Papers

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L^2_{loc}$ and $V \in L^1_{loc}$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p \in (1, 2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.

In this paper, we establish sparse bounds for singular integral operators with rough non-translation-invariant Calderón-Zygmund kernels. Our approach verifies the localized hypothesis in the sparse domination framework introduced by Conde-Alonso, Di Plinio et al. The proof refines Seeger's microlocal decomposition using a Littlewood-Paley analysis adapted to the non-translation-invariant setting. As applications, we obtain quantitative weighted estimates for Calderón-type commutators and other singular integrals with rough kernels.

Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group $\mathcal{C}_R$ and analyze the structure of its algebraic analogue $\mathcal{AC}_R$ modulo torsion. Using Milnor signatures, we show that $\mathcal{AC}_R$ has infinite rank for every $R$. We also show that any prescribed finite set of natural numbers can be realized as orders of elements in a single relative concordance group.

We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\le1$, $(a,q)=1$, $|\alpha-a/q|\le q^{-2}$ and $q\le N/(\log N)^2$, then \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha)\ll\frac N{\log N}+\frac N{\sqrt q}. \] Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which $q$ is replaced by $q(1+N|\alpha-a/q|)$, when this quantity is at most $N/(\log N)^2$, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative $f$ satisfying only $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$, for every fixed $A>1$.

Let $\LL = -\Delta + V$ be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform $\nabla \LL^{-\sfrac{1}{2}}$ is bounded on $L^p(\RR^d)$ for $p \in (1,2]$, with a bound independent of $V$. We prove more, namely that the set \[ \{ \nabla \LL^{-\sfrac{1}{2}},\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) \} \] is ${\mathcal R}$-bounded on $L^p(\RR^d)$ for $p \in (1,2]$. We also prove the ${\mathcal R}$-boundedness on $L^p(\RR^d)$, for $p \in (1,\infty)$, of the spectral multipliers \[ { m(\LL),\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) } \] for any $m:[0, \infty) \to \CC$ satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators $-{\bf \Delta} + {\bf V}$ on vector-valued spaces $L^p(\RR^d,\ell^2)$. In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.

We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.

We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors, with a simplified formula for odd lifting factors. A residue characterization identifies additional common factors of the minors and leads to a method for controlling the dimension through polynomial selection. We also give a condition for changing the minor generator without changing the code. Explicit polynomial constructions yield 13 quantum codes with verified minimum distances, including a $[[255,80,12]]$ code of rate $80/255$.

The problem of Private Information Retrieval (PIR) across graph-based replicated systems has seen a surge of interest in recent literature. In such systems, servers are modeled as the vertices of a graph, and each file is represented by an edge connecting the pair of servers that store it. While previous research on Graph-PIR (G-PIR) has focused mostly on perfect information-theoretic privacy, relaxing this constraint has the potential for significantly higher retrieval rates. In this work, we consider the setting of Graph-based Weak Private Information Retrieval (G-WPIR), under an $\epsilon$-privacy metric. We adapt an existing G-WPIR scheme for arbitrary graphs to analyze its performance under the $\epsilon$-privacy metric. Additionally, we establish a lower bound on the expected download cost for arbitrary graphs under the $\epsilon$-privacy metric and derive from it an explicit rate upper bound for complete graphs. Finally, we show the achievable rate-privacy trade-offs for two specific graph classes—complete graphs and complete bipartite graphs.

We introduce the Enhanced Neumann Pólya inequality, which is stronger than Pólya's conjecture, and verify it for many classes of domains, including strip-tiling domains, Euclidean balls, thin product domains, and certain higher-dimensional tiling domains. We develop a geometric removal approach for the Enhanced Neumann Pólya inequality. In particular, our results provide new classes of domains satisfying the Neumann part of Pólya's conjecture.

We study a Patlak–Keller–Segel–Navier–Stokes system on $\mathbb T\times\mathbb R$ with density-dependent diffusion and chemotactic sensitivity growing like $n^{1+\alpha}$, where $0<\alpha<1$. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501–543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density–fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that $\alpha<1$ allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.

We study two explicit constructions of binary Calderbank–Shor–Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for $N=31$ and every $N\ge37$. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.

In 2018, Goddard and Henning conjectured that every cubic graph has four total dominating sets covering each vertex at most twice, and asked the analogous question for transversals of $3$-uniform $3$-regular hypergraphs. We prove the conjecture, which also answers the question in the affirmative.

For a Fourier-Mukai transform whose kernel is a family of Gieseker stable sheaves, we shall give a condition for the preservation of Gieseker stability of sheaves.

Random reshuffling stochastic gradient descent (RR-SGD) is widely used for finite-sum optimization in machine learning. For convex $L$-smooth components, recent work establishes convergence guarantees for an averaged iterate with component stepsizes as large as order $1/L$. However, the final iterate is a natural output in practice, and existing last-iterate guarantees in this setting require stepsizes that scale inversely with the number of components $n$. We establish an expected objective-gap bound for the last iterate of standard RR-SGD on unconstrained problems with a constant component stepsize satisfying $0<\eta\lesssim 1/(L(1 + \log(nK)))$, where $K$ is the number of epochs. By retaining the sampling bias at random comparison points, we remove the explicit $1/n$ factor from the stepsize restriction. Combined with existing small-step bounds, our result matches the corresponding averaged-iterate convergence rate up to logarithmic factors.

We shall study birational maps of moduli spaces of stable sheaves on a K3 surface with Picard number 1 induced by Fourier-Mukai transforms.

We develop and analyze a three-dimensional pure source transfer domain decomposition method (PSTDDM) for the Helmholtz equation in free space. Extending the two-dimensional PSTDDM proposed by [Du and Wu, J. Sci. Comput., 83(2020),83:68], the proposed method transfers sources successively along the $x_1$, $x_2$, and $x_3$ directions, yielding layer-wise, bar-wise, and block-wise PSTDDMs. The main challenge in the convergence analysis is deriving wavenumber-explicit truncation-error estimates. To address this challenge, we establish inf–sup conditions for the uniaxial perfectly matched layer (UPML) problem under relaxed restrictions on the PML parameters and reformulate the source-transfer operators. We then introduce a positive-definite reconstruction step that produces a globally conforming approximation. In addition, a new quantitative lower bound is established for the imaginary part of the complex PML distance, more easily yielding exponential decay estimates in three dimensions. These lead to wavenumber-explicit convergence estimates for all three PSTDDMs. Numerical experiments demonstrate the effectiveness of the proposed methods both as direct solvers and as preconditioners for GMRES.

Let $p$ be an odd prime, and let $\rho\colon G\to \GSp_{2n}(\Z_p)$ be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph $\Gamma_\rho^\GSp$ associated with $\rho$ to be the directed graph whose vertices are the residual factors of $\rho$ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by $\rho$. Using the Euclidean Bruhat–Tits building associated with $\GSp_{2n}(\Q_p)$, under some assumptions, we prove the existence of certain paths in $\Gamma_\rho^\GSp$ and show that $\Gamma_\rho^\GSp$ is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.

We propose a potential projection (PP) method for the nonlinear Schrödinger equation (NLSE) with low-regularity and singular potentials by replacing the potential with its Fourier projection (truncated Fourier series). This gives rise to an NLSE with projected potential (NLSEwPP). We rigorously establish its well-posedness under low-regularity potentials, and its convergence to the original NLSE. The NLSEwPP is then directly discretized by the standard time-splitting Fourier pseudospectral (TS-FP) method with the mesh size chosen according to the potential projection parameter, resulting in the fully discrete TS-PP-FP method. Numerically, TS-PP-FP consistently achieves optimal temporal and spatial convergence rates with respect to the regularity of the exact solution under low-regularity and even singular potentials. Compared with the time-splitting Fourier spectral method, TS-PP-FP attains the same convergence rates at significantly lower computational cost and preserves the mass conservation and time symmetry (that are lost in the Fourier spectral discretization of the time-splitting method); compared with the time-splitting FP method, it is substantially more accurate for low-regularity potentials with the same computational cost.

We study Dirac points for a two-dimensional scalar Helmholtz equation in a periodic dielectric medium with two penetrable inclusions per honeycomb cell. In a joint small-inclusion and high-contrast regime, we reduce the Bloch eigenvalue problem to a two-dimensional interaction matrix and derive a two-term asymptotic expansion for the two relevant eigenvalues, uniformly for quasi-momenta separated from the reciprocal lattice. At a vertex K of the Brillouin zone, rotation and inversion-conjugation symmetries force these eigenvalues to coincide. A Schur-complement analysis of the exact operator pencil, together with a nonvanishing lattice Green-function coefficient, proves that the resulting double eigenvalue has a nonzero conical splitting and gives an asymptotic formula for the conical slope in the scaled eigenvalue. For fixed sufficiently large contrast, symmetry-preserving changes of variables and analytic perturbation theory then continue this Dirac point as the inclusion size varies: it remains nondegenerate except at a relatively closed discrete set of admissible sizes.

Softplus affine\((1,1)\) models keep the conditional mean positive while allowing coefficients of either sign. We study them under Poisson, negative binomial, gamma and related conditional distributions. A stationary solution with moments of all positive orders exists whenever the coefficient sum is less than one, strictly enlarging previously known sufficient regions. At the unit root, the coefficient sum equals one. Under an invertibility condition, a stationary solution exists in this case if and only if the intercept is negative. At a stationary unit root, the conditional mean has a power-law tail with an explicit index, and the observations can exhibit long memory. For a fixed softplus scale, the Poisson quasi-maximum likelihood estimator is consistent and asymptotically normal at the usual \(\sqrt n\) rate below and at the unit root, with the coefficient sum left unrestricted in estimation. At the unit root, these results require only a finite mean and yield standard Wald inference even when the observations have infinite variance or long memory. Simulations and two applications examine finite-sample inference and sensitivity to the softplus scale.

We show that limit-periodic Jacobi matrices in the weak exponential class $EC(\eta)$ have purely absolutely continuous spectral type when $\eta$ is sufficiently large, depending only on the $R$-boundedness constant. The proof first establishes homogeneity of the spectrum from the weak exponential approximation rate. A closure theorem for reflectionless Jacobi matrices then transfers reflectionlessness from the periodic approximants to the limit, and a theorem of Poltoratski and Remling rules out singular spectrum on a homogeneous set.

We prove an endpoint Pucci maximum principle for domains in $\mathbb R^2$, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for $q$-Hessians where $1 < q < 2$. The fractional elementary symmetric functions we use are derived through fractional differentiation. Their signed singular value extension coincides with the Burkholder functional introduced by Burkholder and gives us access to a local Burkholder inequality due to Astala, Iwaniec, Prause and Saksman, leading to a different proof of the Pucci maximum principle from the one in that used quasiconformal maps. We include an elementary real-variable proof of this local Burkholder inequality, obtained by combining a few known techniques from existing literature.

In this paper, we construct normalized periodic kernels from univariate functions satisfying suitable moment conditions and derive sufficient conditions for high-order convolution approximation. Two computable quasi-interpolation schemes are obtained by discretizing the convolution integral using either a full tensor grid or random samples. In each case, the analysis separates the continuous convolution error from the error introduced by discretization. The contribution of kernel energy to the error exposes the dimensional dependence of tensor-product kernels and motivates our weight-adapted ANOVA constructions. Under suitable summability conditions on the weights, we establish convolution error and kernel energy estimates that yield dimension-uniform error bounds for the resulting quasi-interpolants. Numerical experiments corroborate the theoretical convergence rates and illustrate the effect of coordinate weighting on dimensional dependence.

We study symmetric 2-(v, k, {\lambda}) designs admitting a flag-transitive point-primitive subgroup G of automorphisms, where gcd(k, {\lambda}) = p and p is a prime. In this paper, we show that G is of affine or almost simple type.

We introduce a $q$-analogue of the bicolored Eulerian quasisymmetric functions by incorporating an additional grading arising from affine length, obtained by interpreting bicolored permutations as affine permutations. We also provide an intrinsic description of this refinement in terms of bicolored permutations by introducing notions of alignments and crossings, together with an explicit bijection identifying these statistics with the alignments and crossings in the existing literature. Finally, we prove that our refinement is symmetric by expressing it in terms of the Theta operators appearing in the Theta coinvariant Conjecture and the modified Macdonald symmetric functions. Assuming the Theta coinvariant conjecture implies the Schur-positivity of our refinement.

Let $F: \mathbb{R}^2 \to \mathbb{R}^2$ be a polynomial map with nowhere zero Jacobian determinant. It is known that the global invertibility of $F$ is equivalent to the pullback of a parallel vector field being parallel. In this paper, we construct the phase portraits on the Poincaré disc of the pullbacks of the canonical annular, radial, and strip vector fields under non-injective maps $F$ that have recently appeared in the literature. Our arguments use local and global properties of vector fields as well as local and global properties of functions, thereby connecting techniques from dynamical systems and singularity theory.

In this paper we study the behavior of transformer networks generating text over long periods of time. We show that even a three-token, single-layer attention-only transformer can fail to converge. We then study a rich class of transformers with arbitrary finite depth, layer normalization, MLPs, RoPE encodings, and an arbitrary mix of local and global layers. Every transformer in this class has a softmax temperature threshold above which its empirical cache distribution converges to a unique limit, independent of initial conditions. We show further that, while each transformer has a temperature threshold, the supremum of this set is infinite, so there is no global minimal temperature that forces all transformers to converge.

In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity $> 2$ of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.

We study the fluctuations of the free energy of the Potts spin glass with $\kappa$ colors at high temperature. The disorder is symmetric with unit variance and finite fourth moment. For configurations with a balanced, or nearly balanced, color profile, the log-partition function satisfies a central limit theorem whenever $\beta^2<\alpha_c(\kappa)$. Here $\kappa\alpha_c(\kappa)$ is the critical inverse temperature of the Curie-Weiss-Potts model. A comparison of color profiles transfers this limit to the unconstrained model for $\beta<\beta_{\max}(\kappa)$, an explicit threshold. In the same range, the magnetizations of two replicas and their overlap matrix satisfy a joint quenched central limit theorem with independent Gaussian limits. The fourth cumulant of the disorder enters the limiting mean and variance explicitly.

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.