Papers

We study the behavior of strongly facially symmetric spaces under completion and passage to norm-dense subspaces. We introduce a natural face-density condition guaranteeing that the completion of a normed SFS-space remains strongly facially symmetric. We prove that a norm-dense subspace of a neutral strongly facially symmetric space inherits the neutral SFS-structure whenever it is invariant under the ambient generalized Peirce projections. Several examples and counterexamples are presented, including intermediate subspaces of the trace class and the dense subspace $C[0,1]\subset L_1[0,1]$.

This article investigates the effective mass density for acoustic and elastic wave propagation in layered media, with emphasis on its transition between these regimes. Conventionally, it is possible to recover the governing equations of acoustics from those of elasticity via the no-shear limit. However, when considering an effective medium, the anisotropic effective mass density typically obtained for acoustics differs from the isotropic one found in elasticity. Furthermore, direct investigation of this limit is hindered by the fact that the effective mass density is independent of the shear modulus $\mu$. To tackle this problem, a transfer matrix approach is combined with Bloch's theorem to derive the effective wavenumber for a periodic material whose unit cell consists of two layers with different densities. The effective mass density is obtained analytically from the wavenumber, allowing for its evaluation in both regimes. Moreover, for the first time, a description of the transition from isotropic elastic to anisotropic acoustic effective density in a periodic layered medium is provided. Additionally, the existence of band structure in $\mu$-space is demonstrated, along with exceptional points, where compressional and shear modes coalesce and waves become evanescent, which heavily affects the behaviour of the effective density in the elasto-acoustic transition.

Matsumoto's algorithm for computing the radical of a polynomial ideal is generalized to the parametric setting within the framework of symbolic computation. The main tool is a comprehensive Gröbner system over a finite field, also known as a parametric Gröbner basis. As a result, an algorithm for computing the radical of a parametric ideal over a finite field is proposed.

We study a Lévy-Montague reflection scheme $\mathsf{Rfn}$ in second-order arithmetic: for each formula $\varphi$, the scheme asserts that every set belongs to a countable coded $\omega$-model such that $\varphi$ is absolute, at all parameters from the model, between the model and the universe. Our central result is a model extension construction: every countable model of $\mathsf{RCA}_0$ can be extended, without changing its first-order part, to a model of $\mathsf{WKL}_0$ together with the full scheme $\mathsf{Rfn}$. It follows at once that $\mathsf{WKL}_0+\mathsf{Rfn}$ is $\Pi^1_1$-conservative over both $\mathsf{WKL}_0$ and $\mathsf{RCA}_0$, that its first-order part is exactly $\mathrm{I}\Sigma_1$, and that it is $\Pi^0_2$-conservative over $\mathsf{PRA}$. The result opens an avenue for adopting, within a theory conservative over $\mathsf{PRA}$, Feferman's $\mathsf{ZFC}$-formalization of universe-based category-theoretic arguments that was achieved using Lévy-Montague reflection. The conservation proof itself, however, is non-finitary. The extension is the union of an $\omega_1$-tower of forcing extensions, and its uncountable cofinality is what secures reflection. We are only able to prove the conservation in $\mathsf{PRA}+\text{1-Con}(\mathsf{Z}_2)$. The results were obtained with extensive use of Anthropic's large language model Fable 5.

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

This paper will show that the Alexander polynomial of a knot, which is of slice type in an oriented homology 3-sphere, obeys the Fox-Milnor polynomial condition. A relation between Alexander polynomial of concordant knots in an oriented homology 3-sphere is established.

We develop a certified, scalable approximation for high-dimensional Wasserstein distributionally robust portfolio optimization. For expected-utility maximization under order-one Wasserstein ambiguity, standard duality yields a semi-infinite convex program. For long-only portfolios with box support under the one-norm ground metric, an exact sample-specific vertex reformulation provides an exponential-size computational benchmark. We then majorize the utility by supporting hyperplanes and dualize the support subproblems, obtaining a finite hyperplane–dual formulation over compact polyhedral supports. Under the one-norm ground metric and polyhedral portfolio constraints, this formulation is a polynomial-size linear program. The uniform utility-approximation error bounds both the robust-value error and the near-optimality gap for the original robust problem. Experiments validate the certified approximation and demonstrate monthly 476-asset rebalancing and computational scalability to 1,000 assets.

We consider weak solutions of the second-order elliptic equation $\operatorname{div} (\mathbf{M}\nabla u) = \operatorname{div}\mathbf{F}$ in $\Omega\subset {\mathbb R}^d$ with Dirichlet boundary conditions, where $\mathbf{M}$ is a uniformly elliptic real-valued symmetric matrix $\mathbf{M}:\Omega \rightarrow \mathbb{R}^{d\times d}$ such that $\frac{1}{K} |\xi|^2 \leq\langle \mathbf{M}\xi,\xi \rangle \leq K |\xi|^2, \forall \xi \in \mathbb{R}^d, K > 1$. We prove that $\|\nabla u\|_p \leq C\|\mathbf{F}\|_p$ for any $p\in \left(\frac{26K-16}{13K-3}, \frac{26K-16}{13K-13}\right)$.

In this paper, we construct the octonionic Hessian complex, which is the octonionic version of the $\partial\bar \partial$-complex. By proving its ellipticity, we can solve the non-homogeneous octonionic Hessian equations under a compatibility condition. This is the octonionic version of the $\partial\bar\partial$-lemma. We also obtain the boundary version of octonionic Hessian operator and introduce the notion of a tangentially octonionic pluriharmonic function, which allows us to establish the Hartogs–Bochner extension for tangentially octonionic pluriharmonic functions on the boundary of a domain with connected complement. This solves the octonionic version of Poincaré's problem for pluriharmonic extension on complex space. The main difficulty of this construction is the lack of associativity in octonionic case. However, we can overcome it by careful use of weak form of associativity including Moufang identities and alternativity.

We prove that the Fremlin projective tensor product of Banach lattices preserves unbounded norm convergence and unbounded absolute weak convergence of elementary tensors without any additional assumptions. The main tool is a tensor-lattice estimate controlling expressions of the form \[ \lvert x_{\alpha}\otimes y_{\beta}-x\otimes y\rvert \wedge w, \] where $w$ is positive in the tensor product. This provides a direct and simplified proof of permanence results previously obtained under extra hypotheses.

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

This paper extends prior work on linguistic pattern based facility layout optimization by enhancing the LP Alinks framework with an explicit spatial uniformity criterion. While earlier studies demonstrated that linguistic patterns can effectively encode expert knowledge and guide agent based layout emergence, their optimization scope remained limited to cost oriented objectives. To address this gap, we introduce the Normalized Coverage Score (NCS), a scale adjusted measure of spatial evenness that complements the classical flow distance economic objective and enables systematic exploration of cost uniformity trade offs. A full factorial experiment comprising 324 conditions evaluates the combined effects of problem size, link density, virtual force scaling, and two families of membership functions on both objectives. Five way and nested three way ANOVAs reveal that structural factors (number of objects and link density) exert a dominant baseline influence on both economic performance and spatial uniformity. However, while economic cost is overwhelmingly governed by these structural characteristics, uniformity is substantially more sensitive to the linguistic pattern parameters, which control the dispersion compaction dynamics of the emerging layouts within the structural constraints. Based on these interactions, we derive a parameter-selection matrix that prescribes settings for cost minimization, uniformity maximization, or balanced compromise. Comparative analyses with Drezner's method, MDS, and non metric MDS demonstrate that the extended LP Alinks framework consistently attains superior uniformity while maintaining competitive economic performance, making it a robust and interpretable decision support tool for early stage facility layout design.

This paper develops a high-order surface finite element method for elliptic equations posed on a smooth closed surface implicitly defined as the zero level set of a function in three dimensions. In contrast to classical surface finite element methods that start from a prescribed triangulation of the surface, the proposed method constructs the discrete surface space from the intersections between the exact surface and an ambient tetrahedral mesh. More precisely, an active tetrahedral shell is generated around the implicit surface, and each cut tetrahedron contributes either a triangular or a quadrilateral surface patch according to its intersection pattern with the zero level set. The finite element space is first defined on the resulting piecewise planar cut surface and is then lifted to the exact surface by local level-set-based parameterizations. The resulting method combines features of surface FEM and unfitted/trace methods. Like surface FEM, it produces a conforming finite element space on the exact surface after lifting; however, the geometry and finite element space are induced by intersections with background tetrahedra like unfitted/trace methods. We prove that the local lifting maps agree pointwise across common faces and assemble into a global homeomorphism. We also derive explicit formulas for tangent vectors, Gram matrices, mass and stiffness integrals, and prove stability and high-order derivative estimates for the lifting maps. The latter estimates are formulated in broken Sobolev norms and lead to a Cea-type energy-norm error bound for the proposed high-order lifted surface finite element method.

Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.

Stabilizer Rényi entropy quantifies the nonstabilizerness of a quantum state through Rényi moments of its Pauli expectation-value distribution. Its standard form sums over Pauli-string degrees and retains only the total Rényi weight. We introduce a fugacity-resolved partition function for the critical transverse-field Ising chain that resolves this degree in the balanced Majorana representation and generates its full counting statistics. This Ising problem extends beyond a single model: exact decimation identities and the correspondence with the half-filled \(XX\) chain establish it as a common finite-size building block for stabilizer and computational-basis Shannon–Rényi entropies. We map the fugacity-resolved all-minors sum, for every positive Rényi index, to a checkerboard-weighted discrete Selberg ensemble on a half-filled doubled root lattice. For positive integer indices, it reduces to a finite aliased Dyson constant term. At \(\alpha=\tfrac12,1,2\), determinant and Pfaffian compressions yield product formulas. At \(\alpha=4\), the generic-fugacity problem admits an exact inverse Jack–Kostka representation, while at unit fugacity a complementary middle-minor identity relates it to the square of the \(\alpha=2\) result. These generating functions determine the balanced-degree statistics. Complement symmetry makes the distribution symmetric about \(k=L/2\), with an index-dependent width. It is exactly binomial at \(\alpha=1\), has variance proportional to \(L\) at \(\alpha=\tfrac12\), and develops an \(L\log L\) enhancement at \(\alpha=2\). After variance rescaling, the centered distributions converge to Gaussian limits at all three indices. Beyond these solvable cases, finite-size numerics reveal an evolution from a central peak to a symmetric bimodal profile and eventually to endpoint dominance as the Rényi index increases.

A variety of universal algebras is called Cross if it is finitely based, finitely generated, and has only finitely many subvarieties. A monoid is aperiodic if all its subgroups are trivial. In the present article, we show that a variety of aperiodic monoids is Cross if and only if it excludes, as subvarieties, a certain list of 22 almost Cross varieties. Consequently, this list of 22 varieties exhausts all almost Cross varieties of aperiodic monoids.

We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results

We prove that the uniform doubling property holds for left-invariant Riemannian metrics on every step-two Carnot group and the sharp uniform doubling constant is $2^Q$, where $Q$ is the homogeneous dimension. More generally, every such metric satisfies the Bishop–Gromov type estimate.

This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA's effectiveness for both 2D contour and 3D shape analysis.

In this paper, we study the cohomology groups of finite cyclic covers of complexified real arrangement complements. An open problem is whether the torsion in the (co)homology of finite covering spaces of hyperplane arrangement complements, including the classical Milnor fiber, is combinatorially determined. Using the chamber cochain complex constructed by Yoshinaga, we obtain explicit upper bounds for the Betti numbers of these covers over arbitrary fields. Furthermore, we introduce a combinatorial condition analogous to the Cohen-Dimca-Orlik (CDO) condition. Under this condition, we prove that the integral cohomology groups of the finite cyclic covers are torsion-free. This partially generalizes the recent results on complex line arrangements obtained by the author and Liu \cite[Theorem 1.3]{LX26}.

In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f & \text{in }\Omega, u=0 & \text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.

We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\geq 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and scalar curvature satisfying $\mathrm{scal}_g\geq -n(n-1)$. We prove that equality in the upper bound \[ \inf_{\Sigma}H\leq (n-1)\sqrt{1+\operatorname{Rad}(\Sigma)^{-2}} \] given by Brendle, Tsiamis, and Wang holds if and only if $(M,g)$ is isometric to a geodesic ball in hyperbolic space.

Let $\mathrm{CHL}_N$ be the cylindrical Hastings–Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $\lambda>0$, and let $\omega_{N,\lambda}$ be its tree-completion time — the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\mathbb{E}[\omega_{N,\lambda}]\le(1+\varepsilon)(\log N)/(2\lambda)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[\omega_{N,\lambda}]}{\log N}=\frac{1}{2\lambda} \qquad\text{for every fixed }\lambda>0 . \]

In this paper, we propose and study a class of differential stochastic variational inequalities (DSVIs), in which an ordinary differential equation (ODE) is coupled with history-dependent stochastic variational inequalities (SVI). This framework models closed-loop stochastic systems with time-varying random equilibria and includes optimization-constrained ODEs as special cases. Under appropriate technical conditions, we establish uniqueness, measurability, and Lipschitz continuity with respect to the state of the second-stage response, and consequently the existence and uniqueness of the induced state trajectory. Moreover, we construct a sample average approximation (SAA) based on independent sample paths and prove uniform convergence of the approximate trajectories. For transfer between related stochastic environments, we derive a local $1/2$-Hölder estimate for parametric variational inequalities with moving feasible sets and a quantitative trajectory-stability bound in terms of the initial-state difference and the Wasserstein distance between exogenous path laws. Numerical experiments illustrate the SAA convergence and transfer-stability results. We further apply the framework to an elderly-health monitoring system. Similarity-weighted reuse of precomputed responses achieves an accuracy close to the full-recomputation benchmark of 0.97, while reducing the online batch runtime from 86 seconds to less than one second. Perturbation and delayed-update experiments additionally characterize robustness to sensor noise and the trade-off between response freshness, predictive accuracy, and computational cost. These results provide theoretical and computational support for efficient transfer learning in history-dependent DSVI systems.

In this paper, symmetric Hadamard matrices are classified up to Hadamard equivalence for all orders at most 32. In particular, an error in the previous classification of symmetric Hadamard matrices of order 28 is corrected. The number of distinct symmetric Hadamard matrices at each order is also enumerated. The classification and enumeration are carried out using a new algorithm that determines whether a given Hadamard matrix is equivalent to a symmetric one. The extension of the algorithm to weighing matrices is also described, and theoretical results concerning symmetric Hadamard and weighing matrices are established.

This article applies dynamic integral quadratic constraints (IQCs) to the analysis and synthesis of accelerated stochastic gradient algorithms. We consider composite objective functions whose gradient can be approximated via mini-batch sampling and we model the resulting stochastic gradient oracle as a feedback nonlinearity in the spirit of Lur'e systems from robust control. Our first main contribution is a family of IQCs that characterize the second-order statistics of mini-batch gradients, extending the classical Zames–Falb multipliers. Our second contribution is a semidefinite-program for certifying exponential convergence rates of stochastic gradient algorithms, and a complementary condition for bounding the asymptotic variance caused by non-vanishing gradient noise. Our third contribution is a convex synthesis procedure that identifies mini-batch gradient algorithms with the smallest certifiable convergence rate.

A twisted link is a generalization of a virtual link associated with link diagrams on closed surfaces, which may be non-orientable. In this paper, we generalize the notion of the index value for twisted knots. Based on this generalization, we introduce a polynomial invariant for twisted knots, called the $\mathcal{B}$-Index polynomial. Furthermore, we construct a family of twisted knots $\{D_n\}_{n\geq 1}$ with arc shift number $n$ and determine their $\mathcal{B}$-Index polynomials explicitly in terms of $n$. These results demonstrate the effectiveness and sensitivity of the $\mathcal{B}$-Index polynomial as an invariant of twisted knots. We also study the behavior of this polynomial under mirror images and orientation reversal. Furthermore, we conclude this paper by investigating the cosmetic crossing change conjecture and establishing a condition under which a crossing does not admit cosmetic behavior.

This work analyzes a method for time integration of high-dimensional linear Schrödinger-type problems based on hierarchical tensor approximations. In particular, this method provides a balance between error bounds and associated approximation ranks, using a scheme for iterative refinement with soft thresholding of tensors. The practical performance of the method is illustrated by numerical tests on coupled oscillators.

We study the spectral stability of periodic shear flows for the two-dimensional Navier–Stokes equations on the $\beta$-plane in the long-wave regime. While it is known that non-rotating periodic shear flows are generically unstable to sufficiently long-wave perturbations, our results show that planetary rotation suppresses this instability mechanism. Using a perturbative analysis based on Kato's reduction, we derive an asymptotic expansion for the principal eigenvalue of the linearized operator that is uniform in the Rossby parameter $\beta \geq 0$, and establish an explicit stability criterion in terms of the shear profile, the viscosity, and the Rossby parameter. Under the critical scaling where the Rossby parameter $\beta$ and the rescaled longitudinal wavenumber $\varepsilon$ are of comparable size, this criterion extends Yudovich's classical long-wave instability threshold to rotating flows and reveals a sharp transition between stability and instability governed by the range of the ratio $\beta/\varepsilon$. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the $\beta$-plane.