Papers

We study the Wasserstein gradient flow generated by a family of extended generalized variance functionals, defined as the expected squared $n$-dimensional volume of a simplex, which includes the classical variance-type interaction and generalized variance as special cases. The key structural observation is that this functional depends only on the covariance matrix. Consequently, the Wasserstein gradient flow reduces to a finite-dimensional system for the eigenvalues of the covariance matrix, and the full measure-valued solution can be recovered through an explicit linear pushforward representation. Using this representation, we establish global well-posedness for arbitrary initial data in $\mathcal P_2(\mathbb R^d)$ without assuming compact support. We also study the long-time behavior of the flow. For every initial datum in $\mathcal P_2(\mathbb R^d)$, the solution converges to a limiting equilibrium measure whose covariance has rank strictly less than $n$. Moreover, we obtain sharp convergence rates in all spectral regimes: exponential in the non-degenerate case and algebraic with the optimal exponent in the degenerate case.

We study the resurgent structure of short–time heat kernel asymptotics from the viewpoint of Picard–Lefschetz theory. For a real analytic Riemannian manifold, we show the heat kernel admits a 1– Gevrey small–time expansion whose Borel transform detects complex–geometric data beyond the real geodesic sector. We formulate an infinite–dimensional Picard–Lefschetz problem of Morse–Floer type for the holomorphic energy functional on the complexified path space, and propose a heat–kernel analogue of the Picard–Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat–kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. We perform a confirming test of this proposal on the hyperbolic plane $\mathbb{H}_{\R}^2$.

We prove a new combinatorial identity for the determinant of the resistance matrix of a finite graph, which involves counts of spanning trees and forests. This generalizes a result of Graham and Pollak on distances matrices of trees. We make use of Bapat's expression of the resistance matrix determinant as a linear algebraic quantity.

In this paper, we propose a modified projected Walk on Spheres method (MPWoS) for screened Poisson equations on embedded manifolds. The method employs local extensions together with the Green representation in local Euclidean balls, coupled with a closest-point projection that maps the boundary samples back to the manifold. This formulation yields a meshfree and highly parallelizable stochastic recursion in the ambient Euclidean space, rather than a direct discretization of the Laplace-Beltrami operator on the manifold. The proposed approach can be viewed as a high-dimensional extension and modification of the projected Walk on Spheres method introduced for surface PDEs in [Sugimoto et al., SIGGRAPH Asia 2024 Conference Papers, pp. 1-10], with three main distinctions: a compensation term that corrects the discrepancy between the ambient Laplacian applied to the closest-point extension and the intrinsic Laplace-Beltrami operator on the manifold, an adaptive radius strategy determined by local geometric and boundary information, and a rigorous error analysis for the proposed algorithm. Under assumptions on the geometric projection and the prescribed compensation accuracy, we establish mean-square error estimates for the proposed Monte Carlo method in both the boundary and closed-manifold settings. Extensive numerical examples on parametrized, implicit, high-dimensional (up to 1000 dimensions), and point-cloud manifolds are presented to illustrate the convergence and efficiency of the proposed method across different geometries.

Let A be a self-injective algebra over an algebraically closed field. We study the relationship between simple-minded systems and weakly simple-minded systems in A-stmod. We present a necessary and sufficient condition for an orthogonal system to be a simple-minded system over domestic Brauer graph algebras. As a byproduct, we construct a class of simple-minded systems over 2-domestic Brauer graph algebras.

Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

Prudence is a stability property of risk functionals recently introduced by Wang and Zitikis and subsequently studied by Amarante and Liebrich. In this paper, we first establish general relationships between prudence and other stability properties, showing, in particular, that weak prudence and prudence coincide for a broad class of convex, law-invariant functionals. We then prove that prudence is preserved by cash-additive hulls of star-shaped functionals under a simple asymptotic condition, and by inf-convolutions of convex, cash-additive, law-invariant prudent functionals. Our results provide general methods for constructing prudent risk measures from existing prudent functionals.

To provide partition-theoretic interpretations to Watson's the third-order mock theta function $\omega(q)$, Andrews defined the odd Durfee symbols and odd ranks. Motivated by Andrews' work, arithmetic properties of odd ranks are widely studied recently. In this paper, we obtain transformation formulas of the odd rank generating functions, which are used to construct families of weak Maass forms and weakly modular forms. As an application,we provide explicit identities for odd ranks modulo 5, analogous to Ramanujan's classical partition identities.

We demonstrate several common techniques for proving linear recurrences from counting Schreier-type sets. These techniques include formula-based arguments, bijective proofs, mathematical induction, the inclusion-exclusion principle, and the characteristic polynomial method. As new contributions, we examine symmetric maximal Schreier sets, Schreier sets that contain a prescribed integer, and Schreier sets that avoid integers belonging to a fixed arithmetic progression. Along the way, we employ useful techniques for identifying meaningful patterns in data and establishing technical identities. The results presented here, together with the diverse proof techniques employed, are expected to serve as a valuable resource for undergraduate researchers interested in this area.

For an odd prime $p$ and any prime $\ell\neq p$, we study finite directed graphs arising from the set of all equivalence classes of Moret-Bailly families of abelian surfaces in characteristic $p$ together with relative $(\ell,\ell)$-isogenies. We relate these graphs to the space of algebraic modular forms on an inner form of $GSp_4/\mathbb{Q}$ that is compact modulo its center and, using the Jacquet-Langlands correspondence, estimate the eigenvalues of their adjacency matrices. We further investigate a Sarnak-Xue type theorem in this setting, providing a first step toward the study of cut-off phenomena for these graphs.

The performance of traditional CP-OFDM degrades severely in doubly-spread wireless channels due to inter-carrier interference (ICI). In this paper, we propose DD domain sensing based CP-OFDM where we transmit a Zadoff-Chu (ZC) pilot signal overlaid on CP-OFDM data carriers. At the receiver, DD domain signal processing is used to acquire the effective DD domain channel filter which is stationary in the DD domain. From this DD domain estimate, we derive the complete frequency domain (FD) input-output (I/O) relation between CP-OFDM carriers, acquiring which is otherwise difficult with traditional time-frequency signal processing. Using this FD I/O relation, we estimate the received FD pilot signal which is then canceled from the received FD signal, resulting in a data-only signal. Joint detection of all CP-OFDM data carriers from this data-only signal equalizes the effect of ICI. Numerical simulations of the standardized 3GPP TDL-C channel shows that in high mobility scenarios, the proposed DD domain sensing based CP-OFDM achieves significantly better spectral efficiency when compared to that achieved by traditional CP-OFDM.

We develop a novel distributional Difference-in-Differences (DiD) framework to capture treatment heterogeneity across outcome distributions. By leveraging optimal transport, we use the control group to estimate the untreated distributional drift from the pre- to post-treatment period and apply it to the treated group's pre-treatment baseline, constructing a counterfactual distribution under the assumption of no treatment effect. We frame the null hypothesis as a distributional equality between the transported counterfactual distribution and the observed treated post-treatment distribution, and test it using a maximum mean discrepancy statistic in a reproducing kernel Hilbert space (RKHS). The resulting nonparametric omnibus test is sensitive to changes in location, scale, shape, and tail behavior. Under the null, we derive the asymptotic Gaussian quadratic-form limit of the test statistic, while under local alternatives, we provide a unified characterization of power that establishes its Pitman local power and moderate-deviation consistency. Our theory reveals how detectability is shaped by the interaction between transport-induced drift and RKHS geometry. Simulations and an application to the Card–Krueger minimum-wage data demonstrate that the proposed method identifies key distributional treatment effects missed by classical mean-based DiD.

A knot P in the three-sphere is freely 2-periodic if it is preserved setwise by a free order-two action. There is a natural projective quotient knot associated to P. We establish a rank inequality between the knot Floer homologies of P and its quotient as a consequence of Large's generalization of Seidel–Smith's localization spectral sequence associated to order 2 actions in Lagrangian Floer homology. As a corollary we obtain an inequality between the Seifert genus of P and the rational Seifert genus of its quotient. We also implement a program which computes the E2 page of this spectral sequence using a modification of Baldwin–Gillam's grid homology calculator.

Our aim is to use tropical differential algebra to systematically build a combinatorial basis for the study of the set of (formal) power series solutions to nonlinear algebraic ordinary differential equations (over $\mathbb{C}$) expanded around the point $t_0=0\in\mathbb{C}$, which may also be effectively computed using standard tropical algebra. This paper is divided into two parts. First, given an ordinary tropical differential equation in one differential variable $P=P(y)$ of (differential) order $k$, we study the sets $Sol_{\mathbb{B}[\![t^{\Gamma}]\!],k}(P)\supset \mu(Sol_{\mathbb{B}[\![t^{\Gamma}]\!]}(P)\!)$ of tropical elementary $k$-solutions and minimal tropical solutions, respectively; we show that these two sets bear many similarities. We do this for tropical solutions $y=\varphi(t)\in \mathbb{B}[\![t^{\Gamma}]\!]$ (with coefficients in the boolean semifield $\mathbb{B}$) of $P$ having support in different relevant submonoids $\Gamma$ of $(\mathbb{R},+,0)$. Then, given an ordinary algebraic differential equation $\mathfrak{P}$ (with meromorphic coefficients in one differential variable $\mathfrak{P}=\mathfrak{P}(y)$ and of differential order $k$), we consider the set $S(\mathfrak{P},0):=ord_t(Sol_{\mathbb{C}[\![t^{\mathbb{R}}]\!]}(\mathfrak{P})\!)\subset\mathbb{R}$ of $t$-adic orders of formal Hahn solutions of $\mathfrak{P}$, which is an algebraic object that gives information about the nature of the germs of solutions of $\mathfrak{P}$ at the point $t_0=0\in\mathbb{C}$. We show that this set is contained in the set of $t$-adic orders of Hahn elementary $k$-solutions of its tropicalization $P=trop(\mathfrak{P})$, this is $S(\mathfrak{P},0)\subset ord_t(Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P))$. In most cases, the set $Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P)$ is a finite family of univariate tropical polynomials.

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative \(L^2\) error \(O(10^{-3})\) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses \(\mathrm{det} F\) well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase – penalizing the operator against the discrete energy, with no additional solution data – shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4\(\times\) wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

We study exact semidefinite programming (SDP) relaxation for a given sparse quadratically constrained quadratic program (QCQP). The SDP relaxation is exact if, whenever it has an optimal solution, it admits a rank-at-most-one optimal solution that corresponds to an optimal solution of the QCQP. Using the maximal cliques of a chordal extension of the aggregate sparsity pattern graph of the data matrices, we formulate the SDP relaxation in terms of clique-wise matrix variables and develop a local-to-global framework for certifying exactness. For each clique-wise matrix variable, we introduce a local sub-SDP with two parameters: a local right-hand-side vector and a consistency matrix specifying the values of entries shared by overlapping clique-wise matrix variables. In the main theorem, these parameters are determined by an optimal solution of the global clique-wise SDP. The theorem shows that if the resulting local sub-SDPs are exact, then the original SDP relaxation is exact. Under the additional assumption that any two distinct cliques intersect in at most one node, we present three classes of local QCQPs that can be incorporated into this framework: convex local QCQPs, local QCQPs characterized by sign-pattern conditions, and separable local QCQPs with a limited number of constraints. Examples illustrate how these different local QCQP classes can be combined in sparse QCQPs.

Let $\mathcal{O}_S$ be the ring of $S$-integers of a global field $K$ of any characteristic, where $S$ is a finite set of valuations of $K$ (and $S$ contains all of the archimedean valuations if the characteristic is zero). We prove that if $|S| \ge 2$, then every unimodular $(2 \times 2)$-matrix over $\mathcal{O}_S$ is a product of $\leq 7$ elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of $SL_2(\mathcal{O}_S)$ started over 50 years ago.

Let $g$ be a smooth Riemannian metric and $H$ a positive function on $\mathbb{S}^2$. We prove that the Bartnik mass of the triple $(\mathbb{S}^2,g,H)$ is bounded above by $\sqrt{|\mathbb{S}^2|_g /16\pi}$ provided the first eigenvalue $\lambda_1(g)$ of the operator $(-\Delta_g+K_g)$ is non-negative. We prove a similar result assuming $H$ is only nonnegative and $\lambda_1(g)>0$. These eigenvalue conditions, in particular, impose no lower bound on $K_g$ (even under an area constraint \cite{Mantoulidis_2015} \S3) and thereby extend previous results which assume $K_g\geq 0$.

We develop a low-frequency moment–Hankel rank method for identifying the number of time-independent point sources in the heat equation from boundary flux data. The method is formulated in the unit disk, where the Laplace-transformed and normalized boundary flux admits an explicit Fourier moment representation. By taking the low-frequency limit, we obtain a finite exponential-sum moment sequence in which the nodes encode the source locations, the weights encode the source strengths, and the number of terms equals the number of point sources. The associated Hankel matrix admits a Vandermonde factorization, and its rank is exactly the source number under the natural assumptions that the source locations are distinct and the source strengths are nonzero. We also analyze the effect of discrete and noisy boundary data. A uniform moment perturbation bound is propagated to the empirical Hankel matrix, and Weyl's singular-value perturbation inequality yields a sufficient condition for stable numerical rank recovery in terms of the smallest nonzero singular value of the ideal Hankel matrix. Numerical experiments confirm the exact rank pattern in the noiseless case, validate the stability threshold under moment noise, and illustrate the loss of resolution for close or weak sources. After the source number is identified, the same moment sequence can be used for location and strength recovery through an annihilating-polynomial and Vandermonde reconstruction procedure.

We investigate universality properties of continuous, positive-definite invariant kernels on Riemannian symmetric spaces, providing a unified harmonic-analytic characterization across compact and non-compact settings. In the compact symmetric case, we prove that a continuous, positive-definite invariant kernel is $C$-universal if and only if all of its spherical coefficients are strictly positive, a sharpening of classical Bochner-type results. This characterization is extended to compact homogeneous spaces, where universality is shown to be equivalent to the strict positive-definiteness of coefficient matrices arising from a representation-theoretic expansion. In contrast, for non-compact symmetric spaces, we establish that any continuous, positive-definite invariant kernel that is $C_0$ and integrable is automatically $C_0$-universal. Our analysis essentially relies on spectral decompositions using spherical functions and group Fourier transforms, in order to provide alternative, harmonic-analytic formulations of universality conditions. Several examples (including kernels on spheres and compact Lie groups, as well as integrable kernels on hyperbolic spaces and symmetric cones) both illustrate the theory and demonstrate practical criteria for constructing universal kernels.

We study ordinary kernel ridge regression over full Hölder-Zygmund balls, measuring errors in Sobolev norms and in pointwise and uniform losses for mixed derivatives. For targets of smoothness $s$, every RKHS norm equivalent to $H^{s+d/2}$ gives the optimal $H^r$ risk for $0\le r<s$; integer choices of $r$ include integrated squared derivative losses. Pointwise and uniform guarantees require additional kernel structure. A localized wavelet kernel attains the sharp derivative rates with the same regularization orders as for function estimation, whereas equivalent changes of the RKHS norm can destroy the population bias rates. For the ordinary Gaussian kernel with a shrinking bandwidth, we prove a uniform derivative risk bound with the optimal polynomial rate and an explicit, unoptimized logarithmic factor. The results allow bounded measurable design densities bounded away from zero and conditionally sub-Gaussian noise.

Statistical dependence among uncertain input parameters in stochastic energy system optimization models is often ignored, even though this can substantially bias outcomes. To address this gap, we are developing a comprehensive framework for characterizing, modelling, and benchmarking statistical dependence. In this work, we present a copula-based workflow to identify, characterize, and model linear and monotonic correlation structures between input parameters, representing the first development step towards this framework. We demonstrate our workflow using solar generation, day-ahead electricity prices, and electricity demand data in Austria between 2019 and 2025. Our results show substantial linear and monotonic dependence between these variables, and that this dependence is well captured by the copula-based approach. Finally, building on this scalable foundation, we highlight key levers for next steps towards the full framework, including time-dependent and higher-dimensional dependence modelling.

We develop a Banach-space framework for first-order time-dependent mean-field games with local couplings, using monotone operator theory and low-order $p$-Laplacian regularization to avoid high-order elliptic smoothing. Under monotonicity and power-growth assumptions, together with either a Lagrangian lower bound or strict positivity of the initial density, we prove existence of weak variational-inequality solutions by Minty's method. The constructed solutions satisfy uniform $L^\beta$ estimates on the density, $L^\alpha$ estimates on the spatial gradient of the value function, and space-time shift estimates sufficient to identify the limiting PDE system. We prove that any variational-inequality solution satisfying these bounds, regardless of how it is obtained, is a MFG solution satisfying the Hamilton–Jacobi and transport equations in the $BV$ sense. This separates the construction of VI-solutions from the verification of the PDE system, a feature not directly available in the existing stationary Banach-space framework. Finally, for each fixed density $m$, we establish a maximal value function among Hamilton–Jacobi subsolutions; every MFG value function coincides with this maximal representative on $\{m>0\}$ and initially on $\{m_0>0\}$. Under semi-strict monotonicity, the density $m$ itself is unique. Our results apply to non-separable Hamiltonians with power growth and impose no dimension restrictions.

We investigate the dynamics of classical solutions to the dust fluid model under spherical symmetry in Schwarzschild spacetime. According to whether the solution will persist globally or develop a finite-time singularity, a precise classification of initial data is provided. Moreover, a detailed analysis on the exact blowup profile near the singularity is given.

We introduce and study quantum phase operators associated with the Quantum Harmonic Oscillator (QHO). We show that these operators are trace-class perturbations of the Susskind-Glogower operators and examine their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.

This paper studies finite closure operators on overlapping finite universes and gives an exact local-to-global obstruction criterion for conservative globalization. Given a finite family of local closure systems, its atlas-generated closure is obtained by repeatedly applying the local closure operators to the parts visible in each chart. This closure is the least global closure operator extending all chart closures. A chart-visible obstruction is a consequence produced by this global propagation that lies inside a chart but is not validated by that chart's own closure operator. The main theorem proves that a finite closure atlas has a global conservative realization exactly when no such obstruction occurs; in that case the atlas-generated closure itself is the conservative realization. The obstruction condition is finite and directly computable. The paper also records the indexed representation layer motivating the terminology. For a finite closure system, an indexed truth space selects closed theories as contexts and represents each element by the region of selected closed theories containing it. Closure consequence is always sound for region inclusion, and the full indexed space of all closed theories recovers the original closure consequence exactly; reduced indexed spaces can therefore create spurious region consequences by deleting separating closed theories. A formal opposite gives a four-region membership decomposition - only one, only the other, both, and neither - unless additional separation assumptions are imposed. Finally, overlap-compatible local closed theories glue by canonical union under the atlas-generated closure. The framework is finite, structural, and closure-theoretic; the logical terminology is used only as an interpretation of the underlying closure data.

We compare repetition-based ultra-low-rate information reconciliation with dedicated ultra-low-rate codes for CV-QKD. Repetition coding offers a favorable performance-complexity trade-off, incurring only a moderate error-rate penalty while reducing decoding complexity by $2\times$, making it attractive for implementation-constrained systems.

We present a finite semigroup whose pseudovariety has membership problem hard for the class Difference P

We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,\alpha}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.

This paper proves a finite-chain counting theorem for Caffarelli–Kohn–Nirenberg bad scales of suitable weak solutions to the three-dimensional incompressible Navier–Stokes equations. The main standard-PDE result bounds the weighted size of a finite set of CKN-bad scales by nonnegative channel costs consisting of vertical one-component concentration, annular leakage, pressure-tail terms, and pressure–flux–energy residuals. The proved closing mechanism is qualitative one-component compactness under a full local critical bound: small vertical component forces CKN smallness at a smaller radius. The paper then gives a canonical detector realization of the same finite-window counting philosophy. The original abstract detector is not identified with standard PDE channels; instead, a new amended canonical detector is defined using energy, flux, pressure-tail, retained low-pressure-mode, and finite-dimensional residual coordinates. For this amended detector, we prove upper realization, lower audit, CKN extraction, and finite-chain bad-scale counting.