Papers

We prove a conjecture of Bradshaw and Atale asserting a Ramanujan type master theorem for the kernel $\pi^m/\sin^m(\pi s)$, whose poles at the non-positive integers have order $m$. The residue data at these poles are organized by a family of polynomials studied by Airault with coefficients the central factorial numbers of the first kind. The proof reduces the conjecture to a Laurent series identity for $\csc^m$, established by induction from an elementary differential recursion, and the resulting theorem holds under a growth hypothesis strictly weaker than Hardy's. As applications, we obtain integral representations for powers of the cosecant, a closed form for the Mellin convolution powers of the Cauchy kernel $(1+x)^{-1}$, and a family of differential identities satisfied by the Airault polynomials.

We provide an explicit analytical evaluation of the known rational Ramanujan-type series for the theory of signature 2. Focusing on the singular moduli $k_r$ for $r \in \{2, 3, 4, 7\}$, we demonstrate that the underlying elliptic identities can be established through modular transformations of degree 2. In particular, we showcase a family of rational harmonic Ramanujan-type series for $1/\pi$ involving higher-degree polynomials

For a permutation of a finite set, we define a landscape by summing a discounted weight over the times each point returns to itself. The landscape value is determined by the point's least period. We relate the trace of the associated resolvent to fixed-point and exact-period counts, and show that the associated determinant is the reciprocal of the Artin-Mazur dynamical zeta function. We apply this framework to the discrete cat map modulo $N$. For every prime $p\ne5$, all nonzero points have period equal to the order of the cat-map matrix modulo $p$, yielding explicit formulas for the landscape trace and determinant. We prove a period-lifting formula at odd primes $p\ne5$, derive exact fixed-point and period counts modulo $5^a$, and combine prime-power results for composite moduli using the Chinese remainder theorem.

We take the first step toward the study of set-theoretic principles below the axiom of choice $\mathsf{AC}$ and the axiom of foundation $\mathsf{AF}$ by studying Frucht's theorem, an ordinary mathematical theorem which is provable with either $\mathsf{AC}$ or $\mathsf{AF}$ but not provable without both, and its variants. Specifically, we propose a number of such principles, study the relations between these principles and the standard axioms, and prove provability and unprovability results using (infinite) graph-theoretic constructions and permutation models, which draw a preliminary map of this new area of set theory.

For a stratified variety $X$ and a motivic invariant $\mathbb{H}$, we consider the stratified invariant which interpolates the invariant of $X$ and that of its interior $X^{\circ}$. Based on observations in moduli theory, we introduce the notion of an echelon tower of stratified varieties and then prove explicit inductive formulae for the stratified invariants of the moduli spaces $\overline{M}_{0,n}$ of stable curves of genus $0$ and the Fulton-MacPherson varieties $Y[n]$ for any smooth projective variety $Y$. Using these, we show that the mysterious bivariate deformations of the even degree Poincaré polynomials of $\overline{M}_{0,n+1}$ and $\mathbb{P}^1[n]$ in \cite{BercziKiem2026} are nothing but stratified virtual Poincaré polynomials.

Let $(X,Y)$ be independent standard real Gaussian random variables, and let $(P\in\mathbb C[X,Y])$ satisfy $\mathbb E!\left(P(X,Y)^m\right)=0$ for every $(m\ge1)$. Using the complex coordinates $Z=\frac{X+iY}{\sqrt2}, \qquad W=\frac{X-iY}{\sqrt2},$ we prove that the monomial support of $(P)$ is strictly one-sided with respect to the weight $\operatorname{wt}(Z^aW^b)=a-b$. Thus either every monomial occurring in $(P)$ has positive weight, or every monomial has negative weight. It follows that, for every $(Q\in\mathbb C[X,Y])$, $\mathbb E!\left(Q(X,Y)P(X,Y)^m\right)=0$ whenever $(m>\deg Q)$. This proves the two-variable Gaussian Moments Conjecture with the explicit threshold $m\ge\deg Q+1$. The main ingredient is a prime-isolation theorem for exposed faces of the Newton polygon of $(P)$. A $(p)$-adic valuation argument at moment indices of the form $(m=qp)$ separates the contribution of a chosen face from all remaining multinomial strata. Frobenius reduction then forces the constant terms of all positive powers of the associated one-variable Laurent polynomial to vanish. The theorem of Duistermaat and van der Kallen implies that the face has weights of one strict sign, while a planar convex-geometric argument rules out support containing weights of both signs. Combined with the known one-variable case and counterexamples in dimensions $(n\ge3)$, this determines the dimensions in which the Gaussian Moments Conjecture holds.

For symplectic and even orthogonal groups over a $p$-adic field, we determine the number of local Arthur packets containing the local theta lift of a tempered representation at the first occurrence in the going-up tower. These counts show that the local theta lifts may lie in many more local Arthur packets than those predicted by the Adams conjecture.

Given an embedded shrinker $\Sigma$ in $\mathbb{R}^{n+1}$ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with $\mathbb{R}^k$, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on $\Sigma$. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.

We introduce a new stability condition for log Calabi–Yau fibrations over curves. We prove that it gives rise to a proper Deligne–Mumford stack with a projective coarse moduli space, whose boundary still parametrizes flat fibrations over curves.

We provide necessary conditions for pro-$\mathcal{C}$ subgroups to embed into free profinite products, where $\mathcal{C}$ is a variety of finite groups that does not contain all finite groups. Under suitable hypotheses, we extend this result to profinite groups acting $k$-acylindrically on profinite trees. Finally, we show that these results can be applied to important classes of profinite groups.

The hard sphere model is a classical model from statistical physics in which particles are represented by equal-sized spheres. Longstanding predictions from the physics literature indicate that in $\mathbb{R}^2$ and $\mathbb{R}^3$ the system undergoes a phase transition, but it remains a major open problem to confirm this. We prove the existence of a phase transition for this model in the hyperbolic plane.

We prove a partial $\epsilon$-regularity theorem for sequences of integer multiplicity stationary geodesic networks in $\mathbb{R}^n$ converging to a multiplicity $k$ line.

We consider a collection of pairwise commuting quantum observables in the setting of Berezin–Toeplitz quantization of a closed Kähler manifold and assume that the Arnold–Liouville theorem applies to their principal symbols. We use joint eigensections of these observables to define isometric embeddings of the quantum spaces into $L^2(\Lambda_{a_0})$, where $\Lambda_{a_0}$ is a fixed Liouville torus. These embeddings allow a broad class of quantum observables, including some defined by discontinuous functions, to be realized as sequences of operators on $L^2(\Lambda_{a_0})$ that converge strongly to multiplication operators. We discuss the spectral implications of this convergence and give applications to contractions of Lie algebra representations and to pairs of spectral projections of quantum observables.

We give a direct differential-geometric proof of a result of Crainic on the cohomology of pullbacks of Lie algebroids under surjective submersions: If the fibers are connected and have vanishing cohomology through degree $n$, pullback with coefficients in any representation is an isomorphism through degree $n$ and is injective in degree $n+1$. The proof reduces to a vanishing result for leafwise cohomology and uses an exhaustion-and-patching argument due to Buchdahl. No local triviality or finite-type assumption on the fibers is required. We also prove the analogous result for holomorphic Lie algebroids.

Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to conjectures of Wehlau concerning norms. We give faithful four-dimensional representations of $C_2^3$ and $C_2^4$ over $\mathbb F_8$ for which every nonlinear orbit norm is decomposable; in the $C_2^4$ example, the invariant ring is polynomial, thereby disproving both Wehlau's norm conjecture and its unrestricted extension.

We give a new, purely geometric proof of the classical fact that billiard maps in confocal ellipses commute. Existing proofs of this result rely on symplectic geometry and an invariant measure on the space of oriented lines; ours uses only elementary projective and Euclidean geometry. The argument rests on a main theorem describing how a billiard reflection can be constructed geometrically via tangents to confocal ellipses, from which the commutation property follows directly. This also yields, as a byproduct, an incidence result for tangents at four reflection points, revealing a hidden symmetry in the confocal billiard configuration. The main theorem recovers, via purely synthetic means, a fact previously established only by direct computation (Berman et al., 2024).

Let $\lambda>1$ be a real algebraic number. We construct continuum many power series $f(z)=\sum_{k\geq0}a_kz^k$ of radius of convergence exactly one such that every nonzero coefficient $a_k$ is algebraic and has modulus $\lambda^m$ for some $m\geq0$. Moreover, for every integer $s\geq0$, the derivative $f^{(s)}$ takes algebraic values at all algebraic points of the open unit disk and is transcendental over $\mathbb{C}(z)$. The proof combines algebraic polygonal cancellation with a sparse polynomial-block argument. This shows that a multiplicative rank-one restriction on coefficient moduli is compatible with algebraicity of the full analytic jet at every algebraic point once algebraic phases are allowed.

We develop a goal-oriented error-estimation framework for first-order system least-squares (FOSLS) finite element methods based on the physical PDE adjoint rather than the adjoint induced by the least-squares formulation. Identified explicitly from the differential operator, its mixed boundary conditions, and the output, the physical adjoint admits its own first-order flux system and hence a native, built-in least-squares estimator, which the least-squares-induced adjoint does not. Our error identities rest on a primal-dual corrected functional: they follow from the continuous primal and adjoint equations alone, hold for arbitrary conforming approximations, and require no Galerkin orthogonality. For outputs containing a weighted Dirichlet-boundary flux, whose weight becomes the essential datum of the adjoint, we construct two corrected approximations, one from the potentials and one also using the fluxes, and prove product-type error estimates in the primal and adjoint errors. The least-squares functionals then yield computable a posteriori bounds and a balanced marking indicator whose element sum equals the product of the two estimators exactly. Numerical experiments confirm the predicted convergence and the effectiveness of the marking.

Random quantum satisfiability is a natural quantum analogue of random constraint satisfaction and a basic model for frustration-free local Hamiltonians. Despite extensive work on its satisfiable and unsatisfiable regimes, the quantitative location of the random quantum \(k\)-SAT threshold has remained poorly understood, with the best general upper bounds leaving a large gap to the known lower bounds. In this paper we prove a new upper bound on the satisfiability threshold of random quantum \(k\)-SAT. Our result improves the previously known asymptotic upper bound by a factor of order \(k\), giving a bound of order \(2^k/k\). The improvement is also significant at small values of \(k\); in particular, for random quantum \(3\)-SAT we obtain a substantially smaller explicit upper bound than the one previously available. The proof combines the geometric formulation of generic quantum satisfiability with a dimension-decay analysis of the full satisfying subspace. The key input is a multiplicative Shearer-type inequality for tensor-product subspaces, which quantifies how global dimension forces nontrivial local dimension on typical sets of qubits.

The Cayley graph $\text{Cay}(G,S)$ is the graph whose vertex set is the group $G$, where two vertices $x$ and $y$ are adjacent if and only if $xy^{-1}$ or $yx^{-1}$ lies in some fixed subset $S$ of $G$. We call the elements of $S$ generators. A circulant graph is a Cayley graph where $G$ is finite and cyclic. Chromatic numbers of circulant graphs have been studied by many authors. A general formula due to Heuberger for the chromatic number of a circulant graph is known when $S$ has two elements, but no such formula is known when $S$ has three or more elements. We say that an element $x$ of $S$ is indispensable if $S\setminus\{x\}$ does not generate $G$. We say that $S$ is minimal if every element of $S$ is indispensable. By a result of Garcia-Marco and Knauer from 2024, if $G$ is nilpotent and $S$ is minimal, then $\text{Cay}(G,S)$ is $3$-colorable. In this article, we prove three main results. First, we give an upper bound for the chromatic number of a circulant graph with three generators, one of which is indispensable. Second, we present an alternate proof of the theorem of Garcia-Marco and Knauer for the case of abelian groups. Third, we apply these methods to provide a considerably more systematic (and potentially generalizable) proof of Heuberger's theorem for the chromatic number of circulant graphs with two generators. Throughout this paper, our primary tool is the theory of Heuberger matrices, for which we provide a brief primer.

We prove new inequalities for elementary symmetric polynomials (ESPs) for vectors that sum to zero, and for square matrices with zero row and column sums. We apply these results to obtain a unified upper bound on the mean-field approximation guarantee for permutation mixtures, as well as a sharp $\chi^2$ version of the de Finetti theorem for finite sequences over a small alphabet. The main proof ideas were developed by the GPT-5.5 Pro model.

In model predictive control (MPC) with time-varying objectives, predicted signals need to be often incorporated in the cost function, such as prices in energy system operation. These are, however, often difficult to predict from the historical trajectory of these signals alone, as they may depend on other contextual events. We propose LiFT-MPC, an MPC framework that integrates a LiFT (Language-in-the-Loop Feedback Tuning) correction scheme to refine such predictions within the MPC loop. The prediction mechanism is updated online via a control-performance loss function, and we establish a performance guarantee for the resulting closed loop system. Numerical experiments using a realistic example of energy-storage management with real prices and news context to improve predictions, demonstrate an improved economic performance

The article develops further Kolmogorov's Algorithmic Complexity Theory. The definition of Randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as Mutual Information in individual infinite sequences. Applications to several areas, like Probability Theory, Theory of Algorithms, Intuitionistic Logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property.

We prove a conjecture of Monical, Tokcan, and Yong that every fixed positive power of the Vandermonde determinant is non-SNP in all sufficiently many variables, where a polynomial is non-SNP if there is a lattice point in its Newton polytope that does not appear with nonzero coefficient. This means our result proves that for every even power $k\geq4$, there is always such a missing lattice monomial in large enough dimensions. The odd case follows from alternation, and the quadratic case was previously known. For every even power $k\geq4$, we exhibit an explicit lattice point in the Newton polytope of $a_{\delta_k}^k$ whose coefficient vanishes. The vanishing is obtained from a Dyson constant-term identity, proved using the finite-variable Jack scalar product and Macdonald's specialization formula. The key even-power construction and proof strategy arose from prompting with OpenAI Codex (GPT Sol 5.6 Extra High), a large language model; the complete transcript appears in the appendix. The authors subsequently checked and organized the argument. The accompanying Lean formalization is available at https://github.com/steven-le-thien/vandermonde-snp.

We solve a longstanding problem, going back at least to Rivière 1998 and open even in the physically most relevant case $n=3$, by proving a sharp curl-Sobolev inequality on $\mathbb{S}^n$ when $n\equiv 3\pmod 4$: for every $\frac{n-1}{2}$-form $\alpha$, the conformally invariant quotient satisfies (with positive denominator) \[ \frac{\Big(\int_{\mathbb{S}^n}|{\rm curl}\alpha|^{\frac{2n}{n+1}}\,{\rm dV}\Big)^{\frac{n+1}{n}}}{\int_{\mathbb{S}^n}\langle{\rm curl}\alpha,\alpha\rangle\,{\rm dV}} \ge \frac{n+1}{2}\,\omega_n^{\frac1n}. \] We also classify all extremals in terms of Killing forms. By conformal invariance, the same result holds on $\mathbb{R}^n$. We then give geometric and variational applications that settle several open conjectures in geometry and mathematical physics. First, we show that on $\mathbb{S}^n$ the round metric is the unique optimizer for the conformal invariant $\mu([g_{{\rm st}}])$. Second, we prove that the unique minimizers of the $3$-energy $\int_{\mathbb{S}^3}|{\rm d} u|^3$ in the homotopy class of the Hopf map $\pi:\mathbb{S}^3\to\mathbb{S}^2$ are exactly $\pi\circ\Phi$ with $\Phi\in{\rm Conf}^+(\mathbb{S}^3)$, confirming a conjecture of Rivière. Third, for the Faddeev-Skyrme energy $\mathcal{FS}_\rho$ on $\mathbb{S}^3$, we establish global minimality of the Hopf map in the full predicted range: for every coupling constant $\rho\le \sqrt{2}$, the unique global minimizers in its homotopy class are precisely $\pi\circ R$ with $R\in\mathrm{SO}(4)$, as expected since Ward 1999. Fourth, in the presence of Dirac zero modes on $\mathbb{S}^3$, we prove the sharp lower bound $\|{\rm curl} A\|_{3/2} \ge 3\omega_3^{\frac 2 3}$ for the magnetic field and characterize equality in terms of Killing spinors; in particular, this yields a sharp criterion for the existence of zero modes and answers a question of Frank-Loss for $n=3$.

We develop a Dirac deformation theory that interpolates between twisted Dirac geometry and Poisson geometry, and prove that this deformation is compatible with the principal structural operations of Dirac geometry: reduction along strong Dirac maps, integration to quasi-symplectic groupoids, and Morita equivalence of Lie algebroids. The fundamental example is the deformation of the Cartan–Dirac structure $L_G$ on a compact Lie group~$G$ to the Kirillov–Kostant–Souriau Poisson structure on $\mathfrak{g}^*$, and its lift to a deformation of the quasi-symplectic groupoid $D(G)\rightrightarrows G$ to the symplectic groupoid $T^*G\rightrightarrows\mathfrak{g}^*$. As applications, we obtain a uniform deformation theory recovering, as special cases, the deformation of quasi-Hamiltonian to Hamiltonian reduction along conjugacy classes, the Steinberg and Sevostyanov slices to their additive (Kostant, Slodowy) counterparts, the multiplicative parabolic and unipotent reductions, the quasi-Hamiltonian implosion to symplectic implosion, and the multiplicative Moore–Tachikawa varieties to their additive analogues. In the language of quasi-symplectic groupoid presentations of $1$-shifted symplectic stacks, our main reduction theorem yields a smooth deformation of the corresponding reduced spaces.

Interval rings comprise a class of one-dimensional integrally closed local integral domains which are overrings of a two-dimensional regular local ring. We construct them by intersecting carefully chosen valuation rings and prove that they have various nice properties. Our interest in these rings is that they represent a major stepping stone toward classifying all integrally closed overrings of a two-dimensional regular local ring.

This paper investigates the validity of the unilateral approximation in near-field non-line-of-sight (NLoS) propagation for large-scale linear antenna arrays. Utilizing multiport communication theory, the study evaluates whether the electromagnetic interaction from the receiver back to the transmitter can be neglected. The analysis examines both discrete arrays with fixed spacing and continuous arrays with fixed aperture as the number of antennas increases. Results show the approximation is asymptotically satisfied on average for both cases. This provides a theoretical foundation for using correlated Rayleigh fading models in physically consistent next-generation wireless systems.

We prove that every unital separable ASH algebra with subquadratic growth has uniform property $\Gamma$ whenever it has no nonzero finite-dimensional representations. When simple and non-elementary, these algebras therefore satisfy the Toms–Winter regularity conjecture despite the fact that they generally fail its three conjecturally equivalent properties. In light of the second author's recent construction of a unital simple separable AH algebra of quadratic growth which fails uniform property $\Gamma$, we conclude that the quadratic dimension growth scale (equivalently, the 2-norm slow dimension growth scale) is the precise geometric threshold governing the potential failure of uniform property \(\Gamma\). We also extend recent work of Elliott–Niu and Vaccaro to the optimal subquadratic scale by proving that separable unital \(C^*\)-algebras with locally tracially subquadratic RSH approximation have uniform property \(\Gamma\), provided that they have no nonzero finite-dimensional representations.

Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.