Papers

For negative values of the spreading coefficient (that is, in the so-called “partial wetting” regime), we prove that the thin-film equation with zero slip and repulsive potentials $P$ of the form $P(h)\approx h^{1-m}$ as $h\to 0$, $m>1$, admits for any positive speed a one-parameter family of travelling-wave solutions with a contact line and (as in standard slippage models) a logarithmically-corrected linear behaviour as $h\to +\infty$. These waves have locally finite rate of dissipation for any $m>1$ and locally finite energy for any $m\in (1,3)$. The result thus confirms that mildly repulsive potentials effectively resolve the no-slip paradox. The family is parametrized by a thermodynamically consistent contact-line condition which reduces to the classical fixed microscopic contact-angle one if $P\equiv 0$.

A classification of all invariant Hermitian structures with parallel Bismut torsion on most flag manifolds is given. More precisely, we prove that on a flag manifold $M=G/H$ such that $G$ is simple and different from $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$ if $H$ is non-abelian, a $G$-invariant Hermitian structure on $M$ has parallel Bismut torsion if and only if the metric is either Kähler or a multiple of the Killing or standard metric.

This paper presents new differential addition (i.e., the addition of two points with the known difference) and doubling formulas, as the core step in Montgomery scaler multiplication, for twisted Edwards curves. The formulas are provided with cost of 5M+4S+1D, 3M+7S+1D when the given difference point is in affine form. Here, M,S,D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively.

The Proca equations (1936) are used in quantum field theory to describe vector bosons (spin 1) with a nonzero mass. The Proca equations are not gauge invariant. In contrast to Stueckelberg's approach (1938), this article presents a gauge invariant generalization of the Proca equation by introducing an additional vector field into the Proca equation. The results are extended to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class $\mathbf{A}$ for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.

We study a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions, and the solitons that emerge as the contrast $\delta$ tends to zero. Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of $\sqrt{\delta}$. Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in $\sqrt{\delta}$), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix. We illustrate the theory numerically and characterise a symmetry-breaking bifurcation in a symmetric dimer.

We study self-adjoint realizations of conic Laplacians on compact Riemann surfaces with radial conic metrics whose cone angles are integral multiples of \(2\pi\). Their critical asymptotic coefficients span a finite-dimensional space with a nondegenerate skew-Hermitian Green form, whose Lagrangian subspaces parametrize the self-adjoint realizations. We construct the associated Weyl functions and derive Kreĭn's resolvent formula, a resolvent trace identity, and, under explicit zeta-regularity hypotheses, a comparison formula for positive-spectrum zeta determinants. The boundary data of formal solutions define a holomorphic Weyl curve in the Grassmannian, with the Weyl functions as its local graph coordinates. Its real restriction is a positive curve in the Lagrangian Grassmannian, and its tangent form is identified with the \(L^2\)-inner product through the Poisson operator. We further identify the boundary determinants with the transition functions of the determinant line bundle induced by the Weyl curve.

Preferential attachment graphs model networks whose growth produces highly uneven degree distributions, describing many real-world systems. Their adjacency spectra are important because they allow graph-theoretic questions to be studied through the eigenvalues of matrices. We analyze the adjacency matrix of a dense Barabási-Albert (B-A) multigraph, where the number of edges added at each step is proportional to the final number of vertices. First, we compute the large-$n$ limit of the expected adjacency matrix and show that it is described by a rank-one limiting kernel, viewed as a continuous analogue of the adjacency matrix. After centering and scaling, the fluctuations form a random matrix with a computable variance profile. Using the quadratic vector equation approach, we derive the limiting bulk spectral distribution. We also determine the asymptotic location of the leading eigenvalue generated by the rank-one mean component.

Efficient simulation of strongly-interacting fermionic systems on quantum processing units (QPUs) is a challenging task due to nonlocal mode entanglement generation. However, it is not yet well understood how the structure of entanglement governs the hardness of large-scale quantum chemistry simulations or the scaling of distributing such workloads. Here, we introduce an algorithm for estimating the distribution complexity of hybrid quantum-classical simulation for electronic structure Hamiltonians over heterogeneous high-performance architectures. Our algorithm relies on efficient analytical evaluation of the low entanglement boundaries for the orbital rotations and dephasing-induced localization within tensor fragments, in a double-factorized representation. Our entanglement estimation scales as $O(N^3)$ for each fragment, where $N$ is the number of orbitals. When QPUs are communicating via a quantum network, the cost of distribution per fragment is reduced quadratically from $O(N^2)$ to $O(N)$. Similarly, for hybrid quantum-classical approaches, with access to only conventional HPC interconnects, the worst-case cost is reduced from $O(\exp(N^2))$ to $O(\exp(N))$. We show that emergent entanglement patterns are induced by the interplay between coherent Gaussian orbital rotations and disordered Coulomb interactions. We discuss the underlying physical mechanisms that govern distribution complexity and introduce model systems that are tunable based on the localizability of fragments and the overlap of interfragment rotations. We characterize three different regimes of hardness for distribution complexity and classical simulability. The framework introduced here enables novel and more efficient quantum-classical application workflows towards utility-scale quantum computing.

We prove that strong ribbon concordance induces a partial order on links in the 3-sphere, extending a theorem of Agol. Using results from knot Floer homology, we certify minimality under the ribbon partial order for a handful of knots and give the first examples of ribbon minimal knots that are not transfinitely nilpotent, resolving a question of Tagami. Using a mixture of classical techniques and recent Heegaard Floer detection results for links, we give several infinite families of links whose members are minimal under the partial order induced by strong ribbon concordance.

The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes $V^{(2m)}_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of $N=1$ supersymmetric Riemann surfaces of genus $g$ with $n$ Neveu-Schwarz boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms $W^{(2m)}_{g,n}(\{{\hat z}_i\})$ to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.

The Vietoris-Rips complex of a metric space is the simplicial complex whose faces are the subsets of points with pairwise distance bounded above by a given scale $r$. In this paper, we study Vietoris-Rips complexes on the vertex set of the $n$-dimensional hypercube equipped with the Hamming distance. These complexes are stable under the action of the automorphism group of the hypercube graph, also known as the hyperoctahedral group, which therefore acts on their homology groups. Our results completely describe the decomposition of these homology groups into irreducible representations of the hyperoctahedral group at scales $r\leqslant 3$ and $r=n-1$.

This brief, in the form of an e-book, is a collection of notes that cover elementary and medium level number theory with a target audience of primarily computer science students. It can be used in the number theory portion of a discrete mathematics course, or a course on the mathematical foundations of computer science, or as background material for a cryptography course. Thematically it is split into five areas that map to chapters. The first chapter is introductory and covers topics including divisibility, prime numbers, and modular arithmetic including modular linear equations. The second chapter covers additional topics such as Euler's totient function, units and inverses, the Chinese remainder theorem, and Fermat's and Euler's theorems. The following chapter covers primitive roots, quadratic residues, the Jacobi and Legendre symbols, Gauss's lemma and Eisenstein's theorem, and briefly discusses applications of number theory to cryptography. The fourth chapter is focused on traditional primality testing methods covering Miller's algorithms, Rabin's conversion of a Miller algorithm into a probabilistic primality test algorithm, Solovay-Strassen's algorithm and several other peripheral results including Carmichael numbers and the equivalence of Miller's two algorithms. Finally the last brief chapter can be viewed as an introduction to more advanced elements of number theory and its coverage includes multiplicative functions, the Möbius function, Dirichlet products and Dirichlet and Möbius inversions. Different parts of this e-book are for freshman to senior undergraduate students in computing and in particular computer science. Graduate students with limited exposure to number theory can use it to acquire a background suitable for typical cryptography courses at the master's level.

We investigate the long time behaviour of the solutions of the cubic Szeg}o equation on the line in the Sobolev space H^{1/2}(R). We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi-periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem.

For each positive integer $n$, let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency defined by an elementary transfer of one unit between parts, followed by reordering. Let $K_n=\mathrm{Cl}(G_n)$ be its clique complex. Previous work revealed two apparently contrasting features of this family. Locally, the graphs $G_n$ become increasingly rich: local clique dimensions grow, degree landscapes refine, support jumps and simplex layers proliferate, and axial, rear, central, shell, and directional structures become more pronounced. Globally, however, the clique complexes remain homotopically simple: $K_n\simeq \bigvee^{b_n}S^2$, where $b_n=\chi(K_n)-1$. This paper gives a conceptual synthesis explaining why these facts are compatible. The local side is governed by ordered local transfer types, which determine local neighborhood graphs, degrees, local clique numbers, and local simplex dimensions. The global side is governed not by the largest local simplices, but by the overlap pattern of canonical full star/top simplices. These simplices form a good cover $\mathcal C_n$, giving $K_n\simeq N(\mathcal C_n)=N_n$. The nerve admits an anchor-cover and intersection-poset reduction $N_n\simeq \Delta(J_n)$, with $\dim\Delta(J_n)\le 2$. Thus high-dimensional local simplices occur inside contractible containers, while global topology is controlled by a low-dimensional overlap poset. Within this family, the qualitative topological problem therefore collapses to the numerical computation and interpretation of a single integer, the Euler characteristic $\chi(K_n)$, equivalently the bouquet rank $b_n=\chi(K_n)-1$.

This paper introduces a new CR invariant for co-oriented contact structures on closed, orientable 3-manifolds. The invariant, which we denote as $\mu_M(\xi)$, takes values in the Picard group of complex line bundles $\Pic_{\C}(M)$. The construction associates to a contact structure $\xi$ and a supporting open book decomposition an embedding into $\C^3$, where the contact structure becomes the holomorphic line field along the binding. Using Stein theory, the induced holomorphic line bundle extends to all of $\C^3$ but we consider only its restriction to $M$. By Giroux's correspondence, we prove this construction is independent of the choice of open book, yielding a well-defined invariant $\mu_M(\xi) \in \Pic_{\C}(M)$ over the manifold. As an application, we distinguish two tight contact structures on the 3-torus $\T^3$ by showing their first Chern classes are different.

Let $G$ be a graph, and $a$ and $b$ be integers. Suppose there are infinite number of stairs, numbering level $0$, level $1$, etc. How do you place every vertex of the graph $G$ on a level as high as possible such that a vertex placed on level $i$ should have at least $i-b$ neighbors among the vertices placed on level $i$ and above while have at least $i$ neighbors among the vertices placed on level $i-a$ and above? This new game on graphs is referred to as “network pearls climb stairs". In this paper, we develop a more general theory, notably a correspondence between structure and dynamics, which particularly leads to the optimal solution of the game. Moreover, as $a$ and $b$ vary, the corresponding level numbers for a vertex obviously provide a structural profile (or spectrum) for the vertex the applications of which are worthy of further study.

A model is multicalibrated on a collection of group weights $G$ if it is calibrated – i.e. unbiased even conditional on its prediction – not just overall, but also after reweighting contexts by each $g \in G$. It is a useful property for many downstream applications and is a basic desideratum of trustworthy machine learning. Before this work, all predictors known to attain the minimax-optimal $\widetilde O(\varepsilon^{-3})$ sample complexity rate for $\varepsilon$-multicalibration were randomized, while deterministic predictors were known only with substantially worse sample complexity. Whether randomization is necessary for optimal sample complexity in multicalibration was explicitly asked by [CLNR26] and implicitly in several prior works. We resolve this open problem by giving a minimax-optimal multicalibration algorithm that outputs a deterministic predictor. We then generalize the algorithm to produce optimal deterministic predictors that satisfy outcome indistinguishability (OI) with respect to finite or finitely covered collections of tests. As an application, this also gives deterministic omnipredictors and panpredictors with optimal sample complexity, resolving open problems posed by [OKK25] and [BHHLZ25].

We place the attention token on the group: a token is an element $g_i$ of a matrix Lie group $G$ – a bare transformation, with no feature payload and no external action $\rho(g)$ carrying it. To our knowledge this is the first attention construction whose tokens are bare matrix Lie group elements: their score is the closed-form algebra norm of the relative pose rather than a learned kernel, and it reaches the affine full-frame groups that every irrep- or surjective-exp-based method must exclude. We call it Lie-Algebra Attention. Once tokens are group elements, the rest follows with none of the usual representation-theoretic machinery. The relative geometry of a pair is canonical, $g_i^{-1} g_j$, so the pairwise invariant $w_{ij} = \log(g_i^{-1} g_j)$ is intrinsic rather than designed; equivariance under the diagonal $G$-action is tautological, and the cocycle condition holds automatically. The attention score is the negative squared algebra norm, $s_{ij} = -\|\log(g_i^{-1} g_j)\|_\lambda^2/\tau$: the canonical proximity kernel under a block-weighted Frobenius inner product, with no irreducible representations, spherical harmonics, Clebsch-Gordan products, or learned kernel. The construction applies to any matrix Lie group on a chosen logarithm chart containing the relative poses, including the non-compact non-abelian affine groups with scale and shear that no vector-token attention method reaches: neither the irrep tradition nor surjective-exp methods. Three sequence-completion experiments, on SE(2), SO(3), and Aff(2), bear this out: the closed-form score matches a learned MLP kernel on the same invariant and outperforms it on SE(2), using 50 to 80x fewer score parameters, while a vector-token baseline breaks invariance by five to twelve orders of magnitude.

Bilevel optimization is an indispensable modeling tool for modern machine learning and engineering design. However, the theory and practice for finding second order stationary points in the context of bilevel optimization still remain largely unsettled. Even for bilevel optimization with strongly convex lower-level problem, the hyperfunction it induces is in general nonconvex. Although the Cubic Regularized Newton methods (CRN) famously achieve the optimal $\mathcal{O}(\varepsilon^{-1.5})$ SOSP (second-order stationary point) rate in single-level optimization, it is unclear how to control the accuracy of the hypergradient and hyper-Hessian computations in the context of applying the second-order methods to bilevel problems in order for the overall process to be efficient. In this paper, we set out to answer this question. In particular, we first formulate a double loop CRN baseline that achieves the optimal outer rate but requires repeated lower level solves. Next, we propose a single loop cubic regularized Newton algorithm that combines one lower-level gradient step with one Newton step for the hypergradient, and prove an overall deterministic $\mathcal{O}(\varepsilon^{-1.5})$ total oracle complexity, which is optimal. In addition, we illustrate that some intuitively simple modifications of our method may fail to hold up the convergence result. To the best of our knowledge, this is the first deterministic single loop method for unconstrained NCSC (non-convex upper-level and strongly convex lower-level) bilevel optimization setting that achieves the $\mathcal{O}(\varepsilon^{-1.5})$ optimal convergence rate for finding an $\varepsilon$-SOSP of the hyperfunction.

In light of recent advances in conformal blow-up methods for the positive mass theorem, including He–Shi–Yu, Bi–Hao–He–Shi–Zhu, and Brendle–Wang, we develop a Schoen–Yau type singular dimension descent method for positive scalar curvature obstructions in arbitrary dimensions. We prove obstructions to positive scalar curvature on enlargeable manifolds and establish the corresponding cubical width inequalities and two-systole estimates. The method also applies to enlargeable AM–PI spaces, giving a positive scalar curvature obstruction when the singular set has Assouad codimension greater than \(3-2/n\).

The process matrix formalism describes quantum correlations in scenarios without a fixed causal order between local laboratories. Operational signatures of such correlations can be investigated through causal games. A paradigmatic example is the Guess-Your-Neighbour's-Input game, in which two parties attempt to guess each other's inputs. Correlations compatible with any definite, or probabilistically mixed, causal order cannot achieve a winning probability exceeding $1/2$. The best process-matrix strategy currently known attains a value of approximately $0.6218$ using local dimension $d=5$, while the strongest known dimension-independent upper bound is $0.7592$. In this work, we investigate whether increasing the local dimension beyond $d = 5$ can narrow this gap. To this end, we employ a see-saw optimization scheme in which each step is formulated as a semidefinite program. For scalability, we develop a custom implementation of the SCS solver in which the dominant computational cost, the projection onto the positive-semidefinite cone, is offloaded to a GPU, yielding a six-fold speedup. Using this implementation, we explore local dimensions up to $d = 8$, and we do not find significant improvements over the value at $d=5$. Our results suggest that either qualitatively different strategies are required to approach the known upper bound, or that the bound itself is not tight.

An elastic space curve is a critical point of the bending energy subject to appropriate constraints. An analytic representation, equivalent to the spherical pendulum equation, leads to an 11-parameter description of the space of 3D elastic curve segments. We give a numerically stable method for recovering the 11 parameters from a given elastic curve segment. Using this, we give a fast and stable method to approximate an arbitrary space curve segment by a 3D elastica. Applications include interactive design with exact elastic curves and CAD surface rationalization for robotic hot-blade cutting.

We introduce the Frontier decoder, a pruned dynamic-programming decoder for sparse quantum decoding problems. Frontier processes error variables in a chosen order, merges prefixes with the same residual syndrome and logical label, and approximates logical-coset posterior masses by retaining only a narrow scored frontier. Without pruning, the recursion is exact ordered inference with exponential complexity. In the code-capacity setting, the decoder reaches thresholds close to optimal for the surface code and the color code. In the circuit-level noise model, it achieves state-of-the-art performance with a very small average retained list size: less than 100 for the gross code $[[144,12,12]]$ at a physical error rate of $0.001$. When the list size is constant, the decoder has linear complexity, suggesting the possibility of low-latency implementations.

Throughout this work, we will carry out a rigorous mathematical analysis of a class of control systems that is widely used in applications but still lacks a consistent theoretical foundation for describing the types of limit sets that may arise from its dynamics. There are applications in which, for example, a treatment for a given disease is administered until the level of diseased cells falls below a prescribed threshold C1. At that point, the treatment is suspended in order to allow the patient's organism to recover from its side effects. Subsequently, when the level of diseased cells reaches a second threshold C2 bigger than C1, the treatment is resumed, and the protocol is repeated. To the best of our knowledge, there is not a mathematical classification of such models. In this paper, we initiate what is intended to become a consistent body of literature aimed at determining the limit sets of such models. We begin with the planar case, in which two linear vector fields are active and two switching boundaries are considered. Naturally, in future developments, control systems in higher dimensions, featuring additional vector fields and more general switching manifolds, should also be considered.

We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, \(-V\). Specifically, we analyze the random Schrödinger operator defined by \(H = -\Delta +\ve \sum_{n} \omega_n \chi_n - V\), where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.

We introduce the CLUSTER algorithm (coordinate-level update strategy for trust-region step evaluation refinement) for local derivative-free optimization problems where there is a cost to changing each parameter (or clusters of parameters). For example, this type of cost model is appropriate for optimizing robot-controlled laboratory experiments, in which a robot may incur a separate motion for each parameter cluster to be adjusted. We build off of a class of quadratic-interpolation optimization algorithms by Powell and Conn that are known to perform well for twice-differentiable objectives (e.g. low-noise experiments), and show that the CLUSTER variants improve performance on a variety of test problems (including an optics laboratory experiment) by around 50$\%$, and greatly outperform common competing algorithms for laboratory optimization (Bayesian optimization and Nelder–Mead). We also adapt the convergence proof of the Conn algorithm to obtain a similar convergence guarantee for CLUSTER-Conn.

We present a two-level Schwarz method as an alternative to Algebraic Multigrid method(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poisson equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes equation. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

The minimum positive codegree $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The positive codegree Turán density of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.