Papers

Let $\Omega$ be an irreducible component of the locus of polynomial maps $ F:\mathbb C^n \to \mathbb C^n$ satisfying $\text{\rm deg} F\leq k$ and $\text{det}DF=1$. It is shown that either $\Omega \subset \text{\rm Aut} (\mathbb C^n)$, as claimed by the Jacobian Conjecture, or the general $F\in \Omega$ is not an automorphism.

Public randomness is a security primitive whose deployed behavior is often observable only through a finite transcript. We study black-box auditing of $k$-subset draws from $m$ labels under the exact uniform-without-replacement null. The outcome space has size $\binom{m}{k}$, and unrestricted uniformity testing therefore requires $\Theta(\sqrt{\binom{m}{k}}/\varepsilon^2)$ samples, establishing an information-theoretic limit on transcript-only certification. For structured faults, we construct generator-agnostic audits on the hypersimplex using marginal chi-square, pair maxima, serial overlap, anchored-box discrepancy, and low-dimensional $H_0$/MST geometry, all calibrated under the exact combinatorial null. We also prove a finite-witness guarantee whose sample complexity depends logarithmically on the number of audited witnesses rather than on the full support size. Across observed and reference-source audits, no statistic remains significant after false-discovery correction (minimum BH $q=0.279$). GPU Monte Carlo experiments, using up to 300,000 null and 60,000 alternative replications per condition, show that marginal-preserving deviations can evade one-dimensional tests while remaining detectable through joint geometry. At $n=1{,}956$, a block-cluster alternative of strength 0.04 yields power 0.638 for pair maxima versus 0.051 for marginal chi-square; a band-repulsion alternative of strength 0.08 yields power 0.741 for anchored boxes versus 0.051. These results characterize which structured deviations finite public transcripts can detect and the sample sizes required for doing so.

In this paper, we investigate the Hermitian distance minimization problem for determinantal varieties, the Segre variety, and the Veronese variety. In particular, for binary forms, we obtain upper and lower bounds for the number of critical points that depend linearly on the order, and we determine all possible values in the case of order three.

We consider quasilinear conservative systems \[ (\phi(|\dot q|)\dot q)'=\nabla V(q), \qquad q\in\R^{N}, \] with $\Phi$-growth kinetic term and potential $V\in C^{1}(\R^{N};\R)$. Assuming that for some $c\in\R$ the sublevel set $\{V\le c\}$ splits into two disjoint closed subsets $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, we prove the existence of trajectories $q_c$ with prescribed energy $-c$ connecting $\mathcal V_c^{-}$ and $\mathcal V_c^{+}$, obtained through an energy-constrained variational method. Although the construction yields weak solutions in an Orlicz-Sobolev setting, minimal $c$-connections are shown to be classical $C^2$ trajectories satisfying the strong energy identity $E_{q_c}\equiv -c$. The resulting entire trajectories include heteroclinic, homoclinic, and brake-type orbits. Applications to double-well, Duffing-type, and multiple pendulum systems are discussed.

We investigate a discrete warped Chern-Simons description of three-dimensional warped gravity based on boundary holonomies and monodromy sectors. Starting from the lower-spin SL(2, R) + U(1) gauge structure associated with warped AdS(3) holography and warped conformal field theories (WCFTs), we construct a discrete boundary framework in which ordered products of link holonomies replace continuous gauge connections along noncontractible cycles. In this setting, boundary monodromies become the primary gauge-invariant observables characterizing the physical sectors of the theory. We show that the hyperbolic, eliliptic, and parabolic sectors naturally arise from the conjugacy classes of the discrete SL(2, R) monodromy, while the additional U(1) holonomy supplies the warped contribution to the boundary charges. Using these monodromy invariants, we derive a discrete entropy relation entirely from boundary holonomy data without relying on a smooth geometric thermal background. The resulting entropy reproduces the characteristic warped black-hole and WCFT structure in the continuum limit. We further demonstrate that the continuum warped holonomy conditions are recovered from the large-lattice limit of the ordered boundary products, establishing a direct correspondence between discrete monodromies and continuous Wilson loops. Our analysis suggests that warped gravitational thermodynamics may be understood from a fundamentally holonomy-based perspective in which boundary monodromy sectors provide an alternative organizational description of the physical states within the discrete warped framework. Keywords: warped Chern-Simons theory, boundary monodromies, holonomy sectors, Wilson loops, warped thermodynamics.

The type-specific knotting probability of an arc diagram is earlier defined by using chord diagrams of ribbon surface-links in 4D space. By modifying this notion, Non-Trivialization probability (simply NT probability) for the arc diagram is introduced and generalized to an arc system diagram. Some properties of the NT probability are shown. The method of transforming a polygonal arc in 3D space into a unique arc diagram up to isomorphisms earlier developed is generalized to a polygonal arc system in 3D space to define the NT probability.

The Maximum Closure Problem asks for a maximum-weight closed subset of a precedence-constrained set of vertices; when vertex weights depend affinely on a scalar parameter $\lambda$, as in open-pit mine scheduling, the goal becomes computing the optimal closed set for every value of $\lambda$ at once. We address this problem when the precedence graph is a directed forest. We present two algorithms (Peel-and-Contract, PaC, and its dual DPaC) that compute the full parametric solution – a canonical sequence of disjoint closure layers inducing the nested optimal closures – by iteratively aggregating vertices along precedence arcs, with time-complexity $\mathcal{O}(n^2)$ for general forests and space-complexity $\mathcal{O}(n)$; heap-based implementations (HPaC, DHPaC) have a worse time-complexity of $\mathcal{O}(n^2\log n)$, but on random instances their computing times empirically grow as $n\log n$. When the forest is restricted to in-trees or out-trees, specialized heap-based variants (HIPaC, HOPaC) achieve time-complexity $\mathcal{O}(n\log n)$. We then present a further aggregation-based algorithm (Rake-and-Compress, RaC) with time- and space-complexity $\mathcal{O}(n\log n)$ also for general forests, which matches the $\Omega(n\log n)$ lower bound for comparison-based algorithms. Computational experiments on random and structured forests with up to $10^5$ vertices characterize the practical behavior of the proposed algorithms, identify the topologies on which HPaC and RaC are preferable, and compare HPaC with the public implementation of the fully parametric pseudoflow algorithm, the state of the art for arbitrary precedence graphs, which it outperforms on forests by one to more than three orders of magnitude.

In this paper, we prove that the Mori cone of Hassett spaces whose universal family is a $\mathbf{P}^1$-bundle is generated by 1-dimensional strata. This extends the case of the symmetric GIT quotient $(\mathbf{P}^1)^n//PGL_2$ established by Bolognesi and Massarenti. Along the way, we review how these spaces are naturally isomorphic to certain GIT quotients $(\mathbf{P}^1)^n//PGL_2$, characterize them as the targets of the birational contractions of the blow-up of $\mathbf{P}^{n-3}$ at $n-1$ general points with $\mathbf{Q}$-factorial image, and deduce that their effective cone is likewise generated by strata.

A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\mathrm{sat}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Cameron and Puleo [Discrete Math. 345 (2022), 112867] showed that $\mathrm{sat}(n,K_1\vee F)\le n-1+\mathrm{sat}(n-1,F)$ for all $n>|V(F)|$. It is natural to ask when the equality holds. Most known results on this bound consider non-empty graphs $F$ without isolated vertices. We study families in which $F$ contains isolated vertices. For $F=K_{p-1}\cup qK_1$, where $p\ge4$ and $q\ge p-1$, we prove that $\mathrm{sat}(n,K_1\vee F)=(p-2)n-\binom{p-1}{2}$ for $n\ge\max\{p+q,\binom p2\}$ and determine the unique extremal graph. We also give the exact result for $q=1$, where the extremal graph has a different structure. For $F=pK_2\cup qK_1$, in the natural range $n\ge2p+q+1$, we prove that $\mathrm{sat}(n,K_1\vee F)=n+3p-4$ holds exactly when $n\ge3p-2$, and the order condition is sharp. Furthermore, we characterize all extremal graphs.

We study a natural map between projective varieties $V(\mathbf{F}_{1})$ over the field with one element and the Cuntz-Krieger algebras $O_A$. Using the $K$-theory of $O_A$, we calculate the Frobenius action and cardinality of the set $V(\mathbf{F}_{1^r})$. It is proved that the zeta function of $V(\mathbf{F}_{1})$ satisfies all Weil's Conjectures except for an analog of the Riemann hypothesis. We use the crossed product structure of $O_A$ to establish a morphism of the schemes $\operatorname{Spec} ~(\mathbf{Z})\to \operatorname{Spec} ~(\mathbf{F}_{1})\simeq \{\operatorname{pt}\}$.

Optimal transport methods have recently attracted a lot of attention in statistics. Their appeal lies in providing a geometric framework for comparing probability measures, leading to new perspectives on classical problems. A central problem in statistics is the construction of valid confidence sets as fundamental inferential tools in practice. A well-known problem is that for complex problems or relatively small samples, their asymptotic approximations often show poor performance. This suggests to apply optimal transport methods when constructing confidence sets for hard problems to improve their coverage properties. We introduce such a procedure, derive the theoretical framework studying consistency and error bounds for the coverage probability of the resulting intervals. To guarantee feasibility in practice, we propose data-driven choices for our hyper parameters. This approach extends classical quantile-based confidence intervals by leveraging optimal couplings to minimize coverage deviations. Simulations demonstrate striking performance in different estimation problems, outperforming standard methods in accuracy and robustness.

Let $H$ be a graph. A graph $G$ is $H$-saturated if $G$ is $H$-free, but adding any edge between two non-adjacent vertices of $G$ yields an $H$-copy as a subgraph. The saturation number $\mathrm{sat}(n, H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. The saturation number for the join of a vertex and a graph $F$, denoted by $K_1\vee F$, has attracted considerable attention. Cameron and Puleo [Discrete Math. 345 (2022), 112867] proved that $\mathrm{sat}(n,K_1 \vee F)\le n-1+\mathrm{sat}(n-1, F)$ for $n > |V(F)|$. A natural question is when the above equality holds. Most existing results impose conditions on $F$ and assume that $F$ has no isolated vertices. Let $K_p^-$ be the graph obtained by deleting one edge from the complete graph $K_p$. In this paper, we investigate the saturation number of $K_1\vee F$ when $F$ contains isolated vertices, and determine the exact value of $\mathrm{sat}(n, K_1\vee F)$ when $F=K^-_{3}\cup sK_1(s\ge 1)$ or $F=K^-_{p-1}\cup K_1(p\ge 5)$. In our results, $\mathrm{sat}(n,K_1 \vee F)= n-1+\mathrm{sat}(n-1, F)$ holds when $F=K^-_{3}\cup sK_1$ for any $s\ge 1$, but fails when $F=K^-_{p-1}\cup K_1$ for $p\ge 5$.

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form thus is not immediate.

Evolutionary entropy measures the temporal organization of reproductive contributions along the life cycle of an age-structured population. We develop a mathematical and empirical framework showing that, in iteroparous animal populations represented by Leslie-type demographic matrices, reproductive windows are frequently organized near the age classes selected by entropy maximization. Evolutionary entropy complements the classical net reproductive number and asymptotic growth rate: whereas these measure lifetime replacement and growth, entropy measures the temporal dispersion of the growth-adjusted reproductive distribution. Our central result is a reduction principle: under Euler–Lotka normalization, evolutionary entropy and generation time are invariant under multiplicative rescaling of survivorship and fertility on the reproductive interval. The relevant entropy is determined not by absolute survivorship, fertility, or juvenile mortality, but by the normalized post-maturity reproductive distribution. We derive explicit entropy functionals for finite and open-group Leslie models, including geometric reproductive tails. For the geometric regime, governed by we prove a sharp critical threshold separating populations with a unique finite entropy-maximizing endpoint from those whose entropy increases toward an asymptotic value in terms solely of the age at first reproduction. The theory is tested on 130 animal species. Entropy-derived predictions, computed from the demographic matrices alone, are compared with independent life-history variables. Predicted and observed reproductive medians coincide exactly for a majority of species, over 90% are predicted within three reproductive classes, and associations remain strong after phylogenetic correction. These results identify a quantitative regularity across taxa, with geometric reproductive distributions playing a central role.

The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.

We show that L2 Wasserstein spaces over Riemannian manifolds are isometrically rigid if and only if their underlying Riemannian manifolds do not admit a Euclidean de Rham factor. We further show that, unless the manifold is isometric to the real line, every isometry of the Wasserstein space is shape-preserving in the sense of Kloeckner. Finally, we demonstrate that two such Wasserstein spaces are isometric if and only if their underlying Riemannian manifolds are isometric.

In this paper, we employ the iterated integral expression of multiple polylogarithms to establish a weighted sum formula for finite multiple mixed values. As applications, we derive a one-parameter weighted sum formula for finite multiple $T$-values of arbitrary even depth, as well as several relations for depth-two, height-zero and height-one finite multiple mixed values.

We study how the effort and success probability of frontier AI systems scale with human difficulty on problems from Project Euler, an online platform of computational mathematics problems. Our dataset, from the MathArena benchmark, consists of 3840 attempts across 50 problems and 26 model configurations, with problem difficulty measured by the site's public human solve times. Motivated by a proposal of Timothy Gowers, we test a power-law relation $t_{\text{machine}} = a \cdot t_{\text{human}}^b$ between generated-token cost per successful answer and human time, and find $b < 1$ for 20 of the 25 models with usable fits, including the strongest base models; this operationalization therefore does not support an earlier prediction that machines scale worse than humans with difficulty. We also investigate whether success probability on the tested problems can be modeled by a simple exponential decay $p_{\text{success}} = e^{c t_{\text{human}}}$, predicting a linear relation between $\log p_{\text{success}}$ and $t_{\text{human}}$. Using a binning approach for data aggregation we find moderate empirical support (median bin-level $R^2 = 0.92$ across the 22 best-covered configurations) for this model. Following METR, we also fit logistic success curves and extract 50% task-length horizons $h_{50}$; the strongest configurations in our 20 April 2026 snapshot reach roughly $2.5$–$4.3$ hours on our fastest-five human baseline, with a log-linear fit through the state-of-the-art frontier giving a descriptive doubling time of about $75$~days for the SOTA $h_{50}$.

A strong version of the Beauville-Voisin conjecture asserts that for hyper-Kähler varieties, the subring of the Chow ring generated by divisors, Chern classes and Lagrangian constant cycle subvarieties should inject into cohomology. We verify this in codimension larger than two for Hilbert squares of K3 surfaces, for Fano varieties of lines in cubic fourfolds, and for double EPW sextics.

We consider a Sturm–Liouville problem in which the spectral parameter appears linearly in one of the boundary conditions. The study focuses on the root functions of the problem, including eigenfunctions and associated functions corresponding to multiple eigenvalues. By employing the characteristic function of the boundary value problem, explicit representations are obtained for the biorthogonal system and for several special associated functions that play a crucial role in the spectral analysis. These representations allow previously established criteria for the basis and minimality properties of the system of root functions to be reformulated directly in terms of the characteristic function and its derivatives at the eigenvalues. As a consequence, the investigation of particular boundary value problems becomes considerably simpler. Several illustrative examples are analyzed to demonstrate the effectiveness of the proposed approach and to show its agreement with known results in the literature.

We study the wave equation in the exterior of a strictly convex bounded domain $K \subset {\mathbb R}^d, d \geq 3,$ odd, with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $0 < \gamma(x) <1, \:\forall x \in \Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ In [10] we established that for $\gamma \equiv const$ and $K = \{x \in {\mathbb R}^3: \:|x| \leq 1\}$ the operator $G$ has no eigenvalues and we conjectured that the same result holds for every strictly convex obstacle. In this paper we prove this conjecture.

We show that the Steinberg algebra construction for ample groupoids is part of a pseudofunctor from the bicategory of ample groupoids and groupoid correspondences to the bicategory of rings with local units and nondegenerate bimodules. We define a covariance ring for diagrams in this bicategory of rings and show that it is a bicategorical limit. We compute the covariance ring for a diagram of “proper” bimodules over an Ore monoid. For diagrams coming from groupoid correspondences, we identify the covariance ring with the Steinberg algebra of its groupoid model.

We study high-dimensional differentially private (DP) covariance estimation in the operator norm, and principal component analysis (PCA), under $k$-row-column sparsity ($k$-RCS) of the covariance matrix. In the non-private setting, it is known that $\mathsf{poly}(k, \log d)$ samples suffice to solve both of these problems. However, the only comparable result known under DP (Wang et al. 2021) requires $\Omega(d)$ samples under standard parameterizations of the problem. We investigate when this curse of dimensionality is inherent for sparse covariance estimation tasks under DP. On the upper bound front, we show that a $\mathsf{poly}(k, \log d)$ sample complexity for PCA is possible under DP, if we also posit sparsity of the leading eigenvector. We complement this result with $\mathsf{poly}(d)$ lower bounds under DP for both sparse covariance estimation and PCA, establishing an exponential gap between the private and non-private variants of these problems when $k = \mathsf{polylog}(d)$. To our knowledge, no such separation has previously been demonstrated for any sparse estimation problems in private high-dimensional statistics. Our techniques are flexible enough that they imply stronger lower bounds even for the well-studied problem of standard DP PCA, without sparsity assumptions.

Accelerated mix design optimization is critical for deploying low-carbon supplementary cementitious materials (SCMs) because traditional experimental approaches require extensive testing campaigns. The authors demonstrate that Bayesian optimization (BO) can identify near-optimal blended cement formulations using an AI-driven approach with minimal experimental data. Starting with only 10 initial experiments and a 2:1 data-to-variable ratio reflecting realistic laboratory constraints, the authors optimized copper slag, limestone, Portland cement systems for 32.5N strength class with respect to 2-day compressive strength, cost, and CO2 emissions. Gaussian process surrogate models guided sequential experimentation through Expected Improvement and Upper Confidence Bound acquisition functions. Within 2-6 iterations, BO identified Pareto non-dominated solutions meeting strength requirements (greater than 10 MPa at 2 days, greater than 32.5 MPa at 28 days) while achieving CO2 emissions below 500 kg CO2/ton; consistent with industry decarbonization targets. A conservative update strategy incorporating uncertainty bounds for 28-day strength enabled rapid iteration without waiting 28 days per cycle. Comparative analysis revealed that linear kernels outperformed nonlinear alternatives in predictive accuracy, though radial basis function kernels were preferred for active learning due to superior uncertainty quantification. This work demonstrates BO as a practical decision-support tool for cement research under severe data constraints.

We present an algorithm for computing Hamiltonian cycles that are invariant under a graph automorphism acting on them as a rotation. We also present an application of this algorithm for computing the Hamiltonian compression factor of a graph, that is, the largest order of an automorphism preserving some Hamiltonian cycle and acting on it as a rotation. As an example, we compute the Hamiltonian compression factors of all cubic edge-transitive graphs on up to $10{,}000$ vertices, with the exception of two graphs, which are not Hamiltonian, and $98$ graphs (the smallest having $2304$ vertices) for which only a lower bound for the compression factor is given. As a byproduct, we obtain shortest LCF codes for each of these graphs (except for the two non-Hamiltonian ones; for the $98$ unresolved graphs, the codes obtained are the shortest among those we found).

In this article, we show that, for the symmetric group $S_n$, the Young subgroup $S_{\lambda}$, with $l(\lambda)\geq 3$, is not a code with respect to a conjugacy class of $S_n$. This provides a partial answer to \cite[Problem 4.1]{FL26}. We then provide a characterization of conjugacy classes $X_i$, such that subgroups $S_{(n-2,1,1)}$, $S_{(n-3,2,1)}$, and $Y_k$ for $k\leq 3$ are codes for $X_1 \cup X_2$. Furthermore, we describe some codes in the finite Coxeter groups of type $B_n$, $C_n$, $D_n$, and in the generalized symmetric group $C_m \wr S_n$.

A sign pattern is a matrix whose entries belong to $\{+, -, 0\}$. Let $\mathcal{P}$ be a symmetric sign pattern and $A$ a real symmetric matrix in its qualitative class. A vertex $v_j$ of the underlying graph of $\mathcal{P}$ is called a $P$-vertex if $ m_{A(j)}(0)-m_A(0)=1,$ where $A(j)$ is the principal submatrix obtained by deleting the $j$-th row and column of $A$, and $m_A(0), ~m_{A(j)}(0)$ denotes the algebraic multiplicity of the eigenvalue $0$ of $A, ~A(j)$, respectively. We say that $\mathcal{P}$ requires full $P$-vertices if every symmetric matrix in its qualitative class has all vertices as $P$-vertices. In this paper, we investigate structural conditions under which symmetric sign patterns require full $P$-vertices. We establish necessary and sufficient conditions for several classes of sign patterns to require full $P$-vertices. In particular, we prove that a tree sign pattern with a $0$-diagonal requires full $P$-vertices if and only if its underlying graph admits a perfect matching. We also derive necessary and sufficient conditions for sign patterns whose underlying graphs contain cycles but no loops to require full $P$-vertices.

In this paper, we develop an off-policy method for continuous-time reinforcement learning (CTRL), where the system dynamics are governed by an unknown stochastic differential equation (SDE) and only discrete-time trajectory data are available. A central challenge is that the classical discrete-time state–action value function loses its action-dependent signal in the continuous-time limit (Baird, 1994; Jia and Zhou, 2023; Tallec et al., 2019). On the other hand, continuous-time control provides local notions such as the instantaneous advantage function $q(s,a)$, but these typically rely on state value function $V(s)$. To obtain a formulation with a single unknown function, we introduce a new definition of the state–action value function $Q(s,a)$ in CTRL and derive its governing equation. Building on the PhiBE approximation (Zhu, 2024; Zhu et al., 2025), we propose iterative algorithms to approximate the optimal $Q$-function in both model-based and model-free settings using only discrete-time off-policy data. Under linear function approximation, we establish error bounds and explicit convergence rates for the algorithm in the popoluation level. The finite-data algorithm is evaluated numerically.

In this paper, we explore the theory of higher dimers on band graphs. First, we provide a combinatorial interpretation for the trace of the $q$-deformed higher continued fraction matrices, by showing that with respect to a $q$-weighting on edges, the trace gives the dimer partition function on the set of good higher dimers, which generalizes the notion of good perfect matchings. We also show that the set of good higher dimer covers form a distributive lattice with respect to face flips on square faces. Finally, we attempt to generalize the symmetry result on circular fence posets to the case of good higher dimers, by showing that the dimer partition on a certain family of band graphs are palindromic, in particular, through an approach fitting in the context of dimer theory.

We introduce a new class of sub-Finsler metrics, called sub-Randers metrics, obtained by adding a one-form $\beta \in \Gamma(\mathcal{D}^*)$ to a sub-Riemannian metric $a$ on a bracket-generating distribution $\mathcal{D} \subset TM$. We define a sub-Randers manifold as a triple $(M, \mathcal{D}, F)$, where $M$ is an $n$-dimensional smooth manifold and $F(v) = \sqrt{a(v,v)} + \beta(v)$, the condition $\|\beta\|_a < 1$ ensures positive definiteness and convexity. Explicit equations for sub-Randers normal geodesics are derived, and we show that normal geodesics depend on $\beta$ while abnormal geodesics are determined solely by the bracket-generating distribution $\mathcal{D}$. Furthermore, we show that Zermelo navigation on $\mathcal{D}$ naturally generates sub-Randers normal geodesics. Finally, we prove a Hopf-Rinow type theorem which guarantees the existence of minimizing geodesics despite asymmetry, generalizing classical results to the sub-Randers setting.