arXiv Math @arxiv_math@qoto.org

We consider the relay version of the generalized Hutchinson's equation as a phenomenological model of an isolated neuron. After an exponential substitution, the equation takes the form of a differential-difference equation with a piecewise-constant right-hand side. In the work of A. Yu. Kolesov and others, this equation was studied with a negative continuous initial function, and the existence and orbital stability of a periodic solution with a period longer than the delay were proven. In the present work, all possible solutions with continuous on the interval of the delay length initial functions, containing no more than two zeros, are constructed. It is proven that the equation has a periodically unstable solution, the period of which is shorter than the delay.

We study the exponential stability for the mean field systems in its synchronized state. More precisely, we study the linearized system and we obtain a exponential stable invariant manifold.

We study the dynamics of a piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in $\mathbb{R}^2$. The geometric map of study may be analogized to the billiard map with a different reflection rule so that each iteration is a contraction over the space, thereby providing asymptotic behavior of interest. Our study emphasizes the behavior of periodic orbits generated by the map, with description of their geometry and bifurcation behavior. We establish that for any initial point in the space, the orbit will converge to a fixed point or periodic orbit, and we demonstrate that there exists an infinite variety of periodic orbits the orbits may converge to, dependent on the parameters of the underlying space.

Let $G$ be a nontrivial finite permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and this can happen even if $G$ has only two orbits, both of which have size $(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture when $G$ acts primitively on at least one of the orbits. Equivalently, we conjecture that a finite group is never the union of conjugates of two proper subgroups of the same order, and we prove this conjecture when at least one of the subgroups is maximal. We prove other cases of the conjecture, and we highlight connections our results have with intersecting families of permutations and roots of polynomials modulo primes. Along the way, we also prove a linear variant on Isbell's conjecture regarding derangements of prime-power order.

Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + μB) = w_k(A) + w_k(B)$ for some scalar $μ$ with $|μ| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $μ= 1$. We classify linear bijections on $\mathbb M_n$ and on $\mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $\mathbb H_n$ when $n=2k$.

We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\ge 5$). For all three settings and all $k\in\{1,2,3\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. For general $k \geq 4$ our bounds are of the form $c\sqrt{k}n$, for some explicit constant $c$. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of $2$-- and $3$-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

The time-dependent Ginzburg-Landau (TDGL) model requires the choice of a gauge for the problem to be mathematically well-posed. In the literature, three gauges are commonly used: the Coulomb gauge, the Lorenz gauge and the temporal gauge. It has been noticed [J. Fleckinger-Pellé et al., Technical report, Argonne National Lab. (1997)] that these gauges can be continuously related by a single parameter considering the more general $ω$-gauge, where $ω$ is a non-negative real parameter. In this article, we study the influence of the gauge parameter $ω$ on the convergence of numerical simulations of the TDGL model using finite element schemes. A classical benchmark is first analysed for different values of $ω$ and artefacts are observed for lower values of $ω$. Then, we relate these observations with a systematic study of convergence orders in the unified $ω$-gauge framework. In particular, we show the existence of a tipping point value for $ω$, separating optimal convergence behaviour and a degenerate one. We find that numerical artefacts are correlated to the degeneracy of the convergence order of the method and we suggest strategies to avoid such undesirable effects. New 3D configurations are also investigated (the sphere with or without geometrical defect).

Bilevel optimization has witnessed a resurgence of interest, driven by its critical role in trustworthy and efficient machine learning applications. Recent research has focused on proposing efficient methods with provable convergence guarantees. However, while many prior works have established convergence to stationary points or local minima, obtaining the global optimum of bilevel optimization remains an important yet open problem. The difficulty lies in the fact that unlike many prior non-convex single-level problems, this bilevel problem does not admit a ``benign" landscape, and may indeed have multiple spurious local solutions. Nevertheless, attaining the global optimality is indispensable for ensuring reliability, safety, and cost-effectiveness, particularly in high-stakes engineering applications that rely on bilevel optimization. In this paper, we first explore the challenges of establishing a global convergence theory for bilevel optimization, and present two sufficient conditions for global convergence. We provide algorithm-specific proofs to rigorously substantiate these sufficient conditions along the optimization trajectory, focusing on two specific bilevel learning scenarios: representation learning and data hypercleaning (a.k.a. reweighting). Experiments corroborate the theoretical findings, demonstrating convergence to global minimum in both cases.

Domineering is a partizan game where two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally; the first player who cannot move loses. It has been conjectured that the highest temperature possible in Domineering is 2. We have developed a program that enables a parallel exhaustive search of Domineering positions with temperatures close to or equal to 2 to allow for analysis of such positions

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples.

We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth compactly supported test functions. As an application we show that almost all holomorphic Hecke cusp forms, whose weights are in a short interval, satisfy QUE conjecture on the vertical geodesic.

We study simple Osserman limit linear series (that is, Osserman limit linear series having a simple basis) on curves of compact type with three irreducible components. For compact type curves with two components, every exact limit linear series is simple. But, for the case of three components, this property is no longer true. We study a certain distributivity property related with the existence of a simple basis and we find characterizations of that property at any fixed multidegree. Also, we find a characterization of simple limit linear series among the exact limit linear series. In our approach, we also study other structure similar to a simple basis and we get a certain inequality for any exact limit linear series.

In this paper we prove that there exists a compact perturbation of the Ricci flat Taub-Bolt metric that evolves under the Ricci flow into a finite time singularity modelled on the shrinking solition FIK [5]. Moreover, this perturbation can be made arbitrarily L^2-small with respect to the Taub-Bolt metric. The method of proof closely follows the strategy adopted in [14], where the author constructs, via a Ważewski box argument, Ricci flows on compact manifolds which encounter finite singularities modelled on a given asymptotically conical shrinking soliton.

Let $X$ be a Riemann surface. Using the canonical line bundle $K$ and some holomorphic differentials $\boldsymbol{q}$, Hitchin constructed the $G$-Higgs bundles in the Hitchin section for a split real form $G$ of a complex simple Lie group. We study the ${\mathrm{SO}_0(n,n)}$ case. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the ${\mathrm{SO}_0(n,n)}$-structure for any non-compact hyperbolic Riemann surface. Moreover, these harmonic metrics also weakly dominate $h_X$ which is the natural diagonal harmonic metric induced by the unique complete Kähler hyperbolic metric $g_X$ on $X$. Assuming that these holomorphic differentials are all bounded with respect to the metric $g_X$, we are able to prove the uniqueness of such a harmonic metric.

This paper aims to establish the global well-posedness of the free boundary problem for the incompressible viscous resistive magnetohydrodynamic (MHD) equations. Under the framework of Lagrangian coordinates, a unique global solution exists in the half-space provided that the norm of the initial data in the critical homogeneous Besov space $\dot{B}_{p, 1}^{-1+N/p}(\mathbb{R}_{+}^N)$ is sufficiently small, where $p \in [N, 2N-1)$. Building upon prior work such as (Danchin and Mucha, J. Funct. Anal. 256 (2009) 881--927) and (Ogawa and Shimizu, J. Differ. Equations 274 (2021) 613--651) in the half-space setting, we establish maximal $L^{1}$-regularity for both the Stokes equations without surface stress and the linearized equations of the magnetic field with zero boundary condition. The existence and uniqueness of solutions to the nonlinear problems are proven using the Banach contraction mapping principle.

Let $n\geq 1,0<ρ<1, \max\{ρ,1-ρ\}\leq δ\leq 1$ and $$m_1=ρ-n+(n-1)\min\{\frac 12,ρ\}+\frac {1-δ}{2}.$$ If the amplitude $a$ belongs to the Hörmander class $S^{m_1}_{ρ,δ}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{ϕ,a}$ defined by \begin{align*} T_{ϕ,a}f(x)=\int_{\mathbb{R}^{n}}e^{iϕ(x,ξ)}a(x,ξ)\widehat{f}(ξ)dξ, \end{align*} is bounded from the local Hardy space $h^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$. As a corollary, we can also obtain the corresponding $L^p(\mathbb{R}^n)$-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When $0\leq ρ\leq 1,δ\leq \max\{ρ,1-ρ\}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.

A method for successive synthesis of the Weyl matrix on the square lattice is proposed. It allows one to compute the Weyl matrix of a large graph by adding new edges and solving elementary systems of linear algebraic equations at each step. Synthesis of the Weyl matrix is useful to further study the inverse problems of the square lattice. Moreover, our approach can be extended to other types of periodic lattices.

This work develops an efficient real-time inverse formulation for inferring the aerodynamic surface pressures on a hypersonic vehicle from sparse measurements of the structural strain. The approach aims to provide real-time estimates of the aerodynamic loads acting on the vehicle for ground and flight testing, as well as guidance, navigation, and control applications. Specifically, the approach targets hypersonic flight conditions where direct measurement of the surface pressures is challenging due to the harsh aerothermal environment. For problems employing a linear elastic structural model, we show that the inference problem can be posed as a least-squares problem with a linear constraint arising from a finite element discretization of the governing elasticity partial differential equation. Due to the linearity of the problem, an explicit solution is given by the normal equations. Pre-computation of the resulting inverse map enables rapid evaluation of the surface pressure and corresponding integrated quantities, such as the force and moment coefficients. The inverse approach additionally allows for uncertainty quantification, providing insights for theoretical recoverability and robustness to sensor noise. Numerical studies demonstrate the estimator performance for reconstructing the surface pressure field, as well as the force and moment coefficients, for the Initial Concept 3.X (IC3X) conceptual hypersonic vehicle.

A numerical scheme that uses multi-frequency Newton iterations to reconstruct a rough surface profile between two dielectric media is proposed. At each frequency sample, the scheme employs Newton iterations to solve the nonlinear inverse scattering problem. At every iteration, the Newton step is computed by solving a linear system that involves the Frechet derivative of the integral operator, which represents the scattered fields, and the difference between these fields and the measurements. This linear system is regularized using the Tikhonov method. The multi-frequency data is accounted for in a recursive manner. More specifically, the profile reconstructed at a given frequency is used as an initial guess for the iterations at the next frequency. The effectiveness of the proposed method is validated through numerical examples, which demonstrate its ability to accurately reconstruct surface profiles even in the presence of measurement noise. The results also show the superiority of the multi-frequency approach over single-frequency reconstructions, particularly in terms of handling surfaces with sharp variations.

For all $n \geq 1$, there is a notion of $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of $n$-truncated Barsotti-Tate groups over $R$. We show that the category of $n$-smooth commutative group schemes over $R$ is equivalent to a certain full subcategory of Dieudonné modules over $R$. As a consequence, we show that the moduli stack $\mathrm{Sm}_n$ of $n$-smooth commutative group schemes is smooth over $\mathbb{F}_p$ and that the natural truncation morphism $\mathrm{Sm}_{n+1} \to \mathrm{Sm}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.