Papers

For a finite permutation group, a base is a set of points with trivial pointwise stabiliser, and the generalised Saxl graph records which pairs of points lie together in a base of minimum size. Burness and Giudici conjectured that any two vertices of the Saxl graph of a primitive group of base size two have a common neighbour, and Freedman, Huang, Lee and Rekvényi extended this conjecture to arbitrary base size. We disprove both. For each integer $B\ge2$ we construct infinitely many primitive groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size two, where this is the usual Saxl graph, we obtain three further infinite families, one each of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types; in the affine and product type families the Saxl graphs have diameter exactly three. This answers Problem~21.29 in the Kourovka Notebook in the negative. In the positive direction, we prove the Burness–Giudici conjecture for every primitive affine group whose point stabiliser is almost quasisimple of sporadic type, completing work of Lee and Popiel. We conjecture that no base-two counterexample of almost simple or diagonal type exists.

We introduce and study a new geometric representation for general classical spin models. It consists of two coupled percolation configurations that are the joint FK (random cluster) representation of the Ginibre rotation of two independent copies of the spin model, and can be viewed as an extension of Sheffield's cluster swapping defined in the context of height functions. Our approach combines the advantages of the random current and FK representations of the Ising model: it provides a percolation interpretation for various truncated correlation functions and at the same time satisfies the FKG inequality (for a large subclass of models). We highlight its strength and versatility by establishing two very different results. We first prove the converse of the classical fact that boundary multi-point correlation functions of planar Ising models are given by Pfaffians of the two-point functions. Indeed, we show that if a general spin model on a general graph satisfies the Pfaffian relations, then up to natural local modifications it must actually be an Ising model on a planar graph. In particular, the algebraic Pfaffian relations imply the topological feature of planarity. Our second application is a proof of sharpness of the phase transition for a class of spin models first considered by Ellis, Monroe and Newman, which we do by generalising the celebrated argument of Duminil-Copin and Tassion, replacing the use of the random current with our new representation.

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on $\mathrm{GL}_n$. Although the original problem is on $\mathrm{GL}_n$, its spectral components are described entirely by the automorphic spectrum of $\mathrm{GL}_2$. As an application, we prove a partial reciprocity formula relating the second moment of $\mathrm{GL}_n\times\mathrm{GL}_n$ Rankin–Selberg central $L$-values and a mixed moment of $L$-values of $\mathrm{GL}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin–Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.

Scenario optimization, conformal prediction, and related distribution-free certification methods use finite samples to construct decisions or prediction sets with violation-risk guarantees for fresh observations. In several classical settings, the conditional violation risk follows an exact beta law, whose tail has a beta-binomial representation and whose parameter is a support, calibration, or compression dimension. This paper identifies the deterministic boundary mechanism behind these formulas and derives the corresponding law when the observed boundary size is random. A decision rule is represented by an acceptance set for future observations, together with a boundary map selecting the sample points responsible for that set. The resulting pair is called a proper projective boundary scheme when held-out samples are accepted precisely if the full-sample boundary is retained, and accepted non-boundary samples can be deleted without changing that boundary. For every such scheme, the conditional law of the violation risk given the observed boundary size is determined by the boundary's cross-sample complexity profile. A stable profile yields the usual beta law, whereas a varying profile produces an exact profile correction. The framework covers scalar order-statistic calibration, support-reconstructive scenario programs, cascaded support-removal certificates, coordinatewise envelopes, and Pareto-frontier calibration with vector scores. It also yields conditional probabilistic certificates and a no-go result explaining why observed complexity alone is insufficient.

We construct permutation polynomials of the form x + \gamma Tr^{q^2}_q(h(x)) over the finite field F_{q^2}. More precisely, we present two families ofpermutation polynomials whose coefficients range over all elements of the field rather than being restricted to a proper subfield. We also compute the compositional inverses of both families. Our techniques involve the evaluation of certain Kloosterman sums together with the analysis of some equations over finite fields, and we believe these can be of independent interest.

For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\langle X,\nu\rangle=-\sigma_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

A diblock copolymer is a soft-matter composed of two chemically distinct block of repeating monomers covalently bonded together at an end-to-end junction to form a single polymer chain. In this paper, we establish the existence of infinitely many smooth periodic unbounded domain patterns of Delaunay-type in $\mathbb{R}^3$ that optimize the energy distribution in diblock copolymer melts. We emphasize that pattern domains at the equilibrium correspond to stationary sets of the screened Ohta–Kawasaki free energy functional \begin{align*} \mathcal{P}_\gamma(\Omega) := |\partial\Omega| + \gamma \int_{\Omega}\int_{\Omega} G_{\kappa}(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $\gamma>0$, $\kappa>0$ and $G_\kappa(r)=\frac{1}{r} e^{-\kappa r}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler–Lagrange equation \begin{align*} \mathcal{H}_\Omega (x):= H_{\partial\Omega}(x) + \gamma \int_{\Omega} G_{\kappa}(|x-y|) \mathrm{d}y = Const \quad \text{on } \partial\Omega, \end{align*} where $H_{\partial\Omega}$ denotes the mean curvature of the surface $\partial\Omega$. By analyzing the linearization of $\Omega \mapsto \mathcal{H}_\Omega$ around flat cylinders and applying the Crandall–Rabinowitz bifurcation theorem, for any $\kappa > 0$ and sufficiently small $\gamma > 0$, we prove the existence of non-trivial, $2\pi$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.

The study of the largest prime factor in polynomial sequences can be traced back at least to the late 19th century in the work of Störmer. Mahler (1933) and Chowla (1934) proved that the largest prime factor of $n^2+1$ grows at least as fast as $\log_2 n$. In 2023 we improved this to $(\log_2 n)^2/\log_3 n$. In this note we show the lower bound $(\log_2 n)^2/\log_4 n$, and that when this bound is nearly sharp it also holds for many prime factors of $n^2+1$.

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding–Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity $Q$: the interior mass is bounded by $C(n)Q$, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of $\mathbb{H}^n$. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

We define the totally positive part of the coordinate algebra associated to a Cartan datum of simply laced type. We show its relation to the canonical basis of the corresponding plus part of the enveloping algebra.

Consensus-based optimization (CBO) is an efficient metaheuristic for global optimisation with attractive mathematical properties, allowing global convergence results even in non-convex settings. In practice it suffers greatly from the curse of dimensionality, as do most particle-based optimisers. Different strategies have been proposed to apply CBO even for high-dimensional optimisation problems, the most prominent being the so-called anisotropic noise model. However, a recent work by Bonandin et al. highlighted that this method performs well primarily on separable objective functions, and even small coordinate rotations severely degrade the performance. Motivated by this observation, in this work we study the case where the objective function is separable only in a linearly transformed coordinate system. To leverage the efficiency of anisotropic CBO we propose a reparametrisation scheme, which provably estimates such a coordinate transformation and then applies CBO in the new variables. Furthermore, we show how our scheme can be interpreted as noise adaptation in CBO. Numerical examples highlight the efficacy of the method, demonstrating significant performance improvements on challenging benchmark objectives.

We study simply-laced Coxeter groups whose Coxeter diagram is the line graph $\mathcal{L}\Gamma$ of some simple graph $\Gamma$; such groups act on the set of vertices of the graph by transpositions. Our goal is to describe the point stabilizer of this action in detail. For any tree $\Sigma$, we show that the stabilizer is a finite-index reflection subgroup, and identify its Coxeter diagram, thereby discovering finite-index embeddings of Coxeter groups $W(\mathcal{L}^{[2]}\Sigma) \hookrightarrow W(\mathcal{L}\Sigma)$ for any tree $\Sigma$, including infinitely many cases where the stabilizer itself is simply-laced.

This paper presents a finite-length polynomial-domain formulation of lifted-product quantum low-density parity-check (QLDPC) codes. We formulate the code construction over the quotient ring F 2[D]/(D L + 1), where polynomial base matrices are lifted entrywise to binary circulant blocks. This representation gives a compact algebraic description of the lifted-product parity-check matrices and allows CSS orthogonality to be analyzed before binary expansion. We show that the standard circulant lifting map is compatible with polynomial conjugation, which implies that the resulting binary matrices satisfy the CSS commutation constraint. The construction is illustrated with a constraint length 7, rate 1/2 NASA convolutional code example, and numerical examples are provided from a 3 x 4 polynomial parity-check matrix. The finite-length performance of selected constructed codes is then evaluated over the depolarizing channel using various benchmark decoders. The resulting framework gives a structured method to construct finite-length lifted-product QLDPC codes from small polynomial base matrices.

In this paper, we introduce the controlling $L_{\infty}[1]$-algebra of extended Rota-Baxter algebras. As applications, we obtain the cohomology of extended Rota-Baxter algebras and homotopy extended Rota-Baxter algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we describe the intrinsic relationship among extended Rota-Baxter algebras, extended Rota-Baxter Lie algebras, and the eigenvalues of the extended Rota-Baxter operator. As the generalization of Rota-Baxter algebras with weight and modified Rota-Baxter algebras, the corresponding results also hold for those algebras.

We prove the Virasoro constraints for the relative Gromov–Witten theory of all smooth projective effective orbifold curves, allowing relative conditions at ordinary points. The absolute Jiang–Tseng Virasoro conjecture for smooth projective effective orbifold curves follows as a corollary.

We develop a theory of relative (quasi-)perfect complexes for algebraic spaces. Our main result proves that for a pseudocoherent kernel relatively perfect over both factors, the loci where its fiberwise Fourier–Mukai transform is fully faithful or an equivalence are open. The kernel need not be perfect, and these loci commute with Noetherian base change. Moreover, we provide an explicit autoequivalence for elliptic fibrations, and establish that smoothness and Gorensteinness are derived invariants.

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $\eta>0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\leq n^{-\frac{2}{3}-\eta}$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $\delta>0$ and $\rho\in(0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min\{a,b\}\leq n^{-\frac{1}{2}-\delta}$ and $ab\leq \frac{1}{2}-\rho$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.

We consider non-Hermitian random band matrices with growing bandwidth and study convergence of their empirical spectral distributions to the circular law. Let $N$ denote the matrix size and $W_N$ the bandwidth. From a universality perspective, it is conjectured that the circular law holds whenever $W_N\to\infty$. Previous results have mainly required $W_N\gg N^{1/2}$, with the principal exception of \cite{Han2511}, which proves the $W_N\to\infty$ circular law for an open-boundary block-tridiagonal model. Here we prove the circular law at this optimal threshold for several periodic models and under the near-optimal condition $W_N\gg\log N$ for a genuinely discrete model. For the periodic hard-indicator profile and its uniform and polynomially tapered generalizations, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to\infty$. At the same threshold, we prove the circular law for continuous, integrable profiles locally bounded below on every finite interval, including exponentially and Gaussian decaying profiles, with circular complex Gaussian entries. For the periodic full-block model, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to\infty$. The finite-third-moment assumption in the bounded-density results can be weakened to a finite $(2+\alpha)$-moment assumption. For the same full-block model without a density assumption, we prove the circular law for centered variance-one real subgaussian atoms when $W_N\gg\log N$. The proof uses compositions of random transfer operators. An established high-band circular-law input on a short auxiliary ring calibrates the full exterior coefficient norm, and a boundary-uniform local comparison lifts this calibration to target rings even when $N/W_N$ is arbitrarily large.

We study optimal inflow control for a linear transport equation subject to uncertainty in both time-dependent downstream demand and transport velocity. The random velocity induces a random travel time and thereby changes the relation between an inflow decision and the demand observed at its arrival time. For a fixed downstream observation window, we derive explicit optimal controls in the continuous-time setting and for piecewise constant controls. We decompose the irreducible stochastic error into contributions from demand and velocity uncertainty and show that the latter admits a bound that is linear in the velocity variance. As a computationally attractive alternative, we analyze a deterministic mean-velocity proxy whose additional performance loss admits a higher-order bound. We also establish Lipschitz stability of the optimal control with respect to perturbations of the velocity law in the Wasserstein distance. Numerical experiments illustrate the analytical results and investigate the proxy strategy for an uncertain nonlocal transport model beyond the linear theory.

We apply Anderson acceleration to the improved Arrow–Hurwicz (IAH) method for the finite element solution of the steady-state incompressible Navier–Stokes equations. The IAH scheme avoids saddle-point solves by decoupling the velocity and pressure updates, but can require prohibitively many iterations, particularly at high Reynolds numbers. To place the acceleration on a rigorous footing, we reformulate the IAH iteration as a nonlinear fixed-point operator $G$ for the grad-div augmented discrete formulation induced by the scheme and establish its well-definedness, Lipschitz continuity, and Fréchet differentiability, thereby verifying the required smoothness conditions locally near the fixed point. Numerical experiments on problems with known analytical solutions, lid-driven cavity flow up to $Re = 15{,}000$, and channel flow over a full step demonstrate that the resulting Anderson-accelerated improved Arrow–Hurwicz algorithm substantially reduces iteration counts and CPU time while retaining the reported manufactured-solution convergence rates and centerline-velocity agreement.

Posed by Lebesgue in 1914, the universal covering problem asks for the smallest-area planar convex set containing a congruent copy of every set of diameter at most one. We introduce an exact Reuleaux-type variational hierarchy for this constant: its monotone finite-arc values $\Lambda_M$ satisfy $a_{\mathrm{Leb}}=\lim_{M\to\infty}\Lambda_M$, and each level is a continuous finite-dimensional problem. We prove $0\le a_{\mathrm{Leb}}-\Lambda_M\le C M^{-2}$, giving a controlled finite-arc route to the constant itself. As a certified low-order realization, an outward-rounded interval certificate for a regular finite Reuleaux subtest proves $a_{\mathrm{Leb}}\ge0.834$, improving the lower-bound benchmark established by Brass and Sharifi in 2005.

The Modulo Krinkle tilings of Imura (arXiv:2506.07638) form a family of non-periodic, spiral monohedral tilings parametrized by a reduced fraction $m/k$ and an integer $t\ge 2$. We show that the Farey sum (mediant) $(m_1+m_2)/(k_1+k_2)$ of two Farey-adjacent parameters is realized by an exact geometric operation on prototiles: the lower boundary path of the $(m_1+m_2,k_1+k_2)$-prototile is obtained by concatenating the parents' lower paths after an edge-length-preserving progressive rotation (fan-twist) of their edges. Conversely, every prototile admits exactly one fan-twist splice decomposition – no non-adjacent parameters ever splice – the cut position being $k_1=m^{-1}\bmod k$, and the recursion descends the Stern-Brocot tree to a single unit edge. The combinatorial core of the operation is the classical standard factorization of Christoffel words; the contribution here is its exact edge-isometric realization on circular direction systems and the resulting structure theory for the Modulo Krinkle family, including the recently introduced variants: we prove a separation theorem stating that every variant prototile is the common recursively-generated core plus finitely many direction-invariant decoration edges. As a corollary of Imura's spiral-arm count formula, the two Farey parents are visible in the offset-free tiling itself: the numbers of counterclockwise and clockwise spiral arms are $tk_1$ and $tk_2$.

We prove a Faber-Krahn inequality for the first eigenvalue of the $p$-Hermite operator (the weighted $p$-Laplacian with Gaussian weight) on Lipschitz domains in $\R^n$ under Robin boundary conditions with positive Robin parameter. The main result states that, among all domains of given Gaussian measure, the first eigenvalue is minimized by a half-space, and equality holds only for half-spaces. This extends the classical Faber-Krahn inequalities for the $p$-Laplacian \cite{BucurCV} to the $p$-Hermite operator and generalizes the linear case \cite{ChiacchioMathann} to the full nonlinear regime $p>1$.

Given a system $\mathbf{f}=(f_1,\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\mathbf{f}$. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by $\mathbf{f}$. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by $\mathbf{f}$. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.

We show that a superselction sector describing a single nonabelian anyon violates Haag duality, which is equivalent to the fundamental quantum-information theoretic principle of uniqueness of purifications. We provide a simple physics argument as well as a rigorous proof using sector theory, and instantiate our result concretely in Levin-Wen models. We also show that the associated ground state does not allow for quantum steering despite being a pure state. Finally, we show that this ground state violates approximate Haag duality, disproving the conjecture that all gapped ground states satisfy this condition. Our result implies that there are gapped phases of matter where Haag duality fails at every point in the phase.

Ricci curvature measures the average stability of geodesics under transverse perturbations, but how it should be defined in infinite dimensions is often unclear. This paper proposes a definition of Ricci curvature on the space of Hamiltonian diffeomorphisms on the two-dimensional flat torus $\mathrm{HDiff}(\mathbb{T}^2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\mathrm{HDiff}(\mathbb{T}^2)$ by finite-dimensional Lie groups $\mathrm{SU}(N)$. We derive a formula for the Ricci curvature tensor on $\mathrm{SU}(N)$ and provide numerical evidence for its convergence in the large-$N$ limit to our conjectured finite value. Additionally, we explore potential applications for hydrodynamics through the Lyapunov stability of gravest wave modes and Arnold's tradewind estimates for long-term weather predictability. Our framework extends to a wide range of settings. We demonstrate this by introducing Ricci curvature on the state spaces of fluids on rectangular domains, the Lagrangian averaged Euler equation induced by the $H^1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.

For integers q at least 3 and r at least 1, consider the directed Cayley graph on the cyclic group of order qr whose allowed steps are the integers from 1 through r. A Hamilton cycle H is called a q-layer balanced Hamilton starter if, for every step from 1 through r and every residue class modulo q, H contains exactly one arc of that step whose tail belongs to the given residue class. The r translates of H by successive multiples of q then form a Hamilton decomposition of the digraph. A q-layer balanced Hamilton starter is called chain-compatible if one arc can be selected from each Hamilton cycle in this decomposition so that the selected arcs form a simple directed path. In both constructions, the compatible deletion chain is the arithmetic step-2 path beginning at 0 and ending at 2r. For q equal to 4, a chain-compatible starter is obtained explicitly whenever r is congruent to 1 modulo 4 and r is at least 9, while for q equal to 3 one exists for all sufficiently large r congruent to 5 modulo 6. The proofs are constructive: the four-layer case uses an ABAB step word, while in the three-layer case a directed rotational terrace is lifted to a 3-layer balanced directed 1-factor and a fixed four-arc trade joins its two cycles. Deleting the unique prescribed-path arc from each translated Hamilton cycle gives r Hamilton paths; together with the prescribed path, these form an optimal decomposition of the arc set into r+1 directed paths in both cases.

We introduce the notion of proper proximality for countable measured groupoids, extending the corresponding notion for countable groups. We prove that this groupoid property is equivalent to a strengthened form of relative proper proximality for the associated groupoid von Neumann algebra. We investigate permanence properties and show, in particular, that proper proximality is preserved under finite direct products, formation of transformation groupoids, and measure equivalence. We also identify broad classes of properly proximal groupoids, including transverse measured groupoids and free product groupoids, and prove that inner amenable groupoids are never properly proximal.

We prove global well-posedness for the two-dimensional infinite-depth gravity water wave system coupled to a pair of point vortices of opposite strengths. The vortex dipole is initially placed deep below the free surface and oriented so that its leading motion is away from the interface. We show that the vortices retreat almost linearly, and that the velocity induced on the free boundary is therefore integrable in time. Combining this decay with Wu's cubic formulation, together with a transition-of-derivatives and localization argument, we obtain global bounds for small localized perturbations of the flat surface.

Quantum hardware is beginning to address structured combinatorial optimisation, but two steps still block practical use: mapping real operational models onto hardware-compatible instances, and converting noisy hardware output back into feasible decisions. Here we introduce a backend-agnostic computational interface that compiles the discrete decision layer of stochastic unit commitment into a move-based maximum-weight independent set (MWIS) problem, while retaining continuous dispatch and feasibility recovery in the classical computational layer. We validate the approach in a green hydrogen scheduling setting and deploy it on the QuEra Aquila neutral-atom quantum processor. This is the first end-to-end industrial scheduling workflow that connects real operational decisions to programmable neutral-atom hardware through a solver-agnostic MWIS representation. Across a 15-day hardware campaign on 50-node instances, hardware-generated solutions after classical refinement match or exceed the dispatch margins obtained from exact MWIS on every day. When scaling to 144 nodes, encoding quality remains stable, while the probability that the full atom array survives, rather than graph embedding, emerges as the dominant bottleneck to further scaling. Together, these results establish a hardware-compatible computational pathway toward larger problem scales, and lay the groundwork for exploring regimes in which exact classical optimisation may no longer scale efficiently.