arXiv Math @arxiv_math@qoto.org

We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over a field K -- such functors are sometimes called \textit{generic representations} of linear groups over A with coefficients in K. We are especially interested with finitely generated functors of F(A,K) taking finite dimensional values. We prove that they can, under a mild extra assumption (always satisfied if the ring A is noetherian), be built from much better understood functors, namely polynomial functors (in the sense of Eilenberg-MacLane), or factorising at the source through reduction modulo a cofinite ideal of A. We deduce that such functors are always noetherian et that, if the ring A is finitely generated, they have finitely generated projective resolutions.Our methods rely mainly on the study of weight decompositions of functors and their cross-effects, our recent previous work with Vespa (Ann. ENS 2023) and elementary commutative algebra.

Electric vehicles face significant energy supply challenges due to long charging times and congestion at charging stations. Battery swapping stations (BSSs) offer a faster alternative for energy replenishment, but their deployment costs are considerably higher than those of charging stations. As a result, selecting optimal locations for BSSs is crucial to improve their accessibility and utilization. Most existing studies model the BSS location problem using deterministic and static approaches, often overlooking the impact of stochastic and dynamic factors on solution quality. This paper addresses the facility location problem for BSSs within a city network, considering stochastic battery swapping demand. The objective is to optimize the placement of a given set of BSSs to minimize demand loss. To achieve this, we first develop a mathematical programming model for the problem. Then, we propose a simulation optimization method based on a large neighborhood search framework to handle large-scale instances. To reduce the computational cost of simulations, Bayesian optimization is employed to solve the single-station allocation subproblem during the repair process. Numerical experiments demonstrate the efficiency of the proposed approach and highlight the importance of incorporating dynamic factors in decision-making.

Let $G$ be a countable infinite amenable group, $K$ a finite-dimensional compact metrizable space, and $(K^G,σ)$ the full $G$-shift on $K^G$. For any $r\in [0,{\rm mdim}(K^G,σ))$, we construct a minimal subshift $(X,σ)$ of $(K^G,σ)$ with mdim$(X,σ)=r$. Furthermore, we construct a subshift of $([0,1]^G,σ)$ such that its mean dimension is $1$, and that the set of all attainable values of the mean dimension of its minimal subsystems is exactly the interval $[0,1)$.

In this paper, we consider higher tensor powers of oscillator representations over finite fields. We find a decomposition of their endomorphism algebras into group algebras of orthogonal groups, giving a new version of Howe duality for a certain range of Type I reductive dual pairs. Specifically, we find that all occurring terms arise from parabolic inductions and the eta correspondence of R. Howe and S. Gurevich. This also leads to a version of Howe duality in a certain category "interpolating" the oscillator representations.

We provide a partial answer to a question of Ekholm, Honda, and Kálmán about the relationship between Khovanov homology and decomposable Lagrangian cobordisms. We also utilize previously defined filtered invariants to give obstructions to decomposable Lagrangian cobordisms from Khovanov homology.

We provide the first documented examples of immersions of closed oriented manifolds which are not homologous to embeddings, thus answering a question posed by Zhenhua Liu. In these examples, we show the existence of representing immersions whose double points represent a non-trivial homology class in the source manifold. We also provide examples of Steenrod representable integral homology classes, which are not represented by immersions.

Faster-than-Nyquist (FTN) signaling is a non-orthogonal transmission technique offering a promising solution for future generations of communications. This paper studies the capacity of FTN signaling in multiple-input multiple-output (MIMO) channels for high acceleration factors. In our previous study [1], we found the capacity for MIMO FTN channels if the acceleration factor is larger than a certain threshold, which depends on the bandwidth of the pulse shape used. In this paper, we extend the capacity analysis to acceleration factors smaller than this mentioned threshold. In addition to capacity, we conduct peak-to-average power ratio (PAPR) analysis and simulation for MIMO FTN for varying acceleration factors for both Gaussian and QPSK symbol sets. Our analysis reveals important insights about transmission power and received signal-to-noise ratio (SNR) variation in FTN. As the acceleration factor approaches 0, if the transmission power is fixed, the received SNR diminishes, or if the received SNR is fixed, PAPR at the transmitter explodes.

There are three commonly known definitions of the bridge index. These definitions come from combinatorial knot theory, Morse theory, and geometry. In this paper, we prove that the danceability index is a fourth equivalent definition of the bridge index. We extend the danceability invariant to virtual knots in multiple ways and compare these invariants to two different notions of bridge index for virtual knots.

Let $A$ be a noetherian ring, $I$ an ideal of $A$ and $N\subset M$ finitely generated $A$-modules. The relation type of $I$ with respect to $M$, denoted by ${\bf rt}\,(I;M)$, is the maximal degree in a minimal generating set of relations of the Rees module ${\bf R}(I;M)=\oplus_{n\geq 0}I^nM$. It is a well-known invariant that gives a first measure of the complexity of ${\bf R}(I;M)$. To help to measure this complexity, we introduce the sifted type of ${\bf R}(I;M)$, denoted by ${\bf st}\,(I;M)$, a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of ${\bf R}(I;M)$. Just as the relation type ${\bf rt}\,(I;M/N)$ is closely related to the strong Artin-Rees number ${\bf s}\,(N,M;I)$, it turns out that the sifted type ${\bf st}\,(I;M/N)$ is closely related to the medium Artin-Rees number ${\bf m}\,(N,M;I)$, a new invariant which lies in between the weak and the strong Artin-Rees numbers of $(N,M;I)$. We illustrate the meaning, interest and mutual connection of ${\bf m}\,(N,M;I)$ and ${\bf st}\,(I;M)$ with some examples.

In the Internet of Vehicles (IoV), Age of Information (AoI) has become a vital performance metric for evaluating the freshness of information in communication systems. Although many studies aim to minimize the average AoI of the system through optimized resource scheduling schemes, they often fail to adequately consider the queue characteristics. Moreover, the vehicle mobility leads to rapid changes in network topology and channel conditions, making it difficult to accurately reflect the unique characteristics of vehicles with the calculated AoI under ideal channel conditions. This paper examines the impact of Doppler shifts caused by vehicle speeds on data transmission in error-prone channels. Based on the M/M/1 and D/M/1 queuing theory models, we derive expressions for the Age of Information and optimize the system's average AoI by adjusting the data extraction rates of vehicles (which affect system utilization). We propose an online optimization algorithm that dynamically adjusts the vehicles' data extraction rates based on environmental changes to ensure optimal AoI. Simulation results have demonstrated that adjusting the data extraction rates of vehicles can significantly reduce the system's AoI. Additionally, in the network scenario of this work, the AoI of the D/M/1 system is lower than that of the M/M/1 system.

In this paper, we investigate the global well-posedness and scattering theory for the defocusing nonlinear Schrödinger equation $iu_t + Δ_Ωu = |u|^αu$ in the exterior domain $Ω$ of a smooth, compact and strictly convex obstacle in $\mathbb{R}^3$.It is conjectured that in Euclidean space, if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^3))$ with $s_c := \frac{3}{2} - \frac{2}α \in (0, \frac{3}{2})$, then $u$ is global and scatters. In this paper, assuming that this conjecture holds, we prove that if $u$ is a solution to the nonlinear Schrödinger equation in exterior domain $Ω$ with Dirichlet boundary condition and satisfies $u \in L_t^\infty(I; \dot{H}^{s_c}_D(Ω))$ with $s_c \in \left[\frac{1}{2}, \frac{3}{2}\right)$, then $u$ is global and scatters. The main ingredients in our proof are the linear profile decompositions in the critical space $\dot{H}_D^{s_c}(Ω)$ and embed it to the nonlinear equation. Inspired by Killip-Visan-Zhang[Amer. J. Math. {\bf138} (2016)], we overcome the difficulty caused by the breakdown of the scaling and translation invariance. This allows us to utilize the concentration-compactness/rigidity argument of Kenig and Merle [Invent. Math. {\bf 166} (2006)]. To preclude the minimal counterexamples, we established the long-time Stricharrtz estimates and the frequency-localized Morawetz estimates.

Our goal in this paper is to investigate ergodicity of the white-forced wave equation on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The difficulties in our proof are twofold. On the one hand, compared to stochastic parabolic equation, stochastic wave equation is lack of smoothing effect and strongly dissipative mechanism. On the other hand, the whole line leads to the lack of compactness compared to bounded domain. In order to overcome the above difficulties, our proof is based on a new criterion for polynomial mixing established in [20], a new weight type Foiaş-Prodi estimate of wave equation on the whole line and weight energy estimates for stochastic wave equation.

This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $Γ$-convexity. Here $Γ$ is a tuple of free symmetric polynomials determining the geometry of a $Γ$-convex set. The paper introduces the notions of $Γ$-operator systems and $Γ$-ucp maps and establishes a Webster-Winkler type categorical duality between $Γ$-operator systems and $Γ$-convex sets. Next, a notion of an extreme point for $Γ$-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the $Γ$-extreme points of an operator $Γ$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $Γ(K).$ From this result, a Krein-Milman theorem for $Γ$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $Γ$-convex hull of the matricial positivity domain {(also known as a free semialgebraic set)} $D_p$ of a free symmetric polynomial $p$ is given. The approximation consists of a decreasing family of $Γ$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $Γ$-convex hull of $D_p.$

We study the Born and Inverse Born series for a special case of the second harmonic generation system of PDEs. We give a recursive formula for the forward operators and prove boundedness conditions that guarantee the convergence of the Born and inverse Born series. We also use fixed point theory to give explicit conditions for convergence of the Born series.

This article reviews Witten's gauge-theoretic approach to Khovanov homology from the perspective of Haydys-Witten instanton Floer theory. Expanding on Witten's arguments, we introduce a one-parameter family of instanton Floer homology groups $HF_θ(W^4)$, which, based on physical arguments, are expected to be topological invariants of the four-manifold $W^4$. In analogy to the original Yang-Mills instanton Floer theory, these groups are defined by the solutions of the $θ$-Kapustin-Witten equations on $W^4$ modulo instanton solutions of the Haydys-Witten equations that interpolate between them on the five-dimensional cylinder $\mathbb{R}_s \times W^4$. The relation to knot invariants arises when the four-manifold is the geometric blowup $W^4 = [X^3 \times \mathbb{R}^+, K]$ along a knot $K \subset X^3 \times \{0\}$ embedded in its three-dimensional boundary. The boundaries and corners of this manifold require the specification of boundary conditions that preserve the topological invariance of the construction and are fundamentally linked to various dimensional reductions of the Haydys-Witten equations. We provide a comprehensive discussion of these dimensional reductions and relate them to well-known gauge-theoretic equations in lower dimensions, including the $θ$-Kapustin-Witten equations, twisted extended Bogomolny equations, and twisted octonionic Nahm equations. Along the way, we record novel results on the elliptic regularity of the Haydys-Witten equations with twisted Nahm pole boundary conditions. The upshot of the article is a tentative definition of Haydys-Witten Floer theory and a precise restatement of Witten's conjecture: an equality between the Haydys-Witten Floer homology $HF^\bullet_{π/2}([S^3 \times \mathbb{R}^+, K])$ and Khovanov homology $Kh^\bullet(K)$.

We introduce and study a purely syntactic notion of lax cones and $(\infty,\infty)$-limits on finite computads in \texttt{CaTT}, a type theory for $(\infty,\infty)$-categories due to Finster and Mimram. Conveniently, finite computads are precisely the contexts in \texttt{CaTT}. We define a cone over a context to be a context, which is obtained by induction over the list of variables of the underlying context. In the case where the underlying context is globular we give an explicit description of the cone and conjecture that an analogous description continues to hold also for general contexts. We use the cone to control the types of the term constructors for the universal cone. The implementation of the universal property follows a similar line of ideas. Starting with a cone as a context, a set of context extension rules produce a context with the shape of a transfor between cones, i.e.~a higher morphism between cones. As in the case of cones, we use this context as a template to control the types of the term constructor required for universal property.

These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in São Paulo. We give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theory and combinatorics. These lectures are based on the series of our joint papers with Frits Beukers entitled \emph{Dwork crystals} (\cite{DCI,DCII,DCIII}). These methods also have applications in mathematical physics and arithmetic geometry (\cite{IN,Cartier0}), which we overview here towards the end. I am grateful to the organisers of both schools and to the participants of my courses whose questions stimulated writing these notes.

We present a simple geometric approach to studying the $L^p$ boundedness properties of Stein's spherical maximal operator, which does not rely on the Fourier transform. Using this, we recover a weak form of Stein's spherical maximal theorem.

We show that the action of two infinite commuting invariant rings of endomorphisms of a finite-dimensional virtually connected irreducible bi-module linearizes into a vector space over a definable field. The same holds if the action is merely by strongly commuting endogenies, modulo some finite katakernel.