In this paper, we investigate the existence of the asymptotically almost automorphic solution of the following type of abstract nonlinear integro-dynamic equation \begin{eqnarray*} y^{\Delta}(s) &=&Ay(s)+\mathcal{F}\left(s,y(s),\int\limits_{t_0}^{s}{\mathcal{H}(s,\tau,y(\tau))}\Delta\tau\right),~ s\in\mathbb{T}^k, y(0)&=&y_0 \end{eqnarray*} in the Banach space of continuous function on a time scale $\mathbb{T}$. We apply the Krasnoselskii fixed point theorem to show the existence of an almost automorphic solution of the above dynamic equation.

The fractional diffraction optics theory has been elaborated using the Green function technique. The optics-fractional equation describing the diffraction X-ray scattering by imperfect crystals has been derived as the fractional matrix integral Fredholm--Volterra equation of the second kind. In the paper, to solve the Cauchy problems, the Liouville--Neumann-type series formalism has been used to build up the matrix Resolvent-function solution. In the case when the imperfect crystal-lattice elastic displacement field is the linear function $f({\bf R}) = a x+b$, $a, b = const,$ the explicit solution of the diffraction-optics Cauchy problem has been obtained and analyzed for arbitrary fractional-order-parameter $\alpha$, $\alpha\in (0, 1].$

Spinors are used in physics quite extensively. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Tribonacci spinors using the generalized Tribonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini-like formulas which are given for some series of numbers in mathematics for Tribonacci spinors.

This paper introduces the Token Space framework, a novel mathematical construct designed to enhance the interpretability and effectiveness of deep learning models through the application of category theory. By establishing a categorical structure at the Token level, we provide a new lens through which AI computations can be understood, emphasizing the relationships between tokens, such as grouping, order, and parameter types. We explore the foundational methodologies of the Token Space, detailing its construction, the role of construction operators and initial categories, and its application in analyzing deep learning models, specifically focusing on attention mechanisms and Transformer architectures. The integration of category theory into AI research offers a unified framework to describe and analyze computational structures, enabling new research paths and development possibilities. Our investigation reveals that the Token Space framework not only facilitates a deeper theoretical understanding of deep learning models but also opens avenues for the design of more efficient, interpretable, and innovative models, illustrating the significant role of category theory in advancing computational models.

Both the USA TST 2008 and the ELMO Shortlist 2013 suggested two issues that are connected to fixed points. These problems provide a strong linkage between the various attributes of specific points in a triangle. In this article, we will first investigate various theorems concerning the fixed points that have been presented, and then we will demonstrate how those points are connected to a few triangle centers.

We are convinced of the usefulness of sketches and diagrams during mathematical work but the observation is made in our practices that they are not spontaneously used by students. In order to study the understanding and use of sketches by mathematics students, we designed and then proposed a test at different university levels. The test consists of five exercises.The first concerns different representation registers of a set of numbers, the second on a graphic proof of an implicative algebraic proposition and the last three on the graphic approch to the notions of injectivity, surjectivity, bijectivity in the context of the analysis. The sketches, proposed or requested in each exercise, are intended to be aids to changes of register and reasoning. We present what motivated the choices and developments of the exercises then we analyze the results of these tests. In each case, we see difficulties in understanding and the sketches proposed, which leads to think that the sketch must be the subject of specific work at the university level.

In this paper, we are concerned with the sign-changing solutions of variational inequality problems. In order to give the existence results of the sign-changing solutions for variational inequality problems, we first construct a suitable penalty problem related to the variational inequality, and prove the existence of a sign-changing solution for this penalty problem using the invariant set of descending flow method. Secondly, we perform a series of estimates on the sign-changing solution sequence of the penalty problem, and prove that the limit of convergence of the sign-changing solution sequence is the sign-changing solution of the original variational inequality problem. Variational inequality problems have extensive and important applications. In this paper, we used penalty method and the method of invariant set of descending flow to obtain the existence results for solutions of a variational inequality. In particular, we obtain the existence results for sign-changing solutions of variational inequalities for the first time based on the method of invariant set of descending flow.

In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation*} -\Delta^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation*} without the finite volume constraint for $n \geq 2$ and $\frac{(n+1)}{3} \leq p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent and $-\Delta^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality.

This paper aims to present objective methods for constructing new fuzzy sets from known fuzzy or classical sets, defined over the elements of a finite universe's superstructure. The paper proposes rules for assigning membership functions to these new fuzzy sets, leading to two important findings. Firstly, the property concerning the cardinality of a power set in classical theory has been extended to the fuzzy setting, whereby the scalar cardinality of a fuzzy set $\tilde B$ defined on the power set of a finite universe of a fuzzy set $\tilde A$ satisfies $\text{card}(\tilde B)=2^{\text{card}(\tilde A)}$. Secondly, the novel algorithms allow for an arbitrary membership value to be objectively achieved and represented by a specific binary sequence.

We present an exposition of the *Chain Bounding Lemma*, which is a common generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest ones that we are aware of. As a by-product, we show that for every poset $P$ and a function $f$ from the powerset of $P$ into $P$, there exists a maximal well-ordered chain whose family of initial segments is appropriately closed under $f$. We also provide a "computer formalization" of our main results using the Lean proof assistant.

We provide the problems and their solutions to the 2020 USA Mathematical Olympiad.

We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space), sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in the unit cube of $\mathbb{R}^d$ can be decomposed into a controlled number of subsets that are "well-connected" within the original set, along with a "garbage set" of arbitrarily small measure. Our results are quantitative, i.e., they provide bounds independent of the particular set under consideration.

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(\epsilon x,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2, $$ where $\epsilon>0$ and $V$ is a double-well potential with minima at $t=\alpha$ and $t=\beta$ with $\alpha<\beta$. Here, we consider some class of functions $A(x,y)$ that are oscillatory in the variable $y$ and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity.

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen-Cahn type given by $$ -\left(\phi(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R}, $$ where $V$ is a double-well potential with minima at $t=\pm\alpha$ and $\phi:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $\phi(t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $\phi(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator.

Mean field games (MFGs) model equilibria in games with a continuum of weakly interacting players as limiting systems of symmetric $n$-player games. We consider the finite-state, infinite-horizon problem with ergodic cost. Assuming Markovian strategies, we first prove that any solution to the MFG system gives rise to a $(C/\sqrt{n})$-Nash equilibrium in the $n$-player game. We follow this result by proving the same is true for the strategy profile derived from the master equation. We conclude the main theoretical portion of the paper by establishing a large deviation principle for empirical measures associated with the asymptotic Nash equilibria. Then, we contrast the asymptotic Nash equilibria using an example. We solve the MFG system directly and numerically solve the ergodic master equation by adapting the deep Galerkin method of Sirignano and Spiliopoulos. We use these results to derive the strategies of the asymptotic Nash equilibria and compare them. Finally, we derive an explicit form for the rate functions in dimension two.

Over the past three decades, it has been shown that discrete and continuous media can support topologically nontrivial modes. Recently, it was shown that the same is true of the vacuum, namely, right (R) and left (L) circularly polarized photons are topologically nontrivial. Here, we study the topology of another class of massless particles, namely the gravitons. Working in the transverse-traceless gauge and the limit of weak gravity, we show that the collection of all gravitons forms a rank-two vector bundle over the lightcone. We prove the graviton bundle is topologically trivial, allowing us to discover a globally smooth basis for gravitons. It has often been assumed that there exists such a global basis consisting of linear polarized gravitons. We prove that this stronger assumption is false--the graviton bundle has no linearly polarized subbundles. While the total graviton bundle can be decomposed into trivial line bundles, it also breaks apart into two nontrivial $\mathrm{SO}(3)$ invariant subbundles, consisting of the R and L gravitons. Unlike the bundles in the trivial decomposition, the R and L gravitons are in fact irreducible bundle representations of the Poincar\'{e} group, and are thus elementary particles. The nontrivial topologies of the R and L gravitons are fully characterized by the Chern numbers $\mp 4$. These topologies differ from those of the R and L photons, which are characterized by the Chern numbers $\mp 2$. This nontrivial topology obstructs the splitting of graviton angular momentum into spin and orbital angular momentum.

The goal of this paper is to investigate the existence of saddle solutions for some classes of elliptic partial differential equations of the Allen-Cahn type, formulated as follows: \begin{equation*} -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(x,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2. \end{equation*} Here, the function $A:\mathbb{R}^2\to\mathbb{R}$ exhibits periodicity in all its arguments, while $V:\mathbb{R}\to\mathbb{R}$ characterizes a double-well symmetric potential with minima at $t=\pm\alpha$.

We prove the Central Limit Theorem and superpolynomial mixing for environment viewed for the particle process in quasi periodic Diophantine random environment. The main ingredients are smoothness estimates for the solution of the Poisson equation and local limit asymptotics for certain accelerated walks.

For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are equivalent to $3$-colorings. In this paper we are interested in the case that $H$ is the five-vertex cycle $C_5$. A minimal obstruction to $C_5$-coloring is a graph that does not have a $C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring. In this paper we are interested in minimal obstructions to $C_5$-coloring in $F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of their endvertices. We show that there is only a finite number of minimal obstructions to $C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1}, S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the results of Kami\'nski and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of $P_7$-free minimal obstructions to $C_5$-coloring, and of D\k{e}bski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring. We complement our results with a construction of an infinite family of minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal obstructions to $H$-coloring for other graphs $H$.

We compute the large size limit of the moment formula derived in \cite{DHS} for the Hermitian Jacobi process at fixed time. In particular, we identify the terms contributing to the limit and show they satisfy a double recurrence relation. We also determine explicitly some of them and revisit a special case relying on Carlitz summation identity for terminating $1$-balanced ${}_4F_3$ functions taken at unity.

How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$, the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$. By studying its cohomology ring, we prove that, as long as $n$ is greater than $w$, the topological complexity of $\text{conf}(n,w)$ is $2n-2\big\lceil\frac{n}{w}\big\rceil+1$, providing a lower bound for the minimum number of cases such a program must consider.

Penalty methods relax the incompressibility condition and uncouple velocity and pressure. Experience with them indicates that the velocity error is sensitive to the choice of penalty parameter $\epsilon$. So far, there is no effective \'a prior formula for $\epsilon$. Recently, Xie developed an adaptive penalty scheme for the Stokes problem that picks the penalty parameter $\epsilon$ self-adaptively element by element small where $\nabla \cdot u^h$ is large. Her numerical tests gave accurate fluid predictions. The next natural step, developed here, is to extend the algorithm with supporting analysis to the non-linear, time-dependent incompressible Navier-Stokes equations. In this report, we prove its unconditional stability, control of $\|\nabla \cdot u^h\|$, and provide error estimates. We confirm the predicted convergence rates with numerical tests.

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).

We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements $w$. We construct inductive systems of finite-dimensional $\mathcal{U}_q\mathfrak{b}$-modules twisted by $w$, which provide representations in the category $\mathcal{O}^w$. We also establish a classification of simple modules in these categories $\mathcal{O}^w$. We explore convergent phenomenon of $q$-characters of representations of quantum affine algebras, which conjecturally give the $q$-characters of representations in $\mathcal{O}^w$. Furthermore, we propose a conjecture concerning the relationship between the category $\mathcal{O}$ and the twisted category $\mathcal{O}^w$, and we propose a possible connection with shifted quantum affine algebras.

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

We consider the nonlinear Schr\"odinger equation with nonzero conditions at infinity in $\R^2$. We investigate the existence of traveling waves that are periodic in the direction transverse to the direction of propagation and minimize the energy when the momentum is kept fixed. We show that for any given value of the momentum, there is a critical value of the period such that traveling waves with period smaller than the critical value are one-dimensional, and those with larger periods depend on two variables.

A multi-species Fokker-Planck model for simulating particle collisions in a plasma is presented. The model includes various parameters that must be tuned. Under reasonable assumptions on these parameters, the model satisfies appropriate conservation laws, dissipates an entropy, and satisfies an $\mathcal{H}$-Theorem. In addition, the model parameters provide the additional flexibility that is used to match simultaneously momentum and temperature relaxation formulas derived from the Boltzmann collision operator for a binary mixture with Coulomb potentials. A numerical method for solving the resulting space-homogeneous kinetic equation is presented and two examples are provided to demonstrate the relaxation of species bulk velocities and temperatures to their equilibrium values.

For continuous decision spaces, nonlinear programs (NLPs) can be efficiently solved via sequential quadratic programming (SQP) and, more generally, sequential convex programming (SCP). These algorithms linearize only the nonlinear equality constraints and keep the outer convex structure of the problem intact. The aim of the presented sequential mixed-integer quadratic programming (MIQP) algorithm for mixed-integer nonlinear problems (MINLPs) is to extend the SQP/SCP methodology to MINLPs and leverage the availability of efficient MIQP solvers. The algorithm employs a three-step method in each iterate: First, the MINLP is linearized at a given iterate. Second, an MIQP with its feasible set restricted to a specific region around the current linearization point is formulated and solved. Third, the integer variables obtained from the MIQP solution are fixed, and only an NLP in the continuous variables is solved. The outcome of the third step is compared to previous iterates, and the best iterate so far is used as a linearization point in the next iterate. Crucially, the objective values and derivatives from all previous iterates are used to formulate the polyhedral region in the second step. The linear inequalities that define the region build on concepts from generalized Benders' decomposition for MINLPs. Although the presented MINLP algorithm is a heuristic method without any global optimality guarantee, it converges to the exact integer solution when applied to convex MINLP with a linear outer structure. The conducted numerical experiments demonstrate that the proposed algorithm is competitive with other open-source solvers for MINLP. Finally, we solve two mixed-integer optimal control problems (MIOCPs) transcribed into MINLPs via direct methods, showing that the presented algorithm can effectively deal with nonlinear equality constraints, a major hurdle for generic MINLP solvers.

We consider stochastic optimization problems with functional constraints. If the objective and constraint functions are not convex, the classical stochastic approximation algorithms such as the proximal stochastic gradient descent do not lead to efficient algorithms. In this work, we put forth an accelerated SCA algorithm that utilizes the recursive momentum-based acceleration which is widely used in the unconstrained setting. Remarkably, the proposed algorithm also achieves the optimal SFO complexity, at par with that achieved by state-of-the-art (unconstrained) stochastic optimization algorithms and match the SFO-complexity lower bound for minimization of general smooth functions. At each iteration, the proposed algorithm entails constructing convex surrogates of the objective and the constraint functions, and solving the resulting convex optimization problem. A recursive update rule is employed to track the gradient of the objective function, and contributes to achieving faster convergence and improved SFO complexity. A key ingredient of the proof is a new parameterized version of the standard Mangasarian-Fromowitz Constraints Qualification, that allows us to bound the dual variables and hence establish that the iterates approach an $\epsilon$-stationary point. We also detail a obstacle-avoiding trajectory optimization problem that can be solved using the proposed algorithm, and show that its performance is superior to that of the existing algorithms. The performance of the proposed algorithm is also compared against that of a specialized sparse classification algorithm on a binary classification problem.

We provide two candidates for symplectic Weiss calculus based on two different, but closely related, collections of groups. In the case of the non-compact symplectic groups, i.e., automorphism groups of vector spaces with symplectic forms, we show that the calculus deformation retracts onto unitary calculus as a corollary of the fact that Weiss calculus only depends on the homotopy type of the groupoid core of the diagram category. In the case of the compact symplectic groups, i.e., automorphism groups of quaternion vector spaces, we provide a comparison with the other known versions of Weiss calculus analogous to the comparisons of calculi of the second named author, and classify certain stably trivial quaternion vector bundles over finite cell complexes in a range, using elementary results on convergence of Weiss calculi.

Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the recently developed persistent Laplacians. However, classic mathematical models like simplicial complexes often struggle to provide a localized topological description for interactions or individual elements within a complex system involving a specific set of elements. In this work, we introduce persistent interaction homology and persistent interaction Laplacian that emphasize individual interacting elements in the system. We demonstrate the stability of persistent interaction homology as a persistent module. Furthermore, for a finite discrete set of points in the Euclidean space, we provide the construction of persistent interaction Vietoris-Rips complexes and compute their interaction homology and interaction Laplacians. The proposed methods hold significant promise for analyzing heterogeneously interactive data and emphasizing specific elements in data. Their utility for data science is demonstrated with applications to molecules.

In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final graph with a self-similar structure. The number of spanning trees of a graph is one of the most graph-theoretical parameters, where its applications range from the theory of networks to theoretical chemistry. Two explicit formulas are introduced for the number of spanning trees for the two models. With explicit formulas for some of their topological parameters as well.

In this paper, by using $|x|=2\max\{0,x\}-x$, a class of maximum-based iteration methods is established to solve the generalized absolute value equation $Ax-B|x|=b$. Some convergence conditions of the proposed method are presented. By some numerical experiments, the effectiveness and feasibility of the proposed method are confirmed.

The main object of this paper is the minus class groups associated to CM-fields as Galois modules. In a previous article of the authors, we introduced a notion of equivalence for modules and determined the equivalence classes of the minus class groups. In this paper, we show a concrete application of this result. We also study how large a proportion of equivalence classes can be realized as classes of minus class groups.

This study introduces a new approach to optimize the geometrical parameters of pipe diffusers in centrifugal compressors for Micro Gas Turbines, tailored for a 100 kW unit. The methodology draws insights from optimized airfoil-type diffusers and addresses the unique topological challenges of pipe diffusers, using diffuser maps to enhance design precision. The effectiveness of this method is validated through 3D-RANS based steady CFD simulations, using the ANSYS CFX solver. Comparative performance assessments at 100 percent rotation speed show that the best-performing pipe diffuser slightly trails its airfoil counterpart in efficiency, achieving 82.2 percent total-to-total isentropic efficiency compared to 84.4 percent. However, it offers a reduced frontal area, enhancing compactness. The analysis also reveals a dualistic impact from the leading-edge geometry of the pipe diffuser, which generates two counter-rotating vortices. These vortices have beneficial effects in pseudo and semi-vaneless spaces while introducing destabilizing factors in channel spaces. This investigation highlights potential trade-offs and outlines conditions under which adverse effects dominate, leading to significant flow separation. These insights pave the way for refining diffuser designs to better balance performance with spatial efficiency, marking a critical step forward in compressor technology of micro gas turbine for decentralized power systems.

Our aim in this article is to compute the mixed volume of a matroid. We give two computations. The first one is based on the integration formula for complete fans given by Brion. The second computation is a step-by-step method using deletion of elements in the matroid as in Braden-Huh-Matherne-Proudfoot-Wang.

A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar.

We study the geometry of Calabi-Yau conifold transitions. This deformation process is known to possibly connect a K\"ahler threefold to a non-K\"ahler threefold. We use balanced and Hermitian-Yang-Mills metrics to geometrize the conifold transition and show that the whole operation is continuous in the Gromov-Hausdorff topology.

In this paper, we study the discrete Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+V(x) u=\left(R_{\alpha} *F(u)\right)f(u),\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$ are constants, $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian with $\alpha \in(0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.

We derive the electromagnetic medium equivalent to a collection of all-dielectric nano-particles (enjoying high refractive indices) distributed locally non-periodically in a smooth domain $\Omega$. Such distributions are used to model well known structures in material sciences as the Van-der-Waals heterostructures. Since the nano-particles are all-dielectric, then the permittivity remains unchanged while the permeability is altered by this effective medium. This equivalent medium describes, in particular, the effective medium of 2 dimensional type Van-der-Waals heterostructures. These structures are 3 dimensional which are build as superposition of identical (2D)-sheets each supporting locally non-periodic distributions of nano-particles. An explicit form of this effective medium is provided for the particular case of honeycomb heterostructures. At the mathematical analysis level, we propose a new approach to derive the effective medium when the subwavelength nano-particles are distributed non-periodically. The first step consists in deriving the point interaction approximation, also called the Foldy-Lax approximation. The scattered field is given as a superposition of dipoles (or poles for other models) multiplied by the elements of a vector which is itself solution of an algebraic system. This step is done regardless of the way how the particles are distributed. As a second step, which is the new and critical step, we rewrite this algebraic system according to the way how these nano-particles are locally distributed. The new algebraic system will then fix the related continuous Lippmann Schwinger system which, in its turn, indicates naturally the equivalent medium.

We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). For any positive radial decreasing blow-up solution, we obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$, which takes the form: $$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg), \ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile and the refined space-time profile. As a remarkable fact and completely new observation, our results reveal a {\it structural universality} of the global blow-up profile, being given by the "resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case.

This paper aims to investigate the evolution problem for planar curves with singularities. Motivated by the inverse curvature flow introduced by Li and Wang (Calc. Var. Partial Differ. Equ. 62 (2023), No. 135), we intend to consider the area-preserving and length-preserving inverse curvature flow with nonlocal term for $\ell$-convex Legendre curves. For the area-preserving flow, an $\ell$-convex Legendre curve %of with initial algebraic area $A_0>0$ evolves to a circle of radius $\sqrt{\frac{A_0}{\pi}}$. For the length-preserving flow, an $\ell$-convex Legendre curve %of with initial algebraic length $L_0$ evolves to a circle of radius $\frac{L_0}{2\pi}$. As the by-product, we obtain some geometric inequalities for $\ell$-convex Legendre curves through the length-preserving flow.

We study the Abel-Jacobi image of the Ceresa cycle $W_{k, e}-W_{k, e}^-$, where $W_{k, e}$ is the image of the $k$th symmetric product of a curve $X$ with a base point $e$ on its Jacobian variety. For certain Fermat quotient curves of genus $g$, we prove that for any choice of the base point and $k \leq g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.

In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green's function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t)^{-1}.$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\|\omega_{0}\|_{H^{1}\cap L^1}\leq c_0\nu^{\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $\nu$, then we can reach the conclusion that the solution remains within $O\left( \nu ^{\frac{3}{4}}\right) $ of the Couette flow.

Mond and Pecaric proposed a powerful method, namd as MP method, to deal with operator inequalities. However, this method requires a real-valued function to be convex or concave, and the normalized positive linear map between Hilbert spaces. The objective of this study is to extend the MP method by allowing non-convex or non-concave real-valued functions and nonlinear mapping between Hilbert spaces. The Stone-Weierstrass theorem and Kantorovich function are fundamental components employed in generalizing the MP method inequality in this context. Several examples are presented to demonstrate the inequalities obtained from the conventional MP method by requiring convex function with a normalized positive linear map. Various new inequalities regarding hypercomplex functions, i.e., operator-valued functions with operators as arguments, are derived based on the proposed MP method. These inequalities are applied to approximate hypercomplex functions using ratio criteria and difference criteria. Another application of these new inequalities is to establish bounds for hypercomplex functions algebra, i.e., an abelian monoid for the addition or multiplication of hypercomplex functions, and to derive tail bounds for random tensors ensembles addition or multiplication systematically.

Movable antenna (MA) is an emerging technology that utilizes localized antenna movement to pursue better channel conditions for enhancing communication performance. In this paper, we study the MA-enhanced multicast transmission from a base station equipped with multiple MAs to multiple groups of single-MA users. Our goal is to maximize the minimum weighted signal-to-interference-plus-noise ratio (SINR) among all the users by jointly optimizing the position of each transmit/receive MA and the transmit beamforming. To tackle this challenging problem, we first consider the single-group scenario and propose an efficient algorithm based on the techniques of alternating optimization and successive convex approximation. Particularly, when optimizing transmit or receive MA positions, we construct a concave lower bound for the signal-to-noise ratio (SNR) of each user by applying only the second-order Taylor expansion, which is more effective than existing works utilizing two-step approximations. The proposed design is then extended to the general multi-group scenario. Simulation results demonstrate that significant performance gains in terms of achievable max-min SNR/SINR can be obtained by our proposed algorithm over benchmark schemes. Additionally, the proposed algorithm can notably reduce the required amount of transmit power or antennas for achieving a target level of max-min SNR/SINR performance compared to benchmark schemes.

We prove a bordism version of Gromov's linearity conjecture over a large family of acyclic groups. Since all groups embed into these acyclic groups, it follows that the linear bordism conjecture is true if one allows to enlarge a given group. Our result holds in both PL and smooth categories, and for both oriented and unoriented cases. The method of the proof hinges on quantitative algebraic and geometric techniques over infinite complexes with unbounded local (combinatorial) geometry, which seem interesting on their own. As an application, we prove that there is a universal linear bound for the Cheeger-Gromov $L^2$ $\rho$-invariants of PL $(4k-1)$-manifolds associated with arbitrary regular covers.

We formulate a novel approach to solve a class of stochastic problems, referred to as data-consistent inverse (DCI) problems, which involve the characterization of a probability measure on the parameters of a computational model whose subsequent push-forward matches an observed probability measure on specified quantities of interest (QoI) typically associated with the outputs from the computational model. Whereas prior DCI solution methodologies focused on either constructing non-parametric estimates of the densities or the probabilities of events associated with the pre-image of the QoI map, we develop and analyze a constrained quadratic optimization approach based on estimating push-forward measures using weighted empirical distribution functions. The method proposed here is more suitable for low-data regimes or high-dimensional problems than the density-based method, as well as for problems where the probability measure does not admit a density. Numerical examples are included to demonstrate the performance of the method and to compare with the density-based approach where applicable.

In the field of brain science, data sharing across servers is becoming increasingly challenging due to issues such as industry competition, privacy security, and administrative procedure policies and regulations. Therefore, there is an urgent need to develop new methods for data analysis and processing that enable scientific collaboration without data sharing. In view of this, this study proposes to study and develop a series of efficient non-negative coupled tensor decomposition algorithm frameworks based on federated learning called FCNCP for the EEG data arranged on different servers. It combining the good discriminative performance of tensor decomposition in high-dimensional data representation and decomposition, the advantages of coupled tensor decomposition in cross-sample tensor data analysis, and the features of federated learning for joint modelling in distributed servers. The algorithm utilises federation learning to establish coupling constraints for data distributed across different servers. In the experiments, firstly, simulation experiments are carried out using simulated data, and stable and consistent decomposition results are obtained, which verify the effectiveness of the proposed algorithms in this study. Then the FCNCP algorithm was utilised to decompose the fifth-order event-related potential (ERP) tensor data collected by applying proprioceptive stimuli on the left and right hands. It was found that contralateral stimulation induced more symmetrical components in the activation areas of the left and right hemispheres. The conclusions drawn are consistent with the interpretations of related studies in cognitive neuroscience, demonstrating that the method can efficiently process higher-order EEG data and that some key hidden information can be preserved.

In this paper, we present a novel derivative-free optimization framework for solving unconstrained stochastic optimization problems. Many problems in fields ranging from simulation optimization to reinforcement learning involve settings where only stochastic function values are obtained via an oracle with no available gradient information, necessitating the usage of derivative-free optimization methodologies. Our approach includes estimating gradients using stochastic function evaluations and integrating adaptive sampling techniques to control the accuracy in these stochastic approximations. We consider various gradient estimation techniques including standard finite difference, Gaussian smoothing, sphere smoothing, randomized coordinate finite difference, and randomized subspace finite difference methods. We provide theoretical convergence guarantees for our framework and analyze the worst-case iteration and sample complexities associated with each gradient estimation method. Finally, we demonstrate the empirical performance of the methods on logistic regression and nonlinear least squares problems.

In this paper we study the spectrum of a fundamental differential operator on a Hilbert-P\'olya space. A number is an eigenvalue of this differential operator if and only if it is a nontrivial zero of the Riemann zeta function. An explicit formula is given for the eigenfunction associated with each nontrivial zero of the zeta function. Every eigenfunction is characterized via the Poisson summation formula by a sequence of mysterious functions whose explicit formulas are given.

Let $\pi: X_r \rightarrow \mathbb{P}^2$ be a blow up of $\mathbb{P}^2$ at $r$ distinct points $p_1,p_2,\dots, p_r$. We study lower bounds for Seshadri constants of ample line bundles on $X_r$. First, we consider the case when $p_1,p_2,\dots, p_r$ are on a curve of degree $d\leq 3$. Then we assume that the points are very general and show that $\varepsilon(X_r,L,x)\geq 1$ for any ample line bundle $L$ and any $x\in X_r$ if the Strong SHGH conjecture is true. We explore the relation between bounded negativity and Seshadri constants and study the Seshadri function on $X_r$.

In this paper, we show that the optimal fundamental estimate holds true on a weakly $1$-complete manifold with mild conditions, then we establish the weak Morse inequalities for lower energy forms on the manifold. We also study the case for $q$-convex manifolds.

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.

We give a computational implementation of the Covering Lemma for finite transformation semigroups. The lemma states that given a surjective relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can recover the information lost in the morphism and package that into a third transformation semigroup in such a way that a cascade product (subsemigroup of the wreath product) can emulate $(X,S)$, providing a hierarchical way of understanding its structure and dynamics: $(X,S)\hookrightarrow (Y,T)\wr (Z,U)$. The implementation complements the existing tools for the holonomy decomposition algorithm as it gives an incremental method to get a coarser decomposition when computing the complete skeleton for holonomy is not feasible. Here we describe a simplified and generalized algorithm for the lemma and compare it to the holonomy method. Incidentally, the computational Covering Lemma could be the easiest way of understanding the hierarchical decompositions of transformation semigroups and thus the celebrated Krohn-Rhodes theory.

The mean curvature flow describes the evolution of a surface(a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical method for mean curvature flows by using the Onsager principle as an approximation tool. We first show that the mean curvature flow can be derived naturally from the Onsager variational principle. Then we consider a piecewisely linear approximation of the curve and derive a discrete geometric flow. The discrete flow is described by a system of ordinary differential equations for the nodes of the discrete curve. We prove that the discrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system can be solved by an improved Euler scheme and this leads to an efficient fully discrete scheme. We first consider the method for a simple mean curvature flow and then extend it to volume preserving mean curvature flow and also a wetting problem on substrates. Numerical examples show that the method has optimal convergence rate and works well for all the three problems.

We introduce an interacting particle system which models the inherited sterility method. Individuals evolve on $\mathbb{Z}^d$ according to a contact process with parameter $\lambda>0$. With probability $p \in [0,1]$ an offspring is fertile and can give birth to other individuals at rate $\lambda$. With probability $1-p$, an offspring is sterile and blocks the site it sits on until it dies. The goal is to prove that at fixed $\lambda$, the system survives for large enough $p$ and dies out for small enough $p$. The model is not attractive, since an increase of fertile individuals potentially causes that of sterile ones. However, thanks to a comparison argument with attractive models, we are able to answer our question.

We compute $\delta$-derivations of simple Jordan algebras with values in irreducible bimodules. They turn out to be either ordinary derivations ($\delta = 1$), or scalar multiples of the identity map ($\delta = \frac 12$). This can be considered as a generalization of the "First Whitehead Lemma" for Jordan algebras which claims that all such ordinary derivations are inner. The proof amounts to simple calculations in matrix algebras, or, in the case of Jordan algebras of a symmetric bilinear form, to more elaborated calculations in Clifford algebras.

This paper focuses on multi-agent stochastic differential games for jump-diffusion systems. On one hand, we study the multi-agent game for optimal investment in a jump-diffusion market. We derive constant Nash equilibria and provide sufficient conditions for their existence and uniqueness for exponential, power, and logarithmic utilities, respectively. On the other hand, we introduce a computational framework based on the actor-critic method in deep reinforcement learning to solve the stochastic control problem with jumps. We extend this algorithm to address the multi-agent game with jumps and utilize parallel computing to enhance computational efficiency. We present numerical examples of the Merton problem with jumps, linear quadratic regulators, and the optimal investment game under various settings to demonstrate the accuracy, efficiency, and robustness of the proposed method. In particular, neural network solutions numerically converge to the derived constant Nash equilibrium for the multi-agent game.

We study the principle phi implies box phi, known as `Strength' or `the Completeness Principle', over the constructive version of L\"ob's Logic. We consider this principle both for the modal language with the necessity operator and for the modal language with the Lewis arrow, where L\"ob's Logic is suitably adapted. Central insights of provability logic, like the de Jongh-Sambin Theorem and the de Jongh-Sambin-Bernardi Theorem, take a simple form in the presence of Strength. We present these simple versions. We discuss the semantics of two salient systems and prove uniform interpolation for both. In addition, we sketch arithmetical interpretations of our systems. Finally, we describe the various connections of our subject with Computer Science.

In this paper, we introduce two moduli of w*-semidenting points and characterise the Mazur Intersection Property (MIP) and the Uniform MIP (UMIP) in terms of these moduli. We show that a property slightly stronger than UMIP already implies uniform convexity of the dual. This may lead to a possible approach towards answering the long standing open question whether the UMIP implies the existence of an equivalent uniformly convex renorming. We also obtain the condition for stability of the UMIP under $\ell_p$-sums.

We give a description of finite-zone PT-potentials in terms of explicit theta functional formulas.

We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We work with the dissipative weak solutions that can be seen as a universal closure of consistent approximations. Further, we apply the set-valued version of the Strong law of large numbers for general multivalued mapping with closed range and the Koml\'os theorem on strong converge of empirical averages of integrable functions. Theoretical results are illustrated by a series of numerical simulations obtained by an unconditionally convergent viscosity finite volume method combined with the Monte Carlo method.

This paper introduces a novel approach to analyzing overlapping Schwarz methods for N\'{e}d\'{e}lec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.

In this paper, we study Pohozaev identities, Kelvin transformation and their applications of semilinear Grushin equation. First, we establish two Pohozaev identities generated from translations and determine the location of the concentration point for solution of a kind of Grushin equation by such identities. Next, we establish Pohozaev identity generated from scaling and prove the nonexistence of nontrivial solutions of another kind of Grushin equation by such identity. Finally, we provide the change of Grushin operator by Kelvin transformation and obtain the decay rate of solution at infinity for a critical Grushin equation by Kelvin transformation.

A closed formula for the spectral determinant for the wave equation on a bounded interval, subject to Dirichlet boundary conditions and an $\alpha$-multiple of the Dirac $\delta$-type damping, is derived. Depending on the choice of the branch cut of the logarithm used in its definition, the spectral determinant diverges either for $\alpha =2$ or $\alpha=-2$.

For a Lagrangian system with nonholonomic constraints, we construct extensions of the equations of motion to sets of second-order ordinary differential equations. In the case of a purely kinetic Lagrangian, we investigate the conditions under which the nonholonomic trajectories are geodesics of a Riemannian metric, while preserving the constrained Lagrangian. We interpret the algebraic and PDE conditions of this problem as infinitesimal versions of the relation between the nonholonomic exponential map and the Riemannian metric. We discuss the special case of a Chaplygin system with symmetries and we end the paper with a worked-out example.

In this article we prove analogs of Kawamata's canonical bundle formula, Kawamata subadjunction and plt/lc inversion of adjunction for generalized pairs on Kaehler varieties. We also show that a conjecture of BDPPin dimension n-1 implies that the cone theorem holds for any n-dimensional Kaehler generalized klt pair. Along the way, we obtain more complete versions of some results due to Collins-Tosatti and Cao-Hoering.

This paper is concerned with the Hamiltonian elliptic system in dimension two\begin{equation*}\aligned \left\{ \begin{array}{lll} -\epsilon^2\Delta u+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -\epsilon^2\Delta v+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation*} where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be of exponential growth in the sense of Trudinger-Moser inequality. When $V$ admits one or several local strict minimum points, we show the existence and concentration of single-peak and multi-peak semiclassical states respectively, as well as strong convergence and exponential decay. In addition, positivity of solutions and uniqueness of local maximum points of solutions are also studied. Our theorems extend the results of Ramos and Tavares [Calc. Var. 31 (2008) 1-25], where $f$ and $g$ have polynomial growth. It seems that it is the first attempt to obtain multi-peak semiclassical states for Hamiltonian elliptic system with exponential growth.

For every $k \in \mathbb{N}$ let $f_k:[\frac{1}{k+1}, \frac{1}{k}] \to [0,1]$ be decreasing, linear functions such that $f_k(\frac{1}{k+1}) = 1$ and $f_k(\frac{1}{k}) = 0$, $k = 1, 2, \dots$. We define iterated function system (IFS) $S_n$ by limiting the collection of functions $f_k$ to first n, meaning $S_n = \{f_k \}_{k=1}^n$. Let $J_n$ denote the limit set of $S_n$. Then $\lim\limits_{n\to \infty} \mathcal{P}_{h_n}(J_n) = 2$, where $h_n$ is the packing dimension of $J_n$ and $\mathcal{P}_{h_n}$ is the corresponding packing measure.

We study a fuzzy Boltzmann equation, where particles interact via delocalised collisions, in contrast to classical Boltzmann equations. We discuss the existence and uniqueness of solutions and provide a natural variational characterisation by casting the fuzzy Boltzmann equation into the framework of GENERIC systems (General Equations for Non-Equilibrium Reversible-Irreversible Coupling).

We investigate a geometric criterion for a smooth curve $C$ of genus $14$ and degree $18$ to be described as the zero locus of a section in an Ulrich bundle of rank $3$ on a del Pezzo threefold $V_5 \subset \mathbb{P}^6$. The main challenge is to read off the Pfaffian quadrics defining $V_5$ from geometric structures of $C$. We find that this problem is related to the existence of a special rank-two vector bundle on $C$ with trivial resonance. This answers a question posed by Ciliberto-Flamini-Knutsen, in the case of degree $5$ del Pezzo threefolds. From an explicit calculation of the Betti table of such a curve, we also deduce the uniqueness of the del Pezzo threefold.

We study the regularity properties of H\"older continuous minimizers to non-autonomous functionals satisfying $(p,q)$-growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability and higher differentiability results for solutions to our minimum problem.

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial\Omega$, and $\varepsilon$ is the ellipticity constant of the differential operator in the thin layer $\Sigma$. We study the problem in the limit for $\varepsilon$ going to zero and prove a first-order asymptotic development by $\Gamma$-convergence of the associated energy functional.

We give a general procedure for constructing metric spaces from systems of partitions. This generalises and provides analogues of Sageev's construction of dual CAT(0) cube complexes for the settings of hyperbolic and injective metric spaces. As applications, we produce a ``universal'' hyperbolic action for groups with strongly contracting elements, and show that many groups with ``coarsely cubical'' features admit geometric actions on injective metric spaces. In an appendix with Davide Spriano, we show that a large class of groups have an infinite-dimensional space of quasimorphisms.

We investigate nonexistence of nontrivial nonnegative solutions to a class of semilinear parabolic equations with a positive potential, posed on weighted graphs. Assuming an upper bound on the Laplacian of the distance and a suitable weighted space-time volume growth condition, we show that no global solutions exists. We also discuss the optimality of the hypotheses, thus recovering a critical exponent phenomenon of Fujita type.

Let $\mathbf{E}_n: \mathcal{M} \to \mathcal{M}_n$ and $\mathbf{E}_m: \mathcal{N} \to \mathcal{N}_m$ be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator $\mathcal{E} = (\mathbf{E}_n\otimes \mathbf{E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the two first-named authors, it was show that $\mathcal{E}$ has weak type $(\Phi, \Phi)$ for a family of functions including $\Phi(t) = t \, \log^{2+\varepsilon} t$, for every $\varepsilon > 0$. In this article, we prove that the weak Orlicz type of $\mathcal{E}$ cannot be lowered below $L \log^2 L$, meaning that if $\mathcal{E}$ is of weak type $(\Phi, \Phi)$, then $\Phi(s) \not\in o(s \, \log^2 s)$. Our proof is based on interpolation. Namely, we use recent techniques of Cadilhac/Ricard to formulate a Marcinkiewicz type theorem for maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than $L \log^2 L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than the known optimum as $p \to 1^{+}$.

This paper focuses on studying the long-time dynamics of the subordination process for a range of linear evolution equations, with a special emphasis on the fractional heat equation. By treating inverse subordinators as random time variables and employing the subordination principle to solve forward Kolmogorov equations, we explore the behavior of the solutions over extended periods. We provide a detailed description of the specific classes of subordinators suitable for conducting asymptotic analysis. Our findings not only extend existing research, but also enhance the results previously presented in [9, 10].

Observable operator models (OOMs) offer a powerful framework for modelling stochastic processes, surpassing the traditional hidden Markov models (HMMs) in generality and efficiency. However, using OOMs to model infinite-dimensional processes poses significant theoretical challenges. This article explores a rigorous approach to developing an approximation theory for OOMs of infinite-dimensional processes. Building upon foundational work outlined in an unpublished tutorial [Jae98], an inner product structure on the space of future distributions is rigorously established and the continuity of observable operators with respect to the associated 2-norm is proven. The original theorem proven in this thesis describes a fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space. The presented findings lay the groundwork for future research in approximating observable operators of infinite-dimensional processes, while a remedy to the encountered obstacle is suggested.

We propose a theoretical framework to compute, rapidly and accurately, the signal-to-noise ratio at the output of spatial-division multiplexing (SDM) linear MIMO equalizers with arbitrary numbers of spatial modes and filter taps and demonstrate three orders of magnitude of speed-up compared to Monte Carlo simulations.

In this paper we present a novel approach to the geometric formulation of solid and fluid mechanics within the port-Hamiltonian framework, which extends the standard Hamiltonian formulation to non-conservative and open dynamical systems. Leveraging Dirac structures, instead of symplectic or Poisson structures, this formalism allows the incorporation of energy exchange within the spatial domain or through its boundary, which allows for a more comprehensive description of continuum mechanics. Building upon our recent work in describing nonlinear elasticity using exterior calculus and bundle-valued differential forms, this paper focuses on the systematic derivation of port-Hamiltonian models for solid and fluid mechanics in the material, spatial, and convective representations using Hamiltonian reduction theory. This paper also discusses constitutive relations for stress within this framework including hyper-elasticity, for both finite- and infinite-strains, as well as viscous fluid flow governed by the Navier-Stokes equations.

In this paper, we introduce the generic circular triangle-free graph $\mathbb C_3$ and propose a finite axiomatization of its first order theory. In particular, our main results show that a countable graph $G$ embeds into $\mathbb C_3$ if and only if it is a $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graph. As a byproduct of this result, we obtain a geometric characterization of finite $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graphs, and the (finite) list of minimal obstructions of unit Helly circular-arc graphs with independence number strictly less than three. The circular chromatic number $\chi_c(G)$ is a refinement of the classical chromatic number $\chi(G)$. We construct $\mathbb C_3$ so that a graph $G$ has circular chromatic number strictly less than three if and only if $G$ maps homomorphically to $\mathbb C_3$. We build on our main results to show that $\chi_c(G) < 3$ if and only if $G$ can be extended to a $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graph, and in turn, we use this result to reprove an old characterization of $\chi_c(G) < 3$ due to Brandt (1999). Finally, we answer a question recently asked by Guzm\'an-Pro, Hell, and Hern\'andez-Cruz by showing that the problem of deciding for a given finite graph $G$ whether $\chi_c(G) < 3$ is NP-complete.

We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the computation of syzygies, then apply these techniques to deal with ideal and ring theoretic concepts such as primary decomposition and normalization, and finally use them for geometric case studies which concern curves and surfaces, both from a local and global point of view.

This is a preleminary work. Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many reversible overdamped Langevin dynamics which preserve the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

We investigate the fractional Hardy-H\'enon equation with fractional Brownian noise $$ \partial_t u(t)+(-\Delta)^{\theta/2} u(t)=|x|^{-\gamma} |u(t)|^{p-1}u(t)+\mu \partial_t B^H(t), $$ where $\theta>0$, $p>1$, $\gamma\geq 0$, $\mu \in\mathbb{R}$, and the random forcing $B^H$ is the fractional Brownian motion defined on some complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with Hurst parameter $H\in (0,1)$. We obtain the local existence and uniqueness of mild solutions under tailored conditions on the parameters of the equation.

In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let $P_1(x_1,y_1),\cdots ,P_n(x_n,y_n)$ are $n$ distinct points ($n\geq3$) with $x_1<\cdots<x_n$. Then $\overline{P_1P_2},\cdots \overline{P_nP_1}$ form a convex $n$-gon that lies in the half-space \begin{equation*}{ \underline{\mathbb{H}}=\bigg\{(x,y)\big|\quad x\in\mathbb{R} \quad \mbox{and} \quad y\leq y_1+\bigg(\dfrac{x-x_1}{x_n-x_1}\bigg)(y_n-y_1)\bigg\}\subseteq{\mathbb{R}^{2}} } \end{equation*} if and only if the following inequality holds \begin{equation} \dfrac{y_i-y_{i-1}}{x_i-x_{i-1}} \leq \dfrac{y_{i+1}-y_{i}}{x_{i+1}-x_{i}} \quad \quad \mbox{for all} \quad \quad i\in\{2,\cdots,n-1\} ." \end{equation} Based on this result, we establish a linkage between the property of sequential convexity and convex polygon. We show that in a plane if any $n$ points are scattered in such a way that their horizontal and vertical distances preserve some specific monotonic properties; then those points form a $2$-dimensional convex polytope.

BiHom-superdialgebras are clear generalization of Hom-superdialgebras. The purpose of this note is to describe and to survey structures of BiHom-superdialgebras. Then we derive derivations of BiHomsuperdialgebras.

We present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group $G=M\bowtie_{\gamma} H$, for some $\gamma:M\times M \to H$. By reduction, then, we obtain the Euler-Lagrange type and Hamilton's type equations of the same form for the quotient space $M\cong G/H$, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler-Poincar\'{e} type and Lie-Poisson type equations on the corresponding quotient $\mathfrak{m}\cong \mathfrak{g}/\mathfrak{h}$ of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the $n$th order iterated tangent group $T^{(n)}G$ of a Lie group $G$ as an extension of the $n$th order tangent group $T^nG$ of the same type. More precisely, $\mathfrak{g}$ being the Lie algebra of $G$, $T^{(n)}G \cong \mathfrak{g}^{\times \,2^n-1-n} \bowtie_\gamma T^nG$ for some $\gamma:\mathfrak{g}^{\times \,2^n-1-n} \times \mathfrak{g}^{\times \,2^n-1-n} \to T^nG$. We thus obtain the $n$th order Euler-Lagrange (and then the $n$th order Euler-Poincar\'e) equations over $T^nG$ by reduction from those on $T(T^{n-1}G)$. Finally, we illustrate our results in the realm of the Kepler problem, and the non-linear tokamak plasma dynamics.

In this note, our goal is to describe the concept of generalized derivations in the context of BiHom-supertrialgebras. We provide a comprehensive analysis of the properties and applications of these generalized derivations, including their relationship with other algebraic structures. We also explore various examples and applications of BiHom-supertrialgebras in different fields of mathematics and physics. Our findings contribute to a deeper understanding of the algebraic properties and applications of BiHom-supertrialgebras, and pave the way for further research in this area.

We consider local and nonlocal Cahn-Hilliard equations with constant mobility and singular potentials including, e.g., the Flory-Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. In other words, the separation process takes place even in presence of a pure phase, provided that it is triggered by a convenient reaction term. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we generalize the previously-mentioned concept by examining the essential assumptions required for the reaction term to apply the new strategy. Also, we explore the scenario involving the nonlocal Cahn-Hilliard equation and provide illustrative examples that contextualize within our abstract framework.

We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches.

The aim of the papers is to describe the left regular left quotient ring ${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where $\mS_1=K\langle x,y\, | \, yx=1\rangle$, $\CI_n=K\langle \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\rangle$ is the algebra of scalar integro-differential operators and the Jacobian algebra $\mA_1=K\langle x,\der, (\der x)^{-1}\rangle$. The sets of left and right regular elements of the algebras $\mS_1$, $\CI_1$, $\mA_1$ and $\mI_1=K\langle x, \der,\int\rangle$. A progress is made on the following conjecture, \cite{Clas-lreg-quot}: $${}'Q(\mI_n)\simeq Q(A_n)\;\; {\rm where}\;\; \mI_n =K\bigg\langle x_1,\ldots , x_n, \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\bigg\rangle$$ is the algebra of polynomial integro-differential operators and $Q(A_n)$ is the classical quotient ring (of fractions) of the $n$'th Weyl algebra $A_n$, i.e. a criterion is given when the isomorphism holds. We produce several general constructions of left Ore and left denominator sets that appear naturally in applications and are of independent interest and use them to produce explicit left denominator sets that give the localization ring isomorphic to ${}'Q(\mS_n)$ or ${}'Q(\mI_n)$ or ${}'Q(\mA_n)$ where $\mA_n:=\mA_1^{\t n}$. Several characterizations of one-sided regular elements of a ring are given in module-theoretic and one-sided-ideal-theoretic way.

Let $\lambda$ denote the Liouville function. We show that for all sufficiently large integers $N$, the (non-trivial) convolution sum bound $$ \left|\sum_{1 \leq n < N} \lambda(n) \lambda(N-n)\right| < N-1 $$ holds. This (essentially) answers a question posed at the 2018 AIM workshop on Sarnak's conjecture.

We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $\sigma_\omega$ at $\omega=\zeta_6$ takes essentially minimal values on torus links among all concordance homomorphisms with the same normalisation on the trefoil knot.

This investigation is dedicated to a two-player zero-sum stochastic differential game (SDG), where a cost function is characterized by a backward stochastic differential equation (BSDE) with a continuous and monotonic generator regarding the first unknown variable, which possesses immense applicability in financial engineering. A verification theorem by virtue of classical solution of derived Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation is given. The dynamic programming principle (DPP) and unique weak (viscosity) solvability of HJBI equation are formulated through comparison theorem for BSDEs with monotonic generators and stability of viscosity solution. Some new regularity properties of value function are presented. Finally, we propose three concrete examples, which are concerned with resp., classical, and viscosity solution of HJBI equation, as well as a financial application where an investor with a non-Lipschitzian Epstein-Zin utility deals with market friction to maximize her utility preference.

We give a continued-fraction characterization of Stieltjes moment sequences for which there exists a representing measure with support in $[\xi, \infty)$. The proof is elementary.

Recent works on joint communication and sensing (JCAS) cellular networks have proposed to use time division mode (TDM) and concurrent mode (CM), as alternative methods for sharing the resources between communication and sensing signals. While the performance of these JCAS schemes for object tracking and parameter estimation has been studied in previous works, their performance on target detection in the presence of clutter has not been analyzed. In this paper, we propose a detection scheme for estimating the number of targets in JCAS cellular networks that employ TDM or CM resource sharing. The proposed detection method allows for the presence of clutter and/or temporally correlated noise. This scheme is studied with respect to the JCAS trade-off parameters that allow to control the time slots in TDM and the power resources in CM allocated to sensing and communications. The performance of two fundamental transmit beamforming schemes, typical for JCAS, is compared in terms of the receiver operating characteristics curves. Our results indicate that in general the TDM scheme gives a somewhat better detection performance compared to the CM scheme, although both schemes outperform existing approaches provided that their respective trade-off parameters are tuned properly.

We provide sharp sufficient criteria for an integral $2$-varifold to be induced by a $W^{2,2}$-conformal immersion of a smooth surface. Our approach is based on a fine analysis of the Hausdorff density for $2$-varifolds with critical integrability of the mean curvature and a recent local regularity result by Bi-Zhou. In codimension one, there are only three possible density values below $2$, each of which can be attained with equality in the Li--Yau inequality for the Willmore functional by the unit sphere, the double bubble, and the triple bubble. We show that below an optimal threshold for the Willmore energy, a varifold induced by a current without boundary is in fact a curvature varifold with a uniform bound on its second fundamental form. Consequently, the minimization of the Willmore functional in the class of curvature varifolds with prescribed even Euler characteristic provides smooth solutions for the Willmore problem. In particular, the "ambient" varifold approach and the "parametric" approach are equivalent for minimizing the Willmore energy.

We consider an observed subcritical Galton Watson process $\{Y_n,\ n\in \mathbb{Z} \}$ with correlated stationary immigration process $\{\epsilon_n,\ n\in \mathbb{Z} \}$. Two situations are presented. The first one is when $\mbox{Cov}(\epsilon_0,\epsilon_k)=0$ for $k$ larger than some $k_0$: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when $\{\epsilon_n,\ n\in \mathbb{Z} \}$ has general correlation structure: under mixing assumptions, we exhibit an estimator for the the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.

This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the alternating direction doubling algorithm (ADDA), utilizing the low-rank property of matrices and employing Cholesky factorization for solving. The advantage of the new algorithm lies in computing only the $2^k$-th approximation during the iterative process, instead of every approximation. Its efficient low-rank formula saves storage space and is highly effective from a computational perspective. Finally, the effectiveness of the new algorithm is demonstrated through theoretical analysis and numerical experiments.

We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties. Additionally, some numerical results have demonstrated the effectiveness of this algorithm. Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method. In solving the Riccati equation, we combined inner and outer iterations, first simplifying the Riccati equation into a Lyapunov equation using the Newton method, and then applying the GADI method for solution. Finally, we provided convergence analysis of the method and corresponding numerical results.

Given a geodesic metric space $X$, we construct a corresponding hyperbolic space, which we call the contraction space, that detects all strongly contracting directions in the following sense; a geodesic in $X$ is strongly contracting if and only if its parametrized image in the contraction space is a quasi-geodesic. If a finitely generated group $G$ acts geometrically on $X$, then all strongly-contracting elements act as WPD elements on the contraction space. If the space $X$ is CAT(0), or more generally Morse-dichotomous, that is if all Morse geodesics are strongly-contracting, then all generalized loxodromics act as WPD elements, implying that the action is what we call ``universally WPD''.

Convergence to spatially homogeneous steady states is shown for a chemotaxis model with local sensing and possibly nonlinear diffusion when the intrinsic diffusion rate $\phi$ dominates the inverse of the chemotactic motility function $\gamma$, in the sense that $(\phi\gamma)'\ge 0$. This result encompasses and complies with the analysis and numerical simulations performed in Choi \& Kim (2023). The proof involves two steps: first, a Liapunov functional is constructed when $\phi\gamma$ is non-decreasing. The convergence proof relies on a detailed study of the dissipation of the Liapunov functional and requires additional technical assumptions on $\phi$ and $\gamma$.

We recast the local factors of the Hasse-Weil zeta function at infinity in terms of the Cuntz-Pimsner algebras. The nature of such factors is an open problem studied by Deninger and Serre.

We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $k\Delta_n$ for $k=0,\dots,n$, under the conditions $\Delta_n\to 0$ and $n\Delta_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection.

We consider the topology optimization problem of a 2d permanent magnet synchronous machine in magnetostatic operation with demagnetization. This amounts to a PDE-constrained multi-material design optimization problem with an additional pointwise state constraint. Using a generic framework we can incorporate this additional constraint and compute the corresponding topological derivative. We present and discuss optimization results obtained by a multi-material level set algorithm.

We prove a codimension reduction and congruence theorem for compact $n$-dimensional submanifolds of $\mathbb{S}^{n+p}$ that admit a mean convex isometric embedding into $\mathbb{S}^{n+1}_+$ using a Reilly type formula for space forms.

The availability of cancer measurements over time enables the personalised assessment of tumour growth and therapeutic response dynamics. However, many tumours are treated after diagnosis without collecting longitudinal data, and cancer monitoring protocols may include infrequent measurements. To facilitate the estimation of disease dynamics and better guide ensuing clinical decisions, we investigate an inverse problem enabling the reconstruction of earlier tumour states by using a single spatial tumour dataset and a biomathematical model describing disease dynamics. We focus on prostate cancer, since aggressive cases of this disease are usually treated after diagnosis. We describe tumour dynamics with a phase-field model driven by a generic nutrient ruled by reaction-diffusion dynamics. The model is completed with another reaction-diffusion equation for the local production of prostate-specific antigen, which is a key prostate cancer biomarker. We first improve previous well-posedness results by further showing that the solution operator is continuously Fr\'echet differentiable. We then analyse the backward inverse problem concerning the reconstruction of earlier tumour states starting from measurements of the model variables at the final time. Since this problem is severely ill-posed, only very weak conditional stability of logarithmic type can be recovered from the terminal data. However, by restricting the unknowns to a compact subset of a finite-dimensional subspace, we can derive an optimal Lipschitz stability estimate.

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically, when the lower density $\underline{d}(A) >1/2$ or the upper density $\overline{d}(A)> 3/4$, the existence of such a set $B\subset \mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underline{d}(A) > 3/4$ or $\overline{d}(A)>5/6$, we show that the shift $t$ is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three-coloring of the natural numbers that does not contain a monochromatic $B+B+t$ for any infinite set $B \subset \mathbb{N}$ and number $t \in \mathbb{N}$.

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t-1)K_q)$ for every $2\le p\le q$ and $t\ge 2$.

In the context of the optimization of rotating electric machines, many different objective functions are of interest and considering this during the optimization is of crucial importance. While evolutionary algorithms can provide a Pareto front straightforwardly and are widely used in this context, derivative-based optimization algorithms can be computationally more efficient. In this case, a Pareto front can be obtained by performing several optimization runs with different weights. In this work, we focus on a free-form shape optimization approach allowing for arbitrary motor geometries. In particular, we propose a way to efficiently obtain Pareto-optimal points by moving along to the Pareto front exploiting a homotopy method based on second order shape derivatives.

Mirror Descent is a popular algorithm, that extends Gradients Descent (GD) beyond the Euclidean geometry. One of its benefits is to enable strong convergence guarantees through smooth-like analyses, even for objectives with exploding or vanishing curvature. This is achieved through the introduction of the notion of relative smoothness, which holds in many of the common use-cases of Mirror descent. While basic deterministic results extend well to the relative setting, most existing stochastic analyses require additional assumptions on the mirror, such as strong convexity (in the usual sense), to ensure bounded variance. In this work, we revisit Stochastic Mirror Descent (SMD) proofs in the (relatively-strongly-) convex and relatively-smooth setting, and introduce a new (less restrictive) definition of variance which can generally be bounded (globally) under mild regularity assumptions. We then investigate this notion in more details, and show that it naturally leads to strong convergence guarantees for stochastic mirror descent. Finally, we leverage this new analysis to obtain convergence guarantees for the Maximum Likelihood Estimator of a Gaussian with unknown mean and variance.

Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of $G$. We investigate some properties of $\mathrm{Rad}_u(G)$, and study those $G$ for which $\mathrm{Rad}_u(G)$ is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine $k$-groups.

We study GIT quotients $X_\theta=V\!/\!\!/\!_\theta G$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_\theta$ modulo torsion. Our main result establishes that the Cox ring of $X_\theta$ is isomorphic to the semi-invariant ring of the $\theta$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane.

In this work we provide theoretical estimates for the ranks of the power functions $f(k) = k^{-\alpha}$, $\alpha>1$ in the quantized tensor train (QTT) format for $k = 1, 2, 3, \ldots, 2^{d}$. Such functions and their several generalizations (e.~g. $f(k) = k^{-\alpha} \cdot e^{-\lambda k}, \lambda > 0$) play an important role in studies of the asymptotic solutions of the aggregation-fragmentation kinetic equations. In order to support the constructed theory we verify the values of QTT-ranks of these functions in practice with the use of the TTSVD procedure and show an agreement between the numerical and analytical results.

This paper is devoted to the study of Hopf braces projections in a monoidal setting. Given a cocommutative Hopf brace ${\mathbb H}$ in a strict symmetric monoidal category ${\sf C}$, we define the braided monoidal category of left Yetter-Drinfeld modules over ${\mathbb H}$. For a Hopf brace ${\mathbb A}$ in this category, we introduce the concept of bosonizable Hopf brace and we prove that its bosonization ${\mathbb A}\blacktriangleright\hspace{-0.15cm}\blacktriangleleft {\mathbb H}$ is a new Hopf brace in ${\sf C}$ that gives rise to a projection of Hopf braces satisfying certain properties. Finally, taking these properties into account, we introduce the notions of v$_{i}$-strong projection over ${\mathbb H}$, $i=1,2,3,4$, and we prove that there is a categorical equivalence between the categories of bosonizable Hopf braces in the category of left Yetter-Drinfeld modules over ${\mathbb H}$ and the category of v$_{4}$-strong projections over ${\mathbb H}$.

For the non-gradient exclusion process, we prove its approximation rate of diffusion matrix/conductivity by local functions. The proof follows the quantitative homogenization theory developed by Armstrong, Kuusi, Mourrat and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse-grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. Moreover, the approximation rate of conductivity is uniform with respect to the density via the regularity of the local corrector. As an application, we integrate this result in the work by Funaki, Uchiyama and Yau [IMA Vol. Math. Appl., 77 (1996), pp. 1-40.] and yield a quantitative hydrodynamic limit. In particular, our new approach avoids to show the characterization of closed forms. We also discuss the possible extensions in the presence of disorder on the bonds.

In $n$-dimensional classical field theory one studies maps from $n$-dimensional manifolds in such a way that classical mechanics is recovered for $n=1$. In previous papers we have shown that the standard polysymplectic framework in which field theory is described, is not suitable for variational techniques. In this paper, we introduce for $n=2$ a Lagrange-Hamilton formalism that allows us to define a generalization of Hamiltonian Floer theory. As an application, we prove a cuplength estimate for our Hamiltonian equations that yields a lower bound on the number of solutions to Laplace equations with nonlinearity. We also discuss the relation with holomorphic Floer theory.

This paper is concerned with model order reduction of parametric Partial Differential Equations (PDEs) using tree-based library approximations. Classical approaches are formulated for PDEs on Hilbert spaces and involve one single linear space to approximate the set of PDE solutions. Here, we develop reduced models relying on a collection of linear or nonlinear approximation spaces called a library, and which can also be formulated on general metric spaces. To build the spaces of the library, we rely on greedy algorithms involving different splitting strategies which lead to a hierarchical tree-based representation. We illustrate through numerical examples that the proposed strategies have a much wider range of applicability in terms of the parametric PDEs that can successfully be addressed. While the classical approach is very efficient for elliptic problems with strong coercivity, we show that the tree-based library approaches can deal with diffusion problems with weak coercivity, convection-diffusion problems, and with transport-dominated PDEs posed on general metric spaces such as the $L^2$-Wasserstein space.

Let $K_n$ be a complete graph with $n$ vertices. An embedding of $K_n$ in $S^3$ is called a spatial $K_n$-graph. Knots in a spatial $K_n$-graph corresponding to simple cycles of $K_n$ are said to be constituent knots. We consider the case $n=4$. The boundary of an oriented band surface with zero Seifert form, constructed for a spatial $K_4$, is a four-component associated link. There are obtained relations between normalized Yamada and Jaeger polynomials of spatial graphs and Jones polynomials of constituent knots and the associated link.

We prove a generalization of Istvan F\'ary's celebrated theorem to higher dimension.

Galeotti, Khomskii and V\"a\"an\"aanen recently introduced the notion of the upward L\"owenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal $\kappa$ is the \emph{upward L\"owenheim-Skolem-Tarski number} (ULST) of a logic $\mathcal L$ if it is the least cardinal with the property that whenever $M$ is a model of size at least $\kappa$ satisfying a sentence $\varphi$ in $\mathcal L$, then there are arbitrarily large models satisfying $\varphi$ and having $M$ as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and V\"a\"an\"anen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal. We define the strong ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.

In a recent work, Klartag gave an improved version of Lichnerowicz' spectral gap bound for uniformly log-concave measures, which improves on the classical estimate by taking into account the covariance matrix. We analyze the equality cases in Klartag's bound, showing that it can be further improved whenever the measure has no Gaussian factor. Additionally, we give a quantitative improvement for log-concave measures with finite Fisher information.

Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved.

The demagnetization field in micromagnetism is given as the gradient of a potential which solves a partial differential equation (PDE) posed in R^d. In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem relies on the representation of the potential via the Green's function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green's function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs is obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings; periodic magnetisation, and high-frequency magnetisation. Numerical examples are given to verify the convergence rates.

Conjugate operator $T$ of 1D-harmonic oscillator $N=1/2 (p^2+q^2-1)$ is defined by an operator satisfying canonical commutation relation $[N,T]=-i$ on some domain but not necessarily dense. The angle operator $T_A=1/2 (\arctan q^{-1} p+\arctan pq^{-1} )$ and Galapon operator $T_G=i \sum_{n=0}^\infty(\sum_{m\neq n}\frac{(v_m,\cdot) }{n-m}v_n)$ are examples of conjugate operators, where ${v_n}$ denotes the set of normalized eigenvectors of $N$. Let $T$ be a subset of conjugate operators of $N$. A classification of $T$ are given as $T=T_{\{0\}}\cup T_{D\setminus \{0\}}\cup T_{\partial D}$, and $T_A\in T_{\{0\}}$ and $T_G\in T_{\partial D}$ are shown. Here the classification is specified by a pair of parameters $(\omega,m)\in C\times N$. Finally the time evolution $T_{\omega,m}(t)=e^{itN} T_{\omega,m}e^{-itN}$ for $T_{\omega,m}\in T$ is investigated, and show that $T_{\omega,m}(t)$ is periodic.

In this paper, we study conditions for the existence of an embedding $\widetilde{f} \colon P \to Q \times \mathbb{R}$ such that $f = \mathrm{pr}_Q \circ \widetilde{f}$, where $f \colon P \to Q$ is a piecewise linear map between polyhedra. Our focus is on non-degenerate maps between graphs, where non-degeneracy means that the preimages of points are finite sets. We introduce combinatorial techniques and establish necessary and sufficient conditions for the general case. Using these results, we demonstrate that the problem of the existence of a lifting reduces to testing the satisfiability of a 3-CNF formula. Additionally, we construct a counterexample to a result by V. Po\'{e}naru on lifting of smooth immersions to embeddings. Furthermore, by establishing connections between the stated problem and the approximability by embeddings, we deduce that, in the case of generic maps from a tree to a segment, a weaker condition becomes sufficient for the existence of a lifting.

We prove an explicit form of the Crepant Transformation Conjecture for Grassmannian flops. Our approach uses abelianization to first relate the restrictions of the Lagrangian cones to degree-2 classes, and then deduces the general result using ``explicit reconstruction'' (also known as the method of big I-functions).

We prove the crepant transformation conjecture for relative Grassmann flops over a smooth base $B$. We show that the $I$-functions of the respective GIT quotients are related by analytic continuation and a symplectic transformation. We verify that the symplectic transformation is compatible with Iritani's integral structure, that is, that it is induced by a Fourier-Mukai transform in $K$-theory.

We rigorously establish the existence of many free boundary minimal annuli with boundary in a geodesic sphere of $\mathbb{S}^3$. These arise as compact subdomains of a one-parameter family of complete minimal immersions of $\mathbb{R} \times \mathbb{S}^1$ into $\mathbb{S}^3$ described by do Carmo and Dajczer [1]. While the free boundary minimal annuli we exhibit may have self-intersections and may not be contained in a geodesic ball, we show that there is at least a one-parameter family of embedded examples that are contained in geodesic balls whose radius may be less than, equal to or greater than $\pi$/2. After explaining the connection to Otsuki tori [13], we establish lower bounds on the number of free boundary minimal annuli contained in each Otsuki torus in terms of the corresponding rational number. Finally, we show that some of the recent work of Lee and Seo [7] on isoperimetric inequalities and of Lima and Menezes [10] on index bounds extends to geodesic balls equal to or larger than a hemisphere.

In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved through a different sheafification process, which is the left adjoint functor of the suitable inclusion functor but does not preserve all finite limits. If the monoidal structure in $C$ is given by the categorical product, all constructions coincide with those for Grothendieck toposes. The motivation for such generalization stems from a certain notion of sheaves on quantales that does not form a topos.

We consider the massless Sine-Gordon model in de Sitter spacetime, in the regime $\beta^2 < 4 \pi$ and using the framework of perturbative algebraic quantum field theory. We show that a Fock space representation exists for the free massless field, but that the natural one-parameter family of vacuum-like states breaks the de Sitter boost symmetries. We prove convergence of the perturbative series for the S matrix in this representation, and construct the interacting Haag-Kastler net of local algebras from the relative S matrices. We show that the net fulfills isotony, locality and de Sitter covariance (in the algebraic adiabatic limit), even though the states that we consider are not invariant. We furthermore prove convergence of the perturbative series for the interacting field and the vertex operators, and verify that the interacting equation of motion holds.

We give an explicit description of three operad structures on the species composition $p \circ q$, where $q$ is any given positive operad, and where $p$ is the NAP operad, or a shuffle version of the magmatic operad Mag. No distributive law between $p$ and $q$ is assumed.

Given a finitely presented group $G$ and a surjective homomorphism $G\to \mathbb{Z}^n$ with finitely presented kernel $K$, we give an upper bound on the Dehn function of $K$ in terms of an area-radius pair for $G$. As a consequence we obtain that finitely presented coabelian subgroups of hyperbolic groups have polynomially bounded Dehn function. This generalises results of Gersten and Short and our proof can be viewed as a quantified version of results from Renz' thesis on the second BNSR invariant.

In this paper, we study a broad class of fully nonlinear elliptic equations on Hermitian manifolds. On one hand, under the optimal structural assumptions we derive $C^{2,\alpha}$-estimate for solutions of the equations on closed Hermitian manifolds. On the other hand, we treat the Dirichlet problem. In both cases, we prove the existence theorems with unbounded condition.

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence of perturbations with finite energy of non decaying smooth global solutions.

We construct a martingale observable related to the spin interface for a class of non-integrable variants of the two-dimensional Ising model and express it in terms of Grassmann integrals. Under a conjecture about the scaling limit of this object, which is similar to some results recently obtained using constructive renormalization group methods, this would imply that the distribution of the interface has the same scaling limit as in the integrable model, that is, the Schramm Loewner evolution curve SLE(3).

We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i\}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i\}_{i=1}^{n}$ in $N^\mathcal{U}$ such that $[u_i, w_1]= [w_k, w_{k+1}]=[w_n,u_j]=0$ for all $1\leq k< n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II$_1$ factors in the literature with a unified proof.

Meta-analysis allows rigorous aggregation of estimates and uncertainty across multiple studies. When a given study reports multiple estimates, such as log odds ratios (ORs) or log relative risks (RRs) across exposure groups, accounting for within-study correlations improves accuracy and efficiency of meta-analytic results. Canonical approaches of Greenland-Longnecker and Hamling estimate pseudo cases and non-cases for exposure groups to obtain within-study correlations. However, currently available implementations for both methods fail on simple examples. We review both GL and Hamling methods through the lens of optimization. For ORs, we provide modifications of each approach that ensure convergence for any feasible inputs. For GL, this is achieved through a new connection to entropic minimization. For Hamling, a modification leads to a provably solvable equivalent set of equations given a specific initialization. For each, we provide implementations a guaranteed to work for any feasible input. For RRs, we show the new GL approach is always guaranteed to succeed, but any Hamling approach may fail: we give counter-examples where no solutions exist. We derive a sufficient condition on reported RRs that guarantees success when reported variances are all equal.

The problem of vectorial mesons embedded in an electromagnetic field via Duffin-Kemmer-Petiau (DKP) formalism is reinvestigated. Considering the electromagnetic interaction as a minimal coupling, an incorrect value $(g=1)$ is identified for the gyromagnetic factor ($g$-factor). Furthermore, it is shown that is cumbersome to find analytical solutions due to the presence of the so-called anomalous term for the spin-1 sector of the DKP theory. Suspecting that the anomalous term results from an incomplete version of the DKP equation to describe the electromagnetic interaction, we consider the addition of a non-minimal coupling. This leads to the correct $g$-factor $(g=2)$, and as a consequence, the anomalous term becomes proportional to an external four current. As an application, the DKP equation with a static uniform magnetic field is considered, yielding the corresponding Landau levels.

The Polynomial Affine Gravity is an alternative gravitational model, where the interactions are mediated solely by the affine connection, instead of the metric tensor. In this paper, we explore the space of solutions to the field equations when the torsion fields are turned on, in a homogeneous and isotropic (cosmological) scenario. We explore various metric structures that emerge in the space of solutions.

Contraction theory is a mathematical framework for studying the convergence, robustness, and modularity properties of dynamical systems and algorithms. In this opinion paper, we provide five main opinions on the virtues of contraction theory. These opinions are (i) contraction theory is a unifying framework emerging from classical and modern works, (ii) contractivity is computationally-friendly, robust, and modular stability, (iii) numerous dynamical systems are contracting, (iv) contraction theory is relevant to modern applications, and (v) contraction theory can be vastly extended in numerous directions. We survey recent theoretical and applied research in each of these five directions.

Deep learning has been widely applied to solve partial differential equations (PDEs) in computational fluid dynamics. Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE solver on a coarse mesh. However, end-to-end training of such a PDE correction model over both solver-dependent parameters such as mesh parameters and neural network parameters requires the PDE solver to support automatic differentiation through the iterative numerical process. Such a feature is not readily available in many existing solvers. In this study, we explore the feasibility of end-to-end training of a hybrid model with a black-box PDE solver and a deep learning model for fluid flow prediction. Specifically, we investigate a hybrid model that integrates a black-box PDE solver into a differentiable deep graph neural network. To train this model, we use a zeroth-order gradient estimator to differentiate the PDE solver via forward propagation. Although experiments show that the proposed approach based on zeroth-order gradient estimation underperforms the baseline that computes exact derivatives using automatic differentiation, our proposed method outperforms the baseline trained with a frozen input mesh to the solver. Moreover, with a simple warm-start on the neural network parameters, we show that models trained by these zeroth-order algorithms achieve an accelerated convergence and improved generalization performance.

In this paper, we introduce a novel statistical model for the integrative analysis of Riemannian-valued functional data and high-dimensional data. We apply this model to explore the dependence structure between each subject's dynamic functional connectivity -- represented by a temporally indexed collection of positive definite covariance matrices -- and high-dimensional data representing lifestyle, demographic, and psychometric measures. Specifically, we employ a reformulation of canonical correlation analysis that enables efficient control of the complexity of the functional canonical directions using tangent space sieve approximations. Additionally, we enforce an interpretable group structure on the high-dimensional canonical directions via a sparsity-promoting penalty. The proposed method shows improved empirical performance over alternative approaches and comes with theoretical guarantees. Its application to data from the Human Connectome Project reveals a dominant mode of covariation between dynamic functional connectivity and lifestyle, demographic, and psychometric measures. This mode aligns with results from static connectivity studies but reveals a unique temporal non-stationary pattern that such studies fail to capture.

In this paper, we define a model of non-interacting quantum fields satisfying $(\Delta_g-\lambda^2)\phi=0$ on a Riemannian scattering space $(M,g)$ with two boundary components, i.e. a manifold with two asymptotically conic ends (meaning asymptotic to the "large end" of a cone). Our main result describes a canonical construction of two-point functions satisfying a version of the Hadamard condition.

The standard approach to universal fault-tolerant quantum computing is to develop a general purpose quantum error correction mechanism that can implement a universal set of logical gates fault-tolerantly. Given such a scheme, any quantum algorithm can be realized fault-tolerantly by composing the relevant logical gates from this set. However, we know that quantum computers provide a significant quantum advantage only for specific quantum algorithms. Hence, a universal quantum computer can likely gain from compiling such specific algorithms using tailored quantum error correction schemes. In this work, we take the first steps towards such algorithm-tailored quantum fault-tolerance. We consider Trotter circuits in quantum simulation, which is an important application of quantum computing. We develop a solve-and-stitch algorithm to systematically synthesize physical realizations of Clifford Trotter circuits on the well-known $[\![ n,n-2,2 ]\!]$ error-detecting code family. Our analysis shows that this family implements Trotter circuits with optimal depth, thereby serving as an illuminating example of tailored quantum error correction. We achieve fault-tolerance for these circuits using flag gadgets, which add minimal overhead. The solve-and-stitch algorithm has the potential to scale beyond this specific example and hence provide a principled approach to tailored fault-tolerance in quantum computing.

Variational Physics-Informed Neural Networks (VPINNs) utilize a variational loss function to solve partial differential equations, mirroring Finite Element Analysis techniques. Traditional hp-VPINNs, while effective for high-frequency problems, are computationally intensive and scale poorly with increasing element counts, limiting their use in complex geometries. This work introduces FastVPINNs, a tensor-based advancement that significantly reduces computational overhead and improves scalability. Using optimized tensor operations, FastVPINNs achieve a 100-fold reduction in the median training time per epoch compared to traditional hp-VPINNs. With proper choice of hyperparameters, FastVPINNs surpass conventional PINNs in both speed and accuracy, especially in problems with high-frequency solutions. Demonstrated effectiveness in solving inverse problems on complex domains underscores FastVPINNs' potential for widespread application in scientific and engineering challenges, opening new avenues for practical implementations in scientific machine learning.

The statistics of a chaotic recursion relation can be predicted by constructing an associated sequence of periodic elliptic operators. For such operators, the density of states is well understood, can be computed straightforwardly and explicit formulas can often be derived. The example studied here is a non-linear recursion relation which can be related to a sequence of periodic operators generated by a Fibonacci tiling rule. This link is used to derive an explicit formula for the limiting distribution of orbits of the non-linear recursion relation. This distribution contains characteristic features of the associated operators' densities of states, such as Van Hove singularities near to critical values.

Networks play an ubiquitous role in computer science and real-world applications, offering multiple kind of information that can be retrieved with adequate methods. With the continuous growing in the amount of data available, networks are becoming larger day by day. Consequently, the tasks that were easily achievable on smaller networks, often becomes impractical on huge amount of data, either due to the high computational cost or due to the impracticality to visualise corresponding data. Using distinctive node features to group large amount of connected data into a limited number of clusters, hence represented by a representative per cluster, proves to be a valuable approach. The resulting contracted graphs are more manageable in size and can reveal previously hidden characteristics of the original networks. Furthermore, in many real-world use cases, a definition of cluster is intrinsic with the data, eventually obtained with the injection of some expert knowledge represent by a categorical function. Clusters then results in set of connected vertices taking the same values in a finite set C. In the recent literature, Lombardi and Onofri proposed a novel, fast, and easily parallelisable approach under the name of $\gamma$-contraction to contract a graph given a categorical function. In this work, we formally define such approach by providing a rigorous mathematical definition of the problem, which, to the best of our knowledge, was missing in the existing literature. Specifically, we explore the variadic nature of the contraction operation and use it to introduce the weaker version of the colour contraction, under the name of $\beta$-contraction, that the algorithmic solution exploits. We finally dive into the details of the algorithm and we provide a full assesment on its convergence complexity relying on two constructive proofs that deeply unveil its mode of operation.

We study the Schr\"odinger equation in quantum field theory (QFT) in its functional formulation. In this approach quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schr\"odinger time evolution on Wentzel-Kramers-Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT.

The compact Genetic Algorithm (cGA), parameterized by its hypothetical population size $K$, offers a low-memory alternative to evolving a large offspring population of solutions. It evolves a probability distribution, biasing it towards promising samples. For the classical benchmark OneMax, the cGA has to two different modes of operation: a conservative one with small step sizes $\Theta(1/(\sqrt{n}\log n))$, which is slow but prevents genetic drift, and an aggressive one with large step sizes $\Theta(1/\log n)$, in which genetic drift leads to wrong decisions, but those are corrected efficiently. On OneMax, an easy hill-climbing problem, both modes lead to optimization times of $\Theta(n\log n)$ and are thus equally efficient. In this paper we study how both regimes change when we replace OneMax by the harder hill-climbing problem DynamicBinVal. It turns out that the aggressive mode is not affected and still yields quasi-linear runtime $O(n\cdot polylog (n))$. However, the conservative mode becomes substantially slower, yielding a runtime of $\Omega(n^2)$, since genetic drift can only be avoided with smaller step sizes of $O(1/n)$. We complement our theoretical results with simulations.

Co-channel interference poses a challenge in any wireless communication network where the time-frequency resources are reused over different geographical areas. The interference is particularly diverse in cell-free massive multiple-input multiple-output (MIMO) networks, where a large number of user equipments (UEs) are multiplexed by a multitude of access points (APs) on the same time-frequency resources. For realistic and scalable network operation, only the interference from UEs belonging to the same serving cluster of APs can be estimated in real-time and suppressed by precoding/combining. As a result, the unknown interference arising from scheduling variations in neighboring clusters makes the rate adaptation hard and can lead to outages. This paper aims to model the unknown interference power in the uplink of a cell-free massive MIMO network. The results show that the proposed method effectively describes the distribution of the unknown interference power and provides a tool for rate adaptation with guaranteed target outage.

Semantic communication (SemCom) has emerged as a key technology for the forthcoming sixth-generation (6G) network, attributed to its enhanced communication efficiency and robustness against channel noise. However, the open nature of wireless channels renders them vulnerable to eavesdropping, posing a serious threat to privacy. To address this issue, we propose a novel secure semantic communication (SemCom) approach for image transmission, which integrates steganography technology to conceal private information within non-private images (host images). Specifically, we propose an invertible neural network (INN)-based signal steganography approach, which embeds channel input signals of a private image into those of a host image before transmission. This ensures that the original private image can be reconstructed from the received signals at the legitimate receiver, while the eavesdropper can only decode the information of the host image. Simulation results demonstrate that the proposed approach maintains comparable reconstruction quality of both host and private images at the legitimate receiver, compared to scenarios without any secure mechanisms. Experiments also show that the eavesdropper is only able to reconstruct host images, showcasing the enhanced security provided by our approach.

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Upper Boomerang Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, there are currently no related results for this research. The inverse function is crucial for constructing S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. Therefore, extensive research has been conducted on the inverse function, exploring various properties related to standard attacks. Thanks to the recent advancements in boomerang cryptanalysis, particularly the introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to further investigate the properties of the inverse function $F(x)=x^{2^n-2}$ over $\gf_{2^n}$ for arbitrary $n$. As a consequence, by carrying out certain finer manipulations of solving specific equations over $\gf_{2^n}$, we give all entries of the UBCT, LBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Besides, based on the results of the UBCT and LBCT for the inverse function, we determine that $F(x)$ is hard when $n$ is odd. Furthermore, we completely compute all entries of the DBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Additionally, we provide the precise number of elements with a given entry by means of the values of some Kloosterman sums. Further, we determine the double boomerang uniformity of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Our in-depth analysis of the DBCT of $F(x)$ contributes to a better evaluation of the S-box's resistance against boomerang attacks.

Parallelisation in Bayesian optimisation is a common strategy but faces several challenges: the need for flexibility in acquisition functions and kernel choices, flexibility dealing with discrete and continuous variables simultaneously, model misspecification, and lastly fast massive parallelisation. To address these challenges, we introduce a versatile and modular framework for batch Bayesian optimisation via probabilistic lifting with kernel quadrature, called SOBER, which we present as a Python library based on GPyTorch/BoTorch. Our framework offers the following unique benefits: (1) Versatility in downstream tasks under a unified approach. (2) A gradient-free sampler, which does not require the gradient of acquisition functions, offering domain-agnostic sampling (e.g., discrete and mixed variables, non-Euclidean space). (3) Flexibility in domain prior distribution. (4) Adaptive batch size (autonomous determination of the optimal batch size). (5) Robustness against a misspecified reproducing kernel Hilbert space. (6) Natural stopping criterion.

We study the limiting dynamics of a large class of noisy gradient descent systems in the overparameterized regime. In this regime the set of global minimizers of the loss is large, and when initialized in a neighbourhood of this zero-loss set a noisy gradient descent algorithm slowly evolves along this set. In some cases this slow evolution has been related to better generalisation properties. We characterize this evolution for the broad class of noisy gradient descent systems in the limit of small step size. Our results show that the structure of the noise affects not just the form of the limiting process, but also the time scale at which the evolution takes place. We apply the theory to Dropout, label noise and classical SGD (minibatching) noise, and show that these evolve on different two time scales. Classical SGD even yields a trivial evolution on both time scales, implying that additional noise is required for regularization. The results are inspired by the training of neural networks, but the theorems apply to noisy gradient descent of any loss that has a non-trivial zero-loss set.

We study minimax optimization problems defined over infinite-dimensional function classes. In particular, we restrict the functions to the class of overparameterized two-layer neural networks and study (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural network. As an initial step, we consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space. For this problem, we establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $\mathcal{O}(T^{-1} + \alpha^{-1} ) $ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural network. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $\mathcal{O}(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, and asset pricing.

The area law-like scaling of local quantum entropies is the central characteristic of the entanglement inherent in quantum fields, many-body systems, and spacetime. Whilst the area law is primarily associated with the entanglement structure of the underlying quantum state, we here show that it equally manifests in classical entropies over measurement distributions when vacuum contributions dictated by the uncertainty principle are subtracted. Using the examples of the Gaussian ground and thermal states, but also the non-Gaussian particle state of a relativistic scalar field, we present analytical and numerical area laws for the entropies of various distributions and unveil how quantities of widespread interest such as the central charge and the (local) temperature are encoded in classical observables. With our approach, quantum entropies are no longer necessary to probe quantum phenomena, thereby rendering area laws and other quantum features directly accessible to theoretical models of high complexity as well as state-of-the-art experiments.

We rigorously prove, for the first time, that the non-hydrodynamic sector is gapless in any relativistic kinetic theory whose scattering cross-section decays to zero at large energies. In fact, if particles with very high energy (compared to the temperature) are free streaming, we can use them to build hot non-hydrodynamic waves, which live longer than any hydrodynamic wave. Since many standard cross-sections in quantum field theory vanish at high energies, the existence of these non-thermal long-lived waves is a rather general feature of relativistic systems.

A general theory of stochastic decision forests reconciling two concepts of information flow -- decision trees and refined partitions on the one hand, filtrations from probability theory on the other -- is constructed. The traditional "nature" agent is replaced with a one-shot lottery draw that determines a tree of a given decision forest, while each "personal" agent is equipped with an oracle providing updates on the draw's result and makes partition refining choices adapted to this information. This theory overcomes the incapacity of existing approaches to extensive form theory to capture continuous time stochastic processes like Brownian motion as outcomes of "nature" decision making in particular. Moreover, a class of stochastic decision forests based on paths of action indexed by time is constructed, covering a large fraction of models from the literature and constituting a first step towards an approximation theory for stochastic differential games in extensive form.

Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while concurrently learning the underlying governing equations of the physical system. Specifically, we focus on multi-operator learning by training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations, with potential applications to many physical applications including physics, geology, and biology. More importantly, we provide three extrapolation studies to demonstrate that PROSE-PDE can generalize physical features through the robust training of multiple operators and that the proposed model can extrapolate to predict PDE solutions whose models or data were unseen during the training. Furthermore, we show through systematic numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.

The $k$-parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-parity problem with stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that SGD can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, thus matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. Our theoretical results and findings are supported by empirical evidence, showcasing the efficiency and efficacy of our approach.