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The first author was supported by the Swedish Research Council (SRC, Vetenskapsrådet) under Grant No. 2019-04545. The research has received funding from Knut and Alice Wallenbergs Foundation through the Foundation’s program for mathematics. He is also grateful to the IRIF for their invitation in December 2021, when the the main proof of the last Section was found.

Both the authors are grateful to Paul-André Melliès for his support towards this project. We are also grateful to Eduardo Dubuc (whose previous work [DDS18] was an important inspiration to us) for remarks on the first version of this paper. Similarly, we are grateful to John Bourke for a careful reading of the first version of this paper as well as important remarks.

The original motivation of this work was to find a 2-dimensional notion of presentability encompassing the different finitary first order doctrines corresponding to different fragments of first order logic. Those are 2-categories whose objects are small categories endowed with a certain structure allowing to see them as syntactic categories for first order theories, where one can interpret connectors and inference rules. Prominent examples of doctrines are Lex, for left exact categories, corresponding to cartesian logic; Reg, for regular categories, corresponding to regular logic; Coh for coherent categories, corresponding to coherent logic; but also the 2-categories Ext of extensive categories, Adh of adhesive categories, Ex of exact categories, 𝐏𝐫𝐞𝐭𝐨𝐩ωsubscript𝐏𝐫𝐞𝐭𝐨𝐩𝜔\textbf{Pretop}_{\omega}Pretop start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT of finitary pretopoi, or BoolPretop of boolean finitary pretopoi.

The first author was supported by the Swedish Research Council (SRC, Vetenskapsrådet) under Grant No. 2019-04545. The research has received funding from Knut and Alice Wallenbergs Foundation through the Foundation’s program for mathematics. He is also grateful to the IRIF for their invitation in December 2021, when the the main proof of the last Section was found.

Proving the remaining finitary first order doctrines to be finitely bipresentable could be achieved in several ways. One could directly try to manipulate their axioms as kind of “2-dimensional cartesian predicates” (involving only finitely weighted bilimits or left and right Kan extensions) in a manner reminiscent of Makkai paradigm of injectivity classes. While we believe such a work to be interesting by itself as an occasion to conscientiously perform 2-dimensional logic, such a process would have represented an enormous amount of specific considerations and lemmas for each example. Such a tedious work appeared superfluous thanks to the powerful paradigm of lex colimits introduced by [GL12], which was specifically developed to capture a large class of examples of doctrines defined through different flavours of exactness conditions. Here they are axiomatized as classes of pseudo-algebras for some 2-monad on 𝐋𝐞𝐱𝐋𝐞𝐱{\bf Lex}bold_Lex defined as a free cocompletion under specific colimits weighted in some class of finite weights - this has been described as “cocompleteness in the lex world”. Having proven 𝐋𝐞𝐱𝐋𝐞𝐱{\bf Lex}bold_Lex to be finitely bipresentable, it suffices then to prove the embedding of the category of those pseudo-algebras (which form the corresponding 2-categories of exact categories relative to those weights) to be bifinitary, which is done in Lemma 6.4.4. This captures all the remaining doctrines defined from exactness properties as Reg, Ex, Coh, Adh, Ext and 𝐏𝐫𝐞𝐭𝐨𝐩ωsubscript𝐏𝐫𝐞𝐭𝐨𝐩𝜔\textbf{Pretop}_{\omega}Pretop start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT.

A

2 General position sets of ℳ⁢(G)ℳ𝐺\mathcal{M}(G)caligraphic_M ( italic_G ) containing the root u∗superscript𝑢u^{*}italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT

An independent d𝑑ditalic_d-position set is a subset S⊆V⁢(G)𝑆𝑉𝐺S\subseteq V(G)italic_S ⊆ italic_V ( italic_G ) such that S𝑆Sitalic_S is an independent set and no shortest path of length at most d𝑑ditalic_d passes through three or more vertices of S𝑆Sitalic_S. We denote that largest number of vertices in an independent d𝑑ditalic_d-position set by ipd⁢(G)subscriptip𝑑𝐺{\rm ip}_{d}(G)roman_ip start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_G ).

In this section we use Corollary 9 to derive tight upper and lower bounds for gp⁢(ℳ⁢(G))gpℳ𝐺{\rm gp}(\mathcal{M}(G))roman_gp ( caligraphic_M ( italic_G ) ) and characterise the case of equality with the upper bound. We start with the lower bound. Recall that ip⁢(G)ip𝐺{\rm ip}(G)roman_ip ( italic_G ) is the number of vertices in a largest independent set that is also in general position and gpd⁢(G)subscriptgp𝑑𝐺{\rm gp}_{d}(G)roman_gp start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_G ) is the number of vertices in a largest subset S⊆V⁢(G)𝑆𝑉𝐺S\subseteq V(G)italic_S ⊆ italic_V ( italic_G ) such that no shortest path with length at most d𝑑ditalic_d passes through three or more vertices in S𝑆Sitalic_S. To derive bounds for gp⁢(ℳ⁢(G))gpℳ𝐺{\rm gp}(\mathcal{M}(G))roman_gp ( caligraphic_M ( italic_G ) ) we combine these concepts in the following definition.

Consider the vertices of V3subscript𝑉3V_{3}italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. As V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is a maximum independent 4-position set and (V1,V3)=∅subscript𝑉1subscript𝑉3(V_{1},V_{3})=\emptyset( italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = ∅, it follows for that any vertex w𝑤witalic_w of V3subscript𝑉3V_{3}italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT there is a shortest path P𝑃Pitalic_P of length four containing w𝑤witalic_w and two vertices of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Observe that any pair of vertices from V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are at distance two from each other, since they have a common neighbour in V4subscript𝑉4V_{4}italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT. Also by Condition 2 of Definition 5 each vertex of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is at distance at most three from any vertex of V3subscript𝑉3V_{3}italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. It follows that there is no such path P𝑃Pitalic_P and V3=∅subscript𝑉3V_{3}=\emptysetitalic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = ∅. Similarly, if s>r𝑠𝑟s>ritalic_s > italic_r and v𝑣vitalic_v is a vertex of {vr+1,…,vs}subscript𝑣𝑟1…subscript𝑣𝑠\{v_{r+1},\dots,v_{s}\}{ italic_v start_POSTSUBSCRIPT italic_r + 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT }, then there would have to be a shortest path in G𝐺Gitalic_G of length four that contains v𝑣vitalic_v and two vertices of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. This is impossible, since the vertices of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are at distance two apart and by Condition 2 of Definition 5 each vertex of V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is at distance two from v𝑣vitalic_v. Therefore r=s𝑟𝑠r=sitalic_r = italic_s.

Firstly, we introduce a convention that we use throughout the paper. Given a set S𝑆Sitalic_S of vertices of a graph, we call a geodesic containing at most two vertices of S𝑆Sitalic_S sound and a shortest path containing at least three vertices of S𝑆Sitalic_S unsound.

D

To demonstrate the practical application of the proposed framework of risk-averse MDPs, we also present two examples in Section 7 on optimal liquidation with limit order books and autonomous driving. The optimal liquidation example, posed within a finite horizon context with discrete state and action spaces, underscores the potential of the framework to incorporate latent costs and adjust risk aversion in response to the evolving state. The autonomous driving example, posed within an infinite horizon context, showcases the framework’s capability to handle a large and complex state space, a requirement often necessary for constructing more realistic models. It’s worth noting that the autonomous driving example naturally necessitates certain degenerate transition mechanisms. While these are not compatible with strongly continuous transition kernels in general, they can be easily incorporated into a weakly continuous transition kernel as we do here.

The main goal of this paper is to study finite and infinite horizon risk averse MDPs in a similar framework as above, but with latent costs and randomized actions, under a weakly continuous transition mechanism, subject to state-dependent DRMs. Below we provide a high level discussion on some of the open problems we address in this paper. We would like to note that in some of the earlier risk-averse MDP frameworks, latent costs were not considered. We believe that in many circumstances, latent costs can be used to account for the risk related to factors such as time-discretization and processing delays. To the best of our knowledge, [6] is the first to consider latent costs within risk-averse MDP frameworks. Their DRMs, however, are constructed from compositions of static risk measures. A framework that allows for state-dependent risk measures allows for greater flexibility in adjusting the level of risk aversion depending on where one is in state space (e.g., in the context of portfolio allocation, an investor may become more risk averse as their wealth increase). Apart from the points mentioned above, it is worth questioning whether it is possible to modify the existing risk-averse MDP framework to account for the randomness in randomized actions in a risk-averse manner. We note that randomized actions are allowed in [23], but are evaluated in an arguably risk-neutral manner. Lastly, an adequate discussion on weakly continuous transition kernels is missing from the existing literature on risk averse MDPs. Weakly continuous transition kernels are often preferred in practice, as they provide the flexibility to consider transition dynamics that are partly random and partly deterministic. Moreover, they facilitate data-based modeling by removing the need of working with the density of the kernel or the underlying probability spaces, which typically requires additional model assumptions.

The remainder of the paper is structured as follows. In Section 2, we first introduce our notation, then recall definitions and basic properties of various important concepts, and establish preliminary results, such as the construction of DRMs at the level of distributions. Formulations for risk-averse MDP are organized in the end of the section. Section 3 is devoted to auxiliary results. We introduce some useful operators related to Markovian policy and investigate their regularities. In Section 4, we derive the finite horizon DPP for Markovian actions and argue that Markovian actions can achieve the optimal. The analogous results for infinite horizon are presented in Section 5. In Section 6, we establish a sufficient condition on the optimality of deterministic actions. Section 7 contains a variety of examples that serve to illustrate the concepts proposed in this study. We also provide an example illustrating the necessity of randomized actions and some technical lemmas in Appendix A and Appendix B, respectively. Finally, for reference, Appendix C contains a glossary of notation.

In this section, we present several examples that illustrate the application of the theory developed in the previous sections. In Section 7.1, we discuss a few possible choices for σt,xsubscript𝜎𝑡𝑥\sigma_{t,x}italic_σ start_POSTSUBSCRIPT italic_t , italic_x end_POSTSUBSCRIPT and verify the technical assumptions required for the DPP to be applicable. Subsequently, in Section 7.2, we provide a finite horizon example focusing on optimal liquidation with limit order books. Finally, an infinite horizon example on autonomous driving is presented in Section 7.3.

Here, we focus on the risk-averse MDPs framework proposed in [55]. While a plethora of studies exist on conditional risk measures (CRMs) and dynamic risk measures (DRMs), the problem of risk-averse MDPs and their corresponding DPP cannot be straightforwardly inferred from the established properties of these risk measures. Broadly speaking, DPPs transform a sequential optimization problem into the task of solving an operator Bellman equation. In the context of risk-averse MDPs, this demands separate research, as in general, a readily solvable DPP in terms of operator equations requires additional conditions beyond properties of CRMs and DRMs. This is particularly true when considering the attainability of optimal actions. Below, we offer a concise review of the established DPPs that align with the general framework initiated by [55]. [55] considers deterministic costs, introduces the notion of risk transition mappings, and uses them to construct, in a recursive manner, a class of (discounted) DRMs. The author proceeds to derive both finite and infinite (with bounded costs) time horizon DPPs for such DRMs. We also refer to [56] for the assumptions needed. [58] extends the infinite horizon DPP to unbounded costs as well as for average DRMs. The risk transition mappings involved are assumed to exhibit an analogue of a strong Feller property. [23] studies a similar infinite horizon DPP with unbounded costs under a different set of assumptions. Recently, [6] considers unbounded latent costs and establishes the corresponding finite and infinite horizon DPPs. They also prove sufficiency of Markovian actions against history dependent actions. They construct DRMs, for finite time horizon problems, from iterations of static risk measures that are Fatou and law invariant. The infinite horizon problems require in addition the coherency property. They also require the underlying MDP to exhibit a certain path-wise continuous/semi-continuous transition mechanism. [24, 25] develops a computational approach for optimization with dynamic convex risk measures using deep learning techniques. Finally, it is noteworthy that the concept of risk form is introduced in [28] and is applied to handle two-stage MDP with partial information and decision-dependent observation distribution.

B

\rVert_{B^{\log,a}}\right).≲ divide start_ARG 1 end_ARG start_ARG italic_δ end_ARG divide start_ARG square-root start_ARG italic_κ italic_t end_ARG end_ARG start_ARG roman_log start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ( 2 + divide start_ARG 1 end_ARG start_ARG italic_κ italic_t end_ARG ) end_ARG ( ( ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ∥ ∇ italic_u ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_d italic_s ) start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ∥ italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ∥ italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT roman_log , italic_a end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) .

It remains to combine the Proposition 1 and Lemma 4 to deduce the desired estimate on the rate of strong convergence.

Inserting this estimate for our logarithmic distance into the estimate on the convergence rate (75), we obtain

We make now make use of the periodicity of the problems and the convergence of the geometric series to deduce

In order to translate the estimate on weak convergence into an estimate on strong convergence, we have to make use of an interpolation inequality.

D

{m^{2}}{4}=c_{0}(t).over˙ start_ARG italic_n end_ARG - italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 4 italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , over˙ start_ARG italic_m end_ARG - italic_m italic_n + 2 italic_r = 2 italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_q - divide start_ARG italic_n end_ARG start_ARG 2 end_ARG - divide start_ARG italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 end_ARG = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) .

With this choice the symmetry algebra of the corresponding PDE should be isomorphic to that of the heat equation.

We note that in the case of four-dimensional symmetry algebra, ω=0𝜔0\omega=0italic_ω = 0 (c1=0subscript𝑐10c_{1}=0italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0). So we only have to solve {T,t}=8⁢c2𝑇𝑡8subscript𝑐2\left\{T,t\right\}=8c_{2}{ italic_T , italic_t } = 8 italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Now we can use formula (2.36) to obtain a point transformation mapping to the second canonical form (heat equation with inverse square potential). This gives us the symmetry group of this PDE for the choice

With these values of c2,c1subscript𝑐2subscript𝑐1c_{2},c_{1}italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and c0subscript𝑐0c_{0}italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, it is straightforward to check that q=ℓ⁢x𝑞ℓ𝑥q=\ell xitalic_q = roman_ℓ italic_x is a particular solution of (3.7). The general solution of course can be found. This enables to constructs the most general PDE of the form (3.6) equivalent to the heat equation by a point transformation. The PDE corresponding to the special solution is the celebrated Orstein–Uhlenbeck model.

If, in particular, we require u𝑢uitalic_u to be solution of the heat equation, then Theorem 2.2 provides us with the symmetry group of this PDE (see Theorem 2 of [8] for an alternative derivation of this result). It is sufficient to choose

A

\omega}^{*}R(t))\mathcal{M}_{N}(t)(P(t)E_{\omega}R(t))∫ start_POSTSUBSCRIPT bold_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT over^ start_ARG italic_V end_ARG ( italic_ω ) ( caligraphic_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_t ) ( italic_P ( italic_t ) italic_E start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_R ( italic_t ) ) caligraphic_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_t ) ( italic_P ( italic_t ) italic_E start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT italic_R ( italic_t ) )

−ℳN(t)(R(t)EωP(t))ℳN(t)(R(t)Eω*P(t)))d⁢ω(2⁢π)3\displaystyle-\mathcal{M}_{N}(t)(R(t)E_{\omega}P(t))\mathcal{M}_{N}(t)(R(t)E_{%

−ℳN(t)(R(t)EωP(t))ℳN(t)(P(t)Eω*R(t)+R(t)Eω*P(t)))d⁢ω(2⁢π)3\displaystyle-\mathcal{M}_{N}(t)(R(t)E_{\omega}P(t))\mathcal{M}_{N}(t)(P(t)E_{%

−ℳN(t)(R(t)EωP(t))(ℳN(t)−ℛ(t))(Eω*))dω\displaystyle-\mathcal{M}_{N}(t)(R(t)E_{\omega}P(t))(\mathcal{M}_{N}(t)-%

−ℳN(t)(R(t)EωP(t))ℳN(t)(R(t)Eω*P(t)))d⁢ω(2⁢π)3\displaystyle-\mathcal{M}_{N}(t)(R(t)E_{\omega}P(t))\mathcal{M}_{N}(t)(R(t)E_{%

A

Note the difference between ℱ⁢[S]ℱdelimited-[]𝑆\mathcal{F}[S]caligraphic_F [ italic_S ] and ℱ⁢(S)ℱ𝑆\mathcal{F}(S)caligraphic_F ( italic_S ). In particular, we have ℱ⁢[S]⊂ℱ,ℱdelimited-[]𝑆ℱ\mathcal{F}[S]\subset\mathcal{F},caligraphic_F [ italic_S ] ⊂ caligraphic_F , which is generally not true for ℱ⁢(S).ℱ𝑆\mathcal{F}(S).caligraphic_F ( italic_S ) .

The following result is a vast generalization of Theorem 8 from subfamilies of ([n]k)binomialdelimited-[]𝑛𝑘{[n]\choose k}( binomial start_ARG [ italic_n ] end_ARG start_ARG italic_k end_ARG ) to subfamilies of sufficiently spread families. As in the beginning of Section 3, we present the result for the property of being t𝑡titalic_t-intersecting. This can be extended to several families (and more complicated properties), as we will show in the following sections.

Arguably the most impressive application of the spread approximations method that we found so far is the following result that greatly extends the aforementioned results of Ellis, Friedgut and Pilpel and of Ellis and Lifshitz. The proof is also much simpler and avoids the use of heavy machinery of the previous authors.

We finally note that the approach we used for the Erdős–Sós problem does extend to more general structures, and guarantees that the spread approximation that we obtain not only forbids the same structure, but also all structures that can be obtained from the initial one by replacing common elements of two or more sets with elements that each belong to only one set. It is possible that this can be further strengthened, but requires further research.

Theorem 4 gives something more robust than just a sunflower (actually, the authors of [3] use terminology robust sunflowers): with large probability it guarantees sunflowers that ‘respect’ a random partition. Talagrand [42] observed using duality that the property of a family being r𝑟ritalic_r-spread is roughly equivalent to having a fractional cover that has small r−1superscript𝑟1r^{-1}italic_r start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-biased measure. Frankston, Kahn, Narayanan and Park [29] used this and a slight modification of the proof from [3] to show a general result that relates thresholds and fractional expectation-thresholds for monotone properties.

D

5. Computation of the invariant (ν⊗id)∘Δtensor-product𝜈idΔ(\nu\otimes\operatorname{id})\circ\Delta( italic_ν ⊗ roman_id ) ∘ roman_Δ

Here to unify the terms, when we apply ν𝜈\nuitalic_ν in each case (a)–(e) consisting of two terms, the order of the corresponding components is always adjusted to the components’ order (1,2,3)123(1,2,3)( 1 , 2 , 3 )

The author TK gives an infinitely many nontrivial examples Proposition 2, each of which is distinct from any of the others up to stable equivalence.

We proceed along Cnsubscript𝐶𝑛C_{n}italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT according to the orientation starting from the base point, and we assign labels 1111, …, 2⁢n2𝑛2n2 italic_n to crossings in the encountering order (i.e. from the left to the right in each of the figures of Figure 11).

Each self-intersection is called a crossing 222Only in Section 4, since both curves and link diagrams appear, we call them separately.. A one-component curve with a base point, which is not an intersection, is called pointed. Two curves are stably homeomorphic if there is a homeomorphism of their regular neighborhoods in the ambient surfaces mapping the first curve onto the second one preserving the orientations of the curve and the surface.

B

Both strategies produce higher ratings for larger n𝑛nitalic_n, up to n=Θ⁢(k1/3)𝑛Θsuperscript𝑘13n=\Theta(k^{1/3})italic_n = roman_Θ ( italic_k start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT ), at which point the asymtotic highest rating in k𝑘kitalic_k is Θ⁢(k1/3)Θsuperscript𝑘13\Theta(k^{1/3})roman_Θ ( italic_k start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT ).

Pick any pair of players whose ratings are within δ𝛿\deltaitalic_δ of each other and have the higher rated player beat the lower rated player. Repeat until no two players are within δ𝛿\deltaitalic_δ rating points or until k𝑘kitalic_k games have been played. Note for δ=0𝛿0\delta=0italic_δ = 0, one recovers the original strategy. As before, we may have to flip everyone’s rating to ensure we end with a very high rating, as opposed to a very low rating. We need not assume that the initial ratings are all 0 anymore, but we do need to allow ourselves this possible reflection.

The first strategy is very simple: pick any pair of players with equal rating and have one beat the other. Repeat until all players have a distinct rating or k𝑘kitalic_k games have been played.

This strategy is guaranteed to produce a player of either very high or very low rating. If it produces a player of very low rating, simply re-do the strategy picking the same sequence of pairs of players but have the opposite player win. Since game outcomes are symmetric, this will produce a player of high rating instead.

To the author’s knowledge, this is the first time this question has been posed in print: given n𝑛nitalic_n players starting with equal rating, what is the highest a player could be rated after a total of k𝑘kitalic_k games are played?

B

},\overline{n}\in\overline{N}\}.caligraphic_H start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ∞ , over¯ start_ARG italic_ψ end_ARG end_POSTSUPERSCRIPT = { italic_u ∈ caligraphic_H start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ∞ end_POSTSUPERSCRIPT : italic_u ( italic_σ ( over¯ start_ARG italic_n end_ARG ) italic_φ ) = over¯ start_ARG italic_ψ end_ARG ( over¯ start_ARG italic_n end_ARG ) italic_u ( italic_φ ) for all italic_φ ∈ caligraphic_H start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT , over¯ start_ARG italic_n end_ARG ∈ over¯ start_ARG italic_N end_ARG } .

We remark that the dimension of the space of generalized Whittaker vectors on holomorphic discrete series representations was already determined by Wallach [31]. However, he did not obtain explicit formulas for the Whittaker vectors.

Our first main result is an explicit construction and classification of generalized Whittaker vectors for all holomorphic discrete series representations in three different realizations (bounded domain model, tube domain model and L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-model).

In this section we find explicit expressions for all generalized Whittaker vectors on holomorphic discrete series representations in the three realizations described in Section 1.

In Section 1 we set up the notation and recall the three different models for the holomorphic discrete series. Explicit formulas for generalized Whittaker vectors are obtained in Section 2, thus proving Theorem A. These formulas are used in Section 3 to show Theorem B. This also leads to a formula for the formal dimension. In Section 4 we first show that G⊆P+⁢Kℂ⁢Nℂ𝐺superscript𝑃subscript𝐾ℂsubscript𝑁ℂG\subseteq P^{+}K_{\mathbb{C}}N_{\mathbb{C}}italic_G ⊆ italic_P start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT italic_K start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT, so every g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G can be decomposed as g=p+⁢(g)⁢kℂ⁢(g)⁢nℂ⁢(g)𝑔superscript𝑝𝑔subscript𝑘ℂ𝑔subscript𝑛ℂ𝑔g=p^{+}(g)k_{\mathbb{C}}(g)n_{\mathbb{C}}(g)italic_g = italic_p start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_g ) italic_k start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ( italic_g ) italic_n start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ( italic_g ), and then describe the integral kernel of a holomorphic discrete series embedding, and in particular the embedding of the lowest K𝐾Kitalic_K-types, in terms of the projections kℂ⁢(g)subscript𝑘ℂ𝑔k_{\mathbb{C}}(g)italic_k start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ( italic_g ) and nℂ⁢(g)subscript𝑛ℂ𝑔n_{\mathbb{C}}(g)italic_n start_POSTSUBSCRIPT blackboard_C end_POSTSUBSCRIPT ( italic_g ). Finally, in Section 5 we define the natural Hardy space ℋ2⁢(Ξ,ψ)subscriptℋ2Ξ𝜓\mathcal{H}_{2}(\Xi,\psi)caligraphic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_Ξ , italic_ψ ) and relate it to the holomorphic discrete series contribution of L2⁢(G/N,ψ)superscript𝐿2𝐺𝑁𝜓L^{2}(G/N,\psi)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_G / italic_N , italic_ψ ).

C

[\Psi]&\mbox{otherwise.}\end{array}\right.italic_P ( roman_Ψ ) = { start_ARRAY start_ROW start_CELL [ italic_χ italic_ν start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ⊕ italic_χ italic_ν start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT , ( start_ARG start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW end_ARG ) ] end_CELL start_CELL if roman_ℓ ∣ italic_q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT - 1 and roman_Ψ = roman_Ψ start_POSTSUBSCRIPT roman_St , italic_χ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL [ italic_χ italic_ν start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ⊕ italic_χ italic_ν start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT , ( start_ARG start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW end_ARG ) ] end_CELL start_CELL if roman_ℓ ∣ italic_q start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT + 1 and roman_Ψ = roman_Ψ start_POSTSUBSCRIPT roman_Sp , italic_χ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL [ roman_Ψ ] end_CELL start_CELL otherwise. end_CELL end_ROW end_ARRAY

Now we can gather together all the results of the previous sections to prove a “modified” Prasad conjecture for PGL2subscriptPGL2\operatorname{PGL}_{2}roman_PGL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.

5.5. A modulo ℓℓ\ellroman_ℓ Prasad conjecture for PGL2subscriptPGL2\operatorname{PGL}_{2}roman_PGL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

5. The Prasad conjecture for ℓℓ\ellroman_ℓ-modular representations of PGL2subscriptPGL2\operatorname{PGL}_{2}roman_PGL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

Then we prove a modulo ℓℓ\ellroman_ℓ version of the Prasad conjecture, using our modified injection P𝑃Pitalic_P:

D

}\in S_{x_{2}},y_{1}\geq y_{2}.italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_A ⇒ ∀ italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_S start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , ∀ italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_S start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .

The same conclusion holds if G′⁢(0)=−∞superscript𝐺′0G^{\prime}(0)=-\inftyitalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = - ∞ and 𝒴𝒴\mathcal{Y}caligraphic_Y is a finite set.

(with the convention −∞×0=000-\infty\times 0=0- ∞ × 0 = 0 if G′⁢(0)=−∞superscript𝐺′0G^{\prime}(0)=-\inftyitalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = - ∞ and m=q¯x𝑚superscript¯𝑞𝑥m=\bar{q}^{x}italic_m = over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT). In particular, if G′⁢(0)=−∞superscript𝐺′0G^{\prime}(0)=-\inftyitalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = - ∞, then q¯x≠0superscript¯𝑞𝑥0\bar{q}^{x}\neq 0over¯ start_ARG italic_q end_ARG start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ≠ 0. Rearranging terms yields

The same conclusion holds if G′⁢(0)=−∞superscript𝐺′0G^{\prime}(0)=-\inftyitalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) = - ∞ provided 𝒴𝒴\mathcal{Y}caligraphic_Y is a finite set.

If the function G𝐺Gitalic_G in (24) is such that G′⁢(0)>−∞superscript𝐺′0G^{\prime}(0)>-\inftyitalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) > - ∞, then for every μ∈𝒫⁢(𝒳)𝜇𝒫𝒳\mu\in\mathcal{P}(\mathcal{X})italic_μ ∈ caligraphic_P ( caligraphic_X ) and ν∈𝒫⁢(𝒴)𝜈𝒫𝒴\nu\in\mathcal{P}(\mathcal{Y})italic_ν ∈ caligraphic_P ( caligraphic_Y ), there exists a bounded function f¯¯𝑓\bar{f}over¯ start_ARG italic_f end_ARG on 𝒴𝒴\mathcal{Y}caligraphic_Y such that

C

It is clearly true for x∈Int⁢V.𝑥Int𝑉x\in\mathrm{Int}\ V.italic_x ∈ roman_Int italic_V . As V=∂V∪Int⁢V𝑉𝑉Int𝑉V=\partial{V}\cup\mathrm{Int}\ Vitalic_V = ∂ italic_V ∪ roman_Int italic_V (see [doktoratura, Cor. 1.8.11]), it suffices to prove it for points x∈∂V𝑥𝑉x\in\partial{V}italic_x ∈ ∂ italic_V, which is a consequence of Rem. 1.11.

The first section contains some preliminaries and the notation we use. In Section 2, we work over Berkovich analytic curves, and use the structure of the rings 𝒪∘⁢(U)superscript𝒪𝑈\mathcal{O}^{\circ}(U)caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ( italic_U ) for a well-chosen open U𝑈Uitalic_U to prove a local result on the existence of rational points. The main results here are Theorem 2.3 and Proposition 2.12.

By Lemma 2.4, X𝑋Xitalic_X has a proper smooth model over 𝒪∘⁢(Ux),superscript𝒪subscript𝑈𝑥\mathcal{O}^{\circ}(U_{x}),caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ( italic_U start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) , which we will continue to denote by 𝒳.𝒳\mathcal{X}.caligraphic_X . By Theorem 1.2, 𝒪𝒞,Px^=𝒪∘⁢(π−1⁢(Px))^subscript𝒪𝒞subscript𝑃𝑥superscript𝒪superscript𝜋1subscript𝑃𝑥\widehat{\mathcal{O}_{\mathscr{C},P_{x}}}=\mathcal{O}^{\circ}(\pi^{-1}(P_{x}))over^ start_ARG caligraphic_O start_POSTSUBSCRIPT script_C , italic_P start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG = caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ( italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ), meaning that 𝒪∘⁢(Ux)superscript𝒪subscript𝑈𝑥\mathcal{O}^{\circ}(U_{x})caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ( italic_U start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) is a complete regular local ring of dimension 2.

We will now use the sheaf 𝒪∘superscript𝒪\mathcal{O}^{\circ}caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT (cf. relation (3)).

When there is no risk of ambiguity, we will simply write 𝒪∘superscript𝒪\mathcal{O}^{\circ}caligraphic_O start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT.

C

If M𝑀Mitalic_M is a compact topological manifold of dimension d≥7𝑑7d\geq 7italic_d ≥ 7, M𝑀Mitalic_M is 1111-connected, and π2⁢Msubscript𝜋2𝑀\pi_{2}\,Mitalic_π start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_M is finite, then π0⁢Homeo∂⁢(M)subscript𝜋0subscriptHomeo𝑀\pi_{0}\,{\mathrm{Homeo}}_{\partial}(M)italic_π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT roman_Homeo start_POSTSUBSCRIPT ∂ end_POSTSUBSCRIPT ( italic_M ) is of finite type.

The argument is an induction over the number s𝑠sitalic_s of path components of ∂M𝑀\partial M∂ italic_M, which is finite because M𝑀Mitalic_M is compact. The initial case s=1𝑠1s=1italic_s = 1 is Theorem 4.4. For the induction step for s𝑠sitalic_s to s+1𝑠1s+1italic_s + 1 we will perform a surgery on the boundary. We fix discs Dd−1×{0}superscript𝐷𝑑10D^{d-1}\times\{0\}italic_D start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT × { 0 } and Dd−1×{1}superscript𝐷𝑑11D^{d-1}\times\{1\}italic_D start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT × { 1 } in two different path components of ∂M𝑀\partial M∂ italic_M and consider the space Emb∂Top⁢(Dd−1×I,M)subscriptsuperscriptEmbTopsuperscript𝐷𝑑1𝐼𝑀{\mathrm{Emb}}^{\mathrm{Top}}_{\partial}(D^{d-1}\times I,M)roman_Emb start_POSTSUPERSCRIPT roman_Top end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∂ end_POSTSUBSCRIPT ( italic_D start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT × italic_I , italic_M ) of topological embeddings restricting to the fixed discs on Dd−1×{0,1}superscript𝐷𝑑101D^{d-1}\times\{0,1\}italic_D start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT × { 0 , 1 }. By general position, the map

We prove this in the case that ∂M𝑀\partial M∂ italic_M is 00-connected; the general case is done exactly as in the proof of A by induction over the number of path components.

We can drop the assumption that the path components of ∂M𝑀\partial M∂ italic_M are 1-connected at the expense of additional conditions on M𝑀Mitalic_M; see Corollary 4.5.

Since ∂M𝑀\partial M∂ italic_M is compact, its fundamental group is finitely generated. We do an induction over the number s𝑠sitalic_s of generators, the initial case being a special case of Theorem 4.4. The strategy for the induction step from s𝑠sitalic_s to s+1𝑠1s+1italic_s + 1 is similar to the proof of A, but using discs to perform a surgery on a thickened loop rather than intervals to perform a surgery on two discs.

B

In Section LABEL:sec:sec3 we gather results on ergodicity and SES of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. Such results will be needed to study the Poisson problem (7). In particular in Section LABEL:sec:sec3 we prove exponentially fast decay to μxsuperscript𝜇𝑥\mu^{x}italic_μ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT, with rate of exponential decay which is independent of x𝑥xitalic_x and we produce DE of the form (12) for y𝑦yitalic_y derivatives up to order four of Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. The latter fact will be the key instrument in tackling the issue of the smoothness in x𝑥xitalic_x of the solution of the Poisson equation, when the coefficients of ℒxsuperscriptℒ𝑥\mathcal{L}^{x}caligraphic_L start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT grow super-linearly. Section LABEL:sec:psneqn contains the proof of our main result on Poisson equations, Theorem LABEL:mainthmpois. For readability, and because some of these results are of independent interest, the proof is split in various statements. In particular, Lemma LABEL:lemma:probRep is a well-posedness result for the Poisson problem, and Proposition LABEL:lemma:IntAvgDerRepresentation is the key place where we use SES of Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT to prove smoothness in the parameter x𝑥xitalic_x of the solution of the Poisson equation. Several comments on Proposition LABEL:lemma:IntAvgDerRepresentation are contained in Note LABEL:Rem:der_est_for_Pois_unbounded (which also explains why we need four derivatives of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT). Section LABEL:subsec:detailedproof and Section LABEL:sec:boundsonft1 are devoted to the proof of our UiT averaging result. More specifically, in Section LABEL:subsec:detailedproof we explain our strategy of proof, providing first some heuristics (Section LABEL:subsec:heur), then a strategy of proof for the non fully coupled case (Section (LABEL:subsec:proofofmainthm)) and then explaining how such a strategy can be extended to the fully coupled case (Section LABEL:subsec:fullycoupledsketch). The strategy explained in Section LABEL:subsec:detailedproof clarifies that the two main ingredients in our approach are SES of the averaged semigroup 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and the study of Poisson equations, see Note LABEL:Rem:poisson_for_averaging. Accordingly, in Section LABEL:sec:boundsonft1 we first obtain DEs for 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and then apply the results on Poisson equations stated in previous sections to the Poisson problem at hand. It is important to notice that the DEs for 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT cannot be obtained in the same way as those for Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. Indeed, while 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT are both Markov Semigroups, the coefficients of the SDE associated with Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT are known explicitly, those for the SDE (4) associated to 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT are not, in the sense that if μxsuperscript𝜇𝑥\mu^{x}italic_μ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT is not known explicitly then the coefficient b¯¯𝑏\bar{b}over¯ start_ARG italic_b end_ARG (similarly for σ^^𝜎\hat{\sigma}over^ start_ARG italic_σ end_ARG) is only defined via (5), which is a highly non-linear expression in x𝑥xitalic_x, see Note LABEL:rem:xavgderivs.

The paper is structured as follows. In Section 2 we introduce the necessary notation and state our assumptions. In Section LABEL:sec:mainresults we state our main results on Poisson equations and then on averaging. We state all our main results in the non fully coupled regime first, i.e. in the case in which a=a⁢(x),σ=σ⁢(x)formulae-sequence𝑎𝑎𝑥𝜎𝜎𝑥a=a(x),\sigma=\sigma(x)italic_a = italic_a ( italic_x ) , italic_σ = italic_σ ( italic_x ) (the drift coefficients b𝑏bitalic_b and g𝑔gitalic_g are still allowed to depend on both x𝑥xitalic_x and y𝑦yitalic_y) and then in the fully coupled regime (i.e. in the case in which all the coefficients of the slow-fast system (1)-(2) depend on both x𝑥xitalic_x and y𝑦yitalic_y).

Hence in Section LABEL:sec:mainresults we first state all our results in the non fully coupled case and then in the fully coupled one and the whole paper refers to the non fully coupled case, with the exception of Section LABEL:subsec:fullycoupledresults and Section LABEL:subsec:fullycoupledsketch, where the fully coupled regime is treated.

Section LABEL:sec:numerics contains some numerical experiments. We complement our results with a number of examples, both throughout and in Section LABEL:sec:numerics. Note that when b𝑏bitalic_b is in gradient form, our results can be applied to Langevin-type dynamics, which are ubiquitous in applications, see e.g. [pavliotis2008multiscale] and references therein, and Section LABEL:sec:mainresults.

In Section LABEL:sec:sec3 we gather results on ergodicity and SES of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. Such results will be needed to study the Poisson problem (7). In particular in Section LABEL:sec:sec3 we prove exponentially fast decay to μxsuperscript𝜇𝑥\mu^{x}italic_μ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT, with rate of exponential decay which is independent of x𝑥xitalic_x and we produce DE of the form (12) for y𝑦yitalic_y derivatives up to order four of Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. The latter fact will be the key instrument in tackling the issue of the smoothness in x𝑥xitalic_x of the solution of the Poisson equation, when the coefficients of ℒxsuperscriptℒ𝑥\mathcal{L}^{x}caligraphic_L start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT grow super-linearly. Section LABEL:sec:psneqn contains the proof of our main result on Poisson equations, Theorem LABEL:mainthmpois. For readability, and because some of these results are of independent interest, the proof is split in various statements. In particular, Lemma LABEL:lemma:probRep is a well-posedness result for the Poisson problem, and Proposition LABEL:lemma:IntAvgDerRepresentation is the key place where we use SES of Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT to prove smoothness in the parameter x𝑥xitalic_x of the solution of the Poisson equation. Several comments on Proposition LABEL:lemma:IntAvgDerRepresentation are contained in Note LABEL:Rem:der_est_for_Pois_unbounded (which also explains why we need four derivatives of the semigroup Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT). Section LABEL:subsec:detailedproof and Section LABEL:sec:boundsonft1 are devoted to the proof of our UiT averaging result. More specifically, in Section LABEL:subsec:detailedproof we explain our strategy of proof, providing first some heuristics (Section LABEL:subsec:heur), then a strategy of proof for the non fully coupled case (Section (LABEL:subsec:proofofmainthm)) and then explaining how such a strategy can be extended to the fully coupled case (Section LABEL:subsec:fullycoupledsketch). The strategy explained in Section LABEL:subsec:detailedproof clarifies that the two main ingredients in our approach are SES of the averaged semigroup 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and the study of Poisson equations, see Note LABEL:Rem:poisson_for_averaging. Accordingly, in Section LABEL:sec:boundsonft1 we first obtain DEs for 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and then apply the results on Poisson equations stated in previous sections to the Poisson problem at hand. It is important to notice that the DEs for 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT cannot be obtained in the same way as those for Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT. Indeed, while 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT are both Markov Semigroups, the coefficients of the SDE associated with Ptxsuperscriptsubscript𝑃𝑡𝑥P_{t}^{x}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT are known explicitly, those for the SDE (4) associated to 𝒫¯tsubscript¯𝒫𝑡\bar{\mathcal{P}}_{t}over¯ start_ARG caligraphic_P end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT are not, in the sense that if μxsuperscript𝜇𝑥\mu^{x}italic_μ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT is not known explicitly then the coefficient b¯¯𝑏\bar{b}over¯ start_ARG italic_b end_ARG (similarly for σ^^𝜎\hat{\sigma}over^ start_ARG italic_σ end_ARG) is only defined via (5), which is a highly non-linear expression in x𝑥xitalic_x, see Note LABEL:rem:xavgderivs.

C

Then Rφnsuperscript𝑅superscript𝜑𝑛{{}^{\varphi^{n}}\!}Rstart_FLOATSUPERSCRIPT italic_φ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_FLOATSUPERSCRIPT italic_R is Litchtenbaum for all n≫0much-greater-than𝑛0n\gg 0italic_n ≫ 0 iff dim(R)=0dimension𝑅0\dim(R)=0roman_dim ( italic_R ) = 0.

Localization of a Lichtenbaum module is not necessarily Lichtenbaum, even over 2222-dimensional (regular) rings.

A nonzero R𝑅Ritalic_R-module L𝐿Litalic_L is called (strong) Lichtenbaum if for every (not-necessarily) finitely generated R𝑅Ritalic_R-module F𝐹Fitalic_F, the vanishing of

Suppose 0⁢p⁢t⁢(R)=00𝑝𝑡𝑅00pt(R)=00 italic_p italic_t ( italic_R ) = 0. It may be nice to find {n:R/𝔪n⁢ is not Lichtenbaum}conditional-set𝑛𝑅superscript𝔪𝑛 is not Lichtenbaum\{n:R/\mathfrak{m}^{n}\emph{ is not Lichtenbaum}\}{ italic_n : italic_R / fraktur_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is not Lichtenbaum }.

R/𝔪n⁢ is Lichtenbaum ⟺n=1.⟺𝑅superscript𝔪𝑛 is Lichtenbaum 𝑛1R/\mathfrak{m}^{n}\emph{ is Lichtenbaum }\Longleftrightarrow n=1.italic_R / fraktur_m start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is Lichtenbaum ⟺ italic_n = 1 .

A

\prod\limits_{g=5}^{n}\left(1+\sum\limits_{a=1}^{k}x_{g}^{(a)}\right).- ∑ start_POSTSUBSCRIPT roman_ℓ = italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_d = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_i start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_g = 5 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 1 + ∑ start_POSTSUBSCRIPT italic_a = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_a ) end_POSTSUPERSCRIPT ) .

The contribution of these four linear subdigraphs to det⁢(AΓk,R)detsubscript𝐴subscriptΓ𝑘𝑅\text{det}(A_{\Gamma_{k,R}})det ( italic_A start_POSTSUBSCRIPT roman_Γ start_POSTSUBSCRIPT italic_k , italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) is the same as (10), since

Now, to (10) we associate four linear subdigraphs such that the sum of the signed weights of the four linear subdigraphs is same as (10).

Now to (8), we associate 2(c−1)superscript2𝑐12^{(c-1)}2 start_POSTSUPERSCRIPT ( italic_c - 1 ) end_POSTSUPERSCRIPT linear subdigraphs such that the sum of sign weights of 2(c−1)superscript2𝑐12^{(c-1)}2 start_POSTSUPERSCRIPT ( italic_c - 1 ) end_POSTSUPERSCRIPT linear subdigraphs is (8), where c=s−r.𝑐𝑠𝑟c=s-r.italic_c = italic_s - italic_r . First consider the set

Now the sum of the sign weights of all these 2(c−1)superscript2𝑐12^{(c-1)}2 start_POSTSUPERSCRIPT ( italic_c - 1 ) end_POSTSUPERSCRIPT linear subdigraphs is

B

A𝐴Aitalic_A is invertible if and only if Φ⁢(A)Φ𝐴\Phi(A)roman_Φ ( italic_A ) is invertible. Also, A∈SL⁢(n,ℍ)𝐴SL𝑛ℍA\in\mathrm{SL}(n,\mathbb{H})italic_A ∈ roman_SL ( italic_n , blackboard_H ) if and only if Φ⁢(A)∈SL⁢(2⁢n,ℂ)Φ𝐴SL2𝑛ℂ\Phi(A)\in\mathrm{SL}(2n,\mathbb{C})roman_Φ ( italic_A ) ∈ roman_SL ( 2 italic_n , blackboard_C ).

The following result is also well-known, which classifies strongly AdSL⁢(n,ℂ)subscriptAdSL𝑛ℂ{\rm Ad}_{{\mathrm{SL}}(n,\mathbb{C})}roman_Ad start_POSTSUBSCRIPT roman_SL ( italic_n , blackboard_C ) end_POSTSUBSCRIPT-real semi-simple element in 𝔰⁢𝔩⁢(n,ℂ)𝔰𝔩𝑛ℂ\mathfrak{sl}(n,\mathbb{C})fraktur_s fraktur_l ( italic_n , blackboard_C ).

Let us recall the following results from [GM], which classifies strongly AdSL⁢(n,𝔽)subscriptAdSL𝑛𝔽{\rm Ad}_{{\mathrm{SL}}(n,\mathbb{F})}roman_Ad start_POSTSUBSCRIPT roman_SL ( italic_n , blackboard_F ) end_POSTSUBSCRIPT-real nilpotent element in 𝔰⁢𝔩⁢(n,𝔽)𝔰𝔩𝑛𝔽\mathfrak{sl}(n,\mathbb{F})fraktur_s fraktur_l ( italic_n , blackboard_F ), where 𝔽=ℂ𝔽ℂ\mathbb{F}=\mathbb{C}blackboard_F = blackboard_C or ℍℍ\mathbb{H}blackboard_H.

Next, we recall the well-known Jordan canonical form in M⁢(n,𝔽)M𝑛𝔽\mathrm{M}(n,\mathbb{F})roman_M ( italic_n , blackboard_F ).

Recall that any nilpotent element X∈𝔰⁢𝔩⁢(n,ℂ)𝑋𝔰𝔩𝑛ℂX\in\mathfrak{sl}(n,\mathbb{C})italic_X ∈ fraktur_s fraktur_l ( italic_n , blackboard_C ) has a unique Jordan canonical form N⁢(𝐝⁢(n), 0)N𝐝𝑛 0\mathrm{N}(\mathbf{d}(n),\,0)roman_N ( bold_d ( italic_n ) , 0 ), where 𝐝⁢(n)𝐝𝑛\mathbf{d}(n)bold_d ( italic_n ) is the corresponding partition of n𝑛nitalic_n. This is the correspondence between X𝑋Xitalic_X and the partition 𝐝⁢(n)𝐝𝑛\mathbf{d}(n)bold_d ( italic_n ) of n𝑛nitalic_n.

C

As we know in the smooth case the property of being coisotropic is closed for the C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT topology, and according to [HLS15] even in the C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT case (actually slightly less : to apply [HLS15] we need that the sequence Cksubscript𝐶𝑘C_{k}italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT of coisotro pic to be given by φk⁢(C)subscript𝜑𝑘𝐶\varphi_{k}(C)italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_C ) where the sequence (φk)k≥1subscriptsubscript𝜑𝑘𝑘1(\varphi_{k})_{k\geq 1}( italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ≥ 1 end_POSTSUBSCRIPT C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT converges to φ∈H⁢o⁢m⁢e⁢o⁢(M,ω)𝜑𝐻𝑜𝑚𝑒𝑜𝑀𝜔\varphi\in Homeo(M,\omega)italic_φ ∈ italic_H italic_o italic_m italic_e italic_o ( italic_M , italic_ω )). Also being coisotropic is a property preserved by symplectic reduction.

We shall say that V𝑉Vitalic_V is γ𝛾\gammaitalic_γ-coisotropic if it is non-empty and γ𝛾\gammaitalic_γ-coisotropic at each x∈V𝑥𝑉x\in Vitalic_x ∈ italic_V. It is nowhere γ𝛾\gammaitalic_γ-coisotropic if each point x∈V𝑥𝑉x\in Vitalic_x ∈ italic_V is non-γ𝛾\gammaitalic_γ-coisotropic.

There exists a sequence of Lagrangians such that its Hausdorff limit is non-γ𝛾\gammaitalic_γ-coisotropic.

We shall prove that if the reduction Cxsubscript𝐶𝑥C_{x}italic_C start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is non-γ𝛾\gammaitalic_γ-coisotropic for x𝑥xitalic_x in a dense set, then C𝐶Citalic_C is non-γ𝛾\gammaitalic_γ-coisotropic.

Figure 8. A Hausdorff limit (in red) of Lagrangians (in black) that is not γ𝛾\gammaitalic_γ-coisotropic.

B

In this section we define a series of homological invariants Qisuperscript𝑄𝑖Q^{i}italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT, with Q0superscript𝑄0Q^{0}italic_Q start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT isomorphic to Q𝑄Qitalic_Q in Definition 19. Moreover, we show that Q0superscript𝑄0Q^{0}italic_Q start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is isomorphic to the module of topological point excitations defined in [haah2013commuting] and derive some general properties of Qisuperscript𝑄𝑖Q^{i}italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT. Firstly, we show that Qi⁢(ℭ)=0superscript𝑄𝑖ℭ0Q^{i}(\mathfrak{C})=0italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( fraktur_C ) = 0 for i>D−1𝑖𝐷1i>D-1italic_i > italic_D - 1 (and also for i=D−1𝑖𝐷1i=D-1italic_i = italic_D - 1 for saturated codes). Secondly, we obtain bounds on Krull dimensions of Qi⁢(ℭ)superscript𝑄𝑖ℭQ^{i}(\mathfrak{C})italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( fraktur_C ). We expect Qisuperscript𝑄𝑖Q^{i}italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT to describe i𝑖iitalic_i-dimensional excitations (or defects). This is shown in Section 6 for Lagrangian codes such that all Qisuperscript𝑄𝑖Q^{i}italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT have Krull dimension zero. Computations of Qisuperscript𝑄𝑖Q^{i}italic_Q start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT for certain specific codes are presented in Section 7.

One may ask which quantum phases can be realized as Pauli stabilizer codes. It has been shown [bombin2014structure, haah2021classification] that for codes on ℤ2superscriptℤ2\mathbb{Z}^{2}blackboard_Z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT lattice with prime-dimensional qudits, stacks of toric codes are the only nontrivial phases with a unique ground state in infinite volume. The story is richer for qudits of composite dimension. Namely, it was shown [Z_4] that every abelian anyon model which admits a gapped boundary [kapustin2011topological] may be represented by a Pauli stabilizer code111During final revisions of this manuscript we have learned about [ellison2023pauli], where it is claimed that abelian anyon models without gapped boundaries can also be realized if one uses Pauli subsystem codes.. It was conjectured that the list of models constructed therein is exhaustive (up to finite depth quantum circuits and stabilization). There was even work on algorithmic determination of the corresponding abelian anyon model given a stabilizer code [liang2023extracting]. The proposed classification depends on several assumptions, one of which is that all local excitations in Pauli stabilizer codes are mobile and hence can be created at endpoints of string operators. In this paper we prove this, extending earlier results for prime-dimensional qudits. Stabilizer codes are even richer for D>2𝐷2D>2italic_D > 2 [haah2011local] due to the existence of so-called fractons: local excitations which can not be moved in any direction by acting with local operators. All this shows that mathematical study of stabilizer codes is an interesting and nontrivial problem. It is also closely related to classification of Clifford Quantum Cellular Automata [schlingemann2008structure, haah2021clifford, shirley2022three].

The next proposition shows that our definition of Q⁢(ℭ)𝑄ℭQ(\mathfrak{C})italic_Q ( fraktur_C ) agrees with topological point excitations in [haah2013commuting].

We remark that it follows immediately from our results that for saturated codes ℭℭ\mathfrak{C}fraktur_C with D=2𝐷2D=2italic_D = 2, the module Q⁢(ℭ)𝑄ℭQ(\mathfrak{C})italic_Q ( fraktur_C ) either vanishes or has Krull dimension zero. Together with the discussion in Section 5 it implies that all point excitations are mobile, i.e. they can be transported around by suitable string operators. This result has previously been shown only for codes with qudits of prime dimension [haah2013commuting]. Method adapted therein does not generalize to the case of composite qudit dimension due to the failure of Hilbert’s syzygy theorem, a crucial ingredient of the proof.

Section 2 details the mathematical set-up of translationally invariant stabilizer codes in terms of commutative algebra. Rudiments of symplectic geometry over group rings of ΛΛ\Lambdaroman_Λ are laid out here. Section 3 makes the connection between topological excitations and the functor ExtExt\mathrm{Ext}roman_Ext. Section 4 discusses operations on stabilizer codes, e.g. coarse-graining and stacking. In particular we prove that charge modules are invariant to coarse-graining and that they provide obstructions to obtaining a system from a lower dimensional one by stacking. Section 5 ventures a definition of mobility for excitations in any dimension. We also include a proof for the conjecture that in any 2D code with unique ground state, all excitations are mobile and can be created with string operators. In Section 6, we specialize to codes with only mobile excitations. It is shown that in this case charges may be described by cohomology classes of a certain Čech complex. We show how to obtain interesting operators and physical excitations from Čech cocycles. Moreover, we define braiding in terms of a cup product in the Čech complex and show that our proposal reduces to what is expected for D=2𝐷2D=2italic_D = 2. Several examples are worked out in Section 7. Some known mathematical definitions and facts used in the main text are reviewed in appendices: Gorenstein rings in Appendix A, local cohomology in Appendix B and Čech cohomology in Appendix C.

C

For any polyhedral set C𝐶Citalic_C embedded in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, we denote by KCsubscript𝐾𝐶K_{C}italic_K start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT the convex hull of the set of 00-cells of C𝐶Citalic_C, and denote by K~Csubscript~𝐾𝐶\widetilde{K}_{C}over~ start_ARG italic_K end_ARG start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT the polytopal complex consisting of all the faces of KCsubscript𝐾𝐶K_{C}italic_K start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT (including the set KCsubscript𝐾𝐶K_{C}italic_K start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT itself).

Let 𝒞𝒞\mathcal{C}caligraphic_C be a polyhedral complex embedded in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT in which all cells are pointed. Then the compact part of 𝒞𝒞\mathcal{C}caligraphic_C, com.part⁢(𝒞)com.part𝒞\textrm{com.part}(\mathcal{C})com.part ( caligraphic_C ), is a polytypal complex.

Equipped with the definition of a pointed polyhedral set, we can now define and establish notation for its compact part:

Let 𝒞𝒞\mathcal{C}caligraphic_C be a polyhedral complex embedded in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT in which all cells are pointed.

Definition 5.4 (Compact part of pointed complexes in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT).

D

Now we turn to the proof of Theorem 3.1.1. The main idea of the proof is to deduce it through a series of reductions from the definition of Hodge d𝑑ditalic_d-equivalences.

Now fix i≥0𝑖0i\geq 0italic_i ≥ 0 and suppose we know the claim for i𝑖iitalic_i. The global version of [BL22, Rem. 5.1.2] and the de Rham comparison Theorem (see [BL22, Prop. 5.2.5]) imply222In the formula below, we implicitly use that the Frobenius morphism F:W⁢(k)→W⁢(k):𝐹→𝑊𝑘𝑊𝑘F\colon W(k)\to W(k)italic_F : italic_W ( italic_k ) → italic_W ( italic_k ) is an isomorphism and preserves the ideal (p)⊂W⁢(k)𝑝𝑊𝑘(p)\subset W(k)( italic_p ) ⊂ italic_W ( italic_k ). Therefore, the formula in [BL22, Prop. 5.2.5] can be Frobenius “untwisted”. We also use that the Breuil-Kisin twists can be canonically trivialized for the prism (W⁢(k),(p))𝑊𝑘𝑝(W(k),(p))( italic_W ( italic_k ) , ( italic_p ) ). that

If k𝑘kitalic_k is a perfect field of characteristic p>0𝑝0p>0italic_p > 0 and X𝑋Xitalic_X is a k𝑘kitalic_k-scheme, we denote the (underived) crystalline cohomology by R⁢Γcrys⁢(X/W⁢(k))RsubscriptΓcrys𝑋𝑊𝑘\mathrm{R}\Gamma_{\mathrm{crys}}(X/W(k))roman_R roman_Γ start_POSTSUBSCRIPT roman_crys end_POSTSUBSCRIPT ( italic_X / italic_W ( italic_k ) ). We refer to [BO78, §5] and [Sta21, Tag 07GI] for more details (see also [BL22, Construction F.2]).

The case of μpsubscript𝜇𝑝\mu_{p}italic_μ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT will give us the most difficulty. Here we will find it convenient to pivot to proving a Lefschetz hyperplane theorem for the syntomic cohomology of the Tate twists 𝐙p⁢(i)subscript𝐙𝑝𝑖\mathbf{Z}_{p}(i)bold_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_i ). Using the Nygaard filtration, this will ultimately be reduced to proving a Lefschetz hyperplane theorem for each filtered piece in the conjugate filtration on de Rham cohomology, which has been established in [ABM21]. In particular, we get a Lefschetz hyperplane theorem for the syntomic cohomology of the Tate twists 𝐙p⁢(i)subscript𝐙𝑝𝑖\mathbf{Z}_{p}(i)bold_Z start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_i ) defined in [BMS19] (see also Section 1.2):

For a syntomic k𝑘kitalic_k-scheme X𝑋Xitalic_X, we refer to Corollary 2.1.6, [BL22, Warning 4.6.2, Prop. 5.1.1]111Strictly speaking, [BL22, Prop. 5.1.1] constructs the Nygaard filtration only in the affine case. However, similarly to [BL22, Not. 5.5.23], this can be formally extended to an arbitrary k𝑘kitalic_k-scheme. for the definition of the Nygaard filtration

D

\longrightarrow}}0square-root start_ARG italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG over~ start_ARG italic_T end_ARG start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ⟶ end_ARG start_ARG roman_p end_ARG end_RELOP 0.

Letting n→∞→𝑛n\to\inftyitalic_n → ∞ in (4.2) and recalling that a′⁢(τ)<1superscript𝑎′𝜏1a^{\prime}(\tau)<1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ ) < 1, the first assertion of the corollary follows using (4.26). Thus, A~∙(n)⁢(T~∞,ϵ(n))=T~∞,ϵ(n)−T~0(n)⟶pτsubscriptsuperscript~𝐴𝑛∙subscriptsuperscript~𝑇𝑛italic-ϵsubscriptsuperscript~𝑇𝑛italic-ϵsubscriptsuperscript~𝑇𝑛0superscript⟶p𝜏\tilde{A}^{(n)}_{\bullet}(\tilde{T}^{(n)}_{\infty,\epsilon})=\tilde{T}^{(n)}_{%

It is easily verified that a′⁢(τ)<1superscript𝑎′𝜏1a^{\prime}(\tau)<1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ ) < 1, so (τ,a⁢(τ))𝜏𝑎𝜏(\tau,a(\tau))( italic_τ , italic_a ( italic_τ ) ) is a proper crossing point of the function a⁢(t)𝑎𝑡a(t)italic_a ( italic_t ) and the straight line of gradient one through the origin.

Recall that a′⁢(τ)<1superscript𝑎′𝜏1a^{\prime}(\tau)<1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ ) < 1, so substituting these observations into (4.41) yields after application of Slutsky’s theorem that

τ(n)→τ→superscript𝜏𝑛𝜏\tau^{(n)}\to\tauitalic_τ start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT → italic_τ as n→∞→𝑛n\to\inftyitalic_n → ∞, since τ𝜏\tauitalic_τ is the unique solution in (0,∞)0(0,\infty)( 0 , ∞ ) of a⁢(t)=t𝑎𝑡𝑡a(t)=titalic_a ( italic_t ) = italic_t and a′⁢(τ)<1superscript𝑎′𝜏1a^{\prime}(\tau)<1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_τ ) < 1.

C

The rest of the paper is organized as follows. In Section 2, we provide preliminaries, notation and helicity-conservative finite element scheme. In Section 3, we present a PINN-based algorithm that preserves the helicity. In Section 4, we present numerical results on the convergence and helicity-preserving properties of our algorithms. In Section 5, we give some concluding remarks.

We consider the following system of equations in Ω×(0,𝕋]Ω0𝕋\Omega\times(0,\mathbb{T}]roman_Ω × ( 0 , blackboard_T ]:

We shall consider the following boundary conditions for (1): for all 𝒙∈∂Ω𝒙Ω\bm{x}\in\partial\Omegabold_italic_x ∈ ∂ roman_Ω and t>0𝑡0t>0italic_t > 0,

1:Input: Initial sample points for 𝒖𝒖\bm{u}bold_italic_u, 𝝎𝝎\bm{\omega}bold_italic_ω, p𝑝pitalic_p and sample points in ΩΩ\Omegaroman_Ω for numerical integration and training algorithm, and the number of layers L𝐿Litalic_L, the final time level 𝕋𝕋\mathbb{T}blackboard_T and the time domain of one sequence d𝕋𝕋\mathbb{T}blackboard_T.

1:Input: Initial sample points for 𝒖𝒖\bm{u}bold_italic_u, p𝑝pitalic_p and sample points in ΩΩ\Omegaroman_Ω for numerical integration and training algorithm, and the number of layers L𝐿Litalic_L, the final time level 𝕋𝕋\mathbb{T}blackboard_T and the time domain of one sequence d𝕋𝕋\mathbb{T}blackboard_T.

A

In this section, we present certain desirable properties of the proposed filtration and substantiate our claims in the introduction. In Section 4.1, we discuss how the proposed filtration prolongs persistences of homology classes of high-density regions. Then we discuss, in Section 4.2, the proposed filtration’s scale invariance, which motivates the awkward-looking exponent 1/D1𝐷1/D1 / italic_D in the definition of the RDAD function, and in Section 4.3, its robustness, which is enhanced by the DTM setup. We conclude by giving further mathematical properties of the proposed filtration in Section 4.4. All proofs are delayed to Appendix A.

In this section, we present certain desirable properties of the proposed filtration and substantiate our claims in the introduction. In Section 4.1, we discuss how the proposed filtration prolongs persistences of homology classes of high-density regions. Then we discuss, in Section 4.2, the proposed filtration’s scale invariance, which motivates the awkward-looking exponent 1/D1𝐷1/D1 / italic_D in the definition of the RDAD function, and in Section 4.3, its robustness, which is enhanced by the DTM setup. We conclude by giving further mathematical properties of the proposed filtration in Section 4.4. All proofs are delayed to Appendix A.

We illustrate the results above with corrupted versions of the “Antman" example in Figure 5. We compare the DAD filtration and the RDAD filtration in Figures 6 and 7. The persistence diagrams of RDAD for the corrupted datasets are affected to a lesser extent by the noise and outliers than those of DAD.

The rest of the paper is organized as follows. After reviewing the mathematical background in Section 2, we define the proposed filtration in Section 3 and discuss its properties in Section 4. We discuss bootstrapping in Section 5 and present numerical simulations in Section 6. A discussion and the conclusion are presented in Sections 7 and 8.

In this subsection, we illustrate how the proposed filtration prolongs persistences of homology classes of high-density regions with a numerical example, and we formalize the observations from the example with theorems. For the numerical examples in this and subsequent subsections, parameters are summarized in Table 3 in Appendix B, and implementation details are deferred to Section 6.1.

D

In a broader context of reinforcement learning with partial observability, our work is related to several recent works on POMDPs with special structures. For example, Kwon et al. (2021) considers latent POMDPs, where each process has only one latent state, and the proposed algorithm efficiently infers the latent state using a short trajectory. Kozuno et al. (2021) considers POMDPs having tree-structured states with their positions in certain partitions being the observations. Compared with general POMDPs, these specially structures reduce the complexity of finding the optimal actions, and the corresponding algorithms use techniques closer to those for MDPs. Also, the aforementioned literature only consider tabular POMDPs.

Partial observability poses significant challenges for reinforcement learning, especially when the observation and state spaces are infinite. Given full observability, reinforcement learning is well studied empirically (Mnih et al., 2015; Silver et al., 2016, 2017) and theoretically (Auer et al., 2008; Osband et al., 2016; Azar et al., 2017; Jin et al., 2018; Yang and Wang, 2020; Jin et al., 2020b; Ayoub et al., 2020; Kakade et al., 2020; Du et al., 2021). In particular, for infinite state spaces, neural function approximators achieve remarkable successes empirically (Mnih et al., 2015; Berner et al., 2019; Arulkumaran et al., 2019), while linear function approximators become better understood theoretically (Yang and Wang, 2020; Jin et al., 2020b; Ayoub et al., 2020; Kakade et al., 2020; Du et al., 2021). In contrast, reinforcement learning in partially observed Markov decision processes (POMDPs) is less investigated theoretically despite its prevalence in practice (Cassandra et al., 1996; Hauskrecht and Fraser, 2000; Brown and Sandholm, 2018; Rafferty et al., 2011).

Our work is related to a line of recent work on the sample efficiency of reinforcement learning for POMDPs. In detail, Azizzadenesheli et al. (2016); Guo et al. (2016); Xiong et al. (2021) establish sample complexity guarantees for searching the optimal policy in POMDPs whose models are identifiable and can be estimated by spectral methods. However, Azizzadenesheli et al. (2016) and Guo et al. (2016) add extra assumptions such that efficient exploration of the POMDP can always be achieved by running arbitrary policies. In contrast, the upper bound confidence (UCB) method is used in Xiong et al. (2021) for adaptive exploration. However, they require strictly positive state transition and observation emission kernels to ensure fast convergence to the stationary distribution. The more related work is Jin et al. (2020a), which considers undercomplete POMDPs, in other words, the observations are more than the latent states. Their proposed algorithm can attain the optimal policy without estimating the exact model, but an observable component (Jaeger, 2000; Hsu et al., 2012), which is the same for our algorithm design, while only applies to tabular POMDPs.

More specifically, partial observability poses both statistical and computational challenges. From a statistical perspective, it is challenging to predict future rewards, observations, or states due to a lack of the Markov property. In particular, predicting the future often involves inferring the distribution of the state (also known as the belief state) or its functionals as a summary of the history, which is already challenging even assuming the (observation) emission and (state) transition kernels are known (Vlassis et al., 2012; Golowich et al., 2022). Meanwhile, learning the emission and transition kernels faces various issues commonly encountered in causal inference (Zhang and Bareinboim, 2016). For example, they are generally nonidentifiable (Kallus et al., 2021). Even assuming they are identifiable, their estimation possibly requires a sample size that scales exponentially in the horizon and dimension (Jin et al., 2020a). Such statistical challenges are already prohibitive even for the evaluation of a policy (Nair and Jiang, 2021; Kallus et al., 2021; Bennett and Kallus, 2021), which forms the basis of policy optimization. From a computational perspective, it is known that policy optimization is generally intractable (Vlassis et al., 2012; Golowich et al., 2022). Moreover, infinite observation and state spaces amplify both statistical and computational challenges. On the other hand, most existing results are restricted to the tabular setting (Azizzadenesheli et al., 2016; Guo et al., 2016; Jin et al., 2020a; Xiong et al., 2021), where the observation and state spaces are finite.

In the contexture of reinforcement learning with function approximations, our work is related to a vast body of recent progress (Yang and Wang, 2020; Jin et al., 2020b; Cai et al., 2020; Du et al., 2021; Kakade et al., 2020; Agarwal et al., 2020; Zhou et al., 2021; Ayoub et al., 2020) on the sample efficiency of reinforcement learning for MDPs with linear function approximations. These works characterize the uncertainty in the regression for estimating either the model or value function of an MDP and use the uncertainty as a bonus on the rewards to encourage exploration. However, none of these approaches directly apply to POMDPs due to the latency of the states.

D

A diamond is the resulting graph after deleting an edge from a K4subscript𝐾4K_{4}italic_K start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT.

A wheel Wksubscript𝑊𝑘W_{k}italic_W start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a Cksubscript𝐶𝑘C_{k}italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT plus a universal vertex (see Figure 2).

A gem is a P4subscript𝑃4P_{4}italic_P start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT plus a universal vertex.

An H𝐻Hitalic_H graph is two copies of P3subscript𝑃3P_{3}italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT plus an edge connecting both middle vertices of P3subscript𝑃3P_{3}italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT’s. The latter edge is the middle edge of H𝐻Hitalic_H.

A vertex is universal if it is neighbor of all other vertices of the graph. A vertex is simplicial if the subgraph induced by its neighborhood is complete.

B

𝒟¯={H∈𝔱:|H|≤|H+γ|,γ∈Γ}¯𝒟conditional-set𝐻𝔱formulae-sequence𝐻𝐻𝛾𝛾Γ{\overline{{\mathscr{D}}}}=\{H\in\mathfrak{t}:\,|H|\leq|H+\gamma|,\,\gamma\in\Gamma\}over¯ start_ARG script_D end_ARG = { italic_H ∈ fraktur_t : | italic_H | ≤ | italic_H + italic_γ | , italic_γ ∈ roman_Γ }

We first claim that the cut locus of T⁢p𝑇𝑝Tpitalic_T italic_p is the boundary of 𝒟𝒟{\mathscr{D}}script_D. Indeed, recall that 𝒟¯¯𝒟{\overline{\mathscr{D}}}over¯ start_ARG script_D end_ARG is the set of vectors of 𝔱𝔱\mathfrak{t}fraktur_t with minimal norm among its ΓΓ\Gammaroman_Γ equivalents.

Its Dirichilet domain is 𝒟0subscript𝒟0{\mathscr{D}}_{0}script_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,

Consider the Dirichilet domain 𝒟𝒟{\mathscr{D}}script_D of the lattice ΓΓ\Gammaroman_Γ and the Dirichilet domain 𝒟0subscript𝒟0{\mathscr{D}}_{0}script_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of the fundamental lattice Γ0subscriptΓ0\Gamma_{0}roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (see Section 2.3).

The next result shows that 𝒟𝒟{\mathscr{D}}script_D is a fundamental domain of ΓΓ\Gammaroman_Γ and characterizes its geometry.

D

Then, one can easily verify properties i) and ii) in Definition 1.1. Note that the restriction α<1𝛼1\alpha<1italic_α < 1 is needed in order to deduce that F𝐹Fitalic_F is strictly increasing in each variable. The averaging operator F𝐹Fitalic_F is clearly symmetric.

The case α=1𝛼1\alpha=1italic_α = 1 deserves a special treatment because the corresponding operator does not satisfy the requirements of Definition 1.1, see 2.4.

Let 0≤α<10𝛼10\leq\alpha<10 ≤ italic_α < 1. Any nonnegative α𝛼\alphaitalic_α-harmonious function on ℤdsuperscriptℤ𝑑\mathbb{Z}^{d}blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT is constant. In particular, if α=0𝛼0\alpha=0italic_α = 0, then we retrieve the strong Liouville property for discrete harmonic functions.

An important particular case of the α𝛼\alphaitalic_α-harmonious operator comes up when α≡1𝛼1\alpha\equiv 1italic_α ≡ 1, in which case

where u𝑢uitalic_u is a (continuous) function in the closure of a bounded given domain ΩΩ\Omegaroman_Ω in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT or in a metric measure space (X,d,μ)𝑋𝑑𝜇(X,d,\mu)( italic_X , italic_d , italic_μ ), Bxsubscript𝐵𝑥B_{x}italic_B start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is a ball centered at x𝑥xitalic_x contained in ΩΩ\Omegaroman_Ω and α∈[0,1]𝛼01\alpha\in[0,1]italic_α ∈ [ 0 , 1 ]. Note that when α=0𝛼0\alpha=0italic_α = 0 then we retrieve harmonic functions (see also [1, 2]), whereas for α=1𝛼1\alpha=1italic_α = 1 we obtain the so-called harmonious functions in ΩΩ\Omegaroman_Ω, see [25]. It turns out that, in the Euclidean setting and in the range p≥2𝑝2p\geq 2italic_p ≥ 2, the functional equation (1.5) with α=α⁢(p,d)=p−2p+2⁢d𝛼𝛼𝑝𝑑𝑝2𝑝2𝑑\alpha=\alpha(p,d)=\frac{p-2}{p+2d}italic_α = italic_α ( italic_p , italic_d ) = divide start_ARG italic_p - 2 end_ARG start_ARG italic_p + 2 italic_d end_ARG can be understood as a Dynamic Programming Principle for a stochastic game related to the p𝑝pitalic_p-Laplacian, see [32, 33, 28, 29, 6] and references therein for further details. Functions satisfying (1.5) with α=α⁢(p,d)𝛼𝛼𝑝𝑑\alpha=\alpha(p,d)italic_α = italic_α ( italic_p , italic_d ) have been called p𝑝pitalic_p-harmonious, see [2, 29]. It is then natural to consider discrete versions of equation (1.5). Let α∈[0,1]𝛼01\alpha\in[0,1]italic_α ∈ [ 0 , 1 ]. We will say that u:ℤd→ℝ:𝑢→superscriptℤ𝑑ℝu:\mathbb{Z}^{d}\to\mathbb{R}italic_u : blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R is α𝛼\alphaitalic_α-harmonious if

C

We call a regular oriented matroid co-Eulerian if for each circuit C𝐶Citalic_C, we have |C+|=|C−|superscript𝐶superscript𝐶|C^{+}|=|C^{-}|| italic_C start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | = | italic_C start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |.

By Fact 2.8, a regular oriented matroid M𝑀Mitalic_M is co-Eulerian if and only if its dual is Eulerian.

We call a regular oriented matroid Eulerian, if |(C*)+|=|(C*)−|superscriptsuperscript𝐶superscriptsuperscript𝐶|(C^{*})^{+}|=|(C^{*})^{-}|| ( italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | = | ( italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT | for each cocircuit C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT. The name comes from the fact that for a regular oriented matroid that comes from a directed graph, being Eulerian is equivalent to the digraph being Eulerian.

For an Eulerian digraph G𝐺Gitalic_G, the (oriented) dual M*superscript𝑀M^{*}italic_M start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT of the graphic matroid of G𝐺Gitalic_G is a co-Eulerian regular oriented matroid.

We call a regular oriented matroid co-Eulerian if for each circuit C𝐶Citalic_C, we have |C+|=|C−|superscript𝐶superscript𝐶|C^{+}|=|C^{-}|| italic_C start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | = | italic_C start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |.

A

Without loss of generality we can additionally assume that rank⁢(S~)=nrank~𝑆𝑛\mathrm{rank}(\tilde{S})=nroman_rank ( over~ start_ARG italic_S end_ARG ) = italic_n, since otherwise (11) can be reduced to an equivalent form over a smaller set of parameters 11.

For each 𝒮(h)superscript𝒮ℎ\mathcal{S}^{(h)}caligraphic_S start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT, intersect the set with the associated mp-QP partition from (ii). Assemble a solution over the whole of 𝒮𝒮\mathcal{S}caligraphic_S from the resulting functions.

Note that standard methods for proving that the parametric solution of (10) in x𝑥xitalic_x is PWA continuous can not be applied because the right hand side S⁢(x)𝑆𝑥S(x)italic_S ( italic_x ) of the inequalities is not affine. We can be sure, however, that the problem has a solution for any x∈𝒮𝑥𝒮x\in\mathcal{S}italic_x ∈ caligraphic_S since the associated Minkowski function Ψ⁢(⋅)Ψ⋅\Psi(\cdot)roman_Ψ ( ⋅ ) is a CLF for the polytopic system in (1).

The problem (11) now looks like a standard mp-QP whose solution can be computed parametrically in x𝑥xitalic_x. Putting aside for the moment the issue of possible degeneracy, this parametric solution could even be computed over the whole of 𝒮𝒮\mathcal{S}caligraphic_S. Our proof will therefore proceed as follows (see also Fig. 3 for a schematic representation):

For each 𝒮(h)superscript𝒮ℎ\mathcal{S}^{(h)}caligraphic_S start_POSTSUPERSCRIPT ( italic_h ) end_POSTSUPERSCRIPT, generate a continuous PWA solution to an mp-QP in the form (11) over the whole of 𝒮𝒮\mathcal{S}caligraphic_S.

C

The analysis carried out in the proof of Theorem 1.3 allows us to include negative values for the parameter θ1,subscript𝜃1\theta_{1},italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , namely, θ1∈(−2⁢n,0)subscript𝜃12𝑛0\theta_{1}\in(-2n,0)italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ ( - 2 italic_n , 0 ) (cf. Lemma 1.1 and Remark 3.9). However, we do not know the possible physical meaning in the description of the model.

1.3, we prove the following uniqueness theorem without assuming any smallness condition of the initial data. The existence of solutions for arbitrary large initial data is an open problem. This uniqueness result seems new for chemotaxis problems in the context of critical spaces, including the classical Keller-Segel system (1.1). This issue has been raised in the context of Navier-Stokes equations (see [20] and some references therein).

Motivated by the above considerations, the aim of this paper is to analyze the existence, uniqueness and persistence of global solutions for the spatio-temporal fractional Keller-Segel system (1.2) in the framework of critical Besov spaces without using auxiliary norms. In order to get this aim, we first derive time decay and integral estimates of the Mittag-Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller-Segel system employing only the norm of the natural persistence space. In order to estimate the bilinear operator, in addition to dealing with the action of the fractional heat semigroup, is necessary to prove a product estimate in the homogeneous Besov setting.

in order to choose the function spaces of initial data. Explicitly, we consider the following class of initial data (see the notations in Section 2):

for θ1∈[0,n),subscript𝜃10𝑛\theta_{1}\in[0,n),italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ 0 , italic_n ) , which can be alternatively represented by G⁢(v)⁢f=K⁢(x)∗f,𝐺𝑣𝑓∗𝐾𝑥𝑓G(v)f=K(x)\ast f,italic_G ( italic_v ) italic_f = italic_K ( italic_x ) ∗ italic_f , K⁢(x)∼x|x|n−θ1.similar-to𝐾𝑥𝑥superscript𝑥𝑛subscript𝜃1K(x)\sim\frac{x}{|x|^{n-\theta_{1}}}.italic_K ( italic_x ) ∼ divide start_ARG italic_x end_ARG start_ARG | italic_x | start_POSTSUPERSCRIPT italic_n - italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG . The case θ=2𝜃2\theta=2italic_θ = 2 and θ1=0subscript𝜃10\theta_{1}=0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 corresponds to the classical Keller-Segel system (1.1). For θ1=0,α=1formulae-sequencesubscript𝜃10𝛼1\theta_{1}=0,\alpha=1italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_α = 1 and n=2,𝑛2n=2,italic_n = 2 , in [13] the authors proved a result of local existence and uniqueness of solution for (1.2) in homogeneous Besov spaces by using some estimates of the linear dissipative equation in the framework of mixed temporal-spatial spaces, the Chemin mono-norm methods, the Fourier localization and the Littlewood-Paley theory. Later, in [14], the author proved the existence, uniqueness and stability of solutions for (1.2) in critical Besov spaces under smallness condition on the initial data. The results of [14], are based on the Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-Lqsuperscript𝐿𝑞L^{q}italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT time decay for the semigroup e−t⁢(−Δ)θ/2superscript𝑒𝑡superscriptΔ𝜃2e^{-t(-\Delta)^{\theta/2}}italic_e start_POSTSUPERSCRIPT - italic_t ( - roman_Δ ) start_POSTSUPERSCRIPT italic_θ / 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT in Besov spaces, which leads to use auxiliar norms of Besov type and Kato-time-weighted norms. Some results of global existence and blow-up for the particular case of (1.2) with α=1𝛼1\alpha=1italic_α = 1 and without considering the v𝑣vitalic_v-equation, have been obtained in [15, 16] and some references therein.

A

The cyclic (resp. periodic cyclic) homology of this cyclic module is denoted by H⁢C∗,H⁢(C,M∨)𝐻subscript𝐶∗𝐻𝐶superscript𝑀HC_{\ast,H}(C,M^{\vee})italic_H italic_C start_POSTSUBSCRIPT ∗ , italic_H end_POSTSUBSCRIPT ( italic_C , italic_M start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ) (resp. H⁢P∗,H⁢(C,M∨)𝐻subscript𝑃∗𝐻𝐶superscript𝑀HP_{\ast,H}(C,M^{\vee})italic_H italic_P start_POSTSUBSCRIPT ∗ , italic_H end_POSTSUBSCRIPT ( italic_C , italic_M start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT )), and it is called the (periodic) Hopf-cyclic homology of the H𝐻Hitalic_H-module coalgebra C𝐶Citalic_C, with coefficients in M∨superscript𝑀M^{\vee}italic_M start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT.

Now, given a Hopf-subalgebra K⊆H𝐾𝐻K\subseteq Hitalic_K ⊆ italic_H, let 𝒞𝒞\mathcal{C}caligraphic_C be the left H𝐻Hitalic_H-module coalgebra of (2.1). It then follows at once that

We next record the relative theory. Given a Hopf subalgebra K⊆H𝐾𝐻K\subseteq Hitalic_K ⊆ italic_H, let

Given a left H𝐻Hitalic_H-module algebra A𝐴Aitalic_A, and a right/left SAYD module M𝑀Mitalic_M over H𝐻Hitalic_H, let CHn⁢(A,M∨):=HomH⁢(A⊗(n+1),M∨)assignsubscriptsuperscript𝐶𝑛𝐻𝐴superscript𝑀subscriptHom𝐻superscript𝐴tensor-productabsent𝑛1superscript𝑀C^{n}_{H}(A,M^{\vee}):={\rm Hom}_{H}(A^{\otimes\,(n+1)},M^{\vee})italic_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_A , italic_M start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ) := roman_Hom start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT ⊗ ( italic_n + 1 ) end_POSTSUPERSCRIPT , italic_M start_POSTSUPERSCRIPT ∨ end_POSTSUPERSCRIPT ) be the space of left H𝐻Hitalic_H-linear maps. Then, it follows from [35, Prop. 2.2] that the isomorphisms

Let H𝐻Hitalic_H be a Hopf algebra, and C𝐶Citalic_C a H𝐻Hitalic_H-module coalgebra which acts on a H𝐻Hitalic_H-module algebra A𝐴Aitalic_A. Then, given any SAYD module M𝑀Mitalic_M over H𝐻Hitalic_H, there is a pairing

A

The above discussion raises the question of which definition of δ′superscript𝛿normal-′\delta^{\prime}italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT-type interaction on a graph is motivated physically, i.e. whether it is the one emerging from the analysis of thin networks as the operator 𝒜hom,subscript𝒜normal-hom{\mathcal{A}}_{\rm hom},caligraphic_A start_POSTSUBSCRIPT roman_hom end_POSTSUBSCRIPT , see (16)–(18), or the traditional (see [8]) definition (21)–(22). At first sight, the difference between the two operators is insignificant: it is only in their kernels. However, it can happen to be of paramount importance if, e.g., one considers an ε𝜀\varepsilonitalic_ε-periodic graph with δ′superscript𝛿normal-′\delta^{\prime}italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT-type matching conditions, in which case the homogenisation procedure [16, 17] will lead to drastically different outcomes for the two related setups, as it relies upon a “threshold effect” [11] in the behaviour of the least eigenvalue of the operator on the fundamental cell for small quasimomenta.

In the generic case when 𝒜𝒜\mathcal{A}caligraphic_A is simple (in particular when all lesubscript𝑙𝑒l_{e}italic_l start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT are rationally independent), the claim follows immediately from Lemma 5.5. Indeed, in this case no positive z𝑧zitalic_z can be an eigenvalue of the operator 𝒜𝒜\mathcal{A}caligraphic_A and the same applies to the operator (Amaxδ′)*superscriptsubscriptsuperscript𝐴superscript𝛿′(A^{\delta^{\prime}}_{\max})^{*}( italic_A start_POSTSUPERSCRIPT italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT. Therefore, any reducing self-adjoint “part” of either symmetric operator can only be zero.

The value of the fact that for ϕ∈H1⁢(Γε)italic-ϕsuperscript𝐻1superscriptΓ𝜀\phi\in H^{1}(\Gamma^{\varepsilon})italic_ϕ ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Γ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) one has M⁢(z)⁢ϕ=ME⁢(z)⁢ϕ+MV⁢(z)⁢ϕ𝑀𝑧italic-ϕsubscript𝑀𝐸𝑧italic-ϕsubscript𝑀𝑉𝑧italic-ϕM(z)\phi=M_{E}(z)\phi+M_{V}(z)\phiitalic_M ( italic_z ) italic_ϕ = italic_M start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_z ) italic_ϕ + italic_M start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( italic_z ) italic_ϕ is clear: in contrast to Aε,subscript𝐴𝜀A_{\varepsilon},italic_A start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT , which cannot be additively decomposed into “independent” terms pertaining to the vertex and edge parts of the medium Q𝑄Qitalic_Q owing to the transmission interface conditions on Γε,superscriptΓ𝜀\Gamma^{\varepsilon},roman_Γ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT , the M𝑀Mitalic_M-function is additive (see, e.g., [17, 22], where this property was observed and exploited in the related settings of homogenisation and scattering, respectively). In what follows, we will observe that the resolvent (Aε−z)−1superscriptsubscript𝐴𝜀𝑧1(A_{\varepsilon}-z)^{-1}( italic_A start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT - italic_z ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT can be expressed in terms of M⁢(z)𝑀𝑧M(z)italic_M ( italic_z ) via a version of the celebrated Kreĭn formula, thus reducing the asymptotic analysis of the resolvent to that of the corresponding M𝑀Mitalic_M-function (see, e.g., [1, 7] for alternative approaches to derivation of the Kreĭn formula in our setting).

It was proved in [32, 46, 47] that, within any compact K∈ℂ,𝐾ℂK\in\mathbb{C},italic_K ∈ blackboard_C , the spectra of Aεsubscript𝐴𝜀A_{\varepsilon}italic_A start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT converge in the Hausdorff sense to the spectrum of a graph Laplacian AGsubscript𝐴𝐺A_{G}italic_A start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT. In the book [56], the claimed convergence was enhanced to the norm-resolvent type, with an explicit control of the error as O⁢(εγ),𝑂superscript𝜀𝛾O(\varepsilon^{\gamma}),italic_O ( italic_ε start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ) , where γ>0𝛾0\gamma>0italic_γ > 0 depends on whether the ambient space is two-dimensional. The matching conditions at the vertices of the limiting graph turn out to be either of:

EP/V013025/1. YYE, AVK are grateful to IIMAS–UNAM for the hospitality and financial support during the research visit when part of this work was carried out.

D

But then, since dimΣk⁢(H)<dimΣkdimensionsubscriptΣ𝑘𝐻dimensionsubscriptΣ𝑘\dim\Sigma_{k}(H)<\dim\Sigma_{k}roman_dim roman_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_H ) < roman_dim roman_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, by the inductive hypothesis H𝐻Hitalic_H is k𝑘kitalic_k-Du Bois and hence Σk⁢(H)=∅subscriptΣ𝑘𝐻\Sigma_{k}(H)=\emptysetroman_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_H ) = ∅. This contradicts the statement that Σk⁢(H)≠∅subscriptΣ𝑘𝐻\Sigma_{k}(H)\neq\emptysetroman_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_H ) ≠ ∅.

If X𝑋Xitalic_X is a k𝑘kitalic_k-rational algebraic variety with lci singularities, then X𝑋Xitalic_X is k𝑘kitalic_k-Du Bois.

Suppose either that X𝑋Xitalic_X has isolated singularities or X𝑋Xitalic_X is lci. If X𝑋Xitalic_X is k𝑘kitalic_k-rational, then X𝑋Xitalic_X is k𝑘kitalic_k-Du Bois.

If X𝑋Xitalic_X has lci singularities and X𝑋Xitalic_X is k𝑘kitalic_k-Du Bois, then X𝑋Xitalic_X is (k−1)𝑘1(k-1)( italic_k - 1 )-rational.

Let X𝑋Xitalic_X be a complex algebraic variety. Then X𝑋Xitalic_X is k𝑘kitalic_k-Du Bois if the natural maps

A

We begin with a stylized example to highlight the interaction between an agent’s incentives and the principal’s statistical protocol. Suppose there are two types of pharmaceutical companies: companies with ineffective drugs (θ=0𝜃0\theta=0italic_θ = 0) and companies with effective drugs (θ=1𝜃1\theta=1italic_θ = 1). Further, assume that the company knows its type, while the regulator does not. The company may choose to pay $10 million to run a clinical trial, which results in a statistical test for the null hypothesis that the drug is ineffective. Suppose that the test is carried out so that it has 5% type-I error and 80% power to reject when θ=1𝜃1\theta=1italic_θ = 1; see below.

We report on the expected value of a placebo drug under the three protocols above in Table 1. We find that for typical drugs with $1-10B profit if approved, the standard protocol requiring two trials is incentive-aligned. For extremely profitable drugs earning $100B or more, the protocol ceases to be incentive-aligned. Furthermore, as the protocol is loosened, it ceases to be incentive-aligned even for less profitable drugs. With the high-discretion accelerated protocol, incentive alignment is lost even for a typical drug. In the blockbuster scenario, an organization may have an expected value of hundreds of millions or billions of dollars of profit for running a clinical trial on a placebo.

Is this a good statistical protocol? The answer depends on how much money the pharmaceutical company will make, among other things. In particular, depending on the total profit the company earns when they are approved, even companies with ineffective drugs may be incentivized to run a trial.

We begin with a stylized example to highlight the interaction between an agent’s incentives and the principal’s statistical protocol. Suppose there are two types of pharmaceutical companies: companies with ineffective drugs (θ=0𝜃0\theta=0italic_θ = 0) and companies with effective drugs (θ=1𝜃1\theta=1italic_θ = 1). Further, assume that the company knows its type, while the regulator does not. The company may choose to pay $10 million to run a clinical trial, which results in a statistical test for the null hypothesis that the drug is ineffective. Suppose that the test is carried out so that it has 5% type-I error and 80% power to reject when θ=1𝜃1\theta=1italic_θ = 1; see below.

Conversely, the statistical protocol changes the incentives of the agents. Consider again the large profit case above, where agents receive 100 times their initial investment if they receive approval. Now, however, suppose the principal changes to a stricter protocol such that the probability of approval is only 0.0050.0050.0050.005 when θ=0𝜃0\theta=0italic_θ = 0. With this more stringent standard, the agents of type θ=0𝜃0\theta=0italic_θ = 0 are no longer incentivized to submit their drug for approval; they would lose money, on average, if they chose to run clinical trials for drugs they know to be ineffective. Thus, changing the statistical protocol will affect how the agents are incentivized to act.

B

If γ=0𝛾0\gamma=0italic_γ = 0, α⁢μ>0𝛼𝜇0\alpha\mu>0italic_α italic_μ > 0 and κ∈{0,−1}𝜅01\kappa\in\{0,-1\}italic_κ ∈ { 0 , - 1 }, then there exists a connected component ΣisubscriptΣ𝑖\Sigma_{i}roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT diffeomorphic to a torus.

Let (Mn,g)superscript𝑀𝑛𝑔(M^{n},g)( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_g ) be a compact gradient Einstein-type manifold of constant scalar curvature S=κ⁢n⁢(n−1)𝑆𝜅𝑛𝑛1S=\kappa n(n-1)italic_S = italic_κ italic_n ( italic_n - 1 ), κ∈{−1,0,1}𝜅101\kappa\in\left\{-1,0,1\right\}italic_κ ∈ { - 1 , 0 , 1 }, and with boundary Σ=∪iΣiΣsubscript𝑖subscriptΣ𝑖\Sigma=\cup_{i}\Sigma_{i}roman_Σ = ∪ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. If α⁢μ<0𝛼𝜇0\alpha\mu<0italic_α italic_μ < 0 (or α⁢μ>0𝛼𝜇0\alpha\mu>0italic_α italic_μ > 0) and γ=0𝛾0\gamma=0italic_γ = 0, then

The condition α⁢μ<0𝛼𝜇0\alpha\mu<0italic_α italic_μ < 0 together with Theorem 3 and Gauss-Bonnet theorem immediately implies

It is worth mentioning that similar results can be obtained in the same way as in Theorem 5 and Corollary 3 by analysing α⁢μ>0𝛼𝜇0\alpha\mu>0italic_α italic_μ > 0.

Let (Mn,g)superscript𝑀𝑛𝑔(M^{n},g)( italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_g ) be a compact gradient Einstein-type manifold of constant scalar curvature S=κ⁢n⁢(n−1)𝑆𝜅𝑛𝑛1S=\kappa n(n-1)italic_S = italic_κ italic_n ( italic_n - 1 ), κ∈{−1,0,1}𝜅101\kappa\in\left\{-1,0,1\right\}italic_κ ∈ { - 1 , 0 , 1 }, and with boundary Σ=∪iΣiΣsubscript𝑖subscriptΣ𝑖\Sigma=\cup_{i}\Sigma_{i}roman_Σ = ∪ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. If α⁢μ<0𝛼𝜇0\alpha\mu<0italic_α italic_μ < 0 (α⁢μ>0𝛼𝜇0\alpha\mu>0italic_α italic_μ > 0), then

B

The irreducible q⁢q𝑞𝑞qqitalic_q italic_q-character is factorized into the product of the q2subscript𝑞2q_{2}italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-characters in the other limit q1→1→subscript𝑞11q_{1}\to 1italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → 1.

Hence, the q⁢q𝑞𝑞qqitalic_q italic_q-character is reduced to the degenerated q𝑞qitalic_q-character (q1→1)→subscript𝑞11(q_{1}\to 1)( italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → 1 ) and the ordinary q𝑞qitalic_q-character of weight three (q2→1)→subscript𝑞21(q_{2}\to 1)( italic_q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → 1 ) as follows,

In §3, we consider A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT quiver as a primary example and provide a proof of Conjecture 1.2 in this case.

In §2, as a preliminary, we provide an algorithm to construct the q⁢q𝑞𝑞qqitalic_q italic_q-character.

This is an algorithm to generate q⁢q𝑞𝑞qqitalic_q italic_q-characters geometrically defined as an equivariant integral over the corresponding quiver variety [Nek15, KP22].

C

\smallsetminus$}}})caligraphic_G = ( italic_G , * , ⌢ / , ⌣ ∖ ) is a mixed lattice group then the partial orders defined by 3.1 are group orderings, that is, they satisfy the condition (2.7). To see this, let x≤y𝑥𝑦x\leq yitalic_x ≤ italic_y and z∈G𝑧𝐺z\in Gitalic_z ∈ italic_G. Then x=x⁢

\kern-8.65pt\rotatebox[origin={c}]{-25.0}{$\smallsetminus$}}}yitalic_x = italic_x ⌣ ∖ italic_y and by Definition 3.5 we have x*z=(x⁢

\kern-8.65pt\rotatebox[origin={c}]{-25.0}{$\smallsetminus$}}}x\preccurlyeq yitalic_x = italic_y ⌣ ∖ italic_x ≼ italic_y.

\rotatebox[origin={c}]{-25.0}{$\smallsetminus$}}}y)italic_x ⌣ ∖ italic_y = italic_x ⌢ / ( italic_x ⌣ ∖ italic_y ). By Theorem 3.3(c) we have x⁢

]{-25.0}{$\smallsetminus$}}}yitalic_x ⌣ ∖ ( italic_y ⌣ ∖ italic_z ) ≼ italic_x ⌣ ∖ italic_y by (a). We also have x⁢

A

Suppose they are ℝℝ\mathbb{R}blackboard_R-linear dependent then they are coplanar by the previous result.

A screw with zero resultant is a constant vector field. A screw with non vanishing resultant is also called proper screw. It is invariant under translations along the resultant 𝓈⁢(P+λ⁢s)=𝓈⁢(P)𝓈𝑃𝜆𝑠𝓈𝑃\mathcal{s}(P+\lambda s)=\mathcal{s}(P)caligraphic_s ( italic_P + italic_λ italic_s ) = caligraphic_s ( italic_P ), for each λ∈ℝ𝜆ℝ\lambda\in\mathbb{R}italic_λ ∈ blackboard_R, and is such that the locus {P:𝓈⁢(P)∝s}conditional-set𝑃proportional-to𝓈𝑃𝑠\{P:\mathcal{s}(P)\propto s\}{ italic_P : caligraphic_s ( italic_P ) ∝ italic_s } is a line called the screw axis (the symbol ∝proportional-to\propto∝ means ‘proportional to’). The value of the screw on the axis is called vector invariant 𝓈∥subscript𝓈parallel-to\mathcal{s}_{\parallel}caligraphic_s start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT, and the number p∈ℝ𝑝ℝp\in\mathbb{R}italic_p ∈ blackboard_R such that 𝓈∥=p⁢ssubscript𝓈parallel-to𝑝𝑠\mathcal{s}_{\parallel}=pscaligraphic_s start_POSTSUBSCRIPT ∥ end_POSTSUBSCRIPT = italic_p italic_s is called pitch. Denoting with O𝑂Oitalic_O any point on the axis, we have from Eq. (2)

If two axes are parallel all of them are and so they are coplanar. If no pair of axes are parallel, the three axis intersect a common line. Let P3subscript𝑃3P_{3}italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT be the intersection between z3subscript𝑧3z_{3}italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and the common line. As z3subscript𝑧3z_{3}italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT has zero pitch its screw field vanishes on P3subscript𝑃3P_{3}italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, but −z3=z1+z2subscript𝑧3subscript𝑧1subscript𝑧2-z_{3}=z_{1}+z_{2}- italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the total screw field generated by the other two elements cannot vanish unless they are coplanar and concurrent on P3subscript𝑃3P_{3}italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.

The screw axis of z𝑧zitalic_z is, by definition, the line of u𝑢uitalic_u. Observe that if we take a point A∈E𝐴𝐸A\in Eitalic_A ∈ italic_E, such that u∈A𝑢𝐴u\in Aitalic_u ∈ italic_A (remember that u𝑢uitalic_u belongs to a ∘\circ∘-orthonormal basis and hence to the real 3-dimensional subspace A𝐴Aitalic_A spanned by it) then the screw can be written z=sA+ϵ⁢𝓈⁢(A)𝑧superscript𝑠𝐴italic-ϵ𝓈𝐴z=s^{A}+\epsilon\mathcal{s}(A)italic_z = italic_s start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT + italic_ϵ caligraphic_s ( italic_A ), as z=a⁢u+ϵ⁢b⁢u=a⁢u+ϵ⁢b⁢π⁢(u)𝑧𝑎𝑢italic-ϵ𝑏𝑢𝑎𝑢italic-ϵ𝑏𝜋𝑢z=au+\epsilon bu=au+\epsilon b\pi(u)italic_z = italic_a italic_u + italic_ϵ italic_b italic_u = italic_a italic_u + italic_ϵ italic_b italic_π ( italic_u ) and a⁢u𝑎𝑢auitalic_a italic_u belongs to A𝐴Aitalic_A (since a𝑎aitalic_a is real) we have sA=a⁢usuperscript𝑠𝐴𝑎𝑢s^{A}=auitalic_s start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_a italic_u (thus s=a⁢π⁢(u)𝑠𝑎𝜋𝑢s=a\pi(u)italic_s = italic_a italic_π ( italic_u )) and 𝓈⁢(A)=b⁢π⁢(u)𝓈𝐴𝑏𝜋𝑢\mathcal{s}(A)=b\pi(u)caligraphic_s ( italic_A ) = italic_b italic_π ( italic_u ). This shows that for A𝐴Aitalic_A in the screw axis the screw field 𝓈⁢(A)𝓈𝐴\mathcal{s}(A)caligraphic_s ( italic_A ) is aligned with the resultant and independent of the point in the screw axis. Moreover, Eq. (20) shows that there are no other points where the screw field is aligned with the resultant, thus in terms of the screw field the axis is characterized as the locus where the screw field is proportional to the resultant.

Conversely, if they are coplanar rescale one screw so that the total resultant vanishes, then there is still freedom to change the individual resultants to get a vanishing total screw field over one axis and hence everywhere.

D

_{3}&\psi_{4}\end{pmatrix}.start_ROW start_CELL | italic_ψ ⟩ = ( start_ARG start_ROW start_CELL italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL ⟨ italic_ψ | = divide start_ARG italic_π end_ARG start_ARG 2 end_ARG ( start_ARG start_ROW start_CELL 2 italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL italic_ψ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) . end_CELL end_ROW

Our solution to both of outlined problems is to detach the process of function approximation (interpolation) from the mapping between functional spaces, and explicitly fix the highest possible resolution. To do that, for both domain and codomain of neural operator we consider functions represented by finite series of the form ∑cn⁢fn⁢(x)subscript𝑐𝑛subscript𝑓𝑛𝑥\sum c_{n}f_{n}(x)∑ italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) where fnsubscript𝑓𝑛f_{n}italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT are either Chebyshev or trigonometric polynomials. Now, all functions are equivalent to vectors of finite length, and we can realize mapping between functional spaces with (almost) vanilla feedforward neural network. Benefits of this strategy include: 1) reach set of operations (integration, differentiation, shift) available for Chebyshev and trigonometric polynomials; 2) lossless operations on functions; 3) interpolation on the fine mesh can be performed with arbitrary accuracy; 4) availability of global information about obtained functions (f.e. bounds on functions and their derivatives); 5) well-studied approximation properties; 6) good compression for smooth functions; 7) overall similarity with spectral and pseudospectral methods.

Addition of two functions is equivalent to addition of two vectors with zero padding of the short one:

Chebyshev scalar product of two functions is equivalent to scalar product of vectors with truncation of the long one:

The other important point is that series from Equation 3 are truncated, so coefficients disubscript𝑑𝑖d_{i}italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and bisubscript𝑏𝑖b_{i}italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be stacked in finite-dimensional vectors. However, since these vectors represent sequences, linear algebra becomes slightly unusual. For example, it is possible to add two vectors even if their length does not coincide, or to multiply a vector by a matrix not conformable with it. The simplest way to understand this behavior is to think about sequences as vectors with an infinite number of trailing zeros that represent (zero) coefficients of truncated basis functions. They also can be considered as a simplified version of quasimatrices [TT15]. Standard linear algebra operations for sequences are illustrated in Section 4. It is also explained there, that, in the space of coefficients, linear operators quite appropriately becomes low-rank integral operators.

B

Let 𝒰𝒰\mathcal{U}caligraphic_U be a non-principal ultrafilter on ℕℕ\mathbb{N}blackboard_N. Fix a countable set S={hn}n=1∞𝑆superscriptsubscriptsubscriptℎ𝑛𝑛1S=\{h_{n}\}_{n=1}^{\infty}italic_S = { italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT consisting of distinct elements of G𝐺Gitalic_G.

_{\mathcal{U}}}})( italic_φ start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT ◇ italic_ψ ) ( caligraphic_T ) = italic_ψ ( caligraphic_T start_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT caligraphic_U end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) for all ψ∈X**𝜓superscript𝑋absent\psi\in X^{**}italic_ψ ∈ italic_X start_POSTSUPERSCRIPT * * end_POSTSUPERSCRIPT, 𝒯∈X*𝒯superscript𝑋\mathcal{T}\in X^{*}caligraphic_T ∈ italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT. Notice that for any u∈ℓ1⁢(G)⊗γ𝒜𝑢superscripttensor-product𝛾superscriptℓ1𝐺𝒜u\in\ell^{1}(G)\otimes^{\gamma}\mathcal{A}italic_u ∈ roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_G ) ⊗ start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT caligraphic_A, we have

Notice that, for each τ∈X*𝜏superscript𝑋\tau\in X^{*}italic_τ ∈ italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT, via the identifications illustrated in Theorem 2.1, we have

_{n}))\text{ for all }\tau\in X^{*}.italic_φ start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT ( italic_τ ) = roman_lim start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT italic_τ ( italic_δ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_ρ ( italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) for all italic_τ ∈ italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT .

_{n}))\text{ for all }\tau\in X^{*}.italic_φ start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT ( italic_τ ) = roman_lim start_POSTSUBSCRIPT caligraphic_U end_POSTSUBSCRIPT italic_τ ( italic_δ start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_ρ ( italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) for all italic_τ ∈ italic_X start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT .

B

8⁢ε⁢m8𝜀𝑚8\varepsilon m8 italic_ε italic_m vertices in Vi1∪Vi+1∪Vi+22superscriptsubscript𝑉𝑖1subscript𝑉𝑖1superscriptsubscript𝑉𝑖22V_{i}^{1}\cup V_{i+1}\cup V_{i+2}^{2}italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∪ italic_V start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ∪ italic_V start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Repeating this for every odd i∈[s]𝑖delimited-[]𝑠i\in[s]italic_i ∈ [ italic_s ], we get s/2𝑠2s/2italic_s / 2 loose paths covering

taking the non-covering paths of all individual f⁢(x)⁢f⁢(y)𝑓𝑥𝑓𝑦f(x)f(y)italic_f ( italic_x ) italic_f ( italic_y )-absorbers) and

least 2⁢ε⁢p2𝜀𝑝2\varepsilon p2 italic_ε italic_p, for every i∈[s−1]𝑖delimited-[]𝑠1i\in[s-1]italic_i ∈ [ italic_s - 1 ], i𝑖iitalic_i odd (s+1𝑠1s+1italic_s + 1 is identified with

ti≥(nin−ε⁢h)⁢tsubscript𝑡𝑖subscript𝑛𝑖𝑛𝜀ℎ𝑡t_{i}\geq(\frac{n_{i}}{n}-\varepsilon h)titalic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ ( divide start_ARG italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_n end_ARG - italic_ε italic_h ) italic_t for every i∈[h]𝑖delimited-[]ℎi\in[h]italic_i ∈ [ italic_h ].

A

Third, 𝒵ℓ,m,nsubscript𝒵ℓ𝑚𝑛\mathcal{Z}_{\ell,m,n}caligraphic_Z start_POSTSUBSCRIPT roman_ℓ , italic_m , italic_n end_POSTSUBSCRIPT is the fourth power of a polynomial when ℓ,mℓ𝑚\ell,mroman_ℓ , italic_m and n𝑛nitalic_n are all even and the square of a polynomial when exactly two of ℓ,mℓ𝑚\ell,mroman_ℓ , italic_m and n𝑛nitalic_n are even.

In [Ayy15], a more general model called the loop-vertex model has also been defined for a general graph together with an orientation. The partition function in this case can also be written as a determinant. However, this model depends on the orientation. One of the main motivations for this work is to find natural families of non-planar graphs where the partition function is independent of the orientation, just as in the monopole-dimer model.

We then focus on the special family of grid graphs in higher dimensions. We give an explicit product formula for the partition function of the monopole-dimer model on three-dimensional grid graphs in Theorem 5.1 generalising the expression (1.1). One peculiar feature of this partition function is that it is a fourth power of a polynomial when all side lengths are even. Just as for the partition function of the monopole-dimer model for two-dimensional grids, it would be interesting to obtain a combinatorial interpretation of the fourth root.

Finally, (6.1) tells that the partition function of the monopole-dimer model for even grid lengths is the 2(d−1)superscript2𝑑12^{(d-1)}2 start_POSTSUPERSCRIPT ( italic_d - 1 ) end_POSTSUPERSCRIPT’th power of a polynomial.

Fourth, the formula in Theorem 5.1 coincides with the already known partition function [Ayy15] of the monopole-dimer model for the two-dimensional grid graph when either of ℓ,m,nℓ𝑚𝑛\ell,m,nroman_ℓ , italic_m , italic_n are equal to 1111.

D

Let α∈𝒞1,1∞⁢(X)𝛼subscriptsuperscript𝒞11𝑋\alpha\in{\mathcal{C}}^{\infty}_{1,1}(X)italic_α ∈ caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT ( italic_X ) be a smooth (1,1)11(1,1)( 1 , 1 )-form on X𝑋Xitalic_X.

such that ∂α=0,∂¯⁢α=0formulae-sequence𝛼0normal-¯𝛼0\partial\alpha=0,\bar{\partial}\alpha=0∂ italic_α = 0 , over¯ start_ARG ∂ end_ARG italic_α = 0. We assume moreover that the

We assume that α𝛼\alphaitalic_α is ∂\partial∂ and ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG closed, such that moreover the following properties hold true.

p⋆⁢i⁢∂∂¯⁢ρ=i⁢∂∂¯⁢p⋆⁢ρsubscript𝑝⋆𝑖¯𝜌𝑖¯subscript𝑝⋆𝜌p_{\star}i\partial\bar{\partial}\rho=i\partial\bar{\partial}p_{\star}\rhoitalic_p start_POSTSUBSCRIPT ⋆ end_POSTSUBSCRIPT italic_i ∂ over¯ start_ARG ∂ end_ARG italic_ρ = italic_i ∂ over¯ start_ARG ∂ end_ARG italic_p start_POSTSUBSCRIPT ⋆ end_POSTSUBSCRIPT italic_ρ

[BG13] provided that α𝛼\alphaitalic_α is locally in the image of the ∂∂¯¯\partial\bar{\partial}∂ over¯ start_ARG ∂ end_ARG operator. But in any case we can construct the function φ𝜑\varphiitalic_φ with the properties of (2.1).

B

In particular, note that if two edges share at least two vertices, then these two edges form a cycle of length 2222.

We define the girth of a hypergraph H𝐻Hitalic_H to be the number of edges in a shortest cycle of H𝐻Hitalic_H and denote it by g⁢(H)𝑔𝐻g(H)italic_g ( italic_H ).

We call a copy of H⁢[t]𝐻delimited-[]𝑡H[t]italic_H [ italic_t ] in G⁢[n]𝐺delimited-[]𝑛G[n]italic_G [ italic_n ] canonical if it is the t𝑡titalic_t-blowup of a copy of H𝐻Hitalic_H in G𝐺Gitalic_G.

Given a graph H𝐻Hitalic_H and t∈ℕ𝑡ℕt\in\mathbb{N}italic_t ∈ blackboard_N, the t𝑡titalic_t-blowup of H𝐻Hitalic_H, denoted H⁢[t]𝐻delimited-[]𝑡H[t]italic_H [ italic_t ], is the graph obtained from H𝐻Hitalic_H by replacing each vertex of H𝐻Hitalic_H by an independent set of size t𝑡titalic_t, and replacing every edge of H𝐻Hitalic_H by a complete bipartite graph Kt,tsubscript𝐾𝑡𝑡K_{t,t}italic_K start_POSTSUBSCRIPT italic_t , italic_t end_POSTSUBSCRIPT between the corresponding parts.

Given two graphs H𝐻Hitalic_H and G𝐺Gitalic_G, we define H⁢(G)𝐻𝐺H(G)italic_H ( italic_G ), the hypergraph of copies of H𝐻Hitalic_H in G𝐺Gitalic_G, to be the |E⁢(H)|𝐸𝐻|E(H)|| italic_E ( italic_H ) |-uniform hypergraph with vertex set given by the edges E⁢(G)𝐸𝐺E(G)italic_E ( italic_G ) of G𝐺Gitalic_G, and edge set consisting of each set of edges of G𝐺Gitalic_G which correspond to a copy of H𝐻Hitalic_H in G𝐺Gitalic_G.

A

However, in many real-world applications, due to certain privacy concerns or limitations of the sensor apparatus, the states of the environment cannot be directly stored in the offline datasets. Instead, only partial observations generated from the states of the environments are stored (Dulac-Arnold et al., 2021).

For example, in precision medicine, a physician’s treatment might consciously or subconsciously depend on the patient’s mood and socioeconomic status

but oftentimes offline datasets consisting of past interactions, e.g., treatment records for precision medicine (Chakraborty and Moodie, 2013; Chakraborty and Murphy, 2014) and human driving data for autonomous driving (Sun et al., 2020), are adequately available.

For POMDPs, the learned policy can only depend on the observable information mentioned in Section 2.1.

In particular, both the actions and observations in the offline dataset depend on the unobserved latent states, and thus are confounded (Pearl, 2009).

A

\leq\Upsilon_{m},blackboard_E [ ∥ over¯ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - ∇ italic_f start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∣ bold_italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ] ≤ roman_Υ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ,

We also impose bounded moment conditions on the stochastic estimates g¯tsubscript¯𝑔𝑡{\bar{g}}_{t}over¯ start_ARG italic_g end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and H¯tsubscript¯𝐻𝑡{\bar{H}}_{t}over¯ start_ARG italic_H end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT.

gradient(bounded 2nd moment)::gradientbounded 2nd momentabsent\displaystyle\text{gradient}\quad(\text{bounded 2nd moment}):\quad\quadgradient ( bounded 2nd moment ) :

(bounded 3rd moment)::bounded 3rd momentabsent\displaystyle(\text{bounded 3rd moment}):\quad\quad( bounded 3rd moment ) :

(bounded 4th moment)::bounded 4th momentabsent\displaystyle(\text{bounded 4th moment}):\quad\quad( bounded 4th moment ) :

C

{0},italic_b ( italic_v start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) = italic_b ( over¯ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) = ( italic_q start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ) for all italic_r start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∈ italic_Q start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT and ∥ italic_v start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ ∥ over¯ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≲ ∥ italic_q start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,

Next we will discuss inf-sup conditions for a different pair of norms. First we show a local version of [2, Proposition 1].

First of all, we introduce a series of notation related to the local degrees of freedom of v^∈Qk⁢(K^)d^𝑣subscript𝑄𝑘superscript^𝐾𝑑\hat{v}\in Q_{k}(\hat{K})^{d}over^ start_ARG italic_v end_ARG ∈ italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( over^ start_ARG italic_K end_ARG ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT and v∈V¯K𝑣subscript¯𝑉𝐾v\in\bar{V}_{K}italic_v ∈ over¯ start_ARG italic_V end_ARG start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT (for the analysis only, not for the implementation).

The results in Lemma 1 and Corollary 1 suffice to discuss the discrete inf-sup conditions for the case (a) of pure Dirichlet boundary conditions. If |ΓN|>0subscriptΓ𝑁0|\Gamma_{N}|>0| roman_Γ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT | > 0, we need an additional result, for which we introduce the following notation

The next result is a global version of Theorem 3 and can be seen as an extension of [2, Proposition 1], where the proof was given for k=2𝑘2k=2italic_k = 2 and for meshes made of rectangles for d=2𝑑2d=2italic_d = 2 and bricks for d=3𝑑3d=3italic_d = 3.

A

One might notice that all paths belonging to such induced SP-covers/SP-partitions are also isometric paths of G𝐺Gitalic_G.

Thus, settling Ramsey-type problems for invariants closely related to the chromatic number, one might expect to obtain some essential information and effective techniques for the Gyárfás-Sumner conjecture.

By using the fact, we obtain solutions of Ramsey-type problems for some invariants concerning isometric paths (see Theorem 5.2).

Recently, similar type problems on invariants (other than the order) were obtained as Ramsey-type problems (see, for example, [2, 4, 6, 10, 18]).

In Section 5, we complete the proof of Theorem 1.1, and settle Ramsey-type problems for some invariants (Theorems 5.1 and 5.2).

B

In the sequel, we may sometimes denote ∂/∂xsubscript𝑥\partial/\partial_{x}∂ / ∂ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT by

indeterminates x1subscript𝑥1x_{1}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, …, xnsubscript𝑥𝑛x_{n}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT that defines a diffiety V𝑉Vitalic_V in

so that the order of the diffiety is 𝒪Σsubscript𝒪Σ\mathcal{O}_{\Sigma}caligraphic_O start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT.

A diffiety is a 𝒞∞superscript𝒞\mathcal{C}^{\infty}caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT manifold V𝑉Vitalic_V of

of functions 𝒪⁢(V)𝒪𝑉\mathcal{O}(V)caligraphic_O ( italic_V ) is the ring of 𝒞∞superscript𝒞\mathcal{C}^{\infty}caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT function on V𝑉Vitalic_V

C

According to a general theory in Ref.Kawabata et al. (2019b), there are 38-fold symmetry classes in non-Hermitian systems.

with a unitary operator 𝒮𝒮{\cal S}caligraphic_S, which is distinct from CS in non-Hermitian systems. The presence and/or absence of these symmetries give the 38-fold independent symmetry classes, which is a natural generalization of the Hermitian 10-fold AZ symmetry classes to non-Hermitian systems Kawabata et al. (2019b).

In this case, h±subscriptℎplus-or-minush_{\pm}italic_h start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT does not have any AZ symmetries.

(ii) Furthermore, each of them can host SLS additionally, which defines AZ class + 𝒮𝒮{\cal S}caligraphic_S and AZ† class + 𝒮𝒮{\cal S}caligraphic_S. We also introduce the subindex +(−)+(-)+ ( - ) of 𝒮𝒮{\cal S}caligraphic_S specifying the commutation (anti-commutation) relation between SLS and AZ or AZ† symmetries. For an AZ (AZ†) class having both TRS (TRS†) and PHS (PHS†), 𝒮𝒮{\cal S}caligraphic_S has a double subindex, where the first index specifies the commutation or anticommutation relation between SLS and TRS (TRS†), and the second one specifies those between SLS and PHS (PHS†), respectively.

In addition to AZ† symmetries discussed in the main text, non-Hermitian systems may host the original AZ symmetries defined by the following equations:

D

1}(\Theta).roman_Θ start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT = start_UNDERACCENT roman_Θ ∈ roman_Λ end_UNDERACCENT start_ARG roman_argmax end_ARG italic_Q start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ( roman_Θ ) .

Note that theoretical convergence results of the SAEM algorithm are provided in [11] under the assumption that the model belongs to the curved exponential family. The SSNLME model belongs to this curved exponential family.

An important difference with [34] is that our approach is applied to NLMEM and not to classical linear regression models. Due to the model non-linearity and to the latent nature of the model random effects, the central so-called Q𝑄Qitalic_Q-quantity of the EM algorithm often does not have a closed-form expression and posterior distributions are difficult to compute. To overcome these issues, we propose an inference method using the SAEM algorithm rather than simply the EM algorithm as in [34]. Another important difference is that optimal model selection among the sub-models obtained in the first step does not require the calculation of the marginal posterior of the models for a spike parameter being equal to 0, as in [34], but only of the log-likelihood of the NLMEM taken at the maximum likelihood estimator.

Indeed, as the function g𝑔gitalic_g here is not separable into φisubscript𝜑𝑖\varphi_{i}italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and ψ𝜓\psiitalic_ψ, the model does not belong to the curved exponential family since it is not possible to write Q∼1subscriptsimilar-to𝑄1\overset{\sim}{Q}_{1}over∼ start_ARG italic_Q end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with an exponential form. As a result, the expression for the maximum argument in ψ𝜓\psiitalic_ψ at M-step is not explicit. One solution would be to do numerical optimisation in ψ𝜓\psiitalic_ψ. However, for ease of implementation of the MCMC-SAEM algorithm, following the idea of [22], an extended model belonging to the curved exponential family is used to estimate the parameters by considering:

If the model belongs to the curved exponential family, that is the complete log-posterior can be written as:

D

It is worth noticing that the Mordell–Weil group of Yˇesubscriptˇ𝑌𝑒\check{Y}_{e}overroman_ˇ start_ARG italic_Y end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT is ℤ4subscriptℤ4\mathbb{Z}_{4}blackboard_Z start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT [MP] (see also [SS]*p.102) and thus gives a ℤ4subscriptℤ4\mathbb{Z}_{4}blackboard_Z start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT-action on Yˇesubscriptˇ𝑌𝑒\check{Y}_{e}overroman_ˇ start_ARG italic_Y end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT which descends to the identity on the base of the elliptic fibration.

We further compute explicitly the limiting complex affine manifolds for the case del Pezzo surfaces of degree eight in Section 4.2 and for degree three and four in Section 5.

Recall that the complex affine structure of the the limiting special Lagrangian fibration of del Pezzo surfaces of degree 3333 is determined by Yˇˇ𝑌\check{Y}overroman_ˇ start_ARG italic_Y end_ARG by Theorem 3.1. Since Yˇ→Yˇ9/⟨r⟩→ˇ𝑌subscriptˇ𝑌9delimited-⟨⟩𝑟\check{Y}\rightarrow\check{Y}_{9}/\langle r\rangleoverroman_ˇ start_ARG italic_Y end_ARG → overroman_ˇ start_ARG italic_Y end_ARG start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT / ⟨ italic_r ⟩ is a crepant resolution, the complex affine structure from the limiting special Lagrangian fibration in del Pezzo surface of degree 3333 is exactly the same as that of the special Lagrangian fibration in Xˇ∖{t2=0}ˇ𝑋subscript𝑡20\check{X}\setminus\{t_{2}=0\}overroman_ˇ start_ARG italic_X end_ARG ∖ { italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 }.

In this section, we will describe the limiting complex affine structure for del Pezzo surface of degree 3333 and 4444.

The limiting complex affine structure for del Pezzo surface of degree 3333 is a ℤ3subscriptℤ3\mathbb{Z}_{3}blackboard_Z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT quotient of the complex affine structure of the special Lagrangian in ℙ2superscriptℙ2\mathbb{P}^{2}blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The ℤ3subscriptℤ3\mathbb{Z}_{3}blackboard_Z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT-action is generated by (0−11−1)matrix0111\begin{pmatrix}0&-1\\

C

A natural question is whether Lemmas 19 and 20 are sufficient to get a (4,3)43(4,3)( 4 , 3 )-theorem for P3OP3O\mathrm{P3O}P3O’s, i.e., to decompose the vertices of the corresponding oriented hypergraph into a bounded number of independent sets, i.e., to bound its chromatic number?

As in the first three cases the vertices span less than three vertices, they also cannot fulfill the premise of the interiority property, so it always holds.

We may refer to the vertices of this triangle as the vertices of the hollow, but note that since the hollow is open, its vertices are not a part of it, only of its closure.

The following claim answers this question negatively, as it shows that the size of the largest independent set can be about as small as the square root of the number of vertices.

A natural question is whether Lemmas 19 and 20 are sufficient to get a (4,3)43(4,3)( 4 , 3 )-theorem for P3OP3O\mathrm{P3O}P3O’s, i.e., to decompose the vertices of the corresponding oriented hypergraph into a bounded number of independent sets, i.e., to bound its chromatic number?

C

The paper is divided into three parts. First, in Section 2 we recall some important results and definitions of abstract convexity and clarify the notations used throughout the paper. Then, in Section 3, we generalise several well-known properties, including the sum rule on the subdifferential of the sum of abstract convex functions and the composition rule. Finally, in Section 4, we prove that under suitable conditions the abstract subdifferential is a maximal abstract monotone operator.

The paper is divided into three parts. First, in Section 2 we recall some important results and definitions of abstract convexity and clarify the notations used throughout the paper. Then, in Section 3, we generalise several well-known properties, including the sum rule on the subdifferential of the sum of abstract convex functions and the composition rule. Finally, in Section 4, we prove that under suitable conditions the abstract subdifferential is a maximal abstract monotone operator.

We start by recalling the main definitions and properties of abstract convexity. We will use the notations introduced by [21].

Given a set L𝐿Litalic_L of functions (also known as elementary functions), a function is L𝐿Litalic_L-convex, or abstract convex when it can be expressed as the supremum of functions from L𝐿Litalic_L. Abstract convexity (also known as convexity without linearity) has been extensively studied since the 1970s, and is the subject of at least three monographs [20, 21, 22], and many papers [1, 2, 5, 6, 8, 15] just to name a few. Many notions and results from classical convexity have been generalised, including duality [15], the conjugate function [21], the subdifferential [21] and normal cones [14]. This paper aims to investigate abstract subdifferential calculus, which will be a significant step towards the development of numerical methods based on abstract convexity.

Proposition 2 a generalises the conditions obtained by [3] in the context of the classical lower semi-continuous convex functions (recall that lower semi-continuous convex functions are abstract convex with respect to linear functions, and that abstract convex sets are closed convex sets [21] so that the abstract convex hull of a set is its weak* closure.) They also generalise the results by [14] in the case when the set L𝐿Litalic_L is additive and 0∈L0𝐿0\in L0 ∈ italic_L.

B

\big{)}italic_λ ( italic_t ) ← roman_min ( 0 , italic_λ ( italic_t ) + italic_l ( italic_N start_POSTSUPERSCRIPT on end_POSTSUPERSCRIPT ( italic_t ) - italic_N ) )

We first solve the original problem (III-A) is solved using interior-point methods, and then derive an upper bound by solving the relaxed problem. In Algorithm 1, all multipliers are initialized to 0.05, and the step size is set to 1e-5 with a decaying factor 0.99. Fig. 3 showcases the convergence performance at 8:00, 16:00, and 20:00, respectively. Our results show that the duality gap gradually converges to zero, and the performance remains consistent throughout the day. This confirms that we have obtained the globally optimal solution to the relaxed problem (III-B), and thus established a theoretical upper bound for the original problem (III-A).

In the dual decomposition algorithm, we iteratively solve the subproblem for each zone and update the multipliers λ⁢(t)𝜆𝑡\lambda(t)italic_λ ( italic_t ) until the stopping criterion, either zero duality gap or maximal iterations reached, is met. Details of this procedure are given in Algorithm 1. If the duality gap reduces to zero at termination, constraint (14e) is satisfied and the globally optimal solution to (III-B) is attained. Otherwise, constraint (14e) is violated, but the derived solution is still an upper bound for (III-B) according to the duality theorem. Based on Proposition 1, in either case, we establish a surrogate upper bound for (III-A) that can be used to evaluate its solution optimality.

We derive a theoretical upper bound to evaluate the solution optimality gap of the nonconvex optimization problem. To this end, we develop an integrated relaxation, decomposition, and dynamic programming approach, in which we first relax the original problem through a change of variable, then decouple the relaxed problem using dual decomposition, and finally employ dynamic programming to solve each small-scale subproblem. A tight surrogate upper bound is established, enabling us to measure the distance between the derived solution and the globally optimal solution despite nonconvexity.

The globally optimal solution to the relaxed problem (III-B) is an upper bound for the original problem (III-A).

B

The solution obtained by solving approximately the least-squares problem (5) can be treated as the exact solution of a different least-squares problem, which can be looked at as a yet unknown perturbation of the original problem (5).

By combining the backward stability and the conditioning of the problem, we then provide a forward error estimate at the end of this section (see Corollary 3.14).

When both LHS and RHS of the least-squares are perturbed 33, the optimal backward error to solve the least-squares problem in Equation (35) at a generic iteration k𝑘kitalic_k is

The full-rank property of Rksubscript𝑅𝑘R_{k}italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is not needed to ensure convergence of Full AAR 24. However, we need this property in the following theoretical analysis in order to ensure backward stability. In fact, requiring that Rksubscript𝑅𝑘R_{k}italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is full-rank already excludes the risk of stagnation 24. The same reasoning applies when Rksubscript𝑅𝑘R_{k}italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is replaced by R^ksubscript^𝑅𝑘\hat{R}_{k}over^ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to allow for inaccuracy in the least-squares calculations.

The objective of this section is to estimate the difference between the original least-squares problem (5) and its perturbation. This is important in order to ensure backward stability 33, 34, which is a desired property of the numerical scheme.

D

We study some examples of interpretable groups in the valued fields we considered here and see how our results are reflected in those examples. The examples are, mostly, common to all contexts, but their nature may vary between the different settings.

Let 𝒦=(K,v,…)𝒦𝐾𝑣…{\mathcal{K}}=(K,v,\dots)caligraphic_K = ( italic_K , italic_v , … ) be a sufficiently saturated V-minimal valued field.

Let 𝒦=(K,+,⋅,v,…)𝒦𝐾⋅𝑣…{\mathcal{K}}=(K,+,\cdot,v,\dots)caligraphic_K = ( italic_K , + , ⋅ , italic_v , … ) be an expansion of a valued field, G𝐺Gitalic_G an interpretable group in 𝒦𝒦{\mathcal{K}}caligraphic_K and D𝐷Ditalic_D one of the infinite distinguished sorts.

Let 𝒦=(K,v,…)𝒦𝐾𝑣…{\mathcal{K}}=(K,v,\dots)caligraphic_K = ( italic_K , italic_v , … ) be some expansion of a valued field.

Let 𝒦=(K,v)𝒦𝐾𝑣{\mathcal{K}}=(K,v)caligraphic_K = ( italic_K , italic_v ) be a sufficiently saturated p𝑝pitalic_p-adically closed valued field.

C

(P,Q)roman_Λ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT : roman_Γ → [ start_ATOM - italic_q + 1 end_ATOM . . start_ATOM italic_q - 1 end_ATOM ] , italic_Q ↦ italic_δ ( italic_P , italic_Q ) called its successor marking. The support of ΛPsubscriptΛ𝑃\Lambda_{P}roman_Λ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT consists of the target nodes of outgoing edges from P𝑃Pitalic_P. We denote the marking with the empty support by ∅\emptyset∅.

Let σ⁢(K)={Q1,…,Qk}𝜎𝐾subscript𝑄1…subscript𝑄𝑘\sigma(K)=\left\{\mathinner{Q_{1},\ldots,Q_{k}}\right\}italic_σ ( italic_K ) = { start_ATOM italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ATOM } and σ⁢(L)={R1,…,Rℓ}𝜎𝐿subscript𝑅1…subscript𝑅ℓ\sigma(L)=\left\{\mathinner{R_{1},\ldots,R_{\ell}}\right\}italic_σ ( italic_L ) = { start_ATOM italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_R start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_ATOM }. Because (Γ,δ)Γ𝛿(\Gamma,\delta)( roman_Γ , italic_δ ) is reduced, we can assume that ε⁢(Q1)<ε⁢(Qj)𝜀subscript𝑄1𝜀subscript𝑄𝑗\varepsilon(Q_{1})<\varepsilon(Q_{j})italic_ε ( italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) < italic_ε ( italic_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) and ε⁢(R1)<ε⁢(Rj)𝜀subscript𝑅1𝜀subscript𝑅𝑗\varepsilon(R_{1})<\varepsilon(R_{j})italic_ε ( italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) < italic_ε ( italic_R start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) for all j≠1𝑗1j\neq 1italic_j ≠ 1. Then there are markings U,V𝑈𝑉U,Vitalic_U , italic_V such that ε⁢(K)=ε⁢(U)⋅qε⁢(ΛQ1)𝜀𝐾⋅𝜀𝑈superscript𝑞𝜀subscriptΛsubscript𝑄1\varepsilon(K)=\varepsilon(U)\cdot q^{\varepsilon(\Lambda_{Q_{1}})}italic_ε ( italic_K ) = italic_ε ( italic_U ) ⋅ italic_q start_POSTSUPERSCRIPT italic_ε ( roman_Λ start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT, ε⁢(L)=ε⁢(V)⋅qε⁢(ΛR1)𝜀𝐿⋅𝜀𝑉superscript𝑞𝜀subscriptΛsubscript𝑅1\varepsilon(L)=\varepsilon(V)\cdot q^{\varepsilon(\Lambda_{R_{1}})}italic_ε ( italic_L ) = italic_ε ( italic_V ) ⋅ italic_q start_POSTSUPERSCRIPT italic_ε ( roman_Λ start_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT and q𝑞qitalic_q does not divide ε⁢(U)𝜀𝑈\varepsilon(U)italic_ε ( italic_U ) and ε⁢(V)𝜀𝑉\varepsilon(V)italic_ε ( italic_V ). Because ε⁢(K)ε⁢(L)𝜀𝐾𝜀𝐿\frac{\varepsilon(K)}{\varepsilon(L)}divide start_ARG italic_ε ( italic_K ) end_ARG start_ARG italic_ε ( italic_L ) end_ARG is a power of q𝑞qitalic_q, ε⁢(U)=ε⁢(V)𝜀𝑈𝜀𝑉\varepsilon(U)=\varepsilon(V)italic_ε ( italic_U ) = italic_ε ( italic_V ). Thus, ε⁢(M)=ε⁢(ΛQ1)−ε⁢(ΛR1)𝜀𝑀𝜀subscriptΛsubscript𝑄1𝜀subscriptΛsubscript𝑅1\varepsilon(M)=\varepsilon(\Lambda_{Q_{1}})-\varepsilon(\Lambda_{R_{1}})italic_ε ( italic_M ) = italic_ε ( roman_Λ start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - italic_ε ( roman_Λ start_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ). Now Lemma 11 finishes the proof of the lemma.

Is ε⁢(L)△ε⁢(M)△𝜀𝐿𝜀𝑀\varepsilon(L)\mathrel{\triangle}\varepsilon(M)italic_ε ( italic_L ) △ italic_ε ( italic_M )?

If M𝑀Mitalic_M is a marking, we write −M𝑀-M- italic_M for the marking defined by (−M)⁢(P)=−(M⁢(P))𝑀𝑃𝑀𝑃(-M)(P)=-(M(P))( - italic_M ) ( italic_P ) = - ( italic_M ( italic_P ) ), which clearly evaluates to −ε⁢(M)𝜀𝑀-\varepsilon(M)- italic_ε ( italic_M ).

We define the evaluation ε⁢(P)𝜀𝑃\varepsilon(P)italic_ε ( italic_P ) of a node (ε⁢(M)𝜀𝑀\varepsilon(M)italic_ε ( italic_M ) of a marking resp.)

D

Let H𝐻Hitalic_H be an ordered graph containing an induced subgraph isomorphic to J9subscript𝐽9J_{9}italic_J start_POSTSUBSCRIPT 9 end_POSTSUBSCRIPT, M1subscript𝑀1M_{1}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, M5subscript𝑀5M_{5}italic_M start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT or −M5subscript𝑀5-M_{5}- italic_M start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT. Then the Ordered Graph List-3-Coloring Problem restricted to H𝐻Hitalic_H-free ordered graphs is NP-complete.

Let us emphasize that, as opposed to our notion of “induced subgraph,” our notion of “coloring” for ordered graphs ignores the ordering of vertices. In particular, the Ordered Graph List-3333-Coloring Problem is NP-complete, as so is the usual List-3333-Coloring Problem. We establish a nearly complete dichotomy for the complexity of the Ordered Graph List-3333-Coloring Problem on H𝐻Hitalic_H-free ordered graphs, where H𝐻Hitalic_H is a fixed ordered graph. Our first two theorems, below, give a complete dichotomy for the case |E⁢(H)|≠2𝐸𝐻2|E(H)|\neq 2| italic_E ( italic_H ) | ≠ 2.

Figure 1 depicts ordered graphs H𝐻Hitalic_H for which we prove, in Theorems 5 and 7, that the Ordered Graph List-3333-Coloring Problem restricted to H𝐻Hitalic_H-free ordered graphs can be solved in polynomial time (gray vertices represent arbitrarily many isolated vertices.)

The Ordered Graph List-3-Coloring Problem restricted to H𝐻Hitalic_H-free ordered graphs is NP-complete if H𝐻Hitalic_H has at least three edges..

Note that Theorems 5, 6, 7 and 8, together, determine the complexity of the Ordered Graph List-3333-Coloring Problem restricted to H𝐻Hitalic_H-free ordered graphs for all H𝐻Hitalic_H, except for the following three cases, which remain open (see Figure 2 for a depiction, where gray vertices represent arbitrarily many isolated vertices.)

D

By the proof of Proposition 4.7, we can take Ejsubscript𝐸𝑗E_{j}italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT to be the maximal subobject in 𝒜S≤jsuperscriptsubscript𝒜𝑆absent𝑗\mathcal{A}_{S}^{\leq j}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≤ italic_j end_POSTSUPERSCRIPT. Since 𝒜S≤jsuperscriptsubscript𝒜𝑆absent𝑗\mathcal{A}_{S}^{\leq j}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≤ italic_j end_POSTSUPERSCRIPT is a full subcategory of 𝒜S≤ksuperscriptsubscript𝒜𝑆absent𝑘\mathcal{A}_{S}^{\leq k}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ≤ italic_k end_POSTSUPERSCRIPT for any integers 1≤j<k≤r1𝑗𝑘𝑟1\leq j<k\leq r1 ≤ italic_j < italic_k ≤ italic_r, we get Ej⊂Eksubscript𝐸𝑗subscript𝐸𝑘E_{j}\subset E_{k}italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊂ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, hence we get the unique filtration.

This definition of slicings can be viewed as a refinement of t-structures in triangulated categories.

In this section, we further study the possible applications of Abramovich-Polishchuk’s construction on stability conditions. More explicitly, we want to generalize the construction in [Liu21] to the case when S𝑆Sitalic_S is a higher dimensional projective variety. The idea is to do multiple times of tilts on 𝒜Ssubscript𝒜𝑆\mathcal{A}_{S}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT by induction, and prove suitable inequalities at each inductive step. However, the usual tilted heart (in Remark 2.11) does not satisfy the inequalities we need. In this section, we come up with a finer cut of 𝒜Ssubscript𝒜𝑆\mathcal{A}_{S}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT into torsion pairs (Corollary 5.11), whose tilted heart satisfies the inequalities we need (see Theorem 5.12). This leads us to the notion of l𝑙litalic_l-th level semistable objects (Definition 5.6), which is a generalization of the slope stability and the Gieseker stability. We expect this to be the first step towards the generalization of the construction in [Liu21].

In the meantime, we prove a sequence of positivity results on the coefficients of LE⁢(n)subscript𝐿𝐸𝑛L_{E}(n)italic_L start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_n ) (see also [Liu20, Lemma 4.1]). These positivity results help us to construct weak stability functions on 𝒜Ssubscript𝒜𝑆\mathcal{A}_{S}caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT and its abelian subcategories. This lead us to the notion of l𝑙litalic_l-th level stability, which can be viewed as a generalization of both the slope stability and the Gieseker stability. However, the phase of an object with respect to such stability is no longer a single real number, but a vector of real numbers. Then we show the existence and uniqueness of the filtration of an object E∈𝒜S𝐸subscript𝒜𝑆E\in\mathcal{A}_{S}italic_E ∈ caligraphic_A start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT with respect to such stability, which we call the lexicographic order filtration of E𝐸Eitalic_E.

One can view this as a generalization of the torsion filtration of a sheaf (see [HL10, Definition 1.1.4]).

D

Corollaries 4.1–4.2 are effective examples of the potentials of Theorems 3.1–3.2 and Corollary 3.1, which can be readily specialized to any discrepancy in the ips class. For instance, in the context of mmd with Gaussian and Laplace kernels, Corollary 4.2 ensures informative posterior concentration without requiring additional assumptions on μθsubscript𝜇𝜃\mu_{\theta}italic_μ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT or μ∗superscript𝜇\mu^{*}italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. Similar results can be obtained for all ips discrepancies as long as (III)–(IV) are satisfied and ℜn⁢(𝔉)subscriptℜ𝑛𝔉\mathfrak{R}_{n}(\mathfrak{F})fraktur_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( fraktur_F ) admits explicit upper bounds. For example, when 𝒴𝒴\mathcal{Y}caligraphic_Y is bounded, this is possible for the Wasserstein distance in Example 3.1, leveraging the upper bounds for ℜn⁢(𝔉)subscriptℜ𝑛𝔉\mathfrak{R}_{n}(\mathfrak{F})fraktur_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( fraktur_F ) derived by Sriperumbudur et al. (2012).

Condition (I) is the only assumption made on the data generating process and is present in, e.g., Nguyen et al. (2020) and in the supplementary materials of Bernton et al. (2019). Although the theory we derive in Section 6 relaxes (I) to study convergence and concentration also beyond the i.i.d. context, it shall be emphasized that some of the assumptions considered in the literature may not hold even in i.i.d. settings. Hence, an improved understanding of the abc properties under (I) is crucial to clarify the range of applicability and potential limitations of available existence theory under more complex, possibly non–i.i.d., regimes. In fact, as shown in Section 3.1, certain discrepancies may yield posteriors that are either not well–defined in the limit or lack strong convergence guarantees even in i.i.d. settings. Assumption (II) is standard in Bayesian asymptotics and abc theory (Bernton et al., 2019; Nguyen et al., 2020; Frazier, 2020), and requires that sufficient prior mass is placed on those parameter values yielding models μθsubscript𝜇𝜃\mu_{\theta}italic_μ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT close to μ∗superscript𝜇\mu^{*}italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, under 𝒟𝔉subscript𝒟𝔉\mathcal{D}_{\mathfrak{F}}caligraphic_D start_POSTSUBSCRIPT fraktur_F end_POSTSUBSCRIPT. Finally, (III)–(IV) provide conditions on 𝔉𝔉\mathfrak{F}fraktur_F to ensure that 𝒟𝔉subscript𝒟𝔉\mathcal{D}_{\mathfrak{F}}caligraphic_D start_POSTSUBSCRIPT fraktur_F end_POSTSUBSCRIPT yields meaningful concentration bounds, leveraging Lemma 2.1. Crucially, (III)–(IV) are made directly on the user–selected discrepancy 𝒟𝔉subscript𝒟𝔉\mathcal{D}_{\mathfrak{F}}caligraphic_D start_POSTSUBSCRIPT fraktur_F end_POSTSUBSCRIPT, rather than on the data generating process or on the statistical model, thereby providing general constructive conditions that are more realistic to verify in practice compared to regularity assumptions on the unknown μ∗superscript𝜇\mu^{*}italic_μ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT or on {μθ:θ∈Θ⊆ℝp}conditional-setsubscript𝜇𝜃𝜃Θsuperscriptℝ𝑝\{\mu_{\theta}:\theta\in\Theta\subseteq\mathbb{R}^{p}\}{ italic_μ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT : italic_θ ∈ roman_Θ ⊆ blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT }, while guiding the choice of 𝒟𝔉subscript𝒟𝔉\mathcal{D}_{\mathfrak{F}}caligraphic_D start_POSTSUBSCRIPT fraktur_F end_POSTSUBSCRIPT.

The above results clarify the fundamental relation among the limiting behavior of abc posteriors and the learning properties of the chosen discrepancy, when measured via Rademacher complexity. Moreover, the bounds derived clarify that a sufficient condition to recover a limiting pseudo–posterior with the same threshold–control on the discrepancy among the truths as the one enforced on the corresponding empirical distributions, is that the selected discrepancy has a Rademacher complexity vanishing to zero in the large–data limit. As proved in Section 3.2, this setting also allows constructive derivations of novel, informative and uniform concentration bounds for discrepancy–based abc posteriors in the challenging regime where the threshold shrinks towards zero as both m𝑚mitalic_m and n𝑛nitalic_n diverge. This is facilitated by the existence of meaningful upper bounds for the Rademacher complexity of popular abc discrepancies, along with the availability of constructive conditions for the derivation of these bounds (e.g., Sriperumbudur et al., 2012) which leverage fundamental connections among such a complexity measure and other key quantities in statistical learning theory, such as the Vapnik–Chervonenkis (vc) dimension and the notion of uniform Glivenko–Cantelli classes (see e.g., Wainwright, 2019, Chapter 4). This yields an improved understanding of the factors that govern the concentration of discrepancy–based abc posteriors under a unified perspective that further allows to (i) quantify rates of concentration and (ii) directly translate any advancement on Rademacher complexity into novel abc theory. The former advantage is illustrated within Section 4 through a specific focus on mmd with routinely–implemented bounded and unbounded kernels, whereas the latter is clarified in Section 6, where we extend the theory from Section 3 to non–i.i.d. settings, leveraging results in Mohri & Rostamizadeh (2008) on Rademacher complexity under β𝛽\betaitalic_β–mixing processes (e.g., Doukhan, 1994).

While (III) and (IV) hold for mmd with a bounded kernel without additional assumptions, the currently–available bounds on the Rademacher complexity ensure that mmd with an unbounded kernel meets the above conditions only under specific models and data generating processes, even within the i.i.d. setting. In this context, it is however possible to revisit the results for the Wasserstein case in Proposition 3 of Bernton et al. (2019) under the new Rademacher complexity framework introduced in the present article. In particular, as shown in Proposition 4.1, under mmd with an unbounded kernel, the existence Assumptions 1 and 2 in Bernton et al. (2019) can be directly related to constructive conditions on the kernel, inherently related to our Assumption (IV). This in turn yields informative concentration inequalities that are reminiscent of those in Theorem 3.2 and Corollary 4.2. Notice that these inequalities also hold for summary–based abc with routinely–used unbounded summaries (e.g., moments) as a direct consequence of the discussion in Example 3.3.

Leveraging Theorems 3.1–3.2 and Corollary 3.1 along with the available upper bounds on the Rademacher complexity of mmd — see Example 3.2 — Corollaries 4.1–4.2 substantially refine and expand available knowledge on the limiting properties of mmd–abc with routinely–implemented bounded kernels. Crucially, these discrepancies automatically satisfy (III)–(IV) without additional constraints on the model or on the data generating process. Notice that Corollaries 4.1–4.2 also hold for summary–based distances with bounded summaries as a direct consequence of the discussion in Examples 2.3 and 3.3.

C

We have φNN⁢(𝒙;𝚯n)subscript𝜑NN𝒙superscript𝚯𝑛\varphi_{\rm NN}(\bm{x};{\bm{\Theta}}^{n})italic_φ start_POSTSUBSCRIPT roman_NN end_POSTSUBSCRIPT ( bold_italic_x ; bold_Θ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) as a numerical solution at time tn=n⁢τsuperscript𝑡𝑛𝑛𝜏t^{n}=n\tauitalic_t start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT = italic_n italic_τ.

Following [53], solve an L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-gradient flow associated with the free energy

Algorithm 1 Numerical Algorithm for solving the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-gradient flow

3.1 EVNN scheme for L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-gradient flow

L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-gradient flow that satisfies an energy-dissipation law

B

𝒜g,J,Nsubscript𝒜𝑔𝐽𝑁\mathcal{A}_{g,J,N}caligraphic_A start_POSTSUBSCRIPT italic_g , italic_J , italic_N end_POSTSUBSCRIPT

This is a quasiprojective scheme that parametrizes certain polarized chains of type J𝐽Jitalic_J of g𝑔gitalic_g-dimensional Abelian varieties with full level-N𝑁Nitalic_N structure; its moduli description was first given by de Jong [Jon93] and then in full generality by Rapoport and Zink [RZ96].

Viehmann and Wedhorn [VW10] define a moduli stack of F𝐹Fitalic_F-zips with polarization and endomorphism structure in a PEL-type situation with hyperspecial level structure.

It is roughly given by sending a polarized chain of Abelian varieties to the twisted conjugacy class corresponding to the Frobenius ΦΦ\Phiroman_Φ of the associated rational Dieudonné module, see the work of Oort [Oor02] and also He and Rapoport [HR15].

that parametrizes the Hodge filtration in the de Rham cohomology of a polarized chain of Abelian varieties.

A

χr,r′:Kr→Kr′:subscript𝜒𝑟superscript𝑟′→subscript𝐾𝑟subscript𝐾superscript𝑟′\chi_{r,r^{\prime}}\colon K_{r}\to K_{r^{\prime}}italic_χ start_POSTSUBSCRIPT italic_r , italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT : italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT → italic_K start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT in 𝖣b⁢(𝐤𝕍)superscript𝖣bsubscript𝐤𝕍\mathsf{D}^{\mathrm{b}}({\mathbf{k}}_{\mathbb{V}})sansserif_D start_POSTSUPERSCRIPT roman_b end_POSTSUPERSCRIPT ( bold_k start_POSTSUBSCRIPT blackboard_V end_POSTSUBSCRIPT ). It induces a canonical morphism χr,r′⋆F:Kr⋆F→Kr′⋆F:⋆subscript𝜒𝑟superscript𝑟′𝐹→⋆subscript𝐾𝑟𝐹⋆subscript𝐾superscript𝑟′𝐹\chi_{r,r^{\prime}}\star F\colon K_{r}\star F\to K_{r^{\prime}}\star Fitalic_χ start_POSTSUBSCRIPT italic_r , italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋆ italic_F : italic_K start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ⋆ italic_F → italic_K start_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋆ italic_F.

≃ϕ−′′1⁢R⁢ϕ′′∗⁢R⁢jℒ∗⁢R⁢pℒ∗⁢F\displaystyle\simeq\phi^{{}^{\prime\prime}-1}\mathrm{R}{\phi^{\prime\prime}}_{%

If r≥r′≥0𝑟superscript𝑟′0r\geq r^{\prime}\geq 0italic_r ≥ italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≥ 0, there is a canonical morphism

In particular when r′=0superscript𝑟′0r^{\prime}=0italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 0, we get

for every x∈𝕍𝑥𝕍x\in\mathbb{V}italic_x ∈ blackboard_V and r>0𝑟0r>0italic_r > 0, R⁢Γ⁢(B′⁢(x,r);F)≃R⁢Γ⁢(B′⁢(x,r);G)similar-to-or-equalsRΓsuperscript𝐵′𝑥𝑟𝐹RΓsuperscript𝐵′𝑥𝑟𝐺\mathrm{R}\Gamma(B^{\prime}(x,r);F)\simeq\mathrm{R}\Gamma(B^{\prime}(x,r);G)roman_R roman_Γ ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x , italic_r ) ; italic_F ) ≃ roman_R roman_Γ ( italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x , italic_r ) ; italic_G ),

C

\bullet}(\log D_{Z}))italic_G start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_Y ; blackboard_C ) := roman_Im ( blackboard_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_Z , roman_Ω start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( roman_log italic_D start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) → blackboard_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_Z , roman_Ω start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( roman_log italic_D start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ) )

GkfHk(Y;ℂ):=Im(ℍk(Z,ΩZ∙(logDZ,f1,f2)→ℍk(Z,ΩZ∙(logDZ))\displaystyle G^{f}_{k}H^{k}(Y;\mathbb{C}):=\operatorname{Im}(\mathbb{H}^{k}(Z%

ℍk(Z,ΩZ∙(logDZ,{fi,fj})≅Hk(Y,Yi+Yj),ℍk(ΩZ∙(logDZ,f{i⁢j}))≅Hk(Y,Yi⁢j)\displaystyle\mathbb{H}^{k}(Z,\Omega_{Z}^{\bullet}(\log D_{Z},\{f_{i},f_{j}\})%

Pb⁢ℍk⁢(X,K∙):=Im⁡(ℍk⁢(X,τ≤b𝔭⁢K∙)→ℍk⁢(X,K∙))assignsubscript𝑃𝑏superscriptℍ𝑘𝑋superscript𝐾∙Im→superscriptℍ𝑘𝑋superscriptsubscript𝜏absent𝑏𝔭superscript𝐾∙superscriptℍ𝑘𝑋superscript𝐾∙P_{b}\mathbb{H}^{k}(X,K^{\bullet}):=\operatorname{Im}(\mathbb{H}^{k}(X,{}^{%

Gk+1fHk(Y;ℂ):=Im(ℍk(Z,ΩZ∙(logDZ,f12)→ℍk(Z,ΩZ∙(logDZ))\displaystyle G^{f}_{k+1}H^{k}(Y;\mathbb{C}):=\operatorname{Im}(\mathbb{H}^{k}%

D

If we add max⁡{0,dG⁢(v)−f⁢(v)}0subscript𝑑𝐺𝑣𝑓𝑣\max\{0,d_{G}(v)-f(v)\}roman_max { 0 , italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) - italic_f ( italic_v ) } chips to vertex v𝑣vitalic_v, then we have a chip configuration with at least dG⁢(v)subscript𝑑𝐺𝑣d_{G}(v)italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) chips on each vertex v𝑣vitalic_v, which is certainly a recurrent configuration. Moreover, this addition needs at most the above stated number of chips.

Note that the size of G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT might not be polynomial in the size of G𝐺Gitalic_G as the value of M𝑀Mitalic_M can be large. However, the above reduction will be applied to divisors f𝑓fitalic_f with 0≤f⁢(v)≤dG⁢(v)0𝑓𝑣subscript𝑑𝐺𝑣0\leq f(v)\leq d_{G}(v)0 ≤ italic_f ( italic_v ) ≤ italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_v ) for v∈V𝑣𝑉v\in Vitalic_v ∈ italic_V, in which case the construction is polynomial. In particular, observe that |V′|=|V|+1superscript𝑉′𝑉1|V^{\prime}|=|V|+1| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = | italic_V | + 1.

We define G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT as follows. Let M𝑀Mitalic_M be an integer large enough such that

Let g′superscript𝑔′g^{\prime}italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be an effective divisor such that f′+g′superscript𝑓′superscript𝑔′f^{\prime}+g^{\prime}italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is non-halting on G′superscript𝐺′G^{\prime}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and deg⁡(g′)=distG′nh⁡(f′)degreesuperscript𝑔′subscriptsuperscriptdistnhsuperscript𝐺′superscript𝑓′\deg(g^{\prime})=\operatorname{dist^{nh}_{\mathnormal{G^{\prime}}}}(f^{\prime})roman_deg ( italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = start_OPFUNCTION roman_dist start_POSTSUPERSCRIPT roman_nh end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_OPFUNCTION ( italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). We show that g′⁢(vn⁢e⁢w)=0superscript𝑔′subscript𝑣𝑛𝑒𝑤0g^{\prime}(v_{new})=0italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v start_POSTSUBSCRIPT italic_n italic_e italic_w end_POSTSUBSCRIPT ) = 0, and that for the divisor g≔g′|V≔𝑔evaluated-atsuperscript𝑔′𝑉g\coloneqq g^{\prime}|_{V}italic_g ≔ italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT obtained by restricting g′superscript𝑔′g^{\prime}italic_g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to the set of original vertices, the divisor f+g𝑓𝑔f+gitalic_f + italic_g is recurrent on G𝐺Gitalic_G.

We choose M𝑀Mitalic_M such that M>distGr⁡(f)𝑀subscriptsuperscriptdistr𝐺𝑓M>\operatorname{dist^{r}_{\mathnormal{G}}}(f)italic_M > start_OPFUNCTION roman_dist start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT end_OPFUNCTION ( italic_f ), hence the claim follows by Claim 11.

B

\nu(z_{i})\neq\nu(z_{j})\textrm{ for some }i,j\end{subarray}}\mu_{\nu}.roman_rad ( italic_P ) = roman_rad ( [ italic_z start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT : … : italic_z start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ] ) := divide start_ARG 1 end_ARG start_ARG [ italic_K ( italic_P ) : italic_K ] end_ARG ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_ν ∈ ( italic_M start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT roman_na end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_ν ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≠ italic_ν ( italic_z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) for some italic_i , italic_j end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT .

Another version of the a⁢b⁢c⁢d𝑎𝑏𝑐𝑑abcditalic_a italic_b italic_c italic_d conjecture is described by

Conjecture 2.6 (The a⁢b⁢c⁢d⁢e⁢…𝑎𝑏𝑐𝑑𝑒…abcde\ldotsitalic_a italic_b italic_c italic_d italic_e … conjecture [Voj98, Conjecture 2.3]).

Conjecture 2.7 (The a⁢b⁢c⁢d𝑎𝑏𝑐𝑑abcditalic_a italic_b italic_c italic_d conjecture, version 1111 [Gra07, Section 8]).

Conjecture 2.9 (The a⁢b⁢c⁢d𝑎𝑏𝑐𝑑abcditalic_a italic_b italic_c italic_d conjecture, version 2222 [Loo21a, Conjecture 2.1]).

D

1}+\mathcal{L}^{n}\left(\set{u>0}\cap E\right).caligraphic_F ( italic_u ; italic_E ) = ∫ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT | ∇ italic_u | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_d caligraphic_L start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_β ∫ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∩ italic_E end_POSTSUBSCRIPT ( over¯ start_ARG italic_u end_ARG start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT + under¯ start_ARG italic_u end_ARG start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) italic_d caligraphic_H start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT + caligraphic_L start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( { start_ARG italic_u > 0 end_ARG } ∩ italic_E ) .

We refer to [1, Theorem 4.7, Theorem 4.8, Theorem 5.22] for the proof of this theorem. We now conclude this section with the following proposition whose proof can be found in [5, Lemma 3.1].

In Section 3 we prove that the a priori estimate (1.7) holds for inward minimizers (see Section 3), such an estimate will be crucial in the proof of Theorem 1.1 in Section 4. Finally, in Section 5 we prove Theorem 1.2.

We now state a compactness theorem in SBVSBV\operatorname{SBV}roman_SBV that will be useful in the following.

To prove Theorem 1.2 we will use the following Poincaré-Wirtinger type inequality whose proof can be found in [7, Theorem 3.1 and Remark 3.3].

D

The invariants studied here are all knot concordance invariants. From a modern perspective it would be interesting to prove the analog of Theorem 1 for a family of topologically slice knots.

Acknowledgements. I appreciate helpful feedback from Pat Gilmer and Allison Miller. Comments of a referee of an early version of this paper led to significant improvements.

The key ingredients of the proof of Theorem 1 come from the work of Casson and Gordon [MR900252] and Gilmer [MR656619]. The knot B25subscript𝐵25B_{25}italic_B start_POSTSUBSCRIPT 25 end_POSTSUBSCRIPT appears in [MR900252] as the first example of an algebraically slice knot that is not slice: that is, g4⁢(B25)>0subscript𝑔4subscript𝐵250g_{4}(B_{25})>0italic_g start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_B start_POSTSUBSCRIPT 25 end_POSTSUBSCRIPT ) > 0. A theorem of Jiang [MR620010] demonstrates the linear independence of the set {Bp2}p≥5subscriptsubscript𝐵superscript𝑝2𝑝5\{B_{p^{2}}\}_{p\geq 5}{ italic_B start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_p ≥ 5 end_POSTSUBSCRIPT in concordance, thus implying that g4⁢(n⁢Bp2)>0subscript𝑔4𝑛subscript𝐵superscript𝑝20g_{4}(nB_{p^{2}})>0italic_g start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_n italic_B start_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) > 0 for all p≥5𝑝5p\geq 5italic_p ≥ 5 and all n≥1𝑛1n\geq 1italic_n ≥ 1. It is a theorem from  [MR656619] that permits Jiang’s result to be improved to give a linearly increasing genus bound.

We will refer to the subgroup ℳℳ\mathcal{M}caligraphic_M as a metabolizer. In the statement of Gilmer’s theorem in [MR656619] there is an additional term μ⁢(K,χ)𝜇𝐾𝜒\mu(K,\chi)italic_μ ( italic_K , italic_χ ), but prior to the statement of that theorem he points out that μ⁢(K,χ)=0𝜇𝐾𝜒0\mu(K,\chi)=0italic_μ ( italic_K , italic_χ ) = 0 in the case of characters χ𝜒\chiitalic_χ of prime power order.

The proof of Theorem 1 depends on the following special case of a theorem of Gilmer [MR656619, Theorem 1] that relates values of σ1⁢τ⁢(K,χ)subscript𝜎1𝜏𝐾𝜒\sigma_{1}\tau(K,\chi)italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_τ ( italic_K , italic_χ ) to g4⁢(K)subscript𝑔4𝐾g_{4}(K)italic_g start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_K ).

A

γe∼1/|e|similar-tosubscript𝛾𝑒1𝑒\gamma_{e}\sim 1/\lvert e\rvertitalic_γ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ∼ 1 / | italic_e | away from corners, while being

ah:Vh×Vh→ℝ:subscript𝑎ℎ→subscript𝑉ℎsubscript𝑉ℎℝa_{h}\colon V_{h}\times V_{h}\rightarrow\mathbb{R}italic_a start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT : italic_V start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × italic_V start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT → blackboard_R and

Given a bounded polygonal domain Ω⊂ℝ2Ωsuperscriptℝ2\Omega\subset\mathbb{R}^{2}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT,

bh:Vh×Q̊h→ℝ:subscript𝑏ℎ→subscript𝑉ℎsubscript̊𝑄ℎℝb_{h}\colon V_{h}\times\mathring{Q}_{h}\rightarrow\mathbb{R}italic_b start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT : italic_V start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT × over̊ start_ARG italic_Q end_ARG start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT → blackboard_R

K⊂ℝ2𝐾superscriptℝ2K\subset\mathbb{R}^{2}italic_K ⊂ blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to be

D

{2}italic_y = divide start_ARG 5 end_ARG start_ARG 6 end_ARG italic_x - 5 - roman_log start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_x ) - ⌊ roman_log start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_x + 1 ) ⌋ - 3 ( roman_log start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_x + 1 ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

Note that for k∙=1subscript𝑘∙1k_{\bullet}=1italic_k start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT = 1 or 2222, by [LTXZ-1, Lemmas 5.2 and 5.6], we have

For any fixed k∙≥1subscript𝑘∙1k_{\bullet}\geq 1italic_k start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ≥ 1 and D𝐷Ditalic_D such that D≤k∙+1𝐷subscript𝑘∙1D\leq k_{\bullet}+1italic_D ≤ italic_k start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT + 1, the function f⁢(D,k∙,p)𝑓𝐷subscript𝑘∙𝑝f(D,k_{\bullet},p)italic_f ( italic_D , italic_k start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT , italic_p ) is a decreasing function of p≥11𝑝11p\geq 11italic_p ≥ 11. Hence,

Under the condition k∙≥1subscript𝑘∙1k_{\bullet}\geq 1italic_k start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT ≥ 1, a numerical calculation from Matlab (see Figure 1) shows that

Note that by [LTXZ-1, Lemma 5.2] and the definition of nSwk~,k′subscriptnSsubscript𝑤~𝑘superscript𝑘′\mathrm{nS}_{w_{\tilde{k}},k^{\prime}}roman_nS start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT over~ start_ARG italic_k end_ARG end_POSTSUBSCRIPT , italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, we have

A

This result allows us to capture C0⁢(G(0))subscript𝐶0superscript𝐺0C_{0}(G^{(0)})italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_G start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT )-submodules by open sets in étale groupoids.

we obtain the Galois correspondence result for discrete group coactions on groupoid C*-algebras (Corollary 2.2.8) by analysing discrete group cocycles.

we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras (Corollary 2.2.8).

we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras (Theorem 2.2.1).

Discrete group coactions generalize compact abelian group actions on C*-algebras and are studied in [11], for example.

D

Proof. Consider the g𝑔gitalic_g-rules of type (6). The third, g2⁢n+1,β=δ2⁢n+1,βsubscript𝑔2𝑛1𝛽subscript𝛿2𝑛1𝛽g_{2n+1,\beta}=\delta_{2n+1,\beta}italic_g start_POSTSUBSCRIPT 2 italic_n + 1 , italic_β end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT 2 italic_n + 1 , italic_β end_POSTSUBSCRIPT is the statement that if traffic is injected on the outgoing edge of D𝐷Ditalic_D, it can only be measured on the outgoing edge of D𝐷Ditalic_D (so traffic never flows backwards). The second, gj⁢β=δj⁢β+gj+,βsubscript𝑔𝑗𝛽subscript𝛿𝑗𝛽subscript𝑔superscript𝑗𝛽g_{j\beta}=\delta_{j\beta}+g_{{j^{+}},\beta}italic_g start_POSTSUBSCRIPT italic_j italic_β end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_j italic_β end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_β end_POSTSUBSCRIPT, is the statement that traffic goes through underpasses undisturbed, so gj⁢β=gj+,βsubscript𝑔𝑗𝛽subscript𝑔superscript𝑗𝛽g_{j\beta}=g_{{j^{+}},\beta}italic_g start_POSTSUBSCRIPT italic_j italic_β end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_β end_POSTSUBSCRIPT unless the traffic counter β𝛽\betaitalic_β is placed between j𝑗jitalic_j and j+superscript𝑗{j^{+}}italic_j start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, in which case it measures one unit more if the cars are injected before it, at j𝑗jitalic_j, rather than after it, at j+superscript𝑗{j^{+}}italic_j start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT. Similarly the first of these g𝑔gitalic_g-rules, gi⁢β=δi⁢β+Ts⁢gi+,β+(1−Ts)⁢gj+,βsubscript𝑔𝑖𝛽subscript𝛿𝑖𝛽superscript𝑇𝑠subscript𝑔superscript𝑖𝛽1superscript𝑇𝑠subscript𝑔superscript𝑗𝛽g_{i\beta}=\delta_{i\beta}+T^{s}g_{{i^{+}},\beta}+(1-T^{s})g_{{j^{+}},\beta}italic_g start_POSTSUBSCRIPT italic_i italic_β end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_i italic_β end_POSTSUBSCRIPT + italic_T start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_β end_POSTSUBSCRIPT + ( 1 - italic_T start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ) italic_g start_POSTSUBSCRIPT italic_j start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT , italic_β end_POSTSUBSCRIPT, is the statement of the behaviour of traffic at overpasses. Thus the rules in (6) are obeyed by cars and traffic counters, and as the rules in (6) determine gα⁢βsubscript𝑔𝛼𝛽g_{\alpha\beta}italic_g start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT, the proposition follows. □□\Box□

The quantity ρ1subscript𝜌1\rho_{1}italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is invariant under R3.

The quantity ρ1subscript𝜌1\rho_{1}italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is invariant under R1l.

The quantity ρ1subscript𝜌1\rho_{1}italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is invariant under R2c.

The quantity ρ1subscript𝜌1\rho_{1}italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is invariant under R3.

A

Because of the integration of the two new algorithmic ideas, theoretical frameworks of existing RPMs do not apply, and new analyses are developed.

In particular, we have formulated and proved an error bound-type condition, which plays an instrumental role in the theoretical development of our RPM. We believe that it could be of independent interest in the study of other optimization algorithms.

In fact, under case (i) of Proposition 1, inequality (5) reduces to the Hoffman error bound. Error bound conditions are important subjects in optimization and frequently utilized to study optimization algorithms. For example, as discussed above, conditions similar to (but different from) (5) are employed to study RPMs in [20] and [23]. Going beyond RPMs, error bound conditions also appeared in the study of first-order methods [32, 33], second-order methods [2, 34, 35], and even manifold optimization algorithms [36, 37].

The proposed algorithm features two useful algorithmic ideas that can significantly improve the practical performance: variance reduction and half-space approximation of the complicated subsets. To the best of our knowledge, this is the first time these two ideas are simultaneously incorporated into the framework of RPMs. Furthermore, the proposed RPM enjoys rigorous theoretical guarantees. Under assumptions similar to previous works, we prove that the sequence of iterates generated by our RPM converges to an optimal solution to problem (1).

Finally, we remark that Assumption 3 can also be seen as a generalization of the seminal Hoffman error bound [28], which asserts that the distance of any point to a linear system is linearly bounded by its violation of the linear constraints.

A

The above implies that for any two distinct members X,Y∈𝒳𝑋𝑌𝒳X,Y\in\mathcal{X}italic_X , italic_Y ∈ caligraphic_X there exists a hyperedge in their union intersecting both X𝑋Xitalic_X and Y𝑌Yitalic_Y, and also that for every X∈𝒳𝑋𝒳X\in\mathcal{X}italic_X ∈ caligraphic_X and c∈C𝑐𝐶c\in Citalic_c ∈ italic_C there is a hyperedge contained in X∪{c}𝑋𝑐X\cup\{c\}italic_X ∪ { italic_c } intersecting both X𝑋Xitalic_X and {c}𝑐\{c\}{ italic_c }. Since additionally C𝐶Citalic_C is a clique, this confirms our above claim that 𝒳𝒳\mathcal{X}caligraphic_X together with {c}c∈Csubscript𝑐𝑐𝐶\{c\}_{c\in C}{ italic_c } start_POSTSUBSCRIPT italic_c ∈ italic_C end_POSTSUBSCRIPT yields a complete minor in H𝐻Hitalic_H.

In our third and last application, we give a short reproof of the main result from [16] on coloring digraphs with excluded strong complete minors. The research on this topic was initiated by Axenovich, Girão, Snyder and Weber [3], and then further addressed by Mészáros and the author [16]. To state the result, we adopt the following terminology from [3, 16]:

To prove Theorem 3, we additionally need a structural result about small color-critical hypergraphs established by Stiebitz, Storch and Toft in [22]. We adopt the following terminology from their paper:

The following is a main result from [16]. By considering cycle hypergraphs of digraphs, we will show that it is an immediate consequence of Theorem 1. For t≥1𝑡1t\geq 1italic_t ≥ 1 we denote by Kt⟷⟷subscript𝐾𝑡\overset{\text{\tiny$\longleftrightarrow$}}{K_{t}}over⟷ start_ARG italic_K start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG the complete digraph of order t𝑡titalic_t (containing all possible t⁢(t−1)𝑡𝑡1t(t-1)italic_t ( italic_t - 1 ) arcs).

In this short first section we introduce essential terminology and notation used throughout the paper. The reader familiar with hypergraphs may want to skip this technical section and consult it at a later stage should there be any unclarities.

B

=𝐌b/p1/p⁢(x)⁢Db,p⁢C(α,β).absentsubscriptsuperscript𝐌1𝑝𝑏𝑝𝑥subscript𝐷𝑏𝑝superscript𝐶𝛼𝛽\displaystyle=\mathbf{M}^{1/p}_{b/p}(x)D_{b,p}C^{(\alpha,\beta)}.= bold_M start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b / italic_p end_POSTSUBSCRIPT ( italic_x ) italic_D start_POSTSUBSCRIPT italic_b , italic_p end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT ( italic_α , italic_β ) end_POSTSUPERSCRIPT .

The following is an outline of the paper: We introduce the basic constituents of the JFP method in Sections 2 and 3 (matrix representations of operators on quasimatrices, high-precision floating-point numbers, the JFP basis, etc.). We then focus on the properties (Section 4) and computation (Section 5) of fractional integration operators and matrices acting on the JFP basis. The JFP method is used in Section 6 to solve a variety of FIEs444Julia code for the examples in Section 6 are available at https://github.com/putianyi889/JFP-demo, including an FDE and fractional PDE reformulated as FIEs, and comparisons are made with the sum space method. We conclude the paper with a summary and a discussion of topics for future work. Appendix A is devoted to the above-mentioned pseudo-stabilization of an algorithm for computing fractional integration matrices.

Our work is strongly influenced by the method proposed by Hale and Olver [25], in which direct sums of appropriately weighted Jacobi polynomials are used as basis functions. We shall refer to this method as the sum space method and we note that the basis functions are related to the “generalized Jacobi functions” of [9] and the “polyfractonomials” of [48]. Similar to the ultraspherical spectral method [38] for ODEs, in the sum space method, the domain and range of operators in FDEs and FIEs are represented in different bases to ensure that the matrix representations of operators are banded. This enables a fast algorithm with linear complexity and exponential convergence for a wide range of problems. However, we shall find in Section 6 that the sum space method leads to tremendously large expansion coefficients (larger than 10100superscript1010010^{100}10 start_POSTSUPERSCRIPT 100 end_POSTSUPERSCRIPT) for the solution of an important family of FIEs that arises in the solution to the time-fractional heat/wave equation (𝒟tμ⁢u=𝒟x2⁢usubscriptsuperscript𝒟𝜇𝑡𝑢subscriptsuperscript𝒟2𝑥𝑢\mathcal{D}^{\mu}_{t}u=\mathcal{D}^{2}_{x}ucaligraphic_D start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_u = caligraphic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_u). These large sum space coefficients require very high precision for the accurate computation of solutions and is thus very expensive. By contrast, this family of FIEs poses no difficulties to the JFP method whose solution coefficients are bounded below 1111. We shall clarify this difference in the performance of the two methods by using bounds on Jacobi expansion coefficients for analytic functions to bound the JFP coefficients and by illustrating that the largest expansion coefficient in the sum space basis grows at the same (super-exponential) rate as the largest coefficient of the power series expansion of the solution.

In this section we discuss the following algorithms for computing the entries of fractional integral matrices:

The results derived in this section will motivate the two algorithms for computing fractional integration matrices that are discussed in the next section. The following result gives the action of ℐμsuperscriptℐ𝜇\mathcal{I}^{\mu}caligraphic_I start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT on the JFP basis and is the foundation of the first algorithm to be presented in the following section. The subsequent results in this section will inform the second algorithm.

D

The claim follows by inserting the definition of the curvature tensor of a space form (2.10) into (2.8)

Let γ:I→M:𝛾→𝐼𝑀\gamma\colon I\to Mitalic_γ : italic_I → italic_M be a p𝑝pitalic_p-biharmonic curve parametrized by arclength

Let γ:I→M:𝛾→𝐼𝑀\gamma\colon I\to\ Mitalic_γ : italic_I → italic_M be a smooth curve parametrized by arclength and

Let γ:I→M:𝛾→𝐼𝑀\gamma\colon I\to Mitalic_γ : italic_I → italic_M be a p𝑝pitalic_p-biharmonic curve parametrized by arclength

Let γ:I→M:𝛾→𝐼𝑀\gamma\colon I\to Mitalic_γ : italic_I → italic_M be a p𝑝pitalic_p-biharmonic curve parametrized by arclength.

D

Given f∈MDr⁢(𝔻)𝑓superscriptMD𝑟𝔻f\in\text{MD}^{r}({\mathbb{D}})italic_f ∈ MD start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( blackboard_D ) stable efficient, are all possible tree models semi-conjugate to each other in both directions like in Theorem A?

The answer to this question depends on the type of equivalency we are looking for. For the ergodic type, we show a positive result.

The complexity of mildly dissipative diffeomorphisms exists in between one-dimensional dynamics and surface diffeomorphisms. Let us briefly recall how they are defined. Consider the compact disc 𝔻𝔻{\mathbb{D}}blackboard_D and given r≥1𝑟1r\geq 1italic_r ≥ 1, we use Embr⁡(𝔻)superscriptEmbr𝔻\operatorname{Emb^{r}}({\mathbb{D}})start_OPFUNCTION roman_Emb start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT end_OPFUNCTION ( blackboard_D ) to denote the space of Crsuperscript𝐶𝑟C^{r}italic_C start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT embeddings of 𝔻𝔻{\mathbb{D}}blackboard_D into itself. We say f∈Embr⁡(𝔻)𝑓superscriptEmbr𝔻f\in\operatorname{Emb^{r}}({\mathbb{D}})italic_f ∈ start_OPFUNCTION roman_Emb start_POSTSUPERSCRIPT roman_r end_POSTSUPERSCRIPT end_OPFUNCTION ( blackboard_D ) is dissipative if |detD⁢f⁢(x)|<1,for all ⁢x∈𝔻formulae-sequence𝐷𝑓𝑥1for all 𝑥𝔻|\det Df(x)|<1,\text{for all }x\in{\mathbb{D}}| roman_det italic_D italic_f ( italic_x ) | < 1 , for all italic_x ∈ blackboard_D. This condition implies that for every invariant measure, for almost every point x𝑥xitalic_x, there is a stable manifold Ws⁢(x)superscript𝑊𝑠𝑥W^{s}(x)italic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_x ). If the measure is not supported on a hyperbolic sink, then the stable manifold of these points must have dimension 1. Let us call W𝔻s⁢(x)subscriptsuperscript𝑊𝑠𝔻𝑥W^{s}_{\mathbb{D}}(x)italic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_D end_POSTSUBSCRIPT ( italic_x ) the connected component of Ws⁢(x)∩𝔻superscript𝑊𝑠𝑥𝔻W^{s}(x)\cap{\mathbb{D}}italic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_x ) ∩ blackboard_D that contains x𝑥xitalic_x. We say that f𝑓fitalic_f is mildly dissipative if it is dissipative and if W𝔻s⁢(x)subscriptsuperscript𝑊𝑠𝔻𝑥W^{s}_{\mathbb{D}}(x)italic_W start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_D end_POSTSUBSCRIPT ( italic_x ) splits the disk in two sets for every x𝑥xitalic_x that has a one-dimensional stable manifold. We denote the family of such maps as MDr⁢(𝔻)superscriptMD𝑟𝔻\text{MD}^{r}({\mathbb{D}})MD start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ( blackboard_D ).

A classical way to study dynamical systems is by searching reduced models that capture the main features of the object of study. S. Crovisier and E. Pujals introduced a one-dimensional model in [4] to study a family of surface diffeomorphisms that they labeled as strongly dissipative and was later renamed to mildly dissipative in [6]. The current study examines this model from the perspective of a dynamical invariant. This inquiry is natural as well as interesting because these models are topological objects, yet they are built from an ergodic standpoint of differentiable maps.

A positive answer to this question implies that mildly dissipative maps are in essence a two-dimensional differentiable model of “conjugacy” classes of one-dimensional endomorphisms in trees.

D

Let κ~i=h~i⁢i⁢(X0)subscript~𝜅𝑖subscript~ℎ𝑖𝑖subscript𝑋0\tilde{\kappa}_{i}=\tilde{h}_{ii}(X_{0})over~ start_ARG italic_κ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over~ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) for i=1,…,n𝑖1…𝑛i=1,\ldots,nitalic_i = 1 , … , italic_n.

Therefore, at X0subscript𝑋0X_{0}italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , taking the covariant derivatives twice with respect

be the normal coordinates with respect to X0subscript𝑋0X_{0}italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. We may assume ξ0=e1subscript𝜉0subscript𝑒1\xi_{0}=e_{1}italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and

Therefore, at X0subscript𝑋0X_{0}italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , taking the covariant derivatives twice with respect

Since {hi⁢j⁢(X0)}subscriptℎ𝑖𝑗subscript𝑋0\{h_{ij}(X_{0})\}{ italic_h start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) } is diagonal, as (3.6) we have, at X0subscript𝑋0X_{0}italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,

A

1σmin⁢(A)−σmax⁢(B).1subscript𝜎𝐴subscript𝜎𝐵\displaystyle\frac{1}{\sigma_{\min}(A)-\sigma_{\max}(B)}.divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_A ) - italic_σ start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( italic_B ) end_ARG .

α¯¯𝛼\overline{\alpha}over¯ start_ARG italic_α end_ARG, β¯¯𝛽\overline{\beta}over¯ start_ARG italic_β end_ARG and γ¯¯𝛾\overline{\gamma}over¯ start_ARG italic_γ end_ARG. To

For β¯¯𝛽\overline{\beta}over¯ start_ARG italic_β end_ARG, we have the same as the result presented in

α¯¯𝛼\overline{\alpha}over¯ start_ARG italic_α end_ARG. For β¯¯𝛽\overline{\beta}over¯ start_ARG italic_β end_ARG, its process is

computable estimations for α¯¯𝛼\overline{\alpha}over¯ start_ARG italic_α end_ARG, β¯¯𝛽\overline{\beta}over¯ start_ARG italic_β end_ARG

B

We return to estimating |A2|subscript𝐴2\lvert A_{2}\rvert| italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |. Choosing β=γ𝛽𝛾\beta=\gammaitalic_β = italic_γ and using Hölder’s then Young’s inequalities,

≲(|h|τ+|h|2+|h|1+α∗)⁢(‖u‖W1,γ⁢(Bs)γ+‖f‖B∞α∗,γ′⁢(Bs)γ′).less-than-or-similar-toabsentsuperscriptℎ𝜏superscriptℎ2superscriptℎ1subscript𝛼∗superscriptsubscriptnorm𝑢superscriptW1𝛾subscript𝐵𝑠𝛾superscriptsubscriptnorm𝑓subscriptsuperscript𝐵subscript𝛼∗superscript𝛾′subscript𝐵𝑠superscript𝛾′\displaystyle\lesssim\left(\lvert h\rvert^{\tau}+\lvert h\rvert^{2}+\lvert h%

|A22|≲less-than-or-similar-tosubscript𝐴22absent\displaystyle\lvert A_{22}\rvert\lesssim| italic_A start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT | ≲

≲(|h|γ+|h|2+|h|1+α∗)⁢(‖u‖W1,γ⁢(Bs)γ+‖f‖B∞α∗,γ′⁢(Bs)γ′).less-than-or-similar-toabsentsuperscriptℎ𝛾superscriptℎ2superscriptℎ1subscript𝛼∗subscriptsuperscriptnorm𝑢𝛾superscript𝑊1𝛾subscript𝐵𝑠superscriptsubscriptnorm𝑓subscriptsuperscript𝐵subscript𝛼∗superscript𝛾′subscript𝐵𝑠superscript𝛾′\displaystyle\lesssim\left(\lvert h\rvert^{\gamma}+\lvert h\rvert^{2}+\lvert h%

|A2|≲less-than-or-similar-tosubscript𝐴2absent\displaystyle\lvert A_{2}\rvert\lesssim| italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ≲

D

}{N}\right)^{-1}=\sum_{r=0}^{\infty}\frac{(-1)^{r}}{N^{r}}h_{r}(\lambda),roman_Ω start_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_N end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_λ ) = ∏ start_POSTSUBSCRIPT □ ∈ italic_λ end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_c ( □ ) end_ARG start_ARG italic_N end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_r = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG ( - 1 ) start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_ARG start_ARG italic_N start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT end_ARG italic_h start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_λ ) ,

For any Young diagrams α,β𝛼𝛽\alpha,\betaitalic_α , italic_β with d≤N𝑑𝑁d\leq Nitalic_d ≤ italic_N cells,

exist and are integers for all Young diagrams α,β𝛼𝛽\alpha,\betaitalic_α , italic_β with d≤8𝑑8d\leq 8italic_d ≤ 8.

equivalent to the statement that for all α,β𝛼𝛽\alpha,\betaitalic_α , italic_β with d≤N𝑑𝑁d\leq Nitalic_d ≤ italic_N cells we

For any Young diagrams α,β𝛼𝛽\alpha,\betaitalic_α , italic_β with d≤N𝑑𝑁d\leq Nitalic_d ≤ italic_N cells, we have

A

The first author was supported by SPP 2026 Geometry at infinity funded by the Deutsche Forschungsgemeinschaft.

i.e. the entries of each Misubscript𝑀𝑖M_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are supported in E𝐸Eitalic_E.

The first author was supported by SPP 2026 Geometry at infinity funded by the Deutsche Forschungsgemeinschaft.

The second and third authors were supported by the European Research Council (ERC) grant Rigidity of groups and higher index theory

We begin by recalling briefly the Fox calculus and the associated complex computing group cohomology.

C

\times\mathcal{Y}^{\boxtimes j}(n))/S_{j}.( caligraphic_X ∘ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT caligraphic_Y ) ( italic_m , italic_n ) = ∐ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( caligraphic_X ( italic_m , italic_j ) × caligraphic_Y start_POSTSUPERSCRIPT ⊠ italic_j end_POSTSUPERSCRIPT ( italic_n ) ) / italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .

Finally, given an 𝕊𝕊\mathbb{S}blackboard_S-space 𝒵𝒵\mathcal{Z}caligraphic_Z, we define an 𝕊2superscript𝕊2\mathbb{S}^{2}blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-space Δ⁢𝒵Δ𝒵\Delta\mathcal{Z}roman_Δ caligraphic_Z by the assignment

Given 𝒳𝒳\mathcal{X}caligraphic_X a 𝕊2superscript𝕊2\mathbb{S}^{2}blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-space and 𝒴𝒴\mathcal{Y}caligraphic_Y an 𝕊𝕊\mathbb{S}blackboard_S-space with 𝒴⁢(0)=∅𝒴0\mathcal{Y}(0)=\varnothingcaligraphic_Y ( 0 ) = ∅, we can define two composition operations, ∘1subscript1\circ_{1}∘ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ∘2subscript2\circ_{2}∘ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT:

We define an 𝕊2superscript𝕊2\mathbb{S}^{2}blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-space ℳ¯g⋆subscriptsuperscript¯ℳ⋆𝑔\overline{\mathcal{M}}^{\star}_{g}over¯ start_ARG caligraphic_M end_ARG start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT by

Given an 𝕊2superscript𝕊2\mathbb{S}^{2}blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-space 𝒳𝒳\mathcal{X}caligraphic_X, we define its Hodge–Deligne series by

D

Note that the assumption that the killing rate is uniformly bounded is technical: after using Burkholder-Davis-Gundy inequality, one needs to control the term 𝔼⁢(CTp/2⁢(N+1N)p⁢CT)𝔼superscriptsubscript𝐶𝑇𝑝2superscript𝑁1𝑁𝑝subscript𝐶𝑇\mathbb{E}\left(C_{T}^{p/2}\left(\frac{N+1}{N}\right)^{pC_{T}}\right)blackboard_E ( italic_C start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p / 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_N + 1 end_ARG start_ARG italic_N end_ARG ) start_POSTSUPERSCRIPT italic_p italic_C start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ), however, depending on the particular process at hand, there may be other ways to obtain upper bounds on this quantity which do not require the killing rate to be bounded.

The authors would like to thank the anonymous referee for their constructive comments that improved the exposition and quality of this paper.

where b𝑏bitalic_b is the branching rate, κ𝜅\kappaitalic_κ is the soft killing rate, τ∂subscript𝜏\tau_{\partial}italic_τ start_POSTSUBSCRIPT ∂ end_POSTSUBSCRIPT is the absorption time, ΠTAsuperscriptsubscriptΠ𝑇𝐴\Pi_{T}^{A}roman_Π start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and ΠTBsuperscriptsubscriptΠ𝑇𝐵\Pi_{T}^{B}roman_Π start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT are weights that compensate for the resampling and selection events that occur up to time T𝑇Titalic_T, which we state explicitly in Theorem 2.6, and 𝐄⋅subscript𝐄⋅{\mathbf{E}}_{\cdot}bold_E start_POSTSUBSCRIPT ⋅ end_POSTSUBSCRIPT is the expectation with respect to the law of X𝑋Xitalic_X. As we will later discuss, by choosing particular model parameters, we recover both the classical many-to-one formula [42] and the unbiased estimator proved in [66] for fixed size Moran type genetic algorithms. The second part of Theorem 2.6 then provides a precise bound on the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT distance between the normalised semigroup on the right-hand side of (1) and the normalised occupation measure of the particle system. In particular, it states that this distance converges to zero as the number of initial particles tends to infinity.

Finally, we note that our model fits into the more general class of controlled branching processes introduced by Sevastyanov and Zubkov in [61], where the number of reproductive individuals in each generation depends on the size of the previous generation via a control function. This work was later extended by Yanev [68] to allow for random control functions. We refer the reader to [61, 37, 64, 67] for discrete time versions of this process and to [36] for the continuous time version. In these articles, the authors study the convergence of the survival probability in different regimes, as well as the expected population size. We also note that the control function can be seen as a way to model immigration and emigration of particles, [67]. We refer the reader to [63, 51, 59, 55, 53, 4, 2] for results regarding these latter processes.

This theorem therefore provides a way to build Monte Carlo simulations to estimate the eigenvalue λ*subscript𝜆\lambda_{*}italic_λ start_POSTSUBSCRIPT * end_POSTSUBSCRIPT and the eigenfunctions φ𝜑\varphiitalic_φ and φ~~𝜑\tilde{\varphi}over~ start_ARG italic_φ end_ARG via the NRW. We refer the reader to [18] for further details. However, although simulating a single weighted path has advantages over simulating an entire tree of neutrons, the transience of this process means that many of the particles exit the domain relatively quickly and therefore, a large number of simulations are required in order to obtain information about the system. This also leads to problems with the variance of the estimators. In order to deal with this problem, the notion of ‘hℎhitalic_h-transform’ was developed in [18], where one biases the NRW to prevent it from exiting the domain. This idea is similar to that presented in [60], where the author use hℎhitalic_h-transforms to build Monte-Carlo approximations. Therein, the main result requires that the resulting hℎhitalic_h-transformed process has bounded killing rate. In our case, although this can be theoretically obtained, the natural candidates for function hℎhitalic_h result in a process with unbounded killing rate. We show below that the associated BBMMI is non-explosive, which ensures that Theorem 2.6 applies.

A

=Σ⁢(z→).absentΣ→𝑧\displaystyle={\Sigma}({\overrightarrow{z}})~{}.= roman_Σ ( over→ start_ARG italic_z end_ARG ) .

For a sufficiently large trajectory space 𝒵→→𝒵{\overrightarrow{\mathcal{Z}}}over→ start_ARG caligraphic_Z end_ARG, such as with continuous-time

\overrightarrow{\mathcal{Z}}})}= roman_ln divide start_ARG italic_P ( over→ start_ARG caligraphic_Z end_ARG ) end_ARG start_ARG italic_R ( over→ start_ARG caligraphic_Z end_ARG ) end_ARG

particular system state trajectory as z→→𝑧{\overrightarrow{z}}over→ start_ARG italic_z end_ARG, with z→⁢(t)→𝑧𝑡{\overrightarrow{{z}}(t)}over→ start_ARG italic_z end_ARG ( italic_t ) the

The singleton {z→}→𝑧\{{\overrightarrow{z}}\}{ over→ start_ARG italic_z end_ARG } for any trajectory z→∈𝒵→→𝑧→𝒵{\overrightarrow{z}}\in{\overrightarrow{\mathcal{Z}}}over→ start_ARG italic_z end_ARG ∈ over→ start_ARG caligraphic_Z end_ARG.

A

σα⁢(φ)=∑βi⁢(φ)α∈[0,∞],subscript𝜎𝛼𝜑subscript𝛽𝑖superscript𝜑𝛼0\sigma_{\alpha}(\varphi)=\sum\beta_{i}(\varphi)^{\alpha}\in[0,\infty],italic_σ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_φ ) = ∑ italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_φ ) start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ∈ [ 0 , ∞ ] ,

The barcode ℬ⁢(L,L′)ℬ𝐿superscript𝐿′{\mathcal{B}}(L,L^{\prime})caligraphic_B ( italic_L , italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) of the Floer complex CF⁡(L,L′)CF𝐿superscript𝐿′\operatorname{CF}(L,L^{\prime})roman_CF ( italic_L , italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ), in the

ℬisubscriptℬ𝑖{\mathcal{B}}_{i}caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. This is indeed a distance on the space of un-pinned barcodes,

ℬ⁢(L,L′)ℬ𝐿superscript𝐿′{\mathcal{B}}(L,L^{\prime})caligraphic_B ( italic_L , italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is independent of the choice of a singular decomposition

ℬ⁢(φ)ℬ𝜑{\mathcal{B}}(\varphi)caligraphic_B ( italic_φ ). This is a Floer theoretic variant of the total

D

For a fair comparison with prior rates, we optimize the choice of ϵitalic-ϵ\epsilonitalic_ϵ and η𝜂\etaitalic_η to minimize the factors above and compare with our results. As depicted in Figure 1, the contraction factors we derived are much closer to the tight contraction factors evaluated by the PEP approach.

are strictly smaller (better)than the six contraction factors of [17, Theorem D.6] and are, to the best of our knowledge, the best known rates.

In this sense, our contraction factors are a better proxy of the tight contraction factors than the earlier results in [17].

One possible future direction is to further refine the analysis on the complex plane and obtain tighter contraction factors, as the contraction factors ρ𝜌\rhoitalic_ρ presented in Section 3 are not tight in the sense that

Our contraction factors of Theorems 3.1, 3.2, and 3.3 are smaller (better) than the six prior contraction factors of [17, Theorem D.6].

B

Then, by Lemma 2.2, there exists a cell contained in B𝐵Bitalic_B, and so that cell is weakly g𝑔gitalic_g-wandering.

In order to prove this result in the general setting of polychromatic rearrangement groups, we will need to generalize the results of [Gelander2016InvariableGO].

The aim of this work is to generalize the results from [Gelander2016InvariableGO] and their proofs to the setting of rearrangement groups, proving the following result:

The following Lemma and its proof are the same as Lemma 18 of [Gelander2016InvariableGO]. The only difference is that it applies to cells of a limit space instead of intervals of S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT.

Following the strategy of [Gelander2016InvariableGO], we need to gather some results about the dynamics of rearrangements in order to prove the existence of wandering cells for non-periodic elements.

D

Since Nesterov’s work [Nes05], accelerated first-order convex optimization has become an active research field [Tse08, SV12, WWJ16, FRMP18]. Due to the performance of Nesterov’s discrete scheme and the work [SBC16], a major line of research concerns the understanding and generation of accelerated discrete scheme using continuous-time ODEs, where an adaptive ‘friction term’ causes acceleration. In our approach, acceleration is solely achieved by employing a multilevel problem representation. Acceleration of the first-order Riemannian dynamics is an interesting problem of future research.

A key ingredient of the approach is to replace the feasible set [0,1]nsuperscript01𝑛[0,1]^{n}[ 0 , 1 ] start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT by the interior (0,1)nsuperscript01𝑛(0,1)^{n}( 0 , 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT that is turned into a Riemannian manifold (ℬn,g)superscriptℬ𝑛𝑔(\mathcal{B}^{n},g)( caligraphic_B start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_g ) using information geometry [AN00]. This geometry differs from the more common geometry underlying interior point methods [NT02] (see Section 1.2). Transfer of variables from fine to coarse levels and vice versa are performed by restriction and prolongation mappings, that are adapted to the geometry and whose differentials are used for pushing tangent vectors between levels. Criteria for invoking coarse level computation and certifying resulting search directions are provided. Numerical results illustrate the effectiveness of our approach for the specific instance of (1.1)

In the present paper, we also consider such a Riemannian metric on a simpler structured bounded open convex feasible set, in order to focus on multilevel representation and accelerated first-order optimization that copes with large problem sizes. This necessitates, in particular, to devise restriction and prolongation operators not only for points on the manifold but also for tangent vectors. To this end, we employ information geometry in order to design smooth nonlinear mappings based on geometric averaging that can be efficiently computed in closed form. Numerical experiments demonstrate that our method outperforms the recent state-of-the-art method [HRX21].

Any cluster point of the sequence generated by the Algorithm (2) is a critical point of f𝑓fitalic_f.

[BL89, NT02, AMS08]. The commonly adopted interior point geometry is based on Hessian metrics generated by self-concordant barrier functions, due to the provable optimality in connection with Newton-like second-order optimization [NN94], see e.g. [TP21] for recent related work. Closer to our work is the comprehensive paper by [ABB04] where Hessian metrics generated by convex functions of Legendre type are studied for a class of convex programs that include affine subspace constraints.

D

Recent results in Boolean circuit complexity have demonstrated the existence of monotone functions that a polynomial-sized circuit of bounded depth can compute but require super polynomial monotone circuits to compute [9, 17].

We focused on the threshold activation function. It is an interesting direction to extend our results for other activation functions such as sigmoids. For the universality result of depth 4 monotone networks it seems plausible that one could approximate thresholds by sigmoids and prove that monotone networks of depth 4 with sigmoids are universal approximators of monotone functions. For our lower bounds, based on the matching function m𝑚mitalic_m it appears that new ideas are needed to show a super polynomial separation between the size needed for monotone as opposed to arbitrary networks with sigmoids to approximate m𝑚mitalic_m.

It is reasonable to expect that one could use our machinery to demonstrate monotone real functions that can be computed by a bounded depth circuit yet require super polynomial size to be approximated by monotone networks. Proving separation results between monotone and non-monotone networks with respect to the square loss is another avenue for further research.

Our aim here is to show that if some f:{0,1}d→ℕ:𝑓→superscript01𝑑ℕf:\{0,1\}^{d}\to\mathbb{N}italic_f : { 0 , 1 } start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_N cannot be computed by small monotone circuits, then it cannot be approximated by small monotone networks. We show this by demonstrating a reduction from monotone circuits to monotone networks. Given a monotone network that approximates f^^𝑓\hat{f}over^ start_ARG italic_f end_ARG in [0,1]dsuperscript01𝑑[0,1]^{d}[ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, we show how to construct a monotone circuit of comparable size to calculate f𝑓fitalic_f. The proof uses the following fact established in [5]:

Given the above result, it may seem that, similarly to the case of monotone networks with ReLU activations, the class of monotone networks with threshold activations is too limited, in the sense that it cannot approximate any monotone function with a constant depth (allowing the depth to scale with the dimension was considered in [12], see below). One reason for such a belief is that, for non-monotone networks, depth 2222 suffices to ensure universality. Any continuous function over a bounded domain can be approximated by a depth-2222 network [3, 11, 22] and this universality result holds for networks with threshold or ReLU as activation functions. Our first main result supports the contrary to this belief. We establish a depth separation result for monotone threshold networks and show that monotone networks can interpolate arbitrary monotone data sets by slightly increasing the number of layers. Thereafter, a simple argument shows that monotone networks of bounded depth are universal approximators of monotone functions. As noted, this is in sharp contrast to general neural networks, where adding extra layers can affect the efficiency of the representation [16], but does not change the expressive power.

B

\text{and}\ \lambda_{\chi,f}=\tau\}}\left\langle e_{\chi}\right\rangle.⟨ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ + ∑ start_POSTSUBSCRIPT { italic_χ : italic_χ ∈ roman_Irr ( italic_G ) and italic_λ start_POSTSUBSCRIPT italic_χ , italic_f end_POSTSUBSCRIPT = italic_τ } end_POSTSUBSCRIPT ⟨ italic_e start_POSTSUBSCRIPT italic_χ end_POSTSUBSCRIPT ⟩ .

We are now ready to give a sufficient condition for EKR-module property in terms of spectra of ΓG,ΩsubscriptΓ𝐺Ω\Gamma_{G,\Omega}roman_Γ start_POSTSUBSCRIPT italic_G , roman_Ω end_POSTSUBSCRIPT-compatible matrices.

The subgroup N𝑁Nitalic_N of G𝐺Gitalic_G is a transitive subgroup satisfying the EKR property, and so by Lemma 3.3 of [27], we see that the action of G𝐺Gitalic_G also satisfies the EKR property.

Now by application of Lemma 2.4, we obtain the following sufficient condition for EKR-module property.

As an immediate application, we obtain a significantly shorter proof of Lemma 4.1 and Lemma 4.2 of [2]. The content of these two results is presented as the following corollary. We will use the following technical result in the proof of Theorem 1.7. We would like to mention that it was a key result that led to the “Module Method” described in [2].

C

It would appear from Fig. 8 that this bias is larger for 1qSGD [consider the histogram for estimates of =θ1∗0, which shows that final estimates typically exceed this value]. In fact, there is no theory to predict if 1qSGD is better or worse than 1SPSA in terms of bias. The bias is imperceptible in the stochastic case due to the higher variance combined with the removal of outliers.

Experiments were performed to test the performance of the Lipschitz version of 1qSGD for both constant and vanishing gain algorithms. Two take-aways from the numerical results surveyed below:

Experiments were performed to test the performance of the Lipschitz version of 1qSGD for both constant and vanishing gain algorithms. Two take-aways from the numerical results surveyed below:

Experiments were performed to test the performance of the Lipschitz version of 1qSGD for both constant and vanishing gain algorithms. Two take-aways from the numerical results surveyed below:

Experiments were performed to test the performance of the Lipschitz version of 1qSGD for both constant and vanishing gain algorithms. Two take-aways from the numerical results surveyed below:

A

⟨∇ϕ⁢(y)−∇ϕ⁢(x),y−x⟩≥0∇italic-ϕ𝑦∇italic-ϕ𝑥𝑦𝑥0\langle\nabla\phi(y)-\nabla\phi(x),y-x\rangle\geq 0⟨ ∇ italic_ϕ ( italic_y ) - ∇ italic_ϕ ( italic_x ) , italic_y - italic_x ⟩ ≥ 0, for all x,y∈E𝑥𝑦𝐸x,y\in Eitalic_x , italic_y ∈ italic_E.

The second result regarding the existence of solutions for system (1.3) in the case of bounded constraint sets is stated below.

A direct consequence of the above result is that convex and Gâteaux differentiable functions are in fact lower semicontinuous. Proposition 1.5 also suggests the following generalization of the gradient of a convex function.

Note that the second inequality of (1.3) and (1.9), respectively, have opposite sign, therefore even for particular cases the solution sets of the two problems do not coincide. Moreover, besides the fact that hemivariational inequalities are a generalization of variational inequalities further difficulties occur due to the presence of the nonlinear operators F𝐹Fitalic_F and G𝐺Gitalic_G in each inequality of (1.3) and the fact that the second inequality is of quasi-hemivariational type (since, in general, there does not exist a locally Lipschitz functional T:Y→ℝ:𝑇→𝑌ℝT\colon Y\to\mathbb{R}italic_T : italic_Y → blackboard_R such that ∂CT⁢(σ)=H⁢(σ)⁢∂CJ0⁢(γ2⁢(σ))subscript𝐶𝑇𝜎𝐻𝜎subscript𝐶superscript𝐽0subscript𝛾2𝜎\partial_{C}T(\sigma)=H(\sigma)\partial_{C}J^{0}(\gamma_{2}(\sigma))∂ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_T ( italic_σ ) = italic_H ( italic_σ ) ∂ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_J start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_σ ) )). Consequently, the novelty of our approach comes on the one hand from the fact that this is the first paper dealing with coupled systems consisting of a variational-hemivariational inequality and a quasi-hemivariational inequality and, on the other hand, that the coupling functional is assumed nonlinear (and even in the bilinear case our system is similar but not equivalent to systems studied in the aforementions papers, as the second inequality is reversed.)

The first paper that we are aware of dealing with coupled systems consisting of two hemivariational inequalities is Matei22 where the existence of solution for the following problem is studied:

B

For all A,B∈ℒ⁢(E)𝐴𝐵ℒ𝐸A,B\in\mathcal{L}(E)italic_A , italic_B ∈ caligraphic_L ( italic_E ) and I,J∈ℒ⁢(E)𝐼𝐽ℒ𝐸I,J\in\mathcal{L}(E)italic_I , italic_J ∈ caligraphic_L ( italic_E ) such that I∈max⁡(E,A∩ℬ)𝐼𝐸𝐴ℬI\in\max(E,A\cap{\mathcal{B}})italic_I ∈ roman_max ( italic_E , italic_A ∩ caligraphic_B ) and J∈max⁡(E,B∩ℬ)𝐽𝐸𝐵ℬJ\in\max(E,B\cap{\mathcal{B}})italic_J ∈ roman_max ( italic_E , italic_B ∩ caligraphic_B ), there exists K∈max⁡(E,(A+B)∩ℬ)𝐾𝐸𝐴𝐵ℬK\in\max(E,(A+B)\cap{\mathcal{B}})italic_K ∈ roman_max ( italic_E , ( italic_A + italic_B ) ∩ caligraphic_B ) such that K⊆I+J𝐾𝐼𝐽K\subseteq I+Jitalic_K ⊆ italic_I + italic_J.

(E,ℬ)𝐸ℬ(E,{\mathcal{B}})( italic_E , caligraphic_B ) determines a q𝑞qitalic_q-matroid with collection of bases ℬℬ{\mathcal{B}}caligraphic_B and collection of independent spaces ℐℬsubscriptℐℬ\mathcal{I}_{\mathcal{B}}caligraphic_I start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT.

If ℬℬ\mathcal{B}caligraphic_B satisfies the bases axioms (B1)-(B4) we say that (E,ℬ)𝐸ℬ(E,\mathcal{B})( italic_E , caligraphic_B ) is a collection of bases.

Let (E,ℐ)𝐸ℐ(E,\mathcal{I})( italic_E , caligraphic_I ) be a collection of independent spaces and let (E,ℬ)𝐸ℬ(E,{\mathcal{B}})( italic_E , caligraphic_B ) be a collection of bases.

Let ℬℬ{\mathcal{B}}caligraphic_B be a family of subspaces of E𝐸Eitalic_E that satisfies the axioms (B1), (B2) and (B4). Then ℬℬ{\mathcal{B}}caligraphic_B satisfies (B3).

B

Although the choice of η⁢(𝒚)𝜂𝒚\eta(\boldsymbol{y})italic_η ( bold_italic_y ) is more subtle for the SIPG and IIPG methods, we experienced that too large η⁢(𝒚)𝜂𝒚\eta(\boldsymbol{y})italic_η ( bold_italic_y ) do not have a visual effect on the results in Section 7, while too small η⁢(𝒚)𝜂𝒚\eta(\boldsymbol{y})italic_η ( bold_italic_y ) can lead to wrong approximations. In principle, increasing η𝜂\etaitalic_η adds interelement diffusion to the method. Thus, enlarging η𝜂\etaitalic_η stabilizes the method, while its scaling ensures that convergence rates in hℎhitalic_h do not deteriorate. Too large η𝜂\etaitalic_η values have an adversarial effect on the constants of DG methods, but as long as integrability (concerning 𝒚𝒚\boldsymbol{y}bold_italic_y) is ensured, the QMC methods will work.

In DG, the idea is to modify the variational formulation, and we can no longer exploit conformity to obtain regularity bounds for the DG solutions to (1). This means that the regularity analysis must be rewritten for the DG system, which will also affect the choice of the optimal QMC rule for the computation of the expectation of the stochastic response.

In this study, we assumed the data to allow for solutions u∈H2𝑢superscript𝐻2u\in H^{2}italic_u ∈ italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT concerning the spatial variable. At the cost of some technical but standard extensions of the DG analysis, we can extend our results to the case that u∈H3/2+ϵ𝑢superscript𝐻32italic-ϵu\in H^{3/2+\epsilon}italic_u ∈ italic_H start_POSTSUPERSCRIPT 3 / 2 + italic_ϵ end_POSTSUPERSCRIPT in space. Moreover, using discrete gradients and lifting operators, we believe our results can be transferred to the case of u𝑢uitalic_u having minimal spatial regularity. We dropped extensive discussions on spatial regularity, as detailed in the DG literature. For QMC methods, the advantages of DG are its more regular execution patterns and the additional local mass conservation. In that sense, it preserves more physics than continuous finite element methods. However, we are not aware of any indications that DG works under milder regularity assumptions than continuous finite elements.

The ultimate goal of our analysis is to derive regularity bounds in the flavor of (8) and (9) for the diffusion problem that is discretized with an IPDG method instead of continuous finite elements. To this end, we start with some preliminary considerations before we prove Theorem 5.7 stating that

A common feature of virtually all of the aforementioned QMC literature related to PDE uncertainty quantification is that the QMC rule is designed for the non-discretized PDE problem (1) whereas, in practical computations, one only has access to a discrete approximation of the PDE system. Of course, as long as one uses a conforming finite element (FE) method to perform the discretization of the PDE, the parametric regularity of the non-discretized PDE naturally transfers to the discretized PDE problem and the theory remains sound even for the discretized PDE system. However, in many cases it is preferable to consider non-conforming FE methods such as DG methods (e.g. see the books [11, 16, 32, 34, 47] for a comprehensive overview and [2] for a unified analysis framework for elliptic PDEs using DG) to solve PDEs with uncertain coefficients. Using DG methods, the inter-element continuity constraint of conforming FE methods is dropped and concatenations of arbitrary local polynomials with support in only one element can be used as test and trial functions. This property resembles the finite volume (FV) approach, in which the solution per element is approximated by a local constant. As a consequence, DG methods have similar parallelization properties as FV (but do not need any reconstructions to achieve higher order). Moreover, the problem of hanging nodes is intrinsically bypassed in the DG framework allowing for more general meshes than conforming FE, see [4]. Beyond that, DG directly supports h⁢pℎ𝑝hpitalic_h italic_p refinement, where both the mesh and the local degree of approximating polynomials can be adapted locally, see [4, 5, 45]. Last, DG comes with an intuitive notion of local mass conservation, since it is based on inter-element fluxes (and mass conservative in an element itself). These advantages have made DG methods a popular tool for many applications. In particular, they have been used in the field of computational fluid dynamics (see [52] for a survey—notable mentions are DG methods for incompressible Navier–Stokes [7] and shallow-water equations [1]), multicomponent reactive transport [55], and many more.

C

In the weighted setting, it was shown in [LSST20] that the set of 2222-dimensional w𝑤witalic_w-singular vectors has Hausdorff dimension 2−11+w1211subscript𝑤12-\frac{1}{1+w_{1}}2 - divide start_ARG 1 end_ARG start_ARG 1 + italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG.

The aim of the present article is to extend this 2222-dimensional result to higher dimensional cases regarding the lower bound of the Hausdorff dimension. Our main theorem is as follows.

One of the applications of the variational principle for general flows in [Sol] is to give an upper bound of the Hausdorff dimension of the set

This result was extended to the d𝑑ditalic_d-dimensional singular vectors in [CC16], where the authors proved that the set of d𝑑ditalic_d-dimensional singular vectors has Hausdorff dimension d2/(d+1)superscript𝑑2𝑑1d^{2}/(d+1)italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( italic_d + 1 ).

Historically, the first breakthrough was made in [Che11], where it is proved that the Hausdorff dimension of the set of 2222-dimensional singular vectors is 4/3434/34 / 3.

A

We will use 𝒮n,msubscript𝒮𝑛𝑚\mathcal{S}_{n,m}caligraphic_S start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT to denote the set of all

We will use 𝒮n,msubscript𝒮𝑛𝑚\mathcal{S}_{n,m}caligraphic_S start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT to denote the set of all

(of dimension (n−1)⁢(m−1)𝑛1𝑚1(n-1)(m-1)( italic_n - 1 ) ( italic_m - 1 )) in the linear space of all

for the extremality of A∈𝒮n,m𝐴subscript𝒮𝑛𝑚A\in\mathcal{S}_{n,m}italic_A ∈ caligraphic_S start_POSTSUBSCRIPT italic_n , italic_m end_POSTSUBSCRIPT

the set of extremal arrays in 𝒮n,n+1subscript𝒮𝑛𝑛1\mathcal{S}_{n,n+1}caligraphic_S start_POSTSUBSCRIPT italic_n , italic_n + 1 end_POSTSUBSCRIPT,

B