Papers

Let $G_n$ be the graph on $\{2,\ldots,n\}$ in which two integers are adjacent when they have a common divisor greater than one. We determine the complete asymptotic expansion of its Havel–Hakimi residue $\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's Written on the Wall. If $A=\sum_{k=2}^{\infty}(\log k)/(k^2(k-1))$, then the first two terms are $\R(G_n)=(\zeta(2)-1)n/\log n+(\zeta(2)-1-A)n/\log^2n +O(n/\log^3n)$. More precisely, the difference between $\R(G_n)$ and the prime-vertex contribution to the Caro–Wei sum is $O_\beta(n\exp\{-(\log n)^\beta\})$ for every fixed $0<\beta<1/2$; this estimate yields every coefficient in the expansion. The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques. We also prove that the unlabeled graph determines $\pi(n)$ through its simplicial true-twin classes. Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro–Wei sum. An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.

We study stopping rules in which each family is assigned a positive integer and stops the first time its number of girls equals that integer plus a fixed nonnegative integer multiple of its number of boys. After all families have stopped, we ask for the expected proportion of boys and the expected boy-to-girl ratio in the combined population, and show that both expectations depend on the assigned integers only through their sum. We obtain exact series for both expectations, allowing the probabilities of a boy and a girl to be unequal. Our formulas recover the known formulas for stopping at the first girl and for stopping when girls first outnumber boys. These expectations can also be recovered from earlier general results on Galton–Watson degree profiles and shifted inverse moments, but the resulting series do not appear to have been recorded in these forms. To our knowledge, our two elementary proofs, one bijective and one probabilistic, are also new.

Recursive nonlinear impulse responses require an estimated innovation law whenever the impact shock is normalized by innovation ranks and future innovations are integrated out. The closest semiparametric recursive construction in the literature estimates the relevant innovation quantile functions smoothly and discusses a direct empirical-residual implementation without developing its complete first-order inference theory. We tackle this gap in a finite-dimensional nonlinear structural autoregression with unrestricted continuous marginal innovation distributions and a fixed normal-rank shock. Our estimator replaces each innovation quantile function with the empirical quantile of generated structural residuals and iterates the same structural transition. For any fixed collection of responses, we establish a joint \sqrt{T} asymptotic linear representation with four components: direct transition estimation, the effect of transition estimation on residual order statistics, ordinary innovation-quantile estimation, and the shifted impact quantile. After projection through the recursion, the quantile terms admit a residual-rank-and-spacing representation, yielding feasible inference without innovation-density estimation or quantile smoothing. We then characterize the propagated bias from smoothing, establish validity of a full recursive residual bootstrap, and derive the additional covariance contribution from a finite number of simulated paths, providing bandwidth-free inference for the empirical-residual version of the same normal-rank response used in the smooth recursive construction.

Extending recent work on refractive billiards, we introduce and study the corresponding refractive outer billiards system about a convex polygon. Gutkin and Simanyi showed in 1992 that for regular outer billiards, orbits about a certain class of polygons called quasi-rational polygons are bounded, and that orbits about rational polygons are periodic. Tabachnikov and Culter later proved in 2007 that every outer billiard system about a convex polygon admits a periodic trajectory. We generalize both results to the refractive setting.

For a fixed integer $\ell \ge 2$, we study what values of chromatic index and chromatic number can be attained by some $\ell$-fold cyclic cover of a loopless multigraph. For edge-coloring, we first investigate the density, a fundamental lower bound for the chromatic index, and show that the density of every $\ell$-fold cyclic cover of a graph $G$ is at most that of $G$. We further prove that if $\ell$ is even, then the spectrum of chromatic indices over all $\ell$-fold cyclic covers of $G$ contains every integer between $\Delta(G)$ and $\chi'(G)$. When $\ell$ is odd, the chromatic-index spectrum need not be complete in general; for edge-chromatic critical graphs, we determine exactly which values are attainable. For vertex-coloring, we prove that if $\chi(G)\ge 3$, then the spectrum of chromatic numbers over all $\ell$-fold cyclic covers of $G$ contains every integer between $3$ and $\chi(G)$. Moreover, this spectrum contains $2$ if and only if $G$ is bipartite or $\ell$ is even.

Intensity of attraction quantifies the largest amplitude of a persistent bounded disturbance that an attractor can withstand without loss of controlled confinement in its basin. Although intensity has been formulated separately for flows and maps, its behavior under temporal sampling has remained unresolved. We establish an explicit correspondence between the intensity $\mu(A)$ of a continuous-time attractor and the intensity $\mu_h(A)$ of its exact time-$h$ map. For an $L$-Lipschitz vector field, \[ \frac{\mu(A)}{1+Lh} \leq \frac{\mu_h(A)}{h} \leq \mu(A)\frac{e^{Lh}-1}{Lh}, \] and hence $\mu_h(A)/h\to\mu(A)$. The resulting first-order rate is sharp in general, while smooth scalar escape geometries can exhibit second-order convergence. We extend the framework to one-step numerical methods through a stability theory for block intensity and to attracting invariant graphs over compact invertible nonautonomous bases, obtaining uniform sampling convergence over the forcing phase. For bounded-support random perturbations, normalized discrete intensity is identified with the pathwise safety threshold; above it, finite escape follows under an explicit finite-exit condition, while escape probabilities require additional assumptions on the noise law. We also show that the discrete state–normal boundary map converges to the normalized Pontryagin boundary system governing extremal reachable-set boundaries. Exact scalar benchmarks, a grazing resilience model, planar Duffing escape, anisotropic disturbances, periodic and quasiperiodic forcing, and transfer-operator computations illustrate the theory. These results give intensity estimated from discrete observations or simulations a sampling-independent continuous-time meaning.

Under an explicit formal mapping-space hypothesis, we develop a facewise formulation of the classical AKSZ construction on compact oriented manifolds with ordinary corners. The codimension-$r$ data—a mapping space carrying a closed two-form of degree $r-1$, an action of degree $r$, and a cohomological vector field—and the modified Batalin–Vilkovisky/Batalin–Fradkin–Vilkovisky Hamiltonian identity relating consecutive strata are those of the maximally extended BV–BFV theory of Cattaneo–Mnev–Reshetikhin. What is added here is the organization over the entire face poset: the Hamiltonian defect on a face is the sum of the pullbacks of the primitives on its codimension-one faces, weighted by the orientation incidence numbers, so that the boundary term of the single-stratum identity is resolved into its connected pieces with signs. Organizing these defects by the face incidence complex yields a total-complex theorem: factorially normalized facewise transgression is a cochain map, so closed target forms transgress to cocycles, and the twice-iterated defect vanishes because the signed face differential squares to zero. We verify all four codimension-two cancellations explicitly for four-dimensional BF theory on $M=\Gamma\times[0,1]^2$. We also establish a reduction criterion for singular corner data. If a raw codimension-two descendant is presymplectic and its reduced Dirac structure is the graph of a Poisson bivector, the shifted cotangent construction gives a canonical strict degree-two corner theory. The passage from a reduced Poisson bivector to a strict corner theory is already recorded in \cite{CFT2026}; what is isolated here is the hypothesis under which it applies, and its relation to the face-incidence structure. The construction provides a rigorous ordinary-corner benchmark for extensions of AKSZ descent to Joyce generalized corners.

We give an alternative proof of Yu's Thinning Lemma and of the Law of Thin Numbers using a stochastic variational formula for the relative entropy. We show that this approach yields new convergence rates for the Law of Thin Numbers, complementing and extending previously known results.

Causal inference increasingly extends beyond classical causal effects defined by deterministic treatment assignments, such as the average treatment effect, to stochastic intervention effects that can weaken positivity requirements and offer greater policy relevance. Nonparametric Bayesian models are attractive for estimating these effects due to their flexibility and inherent uncertainty propagation, but this posterior uncertainty need not be well calibrated for the causal effect of interest. We develop a simple post-processing correction that can be applied to posterior samples without changing the prior or fitting algorithm. We prove that, for a broad class of stochastic interventions, the corrected posterior yields asymptotically efficient inference and credible intervals with asymptotically valid frequentist coverage; formally, it satisfies a semiparametric Bernstein-von Mises theorem. The theory covers interventions specified independently of the observed treatment process, as well as interventions that modify it, including incremental propensity score interventions and a new power-tilt intervention. A central contribution is new theory for SoftBART, including conditions under which this flexible tree-based Bayesian model supports calibrated Bayesian inference for stochastic intervention effects. In simulations, the correction reduces bias and improves coverage relative to the uncorrected Bayesian analysis while remaining competitive with frequentist alternatives. We illustrate the method by estimating how expected LDL cholesterol would change under hypothetical increases or decreases in the odds of receiving statin therapy.

Spectral clustering for community detection is analysed in multivariate time series models whose dependence structure is determined by an unobserved stochastic blockmodel. We establish that spectral clustering of the sample covariance matrix achieves exact recovery of the underlying communities. The recovery rates depend explicitly on the network size, sample length, block separation, and degree of data dependence. This demonstrates that exact community recovery under a stochastic blockmodel is possible even when the adjacency matrix is unobserved. Our theory provides extensions of both classical and fine-grained matrix perturbation theory to the setting of dependent data, which may be of independent interest.

We introduce Survival-IDR (S-IDR), a nonparametric estimator of conditional survival distributions under order restrictions, extending Isotonic Distributional Regression (IDR; Henzi et al., 2021) to right-censored outcomes. S-IDR has no tuning parameters and accommodates continuous, discrete, and partially ordered covariates. We first study the direct Kaplan-Meier adaptation of IDR: it is uniformly consistent at the minimax rate, but only when the conditional outcomes are hazard-rate ordered. We trace this restriction to the Kaplan-Meier estimator's failure to satisfy the Cauchy mean value property on non-i.i.d. samples, and use the diagnosis to construct S-IDR. The S-IDR estimator is uniformly consistent under only stochastic dominance of the conditional outcomes, attains the minimax rate when the smoothness of the conditional CDFs is known, and admits a known cross-threshold PAVA acceleration. We further embed S-IDR in a distributional single-index framework on a benchmark suite, and apply it in a case study that validates the MELD score used for liver-transplant wait list management. Accompanying R, Python and Rust packages are available at https://github.com/AlexanderHenzi/isodistrreg.

This paper presents a fully discrete analysis of pressure-robust enriched Galerkin (EG) methods for the time-dependent Stokes equations. The spatial discretization employs a velocity reconstruction operator that preserves the continuous component of the EG space while mapping its enriched discontinuous component into the lowest-order Raviart-Thomas ($\mathcal{R}T_0$) subspace of $H(\mathrm{div})$, ensuring strict local mass conservation and pressure-robustness. Fully discrete schemes are constructed using the backward Euler and Crank-Nicolson methods, with the reconstruction incorporated into both the forcing and discrete time-derivative terms. We prove unconditional stability and derive optimal-order, parameter-explicit a priori error estimates for both the velocity and pressure. In particular, the velocity estimates are independent of the continuous pressure and the irrotational component of the forcing term, contain no inverse-viscosity factors, and explicitly track the spatial mesh size, time-step size, and viscosity. Numerical experiments in two and three dimensions verify the predicted convergence rates, strict local mass conservation, and robustness with respect to the viscosity, while comparisons with non-pressure-robust variants demonstrate the importance of incorporating the reconstruction into the discrete time-derivative term.

We introduce the notion of an F-space - a set equipped with a family of generalized pseudometrics and marked points, and construct the categorical Gromov-Hausdorff distance for F-spaces. Based on this, we propose a new definition of the distance between dynamical systems, understood in a broad sense as parameterized families of maps. The main focus is on local dynamics: we prove the preservation of Lyapunov stability and asymptotic stability under limit transitions with respect to the new distance (in the latter case, assuming a common radius of attraction). It is established that the class of stable systems is a closed and nowhere dense subset in the space of compact dynamical systems. Next, we investigate the properties of the Hausdorff map, which induces an F-space structure on the family of subsets. In the final part, we introduce another modified version of the Gromov-Hausdorff distance for dynamical systems and use it to calculate the exact distance between torus translations for a non-resonant vector.

We study simple inhomogeneous deformations of known two-dimensional classical lattice models described by conformal field theory (CFT) at large distances. The deformations are chosen to vary slowly at lattice scales, while preserving critical behavior. Globally, we find that such systems are described by a CFT in curved space, and identify the underlying space metric. Our two examples are the Ising model and the six vertex model with domain wall boundary conditions. In the latter boundaries are also inhomogeneous, which complicates the analysis. We nevertheless solve the free case using hydrodynamics. In the presence of interactions we determine the arctic curve which separates a critical fluctuating region from an ordered (frozen) phase.

We study the inverse problem of recovering the constitutive tensors of a homogeneous anisotropic electromagnetic medium without magnetoelectric coupling (non-chiral) from its Fresnel surface, the characteristic variety of Maxwell's equations governing electromagnetic wave propagation. For known isotropic permeability, normalized to $\mu = I$, we prove that the Fresnel surface uniquely determines the permittivity tensor $\varepsilon$, and that the associated Fresnel polynomial is reducible precisely when $\varepsilon$ has a repeated eigenvalue. For general, positive-definite symmetric tensors $(\varepsilon,\mu)$, we prove that the Fresnel polynomial is generically irreducible over $\mathbb{C}$ and we identify the natural gauge symmetry under which it is invariant. Using geometric invariant theory, a powerful tool of modern algebraic geometry, we construct an affine quotient of the parameter space by this gauge action and prove that the induced Fresnel-polynomial map is birational onto its image. We deduce that, outside a proper real algebraic exceptional set, the real Fresnel surface determines $(\varepsilon,\mu)$ up to gauge. This establishes generic uniqueness for the inverse problem and, to our knowledge, provides a new application of affine geometric invariant theory to gauge freedom in a PDE inverse problem.

We prove a non-lattice local central limit theorem and Edgeworth expansions for the logarithm of the norms of products of invertible independent random matrices. Our conditions include a contraction assumption, an assumption that supports of the matrices are “large enough" and their distributions are sufficiently regular. As a byproduct of the proof we are also able to provide a different proof to the optimal rates in the CLT proved in \cite{MatBE}. Like in \cite{MatBE} we provide several sufficient conditions for contraction.

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets, the polytope $P_{6,10}$ attributed to Bugaenko. Together with results of Felikson-Tumarkin ($d \ge 7$) and of Burcroff and Ma-Zheng ($d = 4, 5$), this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets. The proof is computer assisted. Affine Gale duality applied to the complete database of order types on 10 points yields 387 combinatorial types, of which Lannér's classification excludes 83. For the remaining 304, an exhaustive search over Coxeter labellings, with no a priori bound on the dihedral angles, leaves a single realizable Gram matrix. Every rejection is certified in exact arithmetic, and completeness of the search is certified independently by DRAT proofs checked by drat-trim. The same code reproduces the known censuses in dimensions 4 and 5. Code, data and certificates are publicly available.

A dominating set $S$ of a graph $G$ is a locating-dominating set (LDS) if, for each pair of distinct vertices not in~$S$, their neighbourhoods in $S$ are distinct. Finding a minimum-cardinality LDS in finite graphs is a well-known NP-hard problem. On infinite graphs, this problem naturally generalises to finding an LDS of minimum density. While density bounds have been widely studied for specific infinite regular grids, no computational complexity results exist for infinite graphs. We prove that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard. This result bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction. Furthermore, our approach can be adapted to establish NP-hardness for related structural problems on infinite periodic graphs.

Let $\mathfrak{B}_n$ denote the hyperoctahedral group. The type $B$ coinvariant rings $R_{\mathfrak{B}_n}^{(k,j)}$ are quotients of the ring of polynomials in $k$ sets of $n$ commuting variables and $j$ sets of $n$ anticommuting variables by the ideal generated by the diagonal $\mathfrak{B}_n$-invariants without constant term. Building upon the work of Kim–Rhoades (2022), we give an explicit formula for the bigraded Frobenius series of $R_{\mathfrak{B}_n}^{(0,2)}$: the bigraded multiplicity of each irreducible $\mathfrak{B}_n$-character is a single Schur polynomial, so $R_{\mathfrak{B}_n}^{(0,2)}$ is multiplicity-free as a $\operatorname{GL}_2 \times \mathfrak{B}_n$-module. We then determine that the trigraded multiplicity of the sign character of $R_{\mathfrak{B}_n}^{(0,3)}$ is given by a single Schur function. Finally, for all $k$ and $j$, we determine the multiplicity of the standard character in the type $A$ coinvariant ring $R_{n}^{(k,j)}$, as well as the multiplicities of the characters indexed by the bipartitions $((n-1),(1))$ and $((n-1,1),\varnothing)$ in $R_{\mathfrak{B}_n}^{(k,j)}$. These are the first nontrivial characters established for all $(k,j)$ in either of types $A$ or $B$.

We construct explicit examples of non-integral variations of $\mathbb{Q}$-Hodge structures. Our approach leverages Fenchel–Nielsen-type parameterizations, due to Kabaya and Maskit, of the Teichmüller component of relative character varieties. Additionally, we discuss various Diophantine results concerning the $\mathcal{O}_{K,S}$-integral points of such character varieties, and give a new proof of Beauville's classical theorem on families of elliptic curves. We conclude by collecting pathological behaviors of the Hodge locus of non-integral $\mathbb{Q}$VHS, in particular the failure of the Cattani–Deligne–Kaplan theorem and the André–Oort conjecture; our results indicate that for $\mathbb{Q}$VHS which are not $\mathbb{Z}$VHS, what can go wrong must go wrong.

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

Let a bounded and smooth domain in the n dimensional Euclidean space be given and consider a boundary value problem of the half order Dirac operator and its power, with f be the value of the half derivative of the solution in the domain and, g be the trace value of the solution over the boundary. We will show that the solution will be a generalized function u in the half order Hilbert space which is the orthogonal sum of the solution that emerged as a domain integral from the domain value of the fractional half Dirac operator and the one that is the boundary integrals that emerges from the boundary or trace value of the solution. We use two different symbols to designate orthogonal sums for subspaces and functions in orthogonal subspaces of the Sobolev space.

We classify graded Keller maps by the signature of their weight vectors and address which hyperbolic weights support counterexamples and how the nonempty loci are stratified. We construct two families. Cyclic cones populate \(K(3,(1,-1,-1))\) at every composite generic fibre degree \(N\geq 6\), including a six-sheeted member over \(\mathbb{Q}(\sqrt{-15})\) and a 33-sheeted member over \(\mathbb{Q}\). A diagonal family gives \(K(3,(1,-p,-p))\neq\varnothing\) for every \(p\geq2\), with \(\det JF_p=-p\) and generic fibre degree \(2p-1\). The geometric monodromy of every member of both families is alternating or symmetric of full degree; hence the solvable case occurs only in generic degree three. At generic degree twelve, normalized quartic cyclic seeds form a smooth plane cubic over \(\mathbb{Q}\), whose points yield pairwise distinct graded-equivalence orbits. Thus the relevant orbit space contains a nonconstant algebraic family, establishing genuine moduli rather than isolated examples."

We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.

We introduce a non-commutative Laplace transform between functionals on path space and formal series in a tensor algebra, under which a natural convolution of path functionals becomes an algebraic product of series. Applying it to a random unitary matrix-valued path development - the quantum path signature - we show that the governing planar loop equations take a non-commutative spectral form. We then extend the loop equations to a $1/N$ genus expansion, organised by topological recursion, and obtain a hierarchy of integral equations on path space for the corrections.

In this essay, I present the case for total opposition to the use of artificial intelligence in mathematics. I offer proposals for how individuals, departments, journals, and institutions can act in concert to make sure that mathematics survives the coming crisis.

We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let $g$ be a generic Finsler metric on $T ^{2}$, sufficiently $C ^{\infty }$-close to a flat metric, and fix a prime $p$ and a nontrivial free homotopy class $\beta $. If there is a class $\beta$ $g$-geodesic string with multiplicity divisible by $p$, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack $[LX/S^1]$.

We classify rank 2 rel 0 arithmetic affine invariant subvarieties in the minimal stratum H(6). The proof follows and extends Apisa's approach in genus three, reducing the analysis to the classification of rank 1 rel 1 cylinder rigid affine invariant subvarieties in lower-genus boundary strata. In particular, together with the classification of algebraically primitive rank 2 rel 0 orbit closures, this gives a complete description of rank 2 affine invariant subvarieties in H(6).

Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless the method terminates, its even and odd normalized residuals converge separately.

This paper considers boundary stabilization problems for a first-order hyperbolic partial integro-differential equation (PIDE) subject to concurrent input and state delays. The coexistence of these two types of delays complicates control design, especially under the case of large input delay that requires to predict more state information. A backstepping-based boundary controller is developed to achieve stabilization and delay compensation. The design relies on two affine Volterra transformations involving both Fredholm- and Volterra-type integral terms, which results in a four-PIDE kernel equations. To establish their well-posedness, the kernel domain is partitioned along characteristic lines into a finite number of triangular subregions, and the kernel solution is constructed successively in the preceding subregion to the next one. The finite-time stability of the resulting closed-loop system is established. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.