In this paper we deal with the quantum dynamics of hyperbolic linear automorphisms of the torus, such as "Arnold's cat map", and study its behavior after Ehrenfest time. We show that, in the basis of wave packets, the associated propagator is well approximed by Birkhoff sums of nilrotations on the torus.

We introduce a novel class of coherent states, termed $\mathcal{W}^{(\bar{\alpha},\bar{\nu})}(z)$-coherent states, constructed using a deformed boson algebra based on the generalized factorial $[n]_{\alpha,\beta,\nu}!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analyzed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies.

This paper introduces a novel theoretical framework for identifying Lagrangian Coherent Structures (LCS) in manifolds with non-constant curvature, extending the theory to Finsler manifolds. By leveraging Riemannian and Finsler geometry, we generalize the deformation tensor to account for geodesic stretching in these complex spaces. The main result demonstrates the existence of invariant surfaces acting as LCS, characterized by dominant eigenvalues of the generalized deformation tensor. We discuss potential applications in astrophysics, relativistic fluid dynamics, and planetary science. This work paves the way for exploring LCS in intricate geometrical settings, offering new tools for dynamical system analysis.

We construct a weak version of the homological first Steenrod square, a natural transformation from the modulo two Chow group to the Chow group modulo two and two-torsion. No assumption is made on the characteristic of the base field. As an application, we generalize a theorem of Nikita Karpenko on the parity of the first Witt index of quadratic forms to the case of a base field of characteristic two.

We prove that the natural functor from the category of Chow motives of smooth projective quadrics with integral coefficients to the category with coefficients modulo 2 induces a bijection on the isomorphism classes of objects.

This paper extends the classical theory of Voronovskaya-type asymptotic expansions to generalized neural network operators defined on non-Euclidean and fractal domains. We introduce and analyze smooth operators activated by modified and generalized hyperbolic tangent functions, extending their applicability to manifold and fractal geometries. Key theoretical results include the preservation of density properties, detailed convergence rates, and asymptotic expansions. Additionally, we explore the role of fractional derivatives in defining neural network operators, which capture non-local behavior and are particularly useful for modeling systems with long-range dependencies or fractal-like structures. Our findings contribute to a deeper understanding of neural network operators in complex, structured spaces, offering robust mathematical tools for applications in signal processing on manifolds and solving partial differential equations on fractals.

This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions can be expanded in sums of pair products of $\,_{1}F_{2}$ functions. In special cases, the $\,_{3}F_{4}$ hypergeometric functions reduce to $\,_{2}F_{3}$ functions. Further special cases allow one to reduce the $\,_{2}F_{3}$ functions to $\,_{1}F_{2}$ functions, and the sums to products of $\,_{0}F_{1}$ (Bessel) and $\,_{1}F_{2}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,_{2}F_{1}$ functions, and $\,_{3}F_{2}$ functions into the realm of $\,_{p}F_{q}$ functions where $p<q$ for both the summand and terms in the series.

We prove that families of Calabi-Yau threefolds (CY3's) admit Bridgeland stability conditions when they are obtained via orbifolding from a family of CY3's admitting Bridgeland stability conditions. In particular, we prove that the quintic mirror admits Bridgeland stability conditions.

We study a class of quantum groups $U^-$ associated with quivers with loops and provide a categorification by constructing their associated Khovanov-Lauda-Rouquier algebras $R$. We prove that the indecomposable projective module over $R$ corresponds to the canonical basis of $U^-$. In the Jordan quiver case, we show that the cyclotomic Khovanov-Lauda-Rouquier algebras $R^\Lambda$ categorify the irreducible highest weight $U$-module $V(\Lambda)$.

In this research, Minkowski type functions which are constructed on certain probability distributions, are introduced. There are investigated differential, integral, and other properties of these functions.

Kinematics of rigid bodies can be analyzed in many different ways. The advantage of using Euler parameters is that the resulting equations are polynomials and hence computational algebra, in particular Gr\"obner bases, can be used to study them. The disadvantage of the Gr\"obner basis methods is that the computational complexity grows quite fast in the worst case in the number of variables and the degree of polynomials. In the present article we show how to simplify computations when the mechanism contains revolute joints. The idea is based on the fact that the ideal representing the constraints of the revolute joint is not prime. Choosing the appropriate prime component reduces significantly the computational cost. We illustrate the method by applying it to the well known Bennett's and Bricard's mechanisms, but it can be applied to any mechanism which has revolute joints.

Chow's Theorem and GAGA are renowned results demonstrating the algebraic nature of projective manifolds and, more broadly, projective analytic varieties. However, determining if a particular manifold is projective is not, generally, a simple task. The Kodaira Embedding Theorem provides an intrinsic characterization of projective varieties in terms of line bundles; in particular, it states that a manifold is projective if and only if it admits a positive line bundle. We prove only the 'if' implication in this paper, giving a sufficient condition for a manifold bundle to be embedded in projective space. Along the way, we prove several other interesting results. Of particular note is the Kodaira-Nakano Vanishing Theorem, a crucial tool for eliminating higher cohomology of complex manifolds, as well as Lemmas 6.2 and 6.1, which provide important relationships between divisors, line bundles, and blowups. Although this treatment is relatively self-contained, we omit a rigorous development of Hodge theory, some basic complex analysis results, and some theorems regarding Cech cohomology (including Leray's Theorem).

We explain the mathematical theory of the Input-Output method for carbon footprints computations.

The numerical computation of equilibrium reward gradients for Markov chains appears in many applications for example within the policy improvement step arising in connection with average reward stochastic dynamic programming. When the state space is large or infinite, one will typically need to truncate the state space in order to arrive at a numerically tractable formulation. In this paper, we derive the first computable a posteriori error bounds for equilibrium reward gradients that account for the error induced by the truncation. Our approach uses regeneration to express equilibrium quantities in terms of the expectations of cumulative rewards over regenerative cycles. Lyapunov functions are then used to bound the contributions to these cumulative rewards and their gradients from path excursions that take the chain outside the truncation set. Our numerical results indicate that our approach can provide highly accurate bounds with truncation sets of moderate size. We further extend our approach to Markov jump processes.

We investigate exploratory randomization for an extended linear-exponential-quadratic-Gaussian (LEQG) control problem in discrete time. This extended control problem is related to the structure of risk-sensitive investment management applications. We introduce exploration through a randomization of the control. Next, we apply the duality between free energy and relative entropy to reduce the LEQG problem to an equivalent risk-neutral LQG control problem with an entropy regularization term, see, e.g. Dai Pra et al. (1996), for which we present a solution approach based on Dynamic Programming. Our approach, based on the energy-entropy duality may also be considered as leading to a justification for the use, in the literature, of an entropy regularization when applying a randomized control.

Advanced societies are crucially dependent on critical infrastructure networks for the reliable delivery of essential goods and services. Hence, well-founded analyses concerning disruptions are needed to guide decisions that seek to ensure the performance of these networks in the face of failures caused by vulnerabilities to external hazards or technical malfunctions. In this setting, we develop a multicriteria decision analysis approach to support the formulation of cost-efficient portfolios of preventive reinforcement actions. Our approach is general in that it (i) allows for multiple objectives, such as those that represent the volume of traffic that is enabled between alternative origin-destination pairs in a transportation network, (ii) uses methods of probabilistic risk assessment to quantify the expected performance of the network, and (iii) solves optimization problems to identify those combinations of reinforcement actions that are cost-efficient in improving the performance of the network, given the available, possibly incomplete information about the relative importance of objectives. Our methodological contributions are illustrated by a case study on the analysis of railway switches at a representative Finnish railway station.

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric \textit{distortion} of the natural map from the former to the latter. This turns out to have both implications for semigroup theory and potential relevance for operator algebras associated to inverse semigroups. Along the way, we also answer a question of Lled\'{o} and Mart\'{i}nez by providing a more direct proof that an $E$-unitary inverse semigroup has Yu's Property A if its maximal group image does.

We present a tensorization algorithm for constructing tensor train representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function and a small set of sample points defining the domain of interest. Thus, it is particularly well-suited for machine learning models, where the domain of interest is naturally defined by the training dataset. We show that this approach can be used to enhance the privacy and interpretability of neural network models. Specifically, we apply our decomposition to (i) obfuscate neural networks whose parameters encode patterns tied to the training data distribution, and (ii) estimate topological phases of matter that are easily accessible from the tensor train representation. Additionally, we show that this tensorization can serve as an efficient initialization method for optimizing tensor trains in general settings, and that, for model compression, our algorithm achieves a superior trade-off between memory and time complexity compared to conventional tensorization methods of neural networks.

In this paper we compute the Chow--Witt rings of the classifying space ${\rm BSL}_n^c$ of quadratically oriented vector bundles of rank $n$. We also discuss the corresponding quadratically-oriented cobordism spectrum ${\rm MSL}^c$ and show that it is equivalent to $\rm{MSL}$ after inverting $\eta$.

This paper establishes an isomorphism between endomorphism algebras from the wrapped Fukaya category of a type of punctured surface, and the class of A-infinity algebras related to bordered knot Floer homology, called star algebras, which the author first constructed in her previous work. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that the star algebras are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.

The accurate estimation of human activity in cities is one of the first steps towards understanding the structure of the urban environment. Human activities are highly granular and dynamic in spatial and temporal dimensions. Estimating confidence is crucial for decision-making in numerous applications such as urban management, retail, transport planning and emergency management. Detecting general trends in the flow of people between spatial locations is neither obvious nor easy due to the high cost of capturing these movements without compromising the privacy of those involved. This research intends to address this problem by examining the movement of people in a SmartStreetSensors network at a fine spatial and temporal resolution using a Transfer Entropy approach.

Anisotropic mesh adaptation with Riemannian metrics has proven effective for generating straight-sided meshes with anisotropy induced by the geometry of interest and/or the resolved physics. Within the continuous mesh framework, anisotropic meshes are thought of as discrete counterparts to Riemannian metrics. Ideal, or unit, simplicial meshes consist only of simplices whose edges exhibit unit or quasi-unit length with respect to a given Riemannian metric. Recently, mesh adaptation with high-order (i.e., curved) elements has grown in popularity in the meshing community, as the additional flexibility of high-order elements can further reduce the approximation error. However, a complete and competitive methodology for anisotropic and high-order mesh adaptation is not yet available. The goal of this paper is to address a key aspect of metric-based high-order mesh adaptation, namely, the adequacy between a Riemannian metric and high-order simplices. This is done by extending the notions of unit simplices and unit meshes, central to the continuous mesh framework, to high-order elements. The existing definitions of a unit simplex are reviewed, then a broader definition involving Riemannian isometries is introduced to handle curved and high-order simplices. Similarly, the notion of quasi-unitness is extended to curved simplices to tackle the practical generation of high-order meshes. Proofs of concept for unit and (quasi-)isometric meshes are presented in two dimensions.

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^1$-topology, are $C^1$ conjugate.

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in $\mathbb R^3$, ellipsoids and bodies of revolution are characterized by the fact that all their planar sections have an axis of reflection. We prove Bezdek's conjecture in arbitrary dimension $n\geq 3$, assuming only that sections passing through a fixed point have an axis of reflection, provided that the complementary invariant subspaces are all parallel to a fixed hyperplane. The result is proven in both orthogonal and affine settings.

We show that every orientation-preserving circle homeomorphism is a composition of two conformal welding homeomorphisms, which implies that conformal welding homeomorphisms are not closed under composition. Our approach uses the log-singular maps introduced by Bishop. The main tool that we introduce are log-singular sets, which are zero capacity sets that admit a log-singular map that maps their complement to a zero capacity set.

In the Bayesian literature, a line of research called resolution of conflict is about the characterization of robustness against outliers of statistical models. The robustness characterization of a model is achieved by establishing the limiting behaviour of the posterior distribution under an asymptotic framework in which the outliers move away from the bulk of the data. The proofs of the robustness characterization results, especially the recent ones for regression models, are technical and not intuitive, limiting the accessibility and preventing the development of theory in that line of research. We highlight that the proof complexity is due to the generality of the assumptions on the prior distribution. To address the issue of accessibility, we present a significantly simpler proof for a linear regression model with a specific prior distribution corresponding to the one typically used. The proof is intuitive and uses classical results of probability theory. To promote the development of theory in resolution of conflict, we highlight which steps are only valid for linear regression and which ones are valid in greater generality. The generality of the assumption on the error distribution is also appealing; essentially, it can be any distribution with regularly varying or log-regularly varying tails. So far, there does not exist a result in such generality for models with regularly varying distributions. Finally, we analyse the necessity of the assumptions.

The problem considered is a multi-objective optimization problem, in which the goal is to find an optimal value of a vector function representing various criteria. The aim of this work is to develop an algorithm which utilizes the trust region framework with probabilistic model functions, able to cope with noisy problems, using inaccurate functions and gradients. We prove the almost sure convergence of the proposed algorithm to a Pareto critical point if the model functions are good approximations in probabilistic sense. Numerical results demonstrate effectiveness of the probabilistic trust region by comparing it to competitive stochastic multi-objective solvers. The application in supervised machine learning is showcased by training non discriminatory Logistic Regression models on different size data groups. Additionally, we use several test examples with irregularly shaped fronts to exhibit the efficiency of the algorithm.

We consider uniformly semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds. In particular, we assume that the manifolds are homeomorphic to each other and that their total absolute curvature is uniformly bounded. The purpose of this paper is a description of the Gromov-Hausdorff limits of such sequences. Our work extends earlier investigations by Burago and Shioya.

Let $C$ be a configuration of $n$ ovals in $\mathbb{S}^2$. We show that there is a Riemannian metric $g$ over $\mathbb{S}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for $n\leq k \leq \alpha_1 n$. We also have that $\lambda\operatorname{Vol}_g\left(\mathbb{S}^2\right) = \Theta(n)$. Additionally, assuming $C$ can be drawn as a topological minor of the $m\times m$ grid graph, we show that there is an infinitesimal perturbation of the round metric on $\mathbb{S}^2$ and a corresponding Laplacian eigenfunction $f$ with eigenvalue $\Theta(m^2)$ such that the zero set of $f$ is equivalent to $C$.

In this paper, we present event constraints as a new modeling paradigm that generalizes joint chance constraints from stochastic optimization to (1) enforce a constraint on the probability of satisfying a set of constraints aggregated via application-specific logic (constituting an event) and (2) to be applied to general infinite-dimensional optimization (InfiniteOpt) problems (i.e., time, space, and/or uncertainty domains). This new constraint class offers significant modeling flexibility in posing InfiniteOpt constraints that are enforced over a certain portion of their domain (e.g., to a certain probability level), but can be challenging to reformulate/solve due to difficulties in representing arbitrary logical conditions and specifying a probabilistic measure on a collection of constraints. To address these challenges, we derive a generalized disjunctive programming (GDP) representation of event constrained optimization problems, which readily enables us to pose logical event conditions in a standard form and allows us to draw from a suite of GDP solution strategies that leverage the special structure of this problem class. We also extend several approximation techniques from the chance constraint literature to provide a means to reformulate certain event constraints without the use of binary variables. We illustrate these findings with case studies in stochastic optimal power flow, dynamic disease control, and optimal 2D diffusion.

This is a chapter for a book in honor of Bernd Sturmfels and his contributions. We describe the contributions by Bernd Sturmfels and his collaborators in harnessing algebraic and combinatorial methods for analyzing chemical reaction networks. Topics explored include the steady-state variety, counting steady states, and the global attractor conjecture. We also recount some personal stories that highlight Sturmfels's long-lasting impact on this research area.

We introduce and study a notion of Castelnuovo-Mumford regularity suitable for scrolls obtained as projectivisations of sums of line bundles on $\mathbb P^m$. We show that this is a natural generalisation of the well known regularity on projective and multiprojective spaces and we prove Horrocks-type splitting criteria for vector bundles.

In this paper, we investigate the rate-distortion-perception function (RDPF) of a source modeled by a Gaussian Process (GP) on a measure space $\Omega$ under mean squared error (MSE) distortion and squared Wasserstein-2 perception metrics. First, we show that the optimal reconstruction process is itself a GP, characterized by a covariance operator sharing the same set of eigenvectors of the source covariance operator. Similarly to the classical rate-distortion function, this allows us to formulate the RDPF problem in terms of the Karhunen-Lo\`eve transform coefficients of the involved GPs. Leveraging the similarities with the finite-dimensional Gaussian RDPF, we formulate an analytical tight upper bound for the RDPF for GPs, which recovers the optimal solution in the "perfect realism" regime. Lastly, in the case where the source is a stationary GP and $\Omega$ is the interval $[0, T]$ equipped with the Lebesgue measure, we derive an upper bound on the rate and the distortion for a fixed perceptual level and $T \to \infty$ as a function of the spectral density of the source process.

We consider the Stokes-Darcy coupled problem, which models the interaction between free-flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element spaces, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both $H(\operatorname{div})$-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioners with respect to the physical parameters and mesh size.

We conduct an analysis of a one-dimensional linear problem that describes the vibrations of a connected suspension bridge. In this model, the single-span roadbed is represented as a thermoelastic Shear beam without rotary inertia. We incorporate thermal dissipation into the transverse displacement equation, following Green and Naghdi's theory. Our work demonstrates the existence of a global solution by employing classical Faedo-Galerkin approximations and three a priori estimates. Furthermore, we establish exponential stability through the application of the energy method. For numerical study, we propose a spatial discretization using finite elements and a temporal discretization through an implicit Euler scheme. In doing so, we prove discrete stability properties and a priori error estimates for the discrete problem. To provide a practical dimension to our theoretical findings, we present a set of numerical simulations.

In this note, we investigate the coboundaries of interval exchange transformations of 3 intervals (3-IETs). More precisely, we show that a differentiable function, whose integral and integral of its derivative is 0, is a coboundary for typical 3-IET if and only if the values at the endpoints of the domain are zero. This contrasts with 2-IETs, i.e., rotations, because every continuous function is a coboundary for every rotation. We also show the existence of rare counterexamples for both cases of possible values at the endpoints of the interval. We obtain our result by studying the properties of associated skew products.

In this paper, we use the modified energy method of Hunter, Ifrim, Tataru, and Wongto prove an improved quintic energy estimate for initial data small in $\dot H^1_x \times L^2_x$ for a wide class of quasilinear wave equations of Kirchhoff type. This allows us to make the first steps towards small data $H^{5/4}_x \times H^{1/4}_x$ local well-posedness. In particular, we prove an enhanced lifespan for corresponding solutions depending only on the $\dot H^{5/4}_x \times \dot H^{1/4}_x$ norm of the initial data as well as the existence of weak solutions for $H^{5/4}_x \times H^{1/4}_x$ initial data, again small in $\dot H^1_x \times L^2_x$. In contrast to previous modified energy results, the nonlinearity in these models depends on an $\dot H^1_x$ norm of the solution. This means a modified energy cannot be deduced algebraically by analyzing resonant interactions between wave packets since all spatial dependence is integrated out in the nonlinearity. Instead, the modified energy is determined as a Taylor series of incremental leading order terms.

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear wave equation in spatial dimension $d = 3$.

We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin (DG) phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this DG-IMEX method.

This paper presents a rigorous theoretical convergence analysis of the Wirtinger Flow (WF) algorithm for Poisson phase retrieval, a fundamental problem in imaging applications. Unlike prior analyses that rely on truncation or additional adjustments to handle outliers, our framework avoids eliminating measurements or introducing extra computational steps, thereby reducing overall complexity. We prove that WF achieves linear convergence to the true signal under noiseless conditions and remains robust and stable in the presence of bounded noise for Poisson phase retrieval. Additionally, we propose an incremental variant of WF, which significantly improves computational efficiency and guarantees convergence to the true signal with high probability under suitable conditions.

he pseudo-Frobenius numbers of a numerical semigroup $H$ are deeply connected to the structure of the defining ideal of its semigroup ring $k[H]$. In this paper, we resolve a certain conjecture related to this connection under the assumption that $k[H]/(t^a)$ is stretched, where $a$ is the multiplicity of $H$. Furthermore, we provide numerical conditions for the tangent cone of $k[H]$ to be Cohen-Macaulay.

Mirzakhani-Zograf proved the large genus asymptotic expansions of Weil-Petersson volumes and showed that the asymptotic coefficients are polynomials in $\mathbb Q[\pi^{-2},\pi^2]$. They also conjectured that these are actually polynomials in $\mathbb Q[\pi^{-2}]$. In this paper, we prove Mirzakhani-Zograf's conjecture.

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set can be represented as the intersection of a unit discrete cube and a face of the convex hull of the given integrally convex set. The first proof resorts to the Fenchel-type duality theorem in discrete convex analysis and the second is more elementary using Farkas' lemma.

2-Chern-Simons theory, or more commonly known as 4d BF-BB theory with gauged shift symmetry, is a natural generalization of Chern-Simons theory to 4-dimensional manifolds. It is part of the bestiary of higher-homotopy Maurer-Cartan theories. In this article, we present a framework towards the combinatorial quantization of 2-Chern-Simons theory on the lattice, taking inspiration from the work of Aleskeev-Grosse-Schomerus three decades ago. The central geometric input is the 2-truncation $\Gamma^2$ of the $\infty$-groupoid of simplices formed by the underlying lattice $\Gamma$. On such a "2-graph", we model states of 2-Chern-Simons holonomies as Crane-Yetter's \textit{measureable fields}. We show that the 2-Chern-Simons action endows the 2-graph states -- as well as their quantum 2-gauge symmetries -- the structure of a Hopf category, and that their associated higher $R$-matriex gives it a comonoidal {\it cobraiding} structure. This is an explicit realization of the categorical ladder proposal of Baez-Dolan, in the context of Lie group lattice 2-gauge theories. Moreover, we will also analyze the lattice 2-algebra on the graph $\Gamma$, and extract the observables of discrete 2-Chern-Simons theory from it.

In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of powers, and solutions of some difference equations, are introduced.

In the present manuscript we give a structure condition that allows for energy approximation and we provide two examples, enjoying such a condition, that cannot be dealt with the results in the literature.

This paper introduces a variation on an identity by Bruckman and Good. Using this identity, we are able to derive various well-known sums involving reciprocals of Fibonacci and Lucas numbers, including the case when the indices form an arithmetic progression. Moreover, we provide generalizations of the Millin series.

This paper presents a reinterpretation of a second-order linear recurrence sequence as a sequence of continuants derived from the convergents to a continued fraction. As a result, we are able to derive the generating function and Binet formula for continuants. Using this result, we provide a continuant-based formulation for well-known identities associated with Lucas sequences.

In a recent paper (2024) Camacho, C\'{a}novas, Mart\'{\i}nez-Legaz and Parra introduced bimonotone operators, i.e., operators $T$ such that both $T$ and $-T$ are monotone, and found some interesting applications to convex feasibility problems, especially in the case the operator is also paramonotone. In the present paper we drop paramonotonicity and examine the question of finding the most general form of a bimonotone operator in a Banach space. We show that any such operator can be reduced in some sense to a single-valued, skew symmetric linear operator. This facilitates the proof of some results involving these operators in applications.

Inspired by Lin-Pan-Wang (Comm. Pure Appl. Math., 65(6): 833-888, 2012), we continue to study the corresponding time-independent case of the Keller-Rubinstein-Sternberg problem. To be precise, we explore the asymptotic behavior of minimizers as $\varepsilon\to0$, for the functional $$\mathbf{E}_\varepsilon(u)= \int_{\Omega}\left(|\nabla u|^{2}+\frac{1}{\varepsilon^{2}} F(u)\right) d x$$under a mass-type constraint $\int_{\Omega}\rho(u)\, dx=m$, where $\rho:\mathbb{R}^k \to \mathbb{R}\in Lip(\mathbb{R}^k)$ is specialized as a density function with $m$ representing a fixed total mass. The potential function $F$ vanishes on two disjoint, compact, connected, smooth Riemannian submanifolds $N^{\pm}\subset\mathbb{R}^k$. We analyze the expansion of $\mathbf{E}_\varepsilon(u_\varepsilon)$ for various density functions $\rho$, identifying the leading-order term in the asymptotic expansion, which depends on the geometry of the domain and the energy of minimal connecting orbits between $N^+$ and $N^-$. Furthermore, we estimate the higher-order term under different geometric assumptions and characterize the convergence $u_{\varepsilon_i}\to v $ in the ${L}^1$ sense.

In this paper, by using the core EP inverse and the Drazin inverse which are two well known generalized inverses, a new class of matrices entitled core EP Drazin matrices (shortly, CEPD matrices) is introduced. This class contains the set of all EP matrices and also the set of normal matrices. Some algebraic properties of these matrices are also investigated. Moreover, some results about the Drazin inverse and the core EP inverse of partial isometries are derived, and using them, some conditions for which partial isometries are CEPD, are obtained. To illustrate the main results, some numerical examples are given.

In this paper, we consider the classical coupon collector problem with uniform probabilities. Since the seminal paper by P. Erd\"os and A. R\'enyi \cite{ErRe}, it is well-known that the renormalized number of attempts required to complete a collection of $n$ items distributed with uniform probability tends to a Gumbel distribution when $n$ goes to infinity. We propose to determine how fast this convergence takes place for a certain distance to be introduced by using the so-called generator approach of Stein's method. To do so, we introduce a semi-group similar to the classical Ornstein-Uhlenbeck semi-group and whose stationary measure is the standard Gumbel distribution. We give its essential properties and apply them to prove that the renormalized number of attempts converges to the Gumbel distribution at rate $\log n/n$.

We prove that the existence of Banach spaces with $L$-orthogonal sequences but without $L$-orthogonal elements is independent of the standard foundation of Mathematics, ZFC. This provides a definitive answer to \cite[Question~1.1]{AvilesMartinezRueda}. Generalizing classical $Q$-point ultrafilters, we introduce the notion of $Q$-measures and provide several results generalizing former theorems by Miller \cite{Miller} and Bartoszynski \cite{Bartoszynski} for $Q$-point ultrafilters.

We construct a Neural Network that approximates the matrix multiplication operator for any activation functions for which there exist a Neural Network which can approximate the scalar multiplication function. In particular, we use the Strassen algorithm for reducing the number of weights and layers needed for such Neural Network. This allows us to define another Neural Network for approximating the inverse matrix operator. Also, by relying on the Galerkin method, we apply those Neural Networks for resolving parametric elliptic PDEs for a whole set of parameters at the same time. Finally, we discuss the improvements with respect to prior results.

Our main objective in the present article is to study the factorization problem for triple-product $p$-adic $L$-functions, particularly in the scenarios when the defining properties of the $p$-adic $L$-functions involved have no bearing on this problem, although Artin formalism would suggest such a factorization. Our analysis, which is guided by the ETNC philosophy, recasts this problem as a comparison of diagonal cycles, Beilinson--Kato elements, and Heegner cycles.

In this short paper, we give a new proof of the classification theorem for noncompact ancient noncollapsed flows in $\mathbb{R}^3$ originally due to Brendle-Choi (Inventiones 2019). Our new proof directly establishes selfsimilarity by combining the fine neck theorem from our joint work with Hershkovits and the rigidity case of Hamilton's Harnack inequality.

We investigate the incompressible inhomogeneous magnetohydrodynamic equations in $\mathbb{R}^3$, under the assumptions that the initial density $\rho_0$ is only bounded, and the initial velocity $u_0$ and magnetic field $B_0$ exhibit critical regularities. In particular, the density is allowed to be piecewise constant with jumps. First, we establish the global-in-time well-posedness and large-time behavior of solutions to the Cauchy problem in the case that $\rho_0$ has small variations, and $u_0$ and $B_0$ are sufficiently small in the critical Besov space $\dot{B}^{3/p-1}_{p,1}$ with $1<p<3$. Moreover, the small variation assumption on $\rho_0$ is no longer required in the case $p=2$. Then, we construct a unique global Fujita-Kato solution under the weaker condition that $u_0$ and $B_0$ are small in $\dot{B}^{1/2}_{2,\infty}$ but may be large in $\dot{H}^{1/2}$. Additionally, we show a general uniqueness result with only bounded and nonnegative density, without assuming the $L^1(0,T;L^{\infty})$ regularity of the velocity. Our study systematically addresses the global solvability of the inhomogeneous magnetohydrodynamic equations with rough density in the critical regularity setting.

In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-\delta})$ (for any $\delta\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence in large dimension ($d\geq 4$) of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. The dimension $3$ appears to be critical. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation.

Low harvested energy poses a significant challenge to sustaining continuous communication in energy harvesting (EH)-powered wireless sensor networks. This is mainly due to intermittent and limited power availability from radio frequency signals. In this paper, we introduce a novel energy-aware resource allocation problem aimed at enabling the asynchronous accumulate-then-transmit protocol, offering an alternative to the extensively studied harvest-then-transmit approach. Specifically, we jointly optimize power allocation and time fraction dedicated to EH to maximize the average long-term system throughput, accounting for both data and energy queue lengths. By leveraging inner approximation and network utility maximization techniques, we develop a simple yet efficient iterative algorithm that guarantees at least a local optimum and achieves long-term utility improvement. Numerical results highlight the proposed approach's effectiveness in terms of both queue length and sustained system throughput.

The following learning problem arises naturally in various applications: Given a finite sample from a categorical or count time series, can we learn a function of the sample that (nearly) maximizes the probability of correctly guessing the values of a given portion of the data using the values from the remaining parts? Unlike the classical task of estimating conditional probabilities in a stochastic process, our approach avoids explicitly estimating these probabilities. We propose a non-parametric guessing function with a learning rate that is independent of the alphabet size. Our analysis focuses on a broad class of time series models that encompasses finite-order Markov chains, some hidden Markov chains, Poisson regression for count process, and one-dimensional Gibbs measures. Additionally, we establish a minimax lower bound for the rate of convergence of the risk associated with our guessing problem. This lower bound matches the upper bound achieved by our estimator up to a logarithmic factor, demonstrating its near-optimality.

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number theory, naturally extend the classical theory. This generalization transitions from the idele class group, which acts as the adelic analog of Galois groups, to a geometric framework associated with schemes and the ring of integers of global fields. This perspective provides a conceptual explanation for the role of the adele class space in the spectral realization of L-function zeros and identifies the idele class group as a generic point in this context. The sector $X_{\mathbb{Q}}$ of the adele class space corresponding to the Riemann zeta function gives the class field counterpart of the scaling topos. The main result is the construction of a functor mapping finite abelian extensions of $\mathbb{Q}$ to finite covers of $X_{\mathbb{Q}}$, with the monodromy of periodic orbits of length $\log p$ under the scaling action corresponding to the Galois action of the Frobenius at the prime p.

This paper focuses on the following problem: {\em what $G_k$-representation polynomials in Conner--Floyd $G_k$-representation algebra arise as fixed point data of $G_k$-manifolds?} where $G_k=(\mathbb{Z}_2)^k$. Using the idea of the GKM theory, we construct a $G_k$-labelled graph from a smooth closed manifold with an effective $G_k$-action fixing a finite set. Then we give an answer to above mentioned problem through two approaches: $G_k$-labelled graphs and $G_k$-representation theory. As an application, we give a complete classification of all 4-dimensional smooth closed manifolds with an effective $G_3$-action fixing a finite set up to equivariant unoriented bordism.

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy H\"{o}rmander conditions of Young type. Specifically, we first establish the pointwise sparse domination for these operators. Then, relying on the dyadic analysis, the Fefferman-Stein inequality with respect to arbitrary weights and the quantitative weighted Coifman-Fefferman inequality are demonstrated. We decouple the relationship between the number of commutators and the index $\varepsilon$, which essentially improved the results of P\'{e}rez and Rivera-R\'{\i}os (Israel J. Math., 2017). As applications, it is shown that all the aforementioned results can be applied to a wide range of operators, such as singular integral operators satisfying the $L^r$-H\"{o}rmander operators, $\omega$-Calder\'{o}n-Zygmund operators with $\omega$ satisfying a Dini condition, Calder\'{o}n commutators, homogeneous singular integral operators and Fourier multipliers.

The palette of a vertex v in a graph G is the set of colors assigned to the edges incident to v. The palette index of G is the minimum number of distinct palettes among the vertices, taken over all proper edge colorings of G. This paper presents results on the palette index of the Cartesian product $G \Box H$, where one of the factor graphs is a path or a cycle. Additionally, it provides exact results and bounds on the palette index of the Cartesian product of two graphs, where one factor graph is isomorphic to a regular or class 1 nearly regular graph.

This paper presents a modified general viscosity iterative process designed to solve variational inclusion and fixed point problems involving multi-valued quasi-nonexpansive and demi-contractive operators. The modified iterative process incorporates a viscosity approximation technique to handle the nonexpansive and contractive mappings, providing a more robust and efficient solution approach. By introducing an additional sequence of iterates, the algorithm iteratively approximates the desired solution by combining fixed point iteration with viscosity approximation. The proposed method has been proven to converge strongly to the solution of the given problem, ensuring the reliability and accuracy of the results.

In [Q. Liao et al., Commun. Math. Sci., 20(2022)], a linear-time Sinkhorn algorithm is developed based on dynamic programming, which significantly reduces the computational complexity involved in solving optimal transport problems. However, this algorithm is specifically designed for the Wasserstein-1 metric. We are curious whether the preceding dynamic programming framework can be extended to tackle optimal transport problems with different transport costs. Notably, two special kinds of optimal transport problems, the Sinkhorn ranking and the far-field reflector and refractor problems, are closely associated with the log-type transport costs. Interestingly, by employing series rearrangement and dynamic programming techniques, it is feasible to perform the matrix-vector multiplication within the Sinkhorn iteration in linear time for this type of cost. This paper provides a detailed exposition of its implementation and applications, with numerical simulations demonstrating the effectiveness and efficiency of our methods.

The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature $\beta_{\mathrm{c}}(q)$, where $q$ is the strength of the transverse field. Bj\"ornberg [Commun. Math. Phys., 232 (2013)] investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing $q$ and the spin-spin coupling $J$ in a proper manner, with fixed temperature. In this paper, we prove that the susceptibility diverges as $(\beta_{\mathrm{c}}(q)-\beta)^{-1}$ as $\beta\uparrow\beta_{\mathrm{c}}(q)$ for $d>4$ assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Bj\"ornberg & Grimmett [J. Stat. Phys., 136 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$).

We consider weak solutions of the equation $$-\Delta_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } \Omega,$$ where $H$ is in some cases called Finsler norm, $\Omega$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and $a(\cdot,u)$, $f(\cdot,u)$ are functions satisfying suitable assumptions. We exploit the Moser iteration technique to prove a Harnack type comparison inequality for solutions of the equation and a Harnack type inequality for solutions of the linearized operator. As a consequence, we deduce a Strong Comparison Principle for solutions of the equation and a strong Maximum Principle for solutions of the linearized operator.

We consider an ensemble of $2\times 2$ normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining probability distributions based on symmetry and statistical independence. The probability densities turn out to be power law with exponents that depend on the boundedness of the domain. For small spacings, $\sigma$, the probability density varies as $\sigma^{\nu}$, $\nu \geq 2$. The degree of level repulsion is a parameter of great interest as it makes a connection to quantum chaos; the lower bound of $\nu$ for our ensemble coincides with the Gaussian Unitary Ensemble. We believe that the systematic development presented here paves the way for further generalizations in the field of random matrix theory for PT-symmetric quantum systems.

We study curvatures of the groups of measure-preserving diffeomorphisms of non-orientable compact surfaces. For the cases of the Klein bottle and the real projective plane we compute curvatures, their asymptotics and the normalized Ricci curvatures in many directions. Extending the approach of V. Arnold, and A. Lukatskii we provide estimates of weather unpredictability for natural models of trade wind currents on the Klein bottle and the projective plane.

In this paper, we formulate an integral period map for the crystalline prismatization of the $p$-integral model of a Shimura variety $\widetilde{S}$ with good reduction. By analyzing reductions of this map, we derive a period map from the mod $p$ fiber $S$ of $\widetilde{S}$ to the moduli stack of 1-1 truncated local $G$-shtukas in the prismatic topology, which refines the zip period map of $S$ within this topology. Furthermore, we show that the pair $(\widetilde{S}, S)$ is associated with a double $G$-zip. Additionally, we introduce a framework of base reduction diagrams.

This paper presents an algorithmic framework for solving unconstrained stochastic optimization problems using only stochastic function evaluations. We employ central finite-difference based gradient estimation methods to approximate the gradients and dynamically control the accuracy of these approximations by adjusting the sample sizes used in stochastic realizations. We analyze the theoretical properties of the proposed framework on nonconvex functions. Our analysis yields sublinear convergence results to the neighborhood of the solution, and establishes the optimal worst-case iteration complexity ($\mathcal{O}(\epsilon^{-1})$) and sample complexity ($\mathcal{O}(\epsilon^{-2})$) for each gradient estimation method to achieve an $\epsilon$-accurate solution. Finally, we demonstrate the performance of the proposed framework and the quality of the gradient estimation methods through numerical experiments on nonlinear least squares problems.

We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato--Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the $\Phi^4_\delta$ measures in dimensions $\delta<\frac{14}{5}$, which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.

The Poisson-Boltzmann (PB) model is a widely used implicit solvent model in protein simulations. Although variants, such as the size modified PB and nonlocal modified PB models, have been developed to account for ionic size effects and nonlocal dielectric correlations, no existing PB variants simultaneously incorporate both, due to significant modeling and computational challenges. To address this gap, in this paper, a nonlocal size modified PB (NSMPB) model is introduced and solved using a finite element method for a protein with a three-dimensional molecular structure and an ionic solution containing multiple ion species. In particular, a novel solution decomposition is proposed to overcome the difficulties caused by the increased nonlinearity, nonlocality, and solution singularities of the model. It is then applied to the development of the NSMPB finite element solver, which includes an efficient modified Newton iterative method, an effective damping parameter selection strategy, and good selections of initial iterations. Moreover, the construction of the modified Newton iterative method is mathematically justified. Furthermore, an NSMPB finite element package is developed by integrating a mesh generation tool, a protein data bank file retrieval program, and the PDB2PQR package to simplify and accelerate its usage and application. Finally, numerical experiments are conducted on an ionic solution with four species, proteins with up to 11439 atoms, and irregular interface-fitted tetrahedral box meshes with up to 1188840 vertices. The numerical results confirm the fast convergence and strong robustness of the modified Newton iterative method, demonstrate the high performance of the package, and highlight the crucial roles played by the damping parameter and initial iteration selections in enhancing the method's convergence. The package will be a valuable tool in protein simulations.

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and expanded in coauthorship with Nicolai Reshetikhin.

Over the past two decades, shrinkage priors have become increasingly popular, and many proposals can be found in the literature. These priors aim to shrink small effects to zero while maintaining true large effects. Horseshoe-type priors have been particularly successful in various applications, mainly due to their computational advantages. However, there is no clear guidance on choosing the most appropriate prior for a specific setting. In this work, we propose a framework that encompasses a large class of shrinkage distributions, including priors with and without a singularity at zero. By reframing such priors in the context of reliability theory and wealth distributions, we provide insights into the prior parameters and shrinkage properties. The paper's key contributions are based on studying the folded version of such distributions, which we refer to as the Gambel distribution. The Gambel can be rewritten as the ratio between a Generalised Gamma and a Generalised Beta of the second kind. This representation allows us to gain insights into the behaviours near the origin and along the tails, compute measures to compare their distributional properties, derive consistency results, devise MCMC schemes for posterior inference and ultimately provide guidance on the choice of the hyperparameters.

This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Although symmetry can be restored by eliminating the partition of unity, this compromises convergence. Alternatively, multiplicative Vanka relaxation updates velocity and pressure sequentially within each patch, propagating updates multiplicatively across the domain and effectively addressing velocity-pressure coupling, ensuring a symmetric relaxation. We demonstrate that the generalized optimal AMG theory consistently provides accurate lower bounds on the convergence rate for SA-AMG applied to Stokes equations. These findings suggest potential avenues for further enhancement in AMG solver design for saddle-point systems.

We prove that there are infinitely many integers, which can represent as sum of a square-free integer and a prime $p$ with $||\alpha p+\beta||<p^{-1/10}$, where $\alpha$ is irrational.

In this work we study Stieltjes differential systems of which the derivators are allowed to change sign. This leads to the definition of the notion of \emph{function of controlled variation}, a characterization of precompact sets of $g$-continuous functions, and an explicit expression of $g$-exponential maps. Finally, we prove a Peano-type existence result and apply it to a model of fluid stratification on buoyant miscible jets and plumes.

In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have one solution under the Osgood condition, or even, the Montel-Tonelli condition. We also explore some results guaranteeing the existence of solution under these conditions. Along the way, we obtain some interesting properties for the Lebesgue-Stieltjes integral associated to a finite sum of nondecreasing and left-continuous maps, as well as a characterization of the pseudometric topologies defined by this type of maps.

We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $\mu$, we show that the notion of $\mu$ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.

This paper delves into the challenging issues in uncertain multi-objective optimization, where uncertainty permeates nonsmooth nonconvex objective and constraint functions. In this context, we investigate highly robust (weakly efficient) solutions, a solution concept defined by efficiency across all scenarios. Our exploration reveals important relationships between highly robust solutions and other robustness notions, including set-based and worst-case notions, as well as connections with proper and isolated efficiency. Leveraging modern techniques from variational analysis, we establish necessary and sufficient optimality conditions for these solutions. Moreover, we explore the robustness of multi-objective optimization problems in the face of various uncertain sets, such as ball, ellipsoidal, and polyhedral sets.

In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition and adjacent twin errors. The three codes he presented are length 3-digit codes with 2 information digits. Existence of a 4-digit code would imply the existence of 10 such disjoint 3-digit codes. Apparently, not even a pair of such disjoint 3-digit codes is known. The code developed herein, has the property that the knowledge of any two digits is sufficient to determine the entire codeword even though their positions were unknown. This fulfills Verhoeff's desire to eliminate "cyclic errors". Phonetic errors, where 2 digit pairs of the forms X0 and 1X are interchanged, are also eliminated.

We study modules and comodules for cohomological Hall algebras equipped with their vertex coproducts arising as objects with classical type stabilizer groups. Specifically we consider how classical type parabolic induction gives rise to actions of CoHAs of quivers with potential, of preprojective algberas, and of dimension zero sheaves on a smooth proper surface. In all cases the CoHA action is compatible with a localised (and vertex) coaction making the module a twisted Yetter-Drinfeld module over the CoHA with its localised braided bialgebra structure. In the case of dimension zero sheaves on a surface the action is related to an approach to the AGT conjecture in classical type using moduli stacks of orthosymplectic perverse coherent sheaves, a compactification of the stack of classical type bundles on a surface.

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.

We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under a fixed normal chart centered at the true parameter, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings-including spheres, the Stiefel manifold, fixed-rank matrices manifolds, and rank-one tensor manifolds-and, for Euclidean submanifolds, introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.

Snapshot compressive imaging (SCI) refers to the recovery of three-dimensional data cubes-such as videos or hyperspectral images-from their two-dimensional projections, which are generated by a special encoding of the data with a mask. SCI systems commonly use binary-valued masks that follow certain physical constraints. Optimizing these masks subject to these constraints is expected to improve system performance. However, prior theoretical work on SCI systems focuses solely on independently and identically distributed (i.i.d.) Gaussian masks, which do not permit such optimization. On the other hand, existing practical mask optimizations rely on computationally intensive joint optimizations that provide limited insight into the role of masks and are expected to be sub-optimal due to the non-convexity and complexity of the optimization. In this paper, we analytically characterize the performance of SCI systems employing binary masks and leverage our analysis to optimize hardware parameters. Our findings provide a comprehensive and fundamental understanding of the role of binary masks - with both independent and dependent elements - and their optimization. We also present simulation results that confirm our theoretical findings and further illuminate different aspects of mask design.

In this paper, we continue the quest to understand the interplay between wrapped Floer homology barcode and topological entropy. Wrapped Floer homology barcode entropy is defined as the exponential growth, with respect to the left endpoints, of the number of not-too-short bars in its barcode. We prove that, in the presence of a topologically transitive, locally maximal hyperbolic set for the Reeb flow on the boundary of a Liouville domain, the barcode entropy is bounded from below by the topological entropy restricted to the hyperbolic set.

The purpose of this article is twofold. Firstly, we use the next-token probabilities given by a language model to explicitly define a $[0,1]$-enrichment of a category of texts in natural language, in the sense of Bradley, Terilla, and Vlassopoulos. We consider explicitly the terminating conditions for text generation and determine when the enrichment itself can be interpreted as a probability over texts. Secondly, we compute the M\"obius function and the magnitude of an associated generalized metric space $\mathcal{M}$ of texts using a combinatorial version of these quantities recently introduced by Vigneaux. The magnitude function $f(t)$ of $\mathcal{M}$ is a sum over texts $x$ (prompts) of the Tsallis $t$-entropies of the next-token probability distributions $p(-|x)$ plus the cardinality of the model's possible outputs. The derivative of $f$ at $t=1$ recovers a sum of Shannon entropies, which justifies seeing magnitude as a partition function. Following Leinster and Schulman, we also express the magnitude function of $\mathcal M$ as an Euler characteristic of magnitude homology and provide an explicit description of the zeroeth and first magnitude homology groups.

This manuscript deals with a hierarchical control problem for Oldroyd equation under the Stackelberg-Nash strategy. The Oldroyd equation model is defined by non-regular coefficients, that is, they are bounded measurable functions. We assume that we can act in the dynamic of the system by a hierarchy of controls, where one main control (the leader) and several additional secondary control (the followers) act in order to accomplish their given tasks: controllability for the leader and optimization for followers. We obtain the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability with respect to the leader control, and the optimality system for leader control.

We study a Stackelberg strategy subject to the evolutionary linearized micropolar fluids equations in domains with moving boundaries, considering a Nash multi-objective equilibrium (non necessarily cooperative) for the "follower players" (as is called in the economy field) and an optimal problem for the leader player with approximate controllability objective. We will obtain the following main results : the existence and uniqueness of Nash equilibrium and its characterization, the approximate controllability of the linearized micropolar system with respect to the leader control and the existence and uniqueness of the Stackelberg-Nash problem, where the optimality system for the leader is given.

This paper is devoted to the study of the approximate controllability for a one-dimensional wave equation in domains with moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to reflection groups and rational Cherednik algebras. In particular, we describe how a twist acts on characters of Coxeter groups of type $B_n$ and $D_n$ and relate them to characters of mystic reflection groups. This is used to characterise twists of standard modules of rational Cherednik algebras as standard modules for certain braided Cherednik algebras. We introduce the coinvariant algebra of a mystic reflection group and use a twist to show that an analogue of Chevalley's theorem holds for these noncommutative algebras. We also discuss several cases where the negative braided Cherednik algebras are, and are not, isomorphic to rational Cherednik algebras.

Let $\dot{z}=f(z)$ be a holomorphic differential equation with center at $p$. In this paper we are concerned about studying the piecewise perturbation systems $\dot{z}=f(z)+\epsilon R^\pm(z,\overline{z}),$ where $R^\pm(z,\overline{z})$ are complex polynomials defined for $\pm\operatorname{Im}(z)> 0.$ We provide an integral expression, similar to an Abelian integral, for the period annulus of $p.$ The zeros of this integral control the bifurcating limit cycles from the periodic orbits of this annular region. This expression is given in terms of the conformal conjugation between $\dot{z}=f(z)$ and its linearization $\dot{z}=f'(p)z$ at $p$. We use this result to control the simultaneous bifurcation of limit cycles of the two annular periods of $\dot{z}={\rm i} (z^2-1)/2$, after both complex and holomorphic piecewise polynomial perturbations. In particular, as far as we know, we provide the first proof of the existence of non nested limit cycles for piecewise holomorphic systems.

We use an infinite $k$-ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with $N$ chips at the root. We describe the stable configurations. We calculate the number of fires for each vertex and the total number of fires. We study a sequence of the number of root fires for a given $k$ as a function of $N$ and study its properties. We do the same for the total number of fires.

In this paper, we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes, we observe that the right ideal structure of the semigroup is `embedded' inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of the Nambooripad's normal category \cite{cross} as our building block, which we call a \emph{connected category}. The main theorem of the paper describes a category equivalence between the category of left reductive regular semigroups and the category of {connected categories}. Then, we specialise our result to describe constructions of $\gl$-unipotent semigroups, right regular bands, inverse semigroups and arbitrary regular monoids. Exploiting the left-right duality of semigroups, we also construct right reductive regular semigroups and use that to describe the more particular subclasses of $\gr$-unipotent semigroups and left regular bands. Finally, we provide concrete (and rather simple) descriptions to the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.

In this paper, we investigate a Liouville-type theorem for the MHD equations using Saint-Venant type estimates. We show that \( (u, B) \) is a trivial solution if the growth of the \( L^s \) mean oscillation of the potential functions for both the velocity and magnetic fields are controlled. Our growth assumption is weaker than those previously known for similar results. The main idea is to refine the Saint-Venant type estimates using the Froullani integral.

In this paper, we investigate the phenomenon of coming down from infinity for (sub)critical cooperative branching processes with pairwise interactions (BPI processes for short) under appropriate conditions. BPI processes are continuous-time Markov chains that extend pure branching dynamics by incorporating additional mechanisms that allow both competition and cooperation events between pairs of individuals. Specifically, we focus on characterising the speed at which BPI processes evolve when starting from a very large initial population in the subcritical and critical cooperative regimes. Further, in the subcritical cooperative regime, we analyse their second-order fluctuations.

This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the system \( A\alpha = \mathbf{b} \), where \( A \) is an \( r \times n \) integral matrix and \( \mathbf{b} \) is an integral vector. We establish a relationship between special constant terms and shifted simplicial cones. This leads to the \texttt{SimpCone[S]} algorithm, which efficiently decomposes polyhedra into simplicial cones. Unlike traditional geometric triangulation methods, this algorithm is versatile for many choices of the strategy \( \texttt{S} \) and can also be applied to parametric polyhedra. The algorithm is useful for efficient volume computation of polytopes and can be applied to address various new research projects. Additionally, we apply our framework to unimodular cone decompositions. This extends the effectiveness of the newly developed \texttt{DecDenu} algorithm from denumerant cones to general simplicial cones.

The growing interest in cislunar space exploration in recent years has driven an increasing demand for efficient low-thrust missions to key cislunar orbits. These missions, typically possessing long thrust arcs, are particularly susceptible to operational uncertainties such as missed thrust events. Addressing these challenges requires efficient robust trajectory design frameworks during the preliminary mission design phase, where it is necessary to explore the solution space at a rapid cadence under evolving operational constraints. However, existing methods for missed thrust design rely on solving high-dimensional nonlinear programs, where generating effective initial guesses becomes challenging. To enhance computational efficiency, quality, and depth of robustness of solutions from global search, we compare two initial guess strategies: a baseline non-conditional global search, which samples from a static distribution with global support, and a conditional global search, which generates initial guesses conditioned on solutions to problems with less depth of robustness. The conditional search provides a sequential procedure for solving increasingly robust problems. We validate the improvements in the conditional approach using a low-thrust case study for the Lunar Gateway Power and Propulsion Element, where our results demonstrate that it significantly improves convergence rate and solution quality, highlighting its potential in preliminary robust trajectory design.

Let R be a commutative ring with identity and M be an R- module. The aim of this paper is to introduce and investigate the notions of nil-M-Noetherian and nil-M-Artinian modules as generalizations of Noetherian and Artinian modules. Also, in this regard we introduce nil versions of some algebraic concepts.

In this paper, we address a crucial but often overlooked issue in applying reinforcement learning (RL) to radio resource management (RRM) in wireless communications: the mismatch between the discounted reward RL formulation and the undiscounted goal of wireless network optimization. To the best of our knowledge, we are the first to systematically investigate this discrepancy, starting with a discussion of the problem formulation followed by simulations that quantify the extent of the gap. To bridge this gap, we introduce the use of average reward RL, a method that aligns more closely with the long-term objectives of RRM. We propose a new method called the Average Reward Off policy Soft Actor Critic (ARO SAC) is an adaptation of the well known Soft Actor Critic algorithm in the average reward framework. This new method achieves significant performance improvement our simulation results demonstrate a 15% gain in the system performance over the traditional discounted reward RL approach, underscoring the potential of average reward RL in enhancing the efficiency and effectiveness of wireless network optimization.

We provide a geometric-combinatorial model for the category of coherent sheaves on the weighted projective line of type (2,2,n) via a cylindrical surface with n marked points on each of its upper and lower boundaries, equipped with an order 2 self-homeomorphism. A bijection is established between indecomposable sheaves on the weighted projective line and skew-curves on the surface. Moreover, by defining a skew-arc as a self-compatible skew-curve and a pseudo-triangulation as a maximal set of distinct pairwise compatible skew-arcs, we show that pseudo-triangulations correspond bijectively to tilting sheaves. Under this bijection, the flip of a skew-arc within a pseudo-triangulation coincides with the tilting mutation. As an application, we prove the connectivity of the tilting graph for the category of coherent sheaves.

We provide a theory of the Fourier-Jacobi expansion for automorphic forms on simple adjoint groups of some general class. This theory respects the Heisenberg parabolic subgroups, whose unipotent radicals are the Heisenberg groups uniformly explained in terms of the notion of cubic norm structures. Based on this theory of the Fourier expansion, we prove that automorphic forms generating quaternionic discrete series representations automatically satisfy the moderate growth condition except for the cases of the group of $G_2$-type and special orthogonal groups of signature $(4,N)$. This should be called ``K\"ocher principle'' verified already for the case of the quaternion unitary group $Sp(1,q)$ for $q>1$ by the author. We also prove that every term of the Fourier expansion with a non-trivial central character for cusp forms generating quaternionic discrete series has no contribution by the discrete spectrum of the Jacobi group, which is a non-reductive subgroup of the Heisenberg parabolic subgroup. This is obtained by showing that generalized Whittaker functions of moderate growth for the Schr\"odinger representations are zero under some assumption of the separation of variables, which suffices for our purpose to establish such consequence.

Sensing and edge artificial intelligence (AI) are envisioned as two essential and interconnected functions in sixth-generation (6G) mobile networks. On the one hand, sensing-empowered applications rely on powerful AI models to extract features and understand semantics from ubiquitous wireless sensors. On the other hand, the massive amount of sensory data serves as the fuel to continuously refine edge AI models. This deep integration of sensing and edge AI has given rise to a new task-oriented paradigm known as integrated sensing and edge AI (ISEA), which features a holistic design approach to communication, AI computation, and sensing for optimal sensing-task performance. In this article, we present a comprehensive survey for ISEA. We first provide technical preliminaries for sensing, edge AI, and new communication paradigms in ISEA. Then, we study several use cases of ISEA to demonstrate its practical relevance and introduce current standardization and industrial progress. Next, the design principles, metrics, tradeoffs, and architectures of ISEA are established, followed by a thorough overview of ISEA techniques, including digital air interface, over-the-air computation, and advanced signal processing. Its interplay with various 6G advancements, e.g., new physical-layer and networking techniques, are presented. Finally, we present future research opportunities in ISEA, including the integration of foundation models, convergence of ISEA and integrated sensing and communications (ISAC), and ultra-low-latency ISEA.

Let $\mathcal{S}_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. A family $\mathcal{F}\subset \mathcal{S}_n$ is called intersecting if for every $\sigma,\pi\in \mathcal{F}$ there exists some $i\in [n]$ such that $\sigma(i)=\pi(i)$. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of permutations in $\mathcal{F}$, which deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for $n\geq 500$ the diversity of an intersecting subfamily of $\mathcal{S}_n$ is at most $(n-3)(n-3)!$, which is best possible.

This article generalizes Venkatesh's structure theorem for the derived Hecke action on the Hecke trivial cohomology of a division algebra over an imaginary quadratic field to division algebras over all number fields. In particular, we show that the Hecke trivial cohomology of a division algebra is a free module for the action of the strict derived Hecke algebra. Moreover, the strict derived Hecke algebra possesses a rational form that preserves the canonical rational structure on the cohomology during the derived Hecke action. The main ingredients in our improvement are a careful study of the congruence classes in the torsion cohomology of the arithmetic manifold and the author's new result on the reduction map in the $K$-theory of the ring of integers in number fields.

We present some new lower bound estimates for certain numbers in Laver table theory and introduce several related structures of interest.

This paper introduces a new type of simulation function within the framework of $b$-metric spaces, leading to the derivation of fixed-point results in this general setting. We explore the theoretical implications of these results and demonstrate their utility through a concrete example.

The paper at hand presents an in-depth investigation into the fatigue behavior of the high-strength aluminum alloy EN AW-7020 T6 using both experimental and numerical approaches. Two types of specimens are investigated: a dog-bone specimen subjected to cyclic loading in a symmetric strain-controlled regime, and a compact tension specimen subjected to repeated loading and unloading, which leads to damage growth from the notch tip. Experimental data from these tests are used to identify the different phases of fatigue. Subsequently, a plastic-damage model is developed, incorporating J2 plasticity with Chaboche-type mixed isotropic-kinematic hardening. A detailed investigation reveals that the Chaboche model must be blended with a suitable isotropic hardening and combined with a proper damage growth model to accurately describe cyclic fatigue including large plastic strains up to failure. Multiple back-stress components with independent properties are superimposed, and exponential isotropic hardening with saturation effects is introduced to improve alignment with experimental results. For damage, different stress splits are tested, with the deviatoric/volumetric split proving successful in reproducing the desired degradation in peak stress and stiffness. A nonlinear activation function is introduced to ensure smooth transitions between tension and compression. Two damage indices, one for the deviatoric part and one for the volumetric part, are defined, each of which is governed by a distinct trilinear damage growth function. The governing differential equation of the problem is regularized by higher-order gradient terms to address the ill-posedness induced by softening. Finally, the plasticity model is calibrated using finite element simulations of the dog-bone test and subsequently applied to the cyclic loading of the compact tension specimen.

In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator $ {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) $ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $\Delta$ and for any bounded continuous function $\varphi $ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=\varphi$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$.

We begin by addressing the time-domain full-waveform inversion using the adjoint method. Next, we derive the scaled boundary semi-weak form of the scalar wave equation in heterogeneous media through the Galerkin method. Unlike conventional formulations, the resulting system incorporates variable density and two additional terms involving its spatial derivative. As a result, the coefficient matrices are no longer constant and depend on the radial coordinate, rendering the common solution methods inapplicable. Thus, we introduce a radial discretization scheme within the framework of the scaled boundary finite element method. We employ finite difference approximation, yet the choice underlying our ansatz is made for demonstration purposes and remains flexible. Next, we introduce an algorithmic condensation procedure to compute the dynamic stiffness matrices on the fly. Therefore, we maneuver around the need to introduce auxiliary unknowns. As a result, the optimization problem is structured in a two-level hierarchy. We obtain the Fr\'echet kernel by computing the zero-lag cross-correlations of the forward and adjoint wavefields, and solve the minimization problems iteratively by moving downhill on the cost function hypersurface through the limited-memory BFGS algorithm. The numerical results demonstrate the effectiveness and robustness of the new formulation and show that using the simplified differential equation along with the conventional formulation is highly inferior to applying the complete form of the differential equation. This approach effectively decomposes the computational load into independent local problems and a single coupled global system, making the solution method highly parallelizable. We demonstrate that, with a simple OpenMP implementation using 12 threads on a personal laptop, the new formulation outperforms the existing approach in terms of computation time.

Recent advancements in smart radio environment technologies aim to enhance wireless network performance through the use of low-cost electromagnetic (EM) devices. Among these, reconfigurable intelligent surfaces (RIS) have garnered attention for their ability to modify incident waves via programmable scattering elements. An RIS is a nearly passive device, in which the tradeoff between performance, power consumption, and optimization overhead depend on how often the RIS needs to be reconfigured. This paper focuses on the metaprism (MTP), a static frequency-selective metasurface which relaxes the reconfiguration requirements of RISs and allows for the creation of different beams at various frequencies. In particular, we address the design of an ideal MTP based on its frequency-dependent reflection coefficients, defining the general properties necessary to achieve the desired beam steering function in the angle-frequency domain. We also discuss the limitations of previous studies that employed oversimplified models, which may compromise performance. Key contributions include a detailed exploration of the equivalence of the MTP to an ideal S-parameter multiport model and an analysis of its implementation using Foster's circuits. Additionally, we introduce a realistic multiport network model that incorporates aspects overlooked by ideal scattering models, along with an ad hoc optimization strategy for this model. The performance of the proposed optimization approach and circuits implementation are validated through simulations using a commercial full-wave EM simulator, confirming the effectiveness of the proposed method.

In this article, we give an explicit construction of the simple modules for both non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras over an algebraically closed field of characteristic not equal to $2$ under certain condition in terms of parameters in defining these algebras. As an application, we obtain a sufficient condition on the semi-simplicty of these cyclotomic Hecke-Clifford superalgebras via a dimension comparison. As a byproduct, both generic non-degenerate and degenerate cyclotomic Hecke-Clifford superalgebras are shown to be semisimple.

In this paper, we consider random walks in Dirichlet random environment (RWDE) on $\mathbb{Z}^2$. We prove that, if the RWDE is recurrent (which is strongly conjectured when the weights are symmetric), then there does not exist any invariant measure for the process viewed from the particle which is absolutely continuous with respect to the static law of the environment. Besides, if the walk is directional transient and under condition $\mathbf{(T')}$, we prove that there exists such an invariant probability measure if the trapping parameter verifies $\kappa > 1$ or after acceleration of the process by a local function of the environment. This gives strong credit to a conjectural classification of cases of existence or non-existence of the invariant measure for two dimensional RWDE. The proof is based on a new identity, stated on general finite graphs, which is inspired by the representation of the $\star$-VRJP, a non-reversible generalization of the Vertex reinforced Jump Process, in terms of random Schr\"odinger operators. In the case of RWDE on 1D graph, the previous identity entails also a discrete analogue of the Matsumoto-Yor property for Brownian motion.

Modularity is a graph parameter measuring how clearly the set of graph vertices may be partitioned into subsets of high edge density. It indicates the presence of community structure in the graph. We study its value for a random preferential attachment model $G_n^h$ introduced by Barab\'asi and Albert in 1999. A graph $G_n^h$ is created from some finite starting graph by adding new vertices one by one. A new vertex always connects to $h\geq1$ already existing vertices and those are chosen with probability proportional to their current degrees. We prove that modularity of $G_n^h$ is with high probability upper bounded by a function tending to $0$ with $h$ tending to infinity. This resolves the conjecture of Prokhorenkova, Pralat and Raigorodskii from 2016. As a byproduct we obtain novel concentration results for the volume and the edge density parameters of subsets of $G_n^h$.

We prove that the category of Nachbin's compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we show that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (i) any finitely accessible category, (ii) any first-order definable class of structures, (iii) any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by D. Mundici: our result asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras. Our proof is independent of Mundici's theorem.

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Dr\`ezet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers.

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$ does not contain a rainbow copy of $F$, but the addition of any nonedge with any color on it would create a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in an $F$-rainbow saturated graph with order $n$. In this paper we proved several results on cycle rainbow saturation. For $n \geq 5$, we determined the exact value of $rsat(n,C_4)$. For $ n \geq 15$, we proved that $\frac{3}{2}n-\frac{5}{2} \leq rsat(n,C_{5}) \leq 2n-6$. For $r \geq 6$ and $n \geq r+3$, we showed that $ \frac{6}{5}n \leq rsat(n,C_r) \leq 2n+O(r^2)$. Moreover, we establish better lower bound on $C_r$-rainbow saturated graph $G$ while $G$ is rainbow.

In this article, we describe primitive quandles with the help of primitive permutation groups. As a consequence, we enumerate finite non-affine primitive quandles up to order $4096$.

In 1694, Gregory and Newton discussed the problem to determine the kissing number of a rigid material ball. This problem and its higher dimensional generalization have been studied by many mathematicians, including Minkowski, van der Waerden, Hadwiger, Swinnerton-Dyer, Watson, Levenshtein, Odlyzko, Sloane and Musin. Recently, Li and Zong introduced and studied the generalized kissing numbers of convex bodies. As a continuation of this project, in this paper we obtain the exact generalized kissing numbers $\kappa_\alpha^*(B^n)$ of the $n$-dimensional balls for $3\le n\le 8$ and $\alpha =2\sqrt{3}-2$. Furthermore, the lattice kissing number of a four-dimensional cross-polytope is determined.

DNA data storage is now being considered as a new archival storage method for its durability and high information density, but still facing some challenges like high costs and low throughput. By reducing sequencing sample size for decoding digital data, minimizing DNA coverage depth helps lower both costs and system latency. Previous studies have mainly focused on minimizing coverage depth in uniform distribution channels under theoretical assumptions. In contrast, our work uses real DNA storage experimental data to extend this problem to log-normal distribution channels, a conclusion derived from our PCR and sequencing data analysis. In this framework, we investigate both noiseless and noisy channels. We first demonstrate a detailed negative correlation between linear coding redundancy and the expected minimum sequencing coverage depth. Moreover, we observe that the probability of successfully decoding all data in a single sequencing run increases and then decreases as coding redundancy rises, when the sample size is optimized for complete decoding. Then we extend the lower bounds of DNA coverage depth from uniform to log-normal noisy channels. The findings of this study provide valuable insights for the efficient execution of DNA storage experiments.

Lately, a novel swarm intelligence model, namely the consensus-based optimization (CBO) algorithm, was introduced to deal with the global optimization problems. Limited by the conditions of Ito's formula, the convergence analysis of the previous CBO finite particle system mainly focuses on the problem with smooth objective function. With the help of smoothing method, this paper achieves a breakthrough by proposing an effective CBO algorithm for solving the global solution of a nonconvex, nonsmooth, and possible non-Lipschitz continuous minimization problem with theoretical analysis, which dose not rely on the mean-field limit. We indicate that the proposed algorithm exhibits a global consensus and converges to a common state with any initial data. Then, we give a more detailed error estimation on the objective function values along the state of the proposed algorithm towards the global minimum. Finally, some numerical examples are presented to illustrate the appreciable performance of the proposed method on solving the nonsmooth, nonconvex minimization problems.

We define the branching ratio of the input tree of a node in a finite directed multigraph, prove that it exists for every node, and show that it is equal to the largest eigenvalue of the adjacency matrix of the induced subgraph determined by all upstream nodes. This real eigenvalue exists by the Perron-Frobenius Theorem for non-negative matrices. We motivate our analysis with simple examples, obtain information about the asymptotics for the limit growth of the input tree, and establish other basic properties of the branching ratio.

We propose an arbitrarily high-order globally divergence-free entropy stable nodal discontinuous Galerkin (DG) method to directly solve the conservative form of the ideal MHD equations using appropriate quadrature rules. The method ensures a globally divergence-free magnetic field by updating it at interfaces with a constraint-preserving formulation [5] and employing a novel least-squares reconstruction technique. Leveraging this property, the semi-discrete nodal DG scheme is proven to be entropy stable. To handle the problems with strong shocks, we introduce a novel limiting strategy that suppresses unphysical oscillations while preserving the globally divergence-free property. Numerical experiments verify the accuracy and efficacy of our method.

We study the interaction between incompressible viscous fluids and multilayered elastic structures in a 3D/2D/3D framework, where a 3D fluid interacts with a 2D thin elastic layer, coupled to a 3D thick elastic solid. The system is driven by time-periodic boundary conditions involving Bernoulli pressure. We prove the existence of at least one time-periodic weak solution when the boundary pressure has a sufficiently small $L^2-$ norm. A key feature of our analysis is the assumption of viscoelasticity in the thick solid, which is crucial for obtaining diffusion estimates and ensuring energy stability. Without this assumption, weak solutions are established for the initial-value problem. Our results extend prior work on 2D/1D/2D configurations to the more complex 3D/2D/3D setting, providing new insights into multilayered fluid-structure interactions.

The uniqueness of global weak solutions to one-dimensional doubly degenerate cross-diffusion system is shown. The equations model the evolution of feeding bacterial populations in a malnourished environment. The key idea of the proof is applying anti-derivative of the sum of weak solutions to the system.

We study moduli spaces and moduli stacks for representations of associative algebras in Azumaya algebras, in rather general settings. We do not impose any stability condition and work over arbitrary ground rings, but restrict attention to the so-called Schur representations, where the only automorphisms are scalar multiplications. The stack comprises twisted representations, which are representations that live on the gerbe of splittings for the Azumaya algebra. Such generalized spaces and stacks appear naturally: For any rational point on the classical coarse moduli space of matrix representations, the machinery of non-abelian cohomology produces a modified moduli problem for which the point acquires geometric origin. The latter are given by representations in Azumaya algebras.

Nitsche's method is a numerical approach that weakly enforces boundary conditions for partial differential equations. In recent years, Nitsche's method has experienced a revival owing to its natural application in modern computational methods, such as the cut and immersed finite element methods. This study investigates Nische's methods based on an anisotropic weakly over-penalized symmetric interior penalty method for Poisson and Stokes equations on convex domains. As our primary contribution, we provide a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relationship between the Crouzeix and Raviart finite element space and the Raviart--Thomas finite element space. We present the error estimates in the energy norm on anisotropic meshes. We compared the calculation results for the anisotropic mesh partitions in the numerical experiments.

The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of $\mathcal{B}$, starting with $P_1$. The goal is to claim a subgraph isomorphic to $H$, with the first player achieving this declared the winner. A fundamental open question, persisting for over three decades, asks whether there exists a graph $H$ such that in the game $R(K_n, H)$, $P_1$ does not have a winning strategy in a bounded number of moves as $n \to \infty$. In this paper, we shift the focus to the variant $R(K_n \sqcup K_n, H)$, introduced by David, Hartarsky, and Tiba, where the board $K_n \sqcup K_n$ consists of two disjoint copies of $K_n$. We prove that there exist infinitely many graphs $H$ such that $P_1$ cannot win in $R(K_n \sqcup K_n, H)$ within a bounded number of moves through a concise proof. This perhaps provides evidence for the existence of examples to the above longstanding open problem.

For a positive integer $n$ let $\mathcal{X}_n$ be either the algebra $M_n$ of $n \times n$ complex matrices, the set $N_n$ of all $n \times n$ normal matrices, or any of the matrix Lie groups $\mathrm{GL}(n)$, $\mathrm{SL}(n)$ and $\mathrm{U}(n)$. We first give a short and elementary argument that for two positive integers $m$ and $n$ there exists a continuous spectrum-shrinking map $\phi : \mathcal{X}_n \to M_m$ (i.e.\ $\mathrm{sp}(\phi(X))\subseteq \mathrm{sp}(X)$ for all $X \in \mathcal{X}_n$) if and only if $n$ divides $m$. Moreover, in that case we have the equality of characteristic polynomials $k_{\phi(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ for all $X \in \mathcal{X}_n$, which in particular shows that $\phi$ preserves spectra. Using this we show that whenever $n \geq 3$, any continuous commutativity preserving and spectrum-shrinking map $\phi : \mathcal{X}_n \to M_n$ is of the form $\phi(\cdot)=T(\cdot)T^{-1}$ or $\phi(\cdot)=T(\cdot)^tT^{-1}$, for some $T\in \mathrm{GL}(n)$. The analogous results fail for the special unitary group $\mathrm{SU}(n)$, and slightly more elaborate versions hold for the spaces of semisimple elements in either $\mathrm{GL}(n)$ or $\mathrm{SL}(n)$, where a qualitatively new (and surprising) phenomenon arises: the map sending $SNS^{-1}$ to $S^{-1}NS$ for positive invertible $S$ and normal $N$ is also an example. As a consequence, we also recover (a strengthened version of) \v{S}emrl's influential characterization of Jordan automorphisms of $M_n$ via preserving properties.

This article studies the fluctuation behaviour of the stochastic point vortex model with common noise. Using the martingale method combined with a localization argument, we prove that the sequence of fluctuation processes converges in distribution to the unique probabilistically strong solution of a linear stochastic evolution equation. In particular, we establish the strong convergence from the stochastic point vortex model with common noise to the conditional McKean Vlasov equation.

We introduce a modification of the Tur\'an density of ordered graphs and investigate this graph parameter.

Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.

We present a stable characterization of on-diagonal upper bounds for heat kernels associated with regular Dirichlet forms on metric measure spaces satisfying the volume doubling property. Our conditions include integral bounds on the jump kernel outside metric balls, a variant of the Faber-Krahn inequality, a cutoff Sobolev inequality, and an integral control of inverse square volumes of balls with respect to the jump kernel. Crucially, we do not assume that the jump kernel has a density, and we show that these assumptions are essentially optimal.

Let $ \mathcal D\equiv G/K $ be an irreducible bounded symmetric domain. Using a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on locally compact Hausdorff groups, we study the non-vanishing of holomorphic automorphic forms on $ \mathcal D $ that are given by Poincar\'e series of polynomial type and correspond via the classical lift to the Poincar\'e series of certain $ K $-finite matrix coefficients of integrable discrete series representations of $ G $. We provide an example application of our results in the case when $ G=\mathrm{SU}(p,q) $ and $ K=\mathrm S(\mathrm U(p)\times\mathrm U(q)) $ with $ p\geq q\geq1 $.

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the following two results for each $\alpha \in \mathbb{R}$: First, for each $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and each $f,g \in L^{\infty}(X)$, for each $\gamma \in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor } f \cdot T^{\lfloor \gamma n \rfloor} g \] converge $\mu$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,\nu,S)$, and each $g \in L^{\infty}(Y)$, there exists a subset $Y_{g} \subset Y$ with $\nu(Y_{g})= 1$ so that for all $\gamma \in \mathbb{Q}$ and $\omega \in Y_{g}$, for any auxiliary $\sigma$-finite measure-preserving system $(X,\mu,T)$, and any $f \in L^{\infty}(X)$, the ``return-times" averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor} f \cdot S^{\lfloor \gamma n \rfloor } g(\omega) \] converge $\mu$-a.e. Moreover, in both cases the sets of convergence are identical for all $\gamma \in \mathbb{Q}$.

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with respect to a prime $ p $ can be explicitly described in terms of the Newton polygon of the polynomial $ g(x) $. Our results provide deeper insights into how the Newton polygon of a polynomial evolves under iteration and composition, with applications to the study of dynamical irreducibility, eventual stability, non-monogenity of tower of number fields, etc.

We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.

Various examples of folding curves in $R^{2}$ have been considered: dragons and other square curves, terdragons and other triangular curves, Peano-Gosper curves based on hexagons. They are self-avoiding. They form coverings of $R^{2}$, by one curve or by a small number of curves, which satisfy the local isomorphism property. They were used to define some fractals. We construct an example with similar properties in $R^{3}$.

We estimate the number of principal ideals $ I $ of norm $ \mathrm{N}(I) \leq x $ in the family of the simplest cubic fields. The advantage of our result is that it provides the correct order of magnitude for arbitrary $ x \geq 1 $, even when $ x $ is significantly smaller than the discriminant. In particular, it shows that there exist surprisingly many principal ideals of small norm.

We use the LLV algebra to describe the action of a finite order automorphism on the total cohomology of a manifold of OG10 type. As an application, we prove that no Enriques manifolds arise as \'etale quotients of hyper-K\"ahler manifolds of OG10 type. This answers a question raised by Pacienza and Sarti.

We consider a continuous-time financial market with an asset whose price is modeled by a linear stochastic differential equation with drift and volatility switching driven by a uniformly ergodic jump Markov process with a countable state space (in fact, this is a Black-Scholes model with Markov switching). We construct a multiplicative scheme of series of discrete-time markets with discrete-time Markov switching. First, we establish that the discrete-time switching Markov chains weakly converge to the limit continuous-time Markov process. Second, having this in hand, we apply conditioning on Markov chains and prove that the discrete-time market models themselves weakly converge to the Black-Scholes model with Markov switching. The convergence is proved under very general assumptions both on the discrete-time net profits and on a generator of a continuous-time Markov switching process.

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. We establish that the property $(\kappa)$ for a Tychonoff space $X$ is equivalent to Baireness of $B_1(X)$ and, hence, the Banakh property for $C_p(X)$ is equivalent to meagerness of $B_1(X)$. Thus, we obtain one characteristic of the Banach property for $C_p(X)$ through the property of space $X$.

The decycling number $\nabla(G)$ of a graph $G$ is the minimum number of vertices that must be removed to eliminate all cycles in $G$. The forest number $f(G)$ is the maximum number of vertices that induce a forest in $G$. So $\nabla(G) + f(G) = |V(G)|$. For the Cartesian product $T \,\square\, T'$ of trees $T$ and $T'$ it is proved that $\nabla(S_n \,\square\, S_{n'}) \leq \nabla(T \,\square\, T')$, thus resolving the conjecture of Wang and Wu asserting that $f(T \,\square\, T') \leq f(S_n \,\square\, S_{n'})$. It is shown that $\nabla(T \,\square\, T') \ge |V(T)| - 1$ and the equality cases characterized. For prisms over trees, it is proved that $\nabla(T\,\square\, K_2) = \alpha'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq \alpha'(G_1) \alpha'(G_2)$, where $\alpha'$ is the matching number.

Counting permutations of $[n]$ by the number of records, i.e. left-to-right maxima, is a classic problem in combinatorial enumeration. In the first volume of ``The Art of Computer Programming", Donald Knuth demonstrated its relevance for analysis of average case complexity of a basic algorithm for determining a maximum in a linear list of numbers. It is well known that the expected, and likely, number of those records in a {\it uniformly\/} random permutation is asymptotic to $\log n$. Cyril Banderier, Rene Beier, and Kurt Mehlhorn studied the case of a non-uniform random permutation, which is obtained from a generic permutation of $[n]$ by selecting its elements one after another independently with probability $p$, and permuting the selected elements uniformly at random. They proved that $E_n(p)$, the largest expected number of the maxima, is between $\text{const}\sqrt{n/p}$ and $O\bigl(\sqrt{(n/p)\log n}\bigr)$ if $p$ is fixed. For $p\gg 1/n$ and simultaneously $1-p\ge \text{const }n^{-1/2}\log n$, we prove that $E_n(p)$ is exactly of order $(1-p)\sqrt{n/p}$.

Data compression plays a key role in reducing storage and I/O costs. Traditional lossy methods primarily target data on rectilinear grids and cannot leverage the spatial coherence in unstructured mesh data, leading to suboptimal compression ratios. We present a multi-component, error-bounded compression framework designed to enhance the compression of floating-point unstructured mesh data, which is common in scientific applications. Our approach involves interpolating mesh data onto a rectilinear grid and then separately compressing the grid interpolation and the interpolation residuals. This method is general, independent of mesh types and typologies, and can be seamlessly integrated with existing lossy compressors for improved performance. We evaluated our framework across twelve variables from two synthetic datasets and two real-world simulation datasets. The results indicate that the multi-component framework consistently outperforms state-of-the-art lossy compressors on unstructured data, achieving, on average, a $2.3-3.5\times$ improvement in compression ratios, with error bounds ranging from $\num{1e-6}$ to $\num{1e-2}$. We further investigate the impact of hyperparameters, such as grid spacing and error allocation, to deliver optimal compression ratios in diverse datasets.

We study the space of conjugacy classes of subgroups of a compact Lie group G whose identity component is a torus, and consider how various invariants of subgroups behave as sheaves over this space. This feeds in to the author's programme to give algebraic models of rational G-equivariant cohomology theories. The methods are illustrated by making the outcome explicit for all toral subgroups of compact connected rank 2 groups.

In this note, pointwise best-possible (lower and upper) bounds on the set of copulas with a given value of the Gini's gamma coefficient are established. It is shown that, unlike the best-possible bounds on the set of copulas with a given value of other known measures such as Kendall's tau, Spearman's rho or Blomqvist's beta, the bounds found are not necessarily copulas, but proper quasi-copulas.

We introduce a scissors congruence $K$-theory spectrum which lifts the equivariant scissors congruence groups for compact $G$-manifolds with boundary, and we show that on $\pi_0$ this is the source of a spectrum level lift of the Burnside ring valued equivariant Euler characteristic of a compact $G$-manifold. We also show that the equivariant scissors congruence groups for varying subgroups assemble into a Mackey functor, which is a shadow of a conjectural higher genuine equivariant structure.

We give tightness criteria for random variables taking values in the space of all compact sets of cadlag real-valued paths, in terms of both the Skorohod J1 and M1 topologies. This extends earlier work motivated by the study of the Brownian web that was concerned only with continuous paths. In the M1 case, we give a natural extension of our tightness criteria which ensures that non-crossing systems of paths have weak limit points that are also non-crossing. This last result is exemplified through a rescaling of heavy tailed Poisson trees and a more general application to weaves.

Discrete Lossless Convexification (DLCvx) formulates a convex relaxation for a specific class of discrete-time non-convex optimal control problems. It establishes sufficient conditions under which the solution of the relaxed problem satisfies the original non-convex constraints at specified time grid points. Furthermore, it provides an upper bound on the number of time grid points where these sufficient conditions may not hold, and thus the original constraints could be violated. This paper extends DLCvx to problems with control pointing constraints. Additionally, it introduces a novel DLCvx formulation for mixed-integer optimal control problems in which the control is either inactive or constrained within an annular sector. This formulation broadens the feasible space for problems with pointing constraints. A numerical example is provided to illustrate its application.

For any arc-weighted oriented graph $D=(V(D), A(D),w)$, we write ${\rm fas}_w(D)$ to denote the minimum weight of a feedback arc set in $D$. In this paper, we consider upper bounds on ${\rm fas}_w(D)$ for arc-weight oriented graphs $D$ with bounded maximum degrees and directed girth. We obtain such bounds by introducing a new parameter ${\rm fasd}(D)$, which is the maximum integer such that $A(D)$ can be partitioned into ${\rm fasd}(D)$ feedback arc sets. This new parameter seems to be interesting in its own right. We obtain several bounds for both ${\rm fas}_w(D)$ and ${\rm fasd}(D)$ when $D$ has maximum degree $\Delta(D)\le \Delta$ and directed girth $g(D)\geq g$. In particular, we show that if $\Delta(D)\leq~4$ and $g(D)\geq 3$, then ${\rm fasd}(D) \geq 3$ and therefore ${\rm fas}_w(D)\leq \frac{w(D)}{3}$ which generalizes a tight bound for an unweighted oriented graph with maximum degree at most 4. We also show that ${\rm fasd}(D)\geq g$ and ${\rm fas}_w(D) \leq \frac{w(D)}{g}$ if $\Delta(D)\leq 3$ and $g(D)\geq g$ for $g\in \{3,4,5\}$ and these bounds are tight. However, for $g=10$ the bound ${\rm fasd}(D)\geq g$ does not always hold when $\Delta(D)\leq 3$. Finally we give some bounds for the cases when $\Delta$ or $g$ are large.

A novel behavior-epidemiology model, which considers $n$ heterogeneous behavioral groups based on level of risk tolerance and distinguishes behavioral changes by social and disease-related motivations (such as peer-influence and fear of disease-related hospitalizations), is developed. In addition to rigorously analyzing the basic qualitative features of this model, a special case is considered where the total population is stratified into two groups: risk-averse (Group 1) and risk-tolerant (Group 2). The two-group behavior model has three disease-free equilibria in the absence of disease, and their stability is analyzed using standard linearization and the properties of Metzler-stable matrices. Furthermore, the two-group model was calibrated and validated using daily hospitalization data for New York City during the first wave, and the calibrated model was used to predict the data for the second wave. Numerical simulations of the calibrated two-group behavior model showed that while the dynamics of the SARS-CoV-2 pandemic during the first wave was largely influenced by the behavior of the risk-tolerant individuals, the dynamics during the second wave was influenced by the behavior of individuals in both groups. It was also shown that disease-motivated behavioral changes had greater influence in significantly reducing SARS-CoV-2 morbidity and mortality than behavior changes due to the level of peer or social influence or pressure. Finally, it is shown that the initial proportion of members in the community that are risk-averse (i.e., the proportion of individuals in Group 1 at the beginning of the pandemic) and the early and effective implementation of non-pharmaceutical interventions have major impacts in reducing the size and burden of the pandemic (particularly the total SARS-CoV-2 mortality in New York City during the second wave).

We prove an elegant structure theorem for log de Rham-Witt sheaves with vanishing along an effective Cartier divisor $D$ defined in arXiv:2403.18763, answering a question of Shuji Saito during the Mainz conference and a question of Yigeng Zhao during a short visit of the author last summer. Our structural result for the log forms also lays the foundation for the study of Milnor $K$-theory with vanishing along $D$ in the paper to come.

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the \(T^{\rtimes}\)-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.

A Liouville-type result for the p-Laplacian on complete Riemannian manifolds is proved. As an application are present some results concerning complete non-compact hypersurfaces immersed in a suitable warped product manifold.

In this paper, we introduce abelian $(n,1)$-categories as an $n$-dimensional analog of abelian categories, where for \( n=1 \), it corresponds to ordinary abelian categories, and for \( n=\infty \), it represents stable DG-categories. We demonstrate that their homotopy categories become extriangulated categories and pretriangulated categories, and we establish a general theory of abelian $(n,1)$-categories. For example, the analog of the existance epi-mono factorizations of morphisms in abelian categories.

This paper proves $L^p$ decay estimates for Schr\"{o}dinger's and wave equations with scalar potentials on three-dimensional Riemannian manifolds. The main result regards small perturbations of a metric with constant negative sectional curvature. We also prove estimates on $\mathbb S^3$, the three-dimensional sphere, and $\mathbb H^3$, the three-dimensional hyperbolic space. Most of the estimates hold for the perturbed Hamiltonian $H=H_0+V$, where $H_0$ is the shifted Laplacian $H_0=-\Delta+\kappa_0$, $\kappa_0$ is the constant (or asymptotic) sectional curvature, and $V$ is a small scalar potential. The results are based on direct estimates of the wave propagator. All results hold in three space dimensions. The metric is required to have four derivatives.

We propose a concept of truncation for arbitrary smooth projective toric varieties and construct explicit cellular resolutions for nef truncations of their total coordinate rings. We show that these resolutions agree with the short resolutions of Hanlon, Hicks, and Lazarev, which were motivated by symplectic geometry, and we use our definition to exhibit nontrivial homology in the commutative algebraic analogue of their construction.

With the rapid expansion of space activities and the escalating accumulation of space debris, Space Domain Awareness (SDA) has become essential for sustaining safe space operations. This paper proposes a decentralized solution using satellite swarms and blockchain, where satellites (nodes) take on the roles of verifiers and approvers to validate and store debris-tracking data securely. Our simulations show that the network achieves optimal performance with around 30 nodes, balancing throughput and response time settling at 4.37 seconds. These results suggest that large-scale networks can be effectively managed by decoupling them into smaller, autonomous swarms, each optimized for specific tasks. Furthermore, we compare the performance of the decentralized swarm architecture with that of a fully shared role model and show significant improvements in scalability and response times when roles are decoupled.

Fluid antenna multiple access (FAMA), enabled by the fluid antenna system (FAS), offers a new and straightforward solution to massive connectivity. Previous results on FAMA were primarily based on narrowband channels. This paper studies the adoption of FAMA within the fifth-generation (5G) orthogonal frequency division multiplexing (OFDM) framework, referred to as OFDM-FAMA, and evaluate its performance in broadband multipath channels. We first design the OFDM-FAMA system, taking into account 5G channel coding and OFDM modulation. Then the system's achievable rate is analyzed, and an algorithm to approximate the FAS configuration at each user is proposed based on the rate. Extensive link-level simulation results reveal that OFDM-FAMA can significantly improve the multiplexing gain over the OFDM system with fixed-position antenna (FPA) users, especially when robust channel coding is applied and the number of radio-frequency (RF) chains at each user is small.

In this paper, we give a new approximate dynamic programming (ADP) method to solve large-scale Markov decision programming (MDP) problem. In comparison with many classic ADP methods which have large number of constraints, we formulate an alternating ADP (AADP) which have both small number of constraints and small number of variables by approximating the decision variables (instead of the objective functions in classic ADP) and write the dual of the exact LP. Also, to get the basis functions, we use kernel approximation instead of empirical choice of basis functions, which can efficiently learn nonlinear functions while retaining the expressive power. By treating option pricing as an large-scale MDP problem, we apply the AADP method to give an empirical proof that American call option will not be exercised earlier if the underlying stock has no dividend payment, which is a classic result proved by Black-Scholes model. We also make comparison of pricing options in high-dimensional with some benchmark option pricing papers which use the classic ADP to give upper and lower bound of the option price.

We consider Banach spaces $X$ that can be linearly lifted into their Lipschitz-free spaces $F(X)$ and, for a group $G$ acting on $X$ by linear isometries, we study the possible existence of $G$-equivariant linear liftings. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $F(X)$ is complemented in its bidual by an equivariant projection.

In this paper, we mainly study the interior $C^2$ estimates for a class of sum Hessian equations. We establish the interior estimates and the Pogorelov type estimates for $0<k<n$. If $k=n$, we derive a weaker Pogorelov type estimates.

In this article, a series of new inequalities involving the $q$-numerical radius for $n\times n$ tridiagonal, and anti-tridiagonal operator matrices has been established. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operator matrices. Additionally, we developed $q$-numerical radius inequalities for $n\times n$ circulant, skew circulant, imaginary circulant, imaginary skew circulant operator matrices. Important examples have been used to illustrate the developed inequalities. In this regard, analytical expressions and a numerical algorithm have also been employed to obtain the $q$-numerical radii. We also provide a concluding section, which may lead to several new problems in this area.

This is the abstract of a series of lectures given during the XIIIth School on Geometry and Physics, Bialystok (Poland), in July 2024. In this minicourse, we first examine the algebraic aspects of barycentric algebras. Then, we focus on various examples and applications, reviewing the pertinence of the barycentric algebra structure.

This paper concerns the isentropic compressible Navier-Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate which is decreasing with respect to the adiabatic exponent $\gamma>1$ provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy are required or boundary effects are ignored, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier-Stokes equations with slip boundary conditions.

As a fundamental and ubiquitous combinatorial notion, species has attracted sustained interest, generalizing from set-theoretical combinatorial to algebraic combinatorial and beyond. The Rota-Baxter algebra is one of the algebraic structures with broad applications from Renormalization of quantum field theory to integrable systems and multiple zeta values. Its interpretation in terms of monoidal categories has also recently appeared. This paper studies species of Rota-Baxter algebras, making use of the combinatorial construction of free Rota-Baxter algebras in terms of angularly decorated trees and forests. The notion of simple angularly decorated forests is introduced for this purpose and the resulting Rota-Baxter species is shown to be free. Furthermore, a twisted bialgebra structure, as the bialgebra for species, is established on this free Rota-Baxter species. Finally, through the Fock functor, another proof of the bialgebra structure on free Rota-Baxter algebras is obtained.

We present PDLP, a practical first-order method for linear programming (LP) designed to solve large-scale LP problems. PDLP is based on the primal-dual hybrid gradient (PDHG) method applied to the minimax formulation of LP. PDLP incorporates several enhancements to PDHG, including diagonal preconditioning, presolving, adaptive step sizes, adaptive restarting, and feasibility polishing. Our algorithm is implemented in C++, available in Google's open-source OR-Tools library, and supports multithreading. To evaluate our method, we introduce a new collection of eleven large-scale LP problems with sizes ranging from 125 million to 6.3 billion nonzeros. PDLP solves eight of these instances to optimality gaps of 1\% (with primal and dual feasibility errors of less than $10^{-8}$) within six days on a single machine. We also compare PDLP with Gurobi barrier, primal simplex, and dual simplex implementations. Gurobi barrier solves only three instances, exceeding our 1TB RAM limit on the other eight. While primal and dual simplex are more memory-efficient than the barrier method, they are slower and solve only three instances within six days. Compared with the conference version of this work (in: Advances in Neural Information Processing Systems 34 (NeurIPS 2021)), the key new contributions are: (i) feasibility polishing, a technique that quickly finds solutions that are approximately optimal but almost exactly feasible (without which only three of the eleven problems can be solved); (ii) a multithreaded C++ implementation available in Google OR-Tools; and (iii) a new collection of large-scale LP problems. Note that the conference version should be referred to for comparisons with SCS and ablation studies, which we do not repeat in this paper.

In this article, we introduce a formal definition of the concept of probability tree and conduct a detailed and comprehensive study of its fundamental structural properties. In particular, we define what we term an inductive probability measure and prove that such trees can be identified with these measures. Furthermore, we prove that probability trees are completely determined by probability measures on the Borel $\sigma$-algebra of the tree's body. We then explore applications of probability trees in several areas of mathematics, including probability theory, measure theory, and set theory. In the first, we show that the cumulative distribution of finitely many dependent and non-identically distributed Bernouli tests is bounded by the cumulative distribution of some binomial distribution. In the second, we establish a close relationship between probability trees and the real line, showing that Borel, measurable sets, and their measures can be preserved, as well as other combinatorial properties. Finally, in set theory, we establish that the null ideal associated with suitable probability trees is Tukey equivalent to the null ideal on $[0, 1]$. This leads to a new elementary proof of the fact that the null ideal of a free $\sigma$-finite Borel measure on a Polish space is Tukey equivalent with the null ideal of $\mathbb{R}$, which supports that the associated cardinal characteristics remain invariant across the spaces in which they are defined.

We introduce a new equivalence relation, named R-equivalence relation, on the set of colorings of an oriented knot diagram by a quandle. We determine the R-equivalence classes of colorings of a diagram of a torus knot by a quandle, called $\mathrm{Rot} \mathbb{E}^{2}$, under a certain condition.

We show that the integer group determinants for the general affine group of degree one, $GA(1,q)$ with $q=p^k$ a prime power, take the form $D=AB^{q-1},$ where $A$ is a $\mathbb Z_{q-1}$ integer group determinant and $B\equiv A \bmod q$. This generalizes the result for $k=1$. When $2^k-1$ is a Mersenne prime we show that this condition is both necessary and sufficient for $GA(1,2^k).$ The same is true for $GA(1,9)$ and $GA(1,27)$.

In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use these dynamical rigidity properties to characterise systems whose factor map to the maximal equicontinuous factor (MEF) is finite-to-one for a residual set, called "almost finite-to-one extensions", or a set of full measure, called "almost surely finite-to-one extensions". In the case of a $\sigma$-compact, locally compact, abelian acting group it is shown that frequently $(m+1)$-stable systems are equivalently characterised as almost $m$-to-one extensions of their MEF. Similarly, it is shown that a system is diam-mean $(m+1)$-equicontinuous if and only if it is an almost surely $m$-to-one extension of its MEF.

In this paper, we investigate receiver design for high frequency (HF) skywave massive multiple-input multiple-output (MIMO) communications. We first establish a modified beam based channel model (BBCM) by performing uniform sampling for directional cosine with deterministic sampling interval, where the beam matrix is constructed using a phase-shifted discrete Fourier transform (DFT) matrix. Based on the modified BBCM, we propose a beam structured turbo receiver (BSTR) involving low-dimensional beam domain signal detection for grouped user terminals (UTs), which is proved to be asymptotically optimal in terms of minimizing mean-squared error (MSE). Moreover, we extend it to windowed BSTR by introducing a windowing approach for interference suppression and complexity reduction, and propose a well-designed energy-focusing window. We also present an efficient implementation of the windowed BSTR by exploiting the structure properties of the beam matrix and the beam domain channel sparsity. Simulation results validate the superior performance of the proposed receivers but with remarkably low complexity.

This research introduces a new approach utilizing optimal control theory (OCT) to assess the Social Optimum (SO) of a vaccination game, navigating the intricate considerations of cost, availability, and distribution policies. By integrating an SIRS/V epidemic model with a behavior model, the study analyzes individual vaccination strategies. A unique optimal control framework, centered on vaccination costs, is proposed, diverging significantly from previous methods. Our findings confirm the effectiveness and feasibility of this approach in managing vaccination strategies. Moreover, we examine the underlying social dilemma of the vaccination game, investigating key parameters. By calculating the Nash equilibrium (NE) through the behavior model and determining the SO using our approach, we measure the Social Efficiency Deficit (SED), quantifying the overall cost gap between the NE and SO. Results indicate that an increased waning immunity rate exacerbates the social dilemma, although higher vaccination costs partially mitigate it. This research provides valuable insights into optimizing vaccination strategies amidst complex societal dynamics.

In problems that involve input parameter information gathered from multiple data sources with varying reliability, incorporating users' trust about different sources in decision-optimization models can potentially improve solution performance and reliability. In this work, we propose a novel multi-reference distributionally robust optimization (MR-DRO) framework, where the model inputs are uncertain and their probability distributions can be statistically inferred from multiple data sources. Via nonparametric data fusion, we construct a Wasserstein ambiguity set to minimize the worst-case expected value of a stochastic objective function, accounting for both uncertainty and unknown reliability of information sources. We reformulate the MR-DRO model as a linear program given linear objective and constraints in the original problem. We also incorporate a dynamic trust update mechanism that adjusts the trust for each source based on its performance over time. In addition, we introduce the concept of probability dominance to identify sources with dominant trust. Via solving instances of resource allocation and portfolio optimization, we demonstrate the effectiveness of the trust-informed MR-DRO approach compared to traditional optimization frameworks relying on a single data source. Our results highlight the significance of integrating (dynamic) user trust in decision making under uncertainty, particularly when given diverse and potentially conflicting input data.

Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $(\mathcal{M}_n)_{k\geq1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leq \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leq p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.

Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gr\"obner cone $\mathcal{C}_{\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\mathcal{A}}$ of relations among the cluster variables of $\mathcal{A}$ to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable $v$, the weight induced by taking compatibility degrees with $v$ belongs to $\mathcal{C}_{\mathcal{A}}$. This allows us to construct an explicit circular term order and prove a conjecture of Ilten, N\'ajera Ch\'avez, and Treffinger. Furthermore, we give explicit descriptions of the rays and lineality spaces of $\mathcal{C}_{\mathcal{A}}$ in terms of combinatorial models for cluster algebras of types $A_n$, $B_n$, $C_n$, $D_n$ with a special choice of frozen variables, and in the case of no frozen variables.

In this paper, we establish that the number of partitions of a natural number with positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say $2n$) and all red parts are even and lie within the interval $(2n,4n]$. This led us to derive a new representation for the third order mock theta function $f_3(q)$ and an analogue of the fundamental identity for the smallest part partition function Spt$(n)$, both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function $\phi_3(q)$.

In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in [Brenner-Constantin-Kadanoff-Schenkel-Venkataramani, 1999], was recently confirmed by [Glogi\'c-Sch\"orkhuber, 2024]. This paper aims to extend the radial stability to the nonradial setting, building on the finite-codimensional stability analysis in our previous work [Li-Zhou, 2024]. The main input is the mode stability of the linearized operator, whose nonlocal nature presents challenges for the spectral analysis. Besides a quantitative perturbative analysis for the high spherical classes, we adapted in the first spherical class the wave operator method of [Li-Wei-Zhang, 2020] for the fluid stability to localize the operator and remove the known unstable mode simultaneously. Our method provides localization beyond the partial mass variable and is independent of the explicit formula of the profile, so it potentially sheds light on other linear nonlocal problems.

This article examines Gaussian processes generated by monotonically modulating stationary kernels. An explicit isometry between the original and the modulated reproducing kernel Hilbert spaces is established, preserving eigenvalues and normalization. The expected number of zeros over the interval $[0,T]$ is shown to be exactly $\sqrt{-\ddot{K}(0)}(\theta(T)-\theta(0))$, where $\ddot{K}(0)$ is the second derivative of the kernel at zero and $\theta$ is the modulating function.

The properties of convex pentagonal monotiles in the 15 Type families and their tilings are summarized. The Venn diagrams of the 15 Type families are also shown.

The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph $G$ does not contain a theta submetric as an embedding, then $G$ has negative type. We show the converse: if a metric graph $G$ contains a theta, then it does not have negative type.

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This manuscript shows that this is indeed the case at least under broad conditions on ramification and deduce from them new properties of the Iwasawa Main Conjecture for modular forms. This manuscript is dedicated to the memory of Jo\"el Bella\"iche.

Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(\Omega))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.

We survey the implications of our joint work with E. Bru\`e and A. Pigati on the structure of blow-downs for a smooth, complete, Riemannian $4$-manifold with nonnegative Ricci curvature and Euclidean volume growth. Very imprecisely, any such manifold looks like a cone over a spherical space form at infinity. We present some open questions and discuss possible future directions along the way.

This work considers a multiobjective version of the unit commitment problem that deals with finding the optimal generation schedule of a firm, over a period of time and a given electrical network. With growing importance of environmental impact, some objectives of interest include CO2 emission levels and renewable energy penetration, in addition to the standard generation costs. Some typical constraints include limits on generation levels and up/down times on generation units. This further entails solving a multiobjective mixed integer optimization problem. The related literature has predominantly focused on heuristics (like Genetic Algorithms) for solving larger problem instances. Our major intent in this work is to propose scalable versions of mathematical optimization based approaches that help in speeding up the process of estimating the underlying Pareto frontier. Our contributions are computational and rest on two key embodiments. First, we use the notion of both epsilon constraints and adaptive weights to solve a sequence of single objective optimization problems. Second, to ease the computational burden, we propose a Mccormick-type relaxation for quadratic type constraints that arise due to the resulting formulation types. We test the proposed computational framework on real network data from [1,50] and compare the same with standard solvers like Gurobi. Results show a significant reduction in complexity (computational time) when deploying the proposed framework.

A defective $k$-coloring is a coloring on the vertices of a graph using colors $1,2, \dots, k$ such that adjacent vertices may share the same color. A $(d_1,d_2)$-\emph{coloring} of a graph $G$ is a defective $2$-coloring of $G$ such that any vertex colored by color $i$ has at most $d_i$ adjacent vertices of the same color, where $i\in\{1,2\}$. A graph $G$ is said to be $(d_1,d_2)$-\emph{colorable} if it admits a $(d_1,d_2)$-coloring. Defective $2$-coloring in planar graphs without $3$-cycles, $4$-cycles, and $6$-cycles has been investigated by Dross and Ochem, as well as Sittitrai and Pimpasalee. They showed that such graphs are $(0,6)$-colorable and $(3,3)$-colorable, respectively. In this paper, we proved that these graphs are also $(2,4)$-colorable.

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq 2e(G)\left(1-\frac{1}{\omega(G)}\right), $ where $\lambda_{1}$ and $\lambda_{2}$ are the largest and the second largest eigenvalues of the adjacency matrix of $G$, respectively. In this paper, we prove that for a sequence of random graphs the conjecture holds true with probability tending to one as the number of vertices tends to infinity.

We tackle the study of a type of local asymptotics, known as Mehler--Heine asymptotics, for some $q$--hypergeometric polynomials. Some consequences about the asymptotic behavior of the zeros of these polynomials are discussed. We illustrate the results with numerical examples.

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study the quasi-Gorensteinness of extriangulated categories. More precisely, we introduce the notion of quasi-$\xi$-projective and quasi-$\xi$-Gorenstein projective objects, investigate some of their properties and their behavior with respect to $\mathbb{E}$-triangles. Moreover, we give some equivalent characterizations of objects with finite quasi-$\xi$-Gorenstein projective dimension. As an application, our main results generalize Mashhad and Mohammadi's work in module categories.

In this paper, we study the global H\"older regularity of solutions to uniformly degenerate parabolic equations. We also study the convergence of solutions as time goes to infinity under extra assumptions on the characteristic exponents of the limit uniformly degenerate elliptic equations.

Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the characteristic polynomial for particular ensemble of $N \times N$ matrices is that it is expressed as the average of the power $N$ of the characteristic polynomial of some other ensemble of random matrices, now of size $k \times k$. With only a few exceptions, such dualities involve (the $\beta$ generalised) classical Gaussian, Laguerre and Jacobi ensembles Hermitian ensembles, the circular Jacobi ensemble, or the various non-Hermitian ensembles relating to Ginibre random matrices. In the case of unitary symmetry in the Hermitian case, they can be studied using the determinantal structure. The $\beta$ generalised case requires the use of Jack polynomial theory, and in particular Jack polynomial based hypergeometric functions. Applications to the computation of the scaling limit of various $\beta$ ensemble correlation and distribution functions are also reviewed. The non-Hermitian case relies on the particular cases of Jack polynomials corresponding to zonal polynomials, and their integration properties when their arguments are eigenvalues of certain matrices. The main tool to study dualities for moments of the spectral density, and generalisations, is the loop equations.

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e inequality, then for every $\varepsilon>0$, there exist a function $f_\varepsilon$ on $\Omega$ and an open set $U_\varepsilon\subset\Omega$ such that the following properties hold true: \begin{enumerate} \item ${\rm Cap}_1(U_\varepsilon)<\varepsilon$; \item $\|f-f_\varepsilon\|_{\BV(\Omega)}< \varepsilon$; \item $f^\vee\equiv f_\varepsilon^\vee$ and $f^\wedge\equiv f_\varepsilon^\wedge$ on $\Omega\setminus U_\varepsilon$; \item $f_\varepsilon^\vee$ is upper semicontinuous on $\Omega$, and $f_\varepsilon^\wedge$ is lower semicontinuous on $\Omega$. \end{enumerate} If the space $X$ is unbounded, then such an approximating function $f_\varepsilon$ can be constructed with the additional property that the uniform limit at infinity of both $f^\vee_\varepsilon$ and $f^\wedge_\varepsilon$ is $0$. Moreover, when $X=\R^d$, we show that the non-centered maximal function of $f_\varepsilon$ is continuous in $\Omega$.

The finite dimensional formulation of Dirichlet's problem for fluid bodies with ellipsoidal shape can be seen as a Hamiltonian system. I apply methods from symmetric bifurcation theory to the solution MacLaurin family of spheroidal solutions. I show that the only possible bifurcations are into types I, $S$ and adjoint $S$ ellipsoids recovering classic results.

Data from Internet of Things (IoT) sensors has emerged as a key contributor to decision-making processes in various domains. However, the quality of the data is crucial to the effectiveness of applications built on it, and assessment of the data quality is heavily context-dependent. Further, preserving the privacy of the data during quality assessment is critical in domains where sensitive data is prevalent. This paper proposes a novel framework for automated, objective, and privacy-preserving data quality assessment of time-series data from IoT sensors deployed in smart cities. We leverage custom, autonomously computable metrics that parameterise the temporal performance and adherence to a declarative schema document to achieve objectivity. Additionally, we utilise a trusted execution environment to create a "data-blind" model that ensures individual privacy, eliminates assessee bias, and enhances adaptability across data types. This paper describes this data quality assessment methodology for IoT sensors, emphasising its relevance within the smart-city context while addressing the growing need for privacy in the face of extensive data collection practices.

For a compact connected Riemannian manifold with smooth boundary, we establish an effective procedure, by which we can calculate all the coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential. In particular, by computing the full symbol of the Dirichlet-to-Neumann map we explicitly give the first four coefficients. They are spectral invariants, which provide precise information concerning the volume, curvatures, drifting function and potential.

Quantum search has emerged as one of the most promising fields in quantum computing. State-of-the-art quantum search algorithms enable the search for specific elements in a distribution by monotonically increasing the density of these elements until reaching a high density. This kind of algorithms demonstrate a theoretical quadratic speed-up on the number of queries compared to classical search algorithms in unstructured spaces. Unfortunately, the major part of the existing literature applies quantum search to problems which size grows exponnentialy with the input size without exploiting any specific problem structure, rendering this kind of approach not exploitable in real industrial problems. In contrast, this work proposes exploiting specific constraints of scheduling problems to build an initial superposition of states with size almost quadraticaly increasing as a function of the problem size. This state space reduction, inspired by the quantum walk algorithm, constructs a state superposition corresponding to all paths in a state-graph embedding spacing constraints between jobs. Our numerical results on quantum emulators highlights the potential of state space reduction approach, which could lead to more efficient quantum search processes by focusing on a smaller, more relevant, solution space.

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.

We study the vectorial stationary Schr\"odinger equation $-\Delta u+a\,U+b\,u=F,$ with a saturated nonlinearity $U=u/|u|$ and with some complex coefficients $(a,b)\in\mathbb{C}^2$. Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on $(a,b)$ and even when the source in the right hand side $F(x)$ is not vanishing for large values of $|x|.$ The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr\"{o}dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of ``solitons with compact support" claimed, without any proof, by several previous authors.

An accelerated deployment of renewable energy sources is crucial for a successful transformation of the current energy system, with wind energy playing a key role in this transition. This study addresses the integrated wind farm layout and cable routing problem, a challenging nonlinear optimization problem. We model this problem as an extended version of the Quota Steiner Tree Problem (QSTP), optimizing turbine placement and network connectivity simultaneously to meet specified expansion targets. Our proposed approach accounts for the wake effect - a region of reduced wind speed induced by each installed turbine - and enforces minimum spacing between turbines. We introduce an exact solution framework in terms of the novel Quota Steiner Tree Problem with interference (QSTPI). By leveraging an interference-based splitting strategy, we develop an advanced solver capable of tackling large-scale problem instances. The presented approach outperforms generic state-of-the-art mixed integer programming solvers on our dataset by up to two orders of magnitude. Moreover, we demonstrate that our integrated method significantly reduces the costs in contrast to a sequential approach. Thus, we provide a planning tool that enhances existing planning methodologies for supporting a faster and cost-efficient expansion of wind energy.

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter %, $\sigma$, which depends on the ratio of {growth to decay rates} of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned with is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this paper, we provide a computationally verifiable sufficient condition such that the inverse matrix can be well approximated in a low-rank TT format. It not only answers what kind of structured matrix whose inverse has the low-rank TT representation but also motivates us to develop an efficient TT-based method to compute the inverse matrix. Furthermore, we prove that the inverse matrix indeed has the low-rank tensor format for a class of large-scale structured matrices induced by differential operators involved in several PDEs, such as the Poisson, Boltzmann, and Fokker-Planck equations. Thus, the proposed algorithm is suitable for solving these PDEs with massive degrees of freedom. Numerical results on the Poisson, Boltzmann, and Fokker-Planck equations validate the correctness of our theory and the advantages of our methodology.

Integrated sensing and communication (ISAC) and ubiquitous connectivity are two usage scenarios of sixth generation (6G) networks. In this context, low earth orbit (LEO) satellite constellations, as an important component of 6G networks, is expected to provide ISAC services across the globe. In this paper, we propose a novel dual-function LEO satellite constellation framework that realizes information communication for multiple user equipments (UEs) and location sensing for interested target simultaneously with the same hardware and spectrum. In order to improve both information transmission rate and location sensing accuracy within limited wireless resources under dynamic environment, we design a multiple-satellite cooperative information communication and location sensing algorithm by jointly optimizing communication beamforming and sensing waveform according to the characteristics of LEO satellite constellation. Finally, extensive simulation results are presented to demonstrate the competitive performance of the proposed algorithms.

Negative refractive index materials have garnered widespread attention due to their anomalous electromagnetic properties. In this paper, we utilize complementing boundary conditions to conduct a priori estimates for Cauchy problems and derive the limiting absorption principle. Consequently, we establish the well-posedness of the transmission problem involving conventional materials and negative refractive index materials within a simulated two-dimensional periodic structure.

Cumulative Games were introduced by Larsson, Meir, and Zick (2020) to bridge some conceptual and technical gaps between Combinatorial Game Theory (CGT) and Economic Game Theory. The partizan ruleset {\sc Robin Hood} is an instance of a Cumulative Game, viz., {\sc Wealth Nim}. It is played on multiple heaps, each associated with a pair of cumulations, interpreted here as wealth. Each player chooses one of the heaps, removes tokens from that heap not exceeding their own wealth, while simultaneously diminishing the other player's wealth by the same amount. In CGT, the {\em temperature} of a {\em disjunctive sum} game component is an estimate of the urgency of moving first in that component. It turns out that most of the positions of {\sc Robin Hood} are {\em hot}. The temperature of {\sc Robin Hood} on a single large heap shows a dichotomy in behavior depending on the ratio of the wealths of the players. Interestingly, this bifurcation is related to Pingala (Fibonacci) sequences and the Golden Ratio $\phi$: when the ratio of the wealths lies in the interval $(\phi^{-1},\phi)$, the temperature increases linearly with the heap size, and otherwise it remains constant, and the mean values has a reciprocal property. It turns out that despite {\sc Robin Hood} displaying high temperatures, playing in the hottest component might be a sub-optimal strategy.

Meixner~(1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal--Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal--Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. In a special case, such a Fock space was already studied by Alpay--J\o rgensen--Seager--Volok~(2013) and Alpay--Porat~(2018). To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences.

We show that $X^\lambda$ is strongly homogeneous whenever $X$ is a non-separable zero-dimensional metrizable space and $\lambda$ is an infinite cardinal. This partially answers a question of Terada, and improves a previous result of the author. Along the way, we show that every non-compact weight-homogeneous metrizable space with a $\pi$-base consisting of clopen sets can be partitioned into $\kappa$ many clopen sets, where $\kappa$ is the weight of $X$. This improves a result of van Engelen.

We give a general proof of the strong consistency of the Maximum Likelihood Estimator for the case of independent non-identically distributed (i.n.i.d) data, assuming that the density functions of the random variables follow a particular set of assumptions. Our proof is based on the works of Wald~\cite{wald1949note}, Goel~\cite{goel1974note}, and Ferguson~\cite{ferguson2017course}. We use this result to prove the strong consistency of a Maximum Likelihood Estimator for Orbit Determination.

Let $k\leq n$ be positive integers and $\mathbb{Z}_{n}$ be the set of integers modulo $n$. A conjecture of Baranyai from 1974 asks for a decomposition of $k$-element subsets of $\mathbb{Z}_{n}$ into particular families of sets called "wreaths". We approach this conjecture from a new algebraic angle by introducing the key object of this paper, the wreath matrix $M$. As our first result, we establish that Baranyai's conjecture is equivalent to the existence of a particular vector in the kernel of $M$. We then employ results from representation theory to study $M$ and its spectrum in detail. In particular, we find all eigenvalues of $M$ and their multiplicities, and identify several families of vectors which lie in the kernel of $M$.

We obtain semiclassical resolvent estimates for the Schr{\"o}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h.Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{\"o}lder with respect to the radial variable and magnetic potentials which are H{\"o}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) .

Let $\iota_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $\iota_{k}(m,l)$ and show that $\iota_{k}(k,l)=\binom{k}{l}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.

Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require multiple decoders, increasing complexity. This work introduces enhanced polar decoding ($\mathsf{PD}^+$), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations in $\mathsf{PD}^+$ include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enable $\mathsf{PD}^+$ to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover, $\mathsf{PD}^+$ is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.

A compilation of new results on the asymptotic behaviour of the Humbert functions $\Psi_1$ and $\Psi_2$, and also on the Appell function $F_2$, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the $1D$ Glauber-Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.

Given a countable discrete amenable group, we investigate the conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and single contractions. Additionally, we develop a relative version of this characterization, identifying conditions under which the additive realization can be chosen within a specified target set. As an application, our results generalize key aspects of thermodynamic formalism, bridging the gap between additive and non-additive frameworks.

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.

This work presents an application of Dynnikov coordinates in geometric group theory. We describe the orbits and dynamics of the action of Dehn twists $t_c$ and $t_d$ in the Dynnikov coordinate plane for a thrice-punctured disc $M$, where $c$ and $d$ are simple closed curves with Dynnikov coordinates $(0,1)$ and $(0,-1)$, respectively. This action has an interesting geometric meaning as a piecewise linear $\mathbb{Z}^{2}$-automorphism preserving the shape of the linearity border fan.

Any anti-associative algebra gives rise to a Jacobi-Jordan algebra by [x, y] = xy + yx. This article aims to introduce the concept of "rhizaform algebras", which offer an approach to addressing anti-associativity. These algebras are defined by two operations whose sum is anti-associative, with the left and right multiplication operators forming bimodules of the sum of anti-associative algebras. This characterization parallels that of dendriform algebras, where the sum of operations preserves associativity. Additionally, the notions of O-operators and Rota-Baxter operators on anti-associative algebras are presented as tools to interpret rhizaform algebras. Notably, anti-associative algebras with nondegenerate Connes cocycles admit compatible rhizaform algebra structures.

We establish partial H\"older continuity of the gradient for equilibrium configurations of vectorial multidimensional variational problems, involving bulk and surface energies. The bulk energy densities are uniformly strictly quasiconvex functions with $p$-growth, $1<p< 2$, without any further structure conditions. The anisotropic surface energy is defined by means of an elliptic integrand $\Phi$ not necessarily regular.

In the $\tau$-tilting theory, there exist two classes of foundamental modules: indecomposable $\tau$-rigid modules and left finite bricks. In this paper, we prove the indecomposable $\tau$-rigid modules and the left finite bricks are uniquely determined by their Newton polytopes spanned by the dimensional vectors of their quotient modules. This is a kind of generalization of Gabriel's result that the indecomposable modules over path algebras of Dynkin quivers are uniquely determined by their dimensional vectors.

In this paper, we present a mathematical study of wave scattering by a hard elastic obstacle embedded in a soft elastic body in three dimensions. Our contributions are threefold. First, we characterize subwavelength resonances using the Dirichlet-to-Neumann map and an auxiliary variational form, showing that these resonances occur when the determinant of a specific matrix vanishes. Second, employing Gohberg-Sigal theory and Puiseux series expansions for multi-valued functions, we derive the asymptotic expansions of subwavelength resonant frequencies in the low-frequency regime through this explicit characterization. Finally, we provide a representation of the scattered field in the interior domain, where the enhancement coefficients are governed by the imaginary parts of the resonant frequencies. Additionally, we establish the transversal and longitudinal far-field patterns for the scattered field in the exterior domain.

In this paper, we propose an integrated sensing and communication (ISAC) system aided by the movable-antenna (MA) array, which can improve the communication and sensing performance via flexible antenna movement over conventional fixed-position antenna (FPA) array. First, we consider the downlink multiuser communication, where each user is randomly distributed within a given three-dimensional zone with local movement. To reduce the overhead of frequent antenna movement, the antenna position vector (APV) is designed based on users' statistical channel state information (CSI), so that the antennas only need to be moved in a large timescale. Then, for target sensing, the Cramer-Rao bounds (CRBs) of the estimation mean square error for different spatial angles of arrival (AoAs) are derived as functions of MAs' positions. Based on the above, we formulate an optimization problem to maximize the expected minimum achievable rate among all communication users, with given constraints on the maximum acceptable CRB thresholds for target sensing. An alternating optimization algorithm is proposed to iteratively optimize one of the horizontal and vertical APVs of the MA array with the other being fixed. Numerical results demonstrate that our proposed MA arrays can significantly enlarge the trade-off region between communication and sensing performance compared to conventional FPA arrays with different inter-antenna spacing. It is also revealed that the steering vectors of the designed MA arrays exhibit low correlation in the angular domain, thus effectively reducing channel correlation among communication users to enhance their achievable rates, while alleviating ambiguity in target angle estimation to achieve improved sensing accuracy.

Let $G$ be a finite simple graph, and let $I(G)$ denote its edge ideal. In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\text{HS}_i(I(G)^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\text{Ass}\,\text{HS}_i(I)\subseteq\text{Ass}\,\text{HS}_i(I^2)\subseteq\text{Ass}\,\text{HS}_i(I^3) \subseteq\cdots$. We prove that edge ideals possess both the 0th and 1st homological strong persistence properties. To this end, we explicitly describe the first homological shift algebra of $I(G)$ and show that $\text{HS}_1(I(G)^{k+1}) = I(G) \cdot \text{HS}_1(I(G)^k)$ for all $k \ge 1$. Finally, we conjecture that if $I(G)$ has a linear resolution, then $\text{HS}_i(I(G)^k)$ also has a linear resolution for all $k \gg 0$, and we present partial results supporting this conjecture.

This work focuses on developing high-order energy-stable schemes for wave-dominated problems in closed domains using staggered finite-difference summation-by-parts (SBP FD) operators. We extend the previously presented uniform staggered grid SBP FD approach to non-orthogonal curvilinear multi-block grids and derive new higher-order approximations. The combination of Simultaneous-Approximation-Terms (SAT) and projection method is proposed for the treatment of interface conditions on a staggered grid. This reduces approximation stiffness and mitigates stationary wave modes of pure SAT approach. Also, energy-neutral discrete Coriolis terms operators are presented. The proposed approach is tested using the linearized shallow water equations on a rotating sphere, a testbed relevant for ocean and atmospheric dynamics. Numerical experiments show significant improvements in capturing wave dynamics compared to collocated SBP FD methods.

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness of the corresponding stationary solutions. Second, by the weak convergence approach, we show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, we obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.

In ``Monk Algebras and Ramsey Theory,'' \emph{J. Log. Algebr. Methods Program.} (2022), Kramer and Maddux prove various representability results in furtherance of the goal of finding the smallest weakly representable but not representable relation algebra. They also pose many open problems. In the present paper, we address problems and issues raised by Kramer and Maddux. In particular, we prove that their Proposition 7 does not generalize, and we answer Problem 1.1 in the negative: relation algebra $1311_{1316}$ is not representable. Thus $1311_{1316}$ is a good candidate for the smallest weakly representable but not representable relation algebra. Finally, we give the first known finite cyclic group representations for relation algebras $31_{37}$, $32_{65}$, $1306_{1314}$, and $1314_{1316}$.

In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1} \left\{ \begin{aligned} \mathcal{L}u &= g(u), \quad u > 0 \quad \text{in} \quad \Omega, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial \nu} &= 0 \quad \text{in} \quad \partial \Omega \cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation*} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{\Omega }}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\bar{\Omega}}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $\Omega\cup \mathcal{N}$ is a bounded set with smooth boundary, $\lambda >0$ is a real parameter and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= \lambda u^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_\lambda)$ to derive the existence of weak solutions along with its $L^\infty$-regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established.

We show that if a locally finite Borel graph with quasitransitive amenable components admits a fractional perfect matching, it will admit a Borel fractional perfect matching. In particular, if a countable amenable quasitransitive graph admits a fractional perfect matching then its Bernoulli graph admits a Borel fractional perfect matching.

In this contribution, we are interested in the analysis of a semi-implicit time discretization scheme for the approximation of a parabolic equation driven by multiplicative colored noise involving a $p$-Laplace operator (with $p\geq 2$), nonlinear source terms and subject to Neumann boundary conditions. Using the Minty-Browder theorem, we are able to prove the well-posedness of such a scheme.

In this paper, we establish a general criterion for the positivity of the variance of the second chaotic component of local functionals of stationary vector-valued Gaussian fields. This criterion is formulated in terms of the spectral properties of the covariance function, without requiring assumptions of integrability or isotropy. It offers a simple and robust framework for analyzing variance asymptotics in such models. We apply this approach to the study of the nodal volume and the number of critical points of a Gaussian field, proving the positivity of the limiting variance under mild conditions on the covariance function. Additionally, we examine the asymptotics of the particular model of Euclidean random waves, deriving the central limit theorem through an analysis of the second and fourth chaotic components, thereby resolving an open question in large dimensions. As a byproduct, we condense and generalize many existing results on the volume of intersections of random waves, bypassing the need for traditional, intricate variance computations. Our findings shed new light on the second-chaos cancellation phenomenon from a spectral perspective and can be extended to any local, possibly singular, functional of Gaussian fields.

We introduce several families of entanglement-assisted (EA) Calderbank-Shor-Steane (CSS) codes derived from two distinct classes of low-density parity-check (LDPC) codes. We derive two families of EA quantum QC-LDPC codes, namely, the spatially coupled (SC) and the non-spatially coupled cases. These two families are constructed by tiling permutation matrices of prime and composite orders. We establish several code properties along with conditions for guaranteed girth for the proposed code families. The Tanner graphs of the proposed EA quantum QC-LDPC and EA quantum QC-SC-LDPC codes have girths greater than four, which is required for good error correction performance. Some of the proposed families of codes require only \textit{minimal} Bell pairs to be shared across the quantum transceiver. Furthermore, we construct two families of EA quantum QC-LDPC codes based on a single classical code, with Tanner graphs having girths greater than six, further improving the error correction performance. We evaluate the performance of these codes using both depolarizing and Markovian noise models to assess the random and burst error performance. Using a modified version of the sum-product algorithm over a quaternary alphabet, we show how correlated Pauli errors can be handled within the decoding setup. Simulation results show that nearly an order of improvement in the error correction performance can be achieved with quaternary decoder compared to binary decoder over the depolarizing and Markovian error channels, thereby generalizing the approach of EA quantum QC-LDPC code designs to work with both random and burst quantum error models, useful in practice.

June Huh and Matthew Stevens conjectured that the Hilbert-Poincar\'e series of the Chow ring of any matroid is a polynomial with only real zeros. We prove this conjecture for the class of uniform matroids. We also prove that the Chow polynomial and the augmented Chow polynomial of any maximally ranked poset has only real zeros.

A new method, dual-space cluster expansion, is proposed to study classical phases transitions in the continuum. It relies on replacing the particle positions as integration variables by the momenta of the relative displacements of particle pairs. Due to the requirement that the particles must be static, coupling via the momenta partitions the set of particles into a set of clusters, and transforms the partition function into a sum over the different cluster decompositions. This allows us to derive a formula for the density that finite clusters can carry in the infinite system. In a simplified example, we then demonstrate that in two and higher dimensions this density has a threshold, beyond which the particles form infinite clusters. The transition is accompanied by a singularity in the free energy. We also show that infinite clusters are always present in condensed phases, most likely submacroscopic in liquids and macroscopic in crystals.

The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Population genetics models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to model the complexity of real systems, and do not explicitly consider the effect of population heterogeneity. In order to allow for population heterogeneity, transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species.

For mathematical programs with complementarity constraints (MPCC), we refine the convergence analysis of the Scholtes regularization. Our goal is to relate nondegenerate C-stationary points of MPCC with nondegenerate Karush-Kuhn-Tucker points of its Scholtes regularization. We detected the following anomalies: (i) in a neighborhood of a nondegenerate C-stationary point there could be degenerate Karush-Kuhn-Tucker points of the Scholtes regularization; (ii) even if nondegenerate, they might be locally non-unique; (iii) if nevertheless unique, their quadratic index potentially differs from the C-index of the C-stationary point under consideration. Thus, a change of the topological type for Karush-Kuhn-Tucker points of the Scholtes regularization is possible. In particular, a nondegenerate minimizer of MPCC might be approximated by saddle points. In order to bypass the mentioned anomalies, an additional generic condition for nondegenerate C-stationary points of MPCC is identified. Then, we uniquely trace nondegenerate Karush-Kuhn-Tucker points of the Scholtes regularization and successively maintain their topological type.

In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.

In this paper, we focus on finding the global minimizer of a general unconstrained nonsmooth nonconvex optimization problem. Taking advantage of the smoothing method and the consensus-based optimization (CBO) method, we propose a novel smoothing iterative consensus-based optimization (SICBO) algorithm. First, we prove that the solution process of the proposed algorithm here exponentially converges to a common stochastic consensus point almost surely. Second, we establish a detailed theoretical analysis to ensure the small enough error between the objective function value at the consensus point and the optimal function value, to the best of our knowledge, which provides the first theoretical guarantee to the global optimality of the proposed algorithm for nonconvex optimization problems. Moreover, unlike the previously introduced CBO methods, the theoretical results are valid for the cases that the objective function is nonsmooth, nonconvex and perhaps non-Lipschitz continuous. Finally, several numerical examples are performed to illustrate the effectiveness of our proposed algorithm for solving the global minimizer of the nonsmooth and nonconvex optimization problems.

Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.

Let $K$ be a field and $\phi(z)\in K[z]$ be a polynomial. Define $\Phi(z) := \frac{1}{\phi(z)} \in K(z).$ For $n \in\mathbb{N}^* $, let the $n$-th iterate of $\Phi(z)$ be defined as $\Phi^{(n)}(z) = \underbrace{\Phi \circ \Phi \circ \cdots \circ \Phi}_{n \text{ times}}(z).$ We express the \(\Phi^{(n)}(z)\) in its reduced form as \( \Phi^{(n)}(z) = \frac{f_{n,\phi}(z)}{g_{n,\phi}(z)}, \) where \(f_{n,\phi}(z)\) and \(g_{n,\phi}(z)\) are coprime polynomials in \(K[z]\). A polynomial $\phi(z) \in K[z]$ is called inversely stable over $K$ if every $g_{n,\phi}(z)$ in the sequence $\{g_{n,\phi}(z)\}_{n=1}^\infty$ is irreducible in $K[z]$. This paper investigates the inverse stability of the binomials $\phi(z) = z^d + c$ over $K$.

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with respect to boundary deformation, and we illustrate how to find a domain in which the Neumann problem can be solved for any energy, if there is some freedom in the choice of the domain. This work is motivated by a Runge approximation result in the context of an inverse problem in point-source scattering with partial data.

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lov{\'a}sz--Schrijver SDP operator $\text{LS}_+$ applied to the fractional stable set polytope. In particular, we show that for every positive integer $\ell$, the smallest possible graph with $\text{LS}_+$-rank $\ell$ contains $3\ell$ vertices. This result is sharp and settles a conjecture posed by Lipt{\'a}k and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer $\ell$ there exists a vertex-transitive graph on $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.

Using techniques from many-body quantum theory, we propose a framework for representing the evolution of observables of measure-preserving ergodic flows through infinite-dimensional rotation systems on tori. This approach is based on a class of weighted Fock spaces $F_w(\mathcal H_\tau)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_\tau$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_\tau)$ and show that their Banach algebra spectra, $\sigma(F_w(\mathcal H_\tau))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_\tau$ of the Koopman operator acting on $\mathcal H_\tau$ are then lifted to rotation systems on these tori akin to the topological models of ergodic systems with pure point spectra in the Halmos--von Neumann theorem. Our scheme also employs a procedure for representing observables of the original system by polynomial functions on finite-dimensional tori in $\sigma(F_w(\mathcal H_\tau))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_\tau$. This leads to models for the Koopman evolution of observables on $L^2$ built from tensor products of finite collections of approximate Koopman eigenfunctions. Numerically, the scheme is amenable to consistent data-driven implementation using kernel methods. We illustrate it with applications to Stepanoff flows on the 2-torus and the Lorenz 63 system. Connections with quantum computing are also discussed.

The transition to a carbon-neutral energy system requires massive installation of renewable energy sources and economically feasible energy storage solutions. This study addresses these challenges by optimizing the design and control strategies of an energy system that meets the heat and electricity demands of a community. The proposed system integrates solar and wind power with energy storage, including seasonal thermal energy storage (STES) and battery, coupled via a heat pump. This approach enhances self-sufficiency and effectively mitigates seasonal mismatches. To model heat transfer between the storage and the ground in the STES system, we employ a multi-node lumped-parameter method. The optimization problem is formulated as a periodic optimal control problem, which is then transcribed into a nonlinear programming problem. To reduce computational complexity, we apply the averaging method, which significantly lowers the effort required to solve the problem. We apply this approach to a case study, where the economically optimized configuration results in a projected total energy cost per household of approximately 75 EUR/month over 30 years for both heat and electricity. This study demonstrates the feasibility of designing economically viable, autonomous energy communities in real-world scenarios, and provides a comprehensive optimization framework for designing system components and control strategies.

In this paper, a distance between the Gaussian Mixture Models(GMMs) is obtained based on an embedding of the K-component Gaussian Mixture Model into the manifold of the symmetric positive definite matrices. Proof of embedding of K-component GMMs into the manifold of symmetric positive definite matrices is given and shown that it is a submanifold. Then, proved that the manifold of GMMs with the pullback of induced metric is isometric to the submanifold with the induced metric. Through this embedding we obtain a general lower bound for the Fisher-Rao metric. This lower bound is a distance measure on the manifold of GMMs and we employ it for the similarity measure of GMMs. The effectiveness of this framework is demonstrated through an experiment on standard machine learning benchmarks, achieving accuracy of 98%, 92%, and 93.33% on the UIUC, KTH-TIPS, and UMD texture recognition datasets respectively.

Given a countable group $A$ with trivial bounded cohomology for all semi-separable coefficients and any group $\Gamma$, we prove that the bounded cohomology of certain restricted verbal wreath products $\Gamma \wr^{_W} A$ vanishes in positive degrees, for all semi-separable coefficients. Examples of such verbal wreath products include the standard restricted wreath product (extending a recent result by Monod for lamplighters groups) as well as verbal wreath products arising from $n$-solvable, $n$-nilpotent and $k$-Burnside ($k = 2,3,4,6$) verbal products. As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in degree $n\geq 1$ for all semi-separable coefficients. Finally, we prove that the stable commutator length of any verbal wreath product $\Gamma \wr^{_W} A$ vanishes.

This paper addresses the stabilization of dynamical systems in the infinite horizon optimal control setting using nonlinear feedback control based on State-Dependent Riccati Equations (SDREs). While effective, the practical implementation of such feedback strategies is often constrained by the high dimensionality of state spaces and the computational challenges associated with solving SDREs, particularly in parametric scenarios. To mitigate these limitations, we introduce the Dynamical Low-Rank Approximation (DLRA) methodology, which provides an efficient and accurate framework for addressing high-dimensional feedback control problems. DLRA dynamically constructs a compact, low-dimensional representation that evolves with the problem, enabling the simultaneous resolution of multiple parametric instances in real-time. We propose two novel algorithms to enhance numerical performances: the cascade-Newton-Kleinman method and Riccati-based DLRA (R-DLRA). The cascade-Newton-Kleinman method accelerates convergence by leveraging Riccati solutions from the nearby parameter or time instance, supported by a theoretical sensitivity analysis. R-DLRA integrates Riccati information into the DLRA basis construction to improve the quality of the solution. These approaches are validated through nonlinear one-dimensional and two-dimensional test cases showing transport-like behavior, demonstrating that R-DLRA outperforms standard DLRA and Proper Orthogonal Decomposition-based model order reduction in both speed and accuracy, offering a superior alternative to Full Order Model solutions.

Numerical simulation of fracture contact poromechanics is essential for various applications, including CO2 sequestration, geothermal energy production and underground gas storage. Modeling this problem accurately presents significant challenges due to the complex physics involved in strongly coupled poromechanics and frictional contact mechanics of fractures. The robustness and efficiency of the simulation heavily depends on a preconditioner for the linear solver, which addresses the Jacobian matrices arising from Newton's method in fully implicit time-stepping schemes. Developing an effective preconditioner is difficult because it must decouple three interdependent subproblems: momentum balance, fluid mass balance, and contact mechanics. The challenge is further compounded by the saddle-point structure of the contact mechanics problem, a result of the Augmented Lagrange formulation, which hinders the direct application of the well-established fixed stress approximation to decouple the poromechanics subproblem. In this work, we propose a preconditioner hat combines nested Schur complement approximations with a linear transformation, which addresses the singular nature of the contact mechanics subproblem. This approach extends the fixed stress scheme to both the matrix and fracture subdomains. We investigate analytically how the contact mechanics subproblem affects the convergence of the proposed fixed stress-based iterative scheme and demonstrate how it can be translated into a practical preconditioner. The scalability and robustness of the method are validated through a series of numerical experiments.

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podle\'s sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.

This paper merely explores the proof by J. Bourgain, H. Furstenberg, Y. Katznelson and D.S. Ornstein of their return times theorem [2] and lights a corner in it regarding the role of auto-correlation (so the proof presented is by no means an original work of the author). This yields a more inherent characterization of the sequences satisfying that theorem. Some further investigation is done, examples provided and questions raised.

We introduce the strong positive recurrence (SPR) property for diffeomorphisms on closed manifolds with arbitrary dimension, and show that it has many consequences and holds in many cases. SPR diffeomorphisms can be coded by countable state Markov shifts whose transition matrices act with a spectral gap on a large Banach space, and this implies exponential decay of correlations, almost sure invariance principle, large deviations, among other properties of the ergodic measures of maximal entropy. Any $C^\infty$ smooth surface diffeomorphism with positive entropy is SPR, and there are many other examples with lesser regularity, or in higher dimension.

A projective structure is Weyl metrizable if it has a representative that preserves a conformal structure. We interpret Weyl metrizability of 3-dimensional projective structures as certain 5-dimensional nondegenerate CR submanifolds in a class of 7-dimensional 2-nondegenerate CR structures. As a corollary, it follows that in dimension three Beltrami's theorem extends to conformal structures, i.e. a flat projective structure is Weyl metrizable exclusively with respect to a flat conformal structure. In higher dimensions it is shown that conformal Beltrami theorem remains true as well.

In this paper we propose a framework to analyze iterative first-order optimization algorithms for time-varying convex optimization. We assume that the temporal variability is caused by a time-varying parameter entering the objective, which can be measured at the time of decision but whose future values are unknown. We consider the case of strongly convex objective functions with Lipschitz continuous gradients and address the class of running algorithms where only one iteration per time change is performed. We model these algorithms as discrete-time linear parameter varying (LPV) systems in feedback with a time-varying gradient. We leverage the approach of analyzing algorithms as uncertain control interconnections with integral quadratic constraints (IQCs) and generalize that framework to the time-varying case. We propose novel IQCs that are capable of capturing the behavior of time-varying nonlinearities and leverage techniques from the LPV literature to establish novel bounds on the tracking error. Quantitative bounds can be computed by solving a semi-definite program and can be interpreted as an input-to-state stability result with respect to a disturbance signal which increases with the temporal variability of the problem. As a departure from results in this research area, our bounds introduce terms that can be interpreted as a temporal rate of change in the cost function and the optimal value. We exemplify our main results with numerical experiments that showcase how our analysis framework is able to capture convergence rates of different first-order algorithms for time-varying optimization through the choice of IQC and rate bounds.

We study a game in which one keeps flipping a coin until a given finite string of heads and tails occurs. We find the expected number of coin flips to end the game when the ending string consists of at most four maximal runs of heads or tails or alternates between heads and tails. This leads to some summation identities involving certain generalizations of the Fibonacci numbers.

There is increased interest in time-dependent (non-autonomous) Hamiltonians, stemming in part from the active field of Floquet quantum materials. Despite this, dispersive time-decay bounds, which reflect energy transport in such systems, have received little attention. We study the dynamics of non-autonomous, time-periodically forced, Dirac Hamiltonians: $i\partial_t\alpha =D(t)\alpha$, where $D(t)=i\sigma_3\partial_x+ \nu(t)$ is time-periodic but not spatially localized. For the special case $\nu(t)=m\sigma_1$, which models a relativistic particle of constant mass $m$, one has a dispersive decay bound: $\|\alpha(t,x)\|_{L^\infty_x}\lesssim t^{-\frac12}$. Previous analyses of Schr\"odinger Hamiltonians suggest that this decay bound persists for small, spatially-localized and time-periodic $\nu(t)$. However, we show that this is not necessarily the case if $\nu(t)$ is not spatially localized. Specifically, we study two non-autonomous Dirac models whose time-evolution (and monodromy operator) is constructed via Fourier analysis. In a rotating mass model, the dispersive decay bound is of the same type as for the constant mass model. However, in a model with a periodically alternating sign of the mass, the results are quite different. By stationary-phase analysis of the associated Fourier representation, we display initial data for which the $L^\infty_x$ time-decay rate are considerably slower: $\mathcal{O}(t^{-1/3})$ or even $\mathcal{O}(t^{-1/5})$ as $t\to\infty$.

We construct an explicit inversion formula for Guillarmou's normal operator on closed surfaces of constant negative curvature. This normal operator can be defined as a weak limit for an "attenuated normal operator", and we prove this inversion formula by first constructing an additional inversion formula for this attenuated normal operator on both the Poincar\'e disk and closed surfaces of constant negative curvature. A consequence of the inversion formula is the explicit construction of invariant distributions with prescribed pushforward over closed hyperbolic manifolds.

The study of numerical rounding errors is often greatly simplified in the analytical treatment of mathematical problems, or even entirely separated from it. In sampling theory, for instance, it is standard to assume the availability of an orthonormal basis for computations, ensuring that numerical errors are negligible. In reality, however, this assumption is often unmet. In this paper, we discard it and demonstrate the advantages of integrating numerical insights more deeply into sampling theory. To clearly pinpoint when the numerical phenomena play a significant role, we introduce the concept of numerical redundancy. A set of functions is numerically redundant if it spans a lower-dimensional space when analysed numerically rather than analytically. This property makes it generally impossible to compute the best approximation of a function in its span using finite precision. In contrast, $\ell^2$-regularized approximations are computable and, therefore, form the foundation of many practical methods. Regularization generally reduces accuracy compared to the best approximation, but our analysis shows that there is a benefit: it also significantly reduces the amount of data needed for accurate approximation. Furthermore, we present a constructive method for optimally selecting data points for $L^2$-approximations, explicitly accounting for the effects of regularization. The results are illustrated for two common scenarios that lead to numerical redundancy: (1) approximations on irregular domains and (2) approximations that incorporate specific features of the function to be approximated. In doing so, we obtain new results on random sampling for Fourier extension frames. Finally, we establish that regularization is implicit in numerical orthogonalization of a numerically redundant set, indicating that its analysis cannot be bypassed in a much broader range of methods.

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the $\infty$-categorical context, we establish a fully faithful embedding of the $\infty$-category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived $\infty$-category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.

We study the hyperbolic components of the family $\mathrm{Sk}(p,d)$ of regular polynomial skew-products of $\mathbb{C}^2$ of degree $d\geq2$, with a fixed base $p\in\mathbb{C}[z]$. Using a homogeneous parametrization of the family, we compute the accumulation set $E$ of the bifurcation locus on the boundary of the parameter space. Then in the case $p(z)=z^d$, we construct a map $\pi_0(\mathcal{D}')\to AB_d$ from the set of unbounded hyperbolic components that do not fully accumulate on $E$, to the set of algebraic braids of degree $d$. This map induces a second surjective map $\pi_0(\mathcal{D}')\to\mathrm{Conj}(\mathfrak{S}_d)$ towards the set of conjugacy classes of permutations on $d$ letters. This article is a continuation in higher degrees of the work of Astorg-Bianchi in the quadratic case $d=2$, for which they provided a complete classification of the hyperbolic components belonging to $\pi_0(\mathcal{D}')$.

Let $G$ be a graph on $n \ge 3$ vertices, whose adjacency matrix has eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$. The problem of bounding $\lambda_k$ in terms of $n$ was first proposed by Hong and was studied by Nikiforov, who demonstrated strong upper and lower bounds for arbitrary $k$. Nikiforov also claimed a strengthened upper bound for $k \ge 3$, namely that $\frac{\lambda_k}{n} < \frac{1}{2\sqrt{k-1}} - \varepsilon_k$ for some positive $\varepsilon_k$, but omitted the proof due to its length. In this paper, we give a proof of this bound for $k = 3$. We achieve this by instead looking at $\lambda_{n-1} + \lambda_n$ and introducing a new graph operation which provides structure to minimising graphs, including $\omega \le 3$ and $\chi \le 4$. Then we reduce the hypothetical worst case to a graph that is $n/2$-regular and invariant under said operation. By considering a series of inequalities on the restricted eigenvector components, we prove that a sequence of graphs with $\frac{\lambda_{n-1} + \lambda_n}{n}$ converging to $-\frac{\sqrt{2}}{2}$ cannot exist.

A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that the singular locus of a Vec-variety is a proper closed Vec-subvariety, analogously to the situation for ordinary finite-dimensional varieties. Via earlier work of the third author, this implies that these singular loci admit a description by finitely many polynomial equations. A natural follow-up question to our main result is whether a Vec-variety also admits a suitably functorial resolution of singularities. We establish some preliminary results in this direction in the regime where the dimension of evaluations of a Vec-variety grows linearly with that of the input vector space.

This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems and two schemes for constrained control problems. For the unconstrained control problem, while standard finite elements achieve a convergence rate of \( k+1 \) (with \( k \) representing the polynomial degree), our approach enhances this rate to \( k+2 \) by selecting the control from a carefully constructed reconstruction space. For the box-constrained problem, we demonstrate that using lowest-order elements (\( \mathbb{P}_0 \)) yields linear convergence, in contrast to finite element methods (FEM) that require linear elements to achieve comparable results. Furthermore, we derive a cubic convergence rate for control in the variational discretization scheme. Numerical experiments are provided to validate the theoretical findings.

We study a particular system of partial differential equations in which the harmonic, the divergence and the gradient operators of the unknown functions appear (harmonic-divgrad system). Using the Killing Hopf theorem and leveraging the properties of Riemannian manifolds with constant sectional curvature we establish the conditions under which these equations admit only the trivial solutions proving their trivialization on positive curvature space forms. The analysis of this particular system is motivated by its occurrence in the study of asymptotic symmetries in $p$-form gauge theories and in mixed symmetry tensor gauge theories.

Motivated by applications in physics (e.g., turbulence intermittency) and financial mathematics (e.g., rough volatility), this paper examines a family of integrated stochastic Volterra processes characterized by a small Hurst parameter $H<\tfrac{1}{2}$. We investigate the impact of kernel approximation on the integrated process by examining the resulting weak error. Our findings quantify this error in terms of the $L^1$ norm of the difference between the two kernels, as well as the $L^1$ norm of the difference of the squares of these kernels. Our analysis is based on a path-dependent Feynman-Kac formula and the associated partial differential equation (PPDE), providing a robust and extendible framework for our analysis.

Using the Tannakian formalism, one can attach to a principally polarized abelian variety a reductive group, along with a representation. We show that this group and the representation characterize Jacobians in genus up to $5$. More generally, our results hold on the bielliptic Prym locus in all genera. This gives the first evidence towards a recent conjecture by Weissauer and Kr\"amer. The main tool in our proof is a criterion for detecting Jacobians relying on Chern-Mather classes.

A simplicial cochain complex can be derived from a locally small poset by taking the nerve of the poset viewed as a category. We show that the simplicial cochain complex and a relative Hochschild cochain complex of the incidence algebra of the poset are isomorphic as operads with multiplications. This result implies that the hG-algebras derived from those operads are isomorphic, which is a generalization of the Gerstenhaber-Shack theorem. The isomorphism also induces a differential graded Lie algebra isomorphism, which we use to compute the moduli space of formal deformations of the incidence algebra.

A closed 4-manifold is symplectic Calabi--Yau (SCY) if its canonical class is trivial. Friedl and Vidussi proved that Thompson's group $F$ cannot be the fundamental group of any SCY manifold. In this paper, we show that its generalizations, called the Brown--Thompson group and the $n$-adic Lodha--Moore groups, cannot be also the fundamental group of any SCY manifold by using their method. From this proof, we also show that there exist non-trivial infinitely many examples which satisfy Geoghegan's conjecture.

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples.

We study the regularity properties of the minimisers of entropic optimal transport providing a natural analogue of the $\varepsilon$-regularity theory of quadratic optimal transport in the entropic setting. More precisely, we show that if the minimiser of the entropic problem satisfies a gradient BMO-type estimate at some scale, the same estimate holds all the way down to the natural length-scale associated to the entropic regularisation. Our result follows from a more general $\varepsilon$-regularity theory for optimal transport costs which can be viewed as perturbations of quadratic optimal transport. We consider such a perturbed cost and require that, under a certain class of admissible affine rescalings, the minimiser remains a local quasi-minimiser of the quadratic problem (in an appropriate sense) and that the cost of "long trajectories" of minimisers (and their rescalings) is small. Under these assumptions, we show that the minimiser satisfies an appropriate $C^{2,\alpha}$ Morrey$\unicode{x2013}$Campanato-type estimate which is valid up to the scale of quasi-minimality.

Let $\rho$ be a completely isometric representation of an operator system $S$ on some Hilbert space $H$. A $C^*$-support of $\rho$ is the $C^*$-algebra generated by $\rho(S)$ inside an injective operator system acting on $H$, equipped with its Choi--Effros product. By leveraging Hamana's theory, we show that such a $C^*$-support is unique precisely when $C^*(\rho(S))$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^*$-supports can be used to give new characterizations of the unique extension property for $*$-representations and of the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^*$-supports of $\rho$ to describe the subspace generated by the so-called abnormalities of $\rho$, thereby complementing a result of Kakariadis.

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for, some new dynamical behavior. In particular, we describe a large class of maps whose Julia sets contain both infinitely many homeomorphic copies of quadratic Julia sets and infinitely many subsets homeomorphic to a set which is obtained by starting with a quadratic Julia set, then changing a finite number of pairs of external ray landing point identifications, following an algorithm we will describe.

We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in both the spatial and temporal dimensions simultaneously. We also propose a novel wavelet-based recursive algorithm to reduce the system sensitivity stemming from steep initial and/or boundary conditions. The resulting nonlinear equations are solved using the Newton-Raphson method. We parallelize the construction of the tangent operator along with the solution of the system of algebraic equations. We perform rigorous verification studies using the nonlinear Burgers' equation. The application of the method is demonstrated solving Sod shock tube problem using the Navier-Stokes equations. The numerical results of the method reveal high-order convergence rates for the function as well as its spatial and temporal derivatives. We solve problems with steep gradients in both the spatial and temporal directions with a priori error estimates.

We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\mathbb{R}$-divisor, if $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part, and either if $M'$ or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension $3$. This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results.

The free Jacobi process is the radial part of the compression of the free unitary Brownian motion by two free orthogonal projections. In this paper, we determine the characteristic curves of the partial differential equation satisfied by its spectral distribution when both projections have the same rank $\alpha \in (0,1)$. Doing so leads for any fixed time $t >0$ to an expression of the moment generating function in a neighborhood of the origin, extending our previous results valid for $\alpha = 1/2$. Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the $\xi$-transform of the spectral distribution of the free unitary Brownian motion, of which we study mapping properties. We also prove a dynamical version of a recent identity pointed out by T. Kunisky and relating the stationary distributions of the free Jacobi processes corresponding to the set of parameters $(\alpha, \alpha)$ and $(1/2,\alpha)$. Actually, our dynamical version relates the Cauchy-Stieltjes transforms of the densities of the finite time spectral distributions.

We present new a stability result for periodic solutions of the periodic predator prey Lotka Volterra model, based on boundaries for the average of the coexistence states. Our result complements previous one in the literature.

We propose a two-stage concatenated coding scheme for reliable and information-theoretically secure communication over intersymbol interference wiretap channels. Motivated by the theoretical coding strategies that achieve the secrecy capacity, our scheme integrates low-density parity-check (LDPC) codes in the outer stage, forming a nested structure of wiretap codes, with trellis codes in the inner stage to improve achievable secure rates. The trellis code is specifically designed to transform the uniformly distributed codewords produced by the LDPC code stage into a Markov process, achieving tight lower bounds on the secrecy capacity. We further estimate the information leakage rate of the proposed coding scheme using an upper bound. To meet the weak secrecy criterion, we optimize degree distributions of the irregular LDPC codes at the outer stage, essentially driving the estimated upper bound on the information leakage rate to zero.

Let $M_{n, m}(\mathbb{R})$ be the space of $n\times m$ real matrices. Define $\mathcal{K}_o^{n,m}$ as the set of convex compact subsets in $M_{n,m}(\mathbb{R})$ with nonempty interior containing the origin $o\in M_{n, m}(\mathbb{R})$, and $\mathcal{K}_{(o)}^{n,m}$ as the members of $\mathcal{K}_o^{n,m}$ containing $o$ in their interiors. Let $\Phi: M_{1, m}(\mathbb{R}) \rightarrow [0, \infty)$ be a convex function such that $\Phi(o)=0$ and $\Phi(z)+\Phi(-z)>0$ for $z\neq o.$ In this paper, we propose the $m$th order Orlicz projection operator $\Pi_{\Phi}^m: \mathcal{K}_{(o)}^{n,1}\rightarrow \mathcal{K}_{(o)}^{n,m}$, and study its fundamental properties, including the continuity and affine invariance. We establish the related higher-order Orlicz-Petty projection inequality, which states that the volume of $\Pi_{\Phi}^{m, *}(K)$, the polar body of $\Pi_{\Phi}^{m}(K)$, is maximized at origin-symmetric ellipsoids among convex bodies with fixed volume. Furthermore, when $\Phi$ is strictly convex, we prove that the maximum is uniquely attained at origin-symmetric ellipsoids. Our proof is based on the classical Steiner symmetrization and its higher-order analogue. We also investigate the special case for $\Phi_{Q}=\phi\circ h_Q$, where $h_Q$ denotes the support function of $Q\in \mathcal{K}^{1, m}_o$ and $\phi: [0, \infty)\rightarrow [0, \infty)$ is a convex function such that $\phi(0)=0$ and $\phi$ is strictly increasing on $[0, \infty).$ We establish a higher-order Orlicz-Petty projection inequality related to $\Pi_{\Phi_Q}^{m, *} (K)$. Although $\Phi_Q$ may not be strictly convex, we characterize the equality under the additional assumption on $Q$ and $\phi$, such as $Q\in \mathcal{K}_{(o)}^{1,m}$ and the strict convexity of $\phi$.

The topic of nonparametric estimation of smooth boundaries is extensively studied in the conventional setting where pairs of single covariate and response variable are observed. However, this traditional setting often suffers from the cost of data collection. Recent years have witnessed the consistent development of learning algorithms for binary classification problems where one can instead observe paired covariates and binary variable representing the statistical relationship between the covariates. In this work, we theoretically study the question of whether multiple smooth boundaries are learnable if the pairwise binary classification setting is considered. We investigate the question with the statistical dependence of paired covariates to develop a learning algorithm using vector-valued functions. The main theorem shows that there is an empirical risk minimization algorithm in a class of deep ReLU networks such that it produces a consistent estimator for indicator functions defined with smooth boundaries. We also discuss how the pairwise binary classification setting is different from the conventional settings, focusing on the structural condition of function classes. As a by-product, we apply the main theorem to a multiclass nonparametric classification problem where the estimation performance is measured by the excess risk in terms of misclassification.

We give direct bijective proofs of the symmetry of the distributions of the number of ascents and descents over standard Young tableaux of shape $\lambda$, where $\lambda$ is a rectangle $(n,n,\dots,n)$ or a truncated staircase $(n,n-1,\dots,n-k+1)$. These can be viewed as instances of the more general symmetry of the distribution of descents over linear extensions of graded posets, for which previous proofs by Stanley and Farley were based on the theory of $P$-partitions and the involution principle, respectively. In the case of two-row rectangles $(n,n)$, our bijection is equivalent to the Lalanne--Kreweras involution on Dyck paths, which bijectively proves the symmetry of the Narayana numbers. Our bijections are defined in terms of certain arrow encodings of standard Young tableaux. This setup allows us to construct other statistic-preserving involutions on tableaux of rectangular shape, providing a simple proof of the fact that ascents and descents are equidistributed up to a shift, and proving a conjecture of Sulanke about certain statistics in the case of three rows. Finally, we use our bijections to define a possible notion of rowmotion on standard Young tableaux of rectangular shape, and to give a bijective proof of the symmetry of the number of descents on canon permutations, which have been recently studied as a variation of Stirling and quasi-Stirling permutations.

The orthogonal momentum amplituhedron O_k was introduced in 2021 by Huang, Kojima, Wen, and Zhang in the study of scattering amplitudes of ABJM theory. It was conjectured that it admits a decomposition into BCFW cells. We prove this conjecture.

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (Rudich 1997), and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: *Demi-bit stretch*: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests (Rudich 1997). They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier (Rudich and Razborov 1997). Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit $b:\{0,1\}^n\to \{0,1\}^{n+1}$ can be stretched into sublinear many demi-bits $b':\{0,1\}^{n}\to \{0,1\}^{n+n^{c}}$, for every constant $0<c<1$. >>> see rest of abstract in paper.

In this work, we study the effectiveness of employing archetypal aperiodic sequencing -- namely Fibonacci, Thue-Morse, and Rudin-Shapiro -- on the Parrondian effect. From a capital gain perspective, our results show that these series do yield a Parrondo's Paradox with the Thue-Morse based strategy outperforming not only the other two aperiodic strategies but benchmark Parrondian games with random and periodical ($AABBAABB\ldots$) switching as well. The least performing of the three aperiodic strategies is the Rudin-Shapiro. To elucidate the underlying causes of these results, we analyze the cross-correlation between the capital generated by the switching protocols and that of the isolated losing games. This analysis reveals that a strong anticorrelation with both isolated games is typically required to achieve a robust manifestation of Parrondo's effect. We also study the influence of the sequencing on the capital using the lacunarity and persistence measures. In general, we observe that the switching protocols tend to become less performing in terms of the capital as one increases the persistence and thus approaches the features of an isolated losing game. For the (log-)lacunarity, a property related to heterogeneity, we notice that for small persistence (less than 0.5) the performance increases with the lacunarity with a maximum around 0.4. In respect of this, our work shows that the optimization of a switching protocol is strongly dependent on a fine-tuning between persistence and heterogeneity.

In the fields of computation and neuroscience, much is still unknown about the underlying computations that enable key cognitive functions including learning, memory, abstraction and behavior. This paper proposes a mathematical and computational model of learning and memory based on a small set of bio-plausible functions that include coincidence detection, signal modulation, and reward/penalty mechanisms. Our theoretical approach proposes that these basic functions are sufficient to establish and modulate an information space over which computation can be carried out, generating signal gradients usable for inference and behavior. The computational method used to test this is a structurally dynamic cellular automaton with continuous-valued cell states and a series of recursive steps propagating over an undirected graph with the memory function embedded entirely in the creation and modulation of graph edges. The experimental results show: that the toy model can make near-optimal choices to re-discover a reward state after a single training run; that it can avoid complex penalty configurations; that signal modulation and network plasticity can generate exploratory behaviors in sparse reward environments; that the model generates context-dependent memory representations; and that it exhibits high computational efficiency because of its minimal, single-pass training requirements combined with flexible and contextual memory representation.

Normal mode analysis (NMA) provides a mathematical framework for exploring the intrinsic global dynamics of molecules through the definition of an energy function, where normal modes correspond to the eigenvectors of the Hessian matrix derived from the second derivatives of this function. The energy required to 'trigger' each normal mode is proportional to the square of its eigenvalue, with six zero-eigenvalue modes representing universal translation and rotation, common to all molecular systems. In contrast, modes associated with small non-zero eigenvalues are more easily excited by external forces and are thus closely related to molecular functions. Inspired by the anisotropic network model (ANM), this work establishes a novel connection between normal mode analysis and sheaf theory by introducing a cellular sheaf structure, termed the anisotropic sheaf, defined on undirected, simple graphs, and identifying the conventional Hessian matrix as the sheaf Laplacian. By interpreting the global section space of the anisotropic sheaf as the kernel of the Laplacian matrix, we demonstrate a one-to-one correspondence between the zero-eigenvalue-related normal modes and a basis for the global section space. We further analyze the dimension of this global section space, representing the space of harmonic signals, under conditions typically considered in normal mode analysis. Additionally, we propose a systematic method to streamline the Delaunay triangulation-based construction for more efficient graph generation while preserving the ideal number of normal modes with zero eigenvalues in ANM analysis.

Optimal Transport (OT) has established itself as a robust framework for quantifying differences between distributions, with applications that span fields such as machine learning, data science, and computer vision. This paper offers a detailed examination of the OT problem, beginning with its theoretical foundations, including the classical formulations of Monge and Kantorovich and their extensions to modern computational techniques. It explores cutting-edge algorithms, including Sinkhorn iterations, primal-dual strategies, and reduction-based approaches, emphasizing their efficiency and scalability in addressing high-dimensional problems. The paper also highlights emerging trends, such as integrating OT into machine learning frameworks, the development of novel problem variants, and ongoing theoretical advancements. Applications of OT are presented across a range of domains, with particular attention to its innovative application in time series data analysis via Optimal Transport Warping (OTW), a robust alternative to methods like Dynamic Time Warping. Despite the significant progress made, challenges related to scalability, robustness, and ethical considerations remain, necessitating further research. The paper underscores OT's potential to bridge theoretical depth and practical utility, fostering impactful advancements across diverse disciplines.

This study aims to benchmark candidate strategies for embedding neural network (NN) surrogates in nonlinear model predictive control (NMPC) formulations that are subject to systems described with partial differential equations and that are solved via direct transcription (i.e., simultaneous methods). This study focuses on the use of physics-informed NNs and physics-informed convolutional NNs as the internal (surrogate) models within the NMPC formulation. One strategy embeds NN models as explicit algebraic constraints, leveraging the automatic differentiation (AD) of an algebraic modelling language (AML) to evaluate the derivatives. Alternatively, the solver can be provided with derivatives computed external to the AML via the AD routines of the machine learning environment the NN is trained in. The three numerical experiments considered in this work reveal that replacing mechanistic models with NN surrogates may not always offer computational advantages when smooth activation functions are used in conjunction with a local nonlinear solver (e.g., Ipopt), even with highly nonlinear systems. Moreover, in this context, the external function evaluation of the NN surrogates often outperforms the embedding strategies that rely on explicit algebraic constraints, likely due to the difficulty in initializing the auxiliary variables and constraints introduced by explicit algebraic reformulations.

Current wireless infrastructure is optimized to support downlink applications. This paper anticipates the emergence of applications where engineering focus shifts from downlink to uplink. The current paradigm of scheduling users on reserved uplink resources is not able to deal efficiently with unpredictable traffic patterns. As a result, 3GPP introduced the 2-step RACH as a mechanism to enable grant-free (random) initial access. The first of the two steps is preamble detection in a RACH slot, and in this paper we describe a low-complexity algorithm for simultaneous detection of multiple preambles in the presence of mobility and delay spread. We provide a pathway to standards adoption by choosing ZC sequences as preambles, as ZC sequences already appear in 5G standards. We construct preambles by using the discrete Zak transform to pass from a ZC sequence of length MN in the TD to a quasi-periodic MxN array in the DD domain. There are MN quasi-periodic Dirac pulses, each corresponding to a Zak-OTFS carrier waveform, and the ZC preamble is simply the corresponding sum of Zak-OTFS carrier waveforms. We detect multiple preambles in the presence of mobility and delay spread by sampling the received signal on the MxN period grid in the DD domain. We approach detection as a compressed sensing problem. We represent a preamble as a column of length MN in the DD domain and apply discrete shifts in delay and Doppler to produce a block with O(MN) columns in the compressed sensing matrix. The superposition of multiple preambles determines a block sparse sum of columns in the sensing matrix. The correlation properties of ZC sequences result in a highly structured compressed sensing matrix, making it possible to identify constituent preambles using OST, which has complexity O(M^3N^3). In this paper, we describe an algorithm with complexity that is O(M^2N^2) in the size of an individual column.

We define a digital twin (DT) of a physical system governed by partial differential equations (PDEs) as a model for real-time simulations and control of the system behavior under changing conditions. We construct DTs using the Karhunen-Lo\`{e}ve Neural Network (KL-NN) surrogate model and transfer learning (TL). The surrogate model allows fast inference and differentiability with respect to control parameters for control and optimization. TL is used to retrain the model for new conditions with minimal additional data. We employ the moment equations to analyze TL and identify parameters that can be transferred to new conditions. The proposed analysis also guides the control variable selection in DT to facilitate efficient TL. For linear PDE problems, the non-transferable parameters in the KL-NN surrogate model can be exactly estimated from a single solution of the PDE corresponding to the mean values of the control variables under new target conditions. Retraining an ML model with a single solution sample is known as one-shot learning, and our analysis shows that the one-shot TL is exact for linear PDEs. For nonlinear PDE problems, transferring of any parameters introduces errors. For a nonlinear diffusion PDE model, we find that for a relatively small range of control variables, some surrogate model parameters can be transferred without introducing a significant error, some can be approximately estimated from the mean-field equation, and the rest can be found using a linear residual least square problem or an ordinary linear least square problem if a small labeled dataset for new conditions is available. The former approach results in a one-shot TL while the latter approach is an example of a few-shot TL. Both methods are approximate for the nonlinear PDEs.

The application of the standard quasi-steady-state approximation to the Michaelis--Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, $e_0$, to the Michaelis constant, $K_M$, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation. Using standard tools from the analysis of ordinary differential equations, we show that while $e_0/K_M$ provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of $e_0$ to the Van Slyke-Cullen constant, $K$. We conclude that the magnitude of $e_0/K$ offers the most accurate measure of the validity of the standard quasi-steady-state approximation.

Cooperative behaviors are deeply embedded in structured biological and social systems. Networks are often employed to portray pairwise interactions among individuals, where network nodes represent individuals and links indicate who interacts with whom. However, it is increasingly recognized that many empirical interactions often involve triple or more individuals instead of the massively oversimplified lower-order pairwise interactions, highlighting the fundamental gap in understanding the evolution of collective cooperation for higher-order interactions with diverse scales of the number of individuals. Here, we develop a theoretical framework of evolutionary game dynamics for systematically analyzing how cooperation evolves and fixates under higher-order interactions. Specifically, we offer a simple condition under which cooperation is favored under arbitrary combinations of different orders of interactions. Compared to pairwise interactions, our findings suggest that higher-order interactions enable lower thresholds for the emergence of cooperation. Surprisingly, we show that higher-order interactions favor the evolution of cooperation in large-scale systems, which is the opposite for lower-order scenarios. Our results offer a new avenue for understanding the evolution of collective cooperation in empirical systems with higher-order interactions.

It is a folklore belief in the theory of spin glasses and disordered systems that out-of-equilibrium dynamics fail to find stable local optima exhibiting e.g. local strict convexity on physical time-scales. In the context of the Sherrington--Kirkpatrick spin glass, Behrens-Arpino-Kivva-Zdeborov\'a and Minzer-Sah-Sawhney have recently conjectured that this obstruction may be inherent to all efficient algorithms, despite the existence of exponentially many such optima throughout the landscape. We prove this search problem exhibits strong low degree hardness for polynomial algorithms of degree $D\leq o(N)$: any such algorithm has probability $o(1)$ to output a stable local optimum. To the best of our knowledge, this is the first result to prove that even constant-degree polynomials have probability $o(1)$ to solve a random search problem without planted structure. To prove this, we develop a general-purpose enhancement of the ensemble overlap gap property, and as a byproduct improve previous results on spin glass optimization, maximum independent set, random $k$-SAT, and the Ising perceptron to strong low degree hardness. Finally for spherical spin glasses with no external field, we prove that Langevin dynamics does not find stable local optima within dimension-free time.

We show that the fermion, in the context of a system that comprises many such entities - which, by virtue of the Pauli exclusion principle, possesses a Fermi surface at zero temperature - may itself be thought of as a collection of non-local particle-hole excitations across this Fermi surface. This result is purely kinematical and completely general - not being restricted to any specific dimension, applicable to both continuum and lattice systems. There is also no implication that it is applicable only to low-energy phenomena close to the Fermi surface. We are able to derive the full single-particle dynamical Green function of this fermion at finite temperature by viewing it as a collection of these non-local particle-hole excitations. The Green function of the fermion then manifests itself as a solution to a first-order differential equation in a parameter that controls the number of particle-hole pairs across the Fermi surface, and this equation itself reveals variable coefficients that may be identified with a Bose-Einstein distribution - implying that there is a sense in which the non-local particle-hole excitations have bosonic qualities while not being exact bosons at the level of operators. We also recall the definition of the non-local particle-hole operator that may be used to diagonalize the kinetic energy of free fermions of the sort mentioned above. Number-conserving products of creation and annihilation operators of fermions are expressible as a (rather complicated) combination of these non-local particle-hole operators.

Piecewise-deterministic Markov processes combine continuous in time dynamics with jump events, the rates of which generally depend on the continuous variables and thus are not constants. This leads to a problem in a Monte-Carlo simulation of such a system, where, at each step, one must find the time instant of the next event. The latter is determined by an integral equation and usually is rather slow in numerical implementation. We suggest a reformulation of the next event problem as an ordinary differential equation where the independent variable is not the time but the cumulative rate. This reformulation is similar to the H\'enon approach to efficiently constructing the Poincar\'e map in deterministic dynamics. The problem is then reduced to a standard numerical task of solving a system of ordinary differential equations with given initial conditions on a prescribed interval. We illustrate the method with a stochastic Morris-Lecar model of neuron spiking with stochasticity in the opening and closing of voltage-gated ion channels.

We aim to assist image-based myopia screening by resolving two longstanding problems, "how to integrate the information of ocular images of a pair of eyes" and "how to incorporate the inherent dependence among high-myopia status and axial length for both eyes." The classification-regression task is modeled as a novel 4-dimensional muti-response regression, where discrete responses are allowed, that relates to two dependent 3rd-order tensors (3D ultrawide-field fundus images). We present a Vision Transformer-based bi-channel architecture, named CeViT, where the common features of a pair of eyes are extracted via a shared Transformer encoder, and the interocular asymmetries are modeled through separated multilayer perceptron heads. Statistically, we model the conditional dependence among mixture of discrete-continuous responses given the image covariates by a so-called copula loss. We establish a new theoretical framework regarding fine-tuning on CeViT based on latent representations, allowing the black-box fine-tuning procedure interpretable and guaranteeing higher relative efficiency of fine-tuning weight estimation in the asymptotic setting. We apply CeViT to an annotated ultrawide-field fundus image dataset collected by Shanghai Eye \& ENT Hospital, demonstrating that CeViT enhances the baseline model in both accuracy of classifying high-myopia and prediction of AL on both eyes.

This article explores the dynamics of savanna ecosystems with grazing, browsing, and migration effects. Covering over one-eighth of the Earth's land area and supporting about one-fifth of the global population, the savanna is an ecological system whose importance has only recently garnered significant attention from biologists. The interactions between organisms in this ecosystem are highly complex, and fundamental mathematical issues remain unresolved. We rigorously analyze traveling waves in savanna systems and focus on whether trees, grass, grazers, and browsers coexist. We demonstrate the existence of various traveling waves, including waves transitioning from extinction to co-existence and waves from a grass-vegetation state (where only grass and grazers exist) to co-existence. Due to the biodiversity of species in grassland ecosystems, it is not appropriate to consider overly simplified models of competition between grasses and trees. From both a biological and mathematical perspective, factors such as animal grazing, browsing, and migration (which facilitates seed dispersal) play a crucial role in promoting ecological stability and coexistence. Additionally, we estimate the nonzero minimum value of the total plant biomass within the savanna dynamic system to better understand the persistence and stability of sustainable development within the ecosystem.

The advent of next-generation ultra-reliable and low-latency communications (xURLLC) presents stringent and unprecedented requirements for key performance indicators (KPIs). As a disruptive technology, non-orthogonal multiple access (NOMA) harbors the potential to fulfill these stringent KPIs essential for xURLLC. However, the immaturity of research on the tail distributions of these KPIs significantly impedes the application of NOMA to xURLLC. Stochastic network calculus (SNC), as a potent methodology, is leveraged to provide dependable theoretical insights into tail distribution analysis and statistical QoS provisioning (SQP). In this article, we develop a NOMA-assisted uplink xURLLC network architecture that incorporates an SNC-based SQP theoretical framework (SNC-SQP) to support tail distribution analysis in terms of delay, age-of-information (AoI), and reliability. Based on SNC-SQP, an SQP-driven power optimization problem is proposed to minimize transmit power while guaranteeing xURLLC's KPIs on delay, AoI, reliability, and power consumption. Extensive simulations validate our proposed theoretical framework and demonstrate that the proposed power allocation scheme significantly reduces uplink transmit power and outperforms conventional schemes in terms of SQP performance.

We demonstrate that Col is PSPACE-complete on triangular grid graphs via a reduction from Bounded Two-Player Constraint Logic. This is the most structured graph family that Col is known to be computationally hard for.

This paper investigates the investment problem of constructing an optimal no-short sequential portfolio strategy in a market with a latent dependence structure between asset prices and partly unobservable side information, which is often high-dimensional. The results demonstrate that a dynamic strategy, which forms a portfolio based on perfect knowledge of the dependence structure and full market information over time, may not grow at a higher rate infinitely often than a constant strategy, which remains invariant over time. Specifically, if the market is stationary, implying that the dependence structure is statistically stable, the growth rate of an optimal dynamic strategy, utilizing the maximum capacity of the entire market information, almost surely decays over time into an equilibrium state, asymptotically converging to the growth rate of a constant strategy. Technically, this work reassesses the common belief that a constant strategy only attains the optimal limiting growth rate of dynamic strategies when the market process is identically and independently distributed. By analyzing the dynamic log-optimal portfolio strategy as the optimal benchmark in a stationary market with side information, we show that a random optimal constant strategy almost surely exists, even when a limiting growth rate for the dynamic strategy does not. Consequently, two approaches to learning algorithms for portfolio construction are discussed, demonstrating the safety of removing side information from the learning process while still guaranteeing an asymptotic growth rate comparable to that of the optimal dynamic strategy.

Modern software systems are typically configurable, a fundamental prerequisite for wide applicability and reusability. This flexibility poses an extraordinary challenge for quality assurance, as the enormous number of possible configurations makes it impractical to test each of them separately. This is where t-wise interaction sampling can be used to systematically cover the configuration space and detect unknown feature interactions. Over the last two decades, numerous algorithms for computing small interaction samples have been studied, providing improvements for a range of heuristic results; nevertheless, it has remained unclear how much these results can still be improved. We present a significant breakthrough: a fundamental framework, based on the mathematical principle of duality, for combining near-optimal solutions with provable lower bounds on the required sample size. This implies that we no longer need to work on heuristics with marginal or no improvement, but can certify the solution quality by establishing a limit on the remaining gap; in many cases, we can even prove optimality of achieved solutions. This theoretical contribution also provides extensive practical improvements: Our algorithm SampLNS was tested on 47 small and medium-sized configurable systems from the existing literature. SampLNS can reliably find samples of smaller size than previous methods in 85% of the cases; moreover, we can achieve and prove optimality of solutions for 63% of all instances. This makes it possible to avoid cumbersome efforts of minimizing samples by researchers as well as practitioners, and substantially save testing resources for most configurable systems.

The conefield variant of a Converse KAM method for 3D vector fields, identifying regions through which no invariant 2-tori pass transverse to a specified direction field, is tested on some helical perturbations of an axisymmetric magnetic field in toroidal geometry. This implementation computes bounds on the slopes of invariant tori of a given class and allows to apply a subsidiary criterion for the extension of the non-existence region, saving significant computation time. The method finds regions corresponding to magnetic islands and chaos for the fieldline flow.

In online betting, the bookmaker can update the payoffs it offers on a particular event many times before the event takes place, and the updated payoffs may depend on the bets accumulated thus far. We study the problem of bookmaking with the goal of maximizing the return in the worst-case, with respect to the gamblers' behavior and the event's outcome. We formalize this problem as the \emph{Optimal Online Bookmaking game}, and provide the exact solution for the binary case. To this end, we develop the optimal bookmaking strategy, which relies on a new technique called bi-balancing trees, that assures that the house loss is the same for all \emph{decisive} betting sequences, where the gambler bets all its money on a single outcome in each round.

Accurate RF propagation modeling in urban environments is critical for developing digital spectrum twins and optimizing wireless communication systems. We introduce OpenGERT, an open-source automated Geometry Extraction tool for Ray Tracing, which collects and processes terrain and building data from OpenStreetMap, Microsoft Global ML Building Footprints, and USGS elevation data. Using the Blender Python API, it creates detailed urban models for high-fidelity simulations with NVIDIA Sionna RT. We perform sensitivity analyses to examine how variations in building height, position, and electromagnetic material properties affect ray-tracing accuracy. Specifically, we present pairwise dispersion plots of channel statistics (path gain, mean excess delay, delay spread, link outage, and Rician K-factor) and investigate how their sensitivities change with distance from transmitters. We also visualize the variance of these statistics for selected transmitter locations to gain deeper insights. Our study covers Munich and Etoile scenes, each with 10 transmitter locations. For each location, we apply five types of perturbations: material, position, height, height-position, and all combined, with 50 perturbations each. Results show that small changes in permittivity and conductivity minimally affect channel statistics, whereas variations in building height and position significantly alter all statistics, even with noise standard deviations of 1 meter in height and 0.4 meters in position. These findings highlight the importance of precise environmental modeling for accurate propagation predictions, essential for digital spectrum twins and advanced communication networks. The code for geometry extraction and sensitivity analyses is available at github.com/serhatadik/OpenGERT/.

Statistical methods for causal inference with continuous treatments mainly focus on estimating the mean potential outcome function, commonly known as the dose-response curve. However, it is often not the dose-response curve but its derivative function that signals the treatment effect. In this paper, we investigate nonparametric inference on the derivative of the dose-response curve with and without the positivity condition. Under the positivity and other regularity conditions, we propose a doubly robust (DR) inference method for estimating the derivative of the dose-response curve using kernel smoothing. When the positivity condition is violated, we demonstrate the inconsistency of conventional inverse probability weighting (IPW) and DR estimators, and introduce novel bias-corrected IPW and DR estimators. In all settings, our DR estimator achieves asymptotic normality at the standard nonparametric rate of convergence. Additionally, our approach reveals an interesting connection to nonparametric support and level set estimation problems. Finally, we demonstrate the applicability of our proposed estimators through simulations and a case study of evaluating a job training program.

A principal curve serves as a powerful tool for uncovering underlying structures of data through 1-dimensional smooth and continuous representations. On the basis of optimal transport theories, this paper introduces a novel principal curve framework constrained by monotonicity with rigorous theoretical justifications. We establish statistical guarantees for our monotone curve estimate, including expected empirical and generalized mean squared errors, while proving the existence of such estimates. These statistical foundations justify adopting the popular early stopping procedure in machine learning to implement our numeric algorithm with neural networks. Comprehensive simulation studies reveal that the proposed monotone curve estimate outperforms competing methods in terms of accuracy when the data exhibits a monotonic structure. Moreover, through two real-world applications on future prices of copper, gold, and silver, and avocado prices and sales volume, we underline the robustness of our curve estimate against variable transformation, further confirming its effective applicability for noisy and complex data sets. We believe that this monotone curve-fitting framework offers significant potential for numerous applications where monotonic relationships are intrinsic or need to be imposed.

Long time-duration low-thrust nonlinear optimal spacecraft trajectory global search is a computationally and time expensive problem characterized by clustering patterns in locally optimal solutions. During preliminary mission design, mission parameters are subject to frequent changes, necessitating that trajectory designers efficiently generate high-quality control solutions for these new scenarios. Generative machine learning models can be trained to learn how the solution structure varies with respect to a conditional parameter, thereby accelerating the global search for missions with updated parameters. In this work, state-of-the-art diffusion models are integrated with the indirect approach for trajectory optimization within a global search framework. This framework is tested on two low-thrust transfers of different complexity in the circular restricted three-body problem. By generating and analyzing a training data set, we develop mathematical relations and techniques to understand the complex structures in the costate domain of locally optimal solutions for these problems. A diffusion model is trained on this data and successfully accelerates the global search for both problems. The model predicts how the costate solution structure changes, based on the maximum spacecraft thrust magnitude. Warm-starting a numerical solver with diffusion model samples for the costates at the initial time increases the number of solutions generated per minute for problems with unseen thrust magnitudes by one to two orders of magnitude in comparison to samples from a uniform distribution and from an adjoint control transformation.

This paper shows that Knill-Laflamme condition, known as a necessary and sufficient condition for quantum error-correction, can be applied to quantum errors where the number of particles changes before and after the error. This fact shows that correctabilities of single deletion errors and single insertion errors are equivalent. By applying Knill-Laflamme condition, we generalize the previously known correction conditions for single insertion and deletion errors to necessary and sufficient level. By giving an example that satisfies this condition, we construct a new single qudit insertion/deletion code and explain its decoding algorithm.

I present an exact solution for the convex roof of the square root threetangle for all states within the Bloch sphere. The working hypothesis is that optimal decompositions contain as many states from the zero-polytope as possible which can be called zero-state locking. The footprint of the measure of entanglement consists in a characteristic pattern for the fixed pure states on the surface which form the optimal solution. The solution is subject to transformation properties due to the SL-invariance of the entanglement measure.

We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on the infinite interval. We obtain the exact solution of an average particle-density profile for the domain wall and the alternating initial conditions via the Bethe ansatz, analytically deriving the asymptotic expressions for the long time dynamics. For the domain wall initial condition, we obtain the scaling form of the average density, elucidating that the diffusive transport always emerges in the long time dynamics if the strength of the dephasing, no matter how small, is positive. For the alternating initial condition, our exact solution leads to the fact that the average density displays oscillatory decay or over-damped decay depending on the strength of the dissipation. Furthermore, we demonstrate that the asymptotic forms approach those of the symmetric simple exclusion process, identifying corrections from it.

This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using L\'evy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.

Semiconductors are widely used in various applications and critical infrastructures. These devices have specified lifetimes and quality targets that manufacturers must achieve. Lifetime estimation is conducted through accelerated stress tests. Electrical parameters are measured at multiple times during a stress test procedure. The change in these Electrical parameters is called lifetime drift. Data from these tests can be used to develop a statistical model predicting the lifetime behavior of the electrical parameters in real devices. These models can provide early warnings in production processes, identify critical parameter drift, and detect outliers. While models for continuous electrical parameters exists, there may be bias when estimating the lifetime of discrete parameters. To address this, we propose a semi-parametric model for degradation trajectories based on longitudinal stress test data. This model optimizes guard bands, or quality guaranteeing tighter limits, for discrete electrical parameters at production testing. It is scalable, data-driven, and explainable, offering improvements over existing methods for continuous underlying data, such as faster calculations, arbitrary non-parametric conditional distribution modeling, and a natural extension of optimization algorithms to the discrete case using Markov transition matrices.

We propose an enhanced zeroth-order stochastic Frank-Wolfe framework to address constrained finite-sum optimization problems, a structure prevalent in large-scale machine-learning applications. Our method introduces a novel double variance reduction framework that effectively reduces the gradient approximation variance induced by zeroth-order oracles and the stochastic sampling variance from finite-sum objectives. By leveraging this framework, our algorithm achieves significant improvements in query efficiency, making it particularly well-suited for high-dimensional optimization tasks. Specifically, for convex objectives, the algorithm achieves a query complexity of O(d \sqrt{n}/\epsilon ) to find an epsilon-suboptimal solution, where d is the dimensionality and n is the number of functions in the finite-sum objective. For non-convex objectives, it achieves a query complexity of O(d^{3/2}\sqrt{n}/\epsilon^2 ) without requiring the computation ofd partial derivatives at each iteration. These complexities are the best known among zeroth-order stochastic Frank-Wolfe algorithms that avoid explicit gradient calculations. Empirical experiments on convex and non-convex machine learning tasks, including sparse logistic regression, robust classification, and adversarial attacks on deep networks, validate the computational efficiency and scalability of our approach. Our algorithm demonstrates superior performance in both convergence rate and query complexity compared to existing methods.

The collective ground state for a spherical symmetric dust ball has been investigated recently in [R. Casadio, Phys. Lett. B 843 (2023) 138055]. In this study, we refine that model by obtaining a mass distribution that accounts for the superposition of wavefunctions across different layers. The refined mass distribution shows significant deviations from the approximation without quantum superpositions. Specifically, the new nearly parabolic distribution replaces the linear mass profile of the original work, featuring an overall downward concavity, which leads to a non-vanishing tension. Notably, the regularity of the metric and causal structure are preserved in the refined analysis.

Machine Learning (ML) models have become a very powerful tool to extract information from large datasets and use it to make accurate predictions and automated decisions. However, ML models can be vulnerable to external attacks, causing them to underperform or deviate from their expected tasks. One way to attack ML models is by injecting malicious data to mislead the algorithm during the training phase, which is referred to as a poisoning attack. We can prepare for such situations by designing anticipated attacks, which are later used for creating and testing defence strategies. In this paper, we propose an algorithm to generate strong poisoning attacks for a ridge regression model containing both numerical and categorical features that explicitly models and poisons categorical features. We model categorical features as SOS-1 sets and formulate the problem of designing poisoning attacks as a bilevel optimization problem that is nonconvex mixed-integer in the upper-level and unconstrained convex quadratic in the lower-level. We present the mathematical formulation of the problem, introduce a single-level reformulation based on the Karush-Kuhn-Tucker (KKT) conditions of the lower level, find bounds for the lower-level variables to accelerate solver performance, and propose a new algorithm to poison categorical features. Numerical experiments show that our method improves the mean squared error of all datasets compared to the previous benchmark in the literature.

Quantum relative entropy, a quantum generalization of the well-known Kullback-Leibler divergence, serves as a fundamental measure of the distinguishability between quantum states and plays a pivotal role in quantum information science. Despite its importance, efficiently estimating quantum relative entropy between two quantum states on quantum computers remains a significant challenge. In this work, we propose the first quantum algorithm for estimating quantum relative entropy and Petz R\'{e}nyi divergence from two unknown quantum states on quantum computers, addressing open problems highlighted in [Phys. Rev. A 109, 032431 (2024)] and [IEEE Trans. Inf. Theory 70, 5653-5680 (2024)]. This is achieved by combining quadrature approximations of relative entropies, the variational representation of quantum f-divergences, and a new technique for parameterizing Hermitian polynomial operators to estimate their traces with quantum states. Notably, the circuit size of our algorithm is at most 2n+1 with n being the number of qubits in the quantum states and it is directly applicable to distributed scenarios, where quantum states to be compared are hosted on cross-platform quantum computers. We validate our algorithm through numerical simulations, laying the groundwork for its future deployment on quantum hardware devices.

We propose a novel Bregman descent algorithm for minimizing a convex function that is expressed as the sum of a differentiable part (defined over an open set) and a possibly nonsmooth term. The approach, referred to as the Variable Bregman Majorization-Minimization (VBMM) algorithm, extends the Bregman Proximal Gradient method by allowing the Bregman function used in the divergence to adaptively vary at each iteration, provided it satisfies a majorizing condition on the objective function. This adaptive framework enables the algorithm to approximate the objective more precisely at each iteration, thereby allowing for accelerated convergence compared to the traditional Bregman Proximal Gradient descent. We establish the convergence of the VBMM algorithm to a minimizer under mild assumptions on the family of metrics used. Furthermore, we introduce a novel application of both the Bregman Proximal Gradient method and the VBMM algorithm to the estimation of the multidimensional parameters of a Dirichlet distribution through the maximization of its log-likelihood. Numerical experiments confirm that the VBMM algorithm outperforms existing approaches in terms of convergence speed.

In this paper, we explore the $\mathbb{Z}_2^n$-graded Lie (super)algebras as novel possible generators of symmetries of $S$-matrix. As the results, we demonstrate that a $\mathbb{Z}_2^n$-graded extension of the supersymmetric algebra can be a symmetry of $S$-matrix. Furthermore, it turns out that a $\mathbb{Z}_2^n$-graded Lie algebra appears as internal symmetries. They are natural extensions of Coleman-Mandula theorem and Haag-Lopszanski-Sohnius theorem, which are the no-go theorems for generators of symmetries of $S$-matrix.

Predicting dynamic behaviors is one of the goals of science in general as well as essential to many specific applications of human knowledge to real world systems. Here we introduce an analytic approach using the sigmoid growth curve to model the dynamics of individual entities within complex systems. Despite the challenges posed by nonlinearity and unpredictability in system behaviors, we demonstrate the applicability of the sigmoid curve to capture the acceleration and deceleration of growth, predicting an entitys ultimate state well in advance of reaching it. We show that our analysis can be applied to diverse systems where entities exhibit nonlinear growth using case studies of (1) customer purchasing and (2) U.S. legislation adoption. This showcases the ability to forecast months to years ahead of time, providing valuable insights for business leaders and policymakers. Moreover, our characterization of individual component dynamics offers a framework to reveal the aggregate behavior of the entire system. We introduce a classification of entities based upon similar lifepaths. This study contributes to the understanding of complex system behaviors, offering a practical tool for prediction and system behavior insight that can inform strategic decision making in multiple domains.

We develop a conformally invariant (CI) framework in $(1+3)$-dimensional de Sitter (dS) spacetime, that unifies the descriptions of graviton, ``partially massless'' graviton, and photon fields. This framework is grounded in a rigorous group-theoretical analysis in the Wigner sense and employs Dirac's six-cone formalism. Originally introduced by Dirac, the concept of conformal space and the six-cone formalism were used to derive the field equations for spinor and vector fields in $(1+3)$-dimensional Minkowski spacetime in a manifestly CI form. Our framework extends this approach to dS spacetime, unifying the treatment of massless and ``partially massless'' fields with integer spin $s>0$ under conformal symmetry. This unification enhances the understanding of fundamental aspects of gravitational theories in curved backgrounds.

We derive explicit equations governing the cumulative biases and weights in Deep Learning with ReLU activation function, based on gradient descent for the Euclidean cost in the input layer, and under the assumption that the weights are, in a precise sense, adapted to the coordinate system distinguished by the activations. We show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. We provide a detailed discussion of several types of solutions to the gradient flow equations. A main motivation for this work is to shed light on the interpretability question in supervised learning.

Despite their simplicity, quantum harmonic oscillators are ubiquitous in the modeling of physical systems. They are able to capture universal properties that serve as reference for the more complex systems found in nature. In this spirit, we apply a model of a Hamiltonian for open quantum systems in equilibrium with a particle reservoir to ensembles of quantum oscillators. By treating (i) a dilute gas of vibrating particles and (ii) a chain of coupled oscillators as showcases, we demonstrate that the property of varying number of particles leads to a mandatory condition on the energy of the system. In particular, the chemical potential plays the role of a parameter of control that can externally manipulate the spectrum of a system and the corresponding accessible quantum states.

We prove that the class of 3D-grids is cannot be transduced from planar graphs, and more generally, from any class of graphs of bounded Euler genus. To prove our result, we introduce a new structural tool called slice decompositions, and show that every graph class transducible from a class of graphs of bounded Euler genus is a perturbation of a graph class that admits slice decompositions.