbit; see also [16] and [20]. Sufficient conditions for the nonremovability of collections of periodic points under isotopy relative to a general compact invariant set can be found in [4].

Note that if cl⁡(𝒪1)∩𝒪2≠∅clsubscript𝒪1subscript𝒪2\operatorname{cl}(\mathcal{O}_{1})\cap\mathcal{O}_{2}\neq\emptysetroman_cl ( caligraphic_O start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∩ caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ ∅ then 𝒪1subscript𝒪1\mathcal{O}_{1}caligraphic_O start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝒪2subscript𝒪2\mathcal{O}_{2}caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are linked in a trivial way, as there is no Jordan curve around 𝒪2subscript𝒪2\mathcal{O}_{2}caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT separating the two sets. Gambaudo showed that for any C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-embedding f𝑓fitalic_f of a disk, and any periodic orbit O𝑂Oitalic_O of f𝑓fitalic_f, there exists a fixed point p𝑝pitalic_p linked to O𝑂Oitalic_O. In consequence, for the torus flow ϕtsubscriptitalic-ϕ𝑡\phi_{t}italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT suspending f𝑓fitalic_f, the sets {ϕt⁢(O)}≥0subscriptsubscriptitalic-ϕ𝑡𝑂absent0\{\phi_{t}(O)\}_{\geq 0}{ italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_O ) } start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT and {ϕt⁢(p)}≥0subscriptsubscriptitalic-ϕ𝑡𝑝absent0\{\phi_{t}(p)\}_{\geq 0}{ italic_ϕ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_p ) } start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT are linked as knots in 𝕊3superscript𝕊3\mathbb{S}^{3}blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. A similar result was obtained simultanously by B. Kolev in [14], who showed linking of a periodic orbit to a fixed point for orientation preserving C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-diffeomorphisms of ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The result of Gambaudo and Kolev was generalized to orientation reversing homeomorphisms of 𝕊2superscript𝕊2\mathbb{S}^{2}blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT by M. Bonino [1], who showed linking of periodic orbits of period at least 3333, to periodic orbits of least period 2222. Bonino also pointed out that Kolev’s proof, in the orientation preserving case, works in C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT as well, as it is enough to perturb a given homeomorphism slightly, by smoothing it out in a small neighborhood of the periodic orbit and then apply the same proof. In the present paper we improve on the results of Gambaudo and Kolev, by proving that any bounded orbit is linked to a fixed point.

In 1988 J. Franks defined what seems to be a deeper form of linking, for which one requires from a periodic orbit to have a non-zero rotation number around a fixed point. Franks asked whether every periodic orbit of an orientation preserving homeomorphism is linked in that sense to a fixed point [3]. This difficult open problem was resolved in the affirmative by P. Le Calvez in 2006 [17]. It seems that Le Calvez’s result cannot be extended to bounded orbits. A result that seems related to both forms of linking is proven in [12], where a way of locating fixed points in proximity of recurrent orbits is given, by the means of topological hulls of unions of arcs, connecting a finite number of points of an ϵitalic-ϵ\epsilonitalic_ϵ-periodic orbit; see also [16] and [20]. Sufficient conditions for the nonremovability of collections of periodic points under isotopy relative to a general compact invariant set can be found in [4].

I am grateful to Toby Hall for comments that helped to improve the paper, and an anonymous referee who pointed out a necessity to revise the initial proof of the main result. This work was supported in part by the National Science Centre, Poland (NCN), grant no. 2019/34/E/ST1/00237.

Our proof is inspired by M. Brown’s paper [5], the writing of [2], and motivated by [15] where Brown’s approach was used to show existence of periodic points in neighborhoods of adding machines in the plane. Unlike the proofs of results in [1], [11] and [14], our proof does not employ any elements of the Nielsen-Thurston theory, but purely topological arguments concerning covering spaces of open surfaces. We believe that this proof reveals a deeper explanation of the phenomenon described by the result of Gambaudo and Kolev.

C

We choose random data for aj⁢(0)subscript𝑎𝑗0a_{j}(0)italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 0 ) on [0,5]05[0,5][ 0 , 5 ] and for bj⁢(0)subscript𝑏𝑗0b_{j}(0)italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 0 ) on [0,4]04[0,4][ 0 , 4 ]. The numerical results are shown in Figure 3. The evolution of solutions ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and bjsubscript𝑏𝑗b_{j}italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT in Figure 3 (a) and (b) shows that solutions do not remain positive even though the initial data is positive. In contrast to Figure 1 (a) and (b), it takes much longer time for the solutions of (8.40) to converge. We point out that the last modes aNsubscript𝑎𝑁a_{N}italic_a start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT and bNsubscript𝑏𝑁b_{N}italic_b start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT are far apart from other modes after convergence to a steady state; this could be an artifact due to the choice of aN+1subscript𝑎𝑁1a_{N+1}italic_a start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT and bN+1subscript𝑏𝑁1b_{N+1}italic_b start_POSTSUBSCRIPT italic_N + 1 end_POSTSUBSCRIPT. The energy dissipation shown in Figure 3 (d) is also consistent with Figure 3 (a) and (b). It is interesting to notice that the energy spectrum shown in Figure 3 (c) exhibits a scaling law; however, the slope is close to −8.1438.143-\frac{8.14}{3}- divide start_ARG 8.14 end_ARG start_ARG 3 end_ARG which has a big discrepancy with the predicted slop −5353-\frac{5}{3}- divide start_ARG 5 end_ARG start_ARG 3 end_ARG. Many numerical runs with the same parameters show such discrepancy. Further investigation for the model with forward and backward energy cascades will be addressed in future work.

Thus it is plausible to assume εu,j∼εb,jsimilar-tosubscript𝜀𝑢𝑗subscript𝜀𝑏𝑗\varepsilon_{u,j}\sim\varepsilon_{b,j}italic_ε start_POSTSUBSCRIPT italic_u , italic_j end_POSTSUBSCRIPT ∼ italic_ε start_POSTSUBSCRIPT italic_b , italic_j end_POSTSUBSCRIPT. The scaling laws of the magnetic energy spectrum ℰbsubscriptℰ𝑏\mathcal{E}_{b}caligraphic_E start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT and magnetic dissipation wavenumber κdbsuperscriptsubscript𝜅d𝑏\kappa_{\mathrm{d}}^{b}italic_κ start_POSTSUBSCRIPT roman_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT in Conjecture 4.2 are thereby inspired by (4.24) and (4.25) respectively.

M. Dai, M. Hoeller and Q. Peng are partially supported by NSF grants DMS-1815069 and DMS-2009422. X. Zhang is partially supported by NSF grants DMS-1913120 and DMS-2208515.

for j≥0𝑗0j\geq 0italic_j ≥ 0 and a−1=b−1=0subscript𝑎1subscript𝑏10a_{-1}=b_{-1}=0italic_a start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT = italic_b start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT = 0; and another with both forward and backward energy cascades,

In the second setting, we take θ=2𝜃2\theta=2italic_θ = 2 (δ=1𝛿1\delta=1italic_δ = 1) and N=12𝑁12N=12italic_N = 12, and all the other parameters remain unchanged. In this case, the intermittency dimension is smaller and the nonlinearity is stronger. The numerical simulation is shown in Figure 2. The amplitudes of trajectories of ajsubscript𝑎𝑗a_{j}italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and bjsubscript𝑏𝑗b_{j}italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are plotted respectively in Figure 2 (a) and (b).

B

F0⁢ki⁢jsuperscriptsubscript𝐹0𝑘𝑖𝑗\displaystyle{F_{0k}}^{ij}italic_F start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT

=−Fk⁢0i⁢j=−r−2⁢(xi⁢ 1 0 -.25 1δkj−xj⁢ 1 0 -.25 1δki)⁢ψt,absentsuperscriptsubscript𝐹𝑘0𝑖𝑗superscript𝑟2superscript𝑥𝑖superscriptsubscript 1 0 -.25 1δ𝑘𝑗superscript𝑥𝑗superscriptsubscript 1 0 -.25 1δ𝑘𝑖subscript𝜓𝑡\displaystyle=-{F_{k0}}^{ij}=-r^{-2}\big{(}x^{i}\mbox{\hskip 4.44444pt{1 0 -.2%

Aμi⁢j⁢(t,x)=−(xi⁢ 1 0 -.25 1δμj−xj⁢ 1 0 -.25 1δμi)⁢|x|−2⁢ψ⁢(t,|x|)superscriptsubscript𝐴𝜇𝑖𝑗𝑡𝑥superscript𝑥𝑖superscriptsubscript 1 0 -.25 1δ𝜇𝑗superscript𝑥𝑗superscriptsubscript 1 0 -.25 1δ𝜇𝑖superscript𝑥2𝜓𝑡𝑥{A_{\mu}}^{ij}(t,x)=-\big{(}x^{i}{\mbox{\hskip 4.44444pt{1 0 -.25 1}\hbox to0.%

−r−2⁢(xi⁢xk⁢ 1 0 -.25 1δℓj−xi⁢xℓ⁢ 1 0 -.25 1δkj+xj⁢xℓ⁢ 1 0 -.25 1δki−xj⁢xk⁢ 1 0 -.25 1δℓi)⁢(r−1⁢ψr−r−2⁢(2⁢ψ+ψ2)),superscript𝑟2superscript𝑥𝑖subscript𝑥𝑘superscriptsubscript 1 0 -.25 1δℓ𝑗superscript𝑥𝑖subscript𝑥ℓsuperscriptsubscript 1 0 -.25 1δ𝑘𝑗superscript𝑥𝑗subscript𝑥ℓsuperscriptsubscript 1 0 -.25 1δ𝑘𝑖superscript𝑥𝑗subscript𝑥𝑘superscriptsubscript 1 0 -.25 1δℓ𝑖superscript𝑟1subscript𝜓𝑟superscript𝑟22𝜓superscript𝜓2\displaystyle\quad-r^{-2}\big{(}x^{i}x_{k}\mbox{\hskip 4.44444pt{1 0 -.25 1}%

=−r−2⁢( 1 0 -.25 1δki⁢ 1 0 -.25 1δℓj− 1 0 -.25 1δkj⁢ 1 0 -.25 1δℓi)⁢(2⁢ψ+ψ2)absentsuperscript𝑟2superscriptsubscript 1 0 -.25 1δ𝑘𝑖superscriptsubscript 1 0 -.25 1δℓ𝑗superscriptsubscript 1 0 -.25 1δ𝑘𝑗superscriptsubscript 1 0 -.25 1δℓ𝑖2𝜓superscript𝜓2\displaystyle=-r^{-2}\big{(}\mbox{\hskip 4.44444pt{1 0 -.25 1}\hbox to0.0pt{%

A

^{2}],0 = ∫ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT [ roman_Ω ] ∧ [ italic_η ] ∧ [ italic_γ ] = italic_c ⋅ ∫ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT [ roman_Ω ] ∧ [ italic_η start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ,

The above discussions in Sections 3 and 4 are valid for abstract elements on the boundary of ℋ¯¯ℋ\overline{\mathscr{H}}over¯ start_ARG script_H end_ARG, and hence can be identically applied to study Problem 5.1 and give the following results.

We first prove a general result for abstract elements on the boundary of ℋ¯¯ℋ\overline{\mathscr{H}}over¯ start_ARG script_H end_ARG.

(4) We may particularly mention that, given the abstract versions of Hodge index theorem (see [6, 15]), we have some abstract elements on the boundary of ℋ¯¯ℋ\overline{\mathscr{H}}over¯ start_ARG script_H end_ARG.

Example 3.5 (Known elements on the boundary of ℋ¯¯ℋ\overline{\mathscr{H}}over¯ start_ARG script_H end_ARG).

B

Let 𝐚𝐚\mathbf{a}bold_a and 𝐛𝐛\mathbf{b}bold_b be distinct vectors in 𝐑≥0nsubscriptsuperscript𝐑𝑛absent0\mathbf{R}^{n}_{\geq 0}bold_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT.

If P𝑃Pitalic_P and Q𝑄Qitalic_Q are doubly stochastic, then P⁢Q𝑃𝑄PQitalic_P italic_Q is doubly stochastic.

if 𝐚𝐚\mathbf{a}bold_a majorizes 𝐛𝐛\mathbf{b}bold_b, then there is a doubly stochastic matrix P𝑃Pitalic_P such that P⁢(𝐚)=𝐛𝑃𝐚𝐛P(\mathbf{a})=\mathbf{b}italic_P ( bold_a ) = bold_b.

If 𝐛≺𝐚precedes𝐛𝐚\mathbf{b}\prec\mathbf{a}bold_b ≺ bold_a, then there is a doubly stochastic matrix P𝑃Pitalic_P

then there is a doubly stochastic matrix P𝑃Pitalic_P such that 𝐛=P⁢(𝐚)𝐛𝑃𝐚\mathbf{b}=P(\mathbf{a})bold_b = italic_P ( bold_a ).

C

\rho^{-(N+1-2s)}&\text{ as }\rho\to+\infty.\end{dcases}italic_K start_POSTSUBSCRIPT italic_N , - italic_s end_POSTSUBSCRIPT ( italic_ρ ) ∼ { start_ROW start_CELL italic_ρ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL start_CELL as italic_ρ → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL | italic_ρ - 1 | start_POSTSUPERSCRIPT 2 italic_s - 1 end_POSTSUPERSCRIPT end_CELL start_CELL as italic_ρ → 1 , for italic_s ≠ divide start_ARG 1 end_ARG start_ARG 2 end_ARG , end_CELL end_ROW start_ROW start_CELL roman_log divide start_ARG 1 end_ARG start_ARG | italic_ρ - 1 | end_ARG end_CELL start_CELL as italic_ρ → 1 , for italic_s = divide start_ARG 1 end_ARG start_ARG 2 end_ARG , end_CELL end_ROW start_ROW start_CELL italic_ρ start_POSTSUPERSCRIPT - ( italic_N + 1 - 2 italic_s ) end_POSTSUPERSCRIPT end_CELL start_CELL as italic_ρ → + ∞ . end_CELL end_ROW

The error in (5.1) is not necessarily signed. Thus, one needs to argue that if the upper bound for v𝑣vitalic_v (or equivalently for V𝑉Vitalic_V) eventually does not hold, then the error is absorbed in a suitably large region where the ODE argument can be carried out to yield a contradiction.

We remark that while explicit asymptotic constants are not needed for our purposes, they can be obtained from the representation (7.3).

it is remarked in [12, Remark 2.2] that solutions to (6.1) that are locally bounded around the origin do not exist. In 4.3, no boundedness nor any growth condition is assumed. Besides, the equation (for u𝑢uitalic_u) only needs to be satisfied in an exterior domain.

Constants are universal when they depend only on N,k,n𝑁𝑘𝑛N,k,nitalic_N , italic_k , italic_n, s𝑠sitalic_s, ΩΩ\Omegaroman_Ω and ΣΣ\Sigmaroman_Σ. Generic universal constants are denoted by C𝐶Citalic_C (big constants) and c𝑐citalic_c (small constants).

B

We consider an online algorithm similar to (2), but in a setting where: (i) the gradient is corrupted by random errors that follow a sub-Weibull distribution [4]; and, (ii) measurements {𝐲t}t∈ℕ∪{0}subscriptsubscript𝐲𝑡𝑡ℕ0\{{\mathbf{y}}_{t}\}_{t\in\mathbb{N}\cup\{0\}}{ bold_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t ∈ blackboard_N ∪ { 0 } end_POSTSUBSCRIPT are noisy and may not be available at each time t𝑡titalic_t. The latter models processing and communication bottlenecks in the sensing layers of the system (for example, in power grid metering systems and transportation systems). On the other hand, the sub-Weibull model allows us to consider concurrent learning and optimization frameworks where the costs 𝐱↦Ut⁢(𝐱)maps-to𝐱subscript𝑈𝑡𝐱{\mathbf{x}}\mapsto U_{t}({\mathbf{x}})bold_x ↦ italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x ) and 𝐲↦Ct⁢(𝐲)maps-to𝐲subscript𝐶𝑡𝐲{\mathbf{y}}\mapsto C_{t}({\mathbf{y}})bold_y ↦ italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_y ) are learned via Gaussian Process (GP) regression [5], parametric methods [6], non-parametric methods [7], and neural networks [8] from both a set of recorded data and (an infrequent set of) functional evaluations acquired during the execution of the algorithm. In this paper, we particularly highlight the use of GPs [9, 10]. Learning the cost concurrently with the execution of the algorithm finds ample applications in cyber-physical systems with human-in-the-loop [11], where Ut⁢(𝐱)subscript𝑈𝑡𝐱U_{t}({\mathbf{x}})italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x ) models users’ preferences, as well as data-enabled and perception-based optimization [12] where Ct⁢(𝐲)subscript𝐶𝑡𝐲C_{t}({\mathbf{y}})italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_y ) is learned from data.

Although it is not the main focus of this paper, we also acknowledge works on zeroth-order methods (see, e.g., [19, 20]) where gradient errors emerge from single- or multi-point gradient estimation; our framework based on a sub-Weibull model can be applied to derive convergence bounds when the gradient in (2) is estimated via single- or multi-point estimation. Finally, we mention that several convergence results have been derived for classical online algorithms [21, 22], including asynchronous implementations [23]; our results provide extensions to cases with sub-Weibull gradient errors.

We consider an online algorithm similar to (2), but in a setting where: (i) the gradient is corrupted by random errors that follow a sub-Weibull distribution [4]; and, (ii) measurements {𝐲t}t∈ℕ∪{0}subscriptsubscript𝐲𝑡𝑡ℕ0\{{\mathbf{y}}_{t}\}_{t\in\mathbb{N}\cup\{0\}}{ bold_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t ∈ blackboard_N ∪ { 0 } end_POSTSUBSCRIPT are noisy and may not be available at each time t𝑡titalic_t. The latter models processing and communication bottlenecks in the sensing layers of the system (for example, in power grid metering systems and transportation systems). On the other hand, the sub-Weibull model allows us to consider concurrent learning and optimization frameworks where the costs 𝐱↦Ut⁢(𝐱)maps-to𝐱subscript𝑈𝑡𝐱{\mathbf{x}}\mapsto U_{t}({\mathbf{x}})bold_x ↦ italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x ) and 𝐲↦Ct⁢(𝐲)maps-to𝐲subscript𝐶𝑡𝐲{\mathbf{y}}\mapsto C_{t}({\mathbf{y}})bold_y ↦ italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_y ) are learned via Gaussian Process (GP) regression [5], parametric methods [6], non-parametric methods [7], and neural networks [8] from both a set of recorded data and (an infrequent set of) functional evaluations acquired during the execution of the algorithm. In this paper, we particularly highlight the use of GPs [9, 10]. Learning the cost concurrently with the execution of the algorithm finds ample applications in cyber-physical systems with human-in-the-loop [11], where Ut⁢(𝐱)subscript𝑈𝑡𝐱U_{t}({\mathbf{x}})italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_x ) models users’ preferences, as well as data-enabled and perception-based optimization [12] where Ct⁢(𝐲)subscript𝐶𝑡𝐲C_{t}({\mathbf{y}})italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( bold_y ) is learned from data.

The ability of our framework to model various learning settings is grounded on the fact that the sub-Weibull distribution includes sub-Gaussian and sub-exponential errors as sub-cases, as well as random errors whose distribution has a finite support [4, 13].

The parameters ν1,ν2subscript𝜈1subscript𝜈2\nu_{1},\nu_{2}italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT differ by a constant that depends on θ𝜃\thetaitalic_θ; in particular, if property (ii) holds with parameter ν2subscript𝜈2\nu_{2}italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, then property (i) holds with ν1=(2⁢e/θ)θ⁢ν2subscript𝜈1superscript2𝑒𝜃𝜃subscript𝜈2\nu_{1}=\left(2e/\theta\right)^{\theta}\nu_{2}italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( 2 italic_e / italic_θ ) start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Hereafter, we use the short-hand notation X∼subW⁢(θ,ν)similar-to𝑋subW𝜃𝜈X\sim\mathrm{subW}(\theta,\nu)italic_X ∼ roman_subW ( italic_θ , italic_ν ) to indicate that X𝑋Xitalic_X is a sub-Weibull rv according to Definition 1(ii) (i.e., ‖X‖k≤ν⁢kθsubscriptnorm𝑋𝑘𝜈superscript𝑘𝜃\|X\|_{k}\leq\nu k^{\theta}∥ italic_X ∥ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ italic_ν italic_k start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT, ∀k≥1for-all𝑘1\forall\,k\geq 1∀ italic_k ≥ 1). We note that the sub-Weibull class includes sub-Gaussian and sub-exponential rvs as sub-cases; in particular, if θ=1/2𝜃12\theta=1/2italic_θ = 1 / 2 and θ=1𝜃1\theta=1italic_θ = 1 we have sub-Gaussian and sub-exponential rvs, respectively. Furthermore, if a rv has a distribution with finite support, it belongs to the sub-Gaussian class (by Hoeffding’s inequality [13, Theorem 2.2.6]) and, thus, to the sub-Weibull class.

C

Assume the Generalized Riemann Hypothesis for Hecke L𝐿Litalic_L-functions over ℚ⁢(ω)ℚ𝜔\mathbb{Q}(\omega)blackboard_Q ( italic_ω ).

Let W𝑊Witalic_W be a smooth Schwartz function, compactly supported in (0,∞)0(0,\infty)( 0 , ∞ ). Let ℓ∈ℤℓℤ\ell\in\mathbb{Z}roman_ℓ ∈ blackboard_Z.

Let f𝑓fitalic_f be a smooth 1111-periodic function and W𝑊Witalic_W be a smooth function that is compactly supported in (0,∞)0(0,\infty)( 0 , ∞ ). Then we have

Let W𝑊Witalic_W be a smooth function that is compactly supported in (0,∞)0(0,\infty)( 0 , ∞ ). Then as X→∞→𝑋X\rightarrow\inftyitalic_X → ∞ we have

Let W𝑊Witalic_W be a smooth function compactly supported in (0,∞)0(0,\infty)( 0 , ∞ ). Then as X→∞→𝑋X\rightarrow\inftyitalic_X → ∞ we have

B

\mathbf{z}}f(\mathbf{r}).roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT over¯ start_ARG caligraphic_L end_ARG start_POSTSUPERSCRIPT ( bold_z , italic_n ) end_POSTSUPERSCRIPT italic_f ( bold_r ) = roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_n blackboard_E start_POSTSUBSCRIPT ( bold_r , bold_z ) end_POSTSUBSCRIPT [ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∧ italic_τ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT 1 = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( italic_R start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_Z start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) italic_d italic_s ] = ∑ start_POSTSUBSCRIPT 1 = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = caligraphic_L start_POSTSUPERSCRIPT bold_z end_POSTSUPERSCRIPT italic_f ( bold_r ) .

To finish the proof we only need to show that the convergence given in (3.8) holds in the strong sense, this is achieved by an application of Theorem 1.33 in [12].

We start this section by showing that the process 𝐑(z,r)superscript𝐑𝑧𝑟\mathbf{R}^{(z,r)}bold_R start_POSTSUPERSCRIPT ( italic_z , italic_r ) end_POSTSUPERSCRIPT is well-defined as the unique strong solution to (1.2), the proof of this result is defered to Appendix B.

In the next result we show that the sequence of processes {(𝐑¯(𝐳,n))t≥0:n≥1}conditional-setsubscriptsuperscript¯𝐑𝐳𝑛𝑡0𝑛1\{(\overline{\mathbf{R}}^{(\mathbf{z},n)})_{t\geq 0}:n\geq 1\}{ ( over¯ start_ARG bold_R end_ARG start_POSTSUPERSCRIPT ( bold_z , italic_n ) end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_t ≥ 0 end_POSTSUBSCRIPT : italic_n ≥ 1 } converges weakly in the space 𝔻⁢([0,T],[0,1]2)𝔻0𝑇superscript012\mathbb{D}([0,T],[0,1]^{2})blackboard_D ( [ 0 , italic_T ] , [ 0 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) for any T>0𝑇0T>0italic_T > 0, to the process 𝐑(𝐳,𝐫)superscript𝐑𝐳𝐫\mathbf{R}^{(\mathbf{z},\mathbf{r})}bold_R start_POSTSUPERSCRIPT ( bold_z , bold_r ) end_POSTSUPERSCRIPT, which is the unique solution to the stochastic differential equation given in (1.2). By this convergence and the previous construction we can understand the process 𝐑(𝐳,𝐫)superscript𝐑𝐳𝐫\mathbf{R}^{(\mathbf{z},\mathbf{r})}bold_R start_POSTSUPERSCRIPT ( bold_z , bold_r ) end_POSTSUPERSCRIPT as describing the evolution of the frequency of individuals of type x in each island but under the assumption that the total size of the population in the i𝑖iitalic_i-th island is constant and given by zi>0subscript𝑧𝑖0z_{i}>0italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT > 0 for each i=1,2𝑖12i=1,2italic_i = 1 , 2.

The main steps to prove that 𝐑(𝐳,𝐫)superscript𝐑𝐳𝐫\mathbf{R}^{(\mathbf{z},\mathbf{r})}bold_R start_POSTSUPERSCRIPT ( bold_z , bold_r ) end_POSTSUPERSCRIPT arises from the relative frequency associated to two independent two-dimensional Feller diffusions with immigration as a limit when the sampling times become dense in the culling procedure are: first we describe the dynamics of the relative frequency process of type-x individuals (when the total size of each island is allowed to vary in time) as the solution to a martingale problem; we next show that the SDE associated to the process 𝐑(𝐳,𝐫)superscript𝐑𝐳𝐫\mathbf{R}^{(\mathbf{z},\mathbf{r})}bold_R start_POSTSUPERSCRIPT ( bold_z , bold_r ) end_POSTSUPERSCRIPT has a unique strong solution, which is a Feller process. The previous steps allow us to reduce the convergence of the culling procedure to a mere convergence of infinitesimal generators, which is the third step.

A

After this work was completed, we learnt that Schulz very recently proved a partial version of our Theorem A in his Ph.D. thesis [Sch21]. His result establishes Theorem A under a technical extra assumption: the existence of finitely many simple closed geodesics γ1,…,γnsubscript𝛾1…subscript𝛾𝑛\gamma_{1},...,\gamma_{n}italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT such that every γisubscript𝛾𝑖\gamma_{i}italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT intersects another γjsubscript𝛾𝑗\gamma_{j}italic_γ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, the complement M∖(γ1∪…∪γn)𝑀subscript𝛾1…subscript𝛾𝑛M\setminus(\gamma_{1}\cup...\cup\gamma_{n})italic_M ∖ ( italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ … ∪ italic_γ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is simply connected, and any other contractible closed geodesic intersects at least one γisubscript𝛾𝑖\gamma_{i}italic_γ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. These assumptions guarantee the existence of a suitable system of surfaces of section of annulus type for the geodesic flow.

We now consider a broken book decomposition of our closed contact manifold (N,λ)𝑁𝜆(N,\lambda)( italic_N , italic_λ ), as introduced in Section 2.3. Since the basic set ΛΛ\Lambdaroman_Λ contains infinitely many closed Reeb orbits, there exists a page of the broken book Σ⊂NΣ𝑁\Sigma\subset Nroman_Σ ⊂ italic_N such that

In Section 2, after introducing some background on Reeb dynamics and broken book decompositions, we prove two technical statements (Propositions 2.2 and 2.1) concerning the diameter of the images of suitable paths under the arrival map between two pages of a broken book. In Section 3, we briefly recall the needed background from hyperbolic dynamics, and we prove a technical statement (Lemma 3.7) concerning heteroclinic rectangles of suitable Reeb flows. Section 4 is devoted to the proofs of our main theorems: in Section 4.1 we prove Theorem D; in Section 4.2 we prove Theorem A, Corollary 1.1, and Theorems B and C.

The stable and unstable laminations of a hyperbolic invariant set might not be closed subspaces of N𝑁Nitalic_N. In this subsection, we essentially show that they are locally closed at least near the closure of a homoclinic class ΛΛ\Lambdaroman_Λ containing infinitely many closed orbits. We do not know if such a statement holds in the general setting of this section. Our proof will be valid in the contact setting, and will employ a broken book decomposition as in Section 2.3.

We begin by recalling some elements from hyperbolic dynamics, which generalize those already introduced in Section 2.2 for periodic orbits.

B

In this paper, we formulated and analyzed a variant of the classical 𝖯𝖣𝖲𝖯𝖣𝖲\mathsf{PDS}sansserif_PDS problem, where one can only observe a small part of the graph using non-adaptive edge queries. This problem is relevant, for example, when access to the edges (connections) between vertices (individuals) may be scarce due to privacy concerns. For this model, we derived the number of queries necessary and sufficient for detecting the presence of the planted subgraph, up to a constant factor. For the special case of planted cliques, our results are completely tight. This work also has left number of specific problems open, including the following:

[ACV14, BI13, VAC+15, CX16, Mon15, CC18, HWX16]. Nonetheless, the best known algorithms exhibit a peculiar phenomenon: there appears to be a statistical-computational gap between the minimum k𝑘kitalic_k at which this task can be solved and the minimum k𝑘kitalic_k at which it can be solved in polynomial-time. Tight statistical-computational bounds for several parameter regimes of the 𝖯𝖣𝖲𝖯𝖣𝖲\mathsf{PDS}sansserif_PDS were recently established through average-case reductions from the planted clique conjecture [MW15, HWX15, BBH18]. The regimes in which these problems are information-theoretically impossible, statistically possible but computational hard, and admit polynomial-time algorithms appear to have a common structure.

It would be quite interesting to provide any concrete evidence for our conjectured statistical-computational gap either by means of an average-case reduction from the planted clique problem, or failure of classes of powerful algorithms (such as, sum-of-squares hierarchy, low-degree polynomials, etc.), below the computational barrier.

The statistical and computational barriers of the problem in Definition 1 depend on the parameters (n,k,p,q)𝑛𝑘𝑝𝑞(n,k,p,q)( italic_n , italic_k , italic_p , italic_q ). Roughly speaking, if the planted subgraph size k𝑘kitalic_k decreases, or if the “distance” between the densities p𝑝pitalic_p and q𝑞qitalic_q decrease, the distributions under the null and alternative hypotheses become less distinguishable. The statistical limits (i.e., necessary and sufficient conditions) for detecting planted dense subgraphs, without any constraints on the computational complexity, were established in [ACV14, VAC+15]. Interestingly, in the same papers it was observed that state-of-the-art low-complexity algorithms are highly suboptimal. This raised the intriguing question of whether those gaps between the amount of data needed by all computationally efficient algorithms and what is needed for statistically optimal algorithms is inherent. According, quite recently [HWX15, BBH18, BBH19], tight statistical-computational gaps for several parameter regimes of 𝖯𝖣𝖲𝖯𝖣𝖲\mathsf{PDS}sansserif_PDS were established through average-case reductions from the planted clique conjecture (see, Conjecture 1 below).

As can be noticed from the second and third items of Theorem 1, there is a significant gap between the query complexity of the optimal algorithm and that of the computationally efficient one. This observation raises the following intriguing question: what is the sharp condition on (n,k,p,q,𝖰)𝑛𝑘𝑝𝑞𝖰(n,k,p,q,\mathsf{Q})( italic_n , italic_k , italic_p , italic_q , sansserif_Q ) under which the problem admits a computationally efficient test with vanishing risk, and conversely, without which no algorithm can detect the planted dense subgraph reliably in polynomial-time? The gap observed in our problem is common to many contemporary problems in high-dimensional statistics studied over the last few years. Indeed, recently, there has been a success in developing a rigorous notion of what can and cannot be achieved by efficient algorithms. Specifically, a line of work initiated in [BR13] has aimed to explain these statistical-computational gaps by reducing from conjecturally hard average-case problems in computer science, most notably, the planted clique problem, conjectured to be computationally hard in the regime k=o⁢(n)𝑘𝑜𝑛k=o(\sqrt{n})italic_k = italic_o ( square-root start_ARG italic_n end_ARG ). Accordingly, such reductions from planted clique were established to prove tight statistical-computational gaps for a wide verity of detection and recovery problems, e.g., [BR13, MW15, CLR17, HWX15, WBP16, WBS16, GMZ17, BBH18, BBH19, WX20, BB20].

B

As well as introducing the circular systems 𝖢𝖡𝖢𝖡\mathsf{C}\mathsf{B}sansserif_CB and 𝖢𝖭𝖡𝖢𝖭𝖡\mathsf{C}\mathsf{N}\mathsf{B}sansserif_CNB just mentioned, we also develop novel function algebras for 𝐅𝐏𝐓𝐈𝐌𝐄𝐅𝐏𝐓𝐈𝐌𝐄\mathbf{F}\mathbf{PTIME}bold_FPTIME and 𝐅𝐄𝐋𝐄𝐌𝐄𝐍𝐓𝐀𝐑𝐘𝐅𝐄𝐋𝐄𝐌𝐄𝐍𝐓𝐀𝐑𝐘\mathbf{F}\mathbf{ELEMENTARY}bold_FELEMENTARY that allow us to prove the aforementioned complexity characterisations via a ‘sandwich’ technique, cf. Figure 1.

It will be convenient for us to work with generalisations of 𝖡𝖡\mathsf{B}sansserif_B including oracles.

Note that Safe Nested Recursion also admits variants that are not morally ‘nested’ but rather use a form of ‘composition during recursion’:

We give a relativised formulation of 𝖡𝖡\mathsf{B}sansserif_B, i.e. with oracles, that allows us to define a form of safe nested recursion.

First, starting from the analysis of Example 24 and the subsequent system 𝖢𝖭𝖡𝖢𝖭𝖡\mathsf{C}\mathsf{N}\mathsf{B}sansserif_CNB, we shall define a version of 𝖡𝖡\mathsf{B}sansserif_B with safe nested recursion, called 𝖭𝖡𝖭𝖡\mathsf{N}\mathsf{B}sansserif_NB.

C

In this section, we analyze quantization strategies for classical and IF-TEM sampling schemes with FRI signals.

In particular, we show that, in contrast to the classical method, for a specific selection of IF-TEM parameters and fixing the ratio between b𝑏bitalic_b and c𝑐citalic_c, increasing the frequency or energy of the signal increases the quantizer’s resolution [1].

For FRI signals, the quantization step ΔIF-TEMsubscriptΔIF-TEM\Delta_{\text{IF-TEM}}roman_Δ start_POSTSUBSCRIPT IF-TEM end_POSTSUBSCRIPT decreases as the number of input pulses L𝐿Litalic_L increases.

Second, when increasing the number of pulses L𝐿Litalic_L, or raising the rate of innovation, the MSE is further decreased.

We show that as the number of pulses L𝐿Litalic_L increases for FRI signals, the dynamic range of each sample decreases. We, therefore, suggest increasing the resolution of the quantizer as a function of L𝐿Litalic_L.

D

Starting from De Giorgi [De ̵54, De ̵55], who managed to show by using Steiner symmetrization and the compactness theorem of sets of finite perimeter that Euclidean balls are the only isoperimetric sets (global minimizers) among sets of finite perimeter, mathematicians have been working in various context of minimizers to study the isoperimetric problem for decades. Such problem is already found to be very subtle in the context of local minimizers, due to the lack of regularity in the higher dimensional situation (and hence the classical moving plane method fails to be applicable), see [SZ18] for an example of local volume-constrained perimeter minimizer admitting singularities. Despite these obstacles, very recently, Delgadino-Maggi [DM19] solved the very important open problem: the characterization of critical points of the Euclidean isoperimetric problem among sets of finite perimeter. In the weakest assumption (critical points of Euclidean isoperimetric problem), they obtained the following (see also [DKS20] for the anisotropic version, which is solved by using a completely different method with [DM19]).

Starting from De Giorgi [De ̵54, De ̵55], who managed to show by using Steiner symmetrization and the compactness theorem of sets of finite perimeter that Euclidean balls are the only isoperimetric sets (global minimizers) among sets of finite perimeter, mathematicians have been working in various context of minimizers to study the isoperimetric problem for decades. Such problem is already found to be very subtle in the context of local minimizers, due to the lack of regularity in the higher dimensional situation (and hence the classical moving plane method fails to be applicable), see [SZ18] for an example of local volume-constrained perimeter minimizer admitting singularities. Despite these obstacles, very recently, Delgadino-Maggi [DM19] solved the very important open problem: the characterization of critical points of the Euclidean isoperimetric problem among sets of finite perimeter. In the weakest assumption (critical points of Euclidean isoperimetric problem), they obtained the following (see also [DKS20] for the anisotropic version, which is solved by using a completely different method with [DM19]).

It is worth mentioning that aforementioned works on the isoperimetric problems are contextualized in the Euclidean space. In view of [Rei80, Ros87, MR91], a natural question arises: is there any characterization of geodesic balls as the only critical points in the isoperimetric problem that is brought up in space forms?

Among sets of finite perimeter and finite volume, finite unions of balls with equal radii are the unique critical points of the Euclidean isoperimetric problem.

and the Alexandrov theorem for critical points of isoperimeteric problem among sets of finite perimeter Theorem 1.1 (see in particular [DM19, Theorem1, step4]),

C

\end{array}\right\},\leavevmode\nobreak\ \leavevmode\nobreak\ X\in\mathcal{X}.italic_ρ ( italic_X ) = roman_min { italic_ψ ( italic_X ) | start_ARRAY start_ROW start_CELL italic_ψ is a normalized, quasi-convex and end_CELL end_ROW start_ROW start_CELL cash-subadditive risk measure , italic_ψ ⩾ italic_ρ end_CELL end_ROW end_ARRAY } , italic_X ∈ caligraphic_X .

Let (Ω,ℱ,P)Ωℱ𝑃(\Omega,\mathcal{F},P)( roman_Ω , caligraphic_F , italic_P ) be a nonatomic probability space in this section. We present below the representation result of SSD-consistent cash-subadditive risk measures. For this, we define the Expected Shortfall (ES) at level t∈[0,1]𝑡01t\in[0,1]italic_t ∈ [ 0 , 1 ] as

In the following theorem, we establish a representation for an SSD-consistent cash-subadditive risk measure as the lower envelope of some family of law-invariant, quasi-convex and cash-subadditive risk measures.

In this section, we present a representation result, Theorem 4.1, of general cash-subadditive risk measures, which illustrates that a cash-subadditive risk measure is the lower envelope of a family of quasi-convex cash-subadditive risk measures.

ℳℳ\mathcal{M}caligraphic_M represent the subset of ℳfsubscriptℳ𝑓\mathcal{M}_{f}caligraphic_M start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT consisting of all its countably additive elements, i.e., probability measures. Let 𝒳=L∞⁢(Ω,ℱ,P)𝒳superscript𝐿Ωℱ𝑃\mathcal{X}=L^{\infty}(\Omega,\mathcal{F},P)caligraphic_X = italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω , caligraphic_F , italic_P ) be the set of all essentially bounded random variables on (Ω,ℱ,P)Ωℱ𝑃(\Omega,\mathcal{F},P)( roman_Ω , caligraphic_F , italic_P ), where P𝑃Pitalic_P-a.s. equal random variables are treated as identical.222As such, equalities and inequalities should be understood in a P𝑃Pitalic_P-a.s. sense.

A

An obvious relation between g⁢(n)𝑔𝑛g(n)italic_g ( italic_n ) and gk⁢(n)subscript𝑔𝑘𝑛g_{k}(n)italic_g start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_n ) is g⁢(n)=maxk⁡gk⁢(n)𝑔𝑛subscript𝑘subscript𝑔𝑘𝑛g(n)=\max_{k}g_{k}(n)italic_g ( italic_n ) = roman_max start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_n ).

In order to prove the upper bound in Theorem 1 and the limit λ𝜆\lambdaitalic_λ at the same time, we use the following quantity

The limit was actually first proved in [4] by the notion of “linear pattern” (and will get proved again in Section 3 with another proof). Note that the limit λ𝜆\lambdaitalic_λ actually depends on the map and the vector of the considered system, but we do not denote it explicitly by λ∗,ssubscript𝜆𝑠\lambda_{*,s}italic_λ start_POSTSUBSCRIPT ∗ , italic_s end_POSTSUBSCRIPT as they are known from context.

The growth rate λ𝜆\lambdaitalic_λ for s𝑠sitalic_s and ∗*∗ in (3) is the smallest λ0subscript𝜆0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT so that all the vectors obtained by combining any number of instances of s/λ0𝑠subscript𝜆0s/\lambda_{0}italic_s / italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT using applications of ∗*∗ are bounded.

A pair (∗,s)𝑠(*,s)( ∗ , italic_s ) is called a system and the following limit λ𝜆\lambdaitalic_λ is called the growth rate of the system

D

It is interesting to ask whether Corollary 1.2.9 holds for more general domains Ω⊂ℝ2Ωsuperscriptℝ2\Omega\subset\mathbb{R}^{2}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. One can also ask about extensions to higher dimensions, with lattice polygons in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT replaced by lattice paths in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT.

We end this section by proving Corollary 1.2.9, which claims that (in dimension four) 𝔤~k⁢(XΩ)subscript~𝔤𝑘subscript𝑋Ω\tilde{\mathfrak{g}}_{k}(X_{\Omega})over~ start_ARG fraktur_g end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ) is the minimal length ℓΩ⁢(∂P)subscriptℓΩ𝑃\ell_{\Omega}({\partial}P)roman_ℓ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( ∂ italic_P ) of the boundary ∂P𝑃{\partial}P∂ italic_P of a convex lattice polygon P𝑃Pitalic_P such that ∂P𝑃{\partial}P∂ italic_P contains exactly k+1𝑘1k+1italic_k + 1 lattice points.

Note that Corollary 1.2.9 involves arbitrary lattice points, whereas Theorem 1.2.8 involves only nonnegative ones.

We can assume that C′superscript𝐶′C^{\prime}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT involves at least one symplectization level, since otherwise we must have C′=C0superscript𝐶′subscript𝐶0C^{\prime}=C_{0}italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, whence (B1) holds.

We next refine Lemma 4.2.3 so that the minimization involves only words which are elliptic and satisfy ind⁡(w)=2⁢kind𝑤2𝑘{\operatorname{ind}}(w)=2kroman_ind ( italic_w ) = 2 italic_k (rather than ind⁡(w)≥2⁢kind𝑤2𝑘{\operatorname{ind}}(w)\geq 2kroman_ind ( italic_w ) ≥ 2 italic_k).

B

\rVert_{L^{p_{1}}}\lVert g\rVert_{L^{p_{2}}}\lVert h\rVert_{L^{p_{3}}}∥ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_f , italic_g , italic_h ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≲ ∥ overroman_ˇ start_ARG italic_m end_ARG ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_h ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT

∥Tm⁢(f,g,h)∥Lp≲∥mˇ∥L1⁢∥f∥Lp1⁢∥g∥Lp2⁢∥h∥Lp3less-than-or-similar-tosubscriptdelimited-∥∥subscript𝑇𝑚𝑓𝑔ℎsuperscript𝐿𝑝subscriptdelimited-∥∥ˇ𝑚superscript𝐿1subscriptdelimited-∥∥𝑓superscript𝐿subscript𝑝1subscriptdelimited-∥∥𝑔superscript𝐿subscript𝑝2subscriptdelimited-∥∥ℎsuperscript𝐿subscript𝑝3\lVert T_{m}(f,g,h)\rVert_{L^{p}}\lesssim\lVert\check{m}\rVert_{L^{1}}\lVert f%

Tm⁢(f,g,h)⁢(x)=1(2⁢π)3/2⁢∫mˇ⁢(y,z,w)⁢f⁢(x−y−z)⁢g⁢(x−y)⁢h⁢(x−y−w)⁢𝑑y⁢𝑑z⁢𝑑wsubscript𝑇𝑚𝑓𝑔ℎ𝑥1superscript2𝜋32ˇ𝑚𝑦𝑧𝑤𝑓𝑥𝑦𝑧𝑔𝑥𝑦ℎ𝑥𝑦𝑤differential-d𝑦differential-d𝑧differential-d𝑤T_{m}(f,g,h)(x)=\frac{1}{(2\pi)^{3/2}}\int\check{m}(y,z,w)f(x-y-z)g(x-y)h(x-y-%

Note that Theorem 2.1 remains true (with essentially the same proof) if we weaken the hypothesis mˇ∈L1ˇ𝑚superscript𝐿1\check{m}\in L^{1}overroman_ˇ start_ARG italic_m end_ARG ∈ italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT to mˇ∈ℳˇ𝑚ℳ\check{m}\in\mathcal{M}overroman_ˇ start_ARG italic_m end_ARG ∈ caligraphic_M, where ℳℳ\mathcal{M}caligraphic_M is the space of finite measures. In particular, this generalization covers the motivating example m=1𝑚1m=1italic_m = 1 (which has mˇ=(2⁢π)3/2⁢δ∈ℳˇ𝑚superscript2𝜋32𝛿ℳ\check{m}=(2\pi)^{3/2}\delta\in\mathcal{M}overroman_ˇ start_ARG italic_m end_ARG = ( 2 italic_π ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT italic_δ ∈ caligraphic_M).

∫|z|≥R|mˇj⁢(y,z,w)|⁢𝑑y⁢𝑑z⁢𝑑w≲N⟨2j⁢R⟩−Nsubscriptless-than-or-similar-to𝑁subscript𝑧𝑅subscriptˇ𝑚𝑗𝑦𝑧𝑤differential-d𝑦differential-d𝑧differential-d𝑤superscriptdelimited-⟨⟩superscript2𝑗𝑅𝑁\int_{|z|\geq R}|\check{m}_{j}(y,z,w)|\,dydzdw\lesssim_{N}\langle 2^{j}R%

B

In the proof of Proposition 4.1 and the rest of the paper we use notation and results from [BB21b, §2 and §3].

By the way the topology on Cl⁢(Prim⁢(G))ClPrim𝐺{\rm Cl}({\rm Prim}(G))roman_Cl ( roman_Prim ( italic_G ) ) was defined (cf. [BB21b, Def. 2.1]), we directly obtain that the closure of the singleton set {F0}subscript𝐹0\{F_{0}\}{ italic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } in Cl⁢(Prim⁢(G))ClPrim𝐺{\rm Cl}({\rm Prim}(G))roman_Cl ( roman_Prim ( italic_G ) ) is

If H𝐻Hitalic_H is a topological group acting continuously on X𝑋Xitalic_X, then (X/H)≈superscript𝑋𝐻(X/H)^{\approx}( italic_X / italic_H ) start_POSTSUPERSCRIPT ≈ end_POSTSUPERSCRIPT is the subspace of Cl⁢(X)Cl𝑋{\rm Cl}(X)roman_Cl ( italic_X ) consisting of closures of orbits of H𝐻Hitalic_H in X𝑋Xitalic_X, that is,

that is, the topology for which a sub-base of closed sets consists of X𝑋Xitalic_X and the sets ↓F:={F′∣F′∈Cl⁢(X),F′⊆F}↓absent𝐹assignconditional-setsuperscript𝐹′formulae-sequencesuperscript𝐹′Cl𝑋superscript𝐹′𝐹\downarrow F:=\{F^{\prime}\mid F^{\prime}\in{\rm Cl}(X),\ F^{\prime}\subseteq F\}↓ italic_F := { italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∣ italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ roman_Cl ( italic_X ) , italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_F },

In particular, for a topological space X𝑋Xitalic_X, we denote by Cl⁢(X)Cl𝑋{\rm Cl}(X)roman_Cl ( italic_X ) the space of all closed subsets of X𝑋Xitalic_X endowed with the upper topology,

D

Finally, all the regularity properties of ⅁⁢𝚫11⅁subscriptsuperscript𝚫11\Game\mathbf{\Delta}^{1}_{1}⅁ bold_Δ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT sets follow from projective determinacy.

In this section we define the main objects of our study, homomorphism graphs, and establish a couple of their properties.

The paper is structured as follows. In Section 2 we collect the most important definitions and theorems that are going to be used. Then, in Section 3 we establish the basic properties of homomorphism graphs and their various modifications. Section 4 contains Marks’ technique’s adaptation to our context, while in Section 5 we prove our main results. We conclude the paper with a couple of remarks in Section 6.

In the end, we added several new applications of our method that cannot be found in [BCG+21].), we had to circumvent several technical hurdles that, rather surprisingly, lead to the main objects that we study in this paper, homomorphism graphs (defined in Section 3).

In this section we apply the theorems proven before to establish our main results. We will choose a target graph using two prominent notions from descriptive set theory: category and Ramsey property.

A

Let Ω⊂BK,κintΩsubscriptsuperscript𝐵int𝐾𝜅\Omega\subset B^{{\rm int}}_{K,\kappa}roman_Ω ⊂ italic_B start_POSTSUPERSCRIPT roman_int end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_K , italic_κ end_POSTSUBSCRIPT and V𝑉Vitalic_V be as in Theorem 2.1. Assume ΣΣ\Sigmaroman_Σ is convex and satisfies the condition (2.1). Then (2.4) holds.

The study of free boundary surfaces or hypersurface has attracted many attention in recent decades. Inspiring results are contained in a series of papers of Fraser and Schoen [4, 5, 6] about minimal hypersurfaces with free boundary in a ball B𝐵Bitalic_B and the first Steklov eigenvalue. Here “free boundary” means the hypersurface intersects ∂B𝐵\partial B∂ italic_B orthogonally. There have been plenty of works about existence [12, 28, 14], regularity [9, 15], stability [29, 27, 19] of free boundary constant mean curvature (or minimal) hypersurfaces in a ball. We refer to a nice survey paper [16, 30] for more details. In this paper we are interested in obtaining new inequalities for weighted Minkowski inequalities with free boundary. We will use Theorem 1.3 to generalize Theorem 1.2 into the setting of domains with partial umbilical free boundary in space forms.

In the end, we follow the same argument of Theorem 2.2 to generalize De Lellis-Topping type almost Schur lemma [3, 1] into free boundary version in space forms.

Proof of Theorem 2.1. Let f𝑓fitalic_f be the solution to the following mixed boundary value problem:

In this section we will introduce the weight functions in space forms for the free boundary setting. First of all, we have the following crucial proposition that we will be used later.

B

Then t⁢(F)𝑡𝐹t(F)italic_t ( italic_F ) is a normal family if and only if its locally uniformly bounded.

We finish this paper with a proof of a result of Vito Antonio Pereira which guarantees the compactness of all leaves in a compact Kähler manifold provided that it is known that at least ONE of the leaves is compact and has finite holonomy (see [8][Theorem 1]).

Kähler manifolds with all leaves compact, also we prove that a holomorphic foliation in a compact Kähler manifold with at least one compact leaf with finite holonomy, then all of its leaves are compact.

Note that 𝒰𝒰\mathcal{U}caligraphic_U is the union of all saturated open sets with all its leaves compact and with finite holonomy,

Let M𝑀Mitalic_M be a compact connected Kähler manifold of dimension n𝑛nitalic_n and 𝔉𝔉\mathfrak{F}fraktur_F be a holomorphic foliation of codimension q<n𝑞𝑛q<nitalic_q < italic_n for which there is at least one compact leaf with finite holonomy, then all leaves are compact with finite holonomy or equivalently their volumes are uniformly bounded and the leaf space is Hausdorff.

B

2}|n(t\bar{\tau}{\varepsilon}^{-2})-[t{\varepsilon}^{-2}]|| italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) | = italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | ∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n ( italic_t over¯ start_ARG italic_τ end_ARG italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ( italic_F start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∘ italic_ϑ start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - italic_E italic_F start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) | ≤ | italic_J start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_t ) | + 2 ( italic_L over^ start_ARG italic_L end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_n ( italic_t over¯ start_ARG italic_τ end_ARG italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ) - [ italic_t italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ] |

|J2⁢(t)|=ε2⁢|∑k=0n⁢(t⁢τ¯⁢ε−2)(Fi⁢j∘ϑk−E⁢Fi⁢j)|≤|J4⁢(t)|+2⁢(L⁢L^)2⁢ε2⁢|n⁢(t⁢τ¯⁢ε−2)−[t⁢ε−2]|subscript𝐽2𝑡superscript𝜀2superscriptsubscript𝑘0𝑛𝑡¯𝜏superscript𝜀2subscript𝐹𝑖𝑗superscriptitalic-ϑ𝑘𝐸subscript𝐹𝑖𝑗subscript𝐽4𝑡2superscript𝐿^𝐿2superscript𝜀2𝑛𝑡¯𝜏superscript𝜀2delimited-[]𝑡superscript𝜀2|J_{2}(t)|={\varepsilon}^{2}|\sum_{k=0}^{n(t\bar{\tau}{\varepsilon}^{-2})}(F_{%

|Fi⁢j∘ϑk−E(Fi⁢j∘ϑk|ℱk−n,k+n)|\displaystyle|F_{ij}\circ{\vartheta}^{k}-E(F_{ij}\circ{\vartheta}^{k}|{%

andJ3⁢(t)=(t⁢E⁢Fi⁢j−ε2⁢n⁢(t⁢τ¯⁢ε−2)⁢E⁢Fi⁢j).andsubscript𝐽3𝑡𝑡𝐸subscript𝐹𝑖𝑗superscript𝜀2𝑛𝑡¯𝜏superscript𝜀2𝐸subscript𝐹𝑖𝑗\mbox{and}\quad J_{3}(t)=(tEF_{ij}-{\varepsilon}^{2}n(t\bar{\tau}{\varepsilon}%

J4⁢(t)=ε2⁢∑k=0[t⁢ε−2](Fi⁢j∘ϑk−E⁢Fi⁢j).subscript𝐽4𝑡superscript𝜀2superscriptsubscript𝑘0delimited-[]𝑡superscript𝜀2subscript𝐹𝑖𝑗superscriptitalic-ϑ𝑘𝐸subscript𝐹𝑖𝑗J_{4}(t)={\varepsilon}^{2}\sum_{k=0}^{[t{\varepsilon}^{-2}]}(F_{ij}\circ{%

D

If each player’s optimization problem is continuous and convex, equilibrium programming methods can often determine a Nash equilibrium by:

On the one hand, these reformulations require restrictive convexity assumptions on the players’ optimization problems and may not otherwise guarantee convergence. Besides a few exceptions (e.g., Sagratella [48]), to date, the majority of equilibrium programming methods require convexity and, otherwise, develop weaker concepts of equilibrium, for instance, quasi Nash equilibria [43, 40].

. Finally, our algorithm does not require the players to have a bounded set of strategies. On the contrary, it supports unbounded strategy sets and can also certify the non-existence of equilibria. Our primary assumption is that the convex hull of each player’s feasible set and its payoff are polyhedral and separable, respectively.

Historically, convexity played a central role in shedding light on the existence and computation of Nash equilibria, e.g., see Facchinei and Pang [27], Daskalakis [18] and the references therein.

If each player’s optimization problem is continuous and convex, equilibrium programming methods can often determine a Nash equilibrium by:

A

\sigma_{(Q/G)\times(Q/G)})( ( italic_Q / italic_G ) × ( italic_Q / italic_G ) , italic_σ start_POSTSUBSCRIPT ( italic_Q / italic_G ) × ( italic_Q / italic_G ) end_POSTSUBSCRIPT ).

Let 𝒰⊂Q×Q𝒰𝑄𝑄\mathcal{U}\subset Q\times Qcaligraphic_U ⊂ italic_Q × italic_Q be of D𝐷Ditalic_D-type; we define

Let 𝒰⊂Q×Q𝒰𝑄𝑄\mathcal{U}\subset Q\times Qcaligraphic_U ⊂ italic_Q × italic_Q be of D𝐷Ditalic_D-type. A

Let 𝒰⊂Q×Q𝒰𝑄𝑄\mathcal{U}\subset Q\times Qcaligraphic_U ⊂ italic_Q × italic_Q be of D𝐷Ditalic_D-type. A smooth

Let 𝒰⊂Q×Q𝒰𝑄𝑄\mathcal{U}\subset Q\times Qcaligraphic_U ⊂ italic_Q × italic_Q be of D𝐷Ditalic_D-type and

D

An instance is described by a directed graph G=(V,E)𝐺𝑉𝐸G=(V,E)italic_G = ( italic_V , italic_E ) with a source s𝑠sitalic_s and sink t𝑡titalic_t, where each arc is equipped with a transit time τe≥0subscript𝜏𝑒0\tau_{e}\geq 0italic_τ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ≥ 0 and a capacity νe>0subscript𝜈𝑒0\nu_{e}>0italic_ν start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT > 0. At s𝑠sitalic_s we have a constant network inflow rate of u0>0subscript𝑢00u_{0}>0italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0, which begins at time 00.

For technical convenience, we will follow previous works and assume that G𝐺Gitalic_G has no directed cycle consisting of arcs with transit time 00.

(i) for every node v∈V𝑣𝑉v\in Vitalic_v ∈ italic_V there is an s𝑠sitalic_s-v𝑣vitalic_v-path in E′superscript𝐸′E^{\prime}italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT,

Since ℓs′=λs=1subscriptsuperscriptℓ′𝑠subscript𝜆𝑠1\ell^{\prime}_{s}=\lambda_{s}=1roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_λ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = 1, and every node is reachable from s𝑠sitalic_s via an active path, ℓv′≥λvsubscriptsuperscriptℓ′𝑣subscript𝜆𝑣\ell^{\prime}_{v}\geq\lambda_{v}roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≥ italic_λ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT for all v𝑣vitalic_v.

We may assume that every node in G𝐺Gitalic_G is both reachable from s𝑠sitalic_s, and can reach t𝑡titalic_t.

D

ℤ/6⁢ℤ⊕ℤdirect-sumℤ6ℤℤ\mathbb{Z}/6\mathbb{Z}\oplus\mathbb{Z}blackboard_Z / 6 blackboard_Z ⊕ blackboard_Z

and multiplying by −11-1- 1 if necessary, we get x4−y4=w2superscript𝑥4superscript𝑦4superscript𝑤2x^{4}-y^{4}=w^{2}italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT = italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for some w∈𝒪K.𝑤subscript𝒪𝐾w\in\mathcal{O}_{K}.italic_w ∈ caligraphic_O start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT . Note that x4,y4,x4−y4≠0superscript𝑥4superscript𝑦4superscript𝑥4superscript𝑦40x^{4},y^{4},x^{4}-y^{4}\neq 0italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ≠ 0 since E𝐸Eitalic_E is smooth and so x,y,w≠0𝑥𝑦𝑤0x,y,w\neq 0italic_x , italic_y , italic_w ≠ 0. By [20, Proposition 11], there is no nontrivial solution to x4−y4=w2superscript𝑥4superscript𝑦4superscript𝑤2x^{4}-y^{4}=w^{2}italic_x start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_y start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT = italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT except D=−7𝐷7D=-7italic_D = - 7 (see Table 1).

Model for E⁢(α,β),𝐸𝛼𝛽E(\alpha,\beta),italic_E ( italic_α , italic_β ) ,  w=−7𝑤7w=\sqrt{-7}italic_w = square-root start_ARG - 7 end_ARG

[2,−6,1],[1/2⁢(−w+1),1/2⁢(w−3),1]26112𝑤112𝑤31[2,-6,1],[1/2(-w+1),1/2(w-3),1][ 2 , - 6 , 1 ] , [ 1 / 2 ( - italic_w + 1 ) , 1 / 2 ( italic_w - 3 ) , 1 ]

[−2398489/2350089,1/3602686437⁢(110531740⁢w+74197200),1]2398489235008913602686437110531740𝑤741972001[-2398489/2350089,1/3602686437(110531740w+74197200),1][ - 2398489 / 2350089 , 1 / 3602686437 ( 110531740 italic_w + 74197200 ) , 1 ]

D

Our first proof in Section 1 will describe the join and meet operations in the acyclic reorientation lattice of a vertebrate directed acyclic graph.

Our second proof in Section 3.4 will show that the acyclic reorientation lattice of a skeletal directed acyclic graph can be obtained from the acyclic reorientation lattice of its transitive reduction by a sequence of interval doublings in the sense of [Day94].

Our first proof in Section 1 will describe the join and meet operations in the acyclic reorientation lattice of a vertebrate directed acyclic graph.

Our first proof in Section 3.3 will enable us to describe the canonical join and meet representations in the acyclic reorientation lattice of a skeletal directed acyclic graph.

Our second proof in Section 3.4 will show that the acyclic reorientation lattice of a vertebrate directed acyclic graph can be obtained from the acyclic reorientation lattice of its transitive reduction by a sequence of convex doublings in the sense of [Day94].

D

)=\left\|u\right\|_{(2)}\,\ \left(u\in\mathcal{L}^{2}(\mathscr{H})\right).roman_Π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( script_H ) = caligraphic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( script_H ) and italic_π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) = ∥ italic_u ∥ start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT ( italic_u ∈ caligraphic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( script_H ) ) .

A map T:ℰ→ℱ:𝑇→ℰℱT:\mathscr{E}\rightarrow\mathscr{F}italic_T : script_E → script_F is said to be adjointable if there exists a map T∗:ℱ→ℰ:superscript𝑇→ℱℰT^{*}:\mathscr{F}\rightarrow\mathscr{E}italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : script_F → script_E such that ⟨T⁢x,y⟩=⟨x,T∗⁢y⟩𝑇𝑥𝑦𝑥superscript𝑇𝑦\left\langle Tx,y\right\rangle=\left\langle x,T^{*}y\right\rangle⟨ italic_T italic_x , italic_y ⟩ = ⟨ italic_x , italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_y ⟩ for all x∈ℰ𝑥ℰx\in\mathscr{E}italic_x ∈ script_E and y∈ℱ𝑦ℱy\in\mathscr{F}italic_y ∈ script_F. The set of all adjointable maps from ℰℰ\mathscr{E}script_E into ℱℱ\mathscr{F}script_F is denoted by ℒ⁢(ℰ,ℱ)ℒℰℱ\mathcal{L}(\mathscr{E},\mathscr{F})caligraphic_L ( script_E , script_F ), and we write ℒ⁢(ℰ)ℒℰ\mathcal{L}(\mathscr{E})caligraphic_L ( script_E )

A closed submodule ℱℱ\mathscr{F}script_F of ℰℰ\mathscr{E}script_E is said to be orthogonally complemented if ℰ=ℱ⊕ℱ⟂ℰdirect-sumℱsuperscriptℱperpendicular-to\mathscr{E}=\mathscr{F}\oplus\mathscr{F}^{\perp}script_E = script_F ⊕ script_F start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT, where ℱ⟂={y∈ℰ:⟨x,y⟩=0,x∈ℱ}superscriptℱperpendicular-toconditional-set𝑦ℰformulae-sequence𝑥𝑦0𝑥ℱ\mathscr{F}^{\perp}=\left\{y\in\mathscr{E}:\left\langle x,y\right\rangle=0,x%

((ℱn,∥⋅∥n):n∈ℕ)\left(\left({\mathscr{F}}^{n},\left\|\cdot\right\|_{n}\right):n\in\mathbb{N}\right)( ( script_F start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , ∥ ⋅ ∥ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) : italic_n ∈ blackboard_N ) are power-normed spaces. Then the collection of multi-bounded adjointable operators from ℰℰ\mathscr{E}script_E to ℱℱ\mathscr{F}script_F is denoted by ℳ⁢ℒ⁢(ℰ,ℱ)ℳℒℰℱ\mathcal{ML}(\mathscr{E},\mathscr{F})caligraphic_M caligraphic_L ( script_E , script_F ). Since ‖T‖≤‖T‖m⁢bnorm𝑇subscriptnorm𝑇𝑚𝑏\left\|T\right\|\leq\left\|T\right\|_{mb}∥ italic_T ∥ ≤ ∥ italic_T ∥ start_POSTSUBSCRIPT italic_m italic_b end_POSTSUBSCRIPT for each T∈ℒ⁢(ℰ,ℱ)𝑇ℒℰℱT\in\mathcal{L}(\mathscr{E},\mathscr{F})italic_T ∈ caligraphic_L ( script_E , script_F ) and ℳ⁢ℒ⁢(ℰ,ℱ)=ℒ⁢(ℰ,ℱ)∩ℳ⁢(ℰ,ℱ)ℳℒℰℱℒℰℱℳℰℱ\mathcal{ML}(\mathscr{E},\mathscr{F})=\mathcal{L}(\mathscr{E},\mathscr{F})\cap%

This result motivates us to introduce the following class of adjointable operators from ℰℰ\mathscr{E}script_E to ℱℱ\mathscr{F}script_F.

D

Moreover, by F3 we have F⁢(c⁢ln⁡(ω0))≫ω0much-greater-than𝐹𝑐subscript𝜔0subscript𝜔0F\left({c\ln\left({\omega_{0}}\right)}\right)\gg\omega_{0}italic_F ( italic_c roman_ln ( italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ) ≫ italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, so we conclude that Δ02≫ω0much-greater-thansuperscriptsubscriptΔ02subscript𝜔0\Delta_{0}^{2}\gg\omega_{0}roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≫ italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,

In the proof of Proposition 2.13, for simplicity we will allow functions to take the values ±∞plus-or-minus\pm\infty± ∞,

Before proving Proposition 2.13, we prove an auxiliary claim which will be helpful both for this proof and later in the paper.

on whether v𝑣vitalic_v is negated in A𝐴Aitalic_A or not. However, since we do not allow for edges of different types, we must represent this differently.

Recall that in Proposition 2.13, apart from the function F𝐹Fitalic_F and the parameter ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,

A

The second condition also implies that N𝑁Nitalic_N contains no minimal hypersurfaces which are accumulating on a side because such a hypersurface must be a limit of contracting minimal hypersurfaces, see Lemma A.3.

In particular, these following lemmas apply for a weakly Frankel manifold N𝑁Nitalic_N in the case when N𝑁Nitalic_N has no boundary. The case where N𝑁Nitalic_N has nonempty contracting minimal boundary ∂N𝑁\partial N∂ italic_N will be relevant in the later sections.

The first possibility is that there is no minimal hypersurface which is contracting on a side. In this case, we say the manifold (M,g)𝑀𝑔(M,g)( italic_M , italic_g ) is weakly Frankel. Recall that any two minimal hypersurfaces in a Frankel manifold intersect with each other. But in a weakly Frankel manifold, two minimal hypersurfaces can be disjoint, but these hypersurfaces must be connected by some minimal foliation.

A special case of a weakly Frankel manifold when the whole manifold is foliated by closed embedded minimal hypersurfaces. In general, even without the weakly Frankel property, the minimal hypersurfaces satisfy some nice properties.

Let (Nn+1,g)superscript𝑁𝑛1𝑔(N^{n+1},g)( italic_N start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT , italic_g ) be a compact manifold with minimal boundary ∂N𝑁\partial N∂ italic_N. If Σ⊂∂NΣ𝑁\Sigma\subset\partial Nroman_Σ ⊂ ∂ italic_N is a non-contracting boundary component such that

A

E⁢M~e⁢o(𝕍,Δ)subscriptsuperscript~𝐸𝑀𝕍Δ𝑒𝑜\widetilde{EM}^{(\mathbb{V},\Delta)}_{eo}over~ start_ARG italic_E italic_M end_ARG start_POSTSUPERSCRIPT ( blackboard_V , roman_Δ ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e italic_o end_POSTSUBSCRIPT.

In this section, we will show how to give examples of A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-monads and A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-comonads on any K𝐾Kitalic_K-linear Grothendieck category. For basic definitions such as A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebras, A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-coalgebras, A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-modules over A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebras and A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-comodules over A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-coalgebras, we refer the reader to the appendix in Section 10.

We now consider modules over A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-monads and comodules over A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-comonads.

4 Locally finite comodules over A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-comonads

2 A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-monads, A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-comonads and adjoints

C

Property (ii) implies (i) by Corollary 8. For the converse, we prove the contrapositive. Let ℒℒ\mathcal{L}caligraphic_L be a layout that is not one-sided or not sliceable. If ℒℒ\mathcal{L}caligraphic_L is not sliceable, then it is not weakly ARU by Lemma 13, hence not strongly ARU, either. Otherwise ℒℒ\mathcal{L}caligraphic_L is sliceable but not one-sided, and then ℒℒ\mathcal{L}caligraphic_L is not strongly ARU by Lemma 10.

Schramm [31] (see also [20, Chap. 6]) proved that every near-triangulation with an outer 4-cycle is the extended dual graph of a (possibly degenerate or nongeneric) subdivision of a rectangle into squares. The result generalizes to rectangular faces of arbitrary aspect ratios (rather than squares):

In particular, r1,…,r4subscript𝑟1…subscript𝑟4r_{1},\ldots,r_{4}italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_r start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT are squares; denote their side lengths by sisubscript𝑠𝑖s_{i}italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, for i=1,…,4𝑖1…4i=1,\ldots,4italic_i = 1 , … , 4. Note that one side of risubscript𝑟𝑖r_{i}italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT strictly contains a side of ri−1subscript𝑟𝑖1r_{i-1}italic_r start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT for i=1,…,4𝑖1…4i=1,\ldots,4italic_i = 1 , … , 4 (with arithmetic modulo 4). Consequently, s1<s2<s3<s4<s1subscript𝑠1subscript𝑠2subscript𝑠3subscript𝑠4subscript𝑠1s_{1}<s_{2}<s_{3}<s_{4}<s_{1}italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, which is a contradiction.

A slice in a layout ℒℒ\mathcal{L}caligraphic_L is a maximal segment with both endpoints incident to the outer face, and subdivides ℒℒ\mathcal{L}caligraphic_L into two sublayouts. A sliceable layout (also known as slicing floorplan or guillotine rectangulation) is one that can be decomposed into trivial layouts through recursive subdivision with slices; see Fig 1(c) for an example. The recursive subdivision of a layout ℒℒ\mathcal{L}caligraphic_L can be represented by a binary space partition tree (BSP-tree), which is a binary tree where each vertex is associated with a sublayout, which is ℒℒ\mathcal{L}caligraphic_L if the vertex is the root and a rectangular face of ℒℒ\mathcal{L}caligraphic_L if the vertex is a leaf [5]. For a nonleaf vertex, the tree additionally stores a slice; and two sublayouts on each side of the slice are associated with the two children. The root slice is between two opposite sides of the bounding box.

Subdividing a square into squares has fascinated humanity for ages [3, 14, 33]. For example, a perfect square tiling is a tiling with squares with distinct integer side lengths.

D

A potential u∈PSH⁢(X,θ)𝑢PSH𝑋𝜃u\in\mathrm{PSH}(X,\theta)italic_u ∈ roman_PSH ( italic_X , italic_θ ) is model if u=P⁢[u]𝑢𝑃delimited-[]𝑢u=P[u]italic_u = italic_P [ italic_u ], and it is ℐℐ\mathcal{I}caligraphic_I-model if u=P⁢[u]ℐ𝑢𝑃subscriptdelimited-[]𝑢ℐu=P[u]_{\mathcal{I}}italic_u = italic_P [ italic_u ] start_POSTSUBSCRIPT caligraphic_I end_POSTSUBSCRIPT.

For any usc function f:X→[−∞,∞):𝑓→𝑋f:X\to[-\infty,\infty)italic_f : italic_X → [ - ∞ , ∞ ) we define

To prove (44) it is enough to show that for any fixed w∈C∞⁢(X)𝑤superscript𝐶𝑋w\in C^{\infty}(X)italic_w ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_X ) we have

Here usc⁡(⋅)usc⋅\operatorname{usc}(\cdot)roman_usc ( ⋅ ) denotes the least upper semi-continuous envelope. In case K=X𝐾𝑋K=Xitalic_K = italic_X, usc⁡(⋅)usc⋅\operatorname{usc}(\cdot)roman_usc ( ⋅ ) is unnecessary, moreover we have P⁢[u]ℐ⁢(0)=P⁢[u]ℐ𝑃subscriptdelimited-[]𝑢ℐ0𝑃subscriptdelimited-[]𝑢ℐP[u]_{\mathcal{I}}(0)=P[u]_{\mathcal{I}}italic_P [ italic_u ] start_POSTSUBSCRIPT caligraphic_I end_POSTSUBSCRIPT ( 0 ) = italic_P [ italic_u ] start_POSTSUBSCRIPT caligraphic_I end_POSTSUBSCRIPT.

Building on the above, for usc functions f1,…,fNsubscript𝑓1…subscript𝑓𝑁f_{1},\ldots,f_{N}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT we define a notion of rooftop envelope:

A

One can also get some characterizations on bounded Lipschitz domain, whose expressions are less elegant however. See Remark 24.

One can also write down the analogy of Proposition 21 and Corollary 23 similar to (38), (39) and (40), we leave the details to the readers as well.

One can see [YSY10, Sections 6.1.1 and 6.2], [WYY17, Theorem 1.6] and [ST07, Theorem 3.3] for their proof. See also [HT23, Sections 3.4, 4.2 and 4.3]. We remark that because of (8) it is enough to consider the case 0≤τ≤1p0𝜏1𝑝0\leq\tau\leq\frac{1}{p}0 ≤ italic_τ ≤ divide start_ARG 1 end_ARG start_ARG italic_p end_ARG. We leave the details to the readers.

To prove Theorem 2 (also to fix the proof of [YSY15, Proposition 4.15]), simply using the boundedness of EΩsubscript𝐸ΩE_{\Omega}italic_E start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT is not enough.

Similar to [Ryc99, Theorem 2.3], we also have the corresponding characterizations using Peetre maximal functions, see Proposition 21 and Corollary 23.

D

Considering the single equation H1⁢(x,D⁢u,u,u21⁢(x))=0subscript𝐻1𝑥𝐷𝑢𝑢subscriptsuperscript𝑢12𝑥0H_{1}(x,Du,u,u^{1}_{2}(x))=0italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x , italic_D italic_u , italic_u , italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) ) = 0, we can obtain a solution u11subscriptsuperscript𝑢11u^{1}_{1}italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

For Case (a), the existence of solution of (1.12) is given by Perron’s method. This is why (H3) is not needed when Hisubscript𝐻𝑖H_{i}italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is increasing in uisubscript𝑢𝑖u_{i}italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For Case (b), the existence of solutions of (1.12) is given by Proposition A.2. The proof relies on the solution semigroup, so the convexity of H𝐻Hitalic_H in p𝑝pitalic_p is needed. This explains why we need (H3) when Hisubscript𝐻𝑖H_{i}italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is decreasing in uisubscript𝑢𝑖u_{i}italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Different from Proposition A.2, we need the Hamiltonian to be superlinear in p𝑝pitalic_p to get the existence of solutions of (1.1) when there is i∈{1,2}𝑖12i\in\{1,2\}italic_i ∈ { 1 , 2 } such that (H6) holds.

Let ϵ→0+→italic-ϵsuperscript0\epsilon\rightarrow 0^{+}italic_ϵ → 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, we get

We first show the existence of viscosity solutions of (2.2). By Proposition A.2, we tern to consider the following equation

Let this process continue. We get the following iteration procedure for n=0,1,2,…𝑛012…n=0,1,2,\dotsitalic_n = 0 , 1 , 2 , …

D

ℙN,β⁢(j∈J)=2⁢(N⁢ℓN)εN.subscriptℙ𝑁𝛽𝑗𝐽2superscript𝑁subscriptℓ𝑁𝜀𝑁\mathbb{P}_{N,\beta}(j\in J)=\frac{2(N\ell_{N})^{\varepsilon}}{N}.blackboard_P start_POSTSUBSCRIPT italic_N , italic_β end_POSTSUBSCRIPT ( italic_j ∈ italic_J ) = divide start_ARG 2 ( italic_N roman_ℓ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT end_ARG start_ARG italic_N end_ARG .

Since ∫χ1=0subscript𝜒10\int\chi_{1}=0∫ italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0, it follows that

Let χ1subscript𝜒1\chi_{1}italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT be as in (5.31) and let χ2:=χ−χ1assignsubscript𝜒2𝜒subscript𝜒1\chi_{2}:=\chi-\chi_{1}italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := italic_χ - italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.

χ2:=χ−χ1.assignsubscript𝜒2𝜒subscript𝜒1\chi_{2}:=\chi-\chi_{1}.italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := italic_χ - italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .

In view of Propositions 5.3 and 5.4, there remains to bound the variance of FluctN⁢[χ1]subscriptFluct𝑁delimited-[]subscript𝜒1\mathrm{Fluct}_{N}[\chi_{1}]roman_Fluct start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT [ italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] and FluctN⁢[χ2]subscriptFluct𝑁delimited-[]subscript𝜒2\mathrm{Fluct}_{N}[\chi_{2}]roman_Fluct start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT [ italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] for the singular linear statistics χ1,χ2subscript𝜒1subscript𝜒2\chi_{1},\chi_{2}italic_χ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_χ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT defined in (5.31) and (5.32). Let us emphasize that x↦|x|−s2maps-to𝑥superscript𝑥𝑠2x\mapsto|x|^{-\frac{s}{2}}italic_x ↦ | italic_x | start_POSTSUPERSCRIPT - divide start_ARG italic_s end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT is not in H1−s2superscript𝐻1𝑠2H^{\frac{1-s}{2}}italic_H start_POSTSUPERSCRIPT divide start_ARG 1 - italic_s end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT.

A

The actual theorem will be stated and proven in a “relative setting”. The growth types of elements and automorphisms are defined in Chapter III.

Alternatively, Bestvina–Feighn–Handel defined topmost attracting laminations [2, Section 6] that can be used to prove a canonical version of Proposition III.2, thereby making the limit pretrees of Theorem III.3 canonical — this is the approach taken in the sequel.

As a preview for the sequel [19], we end the paper discussing whether the real pretree of Theorem III.3 is canonical.

To complete the analogy with the surface setting, we discuss in the epilogue the extent to which a limit pretree produced by this theorem is canonical — this is proven in the sequel [19].

the real pretree produced by the proof of the theorem does not depend on any choices made in the proof — in a sense, the real pretree was predetermined;

C

This implies that the request to insert numbers in all sequence intervals requires that a single number, 00, must be in the intersection. As a result, the number 00 must be inserted in intervals while fulfilling the imposed request, ensuring the presence of numbers in all intervals.

Proof of Lemma 2: The sequence {In}n=1∞superscriptsubscriptsubscript𝐼𝑛𝑛1\{I_{n}\}_{n=1}^{\infty}{ italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT contracts towards the center point 00 as n𝑛nitalic_n approaches infinity. By the NIP, the intersection of this infinite sequence of nested nonempty closed intervals is non-empty, containing only the number 00. As every selected number at any step n𝑛nitalic_n must come from the intersection of the first n𝑛nitalic_n intervals, and 00 is the only number in the intersection of all intervals, its selection is unavoidable when going through all intervals.

There are many variations of proofs of the uncountability of real numbers. However, fundamentally, all of them share a common basis: they revolve around infinite nested sequences of intervals. We will discuss the critical point that appears in all these proofs, the interpretation of the intersection of these intervals and the nature of the number that resides within this intersection. According to the NIP, the intersection of an infinite nested sequence of closed nonempty intervals must be a zero-size interval, a singleton, or a single number. This number must be one of the elements from the intervals of the sequence, which, in all uncountability proofs, are the numbers of the supposed denumerable sequence assumed to include all numbers. Contrarily, in all these proofs, that number is declared as the number that is outside the denumerable sequence, and its existence is used as the argument that the sequence does not contain all numbers. This discrepancy between the NIP, which posits that the number is a member of the sequence, and the uncountability proofs, which assert the opposite, will be elaborated upon, considering the proofs from 1873 and 1879, and the diagonal argument.

The essence of the NIP, in combination with the contracting nature of the intervals towards 00, dictates that 00 is the only number that lies within the intersection of all intervals, i.e., it is the only number within all intervals, as n𝑛nitalic_n extends towards infinity. It is the only number that satisfies, −1/n<0<1/n1𝑛01𝑛-1/n<0<1/n- 1 / italic_n < 0 < 1 / italic_n for all n𝑛nitalic_n.

In the sequence [−1/n,1/n]1𝑛1𝑛[-1/n,1/n][ - 1 / italic_n , 1 / italic_n ], no specific n𝑛nitalic_n-th interval mandates selecting or inserting 00 instead of 12⁢n12𝑛\frac{1}{2n}divide start_ARG 1 end_ARG start_ARG 2 italic_n end_ARG. However, when considering the collective scope of all intervals, only 00 remains in their intersection, necessitating its selection. Similarly, in the sequence [−1n,0)1𝑛0[-\frac{1}{n},0)[ - divide start_ARG 1 end_ARG start_ARG italic_n end_ARG , 0 ), no single n𝑛nitalic_n leads from a non-empty to an empty intersection. There is no specific n𝑛nitalic_n-th interval where the sequence intersection will change from non-empty to empty. It is the cumulative effect of all intervals as n𝑛nitalic_n approaches infinity that results in this outcome, highlighting the significance of considering the entire sequence collectively. When considering all intervals [−1/n,1/n]1𝑛1𝑛[-1/n,1/n][ - 1 / italic_n , 1 / italic_n ] for every natural number n𝑛nitalic_n, we are essentially looking for the intersection of these intervals. The intersection of a set of nested intervals like {[−1/n,1/n]}n=1∞superscriptsubscript1𝑛1𝑛𝑛1\left\{[-1/n,1/n]\right\}_{n=1}^{\infty}{ [ - 1 / italic_n , 1 / italic_n ] } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is the single point that is common to all intervals, and in this case, it is 00. Even when different numbers can be inserted in each interval, the requirement to insert numbers within every interval [−1/n,1/n]1𝑛1𝑛[-1/n,1/n][ - 1 / italic_n , 1 / italic_n ] across the entire infinite set of such intervals, and the contracting nature of these intervals to a single point, which is 00, necessitates choosing number 00 since it is the only number that fits in every interval of that infinite set.

C

Apply Collatz function to even numbers in set 3⁢i3𝑖3i3 italic_i once the results will stay in set 3⁢i3𝑖3i3 italic_i decreased and will reach to a smaller number;

Apply Collatz function to odd numbers in set 3⁢i+13𝑖13i+13 italic_i + 1 the results will stay in set 3⁢i+23𝑖23i+23 italic_i + 2 increased;

Apply Collatz function to odd numbers in set 3⁢i+23𝑖23i+23 italic_i + 2 once the results will stay in set 3⁢i+23𝑖23i+23 italic_i + 2 increased;

Apply Collatz function to even numbers in set 3⁢i+23𝑖23i+23 italic_i + 2 once the results will be in set 3⁢i+13𝑖13i+13 italic_i + 1 decreased and will reach to a smaller number;

Apply Collatz function to odd numbers in set 3⁢i3𝑖3i3 italic_i once the results will be in set 3⁢i+23𝑖23i+23 italic_i + 2 increased;

D

\end{array}\right).italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL , end_CELL start_CELL 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL , end_CELL start_CELL - 1 end_CELL start_CELL end_CELL end_ROW end_ARRAY ) .

A direct calculation shows that the inter-twiner rζ=r0subscript𝑟𝜁subscript𝑟0r_{\zeta}=r_{0}italic_r start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a matrix with real entries given below

By our hypothesis that ω𝜔\omegaitalic_ω is reflection positive with twist r0subscript𝑟0r_{0}italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, any element in ℳ0subscriptℳ0{\cal M}_{0}caligraphic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is A⁢dγrζ𝐴subscript𝑑subscript𝛾subscript𝑟𝜁Ad_{\gamma_{r_{\zeta}}}italic_A italic_d start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT invariant by Proposition 4.2. So we have the following equality for any 1≤i,j≤dformulae-sequence1𝑖𝑗𝑑1\leq i,j\leq d1 ≤ italic_i , italic_j ≤ italic_d:

A direct commutation shows that r0subscript𝑟0r_{0}italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is given by

reflection positive with twist r0subscript𝑟0r_{0}italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Then we have

C

Because the map appearing in the proposition statement is homotopic to zero, Proposition 3.3 obviously follows.

In this section all homotopy groups are taken, as before, with finite coefficients. Some terminology and notation below is adopted from [4].

In [4], Schlichting constructed for any admissible category 𝒜𝒜\mathscr{A}script_A a (topological) spectrum G⁢W⁢(𝒜)𝐺𝑊𝒜GW(\mathscr{A})italic_G italic_W ( script_A ) and defined Grothendieck–Witt groups of this category with ℤ/mℤ𝑚{\mathbb{Z}}/mblackboard_Z / italic_m-coefficients as homotopy groups

As usual, all homotopy and homology groups are taken here with coefficients ℤ/mℤ𝑚{\mathbb{Z}}/mblackboard_Z / italic_m.

Let M𝑀Mitalic_M be an abelian group taken as a trivial G𝐺Gitalic_G-module. Then, the statement i) below stably implies

A

Here χ⁢(Gn,p)𝜒subscript𝐺𝑛𝑝\chi(G_{n,p})italic_χ ( italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT ) is closely linked to the size α=α⁢(Gn,p)𝛼𝛼subscript𝐺𝑛𝑝\alpha=\alpha(G_{n,p})italic_α = italic_α ( italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT ) of the largest independent set of Gn,psubscript𝐺𝑛𝑝G_{n,p}italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT.

Namely, inspired by [9], it seems plausible that a near-optimal coloring can be obtained by first picking as many vertex-disjoint independent sets of size α𝛼\alphaitalic_α as possible,

it then becomes plausible that χ⁢(Gn,p)𝜒subscript𝐺𝑛𝑝\chi(G_{n,p})italic_χ ( italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT ) should also vary by this amount,

Namely, by iteratively applying Lemma 6 to extract independent sets of size Θ⁢(log1/(1−p)⁡(n))Θsubscript11𝑝𝑛\Theta\bigl{(}\log_{1/(1-p)}(n)\bigr{)}roman_Θ ( roman_log start_POSTSUBSCRIPT 1 / ( 1 - italic_p ) end_POSTSUBSCRIPT ( italic_n ) ) from Gn,p⁢[Z]subscript𝐺𝑛𝑝delimited-[]𝑍G_{n,p}[Z]italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT [ italic_Z ],

many vertices in independent sets of largest size α=α⁢(Gn,p)=r+1𝛼𝛼subscript𝐺𝑛𝑝𝑟1\alpha=\alpha(G_{n,p})=r+1italic_α = italic_α ( italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT ) = italic_r + 1 (which in fact are mostly vertex-disjoint).

A

Let ℱ⁢(Ω)ℱΩ\mathcal{F}(\Omega)caligraphic_F ( roman_Ω ) be the set of real-valued functions defined on ΩΩ\Omegaroman_Ω.

in the symmetric group Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, define the function σ⁢f∈ℱ⁢(Ω)𝜎𝑓ℱΩ\sigma f\in\mathcal{F}(\Omega)italic_σ italic_f ∈ caligraphic_F ( roman_Ω ) by

For every permutation σ∈G𝜎𝐺\sigma\in Gitalic_σ ∈ italic_G, the arithmetic and geometric mean inequality

For every function f∈ℱ⁢(Ω)𝑓ℱΩf\in\mathcal{F}(\Omega)italic_f ∈ caligraphic_F ( roman_Ω ) and every permutation σ𝜎\sigmaitalic_σ

is the function [f]G∈ℱ⁢(Ω)subscriptdelimited-[]𝑓𝐺ℱΩ[f]_{G}\in\mathcal{F}(\Omega)[ italic_f ] start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ∈ caligraphic_F ( roman_Ω ) defined by

C

We choose the standard basis of σm,nsubscript𝜎𝑚𝑛\sigma_{m,n}italic_σ start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT to be {Xi⁢Ym−i;0≤i≤m}superscript𝑋𝑖superscript𝑌𝑚𝑖0𝑖𝑚\{X^{i}Y^{m-i};0\leq i\leq m\}{ italic_X start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_Y start_POSTSUPERSCRIPT italic_m - italic_i end_POSTSUPERSCRIPT ; 0 ≤ italic_i ≤ italic_m }, with the action of ΓΓ\Gammaroman_Γ given by

H:={([a]00[d]),a,d∈𝔽p×}.assign𝐻matrixdelimited-[]𝑎00delimited-[]𝑑𝑎𝑑superscriptsubscript𝔽𝑝H:=\Big{\{}\begin{pmatrix}{[a]}&{0}\\

ϖD↦(01p0),a↦(a00σ⁢(a)),a∈ℚp2.formulae-sequencemaps-tosubscriptitalic-ϖ𝐷matrix01𝑝0formulae-sequencemaps-to𝑎matrix𝑎00𝜎𝑎𝑎subscriptℚsuperscript𝑝2\varpi_{D}\mapsto\begin{pmatrix}{0}&{1}\\

(abcd)⁢Xi⁢Ym−i=(a⁢X+c⁢Y)i⁢(b⁢X+d⁢Y)m−i.matrix𝑎𝑏𝑐𝑑superscript𝑋𝑖superscript𝑌𝑚𝑖superscript𝑎𝑋𝑐𝑌𝑖superscript𝑏𝑋𝑑𝑌𝑚𝑖\begin{pmatrix}{a}&{b}\\

We choose the standard basis of σm,nsubscript𝜎𝑚𝑛\sigma_{m,n}italic_σ start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT to be {Xi⁢Ym−i;0≤i≤m}superscript𝑋𝑖superscript𝑌𝑚𝑖0𝑖𝑚\{X^{i}Y^{m-i};0\leq i\leq m\}{ italic_X start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_Y start_POSTSUPERSCRIPT italic_m - italic_i end_POSTSUPERSCRIPT ; 0 ≤ italic_i ≤ italic_m }, with the action of ΓΓ\Gammaroman_Γ given by

C

Using that the expectations in the statement of Proposition 3.1 are well defined and finite and using the bounds in Section 11 as well as the bounds below, it is easy to prove that expectations in the statement of Theorem 3.5 are well defined and finite.

Note that in the stationary case, covered by Theorem 3.6, the linear response of all the functionals under consideration can be computed explicitly from the 2-time distributions of the stationary time-reversed process.

We now use directly Theorem 5.1 to compute σ⁢(ω)𝜎𝜔\sigma(\omega)italic_σ ( italic_ω ) in the special case given by d=1𝑑1d=1italic_d = 1, N𝑁Nitalic_N even and 2-periodic unperturbed jump rates of the form

The decoupled case follows easily from the general case, hence we focus on the first part of the theorem.

The result for case (1) follows directly from (12) in Proposition 3.1. We use it to deduce the linear response formula for case (2). Indeed, by Fubini’s theorem,

D

As a further application of the proposed autocovariance test, new hierarchical clustering analysis is constructed to cluster a large set of high-dimensional time series, where the dissimilarity between two populations is measured via the p-value of the proposed autocovariance test. The major aim of this clustering analysis is to group high-dimensional time series with similarly projected autocovariances.

To clarify the four steps mentioned above, a flow chart is provided in Figure 1, which summarizes the basic logic and procedure for the autocovariance test.

In this section, a novel test, called the autocovariance test, is proposed to detect the equivalence of spikiness for two high-dimensional time series.

To explain the idea of the autocovariance test and its application on the hierarchical clustering in detail, we will simply revisit the factor structures for high-dimensional time series and introduce the proposed test statistic with its asymptotic properties in Section 4.1. Section 4.2 describes how the hypothesis test can be implemented in practice where a flow chart is also provided to clarify the essential idea of the test procedure.

Another important contribution of this paper is a novel autocovariance test which is built on the developed CLT for {λi}subscript𝜆𝑖\{\lambda_{i}\}{ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT }. It is well known that when the data dimension p𝑝pitalic_p increases with sample size T𝑇Titalic_T, directly comparing and testing the equivalence of two autocovariance matrices is infeasible due to the “curse of dimensionality”. The major idea of the proposed so-called autocovariance test is to compare the autocovariance of the low-dimensional common-factor time series. It is equivalent to testing whether spiked population eigenvalues of two high-dimensional autocovariance matrices are the same. It is worth mentioning that, as the CLT involves some unknown parameters, we propose an AR-sieve bootstrap to derive a feasible test statistic. Furthermore, the proposed test statistic is powerful under some local alternative hypotheses, which are demonstrated via theoretical results and various simulation designs. This autocovariance test is not only in its own interest but also motivates other statistical inferences such as statistical clustering analysis on multi-population high-dimensional time series. In this paper, we construct a new hierarchical clustering approach based on the autocovariance test. It is applied to multi-country mortality data, for which we group those countries with similar low-dimensional autocovariances. The clustering results are consistent with findings in common literature on mortality studies.

C

The majority of the uncovered edges outside G⁢[Uk+1]𝐺delimited-[]subscript𝑈𝑘1G[U_{k+1}]italic_G [ italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ] will belong to a random set of edges R𝑅Ritalic_R, which we set aside before applying our regularization and high-girth process lemmas. This random graph R𝑅Ritalic_R will come in handy in the later steps of the proof of Proposition 10.6, when we need to cover the leftover edges. To be precise, for any outcome of the random data (G,𝒜,ℐ,𝒟)𝐺𝒜ℐ𝒟(G,\mathcal{A},\mathcal{I},\mathcal{D})( italic_G , caligraphic_A , caligraphic_I , caligraphic_D ), let R𝑅Ritalic_R be a random subgraph of G𝐺Gitalic_G obtained by including each edge between Uk+1subscript𝑈𝑘1U_{k+1}italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT and Uk∖Uk+1subscript𝑈𝑘subscript𝑈𝑘1U_{k}\setminus U_{k+1}italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∖ italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT with probability n−θsuperscript𝑛𝜃n^{-\theta}italic_n start_POSTSUPERSCRIPT - italic_θ end_POSTSUPERSCRIPT independently. Here θ>0𝜃0\theta>0italic_θ > 0 is a constant that will be chosen sufficiently small with respect to g,β𝑔𝛽g,\betaitalic_g , italic_β (later, we will require ρ𝜌\rhoitalic_ρ to be sufficiently small with respect to θ𝜃\thetaitalic_θ). Let G∗=G∖(R∪G⁢[Uk+1])superscript𝐺𝐺𝑅𝐺delimited-[]subscript𝑈𝑘1G^{*}=G\setminus(R\cup G[U_{k+1}])italic_G start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_G ∖ ( italic_R ∪ italic_G [ italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ] ) be the graph consisting of the edges we would like to cover to prove Proposition 10.6, other than the edges in R𝑅Ritalic_R.

Let γ>0𝛾0\gamma>0italic_γ > 0 be a constant which is small with respect to g,β,θ,ρ,ν𝑔𝛽𝜃𝜌𝜈g,\beta,\theta,\rho,\nuitalic_g , italic_β , italic_θ , italic_ρ , italic_ν. To summarize what we will prove in this subsection, our random set of triangles ℳ‡⊆𝒜superscriptℳ‡𝒜\mathcal{M}^{\ddagger}\subseteq\mathcal{A}caligraphic_M start_POSTSUPERSCRIPT ‡ end_POSTSUPERSCRIPT ⊆ caligraphic_A will satisfy the following properties.

In this subsection, we handle all the edges of the first type. Namely, using a simple random greedy procedure, we augment ℳ∗superscriptℳ\mathcal{M}^{*}caligraphic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT with a further random set of triangles ℳ†⊆𝒜superscriptℳ†𝒜\mathcal{M}^{\dagger}\subseteq\mathcal{A}caligraphic_M start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊆ caligraphic_A, covering all the remaining edges in G⁢[Uk∖Uk+1]𝐺delimited-[]subscript𝑈𝑘subscript𝑈𝑘1G[U_{k}\setminus U_{k+1}]italic_G [ italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∖ italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ]. In the process, we will also cover a few edges of our reserve graph R𝑅Ritalic_R. To summarize what we will prove in this subsection, our random set of triangles ℳ†⊆𝒜superscriptℳ†𝒜\mathcal{M}^{\dagger}\subseteq\mathcal{A}caligraphic_M start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊆ caligraphic_A will satisfy the following properties.

To summarize what we will prove in this subsection, our random set of triangles ℳ∗⊆𝒜superscriptℳ𝒜\mathcal{M}^{*}\subseteq\mathcal{A}caligraphic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⊆ caligraphic_A will satisfy the following properties.

We now start the proof of Proposition 10.6. So, consider initial data (U0,…,Uℓ,𝔉4,…,𝔉g)subscript𝑈0…subscript𝑈ℓsubscript𝔉4…subscript𝔉𝑔(U_{0},\dots,U_{\ell},\mathfrak{F}_{4},\dots,\mathfrak{F}_{g})( italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_U start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT , fraktur_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , … , fraktur_F start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ), and random data (G,𝒜,ℐ,𝒟)𝐺𝒜ℐ𝒟(G,\mathcal{A},\mathcal{I},\mathcal{D})( italic_G , caligraphic_A , caligraphic_I , caligraphic_D ) for stage k𝑘kitalic_k, as in the lemma statement. Let n=|Uk|𝑛subscript𝑈𝑘n=|U_{k}|italic_n = | italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |. Our goal is to find an appropriate set of triangles ℳℳ\mathcal{M}caligraphic_M covering all the edges of G𝐺Gitalic_G that are not in G⁢[Uk+1]𝐺delimited-[]subscript𝑈𝑘1G[U_{k+1}]italic_G [ italic_U start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ]. In this subsection we use our regularization and high-girth process lemmas to find a random set of triangles ℳ∗⊆𝒜superscriptℳ𝒜\mathcal{M}^{*}\subseteq\mathcal{A}caligraphic_M start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⊆ caligraphic_A covering almost all the desired edges.

C

The case, known as co-recursive, was introduced and studied in [14] by adding a constant to the first coefficient b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . The case of another type of perturbation called co-dilation was presented in [19]. The Stieltjes function and the fourth-order differential equation for the co-dilated polynomials inside the Laguerre-Hahn class were investigated in [20, 40]. The general case, called the generalized co-modification, arising from perturbation of coefficients in (1.5) at any level, was studied in [36] from the point of view of Stieltjes function, fourth-order differential equation, and distribution of zeros. A connection to the Laguerre-Hahn class was also investigated in [36]. The properties of co-modified classical orthogonal polynomials were studied in [40]. Its extension to semi-classical orthogonal polynomials was investigated in [20, 41]. Interlacing properties and some new inequalities for the zeros of co-modified OPRL, called co-polynomials on the real line (COPRL), were studied in [10].

The case, known as co-recursive, was introduced and studied in [14] by adding a constant to the first coefficient b0subscript𝑏0b_{0}italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . The case of another type of perturbation called co-dilation was presented in [19]. The Stieltjes function and the fourth-order differential equation for the co-dilated polynomials inside the Laguerre-Hahn class were investigated in [20, 40]. The general case, called the generalized co-modification, arising from perturbation of coefficients in (1.5) at any level, was studied in [36] from the point of view of Stieltjes function, fourth-order differential equation, and distribution of zeros. A connection to the Laguerre-Hahn class was also investigated in [36]. The properties of co-modified classical orthogonal polynomials were studied in [40]. Its extension to semi-classical orthogonal polynomials was investigated in [20, 41]. Interlacing properties and some new inequalities for the zeros of co-modified OPRL, called co-polynomials on the real line (COPRL), were studied in [10].

The former theorem enables the study of the polynomials when finite number of recurrence coefficients are perturbed. In this regard, the following can be proved using techniques given in [5, 43].

In this direction, finite perturbations were studied in [38]. When the perturbations are large, some information is expected to be lost, which was explored in [31]. For perturbations of recurrence coefficients in the recurrence relations of higher-order and their extensions to Sobolev OPRL, see [32, 33]. Recently, a transfer matrix approach was introduced in [10] to study polynomials perturbed in a generalized co-modified way.

This manuscript aims to study properties of the polynomials that satisfy a recurrence relation as (1.1) with new recurrence coefficients perturbed in a (generalized) co-recursive/co-dilated/co-modified way, i.e.,

C

7.30×10−97.30superscript1097.30\times 10^{-9}7.30 × 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT

8.44×10−88.44superscript1088.44\times 10^{-8}8.44 × 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT

8.01×10−88.01superscript1088.01\times 10^{-8}8.01 × 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT

1.01×10−11.01superscript1011.01\times 10^{-1}1.01 × 10 start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT

1.01×10−81.01superscript1081.01\times 10^{-8}1.01 × 10 start_POSTSUPERSCRIPT - 8 end_POSTSUPERSCRIPT

B

In Section 2, we establish recurrence equations encoding fixed-length concatenations. In Section 3, we solve those recurrence equations and deduce formulas associated with fixed-length concatenations. We will then deduce explicit formulas to compute s⁢(n),sr⁢(n),𝑠𝑛subscript𝑠𝑟𝑛s(n),s_{r}(n),italic_s ( italic_n ) , italic_s start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_n ) , and s∗⁢(n)subscript𝑠𝑛s_{*}(n)italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ). Section 4 is devoted to the proof that our concatenating sequences do not obey linear recurrence equations with polynomial coefficients. Our first proof shows this as a general fact about sequences with linearly independent holonomic representations in infinitely many integer intervals. Our second proof arises from divisibility criteria deduced from asymptotic terms of these sequences. Section 5 presents an algorithm to compute Sm⁡(n)Sm𝑛\mbox{\small$\operatorname{Sm}$}(n)roman_Sm ( italic_n ), from which the one of Smr⁡(n)Smr𝑛\mbox{\small$\operatorname{Smr}$}(n)roman_Smr ( italic_n ) can be derived easily. We also present some computations with our implementation in the Maple computer algebra system [9].

The above proof can be used as a template to prove that all terms corresponding to a fixed-length concatenation in (s⁢(n))nsubscript𝑠𝑛𝑛(s(n))_{n}( italic_s ( italic_n ) ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT satisfy a C-finite recurrence equation whose coefficients depend on the length of the concatenation. The following lemma gives the recurrence equations for right-concatenations and left-concatenations of an arithmetic progression. We ignore the case where the common difference disqualifies l𝑙litalic_l-digit concatenations. E.g., in the decimal base, there is no 1111-digit concatenation with 10101010 as the common difference. We assume that the l𝑙litalic_l-digit concatenations involve at least 4444 terms in (s⁢(n))nsubscript𝑠𝑛𝑛(s(n))_{n}( italic_s ( italic_n ) ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, c.f. remark 1.

The non-holonomic character of these sequences can be seen as a consequence of the fact that (51) holds. This condition presents a way to construct non-holonomic sequences. Indeed, since the solutions of the recurrence equations in Section 2 are obtained with the m𝑚mitalic_m-fold hypergeometric solver from [24], it follows that concatenations of terms of an increasing integer-valued hypergeometric-type sequence (see [22]) produce non-holonomic sequences. For instance, concatenations of terms of the sequence of general term 2n+χ{n≡1⁢mod⁢ 2}superscript2𝑛subscript𝜒𝑛1mod22^{n}+\chi_{\{n\equiv 1\ \mathrm{mod}\ 2\}}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_χ start_POSTSUBSCRIPT { italic_n ≡ 1 roman_mod 2 } end_POSTSUBSCRIPT yield a non-holonomic sequence. For observation purposes, one can verify that the guessing algorithms from the GFUN package, [8, 7], or [21] return no differential equation for the corresponding generating function. On the other hand, we can relate the non-holonomic character of these sequences to the apparent non-existence of their generating functions as a differentiable object. Indeed, using Padé approximants, we can accurately approximate their generating functions at every range of concatenations. However, given the “brutal” changes at the endpoints of these ranges, it occurs that if F𝐹Fitalic_F is the generating function of one of these sequences, then

In Section 2, we establish recurrence equations encoding fixed-length concatenations. In Section 3, we solve those recurrence equations and deduce formulas associated with fixed-length concatenations. We will then deduce explicit formulas to compute s⁢(n),sr⁢(n),𝑠𝑛subscript𝑠𝑟𝑛s(n),s_{r}(n),italic_s ( italic_n ) , italic_s start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_n ) , and s∗⁢(n)subscript𝑠𝑛s_{*}(n)italic_s start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ). Section 4 is devoted to the proof that our concatenating sequences do not obey linear recurrence equations with polynomial coefficients. Our first proof shows this as a general fact about sequences with linearly independent holonomic representations in infinitely many integer intervals. Our second proof arises from divisibility criteria deduced from asymptotic terms of these sequences. Section 5 presents an algorithm to compute Sm⁡(n)Sm𝑛\mbox{\small$\operatorname{Sm}$}(n)roman_Sm ( italic_n ), from which the one of Smr⁡(n)Smr𝑛\mbox{\small$\operatorname{Smr}$}(n)roman_Smr ( italic_n ) can be derived easily. We also present some computations with our implementation in the Maple computer algebra system [9].

A holonomic recurrence equation (RE) is a linear homogeneous recurrence equation with polynomial coefficients in the index variable. Although all the recurrence equations in this section have constant coefficients (C-finite), in Section 4, we will see that only one is of minimal order. The other two equations can be reduced to second-order holonomic REs with polynomial coefficients of degree 1111. These observations will serve as a premise for our forthcoming proof in Section 4, which shows that “different holonomicity per range implies non-holonomicity”. Also, the solver used in Section 3 effectively applies to holonomic recurrence equations. As observed in the introduction, from the use of the symbolic-computation algorithm from [24], the formula to compute Smarandache numbers for 1111-digit right-concatenations is obtained from solving a holonomic RE.

D

As Corollary 4.11 and Theorem 7 imply, 𝕊^d−1superscript^𝕊𝑑1\widehat{\mathbb{S}}^{d-1}over^ start_ARG blackboard_S end_ARG start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT is homeomorphic to 𝕊d−1superscript𝕊𝑑1\mathbb{S}^{d-1}blackboard_S start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT. Motivated by this observation, can we prove that M^^𝑀\widehat{M}over^ start_ARG italic_M end_ARG is homeomorphic (or even homotopy equivalent) to M𝑀Mitalic_M when M𝑀Mitalic_M is an arbitrary compact manifold? In other words, does the cMDS procedure preserve the global topology of the input space?

In this setting, we aim to quantify how much distortion will be incurred when trying to embed 𝒳𝒳\mathcal{X}caligraphic_X into Euclidean space of a given dimension via a certain generalized formulation of the cMDS procedure. This extension of cMDS to mm-spaces was first considered in [33, 2] where the authors were able to compute the spectrum of the cMDS operator for the case of the circle 𝕊1superscript𝕊1\mathbb{S}^{1}blackboard_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT with geodesic distance and normalized length measure. They also formulated a number of questions, such as extending the analysis to the case of spheres with arbitrary dimension (with geodesic distance and normalized volume measure) and establishing the stability of the cMDS procedure. The results in the present paper fully answer those and other questions about cMDS on continuous spaces.

In §7,we provide a detailed comparison between our paper and the closely related recent work [34] by Kroshnin, Stepanov and Trevisan. Additionally, we explore several potential avenues for future research that expand upon the concepts presented in this paper.

In [17], de Silva and Tenenbaum slightly modify the argument in [56] in order to show the robustness of landmark MDS, a variant of cMDS which they introduce. Though their setting is quite similar to Sibson’s, there are some improvements. Firstly, the positive eigenvalues of K𝒳subscript𝐾𝒳K_{\mathcal{X}}italic_K start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT, the Gram matrix associated to the input data, are not required to be simple. Secondly, it is not necessary to know the true dimension of the input dataset in order to apply their result. Instead, the target dimension of the embedding is required to satisfy certain conditions involving the eigenvalues of K𝒳subscript𝐾𝒳K_{\mathcal{X}}italic_K start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT.

We are grateful to Henry Adams, Boris Mityagin, and Ery Arias-Castro for helpful conversations related to this work. We also thank Eugene Stepanov for pointing out to imprecisions in the proofs of Propositions 3.15 and 3.25.

D

These three groups arise for the monotone toric fibres in ℂ⁢ℙ3ℂsuperscriptℙ3\mathbb{C}\mathbb{P}^{3}blackboard_C blackboard_P start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, ℂ⁢ℙ2×ℂ⁢ℙ1ℂsuperscriptℙ2ℂsuperscriptℙ1\mathbb{C}\mathbb{P}^{2}\times\mathbb{C}\mathbb{P}^{1}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × blackboard_C blackboard_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, and (ℂ⁢ℙ1)3superscriptℂsuperscriptℙ13(\mathbb{C}\mathbb{P}^{1})^{3}( blackboard_C blackboard_P start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT respectively. We are writing ℤ/2ℤ2\mathbb{Z}/2blackboard_Z / 2 as S2subscript𝑆2S_{2}italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for consistency with what follows.

which provides a bridge between the Hamiltonian dynamics of X𝑋Xitalic_X and the Floer theory of L𝐿Litalic_L, but it has received much less attention.

For general n𝑛nitalic_n one can try to use similar methods to constrain the isomorphism class of ℋLsubscriptℋ𝐿\mathcal{H}_{L}caligraphic_H start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT, but the problem gets much harder because the number of conjugacy classes of finite subgroups of GL⁡(n,ℤ)GL𝑛ℤ\operatorname{GL}(n,\mathbb{Z})roman_GL ( italic_n , blackboard_Z ) grows rapidly: there are 85,3088530885{,}30885 , 308 for n=6𝑛6n=6italic_n = 6 [27]. Using the computer algebra system GAP [32] and the RatProbAlgTori package [14] one can enumerate these conjugacy classes of subgroups for n=4𝑛4n=4italic_n = 4, 5555, or 6666, and rule out many of them using Corollary 4.5 with k=1𝑘1k=1italic_k = 1 or 2222. Our code for this is available at https://github.com/MarcinAugustynowicz/HLM. Combining this with our earlier results for n=2𝑛2n=2italic_n = 2 and 3333, and for toric fibres, we obtain the following.

As n𝑛nitalic_n grows, the problem rapidly becomes much more complicated, but computer experiments suggest the following.

Fix throughout the rest of the paper a monotone Lagrangian n𝑛nitalic_n-torus L𝐿Litalic_L in a symplectic manifold (X,ω)𝑋𝜔(X,\omega)( italic_X , italic_ω ), in which holomorphic curves cannot escape to infinity, as in 1.1.

C

We discuss the case d>1𝑑1d>1italic_d > 1 afterward, most of the arguments being the same. We write σ:=2⁢β−1assign𝜎2superscript𝛽1\sigma:=\sqrt{2\beta^{-1}}italic_σ := square-root start_ARG 2 italic_β start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_ARG. The proof is divided into the following steps.

Because ∇W∇𝑊\nabla W∇ italic_W is Lipschitz, the equation (3.1) has a unique path-wise solution satisfying the following apriori estimate:

Recall that (Xtν)subscriptsuperscript𝑋𝜈𝑡(X^{\nu}_{t})( italic_X start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) denotes the solution of the McKean-Vlasov equation (2.1).

The first step is to show the following apriori estimate on ktνsubscriptsuperscript𝑘𝜈𝑡k^{\nu}_{t}italic_k start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT:

We then define Ωt⁢(h)subscriptΩ𝑡ℎ\Omega_{t}(h)roman_Ω start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_h ) to be the unique solution of the Volterra integral equation:

A

The concept of operator-valued infinitesimal freeness was discussed in the preceding subsection, wherein (scalar-valued) infinitesimal freeness serves as a special case when considering ℬ=ℂℬℂ\mathcal{B}=\mathbb{C}caligraphic_B = blackboard_C.

In addition to Section 5, we review fundamental knowledge of OV and OVI probability theory in Section 2. Moving to Section 3, for each notion of OVI independence, we introduce the corresponding infinitesimal transforms and demonstrate that these transforms satisfy certain multiplicative properties. In Section 4, we provide a differentiable paths approach to study infinitesimal multiplicative convolutions, as outlined in Theorem 4.1.

In this section we apply differentiable paths to compute the infinitesimal multiplicative convolutions, which was obtained in Section 3.

In this subsection, our emphasis shifts to the combinatorial facet of scalar-valued infinitesimal free probability theory, offering essential tools for Section 5.

In this section, we introduce several transforms which play key role to describe the (operator-valued ) infinitesimal multiplicative convolutions for free, Boolean, and monotone respectively. We will apply Proposition 2.4 to reduce our problems to the operator-valued case, and then we can deduce our results by using formulas (2.1), (2.3), and (2.4).

C

However we consider continuous group cohomology, as we shall see in Section 3, we can drop the hypothesis on the degree.

Using the work of Peter Greenberg ([Gre92]) we shall prove in a separate paper [Nar22], new connectivity results for the curious case of PL-foliations in codimension 2222. Then using the version of Mather-Thurston’s theorem for PL homeomorphisms ([Nar23]) due to the author, we prove the perfectness of PL homeomorphisms of surfaces that are isotopic to the identity which answers a question ([Eps70, Section 3]) of Epstein in dimension 2222.

where the first isomorphism is by the Hurewicz theorem and the second is given by the above standard fact about mapping spaces. On the other hand, by the same argument H2⁢(BDiffc⁢(ℝ3)¯;ℤ)subscript𝐻2¯subscriptBDiff𝑐superscriptℝ3ℤH_{2}(\overline{\mathrm{BDiff}_{c}(\mathbb{R}^{3})};\mathbb{Z})italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over¯ start_ARG roman_BDiff start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) end_ARG ; blackboard_Z ) is also isomorphic to π5⁢(B⁢Γ¯3)subscript𝜋5subscript¯BΓ3\pi_{5}(\overline{\mathrm{B}\Gamma}_{3})italic_π start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT ( over¯ start_ARG roman_B roman_Γ end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ). Therefore, for all embeddings of an open disk ℝ3↪M↪superscriptℝ3𝑀\mathbb{R}^{3}\hookrightarrow Mblackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ↪ italic_M, the map

The author was partially supported by NSF CAREER Grant DMS-2239106, NSF DMS-2113828 and Simons Foundation (855209, SN). The author thanks Gael Meigniez for the discussion about Haefliger-Thurston’s conjecture. The author also thanks Mike Freedman for the discussion about Section 3. He is also grateful to Jonathan Bowden, Sander Kupers and Søren Galatius for their comments.

Haefliger introduced and calculated differentiable cohomology of B⁢Γ¯nsubscript¯BΓ𝑛\overline{\mathrm{B}\Gamma}_{n}over¯ start_ARG roman_B roman_Γ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT in [Hae79] and showed that it vanishes up to degree 2⁢n2𝑛2n2 italic_n. And he speculated ([Hae79, Section 6]) the possibility that B⁢Γ¯nsubscript¯BΓ𝑛\overline{\mathrm{B}\Gamma}_{n}over¯ start_ARG roman_B roman_Γ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT might be 2⁢n2𝑛2n2 italic_n-connected. Thurston also stated ([Thu74a]) this range of connectivity for B⁢Γ¯nrsubscriptsuperscript¯BΓ𝑟𝑛\overline{\mathrm{B}\Gamma}^{r}_{n}over¯ start_ARG roman_B roman_Γ end_ARG start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT as a conjecture. Using Mather-Thurston’s theory (see [Mat11, Nar23]), one could equivalently state this conjecture in the context of manifold bundles. Let Diffr⁢(M)superscriptDiff𝑟𝑀\mathrm{Diff}^{r}(M)roman_Diff start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_M ) denote the group of Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-orientation preserving diffeomorphisms of a smooth manifold M𝑀Mitalic_M with the Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT-Whitney topology. We decorate it with superscript δ𝛿\deltaitalic_δ and subscript c𝑐citalic_c if we consider the same group with discrete topology and its subgroup of compactly supported diffeomorphisms respectively. The identity homomorphism between the groups Diffcr⁢(M)δ→Diffcr⁢(M)→superscriptsubscriptDiff𝑐𝑟superscript𝑀𝛿superscriptsubscriptDiff𝑐𝑟𝑀\mathrm{Diff}_{c}^{r}(M)^{\delta}\to\mathrm{Diff}_{c}^{r}(M)roman_Diff start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_M ) start_POSTSUPERSCRIPT italic_δ end_POSTSUPERSCRIPT → roman_Diff start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( italic_M ) induces the map between classifying spaces

C

Authors Contributions Conceptualization, methodology, investigation, writing - original draft, validation, review, editing and reading have been performed by both the authors and agreed to the paper.

Funding Kiran Meena is financial supported by the Department of Atomic Energy, Government of India [Offer Letter No.: HRI/133/1436 Dated 29 November 2022].

Acknowledgments Kiran Meena gratefully acknowledges the research facilities provided by Harish-Chandra Research Institute, Prayagraj, India. In addition, both authors are grateful to the referee for helpful comments that allowed to improve the paper.

Conflict of interest The authors have no Conflict of interest and no financial interests for this article.

The divergence of X𝑋Xitalic_X, denoted by d⁢i⁢v⁢(X)𝑑𝑖𝑣𝑋div(X)italic_d italic_i italic_v ( italic_X ), is given by

A

In [2], the use of randomized integration times destroy the deterministic periodicity issues, which allows for larger integration times, hence reaching a mean free path of order m−1/2superscript𝑚12m^{-1/2}italic_m start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT with η=0𝜂0\eta=0italic_η = 0. The two approaches are thus similar and can be easily combined, for instance by replacing the deterministic step size δ𝛿\deltaitalic_δ by a randomized step-size δ~~𝛿\tilde{\delta}over~ start_ARG italic_δ end_ARG of the same order in order to destroy possible resonances at the frequency 1/δ1𝛿1/\delta1 / italic_δ, then taking K=1𝐾1K=1italic_K = 1 and η=1−m⁢δ𝜂1𝑚𝛿\eta=1-\sqrt{m}\deltaitalic_η = 1 - square-root start_ARG italic_m end_ARG italic_δ, in order to benefit from the robustness of the Langevin scaling with respect to sub-optimal choice of parameters (see Section 3). This is also the spirit of the randomized midpoint method of [54] for undajusted underdamped Langevin. The study of randomized step-size is beyond the scope of the present article.

In this section, we consider P𝑃Pitalic_P the Markov transition operator of the gHMC chain given in (9) and address the question of the optimal choice of the parameters 𝔭=(δ,K,η)𝔭𝛿𝐾𝜂\mathfrak{p}=(\delta,K,\eta)fraktur_p = ( italic_δ , italic_K , italic_η ) for target distributions which are in a given family of Gaussian laws. Gaussian distributions are often used as a theoretical benchmark for comparing MCMC samplers and scaling parameters since convergence rates have explicit expressions. When this is made at the level of continuous time processes (as in [29, 33, 26, 3]), the question of the correct normalisations for a fair comparison between different processes makes the results difficult to interpret. Here we don’t have this problem since we work directly with the numerical schemes. A comparison of the convergence rates (with respect to the number of gradient evaluations) of two schemes is fair if they have the same accuracy on the target measure.

In particular, we see that a suitably tuned Langevin dynamics (with K=1𝐾1K=1italic_K = 1) is competitive for badly-conditioned problems with respect to the optimal gHMC, while being more robust with respect to the choice of parameters. Indeed, if we tune a gHMC with the optimal choices (5) for some m,L𝑚𝐿m,Litalic_m , italic_L and then use it to sample a Gaussian process where ∇2Usuperscript∇2𝑈\nabla^{2}U∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_U has an eigenvalue L′⩾(L+m)2superscript𝐿′superscript𝐿𝑚2L^{\prime}\geqslant(\sqrt{L}+\sqrt{m})^{2}italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⩾ ( square-root start_ARG italic_L end_ARG + square-root start_ARG italic_m end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (which, for ill-conditioned problems, is just slightly above L𝐿Litalic_L), then K⁢δ′⩾π𝐾superscript𝛿′𝜋K\delta^{\prime}\geqslant\piitalic_K italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⩾ italic_π and thus the convergence rate goes from optimal to zero. This does not happen in the Langevin case K=1𝐾1K=1italic_K = 1, as can be checked from the results of Section 2.1, since in this case the optimal parameters are far from the condition K⁢δ′⩾π𝐾superscript𝛿′𝜋K\delta^{\prime}\geqslant\piitalic_K italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⩾ italic_π. The second interesting point enlightened by these results is that, when κ𝜅\kappaitalic_κ is large, we see that the optimal choice of η𝜂\etaitalic_η in (5) is close to 1111, while a classical position HMC chain (with η=0𝜂0\eta=0italic_η = 0) is never competitive in the regime κ→+∞→𝜅\kappa\rightarrow+\inftyitalic_κ → + ∞, being off the optimal convergence rate by a factor π/κ𝜋𝜅\pi/\sqrt{\kappa}italic_π / square-root start_ARG italic_κ end_ARG (this is in particular the case of the unadjusted (overdamped) Langevin algorithm (ULA), for which furthermore the dependency in ε′superscript𝜀′\varepsilon^{\prime}italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT in ρ𝜌\rhoitalic_ρ is of order ε′superscript𝜀′\varepsilon^{\prime}italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT instead of ε′superscript𝜀′\sqrt{\varepsilon^{\prime}}square-root start_ARG italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG); i.e., to put it in different terms, the convergence rate is improved from the order κ−1superscript𝜅1\kappa^{-1}italic_κ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT to the order κ−1/2superscript𝜅12\kappa^{-1/2}italic_κ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT by taking 1−η1𝜂1-\eta1 - italic_η of order κ−1/2⁢K⁢δ′=m⁢K⁢δsuperscript𝜅12𝐾superscript𝛿′𝑚𝐾𝛿\kappa^{-1/2}K\delta^{\prime}=\sqrt{m}K\deltaitalic_κ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT italic_K italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = square-root start_ARG italic_m end_ARG italic_K italic_δ, with respect to the classical case η=0𝜂0\eta=0italic_η = 0 (whenever 1⩽K⩽1/δ′1𝐾1superscript𝛿′1\leqslant K\leqslant 1/\delta^{\prime}1 ⩽ italic_K ⩽ 1 / italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT). Hence, we recover, with a similar scaling of the damping parameter, restricted to Gaussian target measures, a speed-up with respect to classical samplers similar to the one observed recently in [9] in a more general convex case but only at the level of the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT norm for the continuous-time Langevin diffusion. To our knowledge, our analysis in the Gaussian case gives the first result in that direction, with a clear criterion of efficiency involving the discretization error, for a practical generalist algorithm (in the sense that the algorithm is not designed or finely-tuned specifically for Gaussian targets) involving only the computation of the gradient of ln⁡π¯¯𝜋\ln\bar{\pi}roman_ln over¯ start_ARG italic_π end_ARG (without requiring to solve a linear system of some optimization problem, as in implicit or preconditionned schemes as in [28, 30, 48] and references within). Besides, after a first preprint version of the present work was released, a similar conclusion (i.e. a convergence rate of order κ−1/2superscript𝜅12\kappa^{-1/2}italic_κ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT for a practical algorithm in the Gaussian case) was obtained in [2], but now for position HMC (η=0𝜂0\eta=0italic_η = 0) with randomized integration times T=K⁢δ𝑇𝐾𝛿T=K\deltaitalic_T = italic_K italic_δ (also, in total variation rather than Wasserstein distance, but this distinction is not crucial since, following [44], [6] or [43, Proposition 3], a total variation convergence can be obtained from a Wasserstein convergence, with the same rate). An interesting point is that the optimal random integration time in [2] is of order m−1/2superscript𝑚12m^{-1/2}italic_m start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT, in accordance with results in continuous-time which advocates a refreshment rate of order m1/2superscript𝑚12m^{1/2}italic_m start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT, see [35, 18]. This is consistent with our own result, in which we see that K⁢δ/(1−η)𝐾𝛿1𝜂K\delta/(1-\eta)italic_K italic_δ / ( 1 - italic_η ), which is the typical time needed for the process to forget its initial velocity (and thus is the typical distance that would be covered by each coordinate of the position in a flat potential during this time, since each velocity coordinate is of order 1111), is of order m−1/2superscript𝑚12m^{-1/2}italic_m start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT for the optimal choices of parameters, which is the standard deviation of the lowest frequency modes of the target. Illustrating our theoretical analysis in the Gaussian case with numerical experiments in Section 3, we highlight the three following relevant time-scales related to the parameters K,δ,η𝐾𝛿𝜂K,\delta,\etaitalic_K , italic_δ , italic_η:

Motivated by the results in the Gaussian case, our second main contribution is thus a dimension-free non-asymptotic Wasserstein convergence result for the chain in the general strongly convex smooth case, namely when 0<m⁢Id⩽∇2U⩽L⁢Id0𝑚subscript𝐼𝑑superscript∇2𝑈𝐿subscript𝐼𝑑0<mI_{d}\leqslant\nabla^{2}U\leqslant LI_{d}0 < italic_m italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⩽ ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_U ⩽ italic_L italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT but ∇2Usuperscript∇2𝑈\nabla^{2}U∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_U is not necessarily constant (to be clear, here we mean the long-time convergence of the chain towards its biased equilibrium; by contrast, the convergence of the bias to zero as the step-size goes to zero does depend on the dimension, as we discuss below). In practice, this is a restrictive framework which is not much general than the Gaussian case, but the main point is that it means the results are robust in the sense that they do not rely on explicit formulas as in the specific Gaussian case. It can also be thought for multi-modal targets as a way to quantify the sampling of a local mode (a positive lower bound on ∇2Usuperscript∇2𝑈\nabla^{2}U∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_U holding on some balls around the local minimizers of U𝑈Uitalic_U, where the chain spends most of its time between transitions from one mode to another). For these reasons, and because it is convenient to get explicit bounds (in particular in terms of the dimension) this framework has been used in the literature to compare different samplers [7, 12, 13, 14, 17, 19, 23, 24, 38, 51], etc. (of course the non-convex case has also drawn much interest, but this is not the topic of the present work so that we refer the interested reader to e.g. [6, 7, 43] and references within). In fact, our result is a way to revisit similar results such as [44] (for K=1𝐾1K=1italic_K = 1) or [37, 52, 48] (for K=O⁢(δ−1),η=0formulae-sequence𝐾𝑂superscript𝛿1𝜂0K=O(\delta^{-1}),\eta=0italic_K = italic_O ( italic_δ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) , italic_η = 0), i.e. we show that the Langevin and HMC samplers can be treated in a unified framework, in a single computation. Besides, the method is the same as in previous works, namely it relies on the synchronous coupling of two chains. A slight difference with all previous works except [44, 7] is that, similarly to those two works, we establish the long-time convergence of the discrete-time Markov chain toward its equilibrium (independently from the question of the numerical bias on the equilibrium) instead of combining the long-time convergence of the continuous-time ideal process toward the true target with a numerical error analysis to get non-asymptotic efficiency bounds (but not convergence rates) for the discretised chain. As explained in [44], this different approach is motivated by the theoretical discussion in [50] on the difficulties of obtaining convergence rates for discretized Markov chains which scale correctly with the time-step. More importantly, that way, we avoid the question of distinguishing whether the limit continuous-time process as δ→0→𝛿0\delta\rightarrow 0italic_δ → 0 is the Langevin diffusion (3) or an ideal HMC process where the Hamiltonian dynamics is solved exactly.

As a conclusion, the analysis for Gaussian target distributions indicates that it should be more efficient in practice to replace the widely used position HMC samplers by kinetic processes where the velocity is only partially refreshed between two steps of Hamiltonian dynamics, i.e. with inertia. This motivates the study of gHMC with inertia beyond the Gaussian case, which is the topic of the next section.

D

Each component has one of the wing centers on its boundary, and thus the transition A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT appears in the dynamics, and also A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT to A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Furthermore, below the plane z=ε𝑧𝜀z=\varepsilonitalic_z = italic_ε the manifold 𝒲s⁢(0)superscript𝒲𝑠0\mathcal{W}^{s}(0)caligraphic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( 0 ) is fairly simple, in contrast to what happens above this plane (as is indeed observed numerically, e.g. in [OSINGA2002VisualizingLorenz]). Points off η𝜂\etaitalic_η return to R𝑅Ritalic_R without intersecting 𝒲s⁢(0)superscript𝒲𝑠0\mathcal{W}^{s}(0)caligraphic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( 0 ), and thus orbits emanating from A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT must enter 𝒫𝒫\operatorname{\mathcal{P}}caligraphic_P through the quadrant {x,y>0}𝑥𝑦0\{x,y>0\}{ italic_x , italic_y > 0 }, and orbits from A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT enter 𝒫𝒫\operatorname{\mathcal{P}}caligraphic_P from the negative quadrant. Furthermore, points close to the entire edge of Aisubscript𝐴𝑖A_{i}italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT along η𝜂\etaitalic_η are mapped close to a single point that is the first point where the seperatrix hits R𝑅Ritalic_R. In this sense the return map send each of Aisubscript𝐴𝑖A_{i}italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to a triangle like region and the dinamics is influenced by the location of the “triangle tip”. However, there is still too much freedom for this symbolic dynamics to be useful.

This is why we next focus on special parameter values allowing us to pinpoint the behaviour of the return map.

Figure 8. The extended geometric model, defined on S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and containing an invariant trefoil [BonattiPinsky2021]. A neighborhood of the trefoil can be removed and the resulting flow is the unique hyperbolic flow on the trefoil complement with the given two to one return map.

We may assume without loss of generality that κ⁢(s)𝜅𝑠\kappa(s)italic_κ ( italic_s ) is not a parameter on the first homoclinic curve for s>0𝑠0s>0italic_s > 0 (that is κ⁢(s)𝜅𝑠\kappa(s)italic_κ ( italic_s ) is transverse to the homoclinic curve and does not return to it). From the linking of the second homoclinic with L±superscript𝐿plus-or-minusL^{\pm}italic_L start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT we know that at k⁢(1)𝑘1k(1)italic_k ( 1 ) the tip of the right half A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT of the cross section R𝑅Ritalic_R is mapped to the left (and then hits the stable manifold along η𝜂\etaitalic_η at its second return), it follows that the return map r⁢(s)𝑟𝑠r(s)italic_r ( italic_s ) for a small positive s𝑠sitalic_s along κ𝜅\kappaitalic_κ takes the triangle tip corresponding to the right half of the cross section to the left of η𝜂\etaitalic_η (into A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT), and the tip of the left part A2subscript𝐴2A_{2}italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of R𝑅Ritalic_R to the left.

Furthermore, there is a one parametric family of flows on S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, deforming continuously the Lorenz equations at the trefoil parameter to the extended geometric model X1subscript𝑋1X_{1}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. For any flow along the deformation there exists a cross section, so that the return map corresponds to a symbolic dynamics on two symbols.

A

The C2 move shown in Figure 3 performs a Finger move on the clearing strand, shifting the lower part of the vertical strand to the right.

It is enough to prove that altering the direction of a single C2 move yields an arc-equivalent result.

It is straightforward to show that each of our moves relating arc diagrams can be expressed as a composition of shadow moves on the associated shadow cones, establishing the following result as a corollary of Theorem 4.1:

A subtlety in the proof of the Reordering Lemma 4.4 is that when two trees on adjacent faces in 𝕊𝕊\mathbb{S}blackboard_S share a root – Case 2 of Remark 4.2 – swapping the order in which they are cleared does change the resulting arc diagram. However, these are arc-equivalent by the Finger Change Lemma 4.3.

As a corollary of the Finger Change Lemma 4.3, we prove the unsurprising result that this choice does not matter:

D

The work [20] considered the extra constraint of private/secure multiplication and the work [21] considered a trade-off between the size of the entries of the matrix and the recovery threshold.

The works [22, 23] considered coding for distributed gradient descent in machine learning applications.

The work [20] considered the extra constraint of private/secure multiplication and the work [21] considered a trade-off between the size of the entries of the matrix and the recovery threshold.

Machine learning has become a dominant tool in the broader computing community. Its success has been due to the availability of large datasets and more recently the use of hardware optimized to perform multi-linear functions such as the GPU and the TPU,

The works [4, 5, 6] established the use of coding theory for distributed coded matrix-vector and vector-vector multiplication; shortly after and concurrently, the works [1, 7] further improved on these techniques by using alignment to come up with efficient constructions for general matrix-matrix multiplication.

A

Over the years, this resource-theoretic viewpoint has further sparked the development of other resource theories [10, 11, 12, 13, 14, 15, 16, 17], aiming, e.g., for the quantification  [18, 19, 20, 21, 22, 23] of resources and their inter-convertibility [24, 25].

Finally, the allowed dynamics are thermal operations that cannot drive such states out of thermal equilibrium (see, e.g., Refs. [26, 28, 27] for details).

These features are shared by many resources, such as entanglement, coherence, athermality, asymmetry, steerability, and nonlocality (see Appendix A.1).

For quantum states (and correlations), examples of such resources include, but not limited to, entanglement [9, 18], coherence [11, 20], athermality [26, 28, 27], asymmetry [29, 30], nonlocality [31], and steering [32, 33].

Over the years, this resource-theoretic viewpoint has further sparked the development of other resource theories [10, 11, 12, 13, 14, 15, 16, 17], aiming, e.g., for the quantification  [18, 19, 20, 21, 22, 23] of resources and their inter-convertibility [24, 25].

C

In this section, we leverage the ubiquitous change-of-measure argument for deriving impossibility results to derive an instance-dependent lower bound on the expected sample complexity 𝔼⁡[τδ]𝔼subscript𝜏𝛿\operatorname{\mathbb{E}}[\tau_{\delta}]blackboard_E [ italic_τ start_POSTSUBSCRIPT italic_δ end_POSTSUBSCRIPT ] for the online clustering problem. The lower bound is closely related to a combinatorial optimization problem.

In Section 4, we derive an instance-dependent (information-theoretic) lower bound on the expected sample complexity for the online clustering problem; this lower bound, however, involves a tricky optimization problem. By exploiting the structure of the problem and leveraging an interesting combinatorial property, we simplify the optimization to a finite convex minimax problem, which can be solved efficiently. Further analyses of the lower bound provide fundamental insights and essential tools for the design of our algorithm.

We refer to D∗⁢(c,𝒰)superscript𝐷𝑐𝒰D^{*}(c,\mathcal{U})italic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_c , caligraphic_U ) as the hardness parameter of the online clustering task in the sequel. The asymptotic version of the instance-dependent lower bound given in Equation (2) in Theorem 3 is tight in view of the expected sample complexity of the efficient algorithm we present in Section 5. Intuitively, any λ∈𝒫M𝜆subscript𝒫𝑀{\lambda\in\mathcal{P}_{M}}italic_λ ∈ caligraphic_P start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT in Equation (1) can be understood as the proportion of arm pulls, which inspires the design of the sampling rule of a δ𝛿\deltaitalic_δ-PAC online clustering algorithm. The agent wishes to find the optimal proportion of arm pulls to distinguish the instance c𝑐citalic_c from the most confusing alternative instances in Alt⁢(c)Alt𝑐\mathrm{Alt}(c)roman_Alt ( italic_c ) (for which c𝑐citalic_c is not a correct partition). Therefore, with the knowledge of the instance (c,𝒰)𝑐𝒰(c,\mathcal{U})( italic_c , caligraphic_U ), the optimization problem embedded in (1) naturally unveils the optimal sampling rule, which is the basic idea behind the design of our sampling rule in Section 5.

A by-product of Proposition 7 is that the outer supremum in Problem (□□\Box□) can be replaced with a maximum. Intuitively, the maximizer of Problem (□□\Box□) in 𝒫Msubscript𝒫𝑀\mathcal{P}_{M}caligraphic_P start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT represents the optimal proportion of arm pulls, which will be of considerable importance in our design of the sampling rule. In fact, there exists a bijection between the solution to the finite convex minimax problem above and Problem (□□\Box□), as shown in Proposition 8 below. Although the finite convex minimax problem is not strictly convex in w∈𝒫K+𝑤superscriptsubscript𝒫𝐾{w\in\mathcal{P}_{K}^{+}}italic_w ∈ caligraphic_P start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, Proposition 8 states that the solution to Problem (□□\Box□) is unique. This, together with Proposition 9 concerning the continuity of the solution to Problem (□□\Box□), guarantees the computationally efficiency and the asymptotic optimality of our sampling rule in Section 5. The proofs of Proposition 8 and Proposition 9 are deferred to Appendices B.6 and B.7, respectively.

Although the optimization in its original form appears to be intractable, we prove an interesting combinatorial property and reformulate the optimization as a finite convex minimax problem. Moreover, we further present some results on the computation and other useful properties (e.g., the continuity of the optimizer and the optimal value) of the optimization problem (and its sub-problem) embedded in the lower bound, which are fundamental and essential in our algorithm design (see Section 5).

D

As a consequence, if ρ0subscript𝜌0\rho_{0}italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is uniformly positive in ΩΩ\Omegaroman_Ω, then ρ𝜌\rhoitalic_ρ is uniformly positive in Ω×(0,∞)Ω0\Omega\times(0,\infty)roman_Ω × ( 0 , ∞ ).

In this case we observe that the solution converges towards the constant steady state given by the average mass ρ∞subscript𝜌\rho_{\infty}italic_ρ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT, see Figure 1. The convergence is faster around the center of the mass distribution, where the solution takes the largest values, and it is slowest close to the vertexes of the rectangular domain Ω=[−1,1]2Ωsuperscript112\Omega=[-1,1]^{2}roman_Ω = [ - 1 , 1 ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. While the solution remains very small around such points for some time, eventually the combination of diffusion in the supercritical region and mass conservation pushes the solution everywhere towards the constant steady state. Such a behavior is to be expected from the analysis result in Theorem 2.

The algebraic rate of convergence with respect to the negative Sobolev seminorm follows from a duality argument and assumption (12), which describes the behavior of f𝑓fitalic_f near the critical value. We wish to point out that the obtained algebraic decay rate resembles closely the result holding for the porous medium equation in the whole space [22], which is in agreement with the intuition that the dynamics of (1) under assumption (12) should resemble a porous medium equation in the region close to the critical value. Finally, in the critical case ρ∞=ρc⁢rsubscript𝜌subscript𝜌𝑐𝑟\rho_{\infty}=\rho_{cr}italic_ρ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_c italic_r end_POSTSUBSCRIPT we are not able to show a suitable Poincaré-Wirtinger-type inequality and thus we cannot derive a decay estimate for the solution. However, on the other hand, we can deduce strong Lp⁢(Ω)superscript𝐿𝑝ΩL^{p}(\Omega)italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( roman_Ω ) convergence towards the steady state for any p<∞𝑝p<\inftyitalic_p < ∞ by exploiting the free energy estimate and mass conservation.

The following lemma yields the exponential convergent rate for the solution in the supercritical case.

In both cases we observe that the evolution of the system is driven by the diffusion in the supercritical region (i.e. the region of ΩΩ\Omegaroman_Ω where the solution is larger than ρc⁢rsubscript𝜌𝑐𝑟\rho_{cr}italic_ρ start_POSTSUBSCRIPT italic_c italic_r end_POSTSUBSCRIPT) together with mass conservation. Indeed, while clearly the diffusion in the supercritical region pushes the mass downwards towards the critical density, this process drives the evolution of the solution in the whole spatial domain thanks to mass conservation: the mass distribution in the subcritical region (where the solution is smaller than ρc⁢rsubscript𝜌𝑐𝑟\rho_{cr}italic_ρ start_POSTSUBSCRIPT italic_c italic_r end_POSTSUBSCRIPT) is pushed aside and upwards by the mass coming down from the supercritical region. We also observe that in the supercritical case these two processes are sufficiently strong to bring the whole mass distribution to the constant steady state ρ∞subscript𝜌\rho_{\infty}italic_ρ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT, while in the subcritical case the evolution of the system stops before achieving this convergence: the solution is simply pushed by the diffusion towards a non-constant profile which lies entirely in the subcritical region, and after such point the solution do not change any more. We remark that these phenomena can be already observed after a rather short simulation time.

C

Counting maximum independent sets is #⁢𝐏normal-#𝐏\mathbf{\#P}# bold_P-complete on line graphs of bipartite graphs.

(the undirected graph underlying) D𝐷Ditalic_D is a cograph, as the join of a complete graph on 𝒞𝒞\mathcal{C}caligraphic_C and a perfect matching on L⊔{α,ω}square-union𝐿𝛼𝜔L\sqcup\{\alpha,\omega\}italic_L ⊔ { italic_α , italic_ω };

Counting maximum independent sets is #⁢𝐏normal-#𝐏\mathbf{\#P}# bold_P-complete on line graphs of bipartite graphs.

An independent set of a graph is maximal when it has no independent superset, and maximum when its cardinality is largest among independent sets. Clearly, the latter entails the former.

Let G𝐺Gitalic_G be a bipartite graph. In polynomial time, we decide whether G𝐺Gitalic_G has perfect matchings by building a largest matching (e.g. by the Ford–Fulkerson algorithm) and checking whether it is perfect. In the affirmative, the perfect matchings of G𝐺Gitalic_G are the maximum independent sets of its line graph. Now an algorithm that counts the latter also counts the former, but counting the perfect matchings of bipartite graphs is #⁢𝐏#𝐏\mathbf{\#P}# bold_P-complete [Val79].

D

Using the transformation rule under biholomorphism and the product property of generalized Bergman kernels (see [10]), one can construct more and more examples from (1), (2) and (3). It would be interesting to find a full characterization for the equality of (7.8).

By elementary calculations in terms of some local trivializations, it is not hard to prove to following lemmas222Following the referees’ suggestions, we have removed the lengthy proofs of Lemma 4.1 and 4.2 in the final version. Interested readers can refer to the previous version (v2) for the detailed proofs..

Acknowledgements The authors would like to thank the referees for their helpful comments and suggestions.

To authors’ knowledge, the only known example for the equality case is the biholomorphic image of a balanced domain (with a possible closed pluripolar set removed).

Using the transformation rule under biholomorphism and the product property of generalized Bergman kernels (see [10]), one can construct more and more examples from (1), (2) and (3). It would be interesting to find a full characterization for the equality of (7.8).

B

α1456−α1234−α2356subscript𝛼1456subscript𝛼1234subscript𝛼2356\displaystyle\alpha_{1456}-\alpha_{1234}-\alpha_{2356}italic_α start_POSTSUBSCRIPT 1456 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT 1234 end_POSTSUBSCRIPT - italic_α start_POSTSUBSCRIPT 2356 end_POSTSUBSCRIPT

𝕋6::subscript𝕋6absent\displaystyle\mathbb{T}_{6}:blackboard_T start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT :

𝕋1::subscript𝕋1absent\displaystyle\mathbb{T}_{1}:blackboard_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT :

𝕋4::subscript𝕋4absent\displaystyle\mathbb{T}_{4}:blackboard_T start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT :

𝕋3::subscript𝕋3absent\displaystyle\mathbb{T}_{3}:blackboard_T start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT :

A

On the other hand, big-M formulations only introduce one binary variable for each disjunct, resulting in smaller problems in terms of both number of variables and constraints. The big-M formulation also preserves the constraint expressions, whereas the extended convex hull builds upon the perspective function, which can result in numerically more challenging constraints. In general, big-M formulations provide weaker continuous relaxations than the convex hull and may require a solver to explore significantly more nodes in a branch-and-bound tree [21, 67].

But the computationally simpler subproblems of the big-M formulation can offset the larger number of explored nodes. Anderson et al. [2] mention a folklore observation that extended convex hull formulations tend to perform worse than expected. The observation is supported by past numerical results [2, 38, 63, 66], as well as this paper.

The numerical results show a great potential of the P𝑃Pitalic_P-split formulations, by providing a good approximation of the convex hull through a computationally simpler problem. For several the test problems, the intermediate formulations result in a similar number of explored nodes as the convex hull formulation, while reducing the total solution time by an order of magnitude. It is worth mentioning that problems in the computational experiments all have the desired structure described in Assumptions 1–4. Without this structure, the P𝑃Pitalic_P-split formulations do not necessarily have an advantage over either the big-M or convex hull formulation.

The results for the clustering problems are also interesting. The P𝑃Pitalic_P-split formulations seem particularly well-suited for this class of problems and clearly outperform both the convex hull and big-M formulations in terms of computational time. For these problems, the n𝑛nitalic_n-split formulation, i.e., |ℐs|=1subscriptℐ𝑠1|\mathcal{I}_{s}|=1| caligraphic_I start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT | = 1, has the same number of variables as the extended convex hull formulation. But, as these problems have nonlinear constraints in the disjuncts, the n𝑛nitalic_n-split formulation will not necessarily be as strong as the convex hull. However, as the n𝑛nitalic_n-split formulation effectively linearizes the disjunctions, we end up with simpler constraints. Specifically, to express the convex hull, we need rotated second-order cone constraints, while the n𝑛nitalic_n-split formulation is represented through linear and convex quadratic constraints.

The results show that the P𝑃Pitalic_P-split formulations can have a computational advantage over both the classical big-M formulation and the convex hull formulation. The advantage comes from the continuous relaxation being tighter than the relaxation obtained by big-M, while the subproblems remain computationally cheaper then those originating from the convex hull. The results support the hypothesis that P𝑃Pitalic_P-split formulations can well-approximate the convex hull, as the number of explored nodes are similar with both types of formulations for many of the problems. For several of the problem classes, there is a clear trade-off where the intermediate P𝑃Pitalic_P-split formulations, that are not as tight as possible but with fewer variables and constraints, result in the best computational performance. For the different classes of problems, the optimal value of P𝑃Pitalic_P varies. However, the various P𝑃Pitalic_P-split formulations perform well overall, and for some classes all the P𝑃Pitalic_P-split formulations we tested outperform both big-M and convex hull formulations.

A

For each i∈I0𝑖subscript𝐼0i\in I_{0}italic_i ∈ italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, define

Yj⁢kx,i=1n⁢tj⁢k→11x,i⁢∑i′ti′→1y,z,j,k,∀(j,k)∈[m]×[l].formulae-sequencesubscriptsuperscript𝑌𝑥𝑖𝑗𝑘1𝑛subscriptsuperscript𝑡𝑥𝑖→𝑗𝑘11subscriptsuperscript𝑖′subscriptsuperscript𝑡𝑦𝑧𝑗𝑘→superscript𝑖′1for-all𝑗𝑘delimited-[]𝑚delimited-[]𝑙Y^{x,i}_{jk}=\frac{1}{n}t^{x,i}_{jk\rightarrow 11}\sum_{i^{\prime}}{t^{y,z,j,k%

=1n⁢m⁢l⁢(∑j,k𝔼⁡[tj⁢k→11x,i]⁢∑i′𝔼⁡[ti′→1y,z,j,k]−α⁢(∑j,k𝔼⁡[tj⁢k→11x,i])⁢(1m⁢l⁢∑i′,j,k𝔼⁡[ti′→1y,z,j,k]))absent1𝑛𝑚𝑙subscript𝑗𝑘𝔼subscriptsuperscript𝑡𝑥𝑖→𝑗𝑘11subscriptsuperscript𝑖′𝔼subscriptsuperscript𝑡𝑦𝑧𝑗𝑘→superscript𝑖′1𝛼subscript𝑗𝑘𝔼subscriptsuperscript𝑡𝑥𝑖→𝑗𝑘111𝑚𝑙subscriptsuperscript𝑖′𝑗𝑘𝔼subscriptsuperscript𝑡𝑦𝑧𝑗𝑘→superscript𝑖′1\displaystyle=\frac{1}{nml}\Big{(}\sum_{j,k}{\operatorname{\mathbb{E}}[t^{x,i}%

1n⁢m⁢l⁢(∑j,k(tj⁢k→11x,i⁢∑i′ti′→1y,z,j,k)−α⁢(∑j,ktj⁢k→11x,i)⁢(1m⁢l⁢∑i′,j,kti′→1y,z,j,k))≥0.1𝑛𝑚𝑙subscript𝑗𝑘subscriptsuperscript𝑡𝑥𝑖→𝑗𝑘11subscriptsuperscript𝑖′subscriptsuperscript𝑡𝑦𝑧𝑗𝑘→superscript𝑖′1𝛼subscript𝑗𝑘subscriptsuperscript𝑡𝑥𝑖→𝑗𝑘111𝑚𝑙subscriptsuperscript𝑖′𝑗𝑘subscriptsuperscript𝑡𝑦𝑧𝑗𝑘→superscript𝑖′10\displaystyle\frac{1}{nml}\Big{(}\sum_{j,k}{\Big{(}t^{x,i}_{jk\rightarrow 11}%

Yj′⁢k′x,i:=1n⁢tj′,k′→11x,i⁢∑i′ti′→1y,z,j′,k′,∀j′∈[m],k′∈[l].formulae-sequenceassignsubscriptsuperscript𝑌𝑥𝑖superscript𝑗′superscript𝑘′1𝑛subscriptsuperscript𝑡𝑥𝑖→superscript𝑗′superscript𝑘′11subscriptsuperscript𝑖′subscriptsuperscript𝑡𝑦𝑧superscript𝑗′superscript𝑘′→superscript𝑖′1formulae-sequencefor-allsuperscript𝑗′delimited-[]𝑚superscript𝑘′delimited-[]𝑙Y^{x,i}_{j^{\prime}k^{\prime}}:=\frac{1}{n}t^{x,i}_{j^{\prime},k^{\prime}%

A

∑k=1n𝟏B⁢(x,rk)⁢(Tk⁢(x))∼∑k=1nμ⁢(B⁢(x,rk))similar-tosuperscriptsubscript𝑘1𝑛subscript1𝐵𝑥subscript𝑟𝑘superscript𝑇𝑘𝑥superscriptsubscript𝑘1𝑛𝜇𝐵𝑥subscript𝑟𝑘\sum_{k=1}^{n}\mathbf{1}_{B(x,r_{k})}(T^{k}(x))\sim\sum_{k=1}^{n}\mu(B(x,r_{k}))∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT bold_1 start_POSTSUBSCRIPT italic_B ( italic_x , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_x ) ) ∼ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_μ ( italic_B ( italic_x , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )

Boshernitzan proved that if (X,T,μ)𝑋𝑇𝜇(X,T,\mu)( italic_X , italic_T , italic_μ ) is a measure preserving

For a measure preserving dynamical system ([0,1],T,μ)01𝑇𝜇([0,1],T,\mu)( [ 0 , 1 ] , italic_T , italic_μ ), we say

measure μ𝜇\muitalic_μ such that correlations for L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT against B⁢V𝐵𝑉BVitalic_B italic_V

we have for all x𝑥xitalic_x that Tαk⁢(x)∈B⁢(x,rk)superscriptsubscript𝑇𝛼𝑘𝑥𝐵𝑥subscript𝑟𝑘T_{\alpha}^{k}(x)\in B(x,r_{k})italic_T start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_x ) ∈ italic_B ( italic_x , italic_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) for

B

Let G𝐺Gitalic_G temporarily denote one of the groups 𝐇𝐇\mathbf{H}bold_H or 𝐆𝐆\mathbf{G}bold_G, and denote by RepF⁡(G)subscriptRep𝐹𝐺\operatorname{Rep}_{F}(G)roman_Rep start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_G ) the category of finite dimensional algebraic representations of G𝐺Gitalic_G defined over F𝐹Fitalic_F. In [Anc15], Ancona constructs an additive functor

μUG:RepF⁡(G)⟶CHMF⁡(ShG⁡(U)),:subscriptsuperscript𝜇𝐺𝑈⟶subscriptRep𝐹𝐺subscriptCHM𝐹subscriptSh𝐺𝑈\mu^{G}_{U}:\operatorname{Rep}_{F}(G)\longrightarrow\operatorname{CHM}_{F}({%

Let G𝐺Gitalic_G temporarily denote one of the groups 𝐇𝐇\mathbf{H}bold_H or 𝐆𝐆\mathbf{G}bold_G, and denote by RepF⁡(G)subscriptRep𝐹𝐺\operatorname{Rep}_{F}(G)roman_Rep start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_G ) the category of finite dimensional algebraic representations of G𝐺Gitalic_G defined over F𝐹Fitalic_F. In [Anc15], Ancona constructs an additive functor

G⁢(𝐐)\(XG×V×(G⁢(𝐀f)/U))→ShG⁡(U)⁢(𝐂).→\𝐺𝐐subscript𝑋𝐺𝑉𝐺subscript𝐀𝑓𝑈subscriptSh𝐺𝑈𝐂G({\bf Q})\backslash(X_{G}\times V\times(G(\mathbf{A}_{f})/U))\to{%

If V𝑉Vitalic_V is the standard representation of G𝐺Gitalic_G, then μUG⁢(V)=h1⁢(𝒜G)subscriptsuperscript𝜇𝐺𝑈𝑉superscriptℎ1subscript𝒜𝐺\mu^{G}_{U}(V)=h^{1}(\mathscr{A}_{G})italic_μ start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_V ) = italic_h start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( script_A start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ), where 𝒜Gsubscript𝒜𝐺\mathscr{A}_{G}script_A start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is the universal abelian scheme over ShG⁡(U)subscriptSh𝐺𝑈{\operatorname{Sh}}_{G}(U)roman_Sh start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_U ).

A

In Section 2, using the Euler’s substitution, we motivate the definition of critical points of ℰℰ\mathcal{E}caligraphic_E in M⁢(ψ)𝑀𝜓M(\psi)italic_M ( italic_ψ ).

By way of the results in Section 3.1 we show the following for critical points of ℰℰ\mathcal{E}caligraphic_E in M⁢(ψ)𝑀𝜓M(\psi)italic_M ( italic_ψ ): Nondegeneracy (Section 3.2); properties of the coincidence set (Section 3.3); optimal regularity (Section 3.4).

As we mentioned in Section 6.1, u1subscript𝑢1u_{1}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and u2subscript𝑢2u_{2}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are also critical points of ℰpsubscriptℰ𝑝\mathcal{E}_{p}caligraphic_E start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT in M⁢(ψ)𝑀𝜓M(\psi)italic_M ( italic_ψ ) in the sense of Definition 1.2.

In Section 3, we study the optimal regularity and qualitative properties of critical points which are introduced in Section 2.

To study the optimal regularity of minimizers of problem (1.5), we introduce a notion of critical points of problem (1.5).

C

𝒪2subscript𝒪2\operatorname{\mathcal{O}}_{2}caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of optimal strategies of the two players are

𝒪1subscript𝒪1\displaystyle\operatorname{\mathcal{O}}_{1}caligraphic_O start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

Explicit LMI descriptions of the sets 𝒪1subscript𝒪1\operatorname{\mathcal{O}}_{1}caligraphic_O start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and

𝒪2subscript𝒪2\displaystyle\operatorname{\mathcal{O}}_{2}caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

Note that 𝒪1subscript𝒪1\mathcal{O}_{1}caligraphic_O start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝒪2subscript𝒪2\mathcal{O}_{2}caligraphic_O start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are spectrahedra in

A

{\alpha}(x^{n})\,d\theta\right\}^{1/\alpha}}.italic_W start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( caligraphic_P ) ≜ roman_sup start_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∈ caligraphic_X start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_log divide start_ARG roman_sup start_POSTSUBSCRIPT italic_θ ∈ roman_Θ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_ARG start_ARG { ∫ start_POSTSUBSCRIPT roman_Θ end_POSTSUBSCRIPT italic_w ( italic_θ ) italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) italic_d italic_θ } start_POSTSUPERSCRIPT 1 / italic_α end_POSTSUPERSCRIPT end_ARG .

Starting from (11) and substituting the definition of α𝛼\alphaitalic_α-NML given by Equation (27), we have

In this paper, we introduced a new class of general predictors dependent on a real parameter α≥1𝛼1\alpha\geq 1italic_α ≥ 1, which is shown to interpolate to mixture predictors and the NML. The idea for this class of predictors comes from the connection between the worst-case regret achieved by the NML predictor, and Sibson’s α𝛼\alphaitalic_α-mutual information. We proved the optimality of α𝛼\alphaitalic_α-NML under α𝛼\alphaitalic_α-regret, a general regret measure linked to Rényi divergence, that interpolates between the classical average and worst-case regrets. Also, we discussed examples that prove the broad applicability of α𝛼\alphaitalic_α-NML also for families of distributions for which the NML does not exist, and we compared α𝛼\alphaitalic_α-NML to other alternatives to NML such as Luckiness NML. Furthermore, we showed that for the popular family of discrete memoryless sources, one is able to derive some simple formulas to compute the probabilities estimated by the new class of predictors, when the parameter α𝛼\alphaitalic_α is a positive integer. For this class of distributions, we also derive an asymptotic expression for the worst-case regret of α𝛼\alphaitalic_α-NML, which interpolates between those of the Krichevsky-Trofimov estimator and the NML.

The remainder of the paper is organized as follows. In Section II we introduce α𝛼\alphaitalic_α-regret, which is defined in terms of Rényi divergence and described from an operational point of view. In Section III, we introduce the class of α𝛼\alphaitalic_α-NML predictors and we prove their optimality under α𝛼\alphaitalic_α-regret. In Section IV, we discuss the advantages of using α𝛼\alphaitalic_α-NML as an alternative to NML, when the latter cannot be used, and we compare it to other alternatives to NML such as Luckiness NML. In Section V, we apply α𝛼\alphaitalic_α-NML to the parametric family of discrete memoryless sources (DMS), deriving some simple closed-form formulae to compute the probabilities estimated by the predictor, and studying its performance in terms of worst-case regret, discussing how much we pay by using α𝛼\alphaitalic_α-NML instead of NML in this setting.

A possible interpolation between the Luckiness NML defined in (42) and the mixture predictor in (71) is again given by α𝛼\alphaitalic_α-NML of Equation (27), if one chooses the proper prior distribution w𝑤witalic_w. In fact, for any given α≥1𝛼1\alpha\geq 1italic_α ≥ 1, one can take the tilted prior distribution

A

\Phi(0)&\Phi(Q)&\Phi(Q)\end{bmatrix}\geq 0.[ start_ARG start_ROW start_CELL roman_Φ ( italic_P - italic_Q ) end_CELL start_CELL roman_Φ ( italic_P - italic_Q ) end_CELL start_CELL roman_Φ ( 0 ) end_CELL end_ROW start_ROW start_CELL roman_Φ ( italic_P - italic_Q ) end_CELL start_CELL roman_Φ ( italic_P ) end_CELL start_CELL roman_Φ ( italic_Q ) end_CELL end_ROW start_ROW start_CELL roman_Φ ( 0 ) end_CELL start_CELL roman_Φ ( italic_Q ) end_CELL start_CELL roman_Φ ( italic_Q ) end_CELL end_ROW end_ARG ] ≥ 0 .

If ε>0𝜀0\varepsilon>0italic_ε > 0 is a real number, then it follows from the positivity of above matrix that

It follows from the positivity of the above matrix and the commutativity of the range of ΦΦ\Phiroman_Φ that

Moreover, If the range of ΦΦ\Phiroman_Φ is commutative, then the positivity of matrix (4.8) immediately implies (4.4).

(2) ⟹⟹\Longrightarrow⟹ (3). It is immediately follows from (2) according to the fact that the involution is

A

So, given a 2222-interval-PCG, Lemma 1 guarantees that one of its witness trees is a full binary tree and Lemma 2 ensures that its edge-weight function and the two interval use integer values.

In [4] an ILP model to determine whether a given graph is a PCG with respect to a given tree structure is provided.

in Section 3 we generalize some results proved for PCGs to let them work for 2222-interval-PCGs, too, and we exploit an ILP model to determine whether a given graph is a 2222-interval-PCG with respect to a given tree structure;

More precisely, given a graph G𝐺Gitalic_G and an unweighted tree T𝑇Titalic_T, the model determines (if they exist) an edge-weight function w𝑤witalic_w for T𝑇Titalic_T,

Since the solution of the ILP model produces an edge-weighted tree and two intervals witnessing that the input graph is 2222-interval-PCG, we apply it to all 7 graphs of Figure 1 with two special tree structures: complete binary trees and caterpillars.

A

Let f⁢(z)=∑n=0∞an⁢zn∈𝕏⁢(ℂ)𝑓𝑧superscriptsubscript𝑛0subscript𝑎𝑛superscript𝑧𝑛𝕏ℂf(z)=\sum\limits_{n=0}^{\infty}a_{n}z^{n}\in\mathbb{X}(\mathbb{C})italic_f ( italic_z ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∈ blackboard_X ( blackboard_C ) be a formal power series over ℂℂ\mathbb{C}blackboard_C. Then there exists such a formal power series g∈𝕏⁢(ℂ)𝑔𝕏ℂg\in\mathbb{X}(\mathbb{C})italic_g ∈ blackboard_X ( blackboard_C ) that f⁢g=S1𝑓𝑔subscript𝑆1fg=S_{1}italic_f italic_g = italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, if and only if a0≠0subscript𝑎00a_{0}\neq 0italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≠ 0.

By Corollary 4.6, we obtain an inverse of a formal Laurent series satisfying (1)-(3) (if it has an inverse). However, we know a formal Laurent series may have more than one inverse. The following result gives the method of finding all inverses of a formal Laurent series f𝑓fitalic_f satisfying (1)-(3):

Formal Laurent series are a natural extension of formal power series. However, one should point out that their properties are quite different from the analogous properties of formal power series (see e.g. [14]). For example, in general, the product of two given formal Laurent series does not have to exist and the inverse in the set of formal Laurent series does not have to be unique, unlike to formal power series (see [12], Thm. 1.1.9.).

However, this case is much different in the set of formal Laurent series. First, we are going to show that if f𝑓fitalic_f is a formal Laurent series, then one of the following situations occurs:

In this paper we are going mainly to focus on the inverse and the composition in the set of formal Laurent series. First, we establish that every formal Laurent series has either no inverse series, a unique inverse series or uncountably many inverse series. We show the connection between this fact and the lack of associativity of multiplication in the set of formal Laurent series (see Prop. 3.5). Next, we provide some necessary and sufficient conditions for a certain type formal Laurent series to have no inverse or to have one, or to have infinitely many inverses. For that purpose we use the theory of infinite systems of linear algebraic equations, investigated for example in [7, 17, 8]. We also define a general composition of formal Laurent series and use it to examine the Right Distributive Law and the Chain Rule, which hold if one considers formal power series ([10]). In the last section of this paper we examine the boundary convergence of formal Laurent series. We extend some results concerning this problem proved in [13].

C

The notation HmG⁢(X;𝐋ℤ⟨j⟩)superscriptsubscript𝐻𝑚𝐺𝑋subscriptsuperscript𝐋delimited-⟨⟩𝑗ℤH_{m}^{G}\left(X;\mathbf{L}^{\langle j\rangle}_{\mathbb{Z}}\right)italic_H start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT ( italic_X ; bold_L start_POSTSUPERSCRIPT ⟨ italic_j ⟩ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_Z end_POSTSUBSCRIPT ) denotes the Davis-Lück equivariant homology as mentioned above whereas the notation Hm⁢(X;L⟨j⟩⁢(ℤ))subscript𝐻𝑚𝑋superscript𝐿delimited-⟨⟩𝑗ℤH_{m}\left(X;L^{\langle j\rangle}\left(\mathbb{Z}\right)\right)italic_H start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_X ; italic_L start_POSTSUPERSCRIPT ⟨ italic_j ⟩ end_POSTSUPERSCRIPT ( blackboard_Z ) ) denotes the generalized homology of X𝑋Xitalic_X with coefficients in the spectrum L⟨j⟩⁢(ℤ)superscript𝐿delimited-⟨⟩𝑗ℤL^{\langle j\rangle}\left(\mathbb{Z}\right)italic_L start_POSTSUPERSCRIPT ⟨ italic_j ⟩ end_POSTSUPERSCRIPT ( blackboard_Z ).

A family of subgroups is a nonempty set ℱℱ\mathcal{{F}}caligraphic_F of subgroups closed under taking subgroups and conjugation.

The primary families we will consider are 𝒱⁢c⁢y⁢c𝒱𝑐𝑦𝑐\mathcal{{V}}cyccaligraphic_V italic_c italic_y italic_c, the collection of virtually cyclic subgroups, and ℱ⁢i⁢nℱ𝑖𝑛\mathcal{{F}}incaligraphic_F italic_i italic_n, the collection of finite subgroups.

Let 𝒫𝒫\mathcal{{P}}caligraphic_P denote the set of conjugacy classes of maximal finite subgroups of ΓΓ\Gammaroman_Γ.

In the theorems below 𝒫𝒫\mathcal{{P}}caligraphic_P will denote the set of conjugacy classes of nontrivial finite subgroups of ΓΓ\Gammaroman_Γ (all of which are isomorphic to ℤ/pℤ𝑝\mathbb{Z}/pblackboard_Z / italic_p).

A

(E)italic_λ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT := italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( caligraphic_O start_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) end_POSTSUBSCRIPT ( 1 ) ) - divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_π start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_E ).

hand, they also exhibit a threefold X𝑋Xitalic_X for which vol(ξ)X{}_{X}(\xi)start_FLOATSUBSCRIPT italic_X end_FLOATSUBSCRIPT ( italic_ξ ) is not piecewise polynomial (although it is locally analytic in this example). It remains a very

We give several examples to illustrate various applications of Theorem 1.1 (see Example 4.2, Example 4.3, Example 4.4 etc).

We now discuss the pseudo-effective cones Eff¯1⁢(ℙX⁢(E))superscript¯Eff1subscriptℙ𝑋𝐸\overline{\operatorname{{Eff}}}^{1}\bigl{(}\mathbb{P}_{X}(E)\bigr{)}over¯ start_ARG roman_Eff end_ARG start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_P start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) ) of some projective bundles ℙX⁢(E)subscriptℙ𝑋𝐸\mathbb{P}_{X}(E)blackboard_P start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) where the base variety X𝑋Xitalic_X has Picard number ρ⁢(X)≥2𝜌𝑋2\rho(X)\geq 2italic_ρ ( italic_X ) ≥ 2 or E𝐸Eitalic_E is not semistable. Note that in these cases ρ⁢(ℙX⁢(E))=ρ⁢(X)+1≥3𝜌subscriptℙ𝑋𝐸𝜌𝑋13\rho\bigl{(}\mathbb{P}_{X}(E)\bigr{)}=\rho(X)+1\geq 3italic_ρ ( blackboard_P start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) ) = italic_ρ ( italic_X ) + 1 ≥ 3 and hence these cones in three dimensional real vector spaces may not be finite polyhedra type in general. We give an example of a projective bundle ℙX⁢(E)subscriptℙ𝑋𝐸\mathbb{P}_{X}(E)blackboard_P start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_E ) having non-polyhedral pseudo-effective cone.

(9). These natural isomorphisms do not depend on the choice of the splitting, and therefore glue together to give the global isomorphism

B

In general, it is not known whether the Weil classes are algebraic, but in the case of abelian fourfolds of Weil type with discriminant one, the following holds:

Let A𝐴Aitalic_A be abelian fourfold of Weil type with discriminant one. Then, the Weil classes on A𝐴Aitalic_A are algebraic. In particular, the Hodge conjecture holds for the general abelian fourfold of Weil type with discriminant one.

Let A𝐴Aitalic_A be a general abelian fourfold of K𝐾Kitalic_K-Weil type with discriminant one. If the Kuga–Satake Hodge conjecture holds for the corresponding K3 surface, then the Weil classes on A𝐴Aitalic_A are algebraic. Thus, the Hodge conjecture holds for A𝐴Aitalic_A and, hence, for all powers Aksuperscript𝐴𝑘A^{k}italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT.

Let X𝑋Xitalic_X be a general K3 surface of Picard number 16161616 as in Theorem 0.2, and denote by A𝐴Aitalic_A the abelian fourfold of Weil type with discriminant one appearing as simple factor of the Kuga–Satake variety of X𝑋Xitalic_X. Then, if the Kuga–Satake correspondence is algebraic for X𝑋Xitalic_X, the Weil classes on A𝐴Aitalic_A are algebraic. Thus, the Hodge conjecture holds for A𝐴Aitalic_A and, hence, for all powers Aksuperscript𝐴𝑘A^{k}italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT.

In general, it is not known whether the Weil classes are algebraic, but in the case of abelian fourfolds of Weil type with discriminant one, the following holds:

A

Aside of various MHD specific constructions, this article combines the return method and the well-prepared dissipation method as described by Coron et al. in [CoronMarbachSueur2020], where the small-time global exact controllability to trajectories has been studied for incompressible Navier–Stokes equations in two- and three-dimensional domains with Navier slip-with-friction conditions. Meanwhile, we shall also extend certain asymptotic expansions, obtained by Iftimie and Sueur in [IftimieSueur2011] for the incompressible Navier–Stokes equations, to the present MHD system. Due to the structure of the induction equation, the return method has to be carefully implemented in order to avoid generating pressure-like and additional forcing terms in the induction equation. To this end, under the assumptions of 1.2, we modify the return method trajectory from [CoronMarbachSueur2020] to be everywhere divergence-free, but allow a nonzero curl in the control region; this approach seems new and might be useful for further studies on the controllability of the ideal MHD equations.

The Navier slip-with-friction boundary conditions, as already proposed by Navier [Navier1823] two centuries ago, are relevant to a range of applications, thus have been studied in the context of the Navier–Stokes equations from various points of view. For instance, in the absence of magnetic fields, inviscid limit problems are treated in [IftimieSueur2011, ClopeauMikelicRobert1998, Kelliher2006, XiaoXin2013], regularity questions are investigated in [AmrouchePenelSeloula2013, AlBabaAmroucheEscobedo2017, Shibata2007, Shimada2007, AlBaba2019] and controllability problems are tackled in [CoronMarbachSueur2020, Guerrero2006, LionsZuazua1998, Coron1996]. Concerning the situation of incompressible viscous MHD, several singular limit problems involving uncoupled Navier slip-with-friction boundary conditions are addressed in [GuoWang2016, XiaoXinWu2009, MengWang2016]; comparing with these references, the here employed boundary conditions are more general in that the shear stresses of the velocity and the magnetic field at the boundary are linearly coupled with tangential velocity and magnetic field contributions. While (1.6) includes the classical Navier slip condition for the velocity, it can capture also more complex interactions in the presence of magnetic fields.

The well-posedness of the linear problem (3.33), (3.34) is analogous to that of the Navier slip-with-friction boundary layers for the Navier–Stokes equations (cf. [IftimieSueur2011, CoronMarbachSueur2020]).

Let us also mention other recent works on global controllability problems for fluids that employ the return- and well-prepared dissipation methods. For instance, an incompressible Boussinesq system with Navier slip-with-friction boundary conditions for the velocity is considered by Chaves-Silva et al. in [ChavesSilva2020SmalltimeGE]. Moreover, the question of smooth controllability for the Navier–Stokes equations with Navier slip-with-friction boundary conditions is investigated in [LiaoSueurZhang2022]. Further, Coron et al. obtain in [CoronMarbachSueurZhang2019] global exact controllability results for the Navier–Stokes equations under the no-slip condition in a rectangular domain.

Aside of various MHD specific constructions, this article combines the return method and the well-prepared dissipation method as described by Coron et al. in [CoronMarbachSueur2020], where the small-time global exact controllability to trajectories has been studied for incompressible Navier–Stokes equations in two- and three-dimensional domains with Navier slip-with-friction conditions. Meanwhile, we shall also extend certain asymptotic expansions, obtained by Iftimie and Sueur in [IftimieSueur2011] for the incompressible Navier–Stokes equations, to the present MHD system. Due to the structure of the induction equation, the return method has to be carefully implemented in order to avoid generating pressure-like and additional forcing terms in the induction equation. To this end, under the assumptions of 1.2, we modify the return method trajectory from [CoronMarbachSueur2020] to be everywhere divergence-free, but allow a nonzero curl in the control region; this approach seems new and might be useful for further studies on the controllability of the ideal MHD equations.

C

In this way, we have the harmonic heat equation plus some nonlinear terms depending also on Δ2⁢usuperscriptΔ2𝑢\Delta^{2}uroman_Δ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u. We thus pass to the Fourier formulation and then the proof reduces to proving a suitable a priori estimates and compactness results.

where u=u⁢(t,x)>0𝑢𝑢𝑡𝑥0u=u(t,x)>0italic_u = italic_u ( italic_t , italic_x ) > 0 represents the height of the film surface. The complete list of the physical parameters involved in (1.1) is contained in [11, Table I𝐼Iitalic_I] but, for the reader’s convenience, we indicate here their most common values.

A possible approach is the one exploited by J.-G. Liu and R. Strain in [13]: here, under the assumption of medium size data belonging to A2superscript𝐴2A^{2}italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the authors first proved the uniqueness of solutions to

The main contributions provided by [1] concern both the proof of the analiticity and the improvement of the smallness condition on the A0superscript𝐴0A^{0}italic_A start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT semi-norm of the initial datum w.r.t. [9], where the same problem is studied.

A similar approach is contained in the work of J.-G. Liu & B. Strain [13], where they set the equation (1.2) in ℝNsuperscriptℝ𝑁\operatorname{\mathbb{R}}^{N}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT and make use of the Fourier transform.

D

Before proving Lemma 4.3, we need to discuss generalized permutation matrices. These matrices are square matrices where each row and column contain precisely one non-zero entry. Unlike classical permutation matrices where the non-zero entry must be 1111, in generalized permutation matrices, this entry can take any non-zero value. Any invertible generalized permutation matrix can be expressed as the product of an invertible diagonal matrix and a permutation matrix. In fact, the group of all such matrices in GLh⁢(ℝ)subscriptGLℎℝ\text{GL}_{h}(\mathbb{R})GL start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( blackboard_R ) is isomorophic to Sh⋊Δh⁢(ℝ)right-normal-factor-semidirect-productsubscript𝑆ℎsubscriptΔℎℝS_{h}\rtimes\Delta_{h}(\mathbb{R})italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ⋊ roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( blackboard_R ), where Shsubscript𝑆ℎS_{h}italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT is the symmetric group, and Δh⁢(ℝ)subscriptΔℎℝ\Delta_{h}(\mathbb{R})roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( blackboard_R ) be the set of diagonal matrices in GLh⁢(ℝ)subscriptGLℎℝ\text{GL}_{h}(\mathbb{R})GL start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( blackboard_R ). The semidirect product above is defined by the natural action of Shsubscript𝑆ℎS_{h}italic_S start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT on Δh⁢(ℝ)subscriptΔℎℝ\Delta_{h}(\mathbb{R})roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( blackboard_R ) by permuting coordinates. With this identification, or simply by matrix multiplications, the reader may note that

(λλ⋱⋱⋱λλ)m×m,subscriptmatrix𝜆missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression𝜆⋱missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression⋱missing-subexpression⋱missing-subexpressionmissing-subexpressionmissing-subexpression𝜆missing-subexpression𝜆𝑚𝑚\begin{pmatrix}~~\lambda&&&&\\

(0ρ⁢(ti)I⋱⋱⋱I0)m⁢hi×m⁢hi.subscriptmatrix0missing-subexpressionmissing-subexpressionmissing-subexpression𝜌subscript𝑡𝑖𝐼⋱missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression⋱missing-subexpression⋱missing-subexpressionmissing-subexpressionmissing-subexpression𝐼missing-subexpression0𝑚subscriptℎ𝑖𝑚subscriptℎ𝑖\begin{pmatrix}~~0&&&&\rho(t_{i})\\

(0AIm⋱⋱⋱Im0)h⁢m×h⁢mh=A⁢Ih⁢m.subscriptsuperscriptmatrix0missing-subexpressionmissing-subexpressionmissing-subexpression𝐴subscript𝐼𝑚⋱missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression⋱missing-subexpression⋱missing-subexpressionmissing-subexpressionmissing-subexpressionsubscript𝐼𝑚missing-subexpression0ℎℎ𝑚ℎ𝑚𝐴subscript𝐼ℎ𝑚\begin{pmatrix}~~0&&&&A\\

(0a1⋱⋱⋱10)h×hh=a⁢Ih,∀a∈ℝ∖{0}.formulae-sequencesubscriptsuperscriptmatrix0missing-subexpressionmissing-subexpressionmissing-subexpression𝑎1⋱missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression⋱missing-subexpression⋱missing-subexpressionmissing-subexpressionmissing-subexpression1missing-subexpression0ℎℎℎ𝑎subscript𝐼ℎfor-all𝑎ℝ0\begin{pmatrix}~~0&&&&a\\

D

Misspecified within groups scenario introduces observation-level mismatch between the model and data.

Figure 8f shows that PPCs are only effective when used with the mean of group 75757575th quantiles statistic while SPCs are effective

In this case the mean of group 75757575th quantiles statistic is most effective at detecting the misspecification.

With the mean of group 75th quantiles statistic, which is used to detect the lower-level misspecification, and the grand mean, only the within-SPCs have good power.

In this scenario the single SPC p𝑝pitalic_p-value is not useful for detecting model misspecification.

B

Given that ρ𝜌\rhoitalic_ρ and ρ~~𝜌\widetilde{\rho}over~ start_ARG italic_ρ end_ARG are Schwartz in s𝑠sitalic_s, they are also Schwartz in x𝑥xitalic_x.

\scriptscriptstyle-}bold_1 start_POSTSUBSCRIPT + end_POSTSUBSCRIPT bold_italic_ρ start_POSTSUBSCRIPT italic_t , italic_x end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT - end_POSTSUBSCRIPT is 𝒞∞superscript𝒞\mathcal{C}^{\infty}caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT in x𝑥xitalic_x with respect to the Hilbert–Schmidt norm; the same property holds for 𝟏−⁢𝝆~t,x⁢𝟏+subscript1subscript~𝝆𝑡𝑥subscript1\mathbf{1}_{\scriptscriptstyle-}\widetilde{\boldsymbol{\rho}}_{t,x}\mathbf{1}_%

Let ∥∥tr\|\ \|_{\mathrm{tr}}∥ ∥ start_POSTSUBSCRIPT roman_tr end_POSTSUBSCRIPT denote the trace norm.

The identity requires 𝐮𝐮\mathbf{u}bold_u to be differentiable in x𝑥xitalic_x with respect to the trace norm, which holds in our application since 𝝆t,xsubscript𝝆𝑡𝑥{\boldsymbol{\rho}}_{t,x}bold_italic_ρ start_POSTSUBSCRIPT italic_t , italic_x end_POSTSUBSCRIPT and 𝝆~t,xsubscript~𝝆𝑡𝑥\widetilde{\boldsymbol{\rho}}_{t,x}over~ start_ARG bold_italic_ρ end_ARG start_POSTSUBSCRIPT italic_t , italic_x end_POSTSUBSCRIPT are 𝒞∞superscript𝒞\mathcal{C}^{\infty}caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT in x𝑥xitalic_x with respect to the Hilbert–Schmidt norm.

Let ∥∥HS\|\ \|_{\mathrm{HS}}∥ ∥ start_POSTSUBSCRIPT roman_HS end_POSTSUBSCRIPT denote the Hilbert–Schmidt norm.

D