Jean-Marie Souriau's book ''Structure des syst\`emes dynamiques'', published in 1970, republished recently by Gabay, translated in English and published under the title ''Structure of Dynamical Systems, a Symplectic View of Physic'', is a work with an exceptional wealth which, fifty years after its publication, is still topical. In this paper, we give a rather detailled description of its content and we intend to highlight the ideas that to us, are the most creative and promising.

In this paper, we introduce the concept of continuous $g-$atomic subspace for a bounded linear operator and gives several useful continuous resolution of the identity operator on a Hilbert space by implies the theory of continuous $g-$fusion frames. Moreover, we introduce the concept of continuous frame operator for a pair of continuous $g-$fusion bessel sequences.

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression $\overset{n}{\underset{i=1}{\sum}}|f(x_i)-f(x_{i-1})|^{r}$ is bounded for all possible partitions like $P$. One of the main result of the paper deals with the generalization of Classical Jordan decomposition theorem. We have shown that for $r\in]0,1]$, a function of $r$-bounded variation can be written as the difference of two monotone functions. While for $r>1$, under minimal assumptions such functions can be treated as approximately monotone function which can be closely approximated by a nondecreasing majorant. We also proved that for $0<r_1<r_2$; the function class of $r_1$-bounded variation is contained in the class of functions satisfying $r_2$-bounded variations. We go through approximately monotone functions and present a possible decomposition for $f:I(\subseteq \mathbb{R_+})\to\mathbb{R}$ satisfying the functional inequality $$f(x)\leq f(x)+(y-x)^{p}\quad (x,y\in I\mbox{ with $x<y$ and $ p\in]0,1[ $}).$$ A generalized structural study has also be done in that specific section. On the other hand for $\ell[a,b]\geq d$; a function satisfying the following monotonic condition under the given assumption will be termed as $d$-periodically increasing $$f(x)\leq f(y)\quad \mbox{for all}\quad x,y\in I\quad\mbox{with}\quad y-x\geq d.$$ we establish that in a compact interval any bounded function can be decomposed as the difference of a monotone and a $d$-periodically increasing function.

In this article, we propose numerical scheme for solving a multi-term time-fractional nonlocal parabolic partial differential equation (PDE). The scheme comprises $L2$-$1_{\sigma}$ scheme on a graded mesh in time and Galerkin finite element method (FEM) in space. We present the discrete fractional Gr$\ddot{{o}}$nwall inequality for $L2$-$1_{\sigma}$ scheme in case of multi-term time-fractional derivative, which is a multi-term analogue of~\cite[Lemma 4.1]{[r16]}. We derive \textit{a priori} bound and error estimate for the fully-discrete solution. The theoretical results are confirmed via numerical experiments. We should note that, though the way of proving the discrete fractional Gr$\ddot{{o}}$nwall inequality is similar to~\cite{[r5]}, the calculation parts are more complicated in this article.

We study two classes of morphisms in infinite type: tamely presented morphisms and morphisms with coherent pullback. These are generalizations of finitely presented morphisms and morphisms of finite Tor-dimension, respectively. The class of tamely presented schemes and stacks is restricted enough to retain the key features of finite-type schemes from the point of view of coherent sheaf theory, but wide enough to encompass many infinite-type examples of interest in geometric representation theory. The condition that a diagonal has coherent pullback is a natural generalization of smoothness to the tamely presented setting, and we show such objects retain many good cohomological properties of smooth varieties. Our results are motivated by the study of convolution products in the double affine Hecke category and related categories in the theory of Coulomb branches.

In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.

Borel-Serre proved that the integral symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ is a virtual duality group of dimension $n^2$ and that the symplectic Steinberg module $\operatorname{St}^\omega_n(\mathbb{Q})$ is its dualising module. This module is the top-dimensional homology of the Tits building associated to $\operatorname{Sp}_{2n}(\mathbb{Q})$. We find a presentation of this Steinberg module and use it to show that the codimension-1 rational cohomology of $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes for $n \geq 2$, $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) \cong 0$. Equivalently, the rational cohomology of the moduli stack $\mathcal{A}_n$ of principally polarised abelian varieties of dimension $2n$ vanishes in the same degree. Our findings suggest a vanishing pattern for high-dimensional cohomology in degree $n^2-i$, similar to the one conjectured by Church-Farb-Putman for special linear groups.

This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.

We classify the minimal number of generators of artinian equigenerated monomial ideals $I$ such that $\Bbbk[x_1,\ldots,x_n]/I$ is forced to have the weak Lefschetz property.

We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their geometric genus using quasi-modular forms. We conjecture that these Severi curves are eventually reducible, and we formulate a precise conjecture for their degrees in $\mathbb{P}^2$, featuring a divisor sum formula for collision multiplicities of branch points.

While constraints arise naturally in many physical models, their treatment in mathematical and numerical models varies widely, depending on the nature of the constraint and the availability of simulation tools to enforce it. In this paper, we consider the solution of discretized PDE models that have a natural constraint on the positivity (or non-negativity) of the solution. While discretizations of such models often offer analogous positivity properties on their exact solutions, the use of approximate solution algorithms (and the unavoidable effects of floating -- point arithmetic) often destroy any guarantees that the computed approximate solution will satisfy the (discretized form of the) physical constraints, unless the discrete model is solved to much higher precision than discretization error would dictate. Here, we introduce a class of iterative solution algorithms, based on the unigrid variant of multigrid methods, where such positivity constraints can be preserved throughout the approximate solution process. Numerical results for one- and two-dimensional model problems show both the effectiveness of the approach and the trade-off required to ensure positivity of approximate solutions throughout the solution process.

We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical configuration spaces. The corresponding toric varieties provide modular compactifications of the algebraic configuration spaces, with boundary parametrising transverse configurations on tropical expansions. The rubber torus, used to identify equivalent configurations, plays a key role. As an application, we obtain a modular interpretation for the bipermutahedral variety.

We give an analog of exceptional polynomials in the matrix valued setting by considering suitable factorizations of a given second order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence is discussed in detail. We give an example of matrix valued exceptional Laguerre polynomials of arbitrary size.

Using skein theory very much in the spirit of the Reshetikhin--Turaev constructions, we define a $(3+1)$-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the category to define this TQFT, namely to be "chromatic non-degenerate". As a by-product, we obtain an invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in the "twist non-degenerate" case, an invariant of 3-manifolds. Our construction generalizes the Crane--Yetter--Kauffman TQFTs in the semi-simple case, and the Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole construction is very elementary, and we can easily characterize invertibility of the TQFTs, study their behavior under connected sum and provide some examples.

We prove $\textsf{NP-hardness}$ results for determining whether quivers are mutation equivalent to quivers with given properties. Specifically, determining whether a quiver is mutation-equivalent to a quiver with exactly $k$ arrows between any two of its vertices is $\textsf{NP-hard}$. Also, determining whether a quiver is mutation equivalent to a quiver with no edges between frozen vertices is $\textsf{strongly NP-hard}$. Finally, we present a characterization of mutation classes of quivers with two mutable vertices.

Decision-making individuals are often considered to be either imitators who copy the action of their most successful neighbors or best-responders who maximize their benefit against the current actions of their neighbors. In the context of coordination games, where neighboring individuals earn more if they take the same action, by means of potential functions, it was shown that populations of all imitators and populations of all best-responders equilibrate in finite time when they become active to update their decisions sequentially. However, for mixed populations of the two, the equilibration was shown only for specific activation sequences. It is therefore, unknown, whether a potential function also exists for mixed populations or if there actually exists a counter example where an activation sequence prevents equilibration. We show that in a linear graph, the number of ``sections'' (a sequence of consecutive individuals taking the same action) serves as a potential function, leading to equilibration, and that this result can be extended to sparse trees. The existence of a potential function for other types of networks remains an open problem.

In this study, we combine two novel methods, the conformable double Laplace-Sumudu transform (CDLST) and the modified decomposition technique. We use the new approach called conformable double Laplace-Sumudu modified decomposition (CDLSMD) method, to solve some nonlinear fractional partial differential equations. We present the essential properties of the CDLST and produce new results. Furthermore, five interesting examples are discussed and analyzed to show the efficiency and applicability of the presented method. The results obtained show the strength of the proposed method in solving different types of problems.

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the $H^{-1}(\Omega)$-norm for the distributed control, the $H^{1/2}(\Gamma)$-norm for the Dirichlet control and the $H^{-1/2}(\Gamma)$-norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard $C^0$ finite elements can be used to solve the resulted variational inequality. We provide preliminary a priori error estimates for the new algorithm for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new algorithm.

In this note we point out some simple sufficient (plausible) conditions for `turbulence' cascades in suitable limits of damped, stochastically-driven nonlinear Schr\"odinger equation in a $d$-dimensional periodic box. Simple characterizations of dissipation anomalies for the wave action and kinetic energy in rough analogy with those that arise for fully developed turbulence in the 2D Navier-Stokes equations are given and sufficient conditions are given which differentiate between a `weak' turbulence regime and a `strong' turbulence regime. The proofs are relatively straightforward once the statements are identified, but we hope that it might be useful for thinking about mathematically precise formulations of the statistically-stationary wave turbulence problem.

We consider a stochastic process $\{(\boldsymbol{X}(t),V(t)), t \geq 0\}$ which describes a particle performing a minimal cyclic random motion with finite velocities in $\mathbb{R}^3$. The particle can take four directions moving with different velocities $\vec{v}_j$, for $1 \leq j \leq4$, so that the diffusion region is a tetrahedron $\mathcal{T}(t)$. Moreover, we assume that the sequence of sojourn times along each velocity $\vec{v}_j$ follows a geometric counting process of intensity $\lambda_j$, $1 \leq j \leq4$. We perform the analysis of the direction vectors $\vec{v}_j$ and of the domain $\mathcal{T}(t)$, and then we obtain the closed-form expressions of the discrete and absolutely continuous components of the probability law of the process when the initial velocity is $\vec{v}_1$. We also investigate the limiting behavior of the probability density of the process when the intensities $\lambda_j$ tend to infinity. Finally, we introduce the first-passage time problem for the first component of $\boldsymbol{X}(t)$ through a constant positive boundary $\beta >0$ providing the bases for future developments.

In this paper we develop a new decomposition framework to deal with Lagrange multipliers of the Karush-Kuhn-Tucker (KKT) system of constrained optimization problems and variational inequalities in Hilbert spaces. It is different from existing frameworks based on separation theorems. We introduce the essential Lagrange multiplier and establish the basic theory of this new multiplier. The essential Lagrange multiplier poses essentially different existence results in finite and infinite-dimensional spaces. It can also be used to give an essential characterization of the convergence of multipliers generated by the classical augmented Lagrangian method. Our analysis reveals that the essential Lagrange multiplier is at the core of both theories and applications of Lagrange multipliers.

We provide a lower bound on the volume of the compactified mapping torus of a strongly irreducible end-periodic homeomorphism f. This result, together with work of Field, Kim, Leininger, and Loving, shows that the volume of the compactified mapping torus of f is comparable to the translation length of f on a connected component of the pants graph, extending work of Brock in the finite-type setting on volumes of mapping tori of pseudo-Anosov homeomorphisms.

In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.

We present an explicit expression of the cohomology complex of a constructible sheaf of abelian groups on the small \'etale site of an irreducible curve over an algebraically closed field, when the torsion of the sheaf is invertible in the field. This expression only involves finite groups, and is functorial in both the curve and the sheaf. In particular, we explain how to compute the Galois action on this complex. We also present an algorithm which computes it and study its complexity.

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Amp\`ere operator on a bounded strongly pseudoconvex domain $\Omega$ in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $\Omega$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Amp\`ere equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck (1985) and B. Guan (1998). Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Amp\`ere operator.

We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics, via solving the high-dimensional Fokker-Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

Let $n \neq 8$ be a positive integer such that $n+1 \neq 2^u$ for any integer $u\geq 2$. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than or equal to $n+1$. Let $a_j(x)$ with $0\leq j\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\deg\phi(x)$. Assume that the content of $(a_na_0(x))$ is not divisible by any prime less than or equal to $n+1$. In this paper, we prove that the polynomial $f(x) = a_n\frac{\phi(x)^n}{(n+1)!}+ \sum\limits_{j=0}^{n-1}a_j(x)\frac{\phi(x)^{j}}{(j+1)!}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{j}}{(j+1)!}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples.

We prove that any connected curve with a rational point $x$ over a field $k$ is an algebraic $K(\pi,1)$, given that all of its irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible \'etale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $\pi_1(X,\bar x)$-module. When $k$ is algebraically closed, we explicitly describe finite quotients of $\pi_1(X,\bar x)$ that allow to compute the cohomology groups of the sheaf, as well as the cup products $H^1\times H^1\to H^2$.

The proalgebraic fundamental group of a connected topological space $X$, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on $X$. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert-van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk-Scholze spaces and compare it to the motivic Galois group.

This letter introduces a novel algorithm for generating codebooks in molecular communications (MC). The proposed algorithm utilizes character entropy to effectively mitigate inter-symbol interference (ISI) during MC via diffusion. Based on Huffman coding, the algorithm ensures that consecutive bit-1s are avoided in the resulting codebook. Additionally, the error-correction process at the receiver effectively eliminates ISI in the time slot immediately following a bit-1. We conduct an ISI analysis, which confirms that the proposed algorithm significantly reduces decoding errors. Through numerical analysis, we demonstrate that the proposed codebook exhibits superior performance in terms of character error rate compared to existing codebooks. Furthermore, we validate the performance of the algorithm through experimentation on a real-time testbed.

In this paper, we study a cold gas of $N \gg 1$ weakly interacting fermions. We describe the time evolution of the momentum distribution of states close to the Fermi ball by simultaneously analyzing the dynamical behavior of excited particles and holes. Our main result states that, for small values of the coupling constant, and for appropriate initial data, the effective dynamics of the above system is driven by an energy-mollified quantum Boltzmann collision operator, plus a an interaction term with virtual bosonized particle-hole pairs around the Fermi surface.

We prove the three candidate Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell from 2005 for correlations $\rho$ satisfying $-1/36<\rho<1/11$: the Plurality function is the most noise stable three candidate election method with small influences, when the corrupted votes have correlation $-1/36<\rho<1/11$ with the original votes. The previous best result of this type only achieved positive correlations at most $10^{-10^{10}}$. Our result follows by solving the three set Standard Simplex Conjecture of Isaksson-Mossel from 2011 for all correlations $-1/36<\rho<1/11$. The Gaussian Double Bubble Problem corresponds to the case $\rho\to1^{-}$, so in some sense, our result is a generalization of the Gaussian Double Bubble Problem. Our result is also notable since it is the first result for any $\rho<0$, which is the only relevant case for computational hardness of MAX-3-CUT. As an additional corollary, we conclude that three candidate Borda Count is stablest for all $-1/36<\rho<1/11$.

We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators.

For the right-angled Artin group action on the extension graph, it is known that the minimal asymptotic translation length is bounded above by 2 provided that the defining graph has diameter at least 3. In this paper, we show that the same result holds without any assumption. This is done by exploring some graph theoretic properties of biconnected graphs, i.e. connected graphs whose complement is also connected.

This paper exhibits a structural strategy to produce new minimal submanifolds in spheres base on two given ones. The method is to spin the given minimal submanifolds by a curve $\gamma\subset \mathbb S^3$ in a balanced way and leads to resulting minimal submanifolds $-$ spiral minimal products, which form a two-dimensional family arising from intriguing pendulum phenomena decided by $C$ and $\tilde C$. With $C=0$, we generalize the construction of minimal tori in $\mathbb S^3$ explained in [Bre13] to higher dimensional situations. When $C=-1$, we recapture previous relative work in [CLU06] and [HK12] for special Legendrian submanifolds in spheres, and moreover, can gain numerous $\mathscr C$-totally real and totally real embedded minimal submanifolds in spheres and in complex projective spaces respectively. A key ingredient of the paper is to apply the beautiful extension result of minimal submanifolds by Harvey and Lawson [HL75] for a rotational reflection principle in our situation to establish curve $\gamma$.

This paper is about the orbifold theory of affine vertex operator superalgebras. Among the main results, we classify the irreducible modules and determine the fusion rules for the orbifold of the simple affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$.

We reveal an intimate connection between the torus link $T_{2s,2t}$ and the logarithmic $(s,t)$ VOA. We show that the singlet character of $(s,t)$-log VOA at the root of unity coincides with the Kashaev invariant and that it has a property of the quantum modularity. Also shown is that the tail of the $N$-colored Jones polynomial gives the character. Furthermore we propose a geometric method to compute the character.

Let $\Phi(x,y)$ denote the number of integers $n\in[1,x]$ free of prime factors $\le y$. We show that but for a few small cases, $\Phi(x,y)<.6x/\log y$ when $y\le\sqrt{x}$.

In this paper, based on an optimization problem, a sketch-and-project method for solving the linear matrix equation AXB = C is proposed. We provide a thorough convergence analysis for the new method and derive a lower bound on the convergence rate and some convergence conditions including the case that the coefficient matrix is rank deficient. By varying three parameters in the new method and convergence theorems, the new method recovers an array of well-known algorithms and their convergence results. Meanwhile, with the use of Gaussian sampling, we can obtain the Gaussian global randomized Kaczmarz (GaussGRK) method which shows some advantages in solving the matrix equation AXB = C. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods.

For $x\ge y>1$ and $u:= \log x/\log y$, let $\Phi(x,y)$ denote the number of positive integers up to $x$ free of prime divisors less than or equal to $y$. In 1950 de Bruijn [1] studied the approximation of $\Phi(x,y)$ by the quantity \[\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right),\] where $\gamma=0.5772156...$ is Euler's constant and \[\mu_y(u):=\int_{1}^{u}y^{t-u}\omega(t)\,dt.\] He showed that the asymptotic formula \[\Phi(x,y)=\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right)+O\left(\frac{xR(y)}{\log y}\right)\] holds uniformly for all $x\ge y\ge2$, where $R(y)$ is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.

A Coefficient Inverse Problem (CIP) of the determination of a coefficient of the Mean Field Games System (MFGS) of the second order is considered. The input data are generated by a single measurement event. Lateral Cauchy data, i.e. Dirichlet and Neumann boundary data are given for solutions of both equations forming the MFGS. In addition, solutions of both these equations are given at a fixed moment of time. H\"older stability estimates are obtained for both complete and incomplete lateral Cauchy data. These estimates imply uniqueness of our CIP. The apparatus of Carleman estimates is the main mathematical tool of this paper.

Motivated by a result of Treibergs, given a smooth function f(y) on the standard sphere S^2, and any positive constant H_0, we construct a spacelike surface with constant mean curvature H_0 in the Schwarzschild spacetime, which is the graph of a function u(y, r) defined on r>r_0 for some r_0>0 in the standard coordinates exterior to the blackhole. Moreover, u has the following asymptotic behavior: |u(y,r)-r_*-(f(y)+r^{-1}\phi(y)+1/2 r^{-2}\psi(y)|\le Cr^{-3} for some C>0, where r_*=r+2m\log(r/(2m)-1). Here \phi, \psi are functions determined by f and H_0. In particular, the surface intersects the future null infinity with the cut given by the function f.

We study the three-dimensional compressible elastic Navier-Stokes-Poisson equations induced by a new bipolar viscoelastic model derived here, which model the motion of the compressible electrically conducting fluids. The various boundary conditions for the electrostatic potential including the Dirichlet and Neumann boundary conditions are considered. By using a unified energy method, we obtain the unique global $H^2$ solution near a constant equilibrium state in the half-space or exterior of an obstacle. The elasticity plays a crucial role in establishing the $L^2$ estimate for the electrostatic field.

A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes known constructions for $4$-vertex-critical $P_7$-free graphs and $5$-vertex-critical $P_5$-free graphs and is in contrast to the fact that there are only finitely many $5$-vertex-critical $(P_5,C_5)$-free graphs. In fact, our construction is actually even more well-structured, being $(2P_2,K_3+P_1,C_5)$-free.

In this paper, we define a function $\mathcal K(x)\ge 0$ on Alexandrov space $X$, which measures the extent to which the metric of $X$ is locally $C^2$ at the point $x$. The measure $\mathcal K(x)\, \operatorname d\mathcal H^{n-1}$ on $X$ is the counterpart of the curvature measure $scal \,\operatorname d {vol}_{g}$ on smooth manifolds. In particular, we have that every point in the $C^2$-singular set $\mathcal KS(X)=\{x\in X\colon \mathcal K(x)>0\}$ is not locally $C^2$. The larger $\mathcal K(x)$ is, the more bending the space would look like near $x$. Examples demonstrate that $\mathcal KS(X)$ can have co-dimension 1. We prove that this is the worst that can be. Moreover, we show that $\int_{B_r(p)} \mathcal K(x)\, \operatorname d\mathcal H^{n-1}\le c(n,\kappa,\nu)r^{n-2}$ for any $n$-dimensional Alexandrov space $(X,p)$ with curv $\ge \kappa$ and $\operatorname{Vol}\left(B_1(p)\right)\ge\nu>0$. This leads to the Hausdorff dimension estimate $\dim_\mathcal H(\mathcal K\mathcal S(X))\le n-1$, and the Hausdorff measure estimate $\mathcal H^{n-1}\left(\mathcal K\mathcal S_\epsilon(X)\cap B_r\right)\le \epsilon^{-1}\cdot c(n,\nu)r^{n-2}$, where $\mathcal KS_\epsilon(X)=\{x\in X\colon \mathcal K(x)\ge\epsilon\}$. These results provide quantitative estimates of $C^2$-singular points in Alexandrov spaces and make progress on Naber's conjecture on the convergence of curvature measures. We also show that if $n$-dimensional Alexandrov spaces $X_i$ Gromov-Hausdorff converge to a smooth manifold without collapsing, then $\mathcal K_i\, \operatorname d\mathcal H^{n-1}\to 0$ as a measure.

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general linear group of degree $m$ over the ring $\frac{\mathbb{F}_{q}[x]}{\left\langle x^{s}-1 \right\rangle}$. In addition, the sizes of some of these families are computed. Finally, several criteria to construct linear permutation polynomials of $\mathbb{F}_{q^{2p}}$ (where $p$ is a prime number) with prescribed coefficients in $\mathbb{F}_{q^{2}}$ are given. Examples illustrating the main results are presented.

The relay autotuning method identifies plant parameters, from oscillations of the plant under relay feedback. To predict the presence and nature of such oscillations, we apply the following two approaches: (a) analysis of the switching dynamics, while using an ideal relay, and (b) bifurcation analysis, while using a smooth approximation of the relay. For stable plants with positive DC gains, our analyses predict that: (i) a periodic orbit is guaranteed, for a class of non-minimum phase plants of relative degree one, whose step response starts with an inverse response, and (ii) for a wider class of plants, whose root locus diagrams cross the imaginary axis at complex conjugate values, limit cycles are merely suggested.

Motivated by the manifold hypothesis, which states that data with a high extrinsic dimension may yet have a low intrinsic dimension, we develop refined statistical bounds for entropic optimal transport that are sensitive to the intrinsic dimension of the data. Our bounds involve a robust notion of intrinsic dimension, measured at only a single distance scale depending on the regularization parameter, and show that it is only the minimum of these single-scale intrinsic dimensions which governs the rate of convergence. We call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and establish MID scaling with no assumptions on the data distributions so long as the cost is bounded and Lipschitz, and for various entropic optimal transport quantities beyond just values, with stronger analogs when one distribution is supported on a manifold. Our results significantly advance the theoretical state of the art by showing that MID scaling is a generic phenomenon, and provide the first rigorous interpretation of the statistical effect of entropic regularization as a distance scale.

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of prime characteristic. We describe the intersection of a subvariety $X$ of $G$ with a finitely generated subgroup of $G(K)$.

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over $\mathbb{F}_{q}$ such that Tr$_{\mathbb{F}_{q^m}/\mathbb{F}_{q}}(\alpha^{-1})=a$, where $f(x) \in \mathbb{F}_{q^m}(x)$ is a rational function with degree sum $n$. In particular, for $q=5^k, ~k \geq 5$ and degree sum $n=4$, we explicitly find at most 11 choices of $(q, m)$ where existence of such pairs is not guaranteed.

In this paper, we consider a Hard-Core $(HC)$ model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove that the category of $\fa$-cominimax modules is a Melkersson subcategory of $R$-modules whenever $\dim R\leq 1$ and is an Abelian subcategory whenever $\dim R\leq 2$. We prove a characterization theorem for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(X,M)$ to be $\fa$-cominimax for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$.

Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infine family of scattered sequences of order three is obtained. Equivalence issues are also considered.

Motivated by the Farrell-Jones Conjecture for group rings, we formulate the $\mathcal{C}$op-Farrell-Jones Conjecture for the K-theory of Hecke algebras of td-groups. We prove this conjecture for (closed subgroups of) reductive p-adic groups G. In particular, the projective class group $K_0(\mathcal{H}(G))$ for a (closed subgroup) of a reductive p-adic group G can be computed as a colimit of projective class groups $K_0(\mathcal{H}(U))$ where U varies over the compact open subgroups of G. This implies that all finitely generated smooth complex representations of a reductive p-adic G admit finite projective resolutions by compactly induced representations. For SL$_n(F)$ we translate the colimit formula for $K_0(\mathcal{H}(G))$ to a more concrete cokernel description in terms of stabilizers for the action on the Bruhat-Tits building. For negative K-theory we obtain vanishing results, while we identify the higher K-groups $K_n(\mathcal{H}(G))$ with the value of G-homology theory on the extended Bruhat-Tits building. Our considerations apply to general Hecke algebras of the form $\mathcal{H}(G;R,\rho,\omega)$, where we allow a central character $\omega$ and a twist by an action $\rho$ of G on R. For the $\mathcal{C}$op-Farrell-Jones Conjecture we need to assume $\mathbb{Q} \subseteq R$ and a regularity assumption. As a key intermediate step we introduce the $\mathcal{C}vcy-Farrell-Jones conjecture. For the latter no regularity assumptions on R are needed.

In this paper, we study discrete-time dynamical systems generated by evolution operator of mosquito population. An invariant set is found and a Lyapunov function with respect to the operator is constructed in this set. Using the Lyapunov function, the global attraction of a fixed point is proved. Moreover, we give some biological interpretations of our results.

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $\langle A\rangle$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=(a,2a+d,2^2a+3d,...,2^ka+(2^k-1)d)$ and obtain formulas for $F(A)$ and $g(A)$ when $a+d\geq k$. Our formulas simplifies further for some special cases, such as Mersenne and Thabit numerical semigroups. We obtain explicit formulas for generalized Mersenne and Thabit numerical semigroups and some more general numerical semigroups.

The purpose of these notes is to discuss the advances in the theory of Lyapunov exponents of linear $\text{SL}_2(\mathbb{R})$ cocycles over hyperbolic maps. The main focus is around results regarding the positivity of the Lyapunov exponent and the regularity of this function with respect to the underlying data.

We study the decay (at infinity) of extremals of Morrey's inequality in $\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla u\|_{L^p(\mathbb{R}^n)} , $$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey's inequality. We prove that if $n \geq 2$ then any extremal has a power decay of order $\beta$ for any $$ \beta<-\frac13+\frac{2}{3(p-1)}+\sqrt{\left(-\frac13+\frac{2}{3(p-1)}\right)^2+\frac13}. $$

Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise cross-intersecting families if $\mathcal{A}_i$ and $\mathcal{A}_j$ are cross-intersecting when $1\le i<j \le t$. Additionally, if $\mathcal{A}_j\ne \emptyset$ for each $j\in [t]$, then we say that $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\mathcal{A}_1\subset{[n]\choose k_1}, \mathcal{A}_2\subset{[n]\choose k_2}, \dots, \mathcal{A}_t\subset{[n]\choose k_t}$ be non-empty pairwise cross-intersecting families with $t\geq 2$, $k_1\geq k_2\geq \cdots \geq k_t$, and $n\geq k_1+k_2$, we determine the maximum value of $\sum_{i=1}^t{|\mathcal{A}_i|}$ and characterize all extremal families. This answers a question of Shi, Frankl and Qian [Combinatorica 42 (2022)] and unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)] and Shi, Frankl and Qian [Combinatorica 42 (2022)]. The key techniques in previous works cannot be extended to our situation. A result of Kruskal-Katona is applied to allow us to consider only families $\mathcal{A}_i$ whose elements are the first $|\mathcal{A}_i|$ elements in lexicographic order. We bound $\sum_{i=1}^t{|\mathcal{A}_i|}$ by a function $f(R)$ of the last element $R$ (in the lexicographic order) of $\mathcal{A}_1$, introduce the concepts `$c$-sequential' and `down-up family', and show that $f(R)$ has several types of local convexities.

Gromov (2003) constructed finitely generated groups whose Cayley graphs contain (in a certain weak sense) all graphs from a given infinite sequence of expander graphs of unbounded girth and bounded diameter-to-girth ratio. These so-called \emph{Gromov monster groups} provide examples of finitely generated groups that do not coarsely embed into Hilbert space, among other interesting properties. This approach was recently refined by Osajda (2020): in his modified onstruction, all the expander graphs in the sequence can be found as isometric subgraphs of the Cayley graph of the resulting group. This in turn leads to the construction of finitely generated groups with even stronger properties. In this short note, we simplify the combinatorial part of Osajda's construction, decreasing the number of generators of the resulting group significantly.

We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

Analyzing the spectral behavior of random matrices with dependency among entries is a challenging problem. The adjacency matrix of the random $d$-regular graph is a prominent example that has attracted immense interest. A crucial spectral observable is the extremal eigenvalue, which reveals useful geometric properties of the graph. According to the Alon's conjecture, which was verified by Friedman, the (nontrivial) extremal eigenvalue of the random $d$-regular graph is approximately $2\sqrt{d-1}$. In the present paper, we analyze the extremal spectrum of the random $d$-regular graph (with $d\ge 3$ fixed) equipped with random edge-weights, and precisely describe its phase transition behavior with respect to the tail of edge-weights. In addition, we establish that the extremal eigenvector is always localized, showing a sharp contrast to the unweighted case where all eigenvectors are delocalized. Our method is robust and inspired by a sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs (Ganguly and Nam, '22), which can also be applied to analyze the spectrum of general random matrices whose entries are dependent.

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally K\"ahler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-K\"ahler almost abelian Lie algebras.

A systematic derivation provides extended series of correlation inequalities in quantum systems. Each order in truncated Taylor expansion of the spectral representation for the Duhamel correlation function gives its lower and upper bounds. The obtained bound on the Duhamel function and the square root interpolation method enable us to derive a variational solution of specific free energy in the transverse field Sherrington-Kirkpatrick model.

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.

Let $L=\delta^n+\sum_{i=0}^{n-1}a_i(z)\delta^i$ be a differential operator with coefficients in $\mathbb{Q}(z)$ of order $n\geq2$, where $\delta=z d/dz$. Suppose that $L$ has maximal unipotent monodromy at zero. In this paper we give a sufficient condition for the canonical coordinate of $L$ belongs to $\mathbb{Z}_p[[z]]$. This sufficient condition relies on the notion of Frobenius structure. As a consequence of the main result we prove that if $n=4$ and $L$ is an irreducible Picard-Fuchs equation with maximal unipotent monodromy at zero having a strong B-incarnation, then there is an integer $N>0$ such that the canonical coordinate of $L$ belongs to $\mathbb{Z}[1/N][[z]]$.

We consider the damped time-harmonic Galbrun's equation, which is used to model stellar oscillations. We introduce a discontinuous Galerkin finite element method (DGFEM) with $H(\operatorname{div})$-elements, which is nonconform with respect to the convection operator. We report a convergence analysis, which is based on the frameworks of discrete approximation schemes and T-compatibility. A novelty is that we show how to interprete a DGFEM as a discrete approximation scheme and this approach enables us to apply compact perturbation arguments in a DG-setting, and to circumvent any extra regularity assumptions on the solution. The advantage of the proposed $H(\operatorname{div})$-DGFEM compared to $H^1$-conforming methods is that we do not require a minimal polynomial order or any special assumptions on the mesh structure. The considered DGFEM is constructed without a stabilization term, which considerably improves the assumption on the smallness of the Mach number compared to other DG methods and $H^1$-conforming methods, and the obtained bound is fairly explicit. In addition, the method is robust with respect to the drastic changes of magnitude of the density and sound speed, which occur in stars. The convergence of the method is obtained without additional regularity assumptions on the solution, and for smooth solutions and parameters convergence rates are derived.

In this paper we consider the singularities on the boundary of limiting $V$-states of the 2-dim incompressible Euler equation. By setting up a Weiss-type monotoncity formula for a sign-changing unstable elliptic free boundary problem, we obtain the classification of singular points on the free boundary: the boundary of vortical domain would form either a right angle ($90^\circ$) or a cusp ($0^\circ$) near these points in the limiting sense. For the first alternative, we further prove the uniformly regularity of the free boundary near these isolated singular points.

Let $S$ be a closed surface of genus $g\geq 1$, furnished with an area form $\omega$. We show that there exists an open and dense set ${\mathcal O_r}$ of the space of Hamiltonian diffeomorphisms of class $C^r$, $1\leq r\leq\infty$, endowed with the $C^r$-topology, such that every $f\in \mathcal O_r$ possesses infinitely many non contractible periodic orbits. We obtain a positive answer to a question asked by Viktor Ginzburg and Ba\c{s}ak G\"{u}rel. The proof is a consequence of recent previous works of the authors [LecSa].

In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal-dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor-corrector adaptivity and parallel performance studies are explored as well.

We consider the Moran model of population genetics with two types, mutation, and selection, and investigate the line of descent of a randomly-sampled individual from a contemporary population. We trace this ancestral line back into the distant past, far beyond the most recent common ancestor of the population (thus connecting population genetics to phylogeny) and analyse the mutation process along this line. To this end, we use the pruned lookdown ancestral selection graph, which consists of the set of potential ancestors of the sampled individual at any given time. A crucial observation is that a mutation on the ancestral line requires that the ancestral line occupy the top position in the graph just `before' the event (in forward time). Relative to the neutral case (that is, without selection), we obtain a general bias towards the beneficial type, an increase in the beneficial mutation rate, and a decrease in the deleterious mutation rate. This sheds new light on previous analytical results. We discuss our findings in the light of a well-known observation at the interface of phylogeny and population genetics, namely, the difference in the mutation rates (or, more precisely, mutation fluxes) estimated via phylogenetic methods relative to those observed in pedigree studies.

A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the gradient of the solution, always blows up with respect to the distance $\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first establish the pointwise upper bound on the gradient possessing the singularity of order $\varepsilon^{-\beta}$ with $\beta=(1-\alpha)/m$ for some $\alpha\geq0$, where $\alpha=0$ if $d=2$ and $\alpha>0$ if $d\geq3$. In particular, we give a quantitative description for the range of horizontal length of the narrow channel in the process of establishing the gradient estimates, which provides a clear understanding for the applied techniques and methods. For $d\geq2$, we further construct a supersolution to sharpen the upper bound with any $\beta>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a subsolution is also constructed to show the almost optimality of the blow-up rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares. This fact reveals a novel dichotomy phenomena that the singularity of the gradient is uniquely determined by one of the convexity parameter $m$ and the nonlinear exponent $p$ except for the critical case of $p=m+1$ in two dimensions.

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright-Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forward in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backward in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by unknown quantities of interest. We employ the so-called Bernoulli factory, a celebrated tool in simulation-based computing, to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright-Fisher diffusions with general drift functions, and Allen-Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

In this paper, we investigate direct and inverse problems for the time-fractional heat equation with a time-dependent diffusion coefficient for positive operators. First, we consider the direct problem, and the unique existence of the generalized solution is established. We also deduce some regularity results. Here, our proofs are based on the eigenfunction expansion method. Second, we consider the inverse problem of determining the diffusion coefficient. The well-posedness of this inverse problem is shown by reducing the problem to an operator equation for the diffusion coefficient.

This note proves, for simplicity for the heat equation, that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates.

In this paper, we study the long-time behavior of the solutions to the two-dimensional incompressible free Navier Stokes equation (without forcing) with small viscosity $\nu$, when the initial data is close to stable monotone shear flows. We prove the asymptotic stability and obtain the sharp stability threshold $\nu^{\frac{1}{2}}$ for perturbations in the critical space $H^{log}_xL^2_y$. Specifically, if the initial velocity $V_{in}$ and the corresponding vorticity $W_{in}$ are $\nu^{\frac{1}{2}}$-close to the shear flow $(b_{in}(y),0)$ in the critical space, i.e., $\|V_{in}-(b_{in}(y),0)\|_{L_{x,y}^2}+\|W_{in}-(-\partial_yb_{in})\|_{H^{log}_xL^2_y}\leq \epsilon \nu^{\frac{1}{2}}$, then the velocity $V(t)$ stay $\nu^{\frac{1}{2}}$-close to a shear flow $(b(t,y),0)$ that solves the free heat equation $(\partial_t-\nu\partial_{yy})b(t,y)=0$. We also prove the enhanced dissipation and inviscid damping, namely, the nonzero modes of vorticity and velocity decay in the following sense $\|W_{\neq}\|_{L^2}\lesssim \epsilon\nu^{\frac{1}{2}}e^{-c\nu^{\frac{1}{3}}t}$ and $\|V_{\neq}\|_{L^2_tL^2_{x,y}}\lesssim \epsilon\nu^{\frac{1}{2}}$. In the proof, we construct a time-dependent wave operator corresponding to the Rayleigh operator $b(t,y)\partial_x-\partial_{yy}b(t,y)\partial_x\Delta^{-1}$, which could be useful in future studies.

Given a graph $G =(V,E)$, a bijection $f: E \rightarrow \{1, 2, \dots,|E|\}$ is called a local antimagic labeling of $G$ if the vertex weight $w(u) = \sum_{uv \in E} f(uv)$ is distinct for all adjacent vertices. The vertex weights under the local antimagic labeling of $G$ induce a proper vertex coloring of a graph $G$. The \textit{local antimagic chromatic number} of $G$ denoted by $\chi_{la}(G)$ is the minimum number of weights taken over all such local antimagic labelings of $G$. In this paper, we investigate the local antimagic chromatic numbers of the union of some families of graphs, corona product of graphs, and necklace graph and we construct infinitely many graphs satisfying $\chi_{la}(G) = \chi(G)$.

In this paper, we study the order of approximation for max-product Kantorovich sampling operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as $L^{p}$-spaces, interpolation spaces and exponential spaces.

This article presents a class of new relaxation modulus-based iterative methods to process the large and sparse implicit complementarity problem (ICP). Using two positive diagonal matrices, we formulate a fixed-point equation and prove that it is equivalent to ICP. Also, we provide sufficient convergence conditions for the proposed methods when the system matrix is a $P$-matrix or an $H_+$-matrix. Keyword: Implicit complementarity problem, $H_{+}$-matrix, $P$-matrix, matrix splitting, convergence

We consider $G_2$ manifolds with a cohomogeneity two $\mathbb{T}^2\times \mathrm{SU}(2)$ symmetry group. We give a local characterization of these manifolds and we describe the geometry, including regularity and singularity analysis, of cohomogeneity one calibrated submanifolds in them. We apply these results to the manifolds recently constructed by Foscolo-Haskins-N\"ordstrom and to the Bryant-Salamon manifold of topology $S^3\times \mathbb{R}^4$. In particular, we describe new large families of complete $\mathbb{T}^2$-invariant associative submanifolds in them.

In this work, we propose an absolute value block $\alpha$-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block $\alpha$-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block $\alpha$-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block $\alpha$-circulant preconditioner, which fills the gap between block $\alpha$-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen $\alpha$ the MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

We show that the $p$-adic Siegel Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level $p$, by applying the theory of mod $p$-power singular forms. As special cases of this result, we derive the result of Nagaoka and Katsurada--Nagaoka.

We consider the Fleming-Viot particle system consisting of $N$ identical particles evolving in $\mathbb{R}_{>0}$ as Brownian motions with constant drift $-1$. Whenever a particle hits $0$, it jumps onto another particle in the interior. It is known that this particle system has a hydrodynamic limit as $N\rightarrow\infty$ given by Brownian motion with drift $-1$ conditioned not to hit $0$. This killed Brownian motion has an infinite family of quasi-stationary distributions (QSDs), with a Yaglom limit given by the unique QSD minimising the survival probability. On the other hand, for fixed $N<\infty$, this particle system converges to a unique stationary distribution as time $t\rightarrow\infty$. We prove the following selection principle: the empirical measure of the $N$-particle stationary distribution converges to the aforedescribed Yaglom limit as $N\rightarrow\infty$. The selection problem for this particular Fleming-Viot process is closely connected to the microscopic selection problem in front propagation, in particular for the $N$-branching Brownian motion. The proof requires neither fine estimates on the particle system nor the use of Lyapunov functions.

We prove that on a general hypersurface in $\mathbb{P}^N$ of degree $d$ and dimension at least $2$, if an arithmetically Cohen-Macaulay (ACM) bundle $E$ and its dual have small regularity, then any non-trivial Hodge class in $H^{n}(X, E\otimes\Omega^n_X)$, $n = \lfloor\frac{N-1}{2}\rfloor$, produces a trivial direct summand of $E$. As a consequence, we prove that there is no universal Ulrich bundle on the family of smooth hypersurfaces of degree $d\geq 3$ and dimension at least $4$. This last statement may be viewed as a Franchetta-type conjecture for Ulrich bundles on smooth hypersurfaces.

Causal regularization was introduced as a stable causal inference strategy in a two-environment setting in \cite{kania2022causal}. We start with observing that causal regularizer can be extended to several shifted environments. We derive the multi-environment casual regularizer in the population setting. We propose its plug-in estimator, and study its concentration in measure behavior. Although the variance of the plug-in estimator is not well-defined in general, we instead study its conditional variance both with respect to a natural filtration of the empirical as well as conditioning with respect to certain events. We also study generalizations where we consider conditional expectations of higher central absolute moments of the estimator. The results presented here are also new in the prior setting of \cite{kania2022causal} as well as in \cite{Rot}.

In covariance matrix estimation, one of the challenges lies in finding a suitable model and an efficient estimation method. Two commonly used approaches in the literature involve imposing linear restrictions on the covariance matrix or its inverse. Another approach considers linear restrictions on the matrix logarithm of the covariance matrix. In this paper, we present a general framework for linear restrictions on different transformations of the covariance matrix, including the mentioned examples. Our proposed estimation method solves a convex problem and yields an M-estimator, allowing for relatively straightforward asymptotic and finite sample analysis. After developing the general theory, we focus on modelling correlation matrices and on sparsity. Our geometric insights allow to extend various recent results in covariance matrix modelling. This includes providing unrestricted parametrizations of the space of correlation matrices, which is alternative to a recent result utilizing the matrix logarithm.

The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.

A sketch-and-select Arnoldi process to generate a well-conditioned basis of a Krylov space is proposed. At each iteration the procedure utilizes randomized sketching to select a limited number of previously computed basis vectors to project out of the current basis vector. The computational cost grows linearly with the dimension of the Krylov basis. The subset selection problem for the projection step is approximately solved with a number of heuristic algorithms and greedy methods used in statistical learning and compressive sensing.

Given graphs $G_1,\ldots,G_s$ all on the same vertex set and a graph $H$ with $e(H) \leq s$, a copy of $H$ is transversal or rainbow if it contains at most one edge from each $G_c$. When $s=e(H)$, such a copy contains exactly one edge from each $G_i$. We study the case when $H$ is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs $H$. Our proofs use weak regularity in the $3$-uniform hypergraph whose edges are those $xyc$ where $xy$ is an edge in the graph $G_c$. We apply our lemma to give a large class of spanning $3$-uniform linear hypergraphs $H$ such that any sufficiently large uniformly dense $n$-vertex $3$-uniform hypergraph with minimum vertex degree $\Omega(n^2)$ contains $H$ as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft.

We study semilinear elliptic inequalities with a potential on infinite graphs. Given a distance on the graph, we assume an upper bound on its Laplacian, and a growth condition on a suitable weighted volume of balls. Under such hypotheses, we prove that the problem does not admit any nonnegative nontrivial solution. We also show that our conditions are optimal.

In this paper we prove that the positive Lyapunov exponents for the billiard flow in an open billiard in $\R^d$ ($d\geq 3$) are all equal. We should stress that we do not make any particular assumptions about the shape and size of the components of our obstacles -- they are just assumed to be strictly convex and compact with $C^3$ boundaries and satisfy the so called no eclipse condition.

Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper, we determine when the clutter $\mathcal{C}$ is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $\tau=2$ Conjectures for this class of clutters.

The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmel\'e (2003) states that the treewidth of $G$ is at most its circumference minus $1$. In case $G$ is $2$-connected, this upper bound also holds for the pathwidth of $G$; in fact, even the treedepth of $G$ is upper bounded by its circumference (Bria\'nski, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em bond} in $G$, an inclusion-wise minimal set of edges $F$ such that $G-F$ has more components than $G$. In matroidal terms, the cocircumference of $G$ is the circumference of the bond matroid of $G$. Our first result is the following `dual' version of Birmel\'e's theorem: The treewidth of a graph $G$ is at most its cocircumference. Our second and main result is an upper bound of $3k-2$ on the pathwidth of a $2$-connected graph $G$ with cocircumference $k$. Contrary to circumference, no such bound holds for the treedepth of $G$. Our two upper bounds are best possible up to a constant factor.

We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of $T$ and $T^*$ are equal for compact operators. Second, for non-compact, bounded linear operators from $X$ to $Y$, we obtain a relationship between certain $s$-numbers of $T$ and $T^*$ under natural conditions on $X$ and $Y$. Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results by comparing the degree of compactness of $T$ with that of its adjoint $T^*$.

Thurston's earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of Thurston's earthquake theorem, using the bi-invariant geometry of the Lie group $\mathrm{PSL}(2,\mathbb R)$, which is also called Anti-de Sitter three-space. The involved techniques are elementary, and no background knowledge is assumed apart from some two-dimensional hyperbolic geometry.

Uniformly valid inference for cointegrated vector autoregressive processes has so far proven difficult due to certain discontinuities arising in the asymptotic distribution of the least squares estimator. We show how asymptotic results from the univariate case can be extended to multiple dimensions and how inference can be based on these results. Furthermore, we show that the novel instrumental variable procedure proposed by [20] (IVX) yields uniformly valid confidence regions for the entire autoregressive matrix. The results are applied to two specific examples for which we verify the theoretical findings and investigate finite sample properties in simulation experiments.

This paper defines and establishes relations among approximation spaces of certain operators called \textit{H-operators}, which generalize the notion of self-adjoint to Banach spaces.

In this paper we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flows given by a homogeneous, symmetric, smooth non-negative function $\gamma$ defined in an open cone $\Gamma\subset\mathbb{R}^n$. The main results are tangential principles, nonexistence theorems for closed and entire solutions, and a uniqueness result that says that any strictly convex $\gamma$-translator defined on a ball with a single end $\mathcal{C}^2$-asymptotic to a cylinder is the ''bowl''-type solution found in the translator paper of S. Rengaswami.

In this paper, we establish a Modica type estimate on bounded solutions to the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=c_0 &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates we will deduce information about the curvature of $\partial \Omega$ under a certain condition on $c_0$ and $f$. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.

We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard scattering if it is higher order. We handle a general nonlinearity without any vanishing condition, requiring that the linearized operator around the solitary wave has neither nonzero eigenvalues, nor threshold resonances. Initial data are chosen in a neighborhood of the solitary waves in the natural space $H^1 \cap L^{2,1}$ (where the latter is the weighted $L^2$ space). The proof relies on the analysis of resonances as seen through the distorted Fourier transform, combined for the first time with modulation and renormalization techniques.

We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts to finding solutions in a rational Krylov space. By mixing these with usual Krylov spaces, we consider least-squares problem in these mixed rational spaces. Applying the Arnoldi method leads to a sparse, pentadiagonal representation of the forward operator, and we introduce the Lanczos method for solving the least-squares problem by factorizing this matrix. Finally, we present an equivalent conjugate-gradient-type method that does not rely on explicit orthogonalization but uses short-term recursions and Tikhonov regularization in each second step. We illustrate the convergence and regularization properties by some numerical examples.

We propose a nonparametric method for estimating the conditional quantile function that admits a generalized additive specification with an unknown link function. This model nests single-index, additive, and multiplicative quantile regression models. Based on a full local linear polynomial expansion, we first obtain the asymptotic representation for the proposed quantile estimator for each additive component. Then, the link function is estimated by noting that it corresponds to the conditional quantile function of a response variable given the sum of all additive components. The observations are supposed to be a sample from a strictly stationary and absolutely regular process. We provide results on (uniform) consistency rates, second order asymptotic expansions and point wise asymptotic normality of each proposed estimator.

Recently, there are a great deal of work done which connects the Legendrian isotopic problem with contact invariants. The isotopic problem of Legendre curve in a contact 3-manifold was studies via the Legendrian curve shortening flow which was introduced and studied by K. Smoczyk. On the other hand, in the SYZ Conjecture, one can model a special Lagrangian singularity locally as the special Lagrangian cones in C^{3}. This can be characterized by its link which is a minimal Legendrian surface in the 5-sphere. Then in these points of view, in this paper we will focus on the existence of the long-time solution and asymptotic convergence along the Legendrian mean curvature flow in higher dimensional {\eta}-Einstein Sasakian (2n+1)-manifolds under the suitable stability condition due to the Thomas-Yau conjecture.

We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles.

Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So \[ ||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}. \] A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is \emph{$(s,k)$-clash-free} if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\leq k<n$, let $\sigma(n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of $\mathbb{Z}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma(n,k)$ in all cases.

In this paper, we study the qualitative behavior at a vortex blow-up point for Chern-Simon-Higgs equation. Roughly speaking, we will establish an energy identity at a each such point, i.e. the local mass is the sum of the bubbles. Moreover, we prove that either there is only one bubble which is a singular bubble or there are more than two bubbles which contains no singular bubble. Meanwhile, we prove that the energies of these bubbles must satisfy a quadratic polynomial which can be used to prove the simple blow-up property when the multiplicity is small. As is well known, for many Liouville type system, Pohozaev type identity is a quadratic polynomial corresponding to energies which can be used directly to compute the local mass at a blow-up point. The difficulty here is that, besides the energy's integration, there is a additional term in the Pohozaev type identity of Chern-Simon-Higgs equation. We need some more detailed and delicated analysis to deal with it.

This paper is concerned with the following logarithmic Schr\"{o}dinger system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \ &\ \int_{\Omega}|u_i|^2\,dx=\rho_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in H_0^1(\Omega;\mathbb R^2),\end{align}\right.$$ where $\Omega=\mathbb{R}^N$ or $\Omega\subset\mathbb R^N(N\geq3)$ is a bounded smooth domain, $\omega_i\in\mathbb R$, $\mu_i,\ \rho_i>0,\ i=1,2.$ Moreover, $p,\ q\geq1,\ 2\leq p+q\leqslant 2^*$, where $2^*:=\frac{2N}{N-2}$. By using a Gagliardo-Nirenberg inequality and careful estimation of $u\log u^2$, firstly, we will provide a unified proof of the existence of the normalized ground states solution for all $2\leq p+q\leqslant 2^*$. Secondly, we consider the stability of normalized ground states solutions. Finally, we analyze the behavior of solutions for Sobolev-subcritical case and pass the limit as the exponent $p+q$ approaches to $2^*$. Notably, the uncertainty of sign of $u\log u^2$ in $(0,+\infty)$ is one of the difficulties of this paper, and also one of the motivations we are interested in. In particular, we can establish the existence of positive normalized ground states solutions for the Br\'{e}zis-Nirenberg type problem with logarithmic perturbations (i.e., $p+q=2^*$). In addition, our study includes proving the existence of solutions to the logarithmic type Br\'{e}zis-Nirenberg problem with and without the $L^2$-mass $\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$ constraint by two different methods, respectively. Our results seems to be the first result of the normalized solution of the coupled nonlinear Schr\"{o}dinger system with logarithmic perturbation.

We introduced the concept of $n$-Lie algebras in the Loday-Pirashvili category. We will study their representation, cohomology, deformation and abelian extension theory.

We prove that for any integral lattice $\mathcal{L} \subset \mathbb{R}^n$ (that is, a lattice $\mathcal{L}$ such that the inner product $\langle \mathbf{y}_1,\mathbf{y}_2 \rangle$ is an integer for all $\mathbf{y}_1, \mathbf{y}_2 \in \mathcal{L}$) and any positive integer $k$, \[ |\{ \mathbf{y} \in \mathcal{L} \ : \ \|\mathbf{y}\|^2 = k\}| \leq 2 \binom{n+2k-2}{2k-1} \; , \] giving a nearly tight reverse Minkowski theorem for integral lattices.

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability as long as the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - SIT^{-1}||_2 \leq \varepsilon$ in at most $O \left( \log(n) \log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of exact arithmetic matrix pencil diagonalization.

This work examines the controllability of planar incompressible ideal magnetohydrodynamics (MHD). Interior controls are obtained for problems posed in doubly-connected regions; simply-connected configurations are driven by boundary controls. Up to now, only straight channels regulated at opposing walls have been studied. Hence, the present program adds to the literature an exploration of interior controllability, extends the known boundary controllability results, and contributes ideas for treating general domains. To transship obstacles stemming from the MHD coupling and the magnetic field topology, a divide-and-control strategy is proposed. This leads to a family of nonlinear velocity-controlled sub-problems which are solved using J.-M. Coron's return method. The latter is here developed based on a reference trajectory in the domain's first cohomology space.

We propose, analyze and implement a virtual element discretization for an interfacial poroelasticity-elasticity consolidation problem. The formulation of the time-dependent poroelasticity equations uses displacement, fluid pressure, and total pressure, and the elasticity equations are written in the displacement-pressure formulation. The construction of the virtual element scheme does not require Lagrange multipliers to impose the transmission conditions (continuity of displacement and total traction, and no-flux for the fluid) on the interface. We show the stability and convergence of the virtual element method for different polynomial degrees, and the error bounds are robust with respect to delicate model parameters (such as Lame constants, permeability, and storativity coefficient). Finally, we provide numerical examples that illustrate the properties of the scheme.

We prove that the biharmonic NLS equation $\Delta^2 u +2\Delta u+(1+\varepsilon)u=|u|^{p-2}u$ in $\mathbb R^d$ has at least $k+1$ different solutions if $\varepsilon>0$ is small enough and $2<p<2_\star^k$, where $2_\star^k$ is an explicit critical exponent arising from the Fourier restriction theory of $O(d-k)\times O(k)$-symmetric functions. This extends the recent symmetry breaking result of Lenzmann-Weth and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of $k$. We further prove that, as $\varepsilon\to 0^+$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.

Let G = Z2 act freely on a finitistic space X. If the mod 2 cohomology of X is isomorphic to the real projective space RP2n+1(resp. complex projective space CP2n+1) then the mod 2 cohomology of orbit spaces of these free actions are RP1 xCPn(resp. RP2xHPn) [11]. In this paper, we have discussed converse of this result. We have showed that if the mod 2 cohomology of the orbit space X/G is RP1xCPn(resp. RP2 x HPn). Similar, results for p > 2, a prime, is also discussed. It is proved that one of the possibilities of orbit spaces of free involutions on product of projective spaces RPn x RP2m+1(resp. CPn x CP2m+1) is RP1 x RPn x CPm(resp. RP2 x CPn x HPm) [12]. The converse of these statements are also discussed.

Let G = S^d, d = 0, 1 or 3, act freely on a finitistic connected space X. This paper gives the cohomology classification of X if a mod 2 or rational cohomology of the orbit space X/G is isomorphic to the product of a projective space and sphere FP^n x S^m, where F = R, C or H, respectively. For a free involution on X, a lower bound of covering dimension of the coincidence set of a continuous map f: X -> R^k is also determined.

The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker-Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.

Several recent papers have addressed modelling of the tissue growth by the multi-phase models where the velocity is related to the pressure by one of the physical laws (Stoke's, Brinkman's or Darcy's). While each of these models has been extensively studied, not so much is known about the connection between them. In the recent paper (arXiv:2303.10620), assuming the linear form of the pressure, the Authors connected two multi-phase models by an inviscid limit: the viscoelastic one (of Brinkman's type) and the inviscid one (of Darcy's type). Here, we prove that the same is true for a nonlinear, power-law pressure. The new ingredient is that we use relation between the pressure $p$ and the Brinkman potential $W$ to deduce compactness in space of $p$ from the compactness in space of $W$.

Recent progress in the deformation theory of Calabi-Yau varieties $Y$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $k$-liminal singularities for $k\ge 1$. The goal of this paper is to show that certain aspects of this study extend naturally to the $0$-liminal case as well, i.e. to Calabi-Yau varieties $Y$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $Y$ in the case of isolated $0$-liminal hypersurface singularities, and extend Namikawa's unobstructedness theorem for deformations of singular Calabi-Yau threefolds $Y$ with canonical singularities to the case where $Y$ has an isolated $0$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples.

We devise an end-to-end communication channel model that describes the performance of RIS-assisted MIMO wireless links. The model borrows the impedance (interaction) matrix formalism from the Method of Moments and provides a physics-based communication model. In configurations where the transmit and receive antenna arrays are distant from the RIS beyond a wavelength, a reduced model provides accurate results for arbitrary RIS unit cell geometry. Importantly, the simplified model configures as a cascaded channel transfer matrix whose mathematical structure is compliant with widely accepted, but less accurate, system level RIS models. A numerical validation of the communication model is presented for the design of binary RIS structures with scatterers of canonical geometry. Attained results are consistent with path-loss models: For obstructed line-of-sight between transmitter and receiver, the channel capacity of the (optimised) RIS-assisted link scales as $R^{-2}$, with $R$ RIS-receiver distance at fixed transmitter position. Our results shows that the applicability of communication models based on mutual impedance matrices is not restricted to canonical minimum scattering RIS unit cells.

Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure $\mu:\mathcal{A}\to[0,\infty]$, we introduce the $U_\mu^I$-topology and the $m_\mu^I$-topology on $\mathcal{M}(X,\mathcal{A})$ as generalized versions of the topology of uniform convergence or the $U$-topology and the $m$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively. With $I=\mathcal{M}(X,\mathcal{A})$, these two topologies reduce to the $U_\mu$-topology and the $m_\mu$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively, already considered before. If $I$ is a countably generated ideal in $\mathcal{M}(X,\mathcal{A})$, then the $U_\mu^I$-topology and the $m_\mu^I$-topology coincide if and only if $X\setminus \bigcap Z[I]$ is a $\mu$-bounded subset of $X$. The components of $0$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_\mu^I$-topology and the $m_\mu^I$-topology are realized as $I\cap L^\infty(X,\mathcal{A},\mu)$ and $I\cap L_\psi(X,\mathcal{A},\mu)$ respectively. Here $L^\infty(X,\mathcal{A},\mu)$ is the set of all functions in $\mathcal{M}(X,\mathcal{A})$ which are essentially $\mu$-bounded over $X$ and $L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}$. It is established that an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is dense in the $U_\mu$-topology if and only if it is dense in the $m_\mu$-topology and this happens when and only when there exists $Z\in Z[I]$ such that $\mu(Z)=0$. Furthermore, it is proved that $I$ is closed in $\mathcal{M}(X,\mathcal{A})$ in the $m_\mu$-topology if and only if it is a $Z_\mu$-ideal in the sense that if $f\equiv g$ almost everywhere on $X$ with $f\in I$ and $g\in\mathcal{M}(X,\mathcal{A})$, then $g\in I$.

We investigate a model of continuous-time simple random walk paths in $\mathbb{Z}^d$ undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d.~Pareto-distributed with parameter $\alpha>d$, and we tune the strength of the interactions in such a way that they both contribute on the same scale as $t\to\infty$. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for $\alpha>2d$: the random-walk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function. The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution and is in the spirit of a standard extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in \cite{KLMS09}.

We consider two classes of non-toric log del Pezzo $\mathbb{C}^*$-surfaces: on the one side the 1/3-log canonical ones and on the other side those of Picard number one and Gorenstein index at most 65. In each of the two classes we figure out the surfaces admitting a K\"ahler-Einstein metric, a K\"ahler-Ricci soliton and those allowing a Sasaki-Einstein metric on the link of their anticanonical cone. We encounter examples that admit a K\"{a}hler-Ricci soliton but no Sasaki-Einstein cone link metric.

We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of $n$ integers as $n$ grows large, establishing asymptotic expansions in powers of $n^{-1/6}$ in the general case and in powers of $n^{-1/3}$ in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy-Widom distributions $F_\beta$ for $\beta=1$ or $\beta=4$, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of $F_\beta$ with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of L$\beta$E are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities. (This paper continues our work arXiv:2301.02022, which established similar results in the case of general permutations and $\beta=2$.)

In this work, we establish the existence and decay of {\em plasmons}, the quantum of Langmuir's oscillatory waves found in plasma physics, for the linearized Hartree equations describing an interacting gas of infinitely many fermions near general translation-invariant steady states, including compactly supported Fermi gases at zero temperature, in the whole space $\RR^d$ for $d\ge 2$. Notably, these plasmons exist precisely due to the long-range pair interaction between the particles. Next, we provide a survival threshold of spatial frequencies, below which the plasmons purely oscillate and disperse like a Klein-Gordon's wave, while at the threshold they are damped by {\em Landau damping}, the classical decaying mechanism due to their resonant interaction with the background fermions. The explicit rate of Landau damping is provided for general radial homogenous equilibria. Above the threshold, the density of the excited fermions is well approximated by that of the free gas dynamics and thus decays rapidly fast for each Fourier mode via {\em phase mixing}. Finally, pointwise bounds on the Green function and dispersive estimates on the density are established.

The aim of this work is to provide a systemic study and generalization of the celebrated weak KAM theory and Aubry-Mather theory in sub-Riemannian setting, or equivalently, on a Carnot-Caratheodory metric space. In this framework we consider an optimal control problem with state equation of sub-Riemannian type, namely, admissible trajectories are solutions of a linear in control and nonlinear in space ODE. Such a nonlinearity is given by a family of smooth vector fields satisfying the Hormander condition which implies the controllability of the system. In this case, the Hamiltonian function associated with the above control problem fails to be coercive and thus the results in the Tonelli setting can not be applied. In order to overcome this issue, our approach is based on metric properties of the geometry induced on the state space by the sub-Riemannian structure.

In this article, we initiate the study of visibility property in the context of quasihyperbolic metric. Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $\partial_{Euc}\Omega$ be its Euclidean boundary. We say that the pair $\{p,q\}\subset\partial_{Euc}\Omega$ has visible quasihyperbolic geodesic if there exist open neighborhoods $U_p$ of $p$ and $U_q$ of $q$ and a compact set $K_{\{p,q\}}\subset \Omega$ such that for any quasihyperbolic geodesic which joins points in $U_p$ and $U_q$ intersects with $K$. If a domain has visible quasihyperbolic geodesic for every pair of points, we say it is a visibility domain. A major part of this paper is devoted to provide a rich collection of visibility domains. In this line we provide a general visibility criteria for a domain to be a visibility domain, and using this we obtain the visibility of Uniform domains, John domains and domains satisfying quasihyperbolic boundary conditions. We also study the visibility of bounded hyperbolic domains with respect to hyperbolic and quasihyperbolic metric. Further, we explore the relation between Gromov hyperbolicity and visibility. As an application of visibility we prove a sufficient condition for the continuous extension of quasihyperbolic isometries and quasi-isometries. At last we provide some useful examples of visibility domains which are neither John domain nor domains satisfying quasihyperbolic boundary conditions.

This paper considers Bayesian inference for the partially linear model. Our approach exploits a parametrization of the regression function that is tailored toward estimating a low-dimensional parameter of interest. The key property of the parametrization is that it generates a Neyman orthogonal moment condition meaning that the low-dimensional parameter is less sensitive to the estimation of nuisance parameters. Our large sample analysis supports this claim. In particular, we derive sufficient conditions under which the posterior for the low-dimensional parameter contracts around the truth at the parametric rate and is asymptotically normal with a variance that coincides with the semiparametric efficiency bound. These conditions allow for a larger class of nuisance parameters relative to the original parametrization of the regression model. Overall, we conclude that a parametrization that embeds Neyman orthogonality can be a useful device for debiasing posterior distributions in semiparametric models.

The bicategory of parameterized spectra has a remarkably rich structure. In particular, it is possible to take traces in this bicategory, which give classical invariants that count fixed points. We can also take equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these traces in general depends on technical statements about the bicategory that can be difficult to verify directly. In this paper, we demonstrate the effectiveness of two tools -- rigidity and deformable functors -- by using them establish the structure we need to take these equivariant traces and to construct periodic-point invariants in a formal way.

We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver $X_7$. We confirm a conjecture of Geiss-Labardini-Schroer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member $W_0$ of this family is infinite-dimensional, whereas that of another member $W_1$ is finite-dimensional, implying that these potentials are not right equivalent. As a consequence, we draw some conclusions on the associated cluster categories, and in particular obtain a representation theoretic proof that there are no reddening mutation sequences for the quiver $X_7$. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of $W_0$ and $W_1$ are both finite-dimensional. Thus $W_0$ seems to be the first known non-degenerate potential with the property that the finite-dimensionality of its Jacobian algebra depends upon the ground field.

In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of $\mathbb{R}^N$. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide non-negativity of the solutions and uniform control of the total mass. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^s u_i$ where $0<s<1$. Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type $u_i\mapsto -d_i\Delta u_i$. On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case $s=1$.

In this paper, we apply techniques of the equivariant geometry to give a positive answer to the conjecture that a generalized Bour's Theorem holds for surfaces that are invariant under the action of a one-parameter group of isometries of a three-dimensional Riemannian manifold.

We describe certain $C^*$-algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain $D_{2} \subset \mathbb{C}^{2}$. Bounded measurable functions of the form $c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2})$ are called nilpotent symbols. In this work we consider symbols of the form $a(\text{Im}\, \zeta_1) b(\text{Im}\, \zeta_2 -|\zeta_1|^{2})$, where both limits $\lim\limits_{s\rightarrow 0^+} b(s)$ and $\lim\limits_{s\rightarrow +\infty} b(s)$ exist, and $a(s)$ belongs to the set of piece-wise continuous functions on $\overline{\mathbb{R}}=[-\infty,+\infty]$ and having one-side limit values at each point of a finite set $D\subset \mathbb{R}$. We prove that the $C^*$-algebra generated by all Toeplitz operators $T_{ab}$ is isomorphic to $C(\overline{\Pi})$, where $\overline{\Pi}=\overline{\mathbb{R}} \times \overline{\mathbb{R}}_+$ and $\overline{\mathbb{R}}_+=[0,+\infty]$.

We propose an optimal destination scheduling scheme to improve the physical layer security (PLS) of a power-line communication (PLC) based Internet-of-Things system in the presence of an eavesdropper. We consider a pinhole (PH) architecture for a multi-node PLC network to capture the keyhole effect in PLC. The transmitter-to-PH link is shared between the destinations and an eavesdropper which correlates all end-to-end links. The individual channel gains are assumed to follow independent log-normal statistics. Furthermore, the additive impulsive noise at each node is modeled by an independent Bernoulli-Gaussian process. Exact computable expressions for the average secrecy capacity (ASC) and the probability of intercept (POI) performance over many different networks are derived. Approximate closed-form expressions for the asymptotic ASC and POI are also provided. We find that the asymptotic ASC saturates to a constant level as transmit power increases. We observe that the PH has an adverse effect on the ASC. Although the shared link affects the ASC, it has no effect on the POI. We show that by artificially controlling the impulsive to background noise power ratio and its arrival rate at the receivers, the secrecy performance can be improved.

Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vop\v enka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. This incidentally provides a natural solution to some classic philosophical problems such as the composition of a continuum, Zeno's paradoxes and sorites. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vop\v enka's theory reverses the process: he models the finite in the infinite.

We introduce graphs associated to transport problems between discrete marginals, that allow to characterize the set of all optimizers given one primal optimizer. In particular, we establish that connectivity of those graphs is a necessary and sufficient condition for uniqueness of the dual optimizers. Moreover, we provide an algorithm that can efficiently compute the dual optimizer that is the limit, as the regularization parameter goes to zero, of the dual entropic optimizers. Our results find an application in a Stackelberg-Cournot-Nash game, for which we obtain existence and characterization of the equilibria.

Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, are encoded by the space of orbits of a flow on a compact space $(Y, \Phi)$ naturally associated with $G$, uniquely defined up to flow-equivalence, that we call the Deroin space of $G$. We show a realisation result: every expansive flow $(Y, \Phi)$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying multiple new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions.

We construct a sequence $\{\Sigma_\ell\}_{\ell=1}^\infty$ of closed, axially symmetric surfaces $\Sigma_\ell\subset \mathbb{R}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every $\ell$, $$ \int_{\Sigma_{\ell}}H_{\Sigma_\ell}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}<0 $$ where $H_{\Sigma_\ell}$ is the mean curvature of $\Sigma_\ell$. This shows that the Minkowski inequality with optimal constant fails even for perturbations of a round sphere that are small in $W^{2,p}\cap C^1$ unless additional convexity assumptions are imposed.

The purpose of this note is to prove that the strong Christ-Goldberg maximal function is bounded. This is a matrix weighted maximal operator appearing in the theory of matrix weighted norm inequalities. Related to this we record the Rubio de Francia extrapolation theorem with bi-parameter matrix weights.

In general, some of the well known results of measure theory dealing with the convergence of sequences of functions such as the Dominated Convergence Theorem or the Monotone Convergence Theorem are not true when we consider arbitrary nets of functions instead of sequences.In this paper, we study the algebraic genericity of families of nets of functions that do not satisfy important results of measure theory, and we also analyze the particular case of uncountable sequences.

In this paper we study the problem of reconstructing the poset structure of a scheme using coverings. For a finite, dominant and separable morphism $X'\to{X}$ of normal, connected Noetherian schemes, we show that the poset structure of $X'$ can be recovered from the poset structure of $X$ and a set of Galois gluing data over $X$. This data consists of a set of local Galois orbits that are glued using relative Galois groups and transfer maps. To find this gluing data in practice, we introduce symbolic multivariate Newton-Puiseux algorithms that calculate adic approximations of the roots of a univariate polynomial over chains of prime ideals. By matching power series over intersecting chains of prime ideals, we are then able to glue the local Galois orbits of the approximations to obtain the global poset structure of the scheme. We show how to apply these algorithms to two types of coverings: coverings of semistable models of curves and sch\"{o}n coverings of very affine varieties. The first gives an algorithm to calculate the Berkovich skeleton of a curve, and the second an algorithm to calculate boundary complexes.

A common model of robustness of a graph against random failures has all vertices operational, but the edges independently operational with probability $p$. One can ask for the probability that all vertices can communicate ({\em all-terminal reliability}) or that two specific vertices (or {\em terminals}) can communicate with each other ({\em two-terminal reliability}). A relatively new measure is {\em split reliability}, where for two fixed vertices $s$ and $t$, we consider the probability that every vertex communicates with one of $s$ or $t$, but not both. In this paper, we explore the existence for fixed numbers $n \geq 2$ and $m \geq n-1$ of an {\em optimal} connected $(n,m)$-graph $G_{n,m}$ for split reliability, that is, a connected graph with $n$ vertices and $m$ edges for which for any other such graph $H$, the split reliability of $G_{n,m}$ is at least as large as that of $H$, for {\em all} values of $p \in [0,1]$. Unlike the similar problems for all-terminal and two-terminal reliability, where only partial results are known, we completely solve the issue for split reliability, where we show that there is an optimal $(n,m)$-graph for split reliability if and only if $n\leq 3$, $m=n-1$, or $n=m=4$.

This note explores the use of Newton polytopes in the study of Lagrangian fillings of Legendrian submanifolds. In particular, we show that Newton polytopes associated to augmented values of Reeb chords can distinguish infinitely many distinct Lagrangian fillings, both for Legendrian links and higher-dimensional Legendrian spheres. The computations we perform work in finite characteristic, which significantly simplifies arguments and also allows us to show that there exist Legendrian links with infinitely many non-orientable exact Lagrangian fillings.

We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. Such problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the decision problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problem over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d>3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

What is the analogous notion of Gromov-Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the traditional definition. One approach offered by Sormani and Vega \cite{SV} is to define a metric space structure on a spacetime by means of the null distance. Then one can define convergence of spacetimes using the usual definition of Gromov-Hausdorff convergence. In this paper we explore this approach by giving many examples of sequences of warped product spacetimes with the null distance converging in the Gromov-Hausdorff sense. In addition, we give an optimal convergence theorem which shows that under natural geometric hypotheses a sequence of warped product spacetimes converge to a specific limiting warped product spacetime. The examples given further serve to show that the hypotheses of this convergence theorem are optimal.

In this note, we present some results on the classification of quantum cellular automata (QCA) in 1D under strong equivalence rather than stable equivalence. Under strong equivalence, we only allow adding ancillas carrying the original on-site representation of the symmetry, while under stable equivalence, we allow adding ancillas carrying any representation of the symmetry. The former may be more realistic, because in physical systems especially in AMO/quantum computing contexts, we would not expect additional spins carrying arbitrary representations of the symmetry to be present. Ref.~\onlinecite{mpu} proposed two kinds of symmetry-protected indices (SPIs) for QCA with discrete symmetries under strong equivalence. In this note, we show that the more refined of these SPIs still only has a one-to-one correspondence to equivalence classes of $\mathbb{Z}_N$ symmetric QCA when $N$ is prime. We show a counter-example for $N=4$. We show that QCA with $\mathbb{Z}_2$ symmetry under strong equivalence, for a given on-site representation, are classified by $\mathbb{Z}^{pq}$ where $p$ is the number of prime factors of the on-site Hilbert space dimension and $q$ is the number of prime factors of the trace of the nontrivial on-site $\mathbb{Z}_2$ element. Finally, we show that the GNVW index has a formulation in terms of a $\mathbb{Z}_2$ SPI in a doubled system, and we provide a direct connection between the SPI formulation of the GNVW index and a second Renyi version of the mutual information formula for the GNVW index.

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially non-linear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present both theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative based Frank-Wolfe~(FW) algorithm for generic non-linear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the non-linear DRO problem. Finally, for the minimum variance portfolio selection problem we analyze the regularity conditions and compute the FW-oracle in various settings, and validate the theoretical results numerically.

Dynamic feature selection, where we sequentially query features to make accurate predictions with a minimal budget, is a promising paradigm to reduce feature acquisition costs and provide transparency into the prediction process. The problem is challenging, however, as it requires both making predictions with arbitrary feature sets and learning a policy to identify the most valuable selections. Here, we take an information-theoretic perspective and prioritize features based on their mutual information with the response variable. The main challenge is learning this selection policy, and we design a straightforward new modeling approach that estimates the mutual information in a discriminative rather than generative fashion. Building on our learning approach, we introduce several further improvements: allowing variable feature budgets across samples, enabling non-uniform costs between features, incorporating prior information, and exploring modern architectures to handle partial input information. We find that our method provides consistent gains over recent state-of-the-art methods across a variety of datasets.

We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on Stone-Weierstrass theorems on weighted spaces, we can prove a global universal approximation result for generalizations of continuous functions going beyond the usual approximation on compact sets. This then applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks. As a further application of the weighted Stone-Weierstrass theorem we prove a global universal approximation result for linear functions of the signature. We also introduce the viewpoint of Gaussian process regression in this setting and show that the reproducing kernel Hilbert space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes. This paves the way towards uncertainty quantification for signature kernel regression.

There is a one-to-one and onto correspondence between the class of numerical semigroups of depth $n$, where $n$ is an integer, and a certain language over the alphabet $\{1,\ldots,n\}$ which call Kunz language of depth $n$. The Kunz language associated with the numerical semigroups of depth $2$ is the regular language $\{1,2\}^*2\{1,2\}^*$. We prove that Kunz languages associated with numerical semigroups of larger depth are context-sensitive but not context-free.

We investigate the role of projection heads, also known as projectors, within the encoder-projector framework (e.g., SimCLR) used in contrastive learning. We aim to demystify the observed phenomenon where representations learned before projectors outperform those learned after -- measured using the downstream linear classification accuracy, even when the projectors themselves are linear. In this paper, we make two significant contributions towards this aim. Firstly, through empirical and theoretical analysis, we identify two crucial effects -- expansion and shrinkage -- induced by the contrastive loss on the projectors. In essence, contrastive loss either expands or shrinks the signal direction in the representations learned by an encoder, depending on factors such as the augmentation strength, the temperature used in contrastive loss, etc. Secondly, drawing inspiration from the expansion and shrinkage phenomenon, we propose a family of linear transformations to accurately model the projector's behavior. This enables us to precisely characterize the downstream linear classification accuracy in the high-dimensional asymptotic limit. Our findings reveal that linear projectors operating in the shrinkage (or expansion) regime hinder (or improve) the downstream classification accuracy. This provides the first theoretical explanation as to why (linear) projectors impact the downstream performance of learned representations. Our theoretical findings are further corroborated by extensive experiments on both synthetic data and real image data.

In federated learning (FL), clients usually have diverse participation probabilities that are unknown a priori, which can significantly harm the performance of FL if not handled properly. Existing works aiming at addressing this problem are usually based on global variance reduction, which requires a substantial amount of additional memory in a multiplicative factor equal to the total number of clients. An important open problem is to find a lightweight method for FL in the presence of clients with unknown participation rates. In this paper, we address this problem by adapting the aggregation weights in federated averaging (FedAvg) based on the participation history of each client. We first show that, with heterogeneous participation probabilities, FedAvg with non-optimal aggregation weights can diverge from the optimal solution of the original FL objective, indicating the need of finding optimal aggregation weights. However, it is difficult to compute the optimal weights when the participation probabilities are unknown. To address this problem, we present a new algorithm called FedAU, which improves FedAvg by adaptively weighting the client updates based on online estimates of the optimal weights without knowing the probabilities of client participation. We provide a theoretical convergence analysis of FedAU using a novel methodology to connect the estimation error and convergence. Our theoretical results reveal important and interesting insights, while showing that FedAU converges to an optimal solution of the original objective and has desirable properties such as linear speedup. Our experimental results also verify the advantage of FedAU over baseline methods.

Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive comparison and description of neural network architectures in terms of ge-ometry and topology. We focus on the internal representation of neural networks and on the dynamics of changes in the topology and geometry of a data manifold on different layers. In this paper, we use the concepts of topological data analysis (TDA) and persistent homological fractal dimension. We present a wide range of experiments with various datasets and configurations of convolutional neural network (CNNs) architectures and Transformers in CV and NLP tasks. Our work is a contribution to the development of the important field of explainable and interpretable AI within the framework of geometrical deep learning.

We present the explicit expressions for the matrix product operator (MPO) representation for the local conserved quantities of the Heisenberg chain. The bond dimension of the MPO grows linearly with the locality of the charges. The MPO has more simple form than the local charges themselves, and their Catalan tree patterns naturally emerge from the matrix products. The MPO representation of local conserved quantities is generalized to the integrable $\mathrm{SU}(N)$ invariant spin chain.

This paper develops and investigates a dual unscented Kalman filter (DUKF) for the joint nonlinear state and parameter identification of commercial adaptive cruise control (ACC) systems. Although the core functionality of stock ACC systems, including their proprietary control logic and parameters, is not publicly available, this work considers a car-following scenario with a human-driven vehicle (leader) and an ACC engaged ego vehicle (follower) that employs a constant time-headway policy (CTHP). The objective of the DUKF is to determine the CTHP parameters of the ACC by using real-time observations of space-gap and relative velocity from the vehicle's onboard sensors. Real-time parameter identification of stock ACC systems is essential for assessing their string stability, large-scale deployment on motorways, and impact on traffic flow and throughput. In this regard, $L_2$ and $L_\infty$ string stability conditions are considered. The observability rank condition for nonlinear systems is adopted to evaluate the ability of the proposed estimation scheme to estimate stock ACC system parameters using empirical data. The proposed filter is evaluated using empirical data collected from the onboard sensors of two 2019 SUV vehicles, namely Hyundai Nexo and SsangYong Rexton, equipped with stock ACC systems; and is compared with batch and recursive least-squares optimization. The set of ACC model parameters obtained from the proposed filter revealed that the commercially implemented ACC system of the considered vehicle (Hyundai Nexo) is neither $L_2$ nor $L_\infty$ string stable.

Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using theBregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance.

Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources.

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.

We analyze continual learning on a sequence of separable linear classification tasks with binary labels. We show theoretically that learning with weak regularization reduces to solving a sequential max-margin problem, corresponding to a special case of the Projection Onto Convex Sets (POCS) framework. We then develop upper bounds on the forgetting and other quantities of interest under various settings with recurring tasks, including cyclic and random orderings of tasks. We discuss several practical implications to popular training practices like regularization scheduling and weighting. We point out several theoretical differences between our continual classification setting and a recently studied continual regression setting.

We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge-Kutta methods. We show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained when inserting the training data in the MIRK formulae, unlocking symmetric and high-order integrators that would otherwise be implicit for initial value problems. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with experiments using data from several (chaotic) Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, supporting the favorable performance of MII.

Anyonic system not only has potential applications in the construction of topological quantum computer, but also presents a unique property known as topological entanglement entropy in quantum many-body systems. How to understand topological entanglement entropy is one of the most concerned problems for physicists. For an anyonic bipartite system, we define an operational measure of topological correlation based on the principle of maximal entropy, where the topological correlation is the information that cannot be accessed by local operations constrained by anyonic superselection rules and classical communication. This measure can be extended to measure non-local resources of other compound quantum systems in the presence of superselection rules. For a given anyonic bipartite state with maximal rank, we prove that its topological correlation is equal to its entropy of anyonic charge entanglement that has been shown in the literature to be able to derive topological entanglement entropy. This measure provides a more refined classification of correlations in a multipartite system with superselection rules and an illuminating approach to topological phase classification.

We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T \subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$, such that for every subset $S \subseteq T$ there is a packing of $\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest.

We propose a Randomized Progressive Training algorithm (RPT) -- a stochastic proxy for the well-known Progressive Training method (PT) (Karras et al., 2017). Originally designed to train GANs (Goodfellow et al., 2014), PT was proposed as a heuristic, with no convergence analysis even for the simplest objective functions. On the contrary, to the best of our knowledge, RPT is the first PT-type algorithm with rigorous and sound theoretical guarantees for general smooth objective functions. We cast our method into the established framework of Randomized Coordinate Descent (RCD) (Nesterov, 2012; Richt\'arik & Tak\'a\v{c}, 2014), for which (as a by-product of our investigations) we also propose a novel, simple and general convergence analysis encapsulating strongly-convex, convex and nonconvex objectives. We then use this framework to establish a convergence theory for RPT. Finally, we validate the effectiveness of our method through extensive computational experiments.

While black-box variational inference is widely used, there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs-namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides the first rigorous guarantee that black-box variational inference converges for realistic inference problems.

In many applications, learning systems are required to process continuous non-stationary data streams. We study this problem in an online learning framework and propose an algorithm that can deal with adversarial time-varying and nonlinear constraints. As we show in our work, the algorithm called Constraint Violation Velocity Projection (CVV-Pro) achieves $\sqrt{T}$ regret and converges to the feasible set at a rate of $1/\sqrt{T}$, despite the fact that the feasible set is slowly time-varying and a priori unknown to the learner. CVV-Pro only relies on local sparse linear approximations of the feasible set and therefore avoids optimizing over the entire set at each iteration, which is in sharp contrast to projected gradients or Frank-Wolfe methods. We also empirically evaluate our algorithm on two-player games, where the players are subjected to a shared constraint.

We generalize the recent result that all analytic quantum dynamics can be represented exactly as the reduction of unitary dynamics generated by a time-dependent Hamiltonian. More precisely, we prove that the partial trace over analytic paths of unitaries can approximate any Lipschitz-continuous quantum dynamics arbitrarily well. We conclude by discussing potential improvements and generalizations of these results, their limitations, and the general challenges one has to overcome when trying to relate dynamics to quantities on the system-environment level.

Over a couple of decades, space junk has increased rapidly, which has caused significant threats to the LEO operation satellites. An Active Debris Removal $(ADR)$ concept continuously evolves for space junk removal. One of the ADR methods is Space Robotics, whose function is to chase, capture and de-orbit the space junk. This paper presents the development of an on-ground space robotics facility in the TCS Research for on-orbit servicing $(OOS)$ like refueling and debris capture experiments. A Hardware in Loop Simulation (HILS) system will be used for integrated system development, testing, and demonstration of on-orbit docking mechanisms. The HiLS test facility of TCS Research Lab will use two URs in which one UR is attached to the RG2 gripper, and the other is attached to a force-torque sensor and with a scaled mock-up model. The first UR5 will be mounted on a 7-axis linear rail and contain the docking probe. First, UR5 with a suitable gripper has to interface its control boxes. The grasping algorithm was run through the ROS interface line to demonstrate and validate the on-orbit operations. The manipulator will be mounted with LIDAR and a camera to visualize the mock-up model, find the target model's pose and rotational velocity estimation, and a gripper that will move relative to the target model. The other manipulator has the UR10 control, providing rotational and random motion to the mockup, enabling a dynamic simulator fed by force-torque data. The dynamic simulator is fed up with the orbit propagator, which will provide the orbiting environment to the target model. For the simulation of the docking and grasping of the target model, a linear rail of a 6m setup is still in the procurement process. Once reaching proximity, the grasping algorithm will be launched to capture the target model after reading the random motion of the mock-up model.

Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously evaluate the error bound of Physics-Informed Neural Networks (PINNs) on most linear ordinary differential equations (ODEs), certain nonlinear ODEs, and first-order linear partial differential equations (PDEs). The error bound is based purely on equation structure and residual information and does not depend on assumptions of how well the networks are trained. We propose algorithms that bound the error efficiently. Some proposed algorithms provide tighter bounds than others at the cost of longer run time.

U-statistics play central roles in many statistical learning tools but face the haunting issue of scalability. Significant efforts have been devoted into accelerating computation by U-statistic reduction. However, existing results almost exclusively focus on power analysis, while little work addresses risk control accuracy -- comparatively, the latter requires distinct and much more challenging techniques. In this paper, we establish the first statistical inference procedure with provably higher-order accurate risk control for incomplete U-statistics. The sharpness of our new result enables us to reveal how risk control accuracy also trades off with speed for the first time in literature, which complements the well-known variance-speed trade-off. Our proposed general framework converts the long-standing challenge of formulating accurate statistical inference procedures for many different designs into a surprisingly routine task. This paper covers non-degenerate and degenerate U-statistics, and network moments. We conducted comprehensive numerical studies and observed results that validate our theory's sharpness. Our method also demonstrates effectiveness on real-world data applications.

We review the state of the field of gravitational wave astrophysics, framing the challenges, current observations, and future prospects within the context of the predictions of Einstein's theory of general relativity.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Fourier Neural Operators (FNOs) have proven to be an efficient and effective method for resolution-independent operator learning in a broad variety of application areas across scientific machine learning. A key reason for their success is their ability to accurately model long-range dependencies in spatio-temporal data by learning global convolutions in a computationally efficient manner. To this end, FNOs rely on the discrete Fourier transform (DFT), however, DFTs cause visual and spectral artifacts as well as pronounced dissipation when learning operators in spherical coordinates since they incorrectly assume a flat geometry. To overcome this limitation, we generalize FNOs on the sphere, introducing Spherical FNOs (SFNOs) for learning operators on spherical geometries. We apply SFNOs to forecasting atmospheric dynamics, and demonstrate stable auto\-regressive rollouts for a year of simulated time (1,460 steps), while retaining physically plausible dynamics. The SFNO has important implications for machine learning-based simulation of climate dynamics that could eventually help accelerate our response to climate change.

The use of von Neumann -- Morgenstern utility is examined in the context of multiple choices between lotteries. Different conclusions are reached if the choices are simultaneous or sequential. It is demonstrated that utility cannot be additive.

Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. However, both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems. Developed in classical probability theory, they extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences requires a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices).

We present a detailed analysis of the phase-space for the field equations in scalar field cosmology with a chameleon cosmology in a spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker spacetime. For the matter source we assume that it is an ideal gas with a constant equation of state parameter, while for the scalar field potential and the coupling function of the chameleon mechanism we consider four different sets which provide four different models. We consider the $H$-normalization approach and we write the field equations with the help of dimensionless variables. The asymptotic solutions are determined from where we find that the theory can describe the main eras of cosmological history and evolution. Future attractors which describe acceleration exist, however we found past acceleration solutions related to the inflationary era, as also the radiation epoch and the matter dominated eras are provided by the dynamics. We conclude that the Chameleon dark energy model can be used as a unified model for the elements which contribute to the dark sector of the universe.

Three-way decision (3WD) is a powerful tool for granular computing to deal with uncertain data, commonly used in information systems, decision-making, and medical care. Three-way decision gets much research in traditional rough set models. However, three-way decision is rarely combined with the currently popular field of machine learning to expand its research. In this paper, three-way decision is connected with SVM, a standard binary classification model in machine learning, for solving imbalanced classification problems that SVM needs to improve. A new three-way fuzzy membership function and a new fuzzy twin support vector machine with three-way membership (TWFTSVM) are proposed. The new three-way fuzzy membership function is defined to increase the certainty of uncertain data in both input space and feature space, which assigns higher fuzzy membership to minority samples compared with majority samples. To evaluate the effectiveness of the proposed model, comparative experiments are designed for forty-seven different datasets with varying imbalance ratios. In addition, datasets with different imbalance ratios are derived from the same dataset to further assess the proposed model's performance. The results show that the proposed model significantly outperforms other traditional SVM-based methods.

We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi-Yau singularities. A key object in our constructions is the equivariant volume, for which we describe two methods of evaluation: the Berline-Vergne fixed-point formula and the Molien- Weyl formula, supplemented by the Jeffrey-Kirwan prescription. We present two applications in supersymmetric field theories. Firstly, we describe a method for integrating the anomaly polynomial of SCFTs on compact toric orbifolds. Secondly, we discuss equivariant orbifold indices that are expected to play a key role in the computation of supersymmetric partition functions. In the context of supergravity, we propose that the equivariant volume can be used to characterise universally the geometry of a large class of supersymmetric solutions. As an illustration, we employ equivariant localization to prove various gravitational block formulas, recovering previous results as well as obtaining generalizations.