Papers

J. Banecki proved that every smooth projective rational variety is algebraically elliptic in the sense of Gromov. Modifying his proof we show that the same is true for every smooth complete rational variety.

Altschuler and Chewi conjecture that the sub-Gaussian Orlicz–Wasserstein shifted Rényi divergence between two isotropic Gaussians of equal covariance is attained by a deterministic shift of the mean, which would give an exact closed form for the shift budget consumed by their analysis. In this note, we show that this conjecture is true when $q=1$, where the Rényi divergence degenerates to the Kullback–Leibler divergence, and false for every $q>1$ outside the degenerate regime.

We establish sharp asymptotic rates for the two Barzilai–Borwein (BB) rules on uniformly positive quadratics and local nonlinear problems. In finite dimensions, for either fixed rule and an arbitrary positive first step, the gradient root factor is bounded by $(b_0-a_0)/(b_0+a_0)$, where $[a_0,b_0]$ is the initially active spectral interval. Hence the worst trajectory factor is $c_H=(\kappa(H)-1)/(\kappa(H)+1)$. When $H$ has at least two distinct eigenvalues, matched initialization and a balanced endpoint trajectory attain this value. Under matched initialization, the same constant is the optimal uniform-envelope threshold. For bounded, self-adjoint, uniformly positive operators on Hilbert space, scalar spectral measures yield the corresponding active-support bound and optimal matched uniform-envelope threshold, including continuous endpoint spectrum. Finally, if the gradient is strictly Fréchet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $\gamma\in(c_*,1)$, where $c_*=(\kappa(A_*)-1)/(\kappa(A_*)+1)$, is a uniform local envelope rate for either pure BB rule. Every well-defined trajectory converging to the stationary point has error and gradient root factors at most $c_*$ and objective-gap root factor at most $c_*^2$. Over the class of objectives with prescribed distinct derivative endpoints $m_*<M_*$, matched endpoint trajectories for quadratic and $C^\infty$ genuinely nonquadratic examples in $\mathbb R^2$ attain these factors.

In sampling problems, gradient-based schemes such as Langevin Monte Carlo (LMC) mix faster than non-gradient-based methods, but their applicability is limited by access to the gradient of the target log-density. In practice, gradients are often unavailable and function evaluations are noisy, e.g., stochastic simulators or black-box simulators, so we propose LMC-SPSA with noise, which approximates the gradient of the target log-density using two noisy function evaluations per iteration. We prove, under noisy gradient estimates, that LMC-SPSA converges in distribution by proving the convergence in Wasserstein distance. Furthermore, we construct a diminishing step-size schedule that still drives the Wasserstein error bound to convergence, extending convergence guarantees beyond the constant-step setting. Further, we sharpen the dominant dimension dependence of the Wasserstein error from $O(p^4)$ to $O(p^2)$ (with $p$ denoting the dimension), and support this analysis with numerical results. We show that LMC-SPSA achieves $W_2$-accuracy $\varepsilon$ with total noisy-oracle complexity of $O(p/\varepsilon^2+\delta^2p^3/\varepsilon^3)$, where $\delta$ is the paired-noise level. This improves the noise-dependent accuracy scaling relative to the ZO-LMC method of Roy et al. We further establish asymptotically vanishing Wasserstein error as the number of iterations $\to\infty$ under diminishing step-size and perturbation sequences and derive an explicit convergence rate for a balanced schedule under noisy zeroth-order feedback. Empirical experiments are conducted to verify the performance of LMC-SPSA with noise. We provide an oracle-budget-matched comparison with the ZO-LMC method, showing smaller empirical sampling errors under the same function-evaluation budget.

The stringy $E$-function is a fundamental invariant of varieties with log-terminal singularities, which was introduced by Batyrev as a singular analogue of the Hodge–Deligne polynomial. Its coefficients (up to an appropriate sign) are the stringy Hodge numbers $h_{\mathrm{str}}^{p,q}(Y)$. We completely determine when these coefficients are universally non-negative:~for each triple $(n,p,q)$, we classify when $h_{\mathrm{str}}^{p,q}(Y)\geq 0$ for every projective Gorenstein canonical $n$-fold $Y$; we answer this question both when we assume $E_{\mathrm{str}}(Y;u,v)$ is a polynomial and when we drop this condition. As a consequence, we settle Batyrev's non-negativity conjecture in all dimensions:~it holds in dimensions at most $4$ and fails in every dimension at least $5$.

Quadratic optimization becomes hard as soon as either the matrix in the quadratic form has an unfavorable curvature or the feasible set is discrete, combinatorial, or otherwise nonconvex. A complementary phenomenon is also well known in the signal-processing and optimization communities: when the matrix in the quadratic form has small rank, some hard-looking quadratic programs admit exact polynomial-time algorithms for fixed rank. We study the common positive-semidefinite geometry behind this phenomenon. If $Q=BB^\top$ is positive semidefinite, the objective depends on $x$ only through the rank-space shadow $y=B^\top x$. Every optimal shadow $y^*$ uniquely maximizes the linear functional defined by its own direction and satisfies a quantitative quadratic margin. Thus nonlinear optimality collapses to a low-dimensional, self-generated linear exposure direction. We call this the rank-collapse principle. The principle alone does not imply a finite candidate set: efficient exact optimization additionally depends on the projected or active geometry of the feasible family. We organize this distinction through projected-shadow scattering and active-structure collapse, relate it explicitly to established zonotope, convex-combinatorial, edge-skeleton, projected-normal-fan, and fixed-rank sparse-PCA methods, and derive tie-safe consequences for binary and finite-phase vectors, cardinality constraints, matroid bases, and sparse PCA. We also give directional-stability and approximately low-rank certificates, together with reproducible experiments. The paper's contribution is a unified, careful framework and a set of quantitative consequences, rather than a claim to originate the known fixed-rank tractability results that motivate it.

We present an analogue of the classical Gittins index for a one-armed decision problem with a continuous spectrum of payoffs. The model assumes that the decision-maker observes independent realizations of a random variable and, at each step, decides whether to accept the current opportunity or continue observing. We show that the optimal strategy takes the form of a threshold rule, while the corresponding reservation index is determined by a one-dimensional fixed-point equation with a direct decision-theoretic interpretation. The model is illustrated with an example of bookmaker betting related to horse racing and the Kelly criterion. This perspective allows the proposed index to be viewed as a threshold of informational advantage. This, in turn, points to potential applications in optimal stopping problems and decision-making under uncertainty.

A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator $L$, coarse channel $C$, and stochastic section $R$ satisfying $CR=I$, define $P=RC$, $Q=I-P$, and $K=CLR$. The exact coarse propagator $T_t=C\mathrm{e}^{tL}R$ obeys a Volterra equation with memory kernel $\mathcal{M}(t)=CLQ\,\mathrm{e}^{tQLQ}\,QLR$. Every memory jet $\mathcal{M}^{(k)}(0)$ is recursively recoverable from the short-time derivatives of $T_t$ alone. The rank of the block Hankel operator $\mathbb{H}_{\infty}=(\mathcal{M}^{(i+j)}(0))_{i,j\geq 0}$ is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho–Kalman factorization and is unique up to similarity. If $d=\min\{k:\mathcal{M}^{(k)}(0)\neq 0\}$, then every smooth classical $f$-divergence generically scales as $t^{2d+4}$. Exact Markov examples realize exponents $4,6,8$ and closure ranks $1,2,3$. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.

We give a new proof by flow decomposition of the known Grigor'yan type integral criterion for stochastic completeness of weighted graphs. The main ingredient is a finite-domain estimate relating the mass of a killed $1$-resolvent and the current reaching the boundary to relative capacities of intrinsic balls. This estimate yields logarithmic criteria in terms of capacity to infinity and relative capacity, and recovers the volume criterion by an intrinsic cutoff, without passing to a metric graph or refining the graph.

We prove that for almost every $\alpha\in\mathbb{R}$, the \textit{fully inhomogeneous $p$-adic Littlewood Conjecture} holds; namely, $$\forall\, \delta\in\mathbb{R},\;\forall\, \kappa\in\mathbb{Z}_p, \quad \liminf_{|q|\rightarrow+\infty}\, q\,\langle q\alpha+\delta\rangle\,|q+\kappa|_p=0.$$ Here, $\langle\cdot\rangle$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic norm. Moreover, we prove this conjecture for every quadratic irrational $\alpha$. This gives an affirmative answer to the $p$-adic version of a question posed by Cassels in [Cas59].

We show that given a Noetherian and $F$-finite scheme, the fixed point functor on Cartier crystals is fully faithful to the category of sheaves on the flat and relatively perfect site defined by Kato, and its essential image consists of the subcategory of dual-constructible sheaves, which we introduce.

We define a discrete Morse function on the open regular cell complex $C_2(\Gamma)$, arising from the $2$-particle unordered configuration space of a graph $\Gamma$. We compute the homology of the associated Morse complex. Using an isomorphism established by Knudson and Scoville between the Borel-Moore homology of $C_2(\Gamma)$ and the homology of the Morse complex, we conclude that the Borel-Moore homology groups of $C_2(\Gamma)$ are completely determined by the first Betti number of $\Gamma$.

A subset $S \subseteq V(G)$ is called a dual-server dominating set if there exists a partition $\pi_S=\{R,B\}$ of $S$ such that every vertex in $V(G)\setminus S$ has at least one neighbour in $R$ and at least one neighbour in $B.$ The minimum cardinality of a DS-dominating set of $G$ is called the dual-server domination number, denoted by $\gamma_{ds}(G)$. In this paper, we investigate the dual-server domination number of several graph operations. We establish exact values for the dual-server domination number of the join, Cartesian product, and corona of graphs. We also determine the dual-server domination number of the splitting graphs of paths, cycles, and complete bipartite graphs. These results extend the study of dual-server domination and provide further insight into its behaviour under graph operations.

It is proved that for every sufficiently large odd integer $n$, the equation $$ n=p_1^2+p_2^2+p_3^3+p_4^4+p_5^5+p_6^6+p_7^8 $$ is solvable in primes. This refines the work of Cai and Mu in 2015.

This paper considers combinatorial and sampling approaches to deriving a priority vector from an incomplete pairwise comparison matrix. It is shown that the informationally significant combinations of matrix elements are in one-to-one correspondence with spanning trees of the graph induced by the matrix. The combinatorial approach is based on the enumeration of all spanning trees and the componentwise aggregation of the priority vectors derived from them. To reduce computational complexity, a sampling Monte Carlo approach with a sequential stopping criterion is proposed, ensuring a prescribed accuracy for estimating the components of the resulting vector.

We prove a derivative-free analytic linear spectral statistics theorem for entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The transform is required to be centered, variance-normalized, and admissible under the small translations generated by the spike: its shifted mean and second moment have first- and second-order profiles, while its centered shifted fourth cumulants and fourth tails are stable. At every microscopic shift we construct an explicit uniformly bounded three-point variable matching the first four centered moments of the target entry exactly. A generalized-Wigner LSS theorem applies to the resulting bounded triangular array, and a global Fourier–Duhamel derivative estimate transfers the analytic statistic back to the rough transform without a common truncation, coefficient-stability assumption, or local law. The order-one mean consists of the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. The centered covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter $\kappa_4^{f,\nu}$. We distinguish the bulk contour statistic from the full trace in the supercritical regime and show that the separated outlier adds exactly $\varphi(\theta+\theta^{-1})$ to the full-trace centering. As a self-contained consequence, Gaussian noise with $f\in L^{4+\epsilon}(\gamma)$ satisfies the theorem without differentiability of $f$; the resulting bulk and full-trace corollary has explicit Hermite coefficients and includes every centered, variance-normalized polynomial-growth transform. Concrete likelihood-ratio, smooth-transform, bounded rough-transform, and atomic criteria are provided, together with obstructions showing that bare $L^4(\nu)$ is insufficient.

For any $n$, there exists finite simple graphs $G$ and $H$ such that $\chi(G),\chi(H) > n$ and $\chi(G \times H) = 4$.

The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin's principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.

We propose a finite difference discrete fracture model for single-phase flow in fractured porous media. The method is derived from a unified energy principle and is implemented on Cartesian grids that need not conform to the fractures. It introduces no additional degrees of freedom, modifies the underlying finite difference scheme only locally, handles both highly conductive fractures and low-permeability barriers, and naturally preserves symmetry and positive definiteness of the discrete system. Barrier-induced pressure jumps and fracture-induced fluxes can be recovered through inexpensive local post-processing, yielding a sharper representation of the interface effects. Numerical experiments based on manufactured solutions and published $2$D and $3$D benchmark problems demonstrate the accuracy, effectiveness, and flexibility of the method for isolated fractures, complex networks, and anisotropic porous media.

We disprove the conjectural converse to Kato's positivity criterion for commutators of functions of the canonical position and momentum operators $Q$ and $P$ by showing that the operator \[ i\,[\,\arctan(P),\,\arctan(Q)\,] \] is nonnegative and nonzero.

We propose a heuristic for the growth rate of the average class number of curves in $\mathcal M_g(\mathbb F_q)$ as $g \to \infty$. We show that this heuristic implies that the zeros of the zeta functions of such curves become equidistributed on the circle as $g\to\infty$. Our heuristic is based on the assumption that the stable cohomology of the universal Jacobian dominates a point count using the trace formula, following the method of Achter et al.

We establish a topological analogue of Ismagilov's theorem concerning the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold \(M\) with volume form \(\Omega\), we consider the group \(\mathbb{G}^{\Omega}(M)\) of homeomorphisms obtained as uniform limits of smooth volume-preserving isotopies. We show that its first continuous cohomology with values in the Banach space of zero-mean continuous functions is isomorphic to the first de Rham cohomology of \(M\). The proof develops a theory of transport for volume-preserving isotopies, producing a topological volume flux homomorphism and its associated transport cocycle. Under the additional hypothesis that \((M,\Omega)\) satisfies the local \(C^0\)-generation property, and assuming the Müller-Sikorav approximation theorem (known for \(n\neq 4\)), the isomorphism extends to the full identity component \(\operatorname{Homeo}_0^\Omega(M)\) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.

In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller–Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya–Babuška–Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller–Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let $\mathcal B$ be a family of subsets containing $\emptyset$ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in $\mathcal B$, then every coordinate set of the smallest solution lies in $\mathcal B$; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra—where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes—then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For \[ V=U\oplus\bigoplus_{i=1}^{k}W_i,\qquad \dim W_i=1,\qquad \Lambda=\bigcup_{i=1}^{k}(U+W_i), \] we prove that the lamination hull $G_{\Lambda}^{(\infty)}(S)$ of every finite $S\subset V$ is semi-algebraic and effectively computable by a quantifier-free formula over the reals.

For every integer $n \geq 1$, we compute the cobordism groups of dihedral $n$-fold branched covers of $S^3$ with oriented and non-oriented branching sets. We show that the groups are cyclic, generated by the $n$-fold connected dihedral covers of $(2,n)$-torus links. The isomorphism types of the groups are detected using explicit cobordism invariants defined in terms of the Seifert forms on the branching sets, generalizing Cappell-Shaneson characteristic knots associated to dihedral covers.

We develop a harmonic-coordinate approach to the identity component $G_\omega(\Sigma_g)$ of the symplectomorphism group of a closed oriented surface of genus $g\ge2$. Using an intrinsic decomposition of the Abel-Jacobi displacement into a global flux part and a zero-average harmonic fluctuation, we introduce the harmonic flux norm and prove its non-degeneracy on the full identity component without Floer theory. We also show the norm is continuous in the $C^0$-topology, deduce that $\mathrm{Ham}(\Sigma_g,\omega)$ is $C^0$-closed inside $G_\omega(\Sigma_g)$, and produce a locally injective harmonic-coordinate chart near the identity. Along the way we derive first-order expansions for the norm, quantitative fixed-point obstructions, and propose a finite-dimensional persistence invariant (the harmonic barcode) associated to the harmonic displacement filtration.

In this paper, we propose a multistage decoding framework that leverages internal information produced by an underlying message-passing decoder. The proposed method targets the failure dynamics caused by both classical trapping sets and degenerate errors supported on symmetric stabilizers, which are among the primary limitations of iterative decoding for QLDPC codes. To identify unreliable variable nodes, we introduce a heuristic metric that combines several dynamical features of the decoder, including variable-node log likelihood reliabilities, hard-decision oscillations, the number of adjacent unsatisfied checks, and the soft information contributed by unsatisfied checks. Based on this ranking metric, the decoder performs guided rewinds by selectively forcing the initial log likelihood ratio values of the most suspicious variable nodes and restarting the message-passing decoder under the corresponding forced configuration. To manage the combinatorial growth of candidate configurations, the search is formulated within a beam- search framework with controlled beam width. In addition, we introduce a pruning metric based on the combination of the residual syndrome weight and a posteriori reliability of the decoder output, thereby retaining only the most promising search paths. Logical error rate results demonstrate that the proposed decoder significantly outperforms the normalized min- sum decoder and achieves competitive performance with belief propagation enhanced by order-10 ordered statistics decoding.

We study when embeddings lift from structures to their ultrapowers, and when an ultrapower embeds into its direct power. We offer a refinement of Blass's theorem on the Rudin-Keisler order and introduce cardinal invariants $\tau_\mathcal{A}$ which imply the embeddability $\mathcal{A}^I / \mathcal{U} \hookrightarrow \mathcal{A}^I$.

In Penrose's nonlinear graviton construction, a self-dual spacetime arises as the moduli space of holomorphic twistor lines in curved twistor space, whose explicit parametrization is concretely given by the incidence relation. We demonstrate a concrete local construction of the incidence relations for self-dual black hole spacetimes in Kerr-Schild coordinates: Eguchi-Hanson, self-dual Taub-NUT, and self-dual Plebański-Demiański. This is facilitated by implementing the Dunajski-Mason recursion in the framework of Plebański's second heavenly equation, for which we explicitly find the Plebański scalar. Our results describe closed-form formulae. Notably, for the self-dual Taub-NUT solution, the exact incidence relation exhibits linear dependence on the gravitational coupling. The construction of zero-rest-mass fields on self-dual black hole backgrounds is briefly sketched as an application.

We develop an algebraic framework for central configurations of the $n$-body problem with homogeneous potentials, grounded in the exterior algebra of the configuration space and the normalized shifted Brehm–Wintner–Conley (BWC) matrix $S$. Relating the kernel of $S$ to the Plücker coordinates of the configuration, we generalize the determinantal equations obtained by Williams (1938) for the planar five-body problem to central configurations of any dimension and any number of bodies, and derive the Dziobek–Williams equations $\det(S_I^J)=\kappa\, z_I z_J$, which exhibit the compound matrix $S^{(t)}$ as a rank-one matrix. Introducing the Brehm–Wintner–Conley dimension $\operatorname{bwc}(x)=\operatorname{rank}(S)$ (an integer invariant that stratifies central configurations and measures vertical degeneracy), together with the Plücker–BWC coordinates attached to it, we prove that each stratum of central configurations with fixed dimension and Brehm–Wintner–Conley dimension admits a base-point-free map into a Veronese variety, factoring through a Grassmannian invariant; on the Dziobek stratum this recovers the Dziobek–Veronese geometry previously introduced by the author. We further describe universal determinantal relations satisfied by the minors of $S$ and expand explicitly the resulting systems for the planar five- and six-body problems. We also interpret the mass-weighted entries of $S$ as an equilibrium stress: under the MacMillan–Bartky sign condition a strictly convex central configuration underlies a cable–strut tensegrity, and a theorem of Connelly then forces $S$ to be negative semidefinite with nullity three, so that vertical degeneracy cannot occur in this regime.