Papers

We prove the Courtade-Kumar conjecture: among all Boolean functions $f\colon \{-1,1\}^n\to\{-1,1\}$, a dictator retains the most information about a uniformly random input observed through independent binary noise.

In this paper, we study the separation and stability properties of radial solutions to a Hénon-type equation with exponential nonlinearity on the hyperbolic space. By transforming the radial hyperbolic equation into a weighted Euclidean equation and applying well-established separation results in Euclidean space, we classify the solution structures or establish sharp alternatives for radial solutions throughout the full range $n\ge2$ and $\alpha>-2$. We also give an affirmative answer to the open question recently posed by Huang and Zhao.

We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels.

Let $C_0=\frac32-\frac1{\sqrt2}\cot\big(\frac1{\sqrt2}\big) =0.67250\ldots$ and $C_1=\frac{C_0+1}{2}= 0.83625\ldots$. Recently, it is obatained by Alpöge and Furman that more than 67.25% of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, by refining the method of Lamzouri, we slightly improve the bounds $C_0$ and $C_1$ in these two results to $C_0+\delta_0$ and $C_1+\frac{\delta_0}2$ with $\delta_0=6.66624\ldots\times10^{-8}$, respectively.

We consider exponential polynomials of the form $F_l(z)=P_1(z)\exp(Q_1(z))+P_2(z)\exp(Q_2(z))+\cdots+P_{l-1}(z)\exp(Q_{l-1}(z))+P_l(z)$, where $l\geq2$, each $Q_i$ is a non-constant polynomial and each $P_i$, $i=1,2,3\ldots,l-1$, is a polynomial (possibly constant), while $P_l$ is a non-constant polynomial. This article studies the topology of the preimages under $F_l$ of the neighborhoods of the essential singularity at infinity. We prove that for each disk $D$ (with respect to the spherical metric) centered at infinity, the set $F_l ^{-1} (D) $ is connected and, in fact, is an infinitely connected domain with all its boundary components bounded. In such a situation, the point at $\infty$ is called the Baker omitted value of the function. Our methods extend those developed by Das and Nayak for the function $\exp(z)+P(z)$ for any non-constant polynomial $P$ (Complex Var. Elliptic Equ. 70 (2025), 1831-1847), providing a broader framework for studying this phenomenon. We further show that these functions do not admit any Baker wandering domain. Finally, we conclude by posing some problems arising out of this work.

Object-valued time series, including distributions, covariance matrices, networks, and compositions, lack the linear structure required by conventional autocovariance-based definitions of long memory. We develop an intrinsic framework for defining and estimating long memory in metric spaces of negative type. An isometric embedding yields a centered Hilbert-valued process, and the trace of its lag-covariance operator provides a signed, additive measure of temporal dependence. Crucially, this trace equals the difference between the marginal mean pairwise distance and expected lagged distance, so the framework and its estimators use only distances between the original objects. We define the memory parameter through the nonsummable decay of this trace and show that it coincides with the usual parameter in Hilbert-valued settings. Estimation uses Bartlett aggregates of distance-based lag measures. Estimating the common marginal distance from the same dependent sample induces a common-centering bias in these aggregates. We derive its finite-sample form and propose iterated block-difference corrections, log-ratio and multi-bandwidth log-slope estimators, and a localized self-consistency refinement. We establish consistency, document substantial bias reduction in simulations, and find long-memory evidence in foreign-exchange return distributions and U.S. electricity-generation compositions.

We study the fixed-$q$ right-edge asymptotics of the ordinary $q$-Krawtchouk ensemble. For $0<q<1$, finite polynomial duality gives a direct proof of convergence of the correlation kernels to an explicit limiting kernel. The local Jacobi limit is a shifted continuous $q^{-1}$-Hermite expression with an indeterminate moment problem, so the limiting Jacobi expression does not determine a unique self-adjoint operator. A single spectral atom, asymptotically saturating the mass bound, selects the realization and yields strong-resolvent convergence. The limiting kernel represents its positive spectral projection and admits spectral-series, Christoffel–Darboux, and partial-theta representations. After a ground-state transformation, the associated dual operator agrees with a two-step spatial Markov operator of Borodin–Corwin. For $q>1$, the limiting Jacobi operator is of Al-Salam–Carlitz~I type and is essentially self-adjoint.

If $f$ is a tuple of functions satisfying an algebraic ODE and $P\in{\mathbb C}(x)[f]$, it is common in applications to transcendental number theory to consider upper bounds for the order of zero of $P(x,f)$ at a given point in terms of $\operatorname{deg}_x P,\operatorname{deg}_f P$. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field $\mathbb C(x)$ is replaced by a number field $K$ and $\operatorname{deg}_x P$ is replaced by the logarithmic height $\operatorname{h}(P)$. Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.

In this paper, we develop a preconditioner that is robust with respect to both the mesh size and the interaction horizon for finite element discretizations of volume-constrained nonlocal diffusion problems. Motivated by the Fourier multiplier of the nonlocal operator, we construct the preconditioner $B_h=G_h+\beta\delta^2D_h^{-1}$, where $G_h$ is a symmetric local Poisson solver, $D_h$ is the lumped mass matrix, and $\beta$ is a kernel-dependent constant. We rigorously prove that the spectrum of the $B_h$-preconditioned system is bounded above and away from zero uniformly with respect to the mesh size and horizon. Moreover, we develop a pair-cell assembly for the finite element matrix that preserves the symmetry and positive definiteness required by the preconditioner and PCG, and establish an $L^2$-error estimate for the finite element discretization. Finally, we use a single symmetric geometric multigrid V-cycle for $G_h$ in the numerical experiments, and the corresponding numerical results confirm the predicted convergence rate and show bounded condition numbers and PCG iteration counts over the tested meshes, horizons, and kernels.

We propose a local-cohomology obstruction to the existence of small Cohen–Macaulay modules over completed section rings of surfaces. Let $Y$ be a smooth connected complex projective surface, and let $H$ be an ample globally generated divisor with $K_Y\equiv4H$ numerically. Put $c(Y,H)=15H^2/8-\chi(\OO_Y)$. The main claim is that, when $c(Y,H)>0$, every nonzero finite reflexive module $M$ over the completed vertex local ring of $\bigoplus_{n\geq0}H^0(Y,\OO_Y(nH))$ satisfies $\dim_\C H^2_\mm(M)\geq c(Y,H)\rk M$. The argument combines elementary transformations of reflexive extensions, Harder–Narasimhan slopes, Bogomolov's inequality, and a two-section Koszul estimate. A fixed-source kernel estimate transfers the resulting cohomology to the punctured spectrum without requiring a grading on $M$. For the degree-six member of the six-line Hirzebruch–Kummer family, $H^2=1080$ and $\chi(\OO_Y)=1926$, so the bound is $99\rk M$. We give an explicit complete-intersection model for this application and spell out the argument leading to the claimed nonexistence of nonzero finite maximal Cohen–Macaulay modules.

We establish the full version of a conjecture of Guth and Zahl, giving a lower bound for the volume of a semialgebraic set that has a large intersection with a collection of $\delta$-separated $\delta$-tubes. Our proof uses o-minimal methods to simplify the proof of Katz and Rogers, who proved the conjecture up to a small factor. We also establish that the constants depend polynomially on the complexity of the semialgebraic set, and more generally in the #o-minimal setting.

Let $K \subset \mathbb{R}^N$ be a convex body containing the origin in its interior. In this work, we study the diameters of random sections of $K$ and derive upper and lower bounds for them in terms of geometric parameters of $K$. Our bounds hold with large probability, and they offer new insights into this widely studied subject. Our upper bound complements the so-called low $M^*$-estimate and in many cases it is much sharper. The two lower bounds that we give are each of a different nature: depending on the body in question each time, either could be better, and in many interesting cases it matches the upper bound too. Subsequently, we apply our results to determine random diameters of $p$-ellipsoids (images of $\ell_p$ balls under diagonal operators), improving upon previously known results and achieving sharp estimates in many cases. One notable application is to Information-Based Complexity Theory, where we manage to establish a simple (and essentially optimal) dichotomy in response to a very natural conjecture posed by Hinrichs, Prochno and Sonnleitner in 2023. Our solution settles precisely when it is useful to replace the optimal information used for the recovery of vectors from a $p$-ellipsoid with random (Gaussian) information, which can be more practical to obtain.

Uncu and Zudilin studied reflections of finite forms of the modulo nine Kanade–Russell identities under $q\mapsto q^{-1}$, leading to ten conjectural sum–product evaluations, including two proposed by Warnaar. Hickerson conjectured four further identities in his study of contiguous relations for the associated parameterized double sums. By constructing finite forms of Hickerson's sums and studying their reflections, Konenkov proposed two additional product identities, whose conjugates give two more. In this paper, we prove all eighteen conjectural identities by combining Nahm-sum dual evaluations established by the author with parameterized divided-difference identities. For the symmetric reflections, Schur-type decompositions and finite rational identities reduce the required evaluations to nonsingular two-by-two linear systems. The asymmetric and cyclotomic cases are treated through $q$-Airy–theta connection formulas, whereas the Hickerson identities are derived from low-order divided differences.

We prove that the solution functor from bounded good filtered complexes of coherent D-modules on a smooth rigid-analytic variety to modules over the positive almost de Rham period sheaf is fully faithful. On filtered vector bundles with flat connection satisfying Griffiths transversality, the construction recovers, up to duality and extension of scalars, Scholze's horizontal sections functor. As an application, we obtain a notion of de Rham Zariski-constructible sheaves, generalising the classical notion of de Rham local systems.

In this paper, we provide a single-letter characterization of the exact common information for a pair of correlated discrete sources, as well as a single-letter characterization of the admissible region for the shared randomness rate and the communication rate in exact channel synthesis.

We establish a new method for the study of imaginary non-CM fields of arbitrary \(2\)-power degree and class number one. In particular, we show that for degrees greater than \(16\), dihedral fields either lie in the ray class field of one of the three imaginary quadratic fields \(\mathbb{Q}(\sqrt{-2})\), \(\mathbb{Q}(\sqrt{-3})\), \(\mathbb{Q}(\sqrt{-67})\), with the conductor of this ray class field supported only on primes dividing \(2\) and with explicit upper bounds on the relevant exponents, or are Hilbert class fields of imaginary quadratic fields. We also give a complete classification of the \(104\) imaginary non-CM fields of degree \(16\) with class number one, providing explicit tables of defining polynomials, Galois groups, base fields, and conductors. These 104 non-CM fields, together with the five CM fields classified by Louboutin and Okazaki \cite{lou7}, complete the classification of all imaginary fields of degree 16 with class number one. For CM fields, the problem reduces to computing relative class numbers via analytic class number formulae, whereas for non-CM fields the absence of a totally real subfield of index \(2\) necessitates a further approach and leads to a richer variety of Galois groups and ramification patterns.

We give a unified study of local derivations on conformal Galilei Lie algebras in three natural settings: the algebra without central extension, the mass central extension, and the exotic central extension. We first determine the structural form of the derivation algebras and isolate the relevant outer derivations. For the mass extension, we complete the one-string rigidity at the two low parameters not covered by the previous one-dimensional treatment, using direct calculations, and combine this with new spatial compatibility arguments to obtain the result in arbitrary spatial dimension. The exotic extension is treated by a self-contained argument from its defining relations. For the algebra without central extension we resolve the remaining algebraic-reflexivity problem completely. If $d\ge2$, every local derivation is again a derivation. In spatial dimension $d=1$, the same conclusion holds when $2\ell$ is even and also for $\ell=\frac12$; however, for odd $2\ell\ge3$ there is an additional $(2\ell-1)$-dimensional space of pure local derivations. This gives a sharp contrast with the mass central extension, where the central Heisenberg pairing removes precisely this extra local freedom. We also determine the Lie algebra structure of the full local-derivation space: in the exceptional one-dimensional non-central case it is an explicit semidirect product with an additional abelian irreducible ideal.

We prove that every triangle-free graph with minimum degree at least $\frac{n}{3}$ is $4$-colorable and thereby settle a problem of Brandt and Thomassé (2005) at the threshold $\frac{n}{3}$. The number four is best possible. For a positive integer-valued function $f(n)=o(n)$, we relate the chromatic number of $f(n)$-vertex subgraphs of the Kneser graph $KG(n,f(n))$ to that of triangle-free graphs with minimum degree at least $\frac{n}{3}-f(n)$. Consequently, for every $0<\delta<1$ and $\varepsilon>0$, and for all sufficiently large $n$, every $n$-vertex triangle-free graph with minimum degree at least $\frac{n}{3}-n^{1-\delta}$ has chromatic number at most $10^{391}+1+\left\lceil{(1+\varepsilon)(1-\delta)}/{\delta}\right\rceil$. We also show that every sufficiently large $n$-vertex maximal triangle-free graph with minimum degree at least $\frac{n}{3}-f(n)$ and chromatic number at least $10^{391}$ contains a bipartite subgraph with parts of orders $\frac{n}{3}-O(f(n))$ and $\frac{2n}{3}-O(f(n))$; the remaining induced subgraph admits a homomorphism to $KG(\frac{n}{3}-O(f(n)),O(f(n)))$. Finally, we connect maximal $K_r$-free graphs with minimum degree at least $\frac{2r-5}{2r-3}n-f(n)$ to $K_{r-1}$-free graphs and extend these results to $K_r$-free graphs. Our proofs employ the recent strong Brandt–Thomassé theorem of Łuczak, Polcyn, and Reiher.

Motivated by results in Kähler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant $8\pi$. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.

Koornwinder moments generalize the Askey–Wilson moments arising in the asymmetric simple exclusion process. Rains conjectured that, under the specialization $t=q$, the minimal numerators of Koornwinder moments have nonnegative integer coefficients. While the one-row case of this conjecture was previously proved by Corteel, Mandelshtam, and Williams using rhombic staircase tableaux, its dual counterpart, the one-column case, has remained open. In this paper, we derive a closed, manifestly positive combinatorial formula for the normalized numerators of the coefficients of the rescaled Askey–Wilson polynomials. This proves Rains' conjecture for one-column partitions, thereby providing the exact dual counterpart to the previous result.

Given two weighted directed graphs of order \(n\), we study how much their overlap can deviate from its random average under relabelling, and how concentrated the distribution of their intersection is when they are placed at random. Bollobás and Scott studied these problems on weighted $k$-uniform hypergraphs and raised the corresponding questions for directed graphs. In this paper, we study these problems for weighted directed graphs and obtain results analogous to those in the weighted hypergraph setting.

Large-scale multipurpose cohort studies and biobanks often omit covariates needed for specific downstream analyses. We study target-population inference for generalized linear regression when key covariates are completely absent from the target data but observed in a related source population. Standard missing covariate methods are not directly applicable because they require at least partial observation of the covariates in the target population. We develop a doubly robust transfer learning framework under a sub-population shift assumption, which allows the distribution of the observed outcome and covariates to differ between populations while requiring the conditional distribution of the missing covariates given the observed variables to be shared. Unlike linear regression, nonlinear target estimating equations require parameter-indexed conditional functionals of the missing covariates rather than a fixed collection of low-order conditional moments. Our estimator combines importance-weighted source estimating equations with imputation of these conditional functionals. Under the identifying assumption, the estimator remains consistent when either the density-ratio model or the conditional-covariate model is correctly specified. Under regularity conditions, it is root-$n$ consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

In this article, we discuss a multivariable analogue of Lehmer's parity problem. We generalise the notions of Lehmer numbers as Lehmer $q$-tuples for a fixed odd prime $q$. We count the number of Lehmer $q$-tuples and derive an asymptotic estimate with explicit error terms. This deals with estimating various character sums and Deligne's bound for incomplete exponential sums.

A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. Let $\mathrm{spex}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs. For integers $t\geq1$ and $l\geq2$, let $P_{t\cdot l}$ be the starlike tree with $t$ branches of length $l-1$. For sufficiently large $n$, Yin, Li, and Meng [arXiv:2504.04364v1] characterized the unique extremal graph for $\mathrm{spex}(n,P_{t\cdot l})$ when $t=1$, $t=2$, or $t\geq4$. They left the case $t=3$ open and proposed a natural candidate for the extremal graph. We show that this candidate is not extremal and determine the unique extremal graph for $\mathrm{spex}(n,P_{3\cdot l})$. For every $l\geq3$ and all sufficiently large $n$, this unique extremal graph is $K_1\vee\bigl(2P_{2l-3}\cup qP_{l-2}\cup P_r\bigr),$ where $q$ and $r$ are integers satisfying $n=2(2l-3)+q(l-2)+r+1,$ $q\geq0,$ $0\leq r<l-2.$

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce $\mathcal D$-subsolutions extending the determinant sublevel condition of Sui–Sun. Using the auxiliary comparison method of Guo–Phong–Tong and pluripotential theory for complex Monge–Ampère equations, we obtain relative $L^\infty$-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree $d$. We characterize the determinant increment bound with exponent $d/n$ by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative $L^\infty$ estimates for the corresponding fully nonlinear equations.

Service organizations use simulation, pilot studies, and historical data to select service policies that balance aggregate performance and fairness. Existing work primarily evaluates the operational cost of fairness, defined as the performance loss caused by restricting the feasible policy set. In this research, we identify and study implementation cost as a distinct and equally important dimension of fairness, defined as the sampling effort required to verify fairness and reliably select the best fair policy. Specifically, we consider fixed-budget policy selection under fairness requirements based on mean performance, under-service risk, quantiles, and upper tail outcomes, and show that different fairness metrics can induce the same fair policy set and the same operational cost while requiring substantially different amounts of evidence because their estimators have different local verification difficulty. We further show that, near a critical fairness tolerance, the required sampling budget for implementation scales inversely with the square of the distance to the boundary, creating a finite-budget implementation gap. These results imply that the fairness metric, the fairness tolerance, and the available sampling budget should be designed jointly, because a population-level fairness requirement may be operationally attractive but not statistically implementable at the available budget. To translate this perspective into practice, we develop a fairness-guided adaptive allocation algorithm that directs samples toward the ranking and fairness verification comparisons governing false selection. Experiments in synthetic, call center, and emergency department settings demonstrate the implementation cost mechanisms identified in this work and show that the proposed allocation uses the sampling budget much more effectively.

We prove quantitative stability and instability estimates for reversible periodic solutions of strictly Legendre-convex Lagrangian systems. If the configuration dimension is $n$, the fixed-period Morse index is $k$, and $k_D,\nu_D$ are the Dirichlet index and nullity on a half-period, set \[ \widetilde k=k-2k_D-\nu_D. \] Then the algebraic number $\nu_+$ of positive real Floquet multipliers satisfies \[ \nu_+\ge 2n-2\widetilde k, \] and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most $2\widetilde k$ Floquet multipliers can lie outside the positive real axis. In particular, when $k=0$ we recover the positive-real-spectrum theorem for reversible minimizers, while for nonconstant autonomous brake orbits the bounds improve by two. If, moreover, $n\ge2$ and the algebraic multiplicity of $1$ is two, an index-one brake orbit is transversely hyperbolic, with $n-1$ stable and $n-1$ unstable directions. Decoupled oscillators show that the refined bounds can be sharp even when the estimate based on $k$ alone is vacuous, and the Hénon–Heiles straight-line orbit provides a nonlinear illustration of the half-period mechanism.

Point cloud-based CFD enables flow analysis directly on discrete points obtained from 3D scanning and medical imaging, bypassing surface reconstruction, geometry cleanup, and boundary-fitted mesh generation. Derived from immersogeometric analysis, the method immerses the point cloud in a background mesh and enforces no-slip conditions on discrete points through a Nitsche-based weak boundary condition (BC). The framework delivers accurate velocity fields, pressure distribution, and integrated loads; however, accurate prediction of the local wall shear stress (WSS) has remained a critical challenge. The geometry intersects the background mesh arbitrarily, producing cut elements that lack the regularity required for consistent gradient evaluation. The issue is compounded by the stabilization term of the weak BC, whose parameter estimation in the symmetric Nitsche formulation is dependent on the cut configuration and affects the variationally consistent definition of traction from which the WSS is computed. In this work, we propose a new method to obtain accurate wall shear stress in immersed flow analysis with application to point cloud-based CFD, using a non-symmetric Nitsche's formulation with near-wall modeling and a patch-based stress recovery approach with traction compatibility. The method is validated on canonical benchmarks and applied to turbulent flow past a sphere and to a patient-specific aorta, showcasing excellent agreement with reference results.

A topological vector space E has the countable intersection property for subspaces if every family of closed subspaces intersecting to $0$ contains a countable subfamily that also intersects to $0$. We prove that every separable F-space satisfies this property, whereas a TVS with an uncountable Markushevich basis does not. Finally, we construct a separable TVS that fails to satisfy this property.