Papers

We study complete stable minimal hypersurfaces in Riemannian manifolds under various curvature assumptions. In an $(n+1)$-dimensional complete oriented manifold with nonnegative scalar curvature, we prove that no complete oriented noncompact stable minimal hypersurface can be conformally equivalent to a bounded domain in an $n$-dimensional manifold with nonpositive scalar curvature. As a consequence, there exists no complete stable minimal hypersurface in $\mathbb{R}^{n+1}$ that is conformally equivalent to a bounded domain in $\mathbb{R}^n$. We also show that compact stable minimal hypersurfaces in manifolds with nonnegative scalar curvature have nonnegative smooth Yamabe invariant and we characterize the equality case. Next, we consider complete two-sided stable minimal hypersurfaces in ambient manifolds with pinched sectional curvature. Under an additional curvature condition that makes the second fundamental form a Codazzi tensor, we obtain a rigidity result under an $L^2$-condition on the second fundamental form and derive an upper bound for the first eigenvalue of the Laplacian. Finally, we establish that any complete noncompact two-sided minimal hypersurface immersed in a warped product manifold is stable, provided that the angle function is positive and the second derivative of the warping function is nonnegative.

We proof the existence of the Integrated Density of States (IDS) for fractional random Schrödinger operators with Gaussian potential based on the theory of isotropic $\alpha$-stable Lévy processes. Further we show that the IDS exhibits Lifshitz tails and determine its asymptotics at the right end of the spectrum. Isotropic$\alpha$-stable Lévy processes are self-similar but not continuous. Thus self-similarity is a fundamental concept for these quantum operators.

We study the problem of minimizing a weighted sum of squared Wasserstein distances with signed coefficients. In the case of a single positive coefficient, we derive two convex dual formulations: one expressed in terms of Brenier potentials, and one as a projection problem in convex order. Such dual formulations arise via Toland duality by exploiting different notions of convexity within the problem. They generalize those recently established for the metric extrapolation problem, which corresponds to signed barycenters between only two measures. For general signed barycenters, we identify necessary optimality conditions that reduce the problem to the case with one positive coefficient. On the numerical side, we propose a grid-free, particle-based algorithm built on entropic regularization and Sinkhorn iterations to compute signed barycenters with arbitrary coefficients.

We classify finite-energy positive solutions to the Yamabe-type equation on the 15-dimensional octonionic Heisenberg group, whose algebra is non-associative. Extremals for the Folland-Stein-Sobolev inequality on this group are explicitly described. This confirms the conjecture of Garofalo and Vassilev [Duke Math. J. 2001] on the $15$ dimensional octonionic Heisenberg group.

Let $\Delta_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.

We study the far-field behavior of solutions of the one-phase Bernoulli free boundary problem in the exterior of a ball, and of entire solutions with a single compactly supported inhomogeneity of the free boundary condition, which we call a defect. For solutions which blow down to a half-plane solution (proper solutions) we establish an asymptotic expansion at infinity: in dimension $d \geq 3$ the free boundary height converges to a limit at rate $|x|^{2-d}$ with a capacity-type coefficient, while in dimension $d=2$ the expansion carries a logarithmic term. A significant novelty is that the expansions are quantitative and uniform over all the proper solutions.

Muon-spin rotation/relaxation is one of the most powerful microscopic probes for understanding magnetic order, spin dynamics, superconductivity, and complex magnetic phases in emerging quantum materials. Quantitative interpretation depends critically on the accurate identification of the muon stopping site, a problem traditionally addressed using density functional theory-based structural relaxation. However, conventional DFT approaches are computationally demanding, time-consuming, and require extensive calculations. Here, we develop MEOWN (Muon Engine for Optimized Weighted Networks), a machine-learning-based, physics-informed computational framework that combines a Polarizable Unperturbed Electrostatic Potential (P-UEP) model with machine-learning-guided optimization and symmetry-driven relaxation to predict energetically favorable muon stopping sites in crystalline solids. MEOWN explicitly incorporates electrostatic interactions, electronic screening, polarization effects, and zero-point motion into the learning workflow, providing physically interpretable predictions with substantially reduced computational cost while maintaining physical consistency and scientific rigor. To validate the framework, we investigated several well-established benchmark materials, including MnSi, CoF2, CaF2, LiF, and NaF. The predicted muon stopping sites and dipolar fields are in close agreement with previously reported DFT+mu results, demonstrating the reliability and transferability of the approach across chemically diverse systems. Notably, MEOWN predicts the equilibrium muon stopping site within a few minutes, offering a significant speedup over conventional DFT-based calculations. This manuscript presents the theoretical foundations, computational methodology, and validation of MEOWN as a general-purpose software for rapid and reliable muon stopping-site prediction in crystalline materials.

We develop an extension of Deligne–Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat–Tits building, which we call paraductive. We then compute explicit lower bounds for the Tate cohomology of representations of paraductive groups, relating these to Shintani descent, Lusztig restriction, and the Glauberman correspondence. As an application, using Feng's modular functoriality and Scholze's independence of $\ell$, we compute the Fargues–Scholze L-parameters of non-singular depth $0$ cuspidal representations of a (possibly wildly ramified) reductive group over a nonarchimedean local field.

We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature $k$. We prove that, under the assumption that $h/f$ is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.

Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calderón–Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \(\mathcal A_2\) weights \(U,V\) and a fixed Riesz transform \(R_j\) on \(\mathbb R^d\), \(d\geq2\), we characterize the Schatten class membership of \(C_{B,R_j}:L^2(U)\to L^2(V)\). We characterize $\mathcal S^r$ membership for $r>d$ and $\mathcal S^{d,\infty}$ membership by $\ell^r$ and weak $\ell^d$ conditions on the two Hilbert–Schmidt cube oscillations. For $0<r\leq d$, $\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.

We introduce forward recursive aggregator systems in Itô-diffusion markets, bridging the theories of recursive utilities and of forward performance criteria. The resulting criteria are constructed forward in time and accommodate arbitrary or rolling horizons. We derive the associated HJB SPDE and sufficient conditions for consistency and admissibility. We also develop a convex duality theory based on state price densities and identify the density generated by the optimal wealth process. For homothetic aggregators, we characterize a class of forward Epstein–Zin recursive utility processes, derive the optimal investment and consumption policies, and obtain explicit solutions in the Black and Scholes and Heston models.

We prove a stronger form of Cowling's $L^p$-integrability conjecture for an admissible class of Kunze–Stein groups containing all $S$-algebraic groups and all closed CCR boundary-transitive subgroups of tree automorphism groups. We also disprove a cyclic version formulated by Samei–Wiersma.

We prove that a general smooth complete intersection of positive dimension and codimension at least 2 over an algebraically closed field has trivial linear automorphism group, except in the case of two quadrics. The proof uses incidence varieties and treats separately prime-to-characteristic and unipotent automorphisms.

Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the intermediate states. We propose a new strategy to design multi-state HLL-type solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, whose compressibility is estimated based on the relations between fast, Alfvén, and slow speeds. Then, following the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfvén, and slow waves, we calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.

We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $\mu$ and condition number $\kappa$, the two methods achieve exact recovery with high probability from $O(\mu nr\log n\log(n\kappa))$ and $O(\mu nr\log n\log(2\mu r\kappa))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.

In this paper, we study the notion of $\mathcal{I}$$^{st}$-convergence of sequences of real numbers which is an extension work of $\mathcal{I}$$^*$-convergence on real sequences. We also define the convergence field of $\mathcal{I}$$^{st}$-convergence and $\mathcal{I}$$^{st}$-limit points and we discuss some properties related to these notions.

The Thom-Levine theorem is a classical method for proving the triviality of deformations by integrating suitable vector fields. In this note, we show that the converse of this criterion does not hold in the bi-Lipschitz category. More precisely, we give an example of a smooth one-parameter deformation of a real function germ in two variables which is bi-Lipschitz trivial but not strongly bi-Lipschitz trivial. The central germ has an isolated critical point, though it is not finitely determined.

Let $F=F(X)$ be a free group of finite rank, with palindromic length taken with respect to the fixed basis $X$. We embed $F$ as the index-two subgroup of the universal Coxeter group $W=F\rtimes_\theta\langle t\mid t^2=1\rangle$, where $\theta(x)=x^{-1}$ for $x\in X$, and prove $\mathrm{pl}(g)=\min{\ell_T(g),\ell_T(gt)}$. Dyer's deletion theorem then identifies reflection length with the minimum number of unmatched positions in a noncrossing equal-label partial matching on a reduced Coxeter word. This gives an $O(n^3)$-time, $O(n^2)$-space algorithm for palindromic length, together with recovery of an optimal palindromic factorization. The matching model also gives a structural characterization. For every ordered full binary tree with $k$ leaves we define a literal word template whose leaves are palindromes and whose internal vertices carry arbitrary words. A reduced word $w$ represents an element of palindromic length at most $k$ if and only if $w$ is a literal instance of one of these templates. Hence the $C_{k-1}$ ordered binary-tree shapes give a complete finite family for each fixed $k$. For $k=4$ the five templates are exactly the five forms proposed by Frid, proving the completeness of that list. A companion Lean 4 development verifies the four-palindrome classification end to end for every finite rank, including the ordinary reduced-word formulation and the literal five-form conclusion.

We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\mathbb{F}_q$. 1. Random linear codes over $\mathbb{F}_q$ with rate $1 - H_q(p) - \epsilon$ are $(p, O(q H_q(p)/\epsilon))$-list decodable with high probability for all values of $p, q, \epsilon$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/\epsilon$ due to Guruswami, Li, Mosheiff, Resch, Silas, and Wootters [IEEE Trans. Inf. Theory 2022]. Our bound is the first uniform improvement for $q > 2$ since Guruswami, Håstad, and Kopparty [STOC 2010]. 2. Linear hash functions over $\mathbb{F}_q$ hashing $n$ balls to $n$ bins achieve maximum load $O(q \ln \ln q / {\ln q}) \cdot \ln n / {\ln \ln n}$, both in expectation and with probability $1-o(1)$. This nearly matches the lower bound of $\ln n / {\ln \ln n}$. Previously, only a polylogarithmic upper bound was known for $q > 2$, due to Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos [J. ACM 1999]. We reduce list decodability and linear hashing to strong Furstenberg set lower bounds, which we prove using a new polynomial method of multiplicity gaps. While previous polynomial methods analyze a set $S$ by studying polynomials that vanish on it, we consider polynomials that vanish everywhere, but with higher multiplicity inside $S$ than outside.

We study strictly localized mixed states in quantum field theory: states that are indistinguishable from vacuum outside some spacetime region. Such states are generated from the vacuum by so-called Licht maps. While the states associated with individual outcomes of local, projective measurements on the vacuum are not strictly localized, the corresponding non-selective states are strictly localized and even admit decompositions into strictly localized pure states. Remarkably, however, there are strictly localized mixed states that cannot be written as mixtures of pure states strictly localized to the same region. Such states can arise from unsharp local measurements. Finally, we extend Knight's theorem to mixed states, showing that strictly localized bosonic mixed states contain terms with arbitrarily high particle numbers.

Let $R$ be a nonzero commutative ring with identity, $n\geq3$, and $\sigma\in GL_n(R)$. We prove that there exist nonidentity transvections $\tau_1,\tau_2\in GL_n(R)$ such that $\big[[\sigma,\tau_1],\tau_2\big]=I_n$. Hence, Kourovka Notebook Problem 10.46 has a stronger affirmative answer over commutative rings with identity. The proof uses rank-one operators and the existence of a nonzero functional annihilating a standard basis vector and its image under $\sigma$.

Under mild assumptions, we prove the strong main conjecture for BDP Selmer groups of elliptic curves with good reduction in the sense of Kurihara's refined Iwasawa theory. In particular, the structure of the initial Fitting ideal of BDP Selmer groups over finite subextensions in the anticyclotomic $\mathbb{Z}_p$-extension is determined by the values of the corresponding $p$-adic modular form at CM points.

Particle swarm optimization (PSO) is a widely used algorithm featured in many state-of-the-art optimization tool-kits. However, rigorous performance guarantees are still lacking. The standard PSO dynamics do not admit a natural mean-field description, which would provide an avenue for theoretical analysis. By modifying the PSO formulation, one can recover the consensus-based optimization (CBO) algorithm with memory, which admits a mean-field limit and facilitates rigorous convergence analysis. These theoretical guarantees rely heavily on the fact that for CBO, the drift and noise strengths can be chosen independently, whereas they are coupled for PSO. We analyze how the PSO parameter coupling affects existing convergence guarantees for CBO and its variant with memory effect. We show, by an explicit construction, that the coupling still leaves a non-empty set of admissible parameters for these convergence guarantees to hold. However, the admissible parameter ranges shrink in the limits used to recover PSO. The resulting convergence guarantees from CBO therefore do not directly extend to the classical PSO model. We provide numerical simulations illustrating the parameter tradeoffs shown in the theoretical analysis.

We show that every classifiable simple C*-algebra, in particular the Jiang-Su algebra, admits outer actions of several discrete quantum groups, such as finite quantum groups and duals of certain free compact quantum groups. As a consequence of the construction, we provide new examples of hyperfinite subfactors with the Temperley-Lieb-Jones standard invariant and show that $\operatorname{SU}_q(2)$ can act minimally on some Araki-Woods factors for infinitely many $q$.

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model and symmetric perturbations of symmetric inhomogeneous stationary states that are naturally associated with a pendulum Hamiltonian flow. We first show the existence of families of such stationary states having compact support inside the separatrix region, constant value near the origin, and satisfying a linear stability (Penrose) criterion. We then describe the nonlinear dynamics of small compactly supported perturbations with finite regularity of this class of equilibria. More precisely, we prove that for large (but finite) times, in a modulated action-angle coordinate system the solution remains close to a modulated stationary state and that the magnetization of the perturbation satisfes decay estimates driven by the linear dynamics.

In this note we announce the limiting shape for the height function of a uniformly distributed alternating sign matrix (equivalently, a six-vertex model with equal weights and domain-wall boundary conditions)

We consider the harmonically weighted Dirichlet spaces $D(\mu)$ induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure $m$ on the unit circle $\mathbb T$. Let $H^2$ and $D$ denote the Hardy space and the Dirichlet space on the unit disc $\mathbb D$, respectively. For any $f\in H^2$, let $m_f$ denote the measure defined by $dm_f(\zeta)=|f(\zeta)|^2 dm(\zeta)$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $|f'(z)|^2=O(\frac{1}{(1-|z|)^{1-\epsilon}})$ for some $\epsilon>0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f'$ is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of $D(m_f)$ in terms of the radial zero set of $f$. As an application, we prove that if $f',g'$ are bounded and the radial zero sets of $f$ and $g$ are different, then the operators $(M_z, D(m_f))$ and $(M_z, D(m_g))$ are not similar. This generalizes a result of Richter showing that the operators $(M_z, D(m_{z-1}))$ and $(M_z, D)$ are not similar. Then, we classify all invariant subspaces $\mathcal M$ of $(M_z, D(\mu))$ for which $M_z|_{\mathcal M}$ is similar to $(M_z, D(\mu))$, where $\mu$ is any measure mutually absolutely continuous with respect to $m$. Finally, we study a special case of the problem of finding all functions $g\in D$ for which $gD(m_g)$ is a closed invariant subspace of $(M_z, D)$. A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.

Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).

We introduce the Unified Separation Condition (USC) - a simple and general constraint qualification for finite-dimensional nonlinear programming. The USC requires that the origin be uniformly separated from the subdifferential of the constraint violation function in a neighborhood of the feasible set. The main result of this paper is two-fold. First, we show that the classical Mangasarian–Fromovitz constraint qualification (MFCQ) for inequalities and the linear independence constraint qualification (LICQ) for equalities imply the USC. Conversely, USC is strictly more general: it remains applicable in nonsmooth settings where classical conditions are not defined, and it may hold even when MFCQ fails. The proof that MFCQ implies USC is based on Gordan's theorem; the implication from LICQ to USC follows from the linear independence of the gradients of the active constraints. We also introduce a local version of USC and discuss its computational verification via convex quadratic programming. The USC provides a unified framework for constraint qualifications, bridging classical smooth theory and nonsmooth optimization.