Papers

We establish a general structure theorem for local quantum Cuntz–Krieger families associated to finite quantum graphs whose relations are analogous to those of the partial isometries generating an ordinary graph $C^*$-algebra, with the quantum edge correspondence governing composition. We apply this structure theorem to the quantum dephasing graph on $M_n$ to obtain an explicit isomorphism between $O_n$ and the local quantum Cuntz–Krieger algebra. More generally, we obtain a unital embedding of $O_n$ in the local quantum Cuntz–Krieger algebra for the dephasing graph on a finite direct sum of $M_n$.

We prove the wall-crossing formula conjectured by Gross–Joyce–Tanaka for equivariant enumerative invariants of CY4 categories equipped with framing functors. We also establish a version for stable pairs with fixed-determinant obstruction theories, as used in earlier applications by the first-named author. The main technical ingredient, which we develop in this work, is a construction of well-behaved CY4 pullback virtual classes using Jouanolou devices.

The probabilistic formulation of principal component analysis promises statistically grounded solutions to the problem of selecting the number of principal components. However, developing tractable model selection methods is complicated by the fact that the probabilistic principal component analysis (PPCA) model exhibits non-standard large sample asymptotics at model singularities, where the Fisher information matrix does not have full rank. In this work, tools from singular learning theory are used to provide a complete description of the marginal likelihood asymptotics for PPCA. The singular Bayesian information criterion (sBIC) along with our asymptotic results provides an effective procedure for selecting the number of principal components. In particular, the sBIC corrects for the overpenalization of the standard dimension-based BIC, which leads to too few principal components being selected, while also selecting the smallest true model with probability converging to one. Our framework extends beyond PPCA, where we also provide expressions for the sBIC that can be used to select between PPCA and factor analysis models. In both simulations and on real data the effectiveness of the proposed sBIC methodology is demonstrated.

We establish a Demailly-Kollár type continuity theorem for plurisubharmonic functions with respect to adapted measures on normal complex analytic klt pairs. As applications, we prove the equality of the analytic and divisorial versions of the alpha and delta invariants on compact normal Kähler klt pairs, thereby completing a program initiated by the second author. We further show that the two local alpha invariants introduced by Guedj and Trusiani for an isolated log terminal singularity coincide and that their common value is the $n$th root of Li's normalized volume. These identities have geometric consequences. The equality for the delta invariant yields a Yau-Tian-Donaldson type divisorial criterion for the solvability of twisted Kähler-Einstein equations in big cohomology classes, without a semipositivity assumption on the twist. The local alpha identity determines algebraically the critical exponent governing the existence of positively curved KE metrics near an isolated log terminal singularity, thus confirming a prediction of Guedj and Trusiani.

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

An element of a finite group is non-vanishing if no irreducible complex character vanishes on it. We prove that every non-vanishing element of a finite solvable group belongs to the Fitting subgroup. The proof combines the reduction of Isaacs, Navarro, and Wolf with an argument using finite symplectic spaces and Clifford theory.

Pantograph arises in electric trains, material modelling, and the modelling of quantum dot lasers. Pantograph integrodifferential equations are integrodifferential equations involving proportional delays; and they appear in fields such as electrodynamics, epidemiology, control theory, astrophysics, economics, and engineering. This research presents an efficient collocation method based on shifted Jacobi polynomials for obtaining numerical solutions of a class of first order pantograph delay integrodifferential equation of Volterra type with an initial condition. The proposed method expresses the solution of the governing equation as a shifted Jacobi polynomial series with expansion coefficients which are to be determined. Collocating at the roots of the shifted Jacobi polynomials, the underlying problem is reduced to a system of algebraic equations in the unknown expansion coefficients of the shifted Jacobi polynomial series solution. Newton's method is subsequently used to solve the resulting system of algebraic equations and numerical values of the expansion coefficients are obtained. The obtained coefficients are substituted into the assumed series solution to obtain the required numerical solutions. The applicability, reliability, efficiency, and accuracy of the shifted Jacobi collocation method are demonstrated through illustrative examples. Results obtained using the proposed method are compared with exact solutions and other published results. Comparisons of errors which are presented in tables reveal that our method approximates the solution better than the methods under comparison.

The capacity of Low Earth Orbit (LEO) to sustain space operations is under mounting pressure from megaconstellations, legacy fragmentation debris, and new payload classes. Existing assessments of orbital capacity and debris evolution are largely deterministic, tracking mean populations of intact satellites and fragments with ordinary differential equations; they cannot capture the inherent randomness of collisions, breakup sizes, and launch schedules. We develop a stochastic extension of the two-species Lotka–Volterra model of Bradley and Wein, formulated as a density-dependent Markov chain, and study its deterministic and stochastic scaling limits. Because intacts and fragments differ by many orders of magnitude, these limits emerge on distinct time-scales, and different pathways to a collisional Kessler cascade become visible only on the appropriate time horizon. On a fast intact time-scale we obtain an ODE approximation and a Gaussian SDE approximation; on an intermediate fragment time-scale we obtain an ODE approximation, a Gaussian SDE approximation, and the critical Kessler threshold, above which the ODE approximation runs away in Kessler syndrome. Crucially, on a third, slow time-scale at the critical threshold, the fragment count converges to a Feller diffusion, in which runaway is triggered purely by fluctuations rather than by the drift—an effect the ODE approximations and their Gaussian SDE approximations cannot see. Debris runaway may occur sooner, and with higher probability, than deterministic models predict: the intact population can appear well-behaved while fragments quietly accumulate risk. Constellation deployment, debris-removal investment, and slot allocation should account for these stochastic effects, and planning for runaway must depend on the variance of the collision dynamics, not on the mean alone.

We study travelling front solutions of a family of degenerate Fisher-KPP equations $u_t=(u^n u_x)_x+u(1-u^n)$, where $n$ is a positive integer. At the minimal wave speed $c_{\rm min}=\tfrac{1}{\sqrt{1+n}}$, these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to $c=\tfrac{1}{\sqrt{1+n}}+\varepsilon$, with $\varepsilon$ sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.

For $n$ odd, we enumerate rank $n$ complex topological vector bundles on $\mathbb CP^{n+2}$ with prescribed complex $K$-theory class. The answer exhibits a 24-fold periodicity and is closely related to the enumeration of corank $2$ stably trivial vector bundles.

We prove that the heat semigroup on the circle has optimal complete modified logarithmic Sobolev constant \(1\). The proof is based on a matrix-valued Wirtinger inequality and yields the stronger Bogoliubov–Kubo–Mori Fisher information contraction with rate \(e^{-2t}\). As a consequence, the complete tensorization and transference principle yields the same sharp constant for the heat semigroups on classical and quantum tori.

Classical results establish that the Lyapunov exponents of an ergodic measure for linear cocycles over hyperbolic systems can be approximated by the Lyapunov exponents of periodic orbits, provided the cocycle is Hölder continuous. A recent counterexample by Bochi demonstrates that this approximation property fails in general if the Hölder assumption is relaxed to mere continuity. In this paper, we introduce a geometric condition that successfully substitutes this analytical regularity hypothesis. More precisely, we prove that if a cocycle - even a discontinuous one - admits a continuous family of invariant holonomies, the periodic approximation of Lyapunov exponents still holds. Our geometric approach yields a proof that is substantially simpler and more direct than existing arguments in the literature, even when applied to classical settings such as fiber-bunched cocycles for which previous results were already available.

We study the optimal Stein error exponent for one-way distributed hypothesis testing under a communication rate constraint. We establish exact rate-exponent tradeoffs for new classes of finite-alphabet sources, including a complete single-letter characterization for every pair of strictly positive binary joint distributions as well as testing-against-dependence. Our converse analysis refines the auxiliary-receiver method through three complementary steps. First, we establish an information-projection identity that identifies the posterior-divergence optimization in an existing converse with Han's exponent functional. Second, we obtain an additional excess-rate branch by retaining a mutual-information term discarded in the usual rate relaxation. Third, we develop a two-telescope upper bound whose auxiliary variables are coupled through a joint rate constraint. Combining these bounds with information-geometric matching conditions yields exact exponents for three classes of sources by matching the achievable bounds of Han and Shimokawa-Han-Amari.

We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on dissipativity of the frozen fast dynamics and therefore permits degenerate diffusion coefficients. We prove a maximal $L^p$-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order $1/2$. Under an additional decay condition on $(\varepsilon_t)_{t \ge 0}$, this estimate implies that the slow variable is almost surely an asymptotic pseudo-trajectory of the averaged ODE. As a consequence, we obtain criteria for the identification of possible limit points and for convergence toward asymptotically stable equilibria for the slow variable by analyzing the dynamical behavior of the averaged equation.

We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki–Kwaśnicki–Małecki–Stós and Kwaśnicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulations of Kaleta–Kwaśnicki–Małecki. Moreover, we prove that the normalized eigenfunctions are bounded uniformly in the eigenvalue index $n$ and the fractional order $\alpha$. This settles the conjecture proposed by Kwaśnicki through numerical experiments. Furthermore, we prove that the $n$-th eigenfunction has exactly $n-1$ zeros in the interval $(-1,1)$ and every zero is simple, and hence there are exactly $n$ nodal domains. A key ingredient in the proof is an explicit representation of the eigenfunction.

Randomized trials support causal inference, but differences between trial and target populations can limit the transportability of treatment effects to real-world settings. Many existing approaches model the propensity of trial participation and can therefore be sensitive to model misspecification and weak overlap of the covariate distributions. Optimal Transport (OT) offers a different route by comparing the trial and target populations directly in covariate space. We develop RICOT, a Riesz-calibrated OT procedure transporting treatment effects to a treated target population. We consider a semi-unbalanced OT with entropic regularization where the source marginals are relaxed. We show that the uncalibrated OT introduces a bias which does not shrink with increasing sample size. RICOT removes this bias by imposing calibration equations directly within the transport problem. With a growing calibration sieve, the calibrated weight consistently estimates the target-to-trial density ratio, equivalently the Riesz representer of the target expectation functional, even when the entropic and source-relaxation parameters remain fixed and positive. Combined with outcome regression, the resulting estimator is doubly robust and attains the semiparametric efficiency bound under suitable rate conditions. Its variance is estimated directly from the influence function, without resampling or repeated OT optimization. Simulations show low bias and near-nominal coverage across a range of overlap and misspecification settings, including settings in which sampling-score methods perform poorly. We illustrate RICOT in a real-world application involving a rare progressive cardiomyopathy, comparing conventional IPW and AIPW estimators with our OT-based IPW and doubly robust estimators for transporting the randomized treatment effect to a real-world population receiving the same treatment.

We study an infinitely many neutral alleles Moran model where the allelic frequencies are recorded in increasing order of the ages of the alleles. Using this representation, we show that in stationarity the number of individuals with the oldest allelic type evolves as a Markov process, even across times when the oldest allele goes extinct (and is immediately replaced by the formerly second oldest allele). We also find the diffusive limit of the oldest allele chain.

Let $(M, \omega, J)$ be a $2n$-dimensional toric variety determined by a Delzant polytope $P$, whose $T^{n}$-symmetry determines a real polarization $\mathcal{P}_{\mathbb{R}}$. Let $K \subset T^{n}$ be a subtorus. By a construction due to Leung and the first author, the $K$-action induces a mixed polarization $\mathcal{P}_{K}$. This paper investigates the relationship between the quantum Hilbert spaces $\mathcal{H}_{K}$ and $\mathcal{H}_{\mathbb{R}}$ associated with the polarizations $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$. Starting from $\mathcal{P}_{K}$, we use an imaginary-time flow to construct a one-parameter family of mixed polarizations $\mathcal{P}_{K,t}$ on $M$ interpolating between $\mathcal{P}_{K}$ and $\mathcal{P}_{\mathbb{R}}$, with $\mathcal{P}_{K,0}=\mathcal{P}_{K}$ and $\lim_{t\to\infty}\mathcal{P}_{K,t}=\mathcal{P}_{\mathbb{R}}$. For the corresponding quantum Hilbert spaces $\mathcal{H}_{K,t}$, we lift the imaginary-time flow to the prequantum line bundle to obtain a $T^{n}$-equivariant isomorphism $\mathcal{H}_{K}\cong\mathcal{H}_{K,t}$. We finally show that $\mathcal{H}_{K,t}$ converges to $\mathcal{H}_{\mathbb{R}}$ as $t\to\infty$.

We show that Makhlin's polytopes in type $B_n$ are unimodularly equivalent to string polytopes for Demazure modules in type $B_{2n-1}$. The proof passes through type $C$, combining Makhlin's diagonal comparison with symplectic FFLV polytopes and the $B/C$ similarity for string cones.

Let $D(G)$ denote the derived category of smooth $G$-representations on $k$-vector spaces, where $G$ is a locally pro-$p$ group and $k$ is a field of characteristic $p$. In this paper we are primarily interested in the graded center of the subcategory of compact objects $Z^*(D(G)^c)$ and variants thereof. When $G$ is a $p$-adic Lie group, without proper open centralizers, we completely determine this center modulo locally nilpotent elements and give various applications.

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.

Let $G$ be a discrete countable infinite group. Given two rank-one measure-preserving $G$-actions whose invariant measures are either both finite or both infinite, we give a direct explicit construction, from their cutting-and-stacking parameters, of a Borel orbit equivalence on invariant conull subsets. We also prove a topological counterpart of this assertion, under additional compatibility assumptions, for continuous $(C,F)$-actions of $G$ on non-compact locally compact Cantor spaces.

In the recent work by T. H. Chan, it was shown that there are no three consecutive powerful numbers with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power. They get it with Pell equations, elliptic curves, and second-order recurrences. In this paper, with results given by Nagell-Ljunggren, Lebesgue, Ko and some elementary methods, we extend the result to similar cases, whose middle term can be perfect powers for infinitely many powers. With this conclusion, we discuss the solutions to some Diophantine equations.

We consider the Sherrington–Kirkpatrick model with inverse temperature $\beta>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb E{\rm tanh}^2 (h+\beta\sqrt q\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida–Thouless region: $\beta^2\mathbb E{\rm sech}^4(h+\beta\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \mathbb E\langle (R_{12}-q)^2\rangle=O(N^{-1}). $$ As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida–Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.

We study extreme values of normalised $r$-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, and prove Weibull and Fréchet limit laws for these functions when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. We adapt the Poisson approximation strategy of Bj$\ddot{\text{o}}$rklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Str$\ddot{\text{o}}$mbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).

The weak set theory $\mathsf{ReR}$ is obtained from Kripke-Platek Set Theory ($\mathsf{KP}$) by replacing the bounded collection scheme with the bounded replacement scheme. We show that $\mathsf{ReR}$ proves $\mathsf{TCo}$, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to $\mathsf{ReR}$ and $\mathsf{KP}$ have the same consequences. Our proof of $\mathsf{TCo}$ relies on the availability of a fragment of class foundation in $\mathsf{ReR}$. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of $\mathsf{ZF}$ that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which $\mathsf{TCo}$ fails.

We use Morse Theory to prove that the Burkholder function is quasiconvex on symmetric matrices, which implies several sharp inequalities for the Beurling–Ahlfors transform when acting on real-valued functions. More generally, we prove that the separating functions introduced by Székelyhidi are quasiconvex on symmetric matrices.

We construct an Orlik–Solomon algebra for any locally geometric poset as a natural generalization of the one for geometric lattices. This algebra has several interesting features, including a combinatorial no-broken-circuit basis that we obtain through Gröbner basis theory. When the poset captures the intersection data of an arrangement of certain subgroups in a complex abelian Lie group, we infuse the Orlik–Solomon algebra with topological information to compute the cohomology of the arrangement complement. In particular, when the Lie group is compact, we present an explicit differential graded algebra whose cohomology is the rational cohomology of the arrangement complement. We further exhibit a family of abelian arrangements whose integer cohomology has torsion.

Transform-based tensor products, including the T-product and its more general form, namely the higher-order tensor-tensor product, have become fundamental tools for multilinear data analysis in applications such as image processing, signal reconstruction, and robotics. While invertible transforms enable tensor computations to be carried out via matrix operations in the transform domain, the resulting storage and computational costs remain prohibitive for high-dimensional, higher-order tensors. To address this challenge, we integrate low-rank tensor decomposition techniques, specifically tensor train decomposition (TTD) and hierarchical Tucker decomposition (HTD), into transform-based multilinear algebra to improve computational and memory efficiency. In particular, we develop TTD- and HTD-based formulations for the T-product and its associated key algebra, such as block diagonalization and tensor singular value decomposition, by operating directly on the factor matrices or tensors of the decompositions. The framework is further generalized to the higher-order tensor-tensor product and applied to multilinear model order reduction problems. We demonstrate the effectiveness and efficiency of our framework with numerical examples.

We classify fixed point homogeneous Riemannian orbifolds with positive sectional curvature up to equivariant diffeomorphism. As a corollary we obtain a classification of positively curved orbifolds with maximal symmetry rank.