Papers

This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings.

We study a numerical method, built on the tensor neural network (TNN) architecture introduced in \cite{wang2022tensor}, for solving nonlocal diffusion models in high-dimensional spaces. The tensor-product structure of the TNN ansatz, combined with the separability of the Gaussian kernel, reduces the high-dimensional integrals in the nonlocal energy to products of low-dimensional integrals, which are evaluated by Gauss–Legendre quadrature; nonseparable source and boundary data are handled by a TNN-based preconditioning step. For the Dirichlet boundary condition, we establish the asymptotically compatible $L^2$ error estimate \[ \|u_{\mathrm{loc}}-u_{\delta,p}\|_{L^2(\Omega)} \le C\!\left(\frac{\varepsilon_f}{\sqrt\delta} +\frac{\varepsilon_g}{\delta} +\frac{\varepsilon_u}{\sqrt\delta} +\eta_{\mathrm{opt}}\right) +C\sqrt\delta, \] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $\eta_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrt\delta+\varepsilon_u +\eta_{\mathrm{opt}}+\delta)$, and an $H^1$ gradient estimate is further obtained through a smoothing post-processing step. Numerical experiments on tensor-product domains up to $d=20$ support the theoretical results, and additional tests on two- and three-dimensional $L$-shaped domains demonstrate the practical robustness of the method beyond the smooth-domain setting covered by the analysis.

We determine the Fourier dimension of dyadic Mandelbrot cascades under the minimal Kahane-Peyriere integrability condition. The interval theorem is proved in a vector-valued dyadic cascade model in which sibling weights may have arbitrary dependence. For every balanced energy-admissible vector law, almost surely on non-extinction, dim_F(mu)=dim_E(mu)=dim_2(mu)=D_E(X). In the canonical scalar case, under W>=0, E W=1, E[W log_2^+ W]<infinity, and E[W log_2 W]<1, the formula becomes dim_F(mu)=dim_E(mu)=dim_2(mu)=sup_{1<q<2} max{0, 2-(2/q)(1+log_2 E[W^q])}, with the convention that the corresponding term is zero when E[W^q]=infinity. In particular, this scalar specialization gives the canonical Mandelbrot-Kahane Fourier-dimension formula under the minimal integrability condition. We also prove the endpoint theorem for the dyadic Mandelbrot cascade on the unit circle. Under the same scalar assumption, almost surely on non-extinction, dim_F(mu_circle)=sup_{q>1} max{0, (q-1-log_2 E[W^q])/q}. The interval and circle formulas share a light-tail/heavy-tail dichotomy but have different mechanisms: energy dimension for the interval, and minimum lower local dimension for the circle. The circle lower bound follows from a finite-moment annular Fourier theorem.

Given an algebra B over a field k, we study conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application of our results, we show that if X is a quasi-affine variety over an arbitrary field k, and if L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If, moreover, k is algebraically closed and of characteristic zero, and X is irreducible and affine, then L is integrable.

Operations research and management science models support decisions that affect patients, workers, citizens, and public institutions. Decision-makers, such as clinicians approving surgical schedules, planners allocating disaster relief resources, or managers designing workforce rotations, increasingly require clear and actionable justifications that bridge the gap between mathematical optimization outputs and the intuitive reasoning stakeholders need to trust, contest, and implement recommended decisions. Yet the field has traditionally evaluated optimization models through computational criteria such as feasibility, optimality, scalability, and solution time, while treating explanation as a secondary concern. Mathematical transparency, provided through access to objectives, constraints, shadow prices, or sensitivity reports, does not automatically offer the forms of justification that stakeholders need to understand, trust, contest, or implement optimization-based decisions. This paper calls for the development of explainable optimization (XOpt) as a distinct interdisciplinary area that moves beyond algorithmic efficiency and incorporates behavioral, cognitive, and pragmatic perspectives to address this explanatory deficit.

Optimal subsampling efficiently selects the most informative data points, enabling accurate statistical inference while significantly reducing computational burden for massive datasets. However, the existing relevant methods can not directly be applied to pairwise loss problems, particularly for convoluted rank regression (CRR), due to the double summation structure in objective function. To this end, we first propose a new BIweighted Poisson Subsampling (BIPS) framework for such problems through designing a proper weight for a pair of observations instead of for a single observation for objective function. Two concrete inverse probability weighting strategies are considered. Secondly, we focus on the CRR models, under which the BIPS estimator (BIPS-CRR) is formulated. We establish consistency and asymptotic normality for BIPS-CRR, derive its optimal Poisson subsampling probabilities under the L-optimality criterion, and provide a practical algorithm to facilitate implementation. Thirdly, we develop a distributed estimator for CRR that incorporates BIPS as a pilot subsampling strategy. This estimation is globally efficient and is robust to both randomly and non-randomly distributed datasets in distributed computing environments. Extensive simulations and a real-world application demonstrate the excellent finite-sample performance of proposed methodology. Additionally, our BIPS can be readily extended to other U-statistics optimization problems and pairwise learning tasks.

We give a short analytic proof of Raposo's signed refinement of Pohst's inequality.

Motivated by polynomial identity testing with exponentials (Li and Wu, ITCS'26), we study uncertainty principles for the number-theoretic transform (NTT). We show that the NTT satisfies strong sparsity tradeoffs: For every fixed prime $q$ and for all but finitely many primes $p \equiv 1 \pmod q$ every nonzero $f\in \mathbb F_p^{\mathbb Z_q}$ and its number-theoretic transform $\hat f$ satisfy \[ |\mathrm{Supp}(f)| + |\mathrm{Supp}(\hat f)| \ge q+1. \] Thus, a $k$-sparse function has transform support at least $q-k+1$. As our main technical contribution, we prove a probabilistic version of the above uncertainty principle, averaged over primes $p$, in the regime $p=q^{O(1)}$. As an application, we obtain a black-box identity test for $k$-sparse exponential polynomials of degree at most $d$ with vanishing soundness error, for $q$ moderately larger than $k$.

In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier–Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier–Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.

An entire slice regular function $f:\mathbb H\to\mathbb H$ can omit four prescribed quaternionic values only in the affine-dependent case. More precisely, four affinely independent omitted values force $f$ to be constant, while the converse follows from the plane-omission theorem of Bisi–Winkelmann. The proof passes to the real-symmetric stem function. For each omitted value a quadratic zero-divisor criterion gives a zero-free entire function $Q_j$, and the component normal to the affine span is governed by a square-discriminant identity. Finite-order data are excluded by Hadamard factorization and a rigidity argument on the real axis. In the general case, logarithmic Bloch–Ochiai places the $Q$-curve in a translated algebraic torus. The Laurent-square case reduces to the finite-order contradiction, and the nonsquare case is excluded by an even-ramification argument together with the level-one truncated Second Main Theorem of Noguchi–Winkelmann–Yamanoi.

We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar.

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -\Delta u=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{i\theta})=(R\cos\theta,R\sin\theta,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_\theta\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis–Coron maps.

We give some necessary conditions satisfied by special perfect polynomials over $\mathbb{F}_2$.

Recent research has developed practical, parallelizable first-order methods for large scale linear programming, but performance is highly dependent on hyperparameter selection. We derive generalization guarantees for hyperparameter tuning within (cu)PDLP, a state-of-the-art first-order LP solver designed for modern hardware. First, we pin down the behavior of PDHG, the primal-dual hybrid gradient algorithm that underlies PDLP, as a function of its step size and primal weight, leading to linear sample complexity guarantees for learning those parameters. We then conduct a structural analysis of PDLP, which augments PDHG with several specialized techniques like preconditioning, adaptive step sizes, averaging, adaptive restarts, and smoothed primal weight updates. Our analysis captures the behavior of the solution trajectory as a function of the hyperparameters and leverages recent advances in data-driven algorithm design to obtain polynomial sample complexity guarantees for learning those hyperparameters. Finally, we conduct proof-of-concept experiments that demonstrate the need for data-driven PDLP parameter tuning. Our results showcase the versatility of the data-driven algorithm design toolkit for principled hyperparameter tuning within solver-grade implementations of complex modern optimization algorithms.

In this paper, we develop a bialgebra theory for diassociative algebras. Inspired by the notion of a quadratic diassociative algebra, we introduce the concept of a Manin triple of diassociative algebras. We then define a diassociative bialgebra, which is shown to be equivalent to a Manin triple of diassociative algebras through a specific matched pair of diassociative algebras. We further formulate the diassociative Yang-Baxter equation (DYBE) in a diassociative algebra, and prove that symmetric solutions of the DYBE give rise to diassociative bialgebras. To construct such solutions, we also introduce relative Rota-Baxter operators and pre-diassociative algebras. As a key application, we lift the known relationships between diassociative algebras and other algebraic structures to the bialgebra level. In particular, we show that every diassociative bialgebra naturally induces a Leibniz bialgebra, thereby extending Loday's classical result that a diassociative algebra gives rise to a Leibniz algebra. Moreover, we provide explicit constructions of Lie bialgebras via tensor products of diassociative bialgebras and quadratic dendriform algebras.

We study Minkowski averages on Riemannian manifolds in which the interpolation is by action-minimizing magnetic geodesics with respect to a given magnetic potential. We establish equivalence between Brunn-Minkowski inequalities for this operation and lower bounds on a magnetic Ricci curvature. We then discuss various examples, including natural magnetic fields on Kähler and Sasakian manifolds, and prove a sharp, undistorted Brunn-Minkowski inequality for contact magnetic geodesics on the Heisenberg group. We also observe that closed magnetic potentials from different cohomology classes may give rise to different geodesic Minkowski averages.

In this paper, we establish two families of $q$-supercongruences modulo the third and fourth powers of a cyclotomic polynomial by employing Watson's ${}_{8}\phi_7$ transformation, the creative microscoping method introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials.

This paper explores a version of the classical Cesàro integral operator for the Lebesgue space $L^p(0, 1)$ where we discuss its norm, spectral properties, cyclicity, and invariant subspaces. The spectrum of the Cesàro operator will be a crescent domain whose geometry depends on $p$. An important tool will be semigroups of weighted composition operators on $L^p(0, 1)$.

We study complex subvarieties in certain non-Kahler holomorphically symplectic manifolds $X$, called the Bogomolov-Guan manifolds. Let $E$ be an elliptic curve, $L$ an ample line bundle on $E$, $S\subset E^2$ a complex curve, and $p_1, p_2$ the corresponding projections of $S$ to $E$. The curve $S$ is called a quasi-diagonal if $p_1^*L\otimes p_2^* L^{-1}$ is a torsion line bundle. We show that there are at most countably many quasi-diagonals for any $(E,L)$. Using the quasi-diagonals, we classify the projective subvarieties in the Bogomolov-Guan manifold. The Bogomolov-Guan manifold is equipped with a Lagrangian fibration $\pi:\; X \to {\Bbb C} P^n$. We show that an irreducible complex subvariety $Z\subset X$ is Moishezon if and only if $\pi(Z)$ is a point or a certain complex curve which is described in terms of quasi-diagonals. This is used to prove that for a general Bogomolov-Guan manifold, any projective subvariety belongs to a fiber of $\pi$.

We prove that level-$\ell$ Kikuchi graphs of random $2r$-uniform hypergraphs spectrally approximate the Kikuchi graph of the complete $2r$-uniform hypergraph at a sampling rate that is sharp up to a logarithmic factor, in the regime $r\leq \ell \leq n/2$. Our proof is based on the matrix Bernstein inequality, but, unlike prior works, we apply it to an appropriate collection of blocks of Johnson eigenspaces. Our analysis relies on a new, simple band-locality property for arbitrary Kikuchi graphs. As an application, we prove that the natural degree-$2\ell$ sum-of-squares relaxation for the Max $2r$-XOR problem is “integral” when the input is a planted noisy $2r$-XOR instance on a random hypergraph with $\gtrsim n \cdot (n/\ell)^{r-1} \log n$ hyperedges.

Let $N$ and $Q$ be a finite groups with $\gcd(|N|,|Q|) = 1$. In this paper we describe normal subgroups of $G = N \rtimes Q$ via normal subgroups of $N$ and $Q$. Let $p$ and $q$ be distinct primes. Let $\mathfrak{A}_p$ be the variety of elementary abelian $p$-groups. Let $\mathfrak{A}_p\mathfrak{A}_q$ be the variety of extensions of a group in $\mathfrak{A}_p$ by a group in $\mathfrak{A}_q$. We also provide a method of determining the normal subgroups of a group in the variety $\mathfrak{A}_p\mathfrak{A}_q$. We also provide the complete list of normal subgroups of a finite group with cyclic Sylow subgroups.

In this work, we study the stochastic Schrödinger–KdV equation driven by additive noise from both analytical and numerical viewpoints. We first establish the evolution laws for the averaged plasmon number, momentum, and energy, together with the conservation of the averaged particle number. Motivated by these intrinsic structures, we develop two temporal discretizations. One is constructed based on the splitting strategy and Crank–Nicolson scheme, and is shown to preserve the discrete evolution laws of the averaged plasmon number and momentum, as well as the discrete conservation law of the averaged particle number. The other is proposed within the constant scalar auxiliary variable framework, in which the nonlinear energy functional is reformulated so that a modified averaged energy law can be preserved at the discrete level. Combining these temporal discretizations with a local discontinuous Galerkin approximation in space yields structure-preserving full discretizations inheriting the corresponding discrete physical laws. Numerical experiments are presented to validate the theoretical results and to demonstrate the accuracy, robustness, and effectiveness of the proposed methods.

Consider a finite particle system in which, at each step, one particle from a vertex with a lower neighboring vertex moves to a neighboring vertex of minimum height. For the cycle C_n, started with n particles at one vertex and no particles elsewhere, we prove that the expected stabilization time is at most a constant multiple of n^3 The deterministic transportation lower bound is of order n^2. Simulations suggest that the true order should be near n^3. Chat GPT was used to reincarnate notes from 2007.

This note is written for the special issue of OSID dedicated to the 50th anniversary of the Gorini–Kossakowski–Lindblad–Sudarshan equation. Its purpose is not to give a comprehensive historical review, but rather to reconstruct, in a self-contained way, one logical route leading to the GKLS generator. The emphasis is placed on Kossakowski's structural insight: the combination of Markovianity, complete positivity, trace preservation, and the infinitesimal structure of quantum dynamical semigroups naturally leads to the celebrated form of the generator. I also include a few personal recollections of Andrzej Kossakowski, under whose guidance I spent one year as a postdoctoral researcher in 2003–2004.

We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T = \begin{pmatrix} A & B \\ 0 & 0 \end{pmatrix}$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U = \begin{pmatrix} U^*AU & B \\ 0 & 0 \end{pmatrix}$ over all unitary operators $U$. Finally, we study matrices $T \in M_n$ satisfying $\|T^m x\| = \|T^m\| = \|T\|$ for a unit vector $x$ and all positive integers $m$. This analysis connects these matrices to the aforementioned block form and provides a counterexample to the conjecture that if $\|T^k\| = \|T\|$ for all $k \ge 1$, then some power of the matrix has a direct summand that is a scalar multiple of an idempotent.

We construct Newton–Okounkov polytopes of Schubert varieties in partial flag varieties of arbitrary type using the cluster structure on a unipotent cell. When the governing cluster algebra is of infinite type, we prove that for any very ample homogeneous line bundle over a simply laced partial flag variety, the resulting family of Newton–Okounkov polytopes contains infinitely many pairwise nonequivalent polytopes up to integral affine transformation. As an application to symplectic geometry, we construct infinitely many distinct monotone Lagrangian tori in a broad class of simply laced partial flag varieties.

In this article, we study the equation $$ (-\Delta_p)^s u = f(u) $$ in a bounded domain $\Omega\subset \mathbb{R}^n$, where $n\geq 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.

An ultimate theory of a phase of matter should describe all its universal properties via quantities that are measurable numerically and experimentally. In this work, we present a microscopic universal theory of symmetry-enriched topological quantum spin liquids (TQSLs) in two spatial dimensions, which directly utilizes microscopically measurable quantities to describe the universal properties. This theory applies to generic TQSLs, which can be Abelian or non-Abelian, chiral or non-chiral. The symmetries are also general, which can include both internal and lattice symmetries, unitary and anti-unitary symmetries, and discrete and continuous symmetries. There can be spin-orbit coupling, the microscopic degrees of freedom may transform linearly or projectively under the symmetries, and the symmetries can permute anyons. The input of the theory is some microscopic states with anyons, operators that control the dynamics of anyons, and symmetry actions in the TQSL, and its output is a set of data characterizing the universal properties, whose underlying mathematical structure is a generalization of category theory. Based on this theory, we find an explicit bijective map between the universal data characterizing a TQSL with a symmetry described by a group $G$, where the symmetry actions may include both lattice and internal symmetries, and the corresponding universal data for a TQSL with only an internal symmetry group $G$, and thus establish a precise crystalline equivalence principle. We demonstrate our theory in symmetry-enriched TQSLs realized on quantum processors based on superconducting qubits, trapped ions, and Rydberg atoms, and in each example we verify the Lieb-Schultz-Mattis anomaly matching condition. Our theory provides a solid basis for identifying and manipulating symmetry-enriched TQSLs, which further paves the way for fault-tolerant quantum computation based on these systems.

We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincaré inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $$\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$$ for all cubes $Q\subset\mathbb{R}^n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}^n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

It is an interesting open problem whether every compact Hausdorff space can be realized as the maximal $d$-spectrum of an arithmetic frame. We approach this problem by generalizing the $d$-nucleus to a stably continuous frame. We use Priestley duality to characterize the resulting $\underline d$-nucleus, which allows us to prove that every locally compact Hausdorff space can be realized as the maximal $\underline d$-spectrum of a continuous regular frame. As a corollary, we obtain that every locally Stone space can be realized as the maximal $d$-spectrum of an algebraic regular frame.