Papers

Suppose $n\ge2$ and $K\subset\C^n$ is compact and polynomially convex. Write $q(z)=(e^{-iz_1},\ldots,e^{-iz_n})$, $X_K=(\C^*)^n\setminus(K\cap(\C^*)^n)$, and $Y_K=\C^n\setminus q^{-1}(K)$. We prove that $X_K$ and its logarithmic covering $Y_K$ are Oka. More strongly, $Y_K$ admits normalized holomorphic families of entire embeddings through prescribed holomorphic sections. The proof combines polynomial pair Hamiltonian transport in multiplicative coordinates, compatible with the coordinate hyperplanes, with a fixed Fatou–Bieberbach model. Applications include Oka slab crosses over arbitrary nonempty connected planar domains, complements of periodic equal-radius Euclidean balls centered at products of planar lattices, and complements of sufficiently small closed metric balls in products of elliptic curves.

We extend the classical trace Paley-Wiener theorem for $p$-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when $E/F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$. These distributions are relative character of the form $$\pi\mapsto I_\pi (f), \ f\in C_c^\infty(\mathrm{GL}_n(E)),$$ as $\pi$ ranges over $\mathrm{GL}_n(F)$-distinguished irreducible tempered representations, and are constructed from a $\mathrm{GL}_n(F)$-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the "generic" Bernstein center of the unitary group $U(n)$.

Let $f(n)$ be the largest integer such that every set of $n$ points in the plane contains $f(n)$ points no two at distance $1$, and let $m_1$ be the supremum of the upper densities of Lebesgue measurable planar sets with no two points at distance $1$. Larman and Rogers observed that $f(n) \ge m_1 n$. We prove that $f(n)/n \to m_1$; that is, $f(n) = (m_1 + o(1))n$. This is the first half of a conjecture of Dúcz and Varga, and with the bound $m_1 \le 0.247$ of Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki it gives $f(n) \le (0.247 + o(1))n$. The proof builds a random independent subset of the algebraic plane, invariant under algebraic translations and rotations, whose probability of containing the origin is at least the worst finite independence ratio, and turns it into a measurable planar set of at least that upper density. Its only non-classical input is the Haar rigidity theorem for wild character laws from the OpenAI preprint The Euclidean plane is not five-colorable, which has a machine-checked Lean proof. The whole argument has been formalized in Lean 4 on top of it.

We study universal unequal error protection of a single special message over discrete memoryless channels. We construct a sequence of codes with a decoding rule that does not depend on the channel transition law. We derive an achievable error exponent for the special message while ensuring reliable decoding of the ordinary messages. The resulting exponent strictly improves on the natural equal-error-protection random-coding baseline and remains positive as the rate approaches capacity. For symmetric channels, it converges to the optimal red-alert exponent at capacity.

Suppose that each of $n$ players rolls a different die and the players are ordered by their rolls. Can the dice use disjoint labels such that ties are impossible, while all $n!$ orders remain equally likely? We give two general constructions of such dice and solve some related problems.

We study the undetected-error exponent of discrete memoryless channels with erasure decoding. Assuming that the undetected-error and erasure probabilities vanish, we derive an upper bound on the undetected-error exponent at positive rates. The bound uses Somekh-Baruch's genie-aided argument where an auxiliary output provides the decoder with a list of codewords. It recovers Telatar's tight rate-zero converse and is at least as tight as the straight-line bound. A numerical example shows that the improvement over the straight-line bound can be strict.

We derive a quantitative central limit theorem under sublinear expectations from a spatial endpoint regularity estimate for the one-dimensional \(G\)-heat equation. For bounded Lipschitz initial data, the second spatial derivative and the time derivative of its solution are Lipschitz in space and \(1/2\)-Hölder in time away from the initial time. In particular, \([u_{xx}(t,\cdot)]_{\mathrm{Lip}}\le C_G\Lip(\phi)/t\). Combining this estimate with the nonlinear Stein identity introduced in Song \cite{Song2020} and a replacement argument yields a bound of order \((1+\log n)/\sqrt n\) for bounded Lipschitz tests when the third absolute moment is finite and the lower variance is positive. The final replacement step is estimated directly, avoiding the nonintegrable singularity at time zero.

Let $h\geq2$ be an integer and let $\mathbb{N}_0=W_1\cup\cdots\cup W_h$ be a partition into infinite sets. For $W\subseteq\mathbb{N}_0$, let $A(W)$ denote the set of all finite nonempty sums of distinct powers $2^w$ with $w\in W$. We give a necessary and sufficient condition for $A=A(W_1)\cup\cdots\cup A(W_h)$ to be a strongly minimal asymptotic basis of order $h$. This provides a complete answer to a problem posed by Nathanson in 1988.

In this paper, we focus on decoding optimal Reed-Solomon codes of dimension two for insdel errors up to the half-Singleton bound. We present a decoding algorithm for the general insertion-and-deletion scenario with a time complexity smaller than brute-force decoding. Our algorithm corrects insdel errors up to the half-Singleton bound, which exceeds the scope of existing methods. Additionally, for the special cases of insertion-only and deletion-only insdel error patterns, we present two different algorithms with a time complexity smaller than our general algorithm.

We introduce upper and lower discrete intermediate dimensions for unbounded subsets of Euclidean space and extend them to metric spaces. The definitions restrict the covering radii within a scale-dependent range of dyadic shells and take a supremum over an auxiliary parameter. The resulting dimensions are monotone in the parameter and recover the Barlow–Taylor discrete Hausdorff dimension at zero. We prove continuity at every positive parameter, with a sharp quantitative bound, and give an example with a jump at zero. At the upper endpoint, the upper dimension equals upper mass dimension for uniformly discrete Euclidean sets; the analogous lower mass identity fails. We establish a mass distribution principle and prove quasi-isometric invariance under a uniform coarse packing condition. For spaces with uniform polynomial volume growth, we construct clustered subsets attaining the continuity bound. Independent thinning truncates these spectra at the resulting mass dimension, almost surely for all parameters simultaneously. Applications include the discrete Heisenberg group with its word metric and the ranges of simple symmetric random walks.

Let $N$ be a 3-dimensional integral homology sphere. We prove the existence of a sequence of 4-dimensional topologically negatively curved manifolds $M_k$, $k=1,2,\dots$, satisfying the following: (1) all aspherical manifolds $M_k$ have the same universal cover $U_N$ depending only on $N$; (2) all fundamental groups $\pi_1(M_k)$ are hyperbolic and share the same Gromov boundary $X_N$ depending only on $N$; (3) the groups $\pi_1(M_k)$ represent infinitely many distinct quasi-isometry classes. The last conclusion, which represents the new feature, shows that negatively curved geometries in dimension 4 exhibit substantially richer structure than those in dimension 3.

We study radial bound states of \[ \Delta u-\abs{u}^{q-1}u+\abs{u}^{p-1}u=0 \quad\text{in }\R^n, \qquad n\ge3,\qquad 1<q<p<\frac{n+2}{n-2}. \] For every integer $k\ge1$, we prove that there is exactly one radial bound state with $k$ simple zeros and $u(0)>0$. Its absolute maximum is attained at the origin. This confirms the zero-mass, Sobolev-subcritical case of Tang's double-power uniqueness conjecture. The proof adapts Tang's phase-transition analysis and combines it with a finite-level separation argument inspired by Cortázar–García-Huidobro–Yarur. Three main difficulties arise in the zero-mass double-power setting: the sign changes of key comparison quantities in the low-amplitude region, the possible failure of pointwise positivity of an auxiliary quantity when $n=3$ and $p+q<3$, and the absence of exponential decay at infinity. We overcome these difficulties by introducing bridge functionals to propagate the required sign information through low amplitudes, establishing a weighted integral positivity estimate in the exceptional three-dimensional range, and combining a radial-flux tail estimate with finite-level inverse-branch comparison. We also establish the strict ordering of the bound-state shooting heights and show that the shooting variation along the \(k\)-node state has exactly \(k+1\) simple zeros.

Wyner's common information $J(p)$ and the common entropy $G(p)$ of a pair of random variables $(X,Y)$ with finite ranges and joint distribution $p$ are both defined by an optimization over the auxiliary random variables $W$ that make $X$ and $Y$ conditionally independent: $J(p)$ is the infimum of the mutual information $I(X,Y;W)$, and $G(p)$ is the infimum of the entropy $H(W)$. We prove that both quantities are NP-hard to compute. For Wyner's common information, deciding whether $J(p)$ is strictly smaller than a given integer $r$ is NP-hard, and the problem is NP-complete on tables that are uniform on their support and have the same number of positive entries in every row. The reduction is from Maximum Edge Biclique and rests on an exact criterion, in terms of rectangles in the support of $p$, for improving on the auxiliary $W=X$; every yes-instance satisfies $J(p)<r-1/(4N^2)$ and every no-instance satisfies $J(p)=r$, where $N$ is the number of positive entries of $p$. For the common entropy, deciding whether $G(p)\le r$ for a rational number $r$ is NP-hard, even for tables with at most four nonzero entries in each row, all taking one of two values. The reduction is from exact cover by $3$-sets and rests on a majorization property of the masses of the nonnegative rank-one decompositions of $p$, combined with a quantitative form of the Schur concavity of entropy. In both cases the separation between yes- and no-instances is inverse-polynomial in the size of $p$, so that approximating $J(p)$ or $G(p)$ within a suitable inverse-polynomial additive error is also NP-hard. The two mechanisms are genuinely different: we exhibit a table on which a large rectangle in the support lowers $J$ below the threshold of our first reduction while leaving $G$ on it.

Stochastic gradient descent (SGD) and randomized block coordinate descent (RBCD) belong to two fundamentally different classes of stochastic iterative regularization methods, and are promising and popular for solving large-scale inverse problems, due to their excellent scalability with respect to data size and signal size, respectively. In this work, we propose a doubly stochastic block gradient descent (DSBGD) method with early stopping, by integrating the updating strategies of both SGD and RBCD, for solving linear inverse problems in infinite-dimensional Hilbert spaces. At each iteration, the method randomly selects one data pair to obtain a stochastic gradient estimate, and then uses it to update one randomly selected block coordinate. We prove the regularizing property of DSBGD under a priori stopping rules and then derive its convergence rates with the additional source condition and a condition on the forward operator. To the best of our knowledge, these are the first convergence rate results of randomized coordinate descent-type iterative methods for inverse problems from the perspective of regularization theory. We also prove that the DSBGD iteration can be stopped by the classical discrepancy principle in finitely many steps with high probability, and that the mean squared error decreases monotonically before stopping. Several numerical experiments are provided to complement the theoretical analysis.

In this paper, we propose and analyse a novel explicit fourth-order symmetric low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula, the employment of a twisted function tailored for trigonometric integrals and the symmetrization procedure of trigonometric integrators. Robust error analysis is conducted, demonstrating that the proposed scheme achieves fourth-order accuracy in the energy space under a weak regularity requirement in $H^{2+\max(\mu,1)}(\mathbb{T}^d)\times H^{1+\max(\mu,1)}(\mathbb{T}^d)$ with $\mu> \frac{d}{2}$. A numerical experiment shows that the proposed low-regularity integrator is much more accurate than some well-known exponential integrators for approximating the Klein-Gordon equation with non-smooth solutions.

Let $\mathcal B=(B_g)_{g\in G}$ be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra $C_r^*(\mathcal B)$. Using finite projective realizations of the homogeneous Morita equivalence $B_e$-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When $B_e$ is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.

It is well known that the matrix structure affects the convergence of iterative methods, particularly those utilizing Krylov subspaces. The matrix structure, determined by its graph, dictates the propagation order of updates to the vector components that are recursively computed during the conjugate gradient method iterations. Along with the propagation of these component updates, round-off errors also propagate. We present a model for the propagation and accumulation of round-off errors and demonstrate that the larger the values of topological characteristics of the matrix graph, such as its diameter, effective diameter, and pseudodiameter, the slower the round-off error accumulates during the method's iterations, and the longer the propagation of round-off errors remains local when computing the components of the search direction, residual, and approximate solution vectors. Round-off errors arising during the computation of these components affect only those components whose corresponding vertices in the matrix graph are at a distance not exceeding the current iteration number. As the iteration number increases, the propagation of round-off errors between the vertices of the matrix graph becomes global, leading to the loss of orthogonality of residuals and conjugacy of search directions. In the context of mixed-precision computations, this implies that for a matrix with a larger graph diameter (or, more practically, effective diameter), a greater number of method iterations can be performed in a lower-precision mode before the onset of convergence stagnation.

This paper characterizes Cohen-Macaulay algebras in the sense of Auslander and Reiten [Progr. Math. 95(1991)] within the following two classes of finite dimensional algebras: (i) truncated path algebras and (ii) Nakayama algebras. For the former, our characterization in terms of ordinary quivers shows that if a Cohen-Macaulay truncated path algebra $\Lambda$ is connected, then $\Lambda$ or $D(\Lambda)$ gives a dualizing module, where $D$ denotes the standard duality. For the latter, we show that every Cohen-Macaulay Nakayama algebra $\Lambda$ is Iwanaga-Gorenstein. In particular, $\Lambda$ always gives a dualizing module. Consequently, there is no non-trivial Cohen-Macaulay algebra among truncated path and Nakayama algebras.

We construct a Hausdorff quasitopological abelian group that is second-countable and semiregular, with a bounded continuous function that does not factor through any regular first-countable semitopological group. Its regular and Tychonoff reflections coincide, have a countable network, and fail even factorizability through arbitrary first-countable semitopological groups. This gives negative answers to \cite[Problems~6.1, 6.2, 6.4, and~6.5]{Tk2026} of Tkachenko. The construction starts with a countable cone group whose common separation reflection is not Fréchet–Urysohn, answering \cite[Problems~6.8 and~6.9]{Tk2026} negatively. In contrast, symmetrization gives affirmative answers to \cite[Problems~6.6 and~6.7]{Tk2026} without separation assumptions. We also show that every countable semitopological group admits simultaneous factorizations of countable families through second-countable Tychonoff semitopological groups.

Let $\mathcal{C}_{r}^+$ be the set of all balanced signed graphs with underlying graphs $C_r$. A signed graph is called $\mathcal{C}_{r}^+$-free if it contains no subgraph isomorphic to any member of $\mathcal{C}_{r}^+$. In this paper, we characterize the extremal graphs that achieve the maximum index among all $\mathcal{C}_{2k+1}^+$-free unbalanced signed graphs of order $n$ when $k\ge 2$ and $n\ge 187k$.

For the first time, we study the effects of mass transfer on alloy droplet impact, solidification, and solute non-homogeneity or MacroSegregation (MS). For this purpose, a multi-physics model, including fully-coupled hydrodynamics of multiphase flow, wetting, conjugate heat transfer, phase change, and mass transfer, is established and validated carefully across several benchmarks. In addition, a concentration-flux-correction algorithm is proposed to conserve the solute mass within the droplet at the numerical discretization level. The validated model is subsequently used to explore the impact and solidification of an alloy droplet. The results show that accounting for mass transfer and solute transport improves the prediction of the dynamics of a solidifying splat, in a consistent manner with experimental observations, while this effect has been neglected in all previous studies. The analysis demonstrates that this improvement results from the flow sensitivity to the mushy region thickness, which is regulated by local solute concentration through the solidus and liquidus temperatures. Furthermore, analyses of the MS map identify several critical segregation regions most susceptible to defect initiation, such as cracking, including a long negatively segregated vein within the lamella, formed by the severe advective flow through the mush in the middle stages of the impact, a positively segregated region near the rim of the splat, appearing as a result of hook formation, and a major positively segregated zone at the top of the rim, which develops following the accumulation of the solute-rich melt sustained by washout from the lamella.

We prove that the optimal constant in the Chang–Marshall inequality is $e$, as conjectured by Andreev and Matheson. More precisely, if $f$ is analytic in the unit disk, vanishes to order $m\geq1$ at the origin, and has normalized Dirichlet integral $d\leq1$, then the boundary mean of $e^{|f|^2}$ is at most $e^{d/m}$. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.

Strong direct-sum theorems relate the information needed to solve many independent instances of a communication problem to the information needed for one instance. For Gap-Majority composition, an optimal information bound was previously known, but preserving the number of communication rounds incurred an additional logarithmic loss. We prove a round-preserving Gap-Majority lemma with constant additive information loss, independent of both the number of instances and the number of rounds. More generally, we establish a variance direct-sum theorem for additive square-integrable observables over product distributions. For protocols with a bounded number of terminal answer values, the extracted single-coordinate protocols preserve the round budget, and their average information cost is at most the original information cost divided by the number of coordinates, plus a term depending only on the answer alphabet and the allowed variance loss. The proof uses centered counterfactual conditional means and a global variance amortization argument. Rather than encoding absolute conditional means for every possible terminal answer, we encode the information not already determined by the common view. The total squared innovation is controlled by a telescoping variance potential, yielding an information loss independent of the number of coordinates. We also derive same-round mean-square estimation bounds for arbitrary terminal outputs.

Let $\zeta^t(\boldsymbol{k})=\sum_{d=1}^{r}S_d(\boldsymbol{k})t^{r-d}$ be the interpolated multiple zeta value of an admissible index $\boldsymbol{k}$ of depth $r$. We show the monotonicity property of coefficients $S_1(\boldsymbol{k})\ge\cdots\ge S_r(\boldsymbol{k})>0$. In the proof, we represent the coefficients as integrals over a common measure space and compares the integrands pointwise. As consequences, we determine the sign and vanishing of $\zeta^{-1}(\boldsymbol{k})$ and locate the zeros of $\zeta^t(\boldsymbol{k})$ in the closed unit disk.

We exhibit a fixed $4\times 4$ binary matrix with no polynomial induced ordered removal bound. At depth $h\geq 1$, let $m=2^h$. Our explicit host has order $(14h+2)m$, exactly $2m^6+\frac{1}{2}(m-2)m^5$ induced copies, and exact editing distance $m^2$ cells. Its normalized editing distance is $\Theta(h^{-2})$, while its induced-copy density is $\Theta(4^{-h}h^{-8})$. The lower bound holds against arbitrary cell edits through an injective family of $m^2$ protected selections. An exhaustive arithmetic classification identifies every induced copy and proves that a matching canonical repair creates none. The original-copy transversal number is exactly $m$, yielding an unbounded gap between hitting existing copies and eliminating all copies. The identities and the obstruction at every depth are formalized in Lean, with verification sources included as ancillary files.

Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension $d \ge 2$, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.

In this paper, we consider the following reaction-diffusion system \begin{equation*} \begin{cases} u_t=\Delta u+F\bigl(x,t,u,v,V(t)\bigr), & (x,t)\in\Om\times(0,T), v_t=\Delta v+G\bigl(x,t,u,v,U(t)\bigr), & (x,t)\in\Om\times(0,T), \mathcal B u=\mathcal B v=0, & (x,t)\in\partial\Om\times(0,T), u(x,0)=u_0(x),\quad v(x,0)=v_0(x), & x\in\Om, \end{cases} \end{equation*} where \begin{equation*} U(t)=\int_{\Om}u(x,t)\dd x,\qquad V(t)=\int_{\Om}v(x,t)\dd x, \end{equation*} and the boundary operator $\mathcal B$ is either \[ \mathcal B_D w=w \qquad\text{or}\qquad \mathcal B_N w=\partial_\nu w, \] $\Om\subset\R^N$ be a bounded domain with sufficiently smooth boundary, $N\ge 1$, and $\nu$ is the outward unit normal on $\partial\Om$. Under certain conditions, we obtain the upper and lower bounds for the blowup time of the solution to the system.

The complex polynomial Bohnenblust–Hille inequality states that, for a $d$-homogeneous polynomial $P:\mathbb C^N\to\mathbb C$, $P(z)=\sum_{|\alpha|=d}c_\alpha z^\alpha$, the coefficient norm $(\sum_{|\alpha|=d}|c_\alpha|^{p_d})^{1/p_d}$ is bounded by $B_d^{\mathrm{pol}}\|P\|_\infty$, where $p_d=2d/(d+1)$ and $B_d^{\mathrm{pol}}$ is independent of $N$. Here $\|P\|_\infty$ is the supremum norm on the unit polydisc. This inequality has a Boolean analogue for the Fourier–Walsh coefficients of functions of degree at most $d$, with the supremum taken over $\{-1,1\}^N$. In this setting, Defant, Mastyło and Pérez obtained subexponential bounds, and Ivanisvili established polynomial growth. Further polynomial estimates were obtained by Slote, Tseng and Volberg in the Boolean setting and by Slote and Volberg for cyclic groups. We study these coefficient inequalities in a broader framework of finite systems of functions that encompasses and extends the Boolean and cyclic settings. This framework also admits a natural $\ell_p$ formulation, leading to Hardy–Littlewood-type inequalities. For orthonormal systems, we obtain polynomial bounds for the Bohnenblust–Hille constants and, for sufficiently large $p$, for the constants in these Hardy–Littlewood-type inequalities. In the Boolean case, the polynomial power is $5/2$, with a cubic logarithmic factor.

For a fixed prime $p$, a group is called a $p'$-group if the orders of all its elements are coprime to $p$. It is well known that in a finite $p'$-group an automorphism of order $p$ is regular if and only if it is a splitting automorphism of period $p$. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two notions coincide for periodic $p'$-groups. We answer this question in the negative by constructing a periodic group of prime period $q$ that admits a regular automorphism of order $p \neq q$ which is not splitting.

We study invariant probabilities for nonlinear two-population branching systems with competition, cooperation, or predator–prey interaction, driven by Brownian noise and spectrally positive stable jumps. Without immigration, every invariant probability of a conservative competitive or predator–prey system is concentrated on the boundary under either boundary convention. With nontrivial axis dynamics, coordinatewise continuation admits only the point mass at the origin. For cooperative systems, we give criteria for boundary concentration and interior stationarity, including a sharp criterion for bilinear cooperation with a common relative noise exponent. With immigration, explicit parameter conditions yield existence and stationary moment bounds. For square-root diffusions with positive immigration, these conditions ensure interior concentration and full support on the closed quadrant. Further lower bounds on immigration rates give uniqueness and convergence of time-averaged transition laws.