Papers

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.

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 [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 for step acceptance. However, instead of a damped Newton step applied in the general nonlinear programming setting, the computation of a trial point involves applying a backward recursion and a forward simulation. We establish the global convergence of FilterDDP by showing that for a subset of constrained optimal control problems, the this backward-forward procedure satisfies the same properties as a Newton step for the purpose of establishing global convergence of a line-search filter method, following the analysis of Wächter and Biegler [SIAM Journal on Optimization, 16 (2005), pp. 1-31].

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 of a sector whose central angle is exactly $\theta$ and whose arc length equals that of the initial curve. This result reveals how a length-preserving fourth-order geometric flow can straighten out a curve's shape while respecting boundary constraints, ultimately driving it toward a unique equilibrium: the circular arc spanning the prescribed angle. This provides a complete description of the long-time behaviour of this fourth-order geometric flow with mixed boundary conditions.

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 integer $n \ge 2$, the connectedness locus $M_n$ contains infinitely many holes. While the $n=2$ case is equivalent to a known theorem by Calegari, Koch, and Walker, this paper establishes the proof for $n \ge 3$. To prove the existence of holes for larger alphabets, we construct a stationary family of finite-capture loops in the geometry of the associated difference attractor. Each loop surrounds a missing-center configuration, and a finite inverse-tree certificate rigorously demonstrates that the enclosed witness parameter lies outside the connectedness locus. Furthermore, we show that the sequence of witness parameters converges to a canonical algebraic boundary point – termed the renormalization point, $\xi_n$ – where infinitely many of these distinct holes accumulate. The paper's finite geometric checks are verified via exact algebraic certificates.

We construct a modular operad $\mathbf{S}$ of mapping class groupoids modelling the higher-genus Teichmüller tower, with compositions and contractions encoding surface gluing. We prove a presentation theorem for maps out of $\mathbf{S}$ using genus-zero and genus-one generators and relations. Applying the work of Nakamura–Schneps, we obtain a faithful action of their subgroup $\widehat{\Gamma}\subseteq \widehat{\mathsf{GT}}$ on $\widehat{\mathbf{S}}$, and hence a faithful Galois action. The genus-zero truncation identifies with the cyclic operad of parenthesized ribbon braids, whose object-fixing profinite automorphism group is $\widehat{\mathsf{GT}}$. The completed classifying spaces form a modular $\infty$-operad whose corolla values are geometric étale homotopy types of moduli stacks of curves with nonzero tangent vectors at their marked points. The $\widehat{\Gamma}$-action induces a faithful action by homotopy automorphisms on the associated tower of profinite homotopy $1$-types, which agrees with the étale model in genus zero and one.

Roblin's theorem asserts that, in rank one, a coamenable normal subgroup has the same critical exponent as its ambient group. Natural higher-rank analogues would predict that coamenability preserves the limit cone and the growth indicator. We show that both assertions fail, even when the quotient is infinite cyclic. For every odd integer $n\geq 3$, we construct a nonempty open family of Zariski-dense Borel–Anosov Schottky subgroups of $\mathrm{SL}_n(\mathbb R)$ admitting cocyclic normal subgroups with strictly smaller limit cones. Moreover, in $\mathrm{SL}_3(\mathbb R)$, we construct cocyclic pairs with the same limit cone but distinct growth indicators at an interior direction. We then identify the precise rigidity that survives. Let $\Gamma$ be a Zariski-dense Borel–Anosov subgroup of a connected semisimple real algebraic group, and let $N\lhd\Gamma$ be coamenable. Then the growth indicators of $N$ and $\Gamma$ agree on the fixed-point set of the opposition involution, and their Riemannian critical exponents are equal. The examples with equal limit cones show that the restriction to opposition-invariant directions is sharp.

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.

The main aim of this paper is to establish the Hölder 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 nonlocal superposition tail, which appears to be the first contribution of its kind in the literature. The obtained results are new even in the classical linear case $p=2$, thereby illustrating the broader applicability of the analytical techniques developed in this work. As intermediate steps toward the proof of the main results, we also establish a logarithmic estimate for weak supersolutions, local boundedness for weak subsolutions, a weak Harnack inequality for weak supersolutions, an expansion of positivity for weak supersolutions, and tail estimates for weak solutions.

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 electrons and external electrostatic potentials, and the estimates on the rate of convergence are explicit and constructive. We remove the close-to-equilibrium assumption on the initial datum and the parabolic band approximation assumption that were required in previous works. The technique is based on the construction of a suitable highly non-linear Lyapunov functional by modifying the relative entropy. The analysis benefits from uniform bounds for the solution in terms of the global equilibrium.

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.

Scientific discovery is not only answer generation but revision of the representational regime in which evidence, artifacts, operations, and verifiers are typed. We develop a category-theoretic account of agentic discovery for materials science. In a fixed regime b with schema category S_b, the system state is a copresheaf I_t: S_b -> Set, and provenance is the category of elements \int_{S_b} I_t. Fixed-regime operation is an update on such states, endofunctorial only when provenance-preserving refinements are specified and preserved. Discovery is instead a verified regime transition u: S_b -> S_b': old artifacts are preserved, transported by the left Kan extension Lan_u I_t, and compared with the post-transition state to identify residual content beyond functorial transport. This separates retrieval, search, and discovery without subjective novelty. We instantiate the framework in two systems. In Builder/Breaker, a protein-mechanics world model is revised under a Minimum Description Length gate; the accepted law expresses within-chain flexibility as all-mode elastic compliance conditioned by slow collective-mode participation, or mode-conditioned compliance. In CategoryScienceClaw, typed skills, artifacts, open needs, workflow mutation, gates, stress tests, and public discourse become a proof-carrying knowledge-computation graph. A fiber-network example records candidate models, rejected alternatives, an AIC gate, perturbation tests, and an accepted orientation-tensor anisotropic stiffness surrogate over an isotropic fiber-count descriptor. Together, the cases show how category theory can be both a mathematical language for discovery and an engineering specification for self-revising AI discovery systems.

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 by the addition of coarse levels. A successful coarse space for such methods, inspired by iterative substructuring, is the generalized Dryja–Smith–Widlund (GDSW) space. A monolithic two-level overlapping Schwarz preconditioner based on a GDSW coarse space has been introduced for the solution of saddle-point problems arising from incompressible fluid problems, and has subsequently been extended to a three-level method. In the present work, for the first time, we consider a monolithic four-level overlapping Schwarz preconditioner, obtained by applying the two-level monolithic GDSW construction recursively three times, so that the second- and third-level coarse problems are themselves treated by overlapping Schwarz and only the smallest fourth-level coarse problem is solved by a sparse direct method. Numerical results are presented for a three-dimensional incompressible stationary Stokes problem with a Carreau-type non-Newtonian viscosity model posed on the complex geometry of an extrusion die, on up to $4\,000$\,MPI ranks, comparing the four-level preconditioner with its two-level and three-level counterparts in both roles. This work is part of the StroemungsRaum project, funded by the German Bundesministerium für Forschung, Technologie und Raumfahrt (BMFTR, formerly BMBF) as part of the SCALEXA program on new methods and technologies for exascale computing.

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 $\mathcal{F}(X)$. Thus, for a connected space $X$, we will show that every point $p\in \mathcal{O}_{cs}(X)$ is not only noncut, but also an endpoint of $X$. Another result of this paper is that in an arbitrary topological space $X$, the closure of the set $\mathcal{O}_{cs}(X)$ is always a totally disconnected subset. Furthermore, we will also show that $\mathcal{O}_{cs}(X)$ is a closed subset of every first countable totally disconnected space $X$.

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 coupling energy accounting for the nematic and stray field interaction. We prove the existence of an energy minimizer for the introduced ferronematic energy functional in a bulk setting. We then provide a reduced local ferronematic energy in a two-dimensional setting via $\Gamma$-convergence.

We give a formula for $X$-products of attractors using $\mathbb{F}_1$-geometry.

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 hand, the following slight generalisation already implies a rather strong logical system, namely the `Big Five' system ATR$_{0}$ which accommodates transfinite recursion. $$ \text{For $f:\mathbb{R}\rightarrow \mathbb{R}$ Riemann integrable on any interval $[-a, a]$ for $a>0$, there is continuous $g:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x)\leq g(x)$ for all $x\in \mathbb{R}$.} $$ As part of the Reverse Mathematics program, we obtain equivalences for the centred statement and variations involving the axiom of numerical choice. A central result is that numerical choice for $\Pi_{1}^{1}$-formulas is equivalent to ATR$_{0}$. We also obtain equivalences involving basic properties of metric spaces and establish connections to Kohlenbach's generalisations of weak König's lemma, Cousin's lemma, and the representation of open sets.

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 parametric equations. A central feature of our construction is the restriction of all coefficients in the Cox ring $R$ to a specific coefficient ideal $J$. In the two-dimensional case, this approach eliminates the need for $\phi$ to be locally a complete intersection at the base points, that is, the determinant of the implicitization matrix is equal to a power of the implicit equation for arbitrary base points. This result generalizes several previous results in surface implicitization.

Post-training quantization is widely used for compressing large neural networks, but aggressive low-bit quantization can significantly degrade model quality. A common remedy is to augment the quantized weights with a low-rank correction, leading to approximations of the form $W\approx Q+LR$. In this paper, we study this low-precision plus low-rank representation through the layer-wise reconstruction objective $\|XW-X(Q+LR)\|_F^2$, where $X$ is a calibration matrix. We establish, to our knowledge, the first information-theoretic lower bounds for this problem under finite-alphabet and bounded low-rank compensation constraints. We then propose GPTQ-intrinsic LoRA, a training-free algorithm that incorporates the low-rank correction directly into a GPTQ-style quantization pass by appropriately augmenting the calibration Hessian. For the choice $L=V_r$, where $V_r$ contains the top right singular vectors of $X$, we prove layer-wise reconstruction error bounds in which the usual GPTQ dependence on $\|X\|_F^2$ is replaced by the rank-$r$ residual $\|X-X_r\|_F^2$, up to regularization terms. Under natural structural assumptions, these bounds match the information-theoretic lower bounds in their dominant scaling, up to constants and mild factors. We also introduce Bid-Up, a fixed-grid quantization refinement step that can be alternated with optimal low-rank compensation with guaranteed non-increasing layer-wise reconstruction error. Experiments on Qwen3 language models and DeiT vision transformers show that GPTQ-intrinsic LoRA improves over GPTQ and GPTQ followed by low-rank compensation, with additional gains from refinement loops.

We give a straightforward, self-contained, and natural construction of the Big Witt ring using the eñe product that is defined through the action on zeros of polynomials. This is in contrast with classical constructions of the Big Witt ring using formulas out of nowhere.

Starting from Maxwell's and linear momentum balance equations, we derive a ferrofluid model using the generalized Onsager's principle. Guided by a discrete perturbation estimate, we design and analyze families of Galerkin schemes that converge to sufficiently regular solutions and derive error estimates. Finally, we numerically explore the model with our proposed method.

We define the analogue of Lie-Rinehart algebras over $C^\infty$-rings. We show that given a Poisson $C^\infty$-ring $\mathcal{A}$ its module $\Omega_{\mathcal{A}}^{1}$ of $C^\infty$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M} \xrightarrow{\rho} C^\infty\mathrm{Der}(\mathcal{A})$ over a $C^\infty$-ring $\mathcal{A}$, there is a natural Poisson bracket on the $C^\infty$-ring $\mathcal{F}(\mathcal{M})$ associated with the $\mathcal{A}$-module $\mathcal{M}$ (the $C^\infty$-ring analogue of an $\mathcal{A}$-algebra freely generated by the module $\mathcal{M}$). In the case where $\mathcal{A}$ is the $C^\infty$-ring of smooth functions on a manifold $M$ and $\mathcal{M}$ is the module $\Gamma(E)$ of sections of a Lie algebroid $E \to M$, the $C^\infty$-ring $\mathcal{F}(\Gamma(E))$ is the ring of functions $C^\infty(E^\vee)$ on the total space of the vector bundle $E^\vee \to M$ dual to the vector bundle $E$.

In 1970, T. M. Apostol introduced the Möbius function $\mu_{k}$ of order $k$ for all positive integer $k$, as a generalization of the Möbius function $\mu = \mu_{1}$. For any integer $k \ge 2$, he proved $\sum_{n \le x} \mu_{k}(n) = A_{k} x + O_{k}(x^{1/k} \log x)$ where $A_{k}$ is a positive constant. In 2001, A. Bege conjectured both the conditional and unconditional estimates for the sum $\sum_{n \le x, (n, q) = 1}\mu_{k}(n)$ for any positive integer $q$. In this paper, we give affirmative solutions to the conditional version of Bege's conjecture completely and the unconditional one partially. We also give a mean square estimate for the error term.

Let $\Delta$ be the Dirichlet Laplacian on a bounded domain $\Omega \subset \mathbb{R}^{N}$, and let $(-\Delta)^\alpha$ be the associated spectral fractional Laplacian with $\alpha \leq 1, \ \rho <2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $\alpha <1/2$, extending a result previously proven on rectangles for $\alpha <1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $\alpha <1$, we prove boundary null-controllability results. For $\alpha <1/2, \ \rho \leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $\alpha =1$, on rectangles, where higher regularity is required for Dirichlet control.

For every $d\geq1$ we give an explicit equal-weight spherical $5$-design in $\mathbb{S}^{d-1}\subset\mathbb{R}^d$ with at most $72d^2$ points. Our approach utilizes recent construction of complex projective $2$-designs based on Sidon sets.

The behavior of representations under restriction is a central theme in Lie theory. We study wide regular subalgebras of symmetrizable Kac–Moody algebras, extending work of Douglas and Repka on semisimple Lie algebras. A subalgebra is wide if every irreducible integrable highest weight module remains indecomposable upon restriction. Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra with Cartan subalgebra $\mathfrak{h}$, root system $\Phi$, simple roots $\Pi$, and root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_\alpha$. Denote by $\Phi_{\operatorname{re}}$ the set of real roots. To a regular subalgebra $\mathfrak{s}$ normalized by $\mathfrak{h}$, we associate a subset $T\subseteq \Phi$ by declaring $\alpha\in T$ if $\mathfrak{s}\cap \mathfrak{g}_\alpha\ne \{0\}$. One of our main results is an extension of Schur's lemma: if $\mathfrak{h}\subseteq \mathfrak{s}$ and the real closure of $(T\cup(-T))\cap \Phi_{\operatorname{re}}$ contains $\Pi$, then $(\operatorname{End} V)^{\mathfrak{s}}=\mathbb{C}\operatorname{Id}_V$ for every irreducible integrable highest weight module $V$. As a consequence, this real-root closure condition yields a sufficient condition for wideness. In the affine case, we establish a necessary condition for wideness: if $\mathfrak{s}$ is wide, then the closure of $T\cup(-T)$ in $\Phi$ is all of $\Phi$, and this implication holds without assuming that $\mathfrak{h}\subseteq \mathfrak{s}$. A key ingredient is a structural result showing that closed subsets of affine root systems are closed under arbitrary finite root sums that remain roots.

In this article, using the notion of discrepancies, we study the generalized Weierstrass semigroup $\widehat{H}(\mathbf{Q})$, where $\mathbf{Q}$ is an $n$-tuple of distinct totally ramified places of degree one in a linearized function field. As a consequence, we characterize and explicitly determine the sets of absolute maximal elements $\widehat{\Gamma}(\mathbf{Q})$ and relative maximal elements $\widehat{\Lambda}(\mathbf{Q})$, generalizing the existing results in the literature. Finally, we apply our results to some classes of algebraic curves.

We prove Cohn–Vossen-type scalar curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. For $n\ge3$, we obtain an $O(r^{n-2})$ normalized growth estimate under a $\mathbb Z^{n-2}$ subgroup condition on $\pi_1(M)$. For locally conformally flat manifolds, we prove the corresponding normalized estimate in the non-$\mathbb R^n$ case and derive polynomial or exponential upper bounds in the conformally Euclidean case. In dimension three, we prove the sharp asymptotic scalar-curvature flux estimate $8\pi$ under quadratic scalar-curvature decay, confirming the Munteanu–Wang conjectural $8\pi$ bound in this setting; we also prove finiteness of the flux for manifolds with a foliated end. Finally, under the Cohn–Vossen-scale scalar-growth hypothesis, we prove weighted analogues for the weighted scalar curvature $\mathrm{Sc}_{\alpha,\beta}$ on weighted Riemannian manifolds with nonnegative Bakry–Émery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry–Émery regimes.

We develop a nonlinear potential theoretic framework for Schauder estimates for vector-valued solutions of a broad class of nonautonomous variational problems at nearly linear growth. Our approach naturally embraces the variable exponent as well as the Double and Multi phase setting, yielding new regularity results in basic models and recovering optimal regularity recently established in specific cases.

We review the kinetic theory of one-dimensional nonlinear oscillator chains, of which the most famous example is the Fermi-Pasta-Ulam-Tsingou equation. We provide detailed, though not rigorous, accounts of the microscopic to mesoscopic, and mesoscopic to macroscopic limits: derivation of the kinetic wave equation and hydrodynamic limit. We also present the state of the art of the mathematical theory, including proofs. We discuss the connection to two famous problems of Mathematical Physics: the Fermi-Pasta-Ulam-Tsingou paradox, and the derivation of Fourier's law. Finally, many open problems and possible directions for future research are proposed.

We consider integrable hierarchies such as KP, modified KP, 2D Toda lattice, BKP (small and large), DKP, Pfaff-Toda and their multi-component generalizations. We work in the framework of the bilinear formalism in which the universal dependent variable is a tau-function satisfying bilinear equations of the Hirota-Miwa type. Our principal interest in this paper is the dispersionless versions of the hierarchies. In the limit of zero dispersion the main object is an $F$-function, which is the limit of properly re-scaled logarithm of the tau-function. We show that in all the cases there exists an algebraic curve built into the structure of the hierarchy. We call it the dynamical curve. For the KP, modified KP and Toda lattice hierarchies, as well as for their multi-component generalizations, the curve is rational (of genus 0) and can be uniformized by rational or trigonometric functions. For hierarchies of the Pfaff type (DKP and Pfaff-Toda) the dynamical curve is in general a smooth elliptic curve (of genus 1), with its modular parameter being a dynamical variable. It is also shown that the large BKP hierarchy admits two different dispersionless versions. In one of them the dynamical curve degenerates to a rational curve while in the other one it remains to be elliptic. We show that a reformulation of the hierarchies based on uniformization of the dynamical curves by elliptic (or trigonometric) functions makes their structure nice and clear, especially in the multi-component case.