Papers

We develop the convergence analysis of the Interface Control Domain Decomposition (ICDD) method, an overlapping domain decomposition method based on an optimal control framework with Dirichlet interface control functions and interface observation. We consider elliptic problems with possible discontinuous coefficients approximated by $hp-$FEM in each subdomain. When the discretizations are conforming on the overlap between 2D domains, we provide theoretical estimates of the number of GMRES iterations needed to solve the non-symmetric interface Schur complement system associated with ICDD. Our results are obtained by combining novel spectral estimates for the Schur complement matrix of ICDD and classical GMRES convergence theory. We prove that the convergence rate behaves as $\mathcal{O}(\delta^{-1} p^{3/2}\log p)$, where $\delta$ denotes the overlap width and $p$ the local polynomial degree, while remaining independent of the mesh size $h$. Numerical experiments verify the theoretical predictions and show the effectiveness of ICDD in the presence of large coefficient jumps in the computational domain. Since in the conforming case, the considered ICDD formulation coincides with the Substructured Restricted Additive Schwarz (SRAS) method, the analysis also provides convergence estimates for SRAS in two dimensions and, through its known equivalence, for the Restricted Additive Schwarz (RAS) method.

We give a deterministic polynomial-time algorithm that, given rational Hermitian matrices $H_1,\dots,H_N$ of rank at most one, finds signs $s\in\{\pm1\}^N$ with $\|\sum_i s_i H_i\|\le 13\|\sum_i H_i^2\|^{1/2}$. As a corollary, for vectors $v_i$ with $\sum_i v_iv_i^*=I$ and $\|v_i\|^2\le\delta$, the signs yield a partition $[N] = S_1 \cup S_2$ such that each part satisfies $\|\sum_{i \in S_j} v_i v_i^* - \frac{I}{2}\| \leq \frac{13}{2}\sqrt\delta$ for $j = 1,2$. This gives a deterministic polynomial-time algorithm for the Kadison–Singer problem, in Weaver's equivalent discrepancy-theoretic $\mathsf{KS}_2$ formulation, with a universal constant.

We study cover times on discrete tori and ranges on $\mathbb{Z}^d$ for elephant random walks with memory parameter $p\in[0,1)$. In dimension one there is a phase transition at $p=3/4$. Away from criticality we determine the exact first-order asymptotics of the mean cover time, while at $p=3/4$ the mean is of order $L^2/\sqrt{\log L}$, a factor $\sqrt{\log L}$ larger than the natural fluctuation scale $L^2/\log L$. We identify the limiting distribution under each of the three natural fluctuation normalizations. In dimensions $d\geq2$, for every fixed $p<1$, the cover time has the same order as for simple random walk, in expectation and with high probability. The leading constants also agree when $p<(2d+1)/(4d)$, and at equality when $d\geq3$. For the range on $\mathbb{Z}^d$, we prove $L^1$ convergence to the simple random walk asymptotics when $d=2$ and $p<5/8$, and for every $p<1$ when $d\geq3$. Finally, we give a cover-time upper bound for generalized step-reinforced random walks on finite groups in terms of a conditional $L^\infty$ mixing profile.

We consider scaling limits of integer-valued random walks whose transition probabilities are translation-invariant everywhere except on a set that we will call a membrane or an interface. If the jumps outside the membrane are i.i.d. mean-zero random variables with finite variance, then possible examples of the corresponding limit processes include reflected Brownian motion, skew Brownian motion, and Brownian motion with a jump-type exit from zero. We also discuss the question of what the possible limits are if the jumps outside the membrane belong to the domain of attraction of a stable law.

We prove rigidity near the Euclidean metric for the anisotropic Calderón problem on smooth bounded connected domains in dimensions three and higher. Any smooth Riemannian metric sufficiently close to the Euclidean metric in a suitable Hölder norm and having the same Dirichlet-to-Neumann map agrees with it up to a diffeomorphism fixing the boundary pointwise. The result applies to general smooth anisotropic perturbations, without analytic or quasianalytic regularity assumptions, a prescribed conformal class, or a transversal product structure, and requires no convexity of the boundary. The proof combines harmonic coordinates with a Carleman estimate for compactly supported correctors to obtain a quadratic Fourier estimate on a frequency range determined by the perturbation. A frequency decomposition then yields rigidity.

Lara, Srinivas and Stix showed that the étale fundamental group of a proper scheme over an algebraically closed field is topologically finitely presented. Building on this, we show that the étale fundamental group, as introduced by Noohi, of a proper algebraic stack with finite inertia over an algebraically closed field is topologically finitely presented.

We study the inverse problem of recovering initial data for fractional heat equations on bounded domains from observations taken on two types of possibly Lebesgue-null sets: the first consists of general thin sets whose Hausdorff dimension exceeds \(n-1\); the second consists of specially constructed sets of zero Hausdorff dimension, built from algebraic irrational points and rapidly accumulating sequences.By extending thin-set observability inequalities from the classical heat equation to the fractional setting and employing a unified spectral inequality with exponent \(\beta\in(0,s)\), we establish a quantitative observability estimate for every fractional exponent \(s>1/2\). For general thin sets one has \(\beta=1/2\); for the special zero-dimensional sets any \(\beta\in(1/2,s)\) is admissible, yet the short-time observability cost always remains of exponential type, with the rate governed by \(\beta\) relative to \(s\). Under an a priori smoothness assumption, we also prove a logarithmic stability estimate.We further design a regularised least-squares reconstruction algorithm and provide a conditional convergence analysis, showing that the reconstruction error is bounded by the sum of a spectral truncation term and a logarithmic term dictated by the continuous stability, and that the error decays logarithmically as the noise level tends to zero. Numerical simulations using very few observation points confirm the feasibility of the proposed approach.

The paper presents a higher-order global multi-trace integral equation for time-harmonic electromagnetic scattering by composite objects. The higher-order multi-trace formulation is preconditioned with a Calderón multiplicative preconditioner using higher-order quasi-Helmholtz projectors. The higher-order quasi-Helmholtz projectors separate the solenoidal and non-solenoidal components of the basis functions. Separate access to the Helmholtz components circumvents the explicit inversion of an ill-conditioned mixed Gram matrix. This enables the application of Calderón multiplicative preconditioning without refining the mesh and resorting to dual basis functions. Furthermore, it enables low-frequency stabilization, as the solenoidal and non-solenoidal components can be rescaled individually. The higher-order quasi-Helmholtz projectors are computed iteratively, enabling iterative solvers to efficiently solve the preconditioned matrix system, yielding accurate solutions in a few dozen iterations. Numerical experiments are conducted for composite dielectric bodies, confirming the effectiveness of the proposed preconditioner for the global multi-trace formulation discretized with higher-order basis functions for dense meshes and at very low frequencies

We consider weighted lattice point enumeration in lecture hall order polytopes. We observe that these half-open polytopes are closed under weight-lifting for a large class of polynomial weights, including monomials and weights given by binomial coefficients. For rooted trees, we prove that the corresponding (weighted) $h^\ast$-polynomial is real-rooted thereby complementing previous results of Savage-Visontai and Brändén-Leander.

We improve the exponents of the $A_p$ constant in the strong and weak-type weighted bounds for the strong maximal function, for every $1 < p <\infty$ and every dimension $d \geq 2$. In dimension two, the strong and weak $L^2(w)$ exponents improve from $2$ to $7/4$ and from $3/2$ to $5/4$, respectively. The proof exploits the geometry of pairwise intersections of antichains of dyadic rectangles.

We consider the nonlinear wave equation (NLW) with a superlinear pure power nonlinearity in the subconformal and conformal cases. Under some conditions on initial data, this equation is known to have solutions which blow up in finite time. Two questions are then relevant: (i) the classification of all possible blow-up behaviors; (ii) the construction of examples of blow-up solutions. As we will show, the situation is entirely settled in the one-dimensional case, where we fully solve the famous soliton resolution conjecture. Various extensions to higher dimensions and to perturbed versions of NLW are given, including the construction of a solution in 2-d, with a nearly pyramidal blow-up graph. Carrying out this program was made possible thanks to a synergy of techniques from the PDE theory, mathematical physics, and analysis, including ODE techniques, spectral theory, and energy methods. Throughout this presentation, we will insist on connections with the study of other types of PDEs, in particular in the parabolic case. Surprisingly enough, in spite of the difference between parabolic and hyperbolic equations at the linear level, the nonlinear nature brings in a strong unity-both in the results and in the methods-between these two important classes of PDEs.

We prove the cohomological Gopakumar-Vafa/Pandharipande-Thomas correspondence for the local plane and quadric in every effective curve class and Euler characteristic. The strict supports of the simple perverse constituents of the stable pair vanishing cycle direct images are closures of loci of transverse unions of smooth connected curves. Wall-crossing and duality leave a finite range of Euler characteristics to consider. In this range, a commutation relation for point modifications on bounded stacks of surface pairs excludes all other supports. Applying the family Macdonald formula on the reduced locus gives the correspondence as an identity of semisimplified perverse direct images on the Chow variety.

We prove the uniform Mordell–Lang conjecture for semiabelian varieties, that is, for a subvariety $X$ of a semiabelian variety, its intersection with a finite-rank subgroup of rational points, can be covered by at most $c$ cosets contained in $X$, where $c$ depends only on the dimension of the semiabelian variety, the degree of $X$, and the rank of the subgroup. Our proof is new even in the abelian case previously proven by Gao–Ge–Kühne.

Ecological and evolutionary communities do not always settle into a single predictable configuration. Models such as Lotka-Volterra or replicator dynamics may predict several admissible states. The multiplicity of such states is central to understanding robustness, alternative community configurations, coexistence, and shifts in biodiversity. Yet, standard practices are often state-centered: they either focus on the properties of typical equilibria or pool states across systems, losing information about how many states can arise in a single system, including cases where no state exists. Here, we introduce a system-centered probabilistic formalism that captures this hidden structure and could be applied across a wide range of ecological and evolutionary dynamical models, drawing inspiration from hurdle and zero-inflated models. Each ecological system is viewed as generating a distribution over its possible outcomes. This allows us to jointly quantify the occurrence of a given state type, its multiplicity within each system, and the number of species co-existing within these states. The formalism is generic and can be applied to different families of dynamical models. To illustrate its scope, we consider two case studies: the generalized Lotka-Volterra and replicator dynamics with random and structured parameters. This unified perspective reveals patterns that remain invisible when equilibria are analyzed individually or pooled across systems. By turning the state space of a model into interpretable probabilistic quantities, our formalism offers a new way for studying the robustness of community structure, the likelihood of alternative outcomes under small changes in initial conditions, and the potential for shifts between states of low and high diversity.

We extend Behrend's theory of complementary polyhedra to include Jordan-Hölder theory. This provides a uniform approach to describing the Jordan-Hölder filtrations and S-equivalence in moduli stacks of bundles, following Zhang. As an application, we give a description of Jordan-Hölder reductions and S-equivalence in the semistable locus of parahoric torsors and parahoric Higgs torsors, coming from their good moduli spaces constructed by Alper-Halpern-Leistner-Heinloth and Réga. This application uses the Rees construction to identify filtrations of moduli stacks with reductions of bundles, and induces a Jordan-Hölder stratification of the moduli stacks.

For high-quality images with sharp edges, a popular choice for edge-preserving regularization is using a general(ized) $\ell_q$-norm of the gradient of the image. This can be implemented efficiently using the $\ell_2$-norm and a sequence of weighted gradients, with weights derived from the current solution estimate. We can solve the resulting sequence of regularized least squares problems using hybrid Krylov subspace methods, which efficiently compute the regularization parameter using the problem projected on the Krylov subspace. However, each update of the regularization operator requires a new Krylov subspace. The majorization-minimization generalized Krylov subspace method (MM-GKS) addresses this problem by using a single, generalized, Krylov subspace (GKS). Unfortunately, for large-scale problems, if convergence is not fast, MM-GKS has overwhelming memory requirements and computational costs. We propose a variant of MM-GKS that alternately compresses and expands the search space while maintaining strict monotonic convergence. We show that our method provably converges to the minimum of the selected functional, even if the search space dimension is kept very small. This substantially improves on previous theoretical results for MM-GKS, where the convergence proof relies on the basis for the solution space (eventually) spanning the full space. We show that our method can solve large-scale problems efficiently both in terms of memory requirements and computational complexity. We further generalize our proposed method to handle streaming problems, where the data is either not all available simultaneously or needs to be treated as such because of the extreme memory requirements. We use numerical examples from image deblurring, dynamic photoacoustic tomography, and streaming X-ray computed tomography (CT) to illustrate the effectiveness of our proposed methods.

We study divergence-form elliptic equations with Dini coefficients, extending the frameworks of \cite{NV} for Lipschitz coefficients and \cite{HJ1} for Hölder coefficients. We first establish an almost monotonicity property for the doubling index under merely continuous coefficients. This leads to weak volume estimates for critical sets and uniform volume estimates for nodal sets. The Dini assumption is essential here since we construct an example with continuous coefficients and bounded doubling index whose nodal set has infinite measure. We also prove polynomial growth estimates for sub-level sets. Using a different approach, we establish quantitative uniqueness of tangent maps and a cone-splitting principle, which yield explicit Minkowski-type estimates for critical sets. Finally, combined with the results of Kenig–Zhao \cite{KZ4}, our estimates give measure bounds for boundary critical sets.

We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fourier density of a finite spectral projection. Combining this cap with the standard $L^2$ Bessel estimate and a radial-capacity bathtub principle gives, on every open set of finite positive measure in $\mathbb R^n$ with $n\geq2$, \[ \lambda_k\geq c_n(2\pi)^2\omega_n^{-2/n}|\Omega|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1. \] The constant $c_n$ is characterized by a scalar equation, with $c_2=0.5383068077\ldots$. This estimate preserves Weyl scaling and strictly improves the individual consequence of the Li-Yau sum inequality, although it does not improve the sharp leading coefficient in that sum inequality. Second, we retain part of the spectral deficit discarded when an eigenvalue sum is bounded by its largest term. A lower bound for the counting function, integrated through the exact first Riesz-mean identity, leads to a strictly monotone scalar equation. Its unique positive root is no weaker than the volume-only bound, and we give necessary and sufficient criteria for strict improvement over both that baseline and any independent lower bound. Quantitative estimates of Jiang-Lin provide an explicit implementation on bounded Lipschitz domains. The final comparisons and numerical example distinguish improvements within this framework from stronger estimates available under additional geometric or spectral assumptions.

We develop an algebraic mechanism based on Laplacian algebras for constructing minimal hypercones and determining their variational properties. Generator counts and degrees organize the constructions; homogeneous multipliers depending on the generators yield subcalibration fields, stability bounds and quantitative area-minimizing inequalities. Two-generator isoparametric algebras give polynomial subcalibrations for their minimizing hypercones. In ambient dimension at least three, strict gradient-square closure forces nonlinear odd degree to be cubic. We then study algebras of degrees $(2,2,g)$ and quadratic compositions, placing OT–FKM splitting quartics and Riedler's pullbacks in a common quadratic-composition framework. It yields two further infinite quartic families with a sharp minimizing threshold, as well as harmonic sextics from complex Gram coordinates. For an irreducible harmonic minimal cubic on its essential space, we prove a sharp bound of five generators for its smallest Laplacian algebra. We determine the minimizing range of Clifford cubics and construct irreducible nonradial harmonic examples in ambient dimensions $8,14,26$; the last is strictly area minimizing. Finally, spectral families whose smallest algebras have arbitrarily many generators provide examples in every degree at least two: real parts of octonionic products, complex Pfaffians and complex determinants. Explicit potentials establish strict area minimality and stability bounds, including across nonisolated singularities. The algebraic identities and certificates are supported by reproducible exact symbolic computations.

We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over $\mathbb F_2$, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.

A finite structural characterization of the $U$-user multiuser channel-capacity region appears here. This region relies upon three concepts: (i) subset-based message atomization, (ii) synchronized receiver chain-rule/Fano reduction, and (iii) successive-decoding achievability via a finite-super-symbol closure. The resulting region is a finite union of order-indexed polytopes parameterized by the synchronized minimum mutual-information vector $\mathcal{I}_{\min}$. The framework removes auxiliary-random-variable proliferation and makes explicit the finite geometric structure underlying general multiuser converses, with the interference channel providing the primary illustration. For the linear matrix Gaussian multiuser-channel special case under per-user trace covariance constraints, Gaussian signaling is capacity-region-achieving within this framework. Also shown is that maximum rate sum for any Gaussian multiple-user-channel derives from simple iterative procedures.

For integers $1\le k < n$ and real numbers $c_1, \ldots, c_k>0$ and $d_1, \ldots, d_n\ge 0$, we investigate the space of all $k\times n$ matrices $F$ such that the product $FF^*$ is equal to the diagonal matrix ${\rm Diag}(c_1, \ldots, c_k)$ and the squared norms of the columns of $F$ are equal to $d_1, \ldots, d_n$ respectively. Depending on the field where the coefficients of $F$ are taken from, which can be of real or of complex numbers, we are mainly interested in determining whether the resulting space is path-connected or even simply connected relative to the subspace topology in the space of all $k \times n$ matrices. The criteria presented are closely related to results previously obtained by Cahill, Mixon, and Strawn (2017), Needham and Shonkwiler (2021), the last two authors together with Caine (2026), and the author of this work (2024 and 2026). Flag manifolds, namely orbits of the canonical conjugation actions of ${\rm O}(n)$ and ${\rm U}(n)$ on the spaces of symmetric real and Hermitian $n\times n$ matrices, respectively, play a central role in our development.

We study the structure and distribution of atoms in the monoid $\mathcal{M}{\rm on}(R)$ of nonzero monomial ideals of the polynomial ring $R=K[X_1,\ldots, X_N]$, with $N\ge 2$. We characterize the atoms with at most four minimal generators and introduce the density $d(N,\mu)$ of atoms among monomial ideals with $\mu$ minimal generators. For $\mu\le 4$, we prove the existence of this density and determine its asymptotic behavior as $N\to\infty$. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of $\mathcal{M}{\rm on}(R)$ generated by equigenerated ideals.

Wavelets and frames that provide strong guarantees for signal representations are a cornerstone in signal processing theory, but have long been restricted to non-causal settings. Recent work extended wavelet analysis to time-causal systems that operate on streaming signals, with exact reconstruction and time-recursive implementations. However, the frame guarantees that make such representations reliable under noise and quantization are uncharacterized. We derive closed-form frame bounds and conditioning on band-limited domains, together with peak frequencies, constant-Q bandwidth tiling, and spectral decay rates for time-causal bandpass wavelets. Our work provides design rules for the scale ratio and channel count, which we validate numerically against discrete implementations of the causal wavelets in terms of recursive filters.

A skew morphism on a finite group $A$ is a permutation $\varphi$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $\pi:A\to\mathbb{Z}_{|\varphi|}$ such that $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$. The kernel of $\varphi$ is the subgroup $\Ker\varphi=\{x\in A\mid \pi(x)=1\}$, and the index $[A:\Ker\varphi]$ is called the skew-type of $\varphi$. In this paper we construct, classify and enumerate the skew morphisms of skew-type four on cyclic $2$-groups. Our main results give explicit formulas for all such skew morphisms and closed-form expressions for their numbers.

We study a state-constrained Bolza optimal control problem governed by a non-local continuity equation in the Wasserstein space $\sP_2(\R^d)$. We develop a Pontryagin maximum principle for these problems in which both the cost functionals and the constraints are only locally Lipschitz, and the initial measure does not necessarily have compact support. To derive the main result, we introduce a notion of a Clarke-type deterministic subdifferential over $\sP_2(\R^d)$. The proof of the main result combines a careful adaptation of the Lasry-Lions regularization to the Wasserstein space coupled with stability results of the Clarke subdifferential that allow us to recover the first-order information. We also show that, for functionals convex along plans, this subdifferential admits a global supporting characterization and provides a first-order characterization of global minimizers.

We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on $\R^d$, $d\ge2$, in the endpoint space \[ \dot B^{-1}_{\infty,1}(\R^d)\times \dot B^{0}_{\infty,1}(\R^d). \] For every sufficiently small $\varepsilon>0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and \[ \norm[\dot B^{-1}_{\infty,1}]{u_0} +\norm[\dot B^{0}_{\infty,1}]{b_0}<\varepsilon, \] such that the corresponding classical solution satisfies \[ \norm[\dot B^{0}_{\infty,1}]{b(t_\varepsilon)}>\varepsilon^{-1} \] for some $0<t_\varepsilon<\varepsilon$. Consequently, no solution map agreeing with classical solutions can be continuous at the origin in the magnetic endpoint topology. The low–high paraproduct mechanism available when the dyadic summability exponent is greater than one gives no gain at the $\ell^1$ endpoint. We instead combine a resonant four-wave velocity interaction with magnetic stretching to produce uniformly sized contributions on a growing family of dyadic shells. Frequency-localized test functionals extract the endpoint lower bound, while uniform Lagrangian estimates in weighted Fourier $L^1$ spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.

Let $K$ be the non-singleton attractor of a finite iterated function system $\Phi=\{\phi_i(x)=a_i O_i x+t_i\}_{i=1}^m$ on $\mathbb{R}^d$, where $0<a_i<1$, $t_i\in\mathbb{R}^d$ and the $O_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that $\Phi$ satisfies the open set condition, but not the strong separation condition. We show that $K$ cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton–Jacobi–Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is \(O(\tau\Delta t^{\gamma/2})\), and the total mass of interpolated branches is at most \(2\tau/\Delta t\), where \(\tau\) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as \(\tau\to0\). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield \[ \|(u-u_h)^+\|_\infty\le C(\Delta t^{1/4}+h\Delta t^{-1/2}), \qquad \|(u_h-u)^+\|_\infty\le C(\Delta t^{1/10}+h^{1/2}\Delta t^{-1/4}). \] Thus \(\Delta t\asymp h^{10/7}\) gives an \(O(h^{1/7})\) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.

There exist three nonequivalent left invariant Lorentzian metrics on the Heisenberg group $H_{2n+1}$, or equivalently, three nonequivalent Lorentzian inner products on the Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$, denoted by $\mu$, $\nu$, and $\phi$. We show that, in a specific case, $\mu$ is an algebraic Ricci soliton that is shrinking. Moreover, $\nu$ is an algebraic Ricci soliton only on the three-dimensional Heisenberg Lie algebra $\mathfrak{h}_3$ and it is shrinking. Finally, we show that $\phi$ is a steady algebraic Ricci soliton on $\mathfrak{h}_{2n + 1}$ for $n > 1$. However, for $n = 1$, $\phi$ is flat.