Papers

We consider the problem of computing the Lebesgue volume of the unit sublevel set of a positive quasi-homogeneous polynomial. Pushing the Lebesgue measure of an ambient bounding box forward through the polynomial reduces this high-dimensional volume to a one-dimensional moment problem. This removes the ambient dimension from the optimization and confines the dimension to a single preprocessing stage, computing the moments of the polynomial over the box, which is polynomial in the ambient dimension for sparse or separable polynomials. We propose two deterministic algorithms for the resulting univariate relaxations, each returning certified upper and lower bounds on the volume. Both bypass semidefinite optimization entirely and rely only on standard numerical linear algebra. The first approximates a piecewise-constant function by a Chebyshev polynomial, so that each relaxation reduces to a fast cosine transform, and converges at a polynomial rate in the relaxation order. The second extracts the volume bounds from a single generalized eigenvalue problem involving moment and localizing matrices whose size grows linearly with the relaxation order, and converges at an exponential rate; the ratio governing this rate is determined by an a priori upper bound on the polynomial over the bounding box. Finally, the univariate polynomials produced by either algorithm are feasible for the multivariate moment-SOS volume hierarchy. The algebraic and geometric rates therefore transfer to the hierarchy itself, improving on its best known convergence rates.

We present a direct and self-contained proof of Khovanskii's Bezout-type bound for the number of nondegenerate solutions of a system of Pfaffian equations. We isolate the ingredients of Khovanskii's original argument and assemble them into a proof that avoids the general theory of integral manifolds developed in his monograph. Our formulation mildly refines the classical statement: rather than depending on the ambient dimension, our bound depends on the maximum number of variables on which any function in the Pfaffian chain depends. As a consequence, we obtain a refined bound on the number of connected components of a Pfaffian set.

Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, On the Construction of Geographical Maps. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word “best” means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.

We show that the assumption of a stress-free boundary condition at a finite intermediate depth, namely, at the bottom of the Ekman layer, in the analysis of wind-driven ocean flows necessarily leads to an unphysical current profile. Indeed, if the $z$-derivative of the fluid velocity vanishes at a given depth, then this depth necessarily corresponds to a minimum of the velocity profile, with the velocity increasing beneath it. Using a WKB ansatz based on the small variations of the ocean's water density at great depths, we also argue that a no-slip condition at the bottom of the ocean, if sufficiently deep, still effectively implies (up to a very small error) the orthogonality of the Ekman transport and the wind-stress.

The theoretical foundation of infinite-width shallow neural networks relies heavily on continuous integral representations and Barron spaces. Recently, harmonic analysis-specifically the Radon and Ridgelet transforms-has emerged as a powerful tool to invert these representations and compute the optimal network weights. However, a major analytical bottleneck remains: standard neural network activation functions exhibit severe spectral singularities at the frequency origin. To bypass this divergence, existing frameworks either restrict the theory to specific activation families or mathematically quotient out the network's affine components, which inherently limits their practical scope. In this paper, we overcome these limitations by introducing a purely distributional framework for the generalized Radon transform R $\sigma$ that operates on a broad class of tempered distribution activation functions. By defining a regularized spectrum formulation g($\rho$) = (i$\rho$) $\alpha$ $\sigma$($\rho$), we rigorously absorb the origin singularities without truncating the underlying functional space. Building upon this exact reconstruction, we show that under a definite parity assumption on the activation, there exists an exact linear isometry between Barron functions and their optimal weight measures.

In this note, we study two simple operations for extending juggling (siteswap) sequences, called first throws and last catches. Both constructions come from a natural question for a juggler: how can one add throws at the beginning or catches at the end of a pattern without creating a collision? Using landing times and the permutation test, we give necessary and sufficient conditions under which these constructions, when applied to an arbitrary valid siteswap, produce another valid siteswap. We complement our analysis with several examples. An interactive visualization of these extensions is available in [Para].

In this paper, we construct a Fleming-Viot particle system for a class of subcritical Bellman-Harris processes. We prove that it selects the Yaglom limit at a polynomial rate in the number of particles. Since lifetimes are non-exponential, the population size is not Markov, and the analysis must therefore be carried out on the space of age configurations. In this setting, the Lyapunov functions used for Galton-Watson processes are no longer norm-like. Nevertheless, we establish a Yaglom theorem that strengthens the classical result: the conditional laws converge in total variation at an exponential rate, with decay rate given by the Malthusian parameter. We also prove that the drift condition, which links the hazard rate to the offspring law, is necessary within a natural class of Lyapunov functions, showing that it is a feature of the measure-valued lift rather than a defect of the estimates. Finally, we illustrate the estimator through an application to livestock epidemic surveillance, where the Yaglom limit is the null distribution of a change-detection test.

We study expected improvement (EI) for minimizing a deterministic function $f$ in the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on a nonempty compact set $\mathcal X\subset\mathbb R^d$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $\sigma^2k$, $\sigma>0$. A weak-EI policy queries a point whose EI is at least a fixed positive fraction of its maximum. We introduce a notion of sequential separation radius relating ranked selected-point innovation norms to Kolmogorov widths, drawing on greedy approximation. Standard power-function estimates from scattered-data approximation and a finite-budget regret argument yield the rates. After $N$ post-initial queries, every weak-EI policy has simple regret $O(N^{-\nu/d})$ for isotropic Matérn kernels of smoothness $\nu>0$ and $O(\exp[-c_1\min\{N,N^{1/d}\log(eN)\}])$ for the isotropic squared-exponential kernel, with $c_1>0$. For $d=1$, the sharper bound $O(\exp[-c_2N\log(eN)])$ holds for exact EI, with $c_2>0$. These bounds are uniform over each fixed RKHS ball. If $\mathcal X$ has nonempty interior and $B>0$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels, even among randomized strategies whose final recommendation need not be a query point. For the squared-exponential kernel, it is minimax-rate optimal up to constants in the exponent among deterministic methods whose final recommendation may be any point of $\mathcal X$.

We describe algorithms for the exhaustive enumeration and classification of simple arrangements of $n$ pseudolines ($n$ odd) maximizing the number of triangular faces. The depth-first search enumerates reduced words for the longest permutation $w_0$ by branching only on the even-indexed generators, using pruning constraints imposed by the geometry of optimal arrangements. The approach handles both perfect arrangements with a regular triangular pattern and unavoidable deviations from it for $n \equiv 1 \pmod 6$. The output is classified into a hierarchy of equivalence classes: by commutation, by Euclidean transformations, and by projective transformations. For each projective class we recover its full symmetry group $G \subseteq S_{n+1}$ together with the orbit-stabilizer profile of its Euclidean subclasses. Completeness of the search and classification is proved: every wiring diagram is reached. We report full enumerations; e.g. for $n=27$, 85,562,064 wiring diagrams partitioned into 56,646 projective classes. For larger $n$ (up to $n=93$), where exhaustive enumeration is out of reach, we report partial (first-hit) results.

Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian–Pask algebra of the associated $2$-graph. The main objective of this paper is to extend textile systems to dimension $3$ so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for $3$-graphs extending the well-known connection between $2$-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and $3$-graphs. In particular, our investigation shows that the conditions required to form a $3$-graph from a $3$-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated $3$-graphs, thus establishing that our construction is homologically consistent with $3$-graphs.

On the $n$-dimensional torus $\mathbb{T}^n$, we consider the Kirchhoff-Poho{\v z}aev equation, which is the only Kirchhoff-type equation known to admit global solutions for small initial data in $H^s \times H^{s-1}$, $s \geq 2$. We study this equation from a dynamical perspective by means of a quasilinear normal form approach. We perform two steps of normal form reduction and show that the resulting cubic and quintic terms do not contribute to the Sobolev energy estimates. This cancellation reveals a special algebraic structure of the equation and represents a first step towards a dynamical explanation of its exceptional global well-posedness. As a further consequence, we obtain improved bounds on the growth of Sobolev norms and improved lower bounds on the existence time for initial data in $H^s \times H^{s-1}$ with $s \in [3/2, 2)$.

Two-derivative Hermite deferred correction combines high collocation order with sequential single-state solves, but the stopped method depends jointly on the nodes and the correction sweep. We co-design these ingredients for diffusion-dominated semilinear problems. For three subintervals, an H4 predictor, and two corrections, the complete order-seven B-series defect has rank one in the 48-dimensional rooted-tree space: two corrections apply two unary graftings to the one-directional order-five predictor defect. Hence one scalar chain coefficient controls every nonlinear principal-error condition. Rational nodes and an isolated algebraic correction parameter cancel this coefficient; coprimality with the order-eight chain polynomial proves classical order exactly seven. A complementary design, Accuracy-P40, retains generic sixth order but reduces the complete principal-error norm to $9.8\%$ of the LGL–L3 value while satisfying $J_{\mathrm{stiff}}<0.40$. We also distinguish convergence of repeated corrections from absolute stability after a fixed number of sweeps: two corrections have finite negative-real stability intervals, whereas a third correction restores far-stiff output damping for the new designs. High-precision nonlinear order tests verify sixth versus seventh order, and Allen–Cahn and Cahn–Hilliard calculations show that Accuracy-P40 reduces correction and Krylov work.

We study small points in radical extensions of algebraic fields. Given an algebraic extension $\mathbb{F}$ of $\mathbb{Q}$, a finitely generated subgroup $\Gamma\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(\Gamma^{p-\mathrm{div}})\setminus \Gamma^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of Rémond, formulated when $\mathbb{F}$ is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of Rémond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in $p$-adic Lie extensions.

In this paper, we give an enriched count for the 10 lines on a smooth del Pezzo surface S of degree 5 as the quadratic degree of the Euler class of a locally free sheaf. To do this, we see S as a section of the quintic del Pezzo threefold V 5 and the lines on S as a sub-scheme of the P 2 of lines on V 5 . We then use some oriented Schubert calculus and Bott residue formula with N SL 2 -action to compute the degree of the Euler class.

Sequent-style tableaux are a one-sided refutation calculus for classical propositional logic, in which each node of the refutation tree carries a finite block of formulae and the structural rules are absorbed into the data structure and the closure criterion. Building on the correspondence between this block calculus and the cut-free sequent calculus, and following the programme of Kamide and Negri, we recast the calculus as a structural-rule-free G3-style sequent calculus $\mathbf{G}_t$ with shared contexts, and we introduce a G0-style sequent calculus $\mathbf{G}_0$ with independent contexts, explicit weakening and contraction, generalized initial sequents, and a primitive explosion rule. A theorem establishing the equivalence between $\mathbf{G}_0$ and $\mathbf{G}_t$ is proved, and the cut-elimination theorem for $\mathbf{G}_0$ is obtained as a consequence. We then introduce a natural deduction system $\mathbf{N}_g$ with general elimination rules matching the left rules of $\mathbf{G}_0$, and we prove a full normalization theorem for $\mathbf{N}_g$. The proof is achieved by means of bidirectional translations between $\mathbf{G}_0$ and $\mathbf{N}_g$: normal derivations correspond to cut-free derivations, and full normal form to the discipline in which every major premiss of an elimination rule is an assumption. We also determine the reach of the formula-succedent fragment, which is shown to have no theorems, so that the equivalence of the three systems is one of consequence and not of theoremhood, and we show that classical logic is recovered on the succedent side, and recovered exactly, by adjoining the rule of indirect proof.

We consider a reaction diffusion equation, where the lower-order nonlinearity allows for a non-smooth part, described by a maximal monotone relation. We show that the global attractor has finite fractal dimension, and also obtain explicit upper estimates in terms of the data.

We propose GLORO, a bilevel solution methodology for robust optimization involving expensive function evaluations. The methodology is carefully designed to achieve satisfactory optimization results in a computationally efficient manner. It is based on Gaussian Process surrogates in the parameter space for approximate computations of the objective / constraints of the robust optimization problem via Bayesian optimization (lower level), and local models constructed in the variable space using these approximate evaluations (upper level). Both the use of Bayesian optimization (in parameter space) and the use of local models (in variable space) are motivated by the idea of guiding the expensive function evaluations to the regions of interest for the optimization process. The methodological work is motivated by and tested on a robust airfoil design optimization problem. This application focuses on the RAE2822 airfoil, optimizing its shape to ensure a robust lift-to-drag ratio under operational uncertainties in Mach number and angle of attack.

Phase-field approaches to fracture, initially designed as regularization of the Griffith model of brittle fracture, are now commonly viewed as gradient-damage models whose regularization length becomes a material property driving crack nucleation. One weakness of this approach is that the strength surface cannot be arbitrary: its shape is dictated by the elastic energy, and its magnitude by the regularization length. We focus on the antiplane version of the model introduced by Bourdin, Marigo, Maurini and Zolesi (arXiv:2506.22558), which handles crack propagation along unknown paths and nucleation governed by an arbitrary convex strength surface by degrading the strength instead of the stiffness. It can be interpreted as a regularization of softening plasticity in which localization bands obey an equivalent cohesive law set by the strength domain and the toughness, while the role of the regularization length, when small compared to the elasto-cohesive length, is purely numerical. Strength, stiffness, and toughness thus become independent material data, and limit analysis, perfect plasticity, cohesive fracture, and brittle fracture merge into a single variational framework. We derive closed-form solutions for a simple shear problem, propose a numerical scheme combining alternate minimization and conic programming, and numerically verify the equivalent cohesive law, its independence of the regularization, and the size effect governed by the elasto-cohesive length. A "surfing" simulation highlights the structure of the propagating crack while a re-entrant V-notch is used to show how the model bridges small-scale yielding, cohesive fracture, and brittle fracture without a priori hypotheses.

We study Shapovalov elements for a symmetrizable, integrable Kac-Moody Lie superalgebra $\mathfrak{g}$. Let $\gamma$ be a positive root of $\mathfrak{g}$ and $m$ a positive integer, with suitable conditions on the pair $(\gamma, m).$ The Shapovalov element $\theta_{\gamma,m}\in U(\mathfrak{b}^-) $ has the important property that if $\lambda$ lies on a certain hyperplane, then $\theta_{\gamma, m} v_\lambda$ is either zero or a highest weight vector of weight $\lambda -m\gamma$ in $M(\lambda)$. We give a closed fomula for all coefficients that arise in the induction step of the proof. The commutative version of this formula relates the leading terms using maps graded algebras that we call of Shapovalov algebras. This suggests a geometric setting for the study of Shapovalov elements.

In the context of the study of the Ohsawa–Takegoshi $L^2$ extension theorem, a "qualitative" extension theorem in terms of adjoint ideal sheaves (i.e. extension result without $L^2$ estimates but with some form of local $L^2$ condition ensured) is proved on compact Kähler manifolds via harmonic theory and residue computations. The arguments are adapted from those in the study of the injectivity theorem by Chan–Choi–Matsumura. The result guarantees, in particular, the existence of extensions of line-bundle-valued holomorphic top forms over the union of any log-canonical centres of the same codimension to top forms over the ambient space under suitable positivity assumptions.

The author studies the inverse spectral problem of Sturm-Liouville operator on a star-like metric graph. At the vertex of this star-like graph, there are attached $m$ edges that imposed with non-local Sturm-Liouville operator satisfying some suitable non-local boundary conditions. At the vertex, we consider one point interaction condition to model a metric graph that fixed on the end of edges of the graph. This models the vibration or flux that changes over time that monitored at the vertex which serves as certain control/regulation center. The author shows that the system is solvable under very necessary conditions. It is crucial to recover the topology of the network/metric graph which the topology is given. To begin the analysis, one constructs the special solution fixed on one end of edges while maintaining continuous at the vertex. This models a string that is vibrating vertically at the vertex according to certain frequencies. The non-local characteristic function plays a role, and then, one tries to find a non-trivial non-local eigenvalue.

We extend the Bures-Wasserstein mean of positive definite matrices to the case of positive definite operators on a Hilbert space. This is done through its defining stationary point operator equation, coming from the gradient of the sum of squared Bures-Wasserstein distances of centered Gaussians represented by positive definite matrices. This gradient is shown to have a Fréchet derivative which induces a bounded linear operator on the space of Hilbert-Schmidt operators with strictly positive real spectrum. This allows us to conclude the existence and uniqueness of this mean by exhibiting the spectral permanence of this operator when extended to general bounded linear operators and also enables the study of its generated ODE semigroups, which enjoy exponential contraction in a Banach-Finsler metric obtained through the construction of equivalent renormings. Using this exponential contractivity of the flow, we prove a `Nodice'-type of theorem and its stochastic variant, a Sturm-type of strong law of large numbers for probability measures with bounded support. We also verify fundamental properties and establish various operator inequalities satisfied by the Wasserstein mean.

We study the projective-geometric criterion of Hwang-Li for isotrivial cone structures admitting characteristic conic connections. We give a complete classification of flag varieties satisfying this criterion, and in the process characterize flag varieties with an injective Gaussian map. As an application, we classify germs of minimal rational curves whose VMRT at a general point is isomorphic to that of a non-homogeneous smooth projective symmetric variety of Picard number one, apart from two exceptional cases.

We present a unified method for evaluating finite sums of powers of cotangent, tangent, cosecant and secant weighted by primitive $k$th roots of unity. The approach relies on constant term extraction for iterated Laurent series, combined with generating functions and partial fraction decomposition. We obtain explicit closed-form expressions for all even-power sums of the four functions, as well as for odd-power cotangent and tangent sums. The formulas are given in terms of Bernoulli polynomials, Euler polynomials, and universal coefficients $r_{n,t}$. As applications, we recover numerous classical identities—including the ordinary and alternating cotangent power sums and Acton's alternating tangent sum—in a systematic and elementary way.

Let $r\ge 2$ be a real number, $d$ a positive integer. A $d$-dimensional nowhere-zero $r$-flow, or $(r,d)$-NZF, on a graph $G$ is an orientation of $G$ together with a function $f\colon E(G)\to \mathbb{R}^d$, such that for all $e\in E(G)$, the Euclidean norm of $f(e)$ lies in the interval $[1,r-1]$, and for every $v\in V(G)$ the sum of all incoming flow values at $v$ equals the sum of all outgoing ones. The $d$-dimensional flow number of $G$ is the parameter $\phi_d(G)=\inf \{r\colon G$ has an $(r,d)$-NZF$\}$. In this paper we provide a lower bound for the $2$-dimensional flow number of the the Flower snark. In particular, together with a previous numerical result by the authors, we prove that $\phi_2(J_n) \in [1 + 2 \sin\frac{5}{22}\pi, 2.387893647]$, where $J_n$ denotes the Flower snark on $4n$ vertices.

We give a self-contained triangular formulation of anisotropic higher-order covering and inverse estimates for smooth mappings and generalized equations. For a $C^p$ mapping into a finite-dimensional target, an indexed target decomposition is fixed, the corresponding triangular factor condition is imposed, and a direction satisfying the associated triangular $p$-kernel condition is chosen. Surjectivity of the resulting triangular $p$-factor operator yields a fixed-scale inclusion in which each nonzero block $Y_i$ is covered at order $t^i$, and hence an inverse estimate with exponent $1/i$ on that block. Zero blocks are allowed, and $p$ denotes the highest active grade. The smooth result is a derivative-level graded formulation: it verifies the model directly from the $C^p$ derivatives and records the blockwise fixed-scale inclusion needed for the set-valued analysis. For generalized equations, an auxiliary exact-model theorem isolates the range-transfer mechanism with acyclic correction fibres. The main sufficient criterion uses a closed convex-process inner approximation whose sum with the smooth triangular factor operator is surjective, allowing the set-valued term to supply missing directions.

We define a non-unital, generally non-associative, commutative semiring structure on the collection of normal subgroups of a group $G$. This viewpoint allows us to recast in ring-theoretic terms Fitting's classical theorem that the join of two nilpotent normal subgroups is nilpotent. From this perspective, the two key inputs are a binomial expansion in a non-associative setting and the fact that the commutator subgroup of two normal subgroups lies in each factor. The development is formalized in Lean, making essential use of Mathlib for the core definitions and results.

We study the fixed-support doubly regularized Wasserstein barycenter problem. Using the semi-dual formulation of entropic optimal transport, we reformulate the problem as a smooth, unconstrained, convex optimization problem in the dual variables. We then derive explicit expressions for the gradient and Hessian and develop an exact Newton method for high-accuracy barycenter computation. To improve scalability, we propose a sparse Newton variant that sparsifies the transport probability matrices, thereby reducing the cost of Hessian-vector products. We establish theoretical results for the proposed methods, including Hessian approximation bounds and convergence results. Experiments on synthetic and real datasets show that the sparse Newton method converges faster than

Let $G$ be a finite group and let $V(\mathbb ZG)$ be the group of normalized units of its integral group ring. We prove that every $N$-prime arc of $V(\mathbb ZG)$ either already occurs in $G$ or admits commuting witnesses of distinct prime orders. Writing $A(\Delta)$ for the arc set of a directed graph $\Delta$, $E(\Delta)$ for the edge set of an undirected graph, and $\operatorname{Sym}(E)$ for the two orientations of the edges in $E$, this is equivalent to the exact formula \[ A\bigl(\Gamma_{\mathrm N}(V(\mathbb ZG))\bigr) = A\bigl(\Gamma_{\mathrm N}(G)\bigr) \cup \operatorname{Sym}\!\bigl( E(\Gamma_{\mathrm{GK}}(V(\mathbb ZG))) \bigr). \] Consequently, the $N$-Prime Graph Question has an affirmative answer for $G$ if and only if the Prime Graph Question does.

Lieberman, Rosický, and Vasey proved that $\mathbf{CompHaus}^{\mathrm{op}}$ - the opposite of the category of compact Hausdorff spaces - is not finitely concrete by a route through Hilbert and Banach spaces, commutative unital $C^*$-algebras, and Gelfand duality. We give a short topological proof. Moreover, we strengthen the result by identifying a specific sequential colimit in $\mathbf{CompHaus}^{\mathrm{op}}$ that no faithful set-valued functor preserves.