Papers

Graphs, maps on surfaces, and abstract polytopes are related combinatorial structures that tend to be studied by different communities using their own tools and databases. Maniplexes provide a unifying framework that captures all of them. A single database built around maniplexes would help researchers recognize shared structures and translate results across fields. Here we present a compact, interoperable format for storing maniplexes as edge-labeled graphs, designed with such a database in mind. As a first step, we connect two existing datasets of regular 4-maniplexes to the House of Graphs and to Potočnik's tetravalent graph censuses, using canonical forms of their flag graphs, 1-skeleton graphs, and 1-coskeleton graphs.

Data-driven dynamics often asks how to linearize a nonlinear system. We ask instead: which observables should be advanced, and where should their futures live? This leads to Petrov Regression Of Nonlinear Evolution (PRONE), a Petrov–Galerkin regression framework based on $ \Psi(\mathbf{X})K \approx \Phi(\mathbf{Y}), $ with distinct trial and test dictionaries. In this form, DMD, EDMD, SINDy, Koopman regression, sparse regression, and low-rank regression become variants of one construction: different dictionaries, weights, and constraints. We keep the linear algebra of Koopman learning, but drop the artificial requirement that a finite model map a dictionary into itself. With this asymmetry, eigenmodes are no longer the right objects. Instead, we use singular modes, which identify the observable combinations captured by the data, their projected futures, and the strength of the coupling between the two spaces. We identify the limiting projected operator and prove $L^2$ convergence of the resulting nonlinear predictor. We give examples from chaotic maps, the double gyre, a pitching-airfoil wake, and Lorenz–63, where PRONE outperforms DeepONets, Fourier neural operators, and reservoir computers with considerably fewer parameters. These examples show the same message: lift once, regress once, and let the singular structure reveal statistics, transport, prediction, and dimension.

We consider finite groups with at least three conjugacy class sizes that are composite numbers and we prove that, in that situation, the number of prime class sizes is bounded by the number of composite class sizes. The analogous result for (non-)prime-power class sizes is also addressed.

We introduce a finite-element framework for simulating thermal fluctuations in compressible fluids governed by the fluctuating Navier-Stokes equations. The method is designed to preserve the fundamental fluctuation-dissipation balance at the discrete level. This is achieved by defining the stochastic forcing term in the weak formulation, ensuring its covariance is proportional to the discrete viscous dissipation operator. A nodal quadrature rule is employed to eliminate unphysical mesh-scale correlations. The time integration is performed using the Crank-Nicolson scheme to maintain numerical stability and accuracy. The proposed approach is numerically validated in one, two, and three spatial dimensions, demonstrating its capability to correctly capture equilibrium fluctuation statistics across various discretisation parameters.

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call $ \delta(G,H)=\min_T |D(T)| $ the transversal difference number of the pair \((G,H)\). This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $\delta(G,H)\geq 2|G/H|-m(G,H), $ where \(m(G,H)\) is the largest order of a subgroup of \(G\) disjoint from \(H\). The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane \[ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. \] For odd \(p\), this case is the technical core of the paper. Here transversals are graphs of functions \(\mathbb F_p^2\to \mathbb F_p^2\), and \(D(T)\) decomposes into carry-corrected finite-field derivative images. We conjecture that \[ \delta(G,H)=(2p-1)^2 \] for all odd primes \(p\), prove the unconditional lower bound \(3p^2-p-1\), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

Score-based generative models and Langevin samplers rely on estimating the score function $\nabla_x\log p_t(x)$ of a forward diffusion. Classically this is tractable when the drift is linear: the marginal density is Gaussian and the score is a global conditional expectation. For a general nonlinear, state-dependent drift the marginal density has no closed form, and existing methods–denoising score matching and global Fokker–Planck residual penalties–resort to global averaging that inflates estimation error in low-density regions precisely where accuracy is most critical. We address this by developing a local Fokker–Planck geometric framework that replaces global conditioning with local parabolic averaging. Our approach rests on three contributions. First, a time change to the cumulative-variance coordinate reduces the variable-coefficient Fokker–Planck equation to a standard inhomogeneous heat equation, on which we extend Evans' classical heat-ball monotonicity method to derive exact local mean-value representations for the score $\nabla_x\log p$ together with the density, log-density, and entropy density; local well-posedness is established under an explicit dimension-dependent drift budget. Second, for high-dimensional Monte Carlo evaluation of the resulting heat-ball integrals, we introduce the $\kappa$-measure and derive its exact factorized sampler with unit per-sample weight, $\chi^2_2$ radial concentration. Third, the $r\to0$ limit of the heat-ball residual recovers the pointwise Fokker–Planck residual, showing that the local framework is a one-parameter generalization of global FP-residual methods, and that the DSM population minimizer is feasible for the heat-ball constraint at every scale. We validate the framework on 2D structured data on 256-dimensional MNIST, and on a dedicated sampler study confirming the concentration laws.

Given a compact zero set of a Fredholm section, our theorem guarantees the existence of a perturbed compact smooth manifold nearby, leaving the original zero set unaltered wherever transversality is already achieved. Such abstract perturbations allow for typical cobordism arguments. We illustrate this by re-proving a well-known theorem of Schwarz asserting the existence of critical points of the Hamiltonian action functional of different action values on symplectically aspherical manifolds.

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\,\overline{w_{\bar z}}) + \mathfrak{a} w + \mathfrak{b} \bar w = \mathfrak{f}$, with $|\mu| < 1$ and $|\Phi| > |\Psi|$, and compute the closed transformation laws of its data under the recombination of unknowns $w \mapsto \varphi w + \psi \bar w$ and under orientation-preserving $C^1$ changes of variables. The 2-form $\Theta = \frac{\bigl|\,\Phi\,\mathfrak{b} - \Psi\,\mathfrak{a} - (\Phi\, L\Psi - \Psi\, L\Phi)\,\bigr|^2}{\bigl(|\Phi|^2 - |\Psi|^2\bigr)^2\,\bigl(1 - |\mu|^2\bigr)}\; dx\, dy$, with $L = \bar\partial - \mu\,\partial$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M} = \int_\Omega \Theta$ is therefore an invariant of the equivalence class. One recombination and one scaling carry any framed equation, in closed form, onto the trivial-frame slice - a Beltrami-Vekua equation over the same $\mu$ - there identifying $\Theta$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $\mu$ be measurable and locally elliptic and that the frame lie in $W^{1,2}_{\mathrm{loc}} \cap L^\infty_{\mathrm{loc}}$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\|\mu\|_\infty < 1$ is quasiconformally equivalent, of equal mass, to one over $\mu = 0$.

For a given Tychonoff space $X$, a point $p\in \beta(X)\setminus X$ is called remote if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain topological ultrafilters, measures, and compact-like properties corresponding to the ideal consisting of nowhere dense sets. It is shown that the space of remote points is homeomorphic to a subspace of the Stone space taken over the smallest Boolean algebra containing all open and nowhere dense sets. Also, we show that the space of remote points of $\mathbb R$ is $\omega$-bounded.

The blowing-up of the projective plane at a finite set of points yields a del Pezzo surface if and only if the points lie in general position. In this note, we generalize this result to Brauer–Severi surfaces over arbitrary ground fields. Using Galois descent, intersection theory and combinatorial arguments, we provide explicit arithmetic and geometric conditions on the centre of the blowing-up.

When researchers need a mathematical model for a research problem, they face a fragmented landscape: relevant formulas, quantities, assumptions, and model variants are scattered across publications and domain-specific conventions. The Mathematical Models Database (MathModDB) addresses this challenge by providing a curated knowledge graph for mathematical models, deployed on the MaRDI Portal as part of the German National Research Data Infrastructure (NFDI). Building on ontology designs presented in earlier work, this paper focuses on MathModDB as a publicly available service. It addresses researchers who use mathematical models in their work – whether in applied mathematics, engineering, or the natural sciences. We describe its deployment on the Wikibase-powered MaRDI Portal, report on its current scale, and demonstrate its practical use through a walkthrough of an electric discharge modeling use case from plasma physics. We further discuss the ecosystem around MathModDB, including its connection to the MathAlgoDB knowledge graph for numerical algorithms and the MaRDMO documentation tool.

We establish a refined Ryu–Takayanagi (RT) relation in AdS$_3$/CFT$_2$ using relative entropy as the central object, in place of the ultraviolet-divergent von Neumann entanglement entropy. Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman–Graham dictionary identifies with the asymptotic data of a Bañados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by $4G_N$. The construction rests only on the Bisognano–Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.

We classify polar homogeneous foliations on rank one symmetric spaces of noncompact type up to orbit equivalence.

We study the coefficients of Ramanujan's third order mock theta function \[ \rho(q)=\sum_{m\geq 0} \frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n. \] Numerical evidence suggests the striking sign pattern \[ r(3n)>0,\qquad r(3n+1)\leq 0,\qquad r(3n+2)\leq 0. \] We prove an asymptotic form of this phenomenon. More precisely, using Watson's relation between $\rho(q)$ and $\omega(q)$, together with a Rademacher-type expansion for the coefficients of $\omega(q)$ and the corresponding expansion for a theta–eta product, we show that \[ r(n)\sim \kappa_{n\bmod 3}\, \frac{2\pi}{(12n+8)^{1/4}} I_{1/2}\!\left(\frac{\pi\sqrt{12n+8}}{18}\right), \] where \[ \kappa_0=\frac13\cos\frac{\pi}{18}>0, \qquad \kappa_1=-\frac13\sin\frac{2\pi}{9}<0, \qquad \kappa_2=-\frac13\sin\frac{\pi}{9}<0. \] Consequently, \[ r(3n)>0, \qquad r(3n+1)<0, \qquad r(3n+2)<0 \] for all sufficiently large $n$.

We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder–Macdonald polynomials. To this end, the methods of Letzter's 2004 paper are translated to modern conventions and right coideal subalgebras, and extended to non-standard cases and cases with non-reduced restricted root systems. For the last elusive Satake type, FII, a conjecture is provided.

We study optimal estimation of the location parameter of a distribution known only to lie in a symmetric moment class $\mathcal C_0$: the mean-zero distributions with bounded moment $\int\phi\, d\mathbb P\le B$ for a fixed even $\phi$. Our main result concerns the fixed-margin regime, where the error margin $\Delta$ is fixed as $n\to\infty$: we give an exact large-deviation characterization of the smallest worst-case probability $\beta_n(\Delta)$ of an error exceeding $\Delta$ that any measurable estimator can guarantee with $n$ observations. Its exponential rate is exactly a two-point Hellinger exponent over the class shifted to means $\pm\Delta$, $r(\Delta)=-\log\sup_{\mathbb P_{\pm\Delta}\in\mathcal C_{\pm\Delta}}\int\sqrt{d\mathbb P_{-\Delta}\, d\mathbb P_{\Delta}}$, achieved non-asymptotically, $\beta_n(\Delta)\le e^{-nr(\Delta)}$, by a monotone $M$-estimator synthesized from a two-parameter convex program. Lagrangian duality collapses the infinite-dimensional search over estimating functions to two multipliers, which determine a pair of envelopes characterizing the optimal estimating functions; the sandwich shape posited ad hoc in prior constructions emerges naturally. For bounded variance ($\phi(x)=x^2$, $B=\sigma^2$) the exponent is $r(\Delta)=\tfrac12\log(1+\Delta^2/\sigma^2)$. In the fixed-confidence regime, holding $\beta$ fixed and letting the optimal margin $\Delta_n(\beta)$ shrink with $n$, the same synthesis stays optimal to leading order for several concrete classes. As $\beta\downarrow0$ it attains the sharp constant $\sqrt2$ of Catoni for bounded variance and the constant $L(\alpha)$ of Lee and Bhatt et al. for bounded $\alpha$-moments, $\alpha\in(1,2)$, thereby shown tight; for slowly varying $\phi$ it is leading-order minimax at every fixed $\beta$. The least-favorable distributions are simple, supported on at most three atoms.

Given a geometric statistic expressible as a sum of scores which depend on local data, \citet{BYY19} established central limit theorems for centered and normalized versions of these statistics, subject to the underlying point process having fast decay of correlations and also subject to a variance growth condition. Building on this, \citet{CX23} derived rates of normal approximation, as measured by the Wasserstein distance, for statistics of point processes exhibiting fast decay of dependence. Here we go further and establish weak mixing conditions yielding rates of normal convergence for geometric statistics of marked point processes. The mixing conditions, which are in terms of verifiable geometric criteria, provide rates of normal approximation in the Wasserstein distance for statistics of a large class of point processes with dependent marks. Examples include statistics of determinantal point processes, unevenly spaced time series, continuum percolation, interacting diffusions on spatial random graphs, as well as local $U$-statistics.

We study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard-Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of smooth projective varieties. This includes varieties $X$ of dimension $n\geq2$ with $p_g=h^{n,0}(X)>0$ and $H^{n-1,0}(X)=\{0\}$, Enriques surfaces, irregular surfaces with maximal Albanese dimension, smooth hyperkähler varieties, and all the smooth not Fano hypersurfaces in $\mathbb{P}^n$. As a consequence, by a result of Beauville, we establish a Lefschetz property for the Jacobian rings of smooth hypersurfaces in $\mathbb{P}^n$ of degree n+1.

In this paper, we study pencils of plane curves of sufficiently large degree $d$ with simple base points, and their reducible curves whose irreducible components have degree at most $k\geq 2$. Combining techniques from algebraic geometry and combinatorics, we establish an explicit upper bound on the number $m_k$ of such curves in the pencils. We prove that, for sufficiently large $d$, pencils with more than six such curves do not exist. Consequently, under the stronger assumption $d\geq\frac{7}{2}k(k-1)-2$, we obtain the bound $m_k\leq 6$, improving the previously known bound for $d \ge 2k$. We also establish restrictions on pencils containing reducible curves consisting of one irreducible component of degree $k$ together with lines, and obtain nonexistence results for certain such pencils.

Kinetic equations are used to model a wide range of phenomena important for real-world applications. Their applications span astrophysics, nuclear physics, engineering, and social sciences. Due to their high-dimensional phase space, modelling and quantifying uncertainties, relevant for applications, poses a significant challenge even for modern computing infrastructure. In recent years, dynamical low-rank approximation (DLRA) has gained popularity for making fine grid simulations of high-dimensional problems feasible by evolving the solution of a time-dependent PDE as a low-rank factorization. This reduces the computational and memory requirements significantly. In this work, we propose a low-rank multilevel Monte Carlo estimator for kinetic equations based on a probabilistic rank-adaptive DLRA time integrator. The level hierarchy of the low-rank multilevel estimator is constructed through spatial refinement and by ensuring that the low-rank error remains below the spatial discretization error. We demonstrate the efficacy of the estimator through several numerical experiments from radiation transport, radiation therapy, and shallow water flow.

In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these sequences have at most two sufficiently large common values. Our proof combines Baker's theory of linear forms in logarithms of algebraic numbers with techniques from function field theory and from Galois theory.

We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.

We introduce a unified framework via Stein's method for bounding the Kolmogorov distance between the generalized Dickman distributions and the distribution of randomly weighted sums of non-negative integer-valued random variables that are conditionally independent given the weights. By utilizing size-bias couplings and a decomposition of the solution to the corresponding Stein equation into distinct bounded and non-increasing components, our approach yields non-asymptotic error bounds governed only by the quality of a discrete-to-continuous coupling. A similar bound is also obtained for the smooth Wasserstein-2 distance. We apply our abstract results to establish concrete, and in some cases optimal, rates of convergence across diverse probabilistic models. These applications range from settings with independent weights, such as randomly weighted sums of log-primes, to models with intricate dependency structures, including geometric sums over the spectra of the Circular Unitary Ensemble (CUE) and random weights generated by independent increments.

On a regular local $p$-Dirichlet space supporting a Poincaré inequality and a capacity upper bound, we characterize the existence of curve families of positive $p$-modulus in terms of the small-scale behaviour of the scaling function. As an application, we show that, on an Ahlfors-regular metric measure space carrying a strongly local regular Dirichlet form with sub-Gaussian heat kernel estimates, the comparison between the walk dimension and $2$ determines whether the given metric attains its Ahlfors-regular conformal dimension.

We study smooth polarized projective varieties $(X,H)$ whose exterior powers of the tangent bundle are Ulrich. We prove that if $\bigwedge^rT_X$ is $H$-Ulrich for some $0<r<\dim X$, then $X$ is Fano and the intersection number $(-K_X)\cdot H^{n-1}$ is determined explicitly. We then classify the Picard number one case: the only example is the Veronese surface $(\mathbb P^2,\mathcal O_{\mathbb P^2}(2))$.

Temporal (or time-evolving) networks provide a natural framework for modeling complex systems with time-dependent interactions, where understanding the evolution of community structures is a central challenge. While random walk-based approaches to community detection in static networks are well established through the spectral analysis of associated transfer operators, extending these ideas to temporal networks is nontrivial due to the inherent time-dependence of the underlying dynamics. In this work, we develop a general framework for community detection in temporal networks that is based on multi-view canonical correlation analysis (mCCA). We show that the proposed formulation admits a spectral characterization via a time-reversible random walk on an augmented space-time network, providing a clear dynamical interpretation of temporal communities as metastable structures of the process. Furthermore, we analyze key spectral properties of the resulting transfer operators and the interplay between spatial and temporal effects, which allows us to distinguish between structural features and artifacts induced by the snapshot coupling. Finally, we derive a reduced-order model, which preserves the essential spectral properties while significantly improving computational efficiency. We show that the proposed approach effectively detects communities in temporal networks and captures their evolution.

We consider the transmission of the outputs of counting queries over a binary symmetric channel (BSC), where Hamming codes are employed as the channel encoder. Since the channel is inherently noisy, this transmission already provides a degree of privacy protection “for free”, albeit at the cost of reduced utility in the form of decoding errors. A natural question is whether this privacy can be further improved (i) without any additional real-time obfuscation of the data, such as injecting artificial noise prior to transmission, and (ii) without increasing the end-to-end error probability. In this work, we answer this question in the affirmative by deriving an optimal codeword arrangement that strictly improves differential privacy guarantees while incurring no real-time computational overhead and no degradation in utility.

Let $M^n_\kappa$ be the simply connected space form of dimension $n\ge2$ and constant sectional curvature $\kappa\in\{-1,1\}$. For every bounded connected smooth domain $\Omega\subset M^n_\kappa$, assume in the case $\kappa=1$ that $\Omega$ is contained in an open hemisphere, and let $B_\Omega$ be a geodesic ball with $|B_\Omega|=|\Omega|$. We prove $$ \sum_{j=1}^n \frac1{\mu_j(\Omega)}\ge \frac{n}{\mu_1(B_\Omega)}, $$ where $\mu_j(\Omega)$ are the positive Neumann eigenvalues of $\Omega$. Equality holds if and only if $\Omega$ is a geodesic ball. This proves a conjecture proposed by Xia and Wang [Math. Ann. 385, 2023, 863-879].

We study the inhomogeneous random graph with preferential attachment kernel and degree distribution with power-law exponent $\tau\in(2,3)$ as a representative of the class of graphs of preferential attachment type with infinite variance degrees. Under bond percolation with a positive retention probability independent of the size $n$ of the graph there is a unique macroscopic component with high probability. We therefore investigate percolation probabilities $p_n\downarrow0$. We identify a moving critical window at $p_c \sim \beta n^{(\tau-3)/(2\tau-2)}$. Above this window, when $p_n \gg p_c$, the maximal component has size of order $n p^{_{(\tau-1)/(3-\tau)}}_{_n}$ and it is unique. Below this window, when $n^{1/(1-\tau)} \ll p_n \ll p_c$, it is non-unique, star-shaped and has size of order $n^{1/(\tau-1)} p_n$. In the critical window itself, the largest component scaled by $\sqrt{n}$ converges in distribution to a positive random variable with a law given in terms of a subcritical Norros-Reittu graph. This behaviour is markedly different from that seen for other classes of scale-free graphs and is conjectured to persist throughout the broad class of growing graphs with infinite variance.

We develop an operator-algebraic framework for change-of-variables formulas on Wiener space, interpreting them as arising from hidden symmetries acting on observables. We show that general transformations can be represented by time-ordered exponentials generated by annihilation and creation operators, and that these admit an explicit factorization into a determinant, a multiplication operator, and a translation operator. Taking expectations recovers the classical formulas, including the Ramer–Kusuoka formula.