Papers

We establish sharp time-decay estimates for the discrete fourth-order Schrödinger equation on lattices of arbitrary dimension, extending the author's previous work \cite{C24}. In higher dimensions, scalar Fourier factorization reduces the analysis to product estimates for one-dimensional oscillatory integrals. Since this method does not yield the sharp decay exponents in low dimensions, we also use Newton polyhedra to obtain the required uniform estimates. In particular, the sharp decay rate for the biharmonic Schrödinger propagator is $|t|^{-d/4}$.

Classical Inverse Optimal Transport (IOT) typically infers a cost function from a single observed transport plan, which may be sensitive to noise and observational biases in practice. To address this limitation, we propose the Multi-Observation Inverse Optimal Transport (M-IOT) framework, which learns a universal cost function from multiple transport plans. We reformulate the resulting bilevel optimization problem into a tractable single-level convex program and characterize the existence and geometric structure of the optimal solution set. Moreover, we propose a Block Coordinate Descent with Anderson-type Extrapolation (BCDwAE) algorithm and establish its Q-linear convergence. Extensive experiments on synthetic data, urban mobility flows, and e-commerce recommendations demonstrate that M-IOT reduces recovery error compared to single-observation baselines and extracts robust structural patterns from noisy multi-view data.

A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph whose vertex set is partitioned into $k$ classes of the same size and whose edges meet every class in exactly one vertex. Lo and Markström (2014) determined the minimum vertex-degree threshold for perfect matchings when $k=3$, and Lu, Wang and Yuan recently determined it when $k=4$. We prove the corresponding exact result for every fixed $k\ge5$ and all sufficiently large class sizes. The close case follows from the general theorem of Lu, Wang and Yuan. For the non-closed case, we extend their stability result from the 3-partite setting to arbitrary partite uniformity, using the probability-tail rigidity theorem of Cao, Liu and Zhang, thereby replacing the earlier weighted lemma.

This paper continues our work on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences. We extend the method developed for the Schmüdgen-type hierarchy to the Putinar-type, Krivine–Stengle-type, extended-Handelman-type, and Hol-Scherer-type hierarchies. The main idea is to lift a pseudo-moment sequence to a simple set, approximate it by a moment sequence, and project the atoms of a representing measure onto the original feasible set. The Łojasiewicz inequality then converts the error in the defining constraints into an error in the moments. With the Łojasiewicz exponent $0<L\leq 1$, we obtain the convergence rates $\mathrm{O}((\log_2 r)^{3L/2}/r^{L})$ for the Putinar-type hierarchy and $\mathrm{O}(1/r^{L/2})$ for the normalized Krivine–Stengle-type and extended-Handelman-type hierarchies. In the matrix setting, we utilize a Chebyshev-type kernel on $[-1,1]^n$ to derive the rate $\mathrm{O}((\log_2 r)^{3L/2}/r^{L})$ for the Hol-Scherer-type hierarchy. Together with our preceding work, the results provide a universal method for studying convergence rates of different types of moment-SOS hierarchies.

We formulate and solve a degenerate elliptic boundary value problem for the $SU(\infty)$-Toda equation on $[0,\frac12)\times\mathbb{S}^2$. This constructs anti-self-dual Poincaré–Einstein metrics with circle symmetry on the $4$-ball and on complex line bundles over $\mathbb{S}^2$.

In this paper we deal with a nonlinear elliptic problem in the whole space $\mathbb{R}^N$ involving the $p$-Laplacian operator with $p>N$. Under suitable assumptions on the potential term and on the nonlinearity, we establish the existence of at least three distinct weak solutions. Our approach is variational and relies on two different three critical points theorems. The results extend to higher dimensions some recent multiplicity theorems for one-dimensional $p$-Laplacian equations on the real line, and provide explicit ranges for the parameter $\lambda > 0$ ensuring the existence of multiple solutions, which are nonnegative under additional sign conditions. The nontriviality of the solutions is also discussed. Finally, applications to problems whose nonlinearities have separated variables are presented.

For any Tychonoff space $X$, let $D(X)$ denote either the space $C(X)$ of all continuous real-valued functions on $X$ or the space $C^*(X)$ of all bounded continuous real-valued functions on $X$. We write $D_p(X)$ when $D(X)$ is endowed with the topology of pointwise convergence. In our recently published paper [8, Theorem 1.6], we obtained the following result: if $T: D_{p}(X) \to D_{p}(Y)$ is a linear continuous surjection, where $X$ is a metrizable space and $Y$ is a perfectly normal space, then $Y$ inherits a given topological property $\mathcal{P}$ from $X$. A linear continuous surjection $T: E_{p}(X)\to E_{p}(Y)$ is said to be densely defined if $E(X)$ and $E(Y)$ are dense linear subspaces of $D_{p}(X)$ and $D_{p}(Y)$, respectively, and $E(X)$ is correct (see Definition 1.5(a)). In the present paper, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, $\sigma$-compactness or strong countable-dimensionality. Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$, where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.

We study well-formed weighted projective threefolds $X=\mathbb P(w_0,w_1,w_2,w_3)$ that admit a $\mathbb Q$-Gorenstein smoothing to a smooth Fano threefold of Picard number one. For del Pezzo threefolds $V_d$ and prime Fano threefolds $Y_g$, we derive three necessary numerical and local conditions: the anticanonical volume equation, an identity obtained from the linear term of the anticanonical Hilbert polynomial, and a global section condition for smoothing the transversal $A$-singularities along coordinate curves. A computer search using these conditions determines all numerical candidates, apart from the known infinite family for $V_5$, with $w_0+w_1+w_2+w_3\leq 5000$. We construct $\mathbb Q$-Gorenstein smoothings to $V_1, V_2, Y_6, Y_{10},$ and $Y_{12}$ from numerical candidates of weighted projective threefolds, and prove that $\mathbb P(2,5,8,25)$, although a numerical candidate for $V_4$, is not $\mathbb Q$-Gorenstein smoothable. To identify the smooth fibers, we also give a vanishing-cycle criterion ensuring that Picard number one is preserved under the smoothing.

Nearly $\mathrm{G}_2$ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, $\mathrm{Spin}(7)$. We study nearly parallel $\mathrm{G}_2$-structures invariant under a cohomogeneity-one action of $\mathrm{SO}(4)$, and construct the first inhomogeneous nearly parallel $\mathrm{G}_2$-structure on the Berger space $\mathrm{SO}(5)/\mathrm{SO}(3)$. The cone over this metric has full holonomy $\mathrm{Spin}(7)$. The existence of the solution is established by a rigorous computer-assisted shooting argument. We also establish a local desingularisation of a finite quotient of the sine cone over $S^3\times S^3$ by gluing in the asymptotically conical $C_7$ bubble constructed by Foscolo–Haskins–Nordström. We show that as the singular orbit of a family of nearly parallel $\mathrm{G}_2$ manifolds collapses, away from the singular orbit the desingularising family converges to the sine cone quotient, while the natural blow-up converges to the AC torsion-free $\mathrm{G}_2$-metric of $C_7$.

Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all completable balanced triangles on any indexed list of at most sixteen nonzero vectors. The second supplies weights for the direction families that can occur in a critical list with no central terms. We give complete coverage arguments, exact certificate files, and separately implemented checking programs. A compression argument removes any need for an induction through smaller list lengths in the sixteen-term potential theorem. The formula for $r\ge4$ remains open.

This paper is concerned with a capillary $L_p$ Christoffel–Minkowski-type problem involving the prescription of a class of $L_p$ geometric measures formed by the $k$-th capillary area measure and the $q$-th dual curvature measure. By establishing a gradient estimate, we prove the existence of an even, smooth, strictly convex solution in the parameter range $1<p<q\leq k+1$, where $1\leq k\leq n$.

Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\Q,S}$ denotes the Galois group of the maximal extension of $\Q$ unramified outside $S$. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine–Mazur conjecture over $\Q$. Furthermore, we show that for an imaginary quadratic field $K$, any continuous conjugate self-dual two-dimensional $p$-adic representation of its pro-$p$ Galois group $G_{K,S}(p)$ has finite image, provided the primes in $S$ satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's $(3,3)$-conjecture on $p$-class field towers for $p>3$. Namely, we prove that if $K$ is an imaginary quadratic field with $p$-class rank two and the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified $p$-extension of $K$ has Zassenhaus type $(3,3)$, then the $p$-class field tower of $K$ is finite. For $p=3$, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.

This paper shows that an almost-isometric Sobolev map from a Lipschitz subset of an oriented manifold $M$ into another oriented manifold $N$ can be approximated by a map of class $C^{1,\alpha}$, with a universal bound on the $W^{2,m}$ norm, for any $m\in(1,\infty)$. This, in turn, implies a universal bound on the $C^{1,\alpha}$ seminorm. The $W^{1,p}$ distance of the approximation from the original map is controlled in terms of the $L^p$-deviation from being an isometry, with the optimal scaling. The manifolds $M$ and $N$ are assumed to be compact, oriented and have equal dimension; $M$ may have a boundary.

Permanent air-quality networks are expensive to install and maintain, yet many decisions depend on one future regional exposure rather than the complete pollution field. We introduce Target-Aware Permanent Sampling (TAPS), a graph-diffusion framework for selecting permanent locations observed repeatedly over time, that can reduce unnecessary sensor installations, maintenance, and cost while preserving the information needed for future air-quality decisions, subject to validation with the estimator that will use the network. We formulate the permanent space-time sampling problem and show that, in the noiseless model, recovery of one prescribed target can require less information than full-state identification, for which we give an explicit lower bound on the permanent-location count. We also derive a greedy rule whose marginal gain factors into raw target response and a finite-update correction. We evaluate TAPS on regulatory air-quality networks in California, Canada, and England. In each case the prescribed regional target becomes numerically recoverable at permanent-location budgets well below those required to identify the retained state, and TAPS attains lower regularized target risk than target-weight, geometric, and design-based placements. A blind prospective study further shows that the criterion can be applied before candidate sites have any ground-monitor history, using exogenous environmental covariates alone.

We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.

We study well-formed weighted projective threefolds that admit $\mathbb Q$-Gorenstein smoothings to $\mathbb P^3$. Two families are known: the $\mathbb P^2$-type and the $Q$-type, and it is conjectured that these are the only possibilities. We derive numerical and local necessary conditions for such a smoothing. In addition to the anticanonical volume equation, constancy of the anticanonical Hilbert polynomial yields a further identity when all codimension two singularities are of $A$-type. We also obtain a semigroup condition governing the existence of global smoothing directions along codimension two curves with transverse $A$-type singularities. We apply these conditions to prove the expected classification in several cases. In particular, for every fixed square-free integer $d$, there are only finitely many $\mathbb Q$-Gorenstein smoothable spaces $\mathbb P(1,a,b,c)$ such that $\gcd(a,b)=d$. Our method reduces the possible weights to a finite exact computation; for every prime $p\le100$, the computation produces only members of the two expected families. Finally, we prove the classification when $\gcd(a,b)=d$ and $a=d^2$.

Given a free group on $n$ generators, $F_n$, and a topological group $G$, $\operatorname{Aut}\left(F_n\right)$ acts on $\operatorname{Epi}\left(F_n,G\right)$, the set of homomorphisms $f:F_n \rightarrow G$ with dense image. $f\in \operatorname{Epi}\left(F_n,G\right)$ is called redundant if there is a proper free factor $A < F_n$ whose image under $f$ is dense in $G$. For $n \ge 3$, $G = \operatorname{PSL}_2 \left(\mathbb{Q}_p\right)$, we prove that every $f\in \operatorname{Epi}\left(F_n,G\right)$ with torsion-free image is redundant, and give sufficient conditions for $f$ to be redundant even when its image is not torsion-free.

We prove that the solution to the Córdoba-Córdoba-Fontelos equation $\theta_t-u\theta_x=0$, $u=H\theta$, on the real line with analytic initial datum $\theta_0(x)=(1+x^2)^{-1}$ develops a finite-time singularity while remaining uniformly bounded in $C^{2/5}$. We also prove that, as the solution approaches the singular time, the $C^{2/3}$ seminorms of both $\theta$ and $u$ diverge. The main ingredient is an integral representation in terms of a probability measure that is preserved by the nonlinear evolution. From this representation we derive a pointwise quadratic estimate for the key nonlinear term in the equation for $u$, using a comparison of positive kernels. Together with evolution identities for the moments of the measure, this estimate yields both the uniform Hölder bound and the divergence of the higher seminorms.

Let $\Delta_{m,n}$ be the chessboard complex, and let $J_{m,n}$ be the Stanley-Reisner ideal of its Alexander dual. We study the symbolic and ordinary powers of $J_{m,n}$, focusing on their homological and algebraic invariants. First, we characterize when $J_{m,n}^{(q)}$ has linear quotients. For the ordinary powers, we give sufficient conditions on $m$ and $n$ for $J_{m,n}^q$ to have linear quotients. We determine the maximal degree of a minimal generator and the Castelnuovo-Mumford regularity of $J_{m,n}^{(q)}$, obtaining explicit formulas for all $q\ge2$ and showing that both invariants have the same asymptotic slope. We also compute the initial degrees and Waldschmidt constant, proving that $\alpha(J_{m,n}^{(q)})$ is a degree-one quasi-polynomial of period two and $\widehat{\alpha}(J_{m,n})=mn/2$. Explicit formulas for the $\mathrm{v}$-numbers of symbolic and ordinary powers are obtained. Finally, for every $m\ge3$, we show that $J_{m,2m-1}$ is not weakly polymatroidal, disproving a conjecture of Lu and Wang in this setting.

Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever $F$ is edge-color-critical with $\chi(F)=r+1$, every non-$r$-partite, $F$-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let $F=K_1\vee\mu(K_3)$, where $\mu(K_3)$ is the Mycielskian of a triangle. This graph is edge-color-critical with $\chi(F)=5$. We prove that $\SPEX_{5}(n,F)\cap\EX_{5}(n,F)=\varnothing$ for all sufficiently large $n$. Thus no graph can simultaneously maximize both the edge count and the spectral radius.

We develop quantum algorithms that construct block-encodings of solution matrices for continuous- and discrete-time algebraic Riccati equations (CAREs and DAREs), differential Riccati equations (DREs), and finite Riccati recursions. The Quantum Weighted Riesz Method, unifying four kinds of Riccati problems, constructs projectors onto the solution graphs using weighted Riesz branching operators and recovers the solution matrices. Under the stated access and normalization assumptions, our algorithms for CAREs and DAREs have near-optimal query complexity $\Theta(\mathcal R\alpha)$ up to logarithmic factors, with the generalized singularity factor $\mathcal R$ that reflects spectral separation, and the output normalization $\alpha$. For DREs and finite-horizon problems, the query complexity has an additional initialization conditioning factor $\kappa_{\rm init}$. We establish product query lower bounds for specified input-oracle families, demonstrating necessary joint dependence on spectral separation and initialization or output scale. We also prove that selected-value promise problems are BQP-complete on explicit circuit-generated families of single-control discrete-time algebraic and zero-terminal differential Riccati equations. Applications include classical feedback evaluation for linear-quadratic control, a query-complexity comparison for stable random-phase approximation Riccati equations in quantum chemistry, and heated boundary control with numerical validation.

We prove dynamical stability in arbitrary dimension for the modified Mullins–Sekerka flow, the gradient flow of the sharp-interface Ohta–Kawasaki energy on the flat torus. Specifically, we show that if an initial set has the same volume as a strictly stable critical set for the energy and is sufficiently close to it in $C^{3,\alpha}$, then the flow exists for all times and converges exponentially fast, in every $C^k$ norm, to a translate of that critical set. The proof relies on a quantitative Alexandrov-type estimate for strictly stable critical sets of the energy.

The semidefinite linear complementarity problem associated with the multiplicative map $M_A:\mathbb{S}^n\to\mathbb{S}^n$ defined by $M_A(X)=AXA^T$, is considered, in the case when $A\in\mathbb{R}^{n\times n}$ has rank one. A linear map is said to have the $Q_0$-property if every feasible instance of the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When $A=xy^T$ with $x,y\in\mathbb{R}^n$ linearly independent, we construct a matrix $Q\in\mathbb{S}^n$ for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that $M_A$ does not have the $Q_0$-property. Conversely, when $A=uu^T$ for some nonzero $u\in\mathbb{R}^n$, we prove that $M_A$ has the $Q_0$-property. These two results provide a complete characterization of the $Q_0$-property within the family of rank-one multiplicative maps.

Spatial regulation suffices to guarantee reliable, low-latency communication using variable-length stop-feedback (VLSF) codes over multiple access channels. The delay performance of VLSF codes under successive interference cancellation is characterized for transmitters within a coverage region. For deployments modeled by hardcore point processes, the signal-to-interference-plus-noise ratio (SINR) admits a positive lower bound for all transmitters within the coverage region. Under a predefined total error probability constraint, these transmitters are decodable within an upper-bounded delay. For each transmitter, the decoding delay is well approximated by an inverse Gaussian distribution parameterized by the codebook size, target error probability, and the SINR lower bound. Numerical results confirm that hardcore regulation provides uniform performance guarantees across links.

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

We study the de Bruijn–Newman family $H_\tau(z)$, with heat-deformation parameter $\tau$ and $H_0(z)=\tfrac18,\xi(\tfrac12+\tfrac{iz}{2})$. The moving zeros considered here are therefore zeros of $H_\tau$, not directly of $\zeta(s)$. Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero $z=x+iy$ of maximal imaginary height, [ \frac{d}{d\tau}y^2\le -2-4y^2\mathcal G_\tau(z), ] where $\mathcal G_\tau\ge0$ is a projected interaction sum. A second exact inequality bounds $\mathcal G_\tau$ below by an off-zero logarithmic derivative $-\operatorname{Im}(H_\tau'/H_\tau)(x+i\eta)$, providing a direct interface with the effective Polymath approximation $H_\tau=B_\tau F_\tau$. At the frontier $X=6000000185827$, $\tau_0=129/800$, and $y_0^2=87677/2500000$, a directed Cauchy certificate gives $\mathcal G_\tau>3/2$ for $\tau_0\le\tau\le0.178$, $x\ge X$, and $|y|\le y_0$. Combining this contraction with certified upper and lower zero-free barriers, a two-envelope argument reduces the maximal nonreal height to $0.08$ by time $0.1747532546428610755\ldots$; the classical de Bruijn contraction then completes the landing. Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ \Lambda\le0.1779532546428610755\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-$x$ certificates are independent contributions.

We prove the equivalence of Hindman's finite sums theorem with a natural extension of Ramsey's theorem.

The paper introduces the R package CovEsts, which implements several nonparametric estimators for the autocovariance function. First, it presents the theoretical foundations of the implemented estimators, their properties, assumptions, and potential limitations. Next, it outlines the structure of the package and its key functions, including several methods for estimating autocovariance functions, constructing corresponding bootstrap confidence regions, and correcting the provided estimators. The package also includes diagnostic tools, such as several metrics for comparing estimators, and additional functions for broader use. The article illustrates a high degree of flexibility of the package in the selection of function parameters and the tuning of the estimators. Applications of selected estimators and package functions are illustrated using simulated data, yearly sunspot counts, and US unemployment increments data.

The conjugate gradient (CG) method is a classical iterative solver for sparse symmetric positive definite (SPD) linear systems, but its convergence strongly depends on the spectral properties of the system matrix. Preconditioning can improve these properties; however, designing preconditioners that generalize across diverse and irregular sparsity patterns remains challenging. We propose BandPC (Band Preconditioner Combinations, where "Band" denotes residual-norm bands), a data-driven framework that leverages graph neural networks (GNNs) to predict multi-stage preconditioning strategies for the flexible conjugate gradient (FCG) method. BandPC partitions the FCG iteration into three residual-norm-based bands and defines a structured search space of 125 candidate combinations over five classical preconditioners. By exploiting FCG's ability to switch preconditioners across iterations, BandPC learns to map matrix structure directly to a promising preconditioner sequence. To improve training and generalization, we introduce a soft-labeling mechanism that retains near-optimal combinations and normalizes their scores into a label distribution, together with a hierarchical feature representation that captures node-level attributes, edge-level algebraic coupling strengths, and global matrix statistics. Experiments on a hybrid benchmark of synthetic SPD problems and diverse matrices from the SuiteSparse Collection show that BandPC achieves the best iteration count and solution time on 41.3% and 33.1% of test matrices, respectively, even when compared with each matrix's individually best traditional preconditioner. These results demonstrate that learned residual-band preconditioner scheduling can effectively accelerate flexible CG solvers.

We introduce sequences of generalized moments that are a generalization of sequences preserving summability and sequences of moments from Balser's theory of moment summability. We construct extensions of Laplace and Borel integral operators for these new sequences. We find integral representations of moment derivatives for generalized moments. We show the characterization of summable solutions of moment-PDEs with generalized moments. We describe singular directions for summable solutions of moment-PDEs with generalized moments and we express jumps across these singular directions in terms of hyperfunctions and in terms of kernel functions associated with sequences preserving summability, which are connected with given sequences of generalized moments.