Papers

Let $\mathcal C$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a \overline L$ which converge to $\mathcal C$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline L$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll1$, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.

Graph convolutional networks (GCNs) have demonstrated significant success in capturing complex user-item relationships for collaborative filtering (CF). However, due to their reliance on extensive model training, training-free graph filtering (GF)-based CF methods have emerged as a promising alternative, offering computational efficiency by smoothing graph signals via matrix operations. In particular, polynomial GF-based approaches demonstrate improved accuracy through their ability to design more expressive and flexible filtering functions. Despite these advantages, existing GF methods suffer from a critical memory bottleneck: they necessitate storing the full item similarity graph, incurring prohibitive memory costs for large-scale datasets, which limits their practical applicability. To tackle this challenge, we propose Mem-GF (Memory-efficient GF), a new GF-based CF method that departs from conventional designs by principally leveraging the structure of Krylov subspaces as a core mechanism for approximating polynomial graph filters without explicitly storing the item similarity graph. We theoretically analyze the minimum Krylov subspace size that guarantees lossless approximation. Through extensive experiments, we demonstrate that Mem-GF achieves up to 5.74$\times$ lower memory usage and 4.38$\times$ speedup in runtime, while consistently exceeding the recommendation accuracy of state-of-the-art GF and GCN-based methods. Mem-GF robustly scales to datasets with tens of millions of interactions, establishing itself as a practically viable and theoretically grounded solution for efficient CF.

A graph $G$ is conformable if it admits a proper $(\Delta(G)+1)$-coloring in which, among the $\Delta(G)+1$ color classes including the empty ones, at most $\sum_{v\in V(G)}(\Delta(G)-d_G(v))$ have parity different from that of $|V(G)|$. The complexity of deciding conformability was left open in recent work, and positive results for several graph classes had suggested that the problem might be polynomial-time solvable. We settle the general problem by proving that Conformability is NP-complete. Hardness holds even for connected regular graphs $G$ of odd order with independence number $\alpha(G)=3$ and maximum degree $\Delta(G)\ge |V(G)|/2$. In particular, NP-completeness persists when every color class is forced to have the parity of the order. The reduction starts from perfect triangle packing in graphs of clique number three, regularizes the source graph while preserving the relevant triangle packings, and then takes the complement. In the complement, conformable color classes correspond to odd cliques of the regularized graph; $K_4$-freeness restricts these cliques to singletons or triangles, and the number of available colors forces exactly the required number of disjoint triangles.

In a 2005 paper, Bebeacua et al. investigated the xrays of permutations, and conjectured a correspondence between binary xrays and score sequences of tournaments. In 2014, Brualdi and Fritscher conjectured a possible correspondence between score sequences of $2$-tournaments and (not necessarily binary) xrays of permutations. In this paper, we first introduce the concept of a transitive tournament decomposition of $k$-tournaments, then present a construction by which a permutation is used to build $1$- and $2$-tournaments whose score sequences agree with the xray of the permutation in the manner outlined by Bebeacua et al. and Brualdi and Fritscher. We close with an investigation of xrays with restricted terms, including binary xrays, and show that the recent conjectures by Bebeacua et al. and Brualdi and Fritscher are special cases of a more general statement, which we conjecture and for which we provide supporting evidence.

We study first-order methods for solving monotone variational inequalities arising in min-max optimization. Classical approaches such as the extragradient method rely on two gradient queries per iteration, which limits their analysis and applicability in the online and stochastic settings. We propose a family of Generalized Optimistic Methods with Anchoring (GOMA), which combine two-time-scale optimistic updates with an anchoring term inspired by Halpern iteration. In the deterministic setting, GOMA achieves the optimal accelerated last-iterate rate $O(1/k^2)$ on the squared gradient norm for monotone Lipschitz operators. In the stochastic setting with unbounded variance, a simplified single-call variant of GOMA achieves a last-iterate convergence rate of $O(1/\sqrt{k})$ on the squared gradient norm. To the best of our knowledge, this is the first such guarantee for stochastic monotone Lipschitz variational inequalities in the unconstrained setting without variance reduction or growing batches.

Spectral initialization is a critical yet challenging step in sparse phase retrieval. Existing spectral design theory is largely tailored to dense phase retrieval, where the objective is eigenvector estimation. In contrast, sparse initialization first requires a statistically distinct support screening step whose design remains much less understood. This paper develops a stage-specific design theory for diagonal support screening. We formulate each coordinate score as a scalar statistic for distinguishing support from non-support coordinates and adopt the deflection criterion as a tractable measure of screening quality. Within a Hilbert-space formulation, we characterize the optimal spectral preprocessors that maximize this criterion. In the Gaussian model, the unique optimum is the centered linear preprocessor. To obtain a bounded implementation, we introduce a spherical normalization and characterize its exact optimal preprocessor. Since the exact spherical optimum exhibits a boundary singularity, we construct a bounded surrogate preprocessor and establish its unique optimality under a surrogate deflection criterion. The surrogate optimum is shown to be the direction-only projection of the Gaussian rule, removing the unbounded radial factor while preserving the same first-order screening structure. We further establish a general finite-sample diagonal bridge that connects the exact and surrogate deflection quotients to the initialization sample complexity, and that replacing the unknown signal energy by its empirical estimate introduces only a lower-order perturbation. Numerical experiments are consistent with the ordering predicted by the design quotients and show that the Gaussian centered rule and its spherical counterpart behave nearly identically at both the screening and initialization levels.

Autonomous vehicles must generate long-horizon and dynamically feasible trajectories in real time-even when operating at the limits of vehicle handling-to ensure safe operation in adverse conditions. However, existing work rarely quantifies the computational demands of generating such trajectories without prior references, warm starts and often defaults to low-fidelity models, compromising accuracy and control authority. We investigate the modeling and solver design choices that enable real-time solution of long-horizon, reference-free optimal control problems (OCPs) using full vehicle dynamics. To this end, we analyze vehicle stiffness properties to justify the OCP's integration scheme and show that lower-order A-stable methods consistently outperform alternatives, with solve time differences reaching two orders of magnitude. We show that robust nonlinear solver performance hinges on understanding barrier parameter update strategies and safeguarding techniques for Hessian indefiniteness, inherent in some interior point methods. Lastly, we propose a computationally efficient method for generating initial guesses using dynamic equilibrium, unlocking real-time performance and reducing initial infeasibility by up to four orders of magnitude. Extensive benchmarking and high-fidelity BeamNG simulation demonstrate compute times as low as 55 ms over a 260 m horizon, including high-speed obstacle avoidance scenarios where drifting emerges as a necessary component of feasible trajectory generation.

This paper identifies and addresses a mathematical oversight in one of the foundational results on the Reformulation-Linearization Technique (RLT) for polynomial optimization. Correctness of the original result can be easily recovered by adding a minor and natural assumption. However, not being aware of this nuance may lead to the loss of convergence guarantees in RLT-based algorithms, especially if they rely on filtering strategies for the construction of the bound-factor constraints.

We study ergodic primal-dual gap rates for first-order primal-dual methods applied to \[ \min_x f(x)+g(x)+h(Ax), \] where $f$ is smooth and convex, $g$ and $h$ are proper, closed, convex functions, and $A$ is linear. Standard gap-rate proofs often impose the halved smooth-stepsize condition $\tau \le 1/L$, even though the corresponding convergence theory allows the larger range $\tau <2/L$. We introduce a residual-to-gap transfer principle: positive residual terms in the one-step gap inequality are controlled by the decrease of a Lyapunov function. This yields $O(1/K)$ ergodic primal-dual gap bounds with the unhalved primal stepsize $\tau <2/L$ for Condat–Vũ, PD3O, AFBA/PDDY, and PAPC/PDFP$^2$O, under their algorithm-dependent product conditions. We also give a two-dimensional counterexample showing that the fully separated rectangle $\tau <2/L$, $\tau\eta\|A\|^2<4/3$ cannot hold in the general three-function setting.

An expository note about the paper Askey wrote on Szego's 60th anniversary about the later's work.

We introduce a new two-variable polynomial invariant \(P^e\) of strongly involutive links, uniquely characterised by equivariant skein relations and naturally viewed as an equivariant analogue of the HOMFLY–PT polynomial. We prove that a specialisation of \(P^e\) recovers the graded Euler characteristic of the third page of the Lobb–Watson \(\mathcal{G}\)-filtration spectral sequence, generalising Couture's polynomial invariant. We further show that, after a change of variables, \(P^e\) reduces modulo \(2\) to the HOMFLY–PT polynomial, up to an explicit power of the skein variable, thereby answering a generalized form of a question of Couture. We use the resulting skein relations to distinguish infinitely many pairs of alternating mutant knots, and show that \(P^e\) is strictly stronger than the refined Lobb–Watson invariants on infinitely many strongly invertible knots.

We establish the calculus structure on Hochschild invariants of open-closed homotopy algebras. We further define the Getzler-Gauss-Manin connection and show that it is flat up to chain homotopy on the open-closed periodic cyclic chain complex.

Let $L_{\infty}$ be a free Lie algebra of countably infinite rank over a field of characteristic $0$ and let $L_{\infty, c}$ be the free nilpotent Lie algebra of countably infinite rank and class $c$. We prove that every automorphism of $L_{\infty, c}$ is induced by an automorphism of $L_{\infty}$.

We consider the classification of toric self-dual Einstein gravitational instantons with negative cosmological constant. As is well known, any Killing vector field on a self-dual Einstein manifold defines a local conformal Kähler structure. We prove that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions constructed by Calderbank, Pedersen and Singer.

We introduce a combinatorial structure $(n,W,T)$ encoding the topological type of a curve transverse to a fixed cellular arrangement of curves on a compact real surface, in terms of intersection numbers, Dyck words and rooted trees. We apply this formalism to analyze a natural generalization of Hilbert's 16th problem to arrangements of curves. We obtain a complete classification of arrangements of three lines and a cubic, and a partial classification of arrangements of three lines and a quartic. This is achieved using Bézout-type obstructions, Viro's patchworking and translations, and by developing the Julia library NWT to handle large databases of curves.

As large language models scale, memory bandwidth for key-value caches and retrieval-augmented generation systems becomes a critical bottleneck. While 1-bit quantization addresses this constraint, recent TurboQuant relies on dense random rotation matrices to condition the vector distribution before quantization. This projection demands millions of floating-point multiplications per embedding, making it difficult to deploy on constrained edge silicon. We introduce Fast-TurboQuant, a multiplier-free projection architecture that replaces the dense matrix with a structured fast Johnson-Lindenstrauss transform. By applying a Rademacher phase inversion followed by a fast Walsh-Hadamard transform (FWHT), the method leverages sub-Gaussian concentration to satisfy the prerequisites of scalar Lloyd-Max quantization without Gaussian projections. This substitution reduces the arithmetic complexity to only additions, eliminating hardware multipliers. Evaluation on DBpedia OpenAI-3 Large embeddings demonstrates a 19.7 times algorithmic speedup under sequential execution. Furthermore, the dimension expansion due to the FWHT zero-padding reduces the mean squared error and improves Recall@10.

The search for extremal geometries is a central theme in several areas of mathematics. Here, we address the following question: among all n-dimensional orthogonal tori of unit volume, which one maximizes the zeta regularized determinant of the Laplacian? We prove that the equilateral torus is the unique maximizer in each dimension n, for all n greater than or equal to 2, validating Sarnak's conjecture in this context. We also investigate the analogous question for the Laplacian on Euclidean boxes with the Neumann and Dirichlet boundary conditions. For orthogonal flat tori of unit volume and dimension n, we show further that the determinant is strictly decreasing with the dimension and tends to zero as the dimension n tends to infinity.

Large-scale optimization problems are often solved repeatedly under similar structural conditions, leading to substantial computational overhead. This occurs in applications such as power systems, transportation, and supply chain networks, where the underlying structure is fixed while parameters frequently vary under perturbations. This paper proposes a Learning to Optimize (LTO) framework that accelerates the solution of large-scale general mixed-integer problems by leveraging the concept of a backdoor, i.e., a subset of variables that drive most of the computational complexity. The proposed BIPC framework consists of three phases. Phase I is an identification procedure that discovers a backdoor for a set of instances in the distribution. Phase II uses supervised learning to develop machine learning models that, given an instance, predict values for bounded-domain backdoor variables and intervals for wide-domain backdoor variables. These predictions define a reduced optimization problem where the predictions constrain the backdoor variables, while the other variables remain free. Phase III optimizes this reduced problem and, if necessary, applies a correction step to restore feasibility or the optimality guarantees. Experiments on real-world, large-scale problems show substantial reductions in solution time with only a limited loss in solution quality. The framework enables organizations to solve large-scale optimization problems efficiently in the presence of frequent perturbations, such as unexpected events, demand fluctuations, or operational changes. Because these changes affect parameters rather than the problem structure, BIPC can quickly provide high-quality, feasible solutions, offering a practical approach to integrating machine learning into existing optimization pipelines.

In this article, we analyse convergence of the averaged Adam optimizer to an attracting zero of the Adam vector field. We provide a central limit theorem that, in particular, quantifies exactly the speed of convergence. The order of convergence is $n^{-1/2}$ in the number of steps of the algorithm which coincides with the order observed for classical stochastic approximation algorithms. The covariance in the central limit theorem is given in terms of properties of the Adam algorithm in the state of the attractor.

In this paper, we attempt to find the weight spectrum of the Reed-Muller codes $RM(m-7,m)$. In the process, we tried to fully determine the weight spectrum of $RM(7,14)$. We found almost all of the weights, except for a select number of them. The remaining eight missing weights are necessary to give a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ for $c=7$.

This paper is the third part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the previous two papers, we continue to focus on gentle trees $(Q,R)$. Via a modified surface model for gentle algebras with finite global dimension, we developed combinatorial, poset, and quiver representation techniques that allow one to calculate all the resolving subcategories of $\mathbb{K}Q/\langle R \rangle$-mod. Furthermore, they enable one to calculate the resolving subcategory generated by any collection of $\mathbb{K}Q/\langle R \rangle$-modules. In this paper, based on those techniques, we give a combinatorial realization of the Auslander–Reiten one-to-one correspondence between resolving subcategories and tilting modules in $\mathbb{K}Q/\langle R \rangle$-mod.

We extend the geometric electromagnetic particle-in-cell (PIC) framework, GEMPICX, to solve the quasineutral hybrid Vlasov-Maxwell equations with drift-kinetic electrons and fully kinetic ions. A structure-preserving finite difference method that employs dual grids is used. The discrete action principle for the hybrid model is derived, using the dual nature of the grids. The dynamical system for this hybrid quasineutral model does not explicitly involve the temporal evolution term for the electric field. A curl-curl equation is therefore used to implicitly obtain the component of the electric field that is parallel to the background magnetic field, at every timestep. The perpendicular component of the electric field is obtained using the quasineutral Ampere's equation without the displacement current, combined with the definition of the current in the drift-kinetic model. The discretized versions of the electric field equations are large, sparse linear systems. A fully explicit time-stepping scheme as well as two implicit-explicit (IMEX) schemes are tested. The numerical model is validated by verifying the various waves obtained from the dispersion relation.

We compute via trace methods the higher algebraic $K$-theory of the group ring $k[\operatorname{SL}_2(\mathbb{F}_q)]$, as well as the related groups $\operatorname{PSL}_2(\mathbb{F}_q)$, $\operatorname{PGL}_2(\mathbb{F}_q)$, and $\operatorname{GL}_2(\mathbb{F}_q)$, where $k$ is a finite field of characteristic $p$ and $q=p^r$. At the core of the computation is the algebraic $K$-theory of the group ring of the Sylow $p$-subgroup, $k[C_p^r]$, which we determine via a theorem of Lück–Reich–Rognes–Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus–Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.

We prove that the Ricci flow preserves positive sectional curvature on homogeneous spheres and complex projective spaces. In conjunction with prior results, this completes the classification of which homogeneous spaces have positively curved metrics flowing outside the set of positively curved metrics and which do not.

An efficient and robust FFT-based solver is proposed for diffusion-type problems with general Neumann and Dirichlet boundary conditions, based on a Chebyshev collocation framework. The method combines Chebyshev polynomial approximations with FFT-based operators to provide a matrix-free implementation of the discrete differential operator at the Chebyshev-Gauss-Lobatto points. The linear system of equations resulting from the Chebyshev discretization is solved using LGMRES. To overcome convergence problems on fine grids, a hierarchical refinement strategy based on modal prolongation is proposed, enabling the solution of very large 3D problems. The methodology is applicable to homogeneous and heterogeneous domains, as well as to linear and nonlinear constitutive equations. The accuracy of the proposed method is analyzed by solving the Poisson equation in homogeneous 1D and 3D domains with general boundary conditions, using manufactured analytical solutions as references. Convergence to the analytical solution is achieved in a few iterations, with smaller errors than those obtained using DCT/DST approaches. Discretizations of up to $256^3$ are achieved thanks to the hierarchical refinement strategy. In the case of heterogeneous domains, the accuracy and efficiency obtained are similar to those of a standard periodic FFT approach. It is found that the computational complexity of the method preserves the FFT scaling, of order $n\log n$, in all the cases studied.

In this article, we study the family of polynomials \[ P_n(s, z)=\sum_{k=0}^n s^{(k)} z^{n-k} \] for $s>0$. We prove that its roots are simple, and provide a precise localisation of them in specific angular sectors.

We show that for certain odd prime powers $q$, the number of conjugacy classes of $\mathrm{PGL}(2,q)$ is greater than the order of its largest nilpotent subgroup. This answers negatively a question of Liebeck and Pyber.

In this paper, we study local parameterizations of surfaces in the Euclidean space $\mathbb{R}^3$ whose coordinate vector fields are harmonic sections with respect to the induced Riemannian metric. After introducing the notion of harmonic basis vector fields on a surface, we derive necessary and sufficient conditions under which the coordinate vector fields $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ are harmonic. We then focus on two important classes of surfaces: surfaces of revolution and translation surfaces. For these families, we obtain a complete classification of all parameterizations admitting harmonic basis vector fields.

Classical Mckean-Vlasov theory concerns high-dimensional particle systems for which the effect of one particle on another is instantaneous. However in many applications, particularly neuroscience, the effect of one particle on another is not instantaneous but delayed. For biophysically-accurate neuroscientific models, for each j to k one can introduce an additional stochastic process that describes the propagation of the signal from j to k. In this paper we determine a self-consistent autonomous SDE that describe the high-dimensional limit of such neural networks. We apply this system to a model of the visual cortex. Performing a bifurcation analysis, we determine conditions under which hopf bifurcations occur in the network.

Given a central hyperplane arrangement $\mathcal{A}$ defined over a field of characteristic zero, we construct matrices of linear forms whose local ask zeta functions are recovered by the Igusa local zeta function of the cone over $\mathcal{A}$. Our construction extends a previously established connection between ask zeta function of hypergraphs and the Igusa local zeta function of Boolean arrangements. From a combinatorial standpoint, we introduce the truncated flag Hilbert-Poincaré series of $\mathcal{A}$, obtained as a rank specialisation of the flag Hilbert-Poincaré series of the cone over $\mathcal{A}$. Whenever $\mathcal{A}$ admits good reduction over the finite field $\mathbb{F}_q$, suitable substitutions of the variables of its truncated flag Hilbert-Poincaré series recover the local ask zeta functions associated with $\mathcal{A}$. Such formulae provide a means to study the analytic properties of the ask zeta functions considered, as well as to derive their reduced and topological relatives.