arXiv Math @arxiv_math@qoto.org

We consider graphs derived from aperiodically ordered tilings of the plane, by treating each corner of each tile as a vertex and each side of each tile as an edge. We calculate the average degree of these graphs. For the Ammann A2 tiling, we present a closed-form formula for the average degree. For the Kite and Dart Penrose tiling, the Rhomb Penrose Tiling, and the Ammann-Beenker tiling we present numerical calculations for the average degree.

Using properties of harmonic functions in multidimensional space, we transform the Hartree-Fock eigenvalue problem into a more tractable eigenvalue problem in which the Laplacian is eliminated. This new formulation may facilitate the development of novel ways of finding approximate solutions of the electronic problem.

Strengthening of two Clarkson-McCarthy inequalities with several operators is established. These not only confirm a conjecture of the author in [Israel J. Math. 2024], but also improve results of Hirazallah-Kittaneh in [Integral Equations Operator Theory 60 (2008)] and Bhatia-Kittaneh in [Bull. London Math. Soc. 36 (2004)]. We also give a generalization of a result for pairs $(A_i,B_i), 1\le i\le n$ and obtain another form of another inequality. The infinite cases of these inequalities are also discussed. The method is an extension of the result obtained by Bourin and Lee in [Linear Algebra Appl. 601 (2020)]. Some related eigenvalue inequalities are also obtained.

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the solvability of the other group. In 2024 Pawel Piwek gave a negative example. He constructed two groups of order $2^{365}\cdot3^{105}\cdot7^{104}\approx7.3\cdot10^{247}$ of the same order type, where only one is solvable. In this note we produce a much smaller example of order $2^{13}\cdot3^4\cdot7^3=227598336$.

In this paper, we construct examples of exact Lagrangians (of "locally conformally symplectic" type) in cotangent bundles of closed manifolds with locally conformally symplectic structures and give conditions under which the projection induces a simple homotopy equivalence between an exact Lagrangian and the $0$-section of the cotangent bundle. This line of questioning follows in the footsteps of Abouzaid and Kragh, and more generally of the Arnol'd conjecture. Notably, we will see that while exact Lagrangians cannot be spheres in this setting, a naive adaptation of the Abouzaid-Kragh theorem does not hold in this generalization.

In this article, we introduce a novel recursive modification to the classical Goemans-Williamson MaxCut algorithm, offering improved performance in vectorized data clustering tasks. Focusing on the clustering of medical publications, we employ recursive iterations in conjunction with a dimension relaxation method to significantly enhance density of clustering results. Furthermore, we propose a unique vectorization technique for articles, leveraging conditional probabilities for more effective clustering. Our methods provide advantages in both computational efficiency and clustering accuracy, substantiated through comprehensive experiments.

Consider two balls with radius $r>0$ whose centers are at a distance $2$, positioned symmetrically with respect to the origin in ${\mathbb{R}}^d$. Suppose that the initial velocities are independent standard normal vectors. When $r\to0$, the collision probability goes to 0 as $r^{d-1}$, and the asymptotic collision location distribution is a (defective) $t$-distribution. This distribution is rotationally symmetric about the origin for no apparent reason.

In this article, we continue our analysis for a novel recursive modification to the Max $k$-Cut algorithm using semidefinite programming as its basis, offering an improved performance in vectorized data clustering tasks. Using a dimension relaxation method, we use a recursion method to enhance density of clustering results. Our methods provide advantages in both computational efficiency and clustering accuracy for grouping datasets into three clusters, substantiated through comprehensive experiments.

By developing a unified approach based on integral representations, we establish sharp quantitative stability estimates for critical points of the fractional Sobolev inequalities induced by the embedding $\dot{H}^s({\mathbb R}^n) \hookrightarrow L^{2n \over n-2s}({\mathbb R}^n)$ in the whole range of $s \in (0,\frac{n}{2})$.

In the article, two implementations of the representation of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $Λ_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of $Λ_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between $Λ_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with no more than $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n$.

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.

In this paper, we study the uniform and couniform dimensions of inverse polynomial modules over skew Ore polynomials.

We present a comprehensive study of algebras satisfying the identity $(xy)z=y(zx),$ named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type $σ$. We study algebras of associative type $σ$ to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set $X$ was constructed. An analog of Wedderburn-Artin's theorem was established. The algebraic and geometric classifications of complex $4$-dimensional shift associative algebras are given. In particular, we proved that the first non-associative shift associative algebra appears only in dimension $5$.

Hexastix is an arrangement of non-overlapping infinite hexagonal prisms in four different directions that cover $\frac{3}{4}$ of space. We consider a possible generalization to $n$ dimensions, based on the permutohedral lattice $A^*_n$. The central lines of the generalized prisms are going to be oriented in $n+1$ different directions (parallel to the shortest non-zero vectors of $A^*_n$). The projection of the lines oriented in any direction along that direction to a hyperplane perpendicular to it is required to be a translation of the corresponding projection of $A^*_n$, and the minimal distance between lines oriented in any two given directions should be maximal. It is shown that this is possible if $n$ is a prime power. Also, the proportion of $n$-space that is covered is calculated for $n \in \{4, 5\}$, and an alternative generalization is briefly considered.

We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\leq S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.

In this paper, we study the spectral gap and mixing time of the simple exclusion process with $k$ particles in the line segment $[1, N]$ with conductances $c(x, x+1)_{1\le x<N}$ where $c(x, x+1)>0$ is the rate of swapping the contents of the two sites $x$ and $ x+1$. Under the IID assumption on $(c(x, x+1))_{x \in \mathbb{N}}$ with unit inverse expectation $\mathbb{E}[1/c(x, x+1)]=1$, we prove that the spectral gap, denoted by $\mathrm{gap}_{N,k}$, of the process satisfies $\mathrm{gap}_{N, k}=(1+o(1))π^2/N^2$ and the eigenfunction $g_N$ with $g_N(1)=1$ corresponding to the spectral gap of one particle case is well approximated by $h_N(x) := \cos\left( (x-1/2)π/N \right)$. Under some further assumptions on $c(x, x+1)_{x \in \mathbb{N} }$ and $k$, we prove that around time $(1+o(1)) (2 π^2)^{-1} N^2 \log k$, the total variation distance to equilibrium drops abruptly from $1$ to $0$.

We study the motion of a rigid body within a compressible, isentropic, and viscous fluid contained in a fixed bounded domain $Ω\subset \mathbb{R}^3$. The fluid's behavior is described by the Navier-Stokes equations, while the motion of the rigid body is governed by ordinary differential equations representing the conservation of linear and angular momentum. We prescribe a time-independent fluid velocity along the boundary of $Ω$ and a time-independent fluid density at the inflow boundary of $Ω$. Additionally, we assume a no-slip boundary condition at the interface between the fluid and the rigid body. We prove existence of a weak solution to the given problem within a time interval where the rigid body does not touch the boundary $\partialΩ$.

Complex dynamic systems are typically either modeled using expert knowledge in the form of differential equations, or via data-driven universal approximation models such as artificial neural networks (ANN). While the first approach has advantages with respect to interpretability, transparency, data-efficiency, and extrapolation, the second approach is able to learn completely unknown functional relations from data and may result in models that can be evaluated more efficiently. To combine the complementary advantages, universal differential equations (UDE) have been suggested, which replace unknown terms in the differential equations with ANN. These hybrid models allow to both encode prior domain knowledge such as first principles and to learn unknown mechanisms from data. Often, data for the training of UDE can only be obtained via costly experiments. We consider optimal experimental design (OED) for the planning of experiments and generation of data needed to train UDE. The number of weights in the embedded ANN usually leads to an overfitting of the regression problem. To make the OED problem tractable for optimization, we propose and compare dimension reduction methods that are based on lumping of weights and singular value decomposition of the Fisher information matrix (FIM), respectively. They result in lower-dimensional variational differential equations which are easier to solve and which yield regular FIM. Our numerical results showcase the advantages of OED for UDE, such as an increased data-efficiency and better extrapolation properties.

It is well-known that the $L^2$ metric on the moduli space of hyperbolic monopoles, defined using the Coulomb gauge-fixing condition, diverges. This article shows that an alternative gauge-fixing condition inspired by supersymmetry cures this divergence. The resulting geometry is a hyperbolic analogue of the hyperkähler geometry of Euclidean monopole moduli spaces.

This paper aims to provide an explicit computation of the spectral torsion associated with the Connes type operator on even dimension compact manifolds.And we also extend the spectral torsion for the Connes type operator to compact manifolds with boundary.