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arXiv:2606.01395v1 Announce Type: new Abstract: We give a straightforward, self-contained, and natural construction of the Big Witt ring using the e\~ne product that is defined through the action on zeros of polynomials. This is in contrast with classical constructions of the Big Witt ring that use formulas out of nowhere.
arXiv:2606.01415v1 Announce Type: new Abstract: We study rational hypersurfaces $\mathscr{S}$ defined as the closure of the image of a generically finite rational map $\phi:\mathscr{X}\rightarrow \mathbb{P}^{n+1}$, where $\mathscr{X}$ is an $n$-dimensional toric variety. We provide matrix representations for the implicitization of $\mathscr{S}$ that are constructed from the coefficients of linear syzygies and quadratic syzygies of the para..
arXiv:2606.01420v1 Announce Type: new Abstract: Riemann integration remains a well-known part of mathematics for both historical and conceptual reasons. We study basic properties like boundedness of Riemann integrable functions and related classes in mathematical logic. On one hand, weak logical systems already establish that a Riemann integrable function on the unit interval is bounded or dominated by a continuous function. On the other h....
arXiv:2606.01422v1 Announce Type: new Abstract: We give a formula for $X$-products of attractors using $\mathbb{F}_1$-geometry.
arXiv:2606.01430v1 Announce Type: new Abstract: We present a variational approach to ferronematics in a three dimensional setting. The ferronematic energy functional is described by two established theories: the Landau-de Gennes energy to explain the nematic part, the micromagnetic energy to explain the magnetic part, and coupling energies between them. We explicitly include the nonlocal stray field energy in a bulk setting and the couplin..
arXiv:2606.01431v1 Announce Type: new Abstract: If $f$ is a continuous selection for the Vietoris hyperspace $\mathcal{F}(X)$ of the nonempty closed subsets of a space $X$, then the point $p=f(X)\in X$ is not as arbitrary as it might seem at first glance. In fact, the set $\mathcal{O}_{cs}(X)$ of all these points reveals certain information about the variety of Vietoris continuous selections for $\mathcal{F}(X)$. Another result of this pap..
arXiv:2606.01433v1 Announce Type: new Abstract: For complex geometries, the coarse problem of geometric multigrid can be too large to be solved by a direct solver. Here, we report on the use of domain decomposition applied to the multigrid coarse problem. Additive overlapping Schwarz methods are domain decomposition methods for the iterative solution of partial differential equations whose numerical and parallel scalability can be improved....
arXiv:2606.01447v1 Announce Type: new Abstract: For real multivariate polynomials $P$ and $Q$ both vanishing at a point, if the zero set of $Q$ is contained in the zero set of $P$, then there exists a rational function of the form $P^{p}/Q^{q}$ which is locally bounded and such that its extension that vanishes on the zero set of $Q$ is discontinuous. The proof uses inequalities of Lojasiewicz.
arXiv:2606.01448v1 Announce Type: new Abstract: The Boltzmann equation describing the transport of electrons in semiconductor devices with an external electrostatic potential is considered when the spatial variable is in a torus and the wave vector is in the Brillouin zone. We prove the exponential time decay of solutions towards the global equilibrium in a weighted $L^2$ space. Our result holds for wide classes of energy functions of elec..
arXiv:2606.01449v1 Announce Type: new Abstract: The main aim of this paper is to establish the H\"older continuity and the Harnack inequality for weak solutions to Dirichlet problems associated with superposition operators of mixed fractional order, thereby complementing our previous work \cite{BGKL2026}. To achieve this, we extend the De Giorgi--Nash--Moser theory to the framework of superposition operators by introducing a novel {\it no....
arXiv:2606.01453v1 Announce Type: new Abstract: We give an algorithm for choosing a distinguished defining polynomial for a p-adic field extension. This algorithm formed an important ingredient in the recent expansion of the database of p-adic fields within the L-functions and modular forms database.
arXiv:2606.01459v1 Announce Type: new Abstract: Roblin's theorem asserts that, in rank one, coamenable normal subgroups have the same critical exponent as the ambient group. We investigate the higher-rank analogue of this rigidity phenomenon. In higher rank, growth is directional, and there is no single analogue of Roblin's theorem. Instead, the answer splits into three complementary phenomena. First, the full directional invariants are n....
arXiv:2606.01466v1 Announce Type: new Abstract: We construct an operadic model for the higher-genus Teichm\"uller tower. More precisely, we define a modular operad $\mathbf{S}$ in groupoids built from mapping class groups, with compositions and contractions encoding gluing operations on surfaces. We prove a presentation theorem for maps out of $\mathbf{S}$, showing that they are determined by a small number of genus-zero and genus-one gene....
arXiv:2606.01467v1 Announce Type: new Abstract: We investigate the topology of connectedness loci, denoted as $M_n$, for a one-parameter family of collinear affine iterated function systems featuring equally spaced translations. These loci are arithmetically equivalent to the closures of roots of monic polynomials whose non-leading coefficients fall within a prescribed finite interval of integers. Our main theorem proves that for every int....
arXiv:2606.01484v1 Announce Type: new Abstract: We study the curve diffusion flow for open planar curves whose endpoints are constrained to lie on two fixed straight lines that intersect at an angle $\theta (\in(0,\pi)) $. For every such angle, we prove that under suitable initial conditions the flow exists globally in time. Moreover, we show that the evolving curve converges - exponentially and in the smooth topology - to the circular arc..
arXiv:2606.01487v1 Announce Type: new Abstract: In this article, we establish the global convergence properties of the FilterDDP algorithm, which extends the discrete-time differential dynamic programming (DDP) algorithm of Mayne and Jacobson [\emph{International Journal of Control}, 3, (1966), pp. 85-95] to handle nonlinear constraints over states and controls, in addition to the dynamics. FilterDDP adopts a line-search filter procedure f....
arXiv:2606.01491v1 Announce Type: new Abstract: We geometrize the Poisson summation formula for the zero locus of a split quadratic form in an even number of variables over number fields. We do so by making explicit the relationship between Schwartz spaces on quadrics defined in two different ways: via Braverman-Kazhdan spaces and via theta lifts.
arXiv:2606.01492v1 Announce Type: new Abstract: In this expository paper, we develop the basic ideas underlying Grothendieck groups and to illustrate their appearance across algebra, topology, representation theory, and homological algebra. Motivated by the universal construction associated to a commutative monoid, we define the Grothendieck groups abelian categories and rings. Along the way we study several fundamental examples, including..
arXiv:2606.01497v1 Announce Type: new Abstract: Rosenbrock's Theorem is a result, originally motivated by engineering applications, that was first proved over the univariate polynomial rings $\mathcal{R} = \mathbb{R}[x]$ and $\mathcal{R}=\mathbb{C}[x]$, and later established to hold for every elementary divisor domain $\mathcal{R}$. Under some coprimality assumptions on certain submatrices, Rosenbrock's Theorem connects the Smith form of a....
arXiv:2606.01499v1 Announce Type: new Abstract: We provide a simple construction which realizes the Birkhoff center depth at an arbitrary ordinal level and relate it to the Cantor-Bendixson depth.
arXiv:2606.01505v1 Announce Type: new Abstract: We consider a general class of ``inexactly smooth'' convex functions, providing a universal model capturing as special cases $L$-smooth, $M$-Lipschitz, and H\"older smooth functions, and any combination thereof. Such functions possess a calculus closely following that of smooth functions. Our main results provide inexactly smooth functions with interpolation theorems that are necessary and su....
arXiv:2606.01506v1 Announce Type: new Abstract: The classical BFGS algorithm performs excellently for convex optimization problems. However, for non-convex problems, the classical BFGS method may fail to converge reliably. To overcome this limitation, researchers have developed modified BFGS methods that are applicable to both convex and non-convex optimization problems. Among these methods, a robust BFGS algorithm has been shown to achiev..
arXiv:2606.01507v1 Announce Type: new Abstract: We show that the top cohomology of any affine Springer fiber, as a Weyl group representation, contains a large part of the total cohomology of certain Springer fibers. The main ingredient of the proof is the construction of a ``perverse filtration'' on the pure part of the cohomology of affine Springer fibers.
arXiv:2606.01511v1 Announce Type: new Abstract: Suppose $\mathcal{V}$ is a class of stationary integral $n$-varifolds in $B^{n+k}_2(0)\subset\mathbb{R}^{n+k}$ which is closed under weak limits, homotheties, rotations, and disjoint decomposition, and suppose that $\mathcal {V}$ satisfies an $\epsilon$-regularity property near planes of (integer) multiplicity $\leq Q\in \{2,3,\dotsc\}$. This last condition, more precisely, requires that ther....
arXiv:2606.01514v1 Announce Type: new Abstract: Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine, under the generalized Riemann hypothesis, the order of magnitude of $\E|\sum_{n \leq x} h(n)\lambda(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, for all real $x, q$ with $1 \leq q \leq c\log x/\log\log x $ and $c>0$ a small..
arXiv:2606.01519v1 Announce Type: new Abstract: Li and Poon proved that every real square matrix is a real linear combination of four real orthogonal matrices. The resulting question, recorded by Zhan, asks whether four terms are necessary. We prove that three real orthogonal matrices always suffice.
arXiv:2606.01523v1 Announce Type: new Abstract: In this paper, we develop the Rubio de Francia extrapolation theorem for the multilinear compactness on mixed-norm Lebesgue spaces. More precisely, if a multilinear operator is bounded on weighted product spaces, then its compactness can be extrapolated from unweighted product spaces to the full range of weighted mixed-norm spaces. This result is mainly based on a multilinear interpolation th..
arXiv:2606.01530v1 Announce Type: new Abstract: We consider approximation of a Gaussian distribution with a mixture of homoscedastic Gaussians of smaller variance. The solution is obtained by minimising the $L^2$ norm between the original Gaussian and the mixture, which is parameterised to reduce the complexity of the optimisation problem. The developed technique is straightforward, sufficiently robust and yields Gaussian Mixtures that rap..
arXiv:2606.01531v1 Announce Type: new Abstract: Dittert's conjecture gives a sharp upper bound for the Dittert functional on nonnegative matrices whose entries sum to \(n\). It extends the van der Waerden permanent problem from the doubly stochastic polytope to a larger simplex in which row and column sums are allowed to vary. We prove the conjecture for every dimension \(n\ge 17\). The proof combines the Knopp--Sinkhorn lower bound for bo..
arXiv:2606.01534v1 Announce Type: new Abstract: For a properly embedded Willmore surface $\Sigma$ in $\mathbb R^3$, we prove that if the scale-invariant second fundamental form is sufficiently small near infinity, the surface has finitely many ends. Moreover, if this scale-invariant quantity vanishes at infinity, or if there is only one end, the total $L^2$-norm of the second fundamental form is finite.
arXiv:2606.01536v1 Announce Type: new Abstract: In this paper, we establish, for the first time in the literature, the convergence of the practical versions of the Inexact Proximal Point Algorithm (IPPA) and the Inexact Tseng Algorithm (ITA) for computing approximate solutions to monotone inclusions in Hilbert spaces under the the presence of nonsummable errors. Our ap- proach relies on Tikhonov regularization, the contraction property of ..
arXiv:2606.01548v1 Announce Type: new Abstract: In this paper, we compute the Minimal Intersection Radius (MIR) of growing, non-homogeneous ellipsoids in arbitrary ambient dimension. We provide a geometric method to find the MIR using techniques from convex optimization, a secondary method using second-order cone programs, and show that the MIR can be phrased as an LP-type problem, where the computation from convex optimization acts as a c..
arXiv:2606.01554v1 Announce Type: new Abstract: This short note proposes a polynomial-time algorithm for near-optimal Euclidean estimation of a signal constrained to lie in the unit ball of a symmetric norm, where the symmetry is with respect to a known basis and the norm is accessible through an evaluation oracle. We further extend the method to a random-design, moderate-dimensional linear regression setting, where the regression paramete..
arXiv:2606.01559v1 Announce Type: new Abstract: We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that..
arXiv:2606.01571v1 Announce Type: new Abstract: Let $X$ be a geometrically irreducible smooth projective curve over a field $k$. Yamazaki et al. define a biadditive symmetric pairing $\langle -,-\rangle$ on the torsion subgroup of the Picard group $\mathrm{Pic}(X)$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. The intrinsic subgroup $\mathrm{Pic}(X)_\mathrm{tors}^\mathrm{is}$ is the kernel of this pairing. When $X$ is an ellipti..
arXiv:2606.01579v1 Announce Type: new Abstract: This paper establishes gradient estimates and Liouville-type theorems for the \(\Phi\)-Laplacian equation \(\Delta_{\Phi}(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = \Lambda_{\Phi}(u)\) on compact Riemannian manifolds. Using a nonlinear \(\Phi\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds un..
arXiv:2606.01586v1 Announce Type: new Abstract: For a given subset $S\subseteq V(G)$ of a graph $G$, the pair $(G,S)$ is \emph{knitted} if for every partition of $S$ into non-empty subsets $S_1, S_2, \ldots, S_t$, there exist pairwise disjoint connected subgraphs $C_1, C_2, \ldots, C_t$ in $G$ such that $S_i\subseteq V(C_i)$ for all $1 \le i \le t$. A graph $G$ is \emph{$\ell$-knitted} if $(G,S)$ is knitted for every subset $S\subseteq V(G....
arXiv:2606.01587v1 Announce Type: new Abstract: Strata of holomorphic $1$-forms have an absolute period foliation given by varying $1$-forms while keeping their integrals along closed loops fixed. In this paper, we classify the leaf closures of this foliation when the stratum is connected, outside of a subvariety of high codimension.
arXiv:2606.01596v1 Announce Type: new Abstract: Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus m....
arXiv:2606.01609v1 Announce Type: new Abstract: Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parame....
arXiv:2606.01614v1 Announce Type: new Abstract: In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator fr....
arXiv:2606.01618v1 Announce Type: new Abstract: In this work, we address the stability of $N$-soliton solutions to the completely integrable modified Camassa--Holm (mCH) equation. Recently, Li, Liu, and Zhu (Math. Ann. 392 (2025), 899--932) established the orbital stability of 2-soliton solutions in $H^4(\mathbb{R})$ with respect to the solution $u$ and highlighted the stability of mCH $N$-soliton solutions remains an urgent challenge. Mot....
arXiv:2606.01622v1 Announce Type: new Abstract: We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively o....
arXiv:2606.01627v1 Announce Type: new Abstract: Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\Delta$ denote the expected average degree of the graph. As is the case for Erd\H{o}s-R\'enyi graphs, there is a critical value $\Delta_c$, depending ....
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