Papers

Let \(X=(X_1,\ldots,X_n)\) be uniform on the Boolean cube, and let \(Y\) be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability \(p\). For a \(k\)-bit quantizer \(Q(X)\), reporting \(k\) input coordinates retains \(k(1-h_2(p))\) bits of information, making coordinate projections a natural quantization benchmark. For \(k=1\), this is the Courtade–Kumar conjecture: every Boolean function \(f\) satisfies \(I(f(X);Y)\leq 1-h_2(p)\), with equality for coordinate functions. We prove the conjecture in every dimension and for every crossover probability, without a balance assumption. The proof tracks the information gap relative to a coordinate as the channel correlation varies. After a monotone rearrangement, an entropy-flow identity relates the derivative of this gap to the edge boundary of the associated Boolean decision set. A sharp local entropy comparison, Fourier analysis, and logarithmic Sobolev estimates for pivotal sets imply that any positive gap must grow as the noise decreases, contradicting its nonpositive value at the noiseless endpoint. For \(k\geq8\), a coding-theoretic phenomenon emerges. We construct a shortened-Hamming-code quantizer that beats the coordinate benchmark for \(k=8\), and a direct-sum argument extends the failure to every \(k\geq8\). Thus structured coding can retain more information than coordinate selection. Exhaustive searches within several structured code families and Monte Carlo searches over random codebooks for \(2\leq k\leq7\) find that structured codebooks outperform random ones but remain below the coordinate benchmark. This motivates our conjecture that coordinate projections are optimal for \(2\leq k\leq7\).

In this paper, we prove the Brauer $A(\infty)$ condition for good non-defining primes. Consequently, we show a conjecture of Navarro on coprime actions and Brauer characters for primes larger than 3.

We construct complete ALH hyperkähler metrics on complements of anticanonical elliptic curves with zero self-intersection in smooth rational surfaces. This answers the existence question of Filip–Tosatti, related to the picture proposed by Haskins–Hein–Nordström. Conversely, we prove that every hyperkähler ALH gravitational instanton admits, for each induced complex structure, a compactification by an anticanonical elliptic curve in a rational surface.

Appell functions are a large class of multi-variable quasi-elliptic functions, which are also instances of higher depth mock modular forms. A central aspect of these functions is their non-holomorphic modular completion. Our previous work developed Appell functions for a general positive definite lattice $\Lambda$, and expressed the non-holomorphic completion in terms of the generalized error functions $M_P$, which are integrals over a $P$-dimensional hyperplane in $\Lambda\otimes \mathbb{C}$. In this sequel, we express $M_P$ as a $P$-dimensional iterated integral of variables $w_j\in \mathbb{H}$, such that the completion of the Appell function takes the form of an iterated Eichler integral of modular forms. We apply this to the case of the root lattice of the $A_N$ Lie algebra, which is of particular interest because of their appearance in partition functions of topological twisted Yang-Mills theories.

Drifting provides a direct route to one-step generative models, but applying it directly to stochastic transition modeling requires multiple samples of the next state conditioned on the same current state. Standard trajectory data, however, typically provide only one realized next state for each observed current state and therefore do not provide an empirical approximation of the corresponding conditional distribution over possible next states. We introduce block-triangular joint drifting, which instead applies a projected drift field to the empirically accessible joint distribution of consecutive states. Importantly, the block-triangular architecture preserves the current-state marginal while making its second component a direct sampler of the conditional distribution of possible next states. The resulting surrogate generates stochastic trajectories with one model evaluation per time step, without auxiliary generative steps between time steps. Numerical experiments demonstrate accurate marginal and trajectory-dependent statistics and favorable accuracy-cost tradeoffs compared with deterministic, diffusion-, flow-, and distillation-based generative surrogate models.

Let $\Omega=G/\mathbb K$ be an irreducible bounded symmetric domain of rank $r$ and dimension $d.$ In this paper, we study the joint subnormality of $d$-tuple of multiplication operators induced by the coordinate functions on reproducing kernel Hilbert spaces of holomorphic functions on $\Omega$ determined by $\mathbb K$-invariant kernels. We introduce the notion of contractive $d$-tuple associated with $\Omega$ and prove that the weighted Bergman shifts on $\Omega$ are contractive $d$-tuple precisely when they are jointly subnormal. A characterization of joint subnormality further leads to the study of a class of moment problems, which we refer to as twisted moment problems. We establish that the twisted moment problem is equivalent to an appropriate Hausdorff moment problem. This equivalence provides a moment-theoretic characterization of joint subnormality for the multiplication operators under consideration.

The KPZ upper-tail field introduced by Liu and Zhang arises as the scaling limit of the narrow-wedge KPZ fixed point in a microscopic neighborhood of a conditioned large height. In this paper, we give a direct probabilistic identification of the upper-tail field. We identify the grounded positive-time sector with the KPZ fixed point started from the two-sided Brownian \(V\)-profile \(B_{\mathrm{ts}}(2x)-2|x|\), while the negative-time sector is represented through an explicit exponential tilt of the same evolution. The argument uses the directed-landscape variational structure and the local Brownian comparison under upper-tail conditioning. As consequences of these representations, we recover probabilistically the positive- and negative-time large-scale limits of the KPZ upper-tail field.

Reliability theory gives a mature language for layered systems, but its formal tools are not yet standard in frontier AI control. We apply them to Google DeepMind's defenses against rogue deployment. The same control stack can have cubic, quadratic, or linear rare-failure suppression depending on its failure domains. Birnbaum importance identifies which component improvements buy the most nominal reliability, while prevention changes the population on which recovery is demanded. These results give concrete guidance about what to separate, improve, measure, and test.

We establish global existence, decay, and scattering for sufficiently small, smooth, and localized perturbations of a constant non-vacuum equilibrium of a relativistic quantum hydrodynamic system in three space dimensions. In logarithmic-amplitude and phase variables, the equations form a semilinear system of coupled wave equations. The skew-symmetric coupling between the time derivatives cancels in the energy identity, yielding a natural derivative energy. The linearized system has two dispersion branches. At low frequencies, the slow branch exhibits Schrödinger-type dispersion, while the fast branch has a spectral gap; at high frequencies, both branches are wave-like. The main nonlinear difficulty arises from quadratic interactions with nontrivial time and space-time resonances. We show that the symbol of every active quadratic interaction contains the corresponding interaction phase as an exact factor. This structural cancellation removes the resonant denominator and allows us to eliminate the quadratic terms by a nonsingular normal-form transformation. Combining this transformation with dispersive and energy estimates, we obtain uniform-in-time bounds for the derivative energy and $\langle t\rangle^{-3/2}$ decay of the first derivatives in $L^\infty$. We further prove that the nonlinear solution scatters to a solution of the linearized system in the high-order derivative energy norm.

A Kähler structure on a real Lie algebra $\mathfrak{g}$ is a pair $(\omega, J)$ of a 2-cocycle and a complex structure on $\mathfrak{g}$ which are compatible in the sense that $J$ is skew-symmetric with respect to $\omega$. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either $\{0\}$, $\mathbb{R}^2$, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.

Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}$. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I ($P_{\rm I}^{2k}$) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the $P_{\rm I}^{2k}$ equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.

A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous geodesic spaces: besides the classical examples, they include the universal metric trees of finite valence. We also prove that every complete all-set-homogeneous length space is geodesic, and hence the same classification holds in this setting.

Let $\Sigma$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $\Sigma$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $\Sigma$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

We study the linearization of the random $p$-adic orthogonal matrix model. Let $A_n$ and $B_n$ be Haar-random $n\times n$ special orthogonal and alternating matrices over $\mathbb{Z}_p$, respectively. For an odd prime $p$, we prove that $\mathrm{cok}(A_n-I_n)$ and $\mathrm{cok}(B_n)$ have the same limiting distribution as $n \to \infty$ through integers of the same parity. The two parity-dependent limits are interchanged for orthogonal matrices of determinant $-1$, while the full orthogonal group gives their equal mixture. For $p=2$, linearization does not preserve the limiting cokernel distribution, and we explicitly determine the limiting distributions for the special orthogonal model in terms of the spinor norm, which controls the parity of the $2$-adic valuation of the order of the torsion subgroup.

A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):=\sum_{\ell\in{C}(G)}1/\ell$ as a measure of the richness of the cycle length spectrum ${C}(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k, n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\leq n/2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\geq 2k$ satisfies $s(G)\geq\sum_{\ell=2}^{k}1/(2\ell)$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.

The concept of weighted universal Value-at-Risk superadditivity (WUVS) was recently introduced by Chen et al. (2026) as a generalization of the question whether for some infinite mean distributions convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. In this short note we prove that the property WUVS can basically never hold for discrete distributions except for the case of comonotonicity. This implies as a corollary that for discrete distributions with infinite mean it can also never hold that convex combinations of i.i.d. random variables can stochastically dominate the parent distribution. This settles an open problem mentioned in Müller (2025).

In this paper we study whether there is a global to local principle for the homology of order complexes of finite lattices. We explore whether the non-vanishing of the homology of a lattice in degree $a+b-2$ forces the existence of a pair of lattice-complements whose open intervals below have non-vanishing homology in degrees $a$ and $b$ respectively. We consider the same question also in the language of multigraded free resolutions. We provide positive evidence for the existence of such pairs and give counterexamples to strengthenings of the question.

We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.

We reexamine F. Wiener's classical proof that the Bohr radius of $H^\infty$ is $1/3$. We replace $H^\infty$ by the multiplier algebras of complete Pick spaces and describe the Bohr set for a wide class of such algebras. In particular, we show that the Bohr set of the multiplier algebra of the Drury–Arveson space $H^2_d$ is the closed $\ell^2$-ball of radius $1/3$. Other cases covered by our description of Bohr sets include Dirichlet-type spaces of analytic functions in the unit disc and a class of Hilbert spaces of ordinary Dirichlet series, studied earlier by McCarthy and Shalit.

These lecture notes are intended as an introduction to the theory of rational Dunkl operators, the associated special functions and related Markov processes with an emphasis on examples which are related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones of positive semidefinite matrices. We start with a comprehensive introduction into Dunkl theory: Dunkl operators, the intertwining operator and its positivity, the Dunkl kernel and the Dunkl transform, the Dunkl Laplacian and the associated heat semigroup. We further give an outline of the connection with Calogero-Moser-Sutherland models and generalized Hermite polynomials. Moreover, of central interest will be product formulas, generalized translations and associated commutative hypergroup structures on closed Weyl chambers. In particular, we explain how Dunkl theory for particular multiplicities is related to Riemannian symmetric spaces of Euclidean type and Bessel hypergroups on the matrix cones, and how this leads to a bunch of multiplicities for which the Weyl-group invariant Dunkl theory admits a probability preserving translation and an associated commutative hypergroup structure on the closed Weyl chamber. We finally discuss Markov processes on R^N which are related with Dunkl theory with an emphasis on connections to random walk on groups and hypergroups. In particular associated martingales, martingale characterizations, moment functions and Appell characters are studied, i.e. diffusion-reflection processes with the Dunkl Laplacians as generators.

We provide the first compact examples of type-changing generalized Calabi–Yau manifolds using Fei's construction of non-Kähler Calabi–Yau manifolds on branched covers of twistor spaces \cite{Fei16}. The structure, which is generically of type 1 and changes to type 3, is constructed on the product of a hyperelliptic surface of genus three and a four-torus, $\Sigma_3 \times \mathbb{T}^4$, with starting point a non-Kähler Calabi–Yau structure.

We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For $\beta=2$, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended $\mathsf{Sine}$ process from arbitrary symmetric initial configurations with power-law counting exponent $q\in(0,2)$. This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For $\beta\geq1$, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.

Floquet bundle theory serves as a natural extension of the eigenvalue theory for elliptic and periodic parabolic operators to the spectral theory of non-autonomous, non-periodic parabolic operators, which provides a useful spectral tool for studying threshold dynamics in more general non-autonomous systems. However, current understanding and applications of the normalized principal Floquet bundle remain rather limited compared with the eigenvalue theory for elliptic or periodic parabolic problems. In this work, we first extend several fundamental properties of the principal eigenvalue to the normalized principal Floquet bundle of non-autonomous, non-periodic parabolic operators, and obtain a series of parallel results. Furthermore, we focus on the asymptotic behavior of the principal Floquet exponent with respect to the generalized frequency and diffusion rate under various typical parameter limits (including both independent and coupled parameter regimes) for the operator under homogeneous Neumann boundary conditions. The main tools are a parabolic comparison principle developed in this work for long-time limit processes, which is used to derive bounds for the principal Floquet exponent via suitably constructed sub- and super-solutions, supplemented by Harnack's inequality and long-time averaging techniques. Finally, we employ the normalized principal Floquet bundle to define two critical numbers (as extensions of the basic reproduction number) for a non-autonomous spatially diffusive SIS epidemic model, examine the extinction and weak persistence of the disease, and investigate the limiting behavior of the basic critical numbers with respect to the generalized frequency and diffusion rate.

We consider right-linear operators $T$ on a right Hilbert module $V$ over the Clifford algebra R_n, whose S-spectrum lies in an acute double sector. For these bisectorial operators, we introduce spectral projectors P_\pm associated with the two cones of the double sector. They decompose the Hilbert module into two submodules V=V_++V_-, and the bisectorial operator T into two sectorial operators T|_\pm. A crucial but non-trivial, cornerstone in this theory is the boundedness of the projectors P_\pm, which is, in turn connected to a bounded H^\infty-functional calculus of the operator T. We provide two practical criteria: either the squared operator admits a bounded H^\infty-functional calculus, or the operator is m-accretive. Finally, we apply these results to the gradient operator \nabla_a with nonconstant coefficients. For the particular gradient with constant coefficients, we are even able to derive explicit representations of the submodules V_\pm and the projectors P_\pm in Fourier space. Moreover, we identify the sign of the gradient operator with the Clifford-Hilbert transform. This sign plays a central role in the fractional powers of vector operators, which are used, for instance, in the non-local Fourier law of heat propagation.

We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension $2n+1$, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order $n+1$, modulo an $L^{2}$ term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.

We develop a new endgame for numerically computing singular solutions to complex analytic systems by means of a homotopy path whose endpoint is the solution to be computed. When the solution is singular, prediction-correction path tracking may fail at the endpoint. Instead of tracking the path all the way to its end, endgames sample nonsingular points on the incoming path and approximate the endpoint by interpolating the samples with a truncated Puiseux series. By collecting sample points along a logarithmic spiral in the complex plane, the proposed spiral endgame balances between the superior numerical conditioning of a circular sample (known as the Cauchy endgame) and the update efficiency of a linear sample (known as the power series endgame), these being tied together as two extremes of the new approach. The numerical conditioning of the methods are analyzed and compared, and computational examples illustrate the effectiveness of the new spiral sampling method compared with the circular sampling method of the Cauchy endgame.

Inspired by the static problem $(\alpha,\beta)$-Dominating Set, we propose a general temporal domination problem, called $(\alpha,\beta)$-Temporal Dominating Set ($(\alpha,\beta)$-TDS). We show that this problem encompasses Temporal Dominating Set, and additionally provides first temporal extensions of problems such as $k$-Dominating Set and $\alpha$-Dominating Set. In this paper, we study the parameterised complexity of $(\alpha,\beta)$-TDS with respect to temporal neighbourhood diversity (TND), temporal modular-width (TMW), and temporal cliquewidth (TCW). We obtain fixed parameter tractability results for all values of $\alpha$ and $\beta$ with respect to TND; W[1]-hardness with respect to TMW and TCW whenever $\beta$ is in the problem input, or whenever $\alpha\in (0,1)$ and $\beta$ is a fixed constant; and para-NP-hardness with respect to TCW when $\alpha = 0$ and $\beta = 1$, or $\alpha = 1$ and $\beta = 0$.

Tensor-based modulation (TBM) schemes are a promising approach for unsourced random access (URA), where user separation relies on decomposing the received signal tensor via the canonical polyadic decomposition (CPD), typically computed with alternating least squares (ALS). Standard ALS, however, treats the factor matrices as unstructured and fails to exploit the discrete structure of the tensor sub-constellations. We propose DVB-ALS, a discrete variational Bayesian CPD framework with specific priors, tailored to a tensor structure with the corresponding encoding strategy, combined with iterative computation of an approximate posterior distribution. A discrete Gaussian mixture prior on one Grassmannian factor softly aligns the estimates toward the constellation points. The remaining factors are jointly modeled with a structured Gaussian prior whose posterior mean is constrained to the Khatri-Rao product manifold and posterior variance upper-bounded to prevent norm divergence during inference. The resulting closed-form coordinate ascent algorithm jointly estimates all latent factors and their uncertainties. We integrate DVB-ALS into DVB-TBM to design a complete URA receiver with single-user demapping, polar decoding with cyclic redundancy check (CRC) verification, and successive interference cancellation (SIC). Simulation results show significant gains over standard ALS-based decomposition and robust detection performance in URA settings, outperforming state-of-the-art schemes under high system loads.