Papers

Let $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$ and $E^{(p)}$ be its conjugate. We give a bound on the minimal degree of an isogeny from $E$ to $E^{(p)}$ depending on $p$, and show that this bound is both asymptotically optimal as well as sharp in many cases. This bound is obtained by developing a new technique to compute the degree of certain isogenies from a supersingular elliptic curve to its conjugate, and we present extensive computations of the successive minima of the lattice containing these isogenies. Following this, we give several conjectures supported by the data we have obtained, including some on the set of primes $p$ for which the bound we give in this article is attained.

For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/\Gamma$, where $\Gamma\subset \mathrm{U}(N)$ is finite and fixed-point-free, is Kähler-Einstein if and only if $\Gamma$ is trivial. Consequently, if $\Omega$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on $\Omega_{\mathrm{reg}}$ is Kähler-Einstein if and only if $\Omega$ is biholomorphic to $\mathbb{B}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.

A spline dimensional decomposition (SDD) surrogate effectively represents high-dimensional engineering responses with localized features and complex nonlinearities in uncertainty quantification (UQ). However, limited training data can make coefficient estimation from function values severely ill-conditioned. We propose gradient-enhanced SDD (GE-SDD), which trains the surrogate using function values and partial derivatives. A diagonal row-weight matrix balances the function and derivative blocks by their Frobenius norms. We solve the balanced system through ridge regression in probability-weighted Sobolev coordinates and select the regularization parameter using grouped K-fold cross-validation to prevent information leakage. Mapping the solution back to the L2-orthonormal SDD basis preserves closed-form mean and variance estimates. We evaluate the proposed GE-SDD on a two-dimensional continuous exponential function, a linear dynamical system with three uncertain parameters, and a 30-dimensional 25-bar truss. GE-SDD is more accurate than standard SDD and uses gradients more robustly than gradient-enhanced Kriging. GE-SDD achieves a median NRMSE of 1.022% on the nonsmooth benchmark, compared with 8.731% for Kriging. For the truss, GE-SDD yields lower NRMSE and more accurate standard-deviation estimates than Kriging at moderate training sizes and above. Overall, the benefits of gradient augmentation depend on input dimension, basis resolution, training size, and the target UQ quantity.

We evaluate the average of the central values $L(1/2,\chi)$ over the generators of the group of Dirichlet characters modulo $q$, as $q\to\infty$ through the primes.

Time-evolving networks, or temporal networks, play a crucial role in modeling dynamic interactions across various domains, including biology, social sciences, and information technology. Unlike static networks, these systems undergo continuous changes in topology and edge weights, influencing processes such as information flow, transportation efficiency, and neural activity. Understanding and controlling these networks are essential for predicting future behavior and optimizing dynamic processes. This work focuses on the problem of dynamic centrality, a measure of node importance in time-dependent networks. Specifically, we address how to steer network centrality to a desired state by making minimal modifications to the network structure. This problem is formulated as an optimal control problem for an ordinary differential equation, either matrix- or vector-based, where the control acts on network edges. The proposed framework generalizes centrality control problems studied in static networks and leverages the Pontryagin Maximum Principle for efficient solutions. For large-scale problems, the required matrix-function actions are approximated by Krylov-type techniques, avoiding the explicit formation of dense matrix functions. Numerical experiments on synthetic and real temporal networks show that the proposed framework can effectively steer receive centrality under prescribed control constraints.

For $g\geq 2$ and nonzero integers $p,q$, let $\mathrm{Mod}(S_g)$ be the mapping class group of a closed oriented surface $S_g$ of genus $g$, and let $\mathrm{BS}(p,q)$ be the Baumslag-Solitar group. We provide necessary and sufficient conditions under which two mapping classes $G,F\in \mathrm{Mod}(S_g)$ generate a subgroup isomorphic to $\mathrm{BS}(p,q)$. In particular, if $\mathrm{BS}(p,q)$ embeds in $\mathrm{Mod}(S_g)$, then $|p|=|q|$ and $G,F$ are reducible mapping classes of infinite order. We also construct subgroups isomorphic to $\mathrm{BS}(p,p)$ and $\mathrm{BS}(p,-p)$ for $p>1$. Finally, we show that every infinite metacyclic subgroup and every Baumslag-Solitar subgroup of $\mathrm{Mod}(S_g)$ lifts to an isomorphic subgroup of $\mathrm{Diff}^+(S_g)$, that is, these subgroups satisfy the generalized Nielsen realization.

In this paper, we study the primitive actions of almost simple groups with socle \(E_7(q)\) or \(E_8(q)\) on an \(s\)-arc-transitive digraph. Our motivation goes back to the question of whether \(s\) is bounded above for finite connected \(G\)-vertex-primitive \(s\)-arc-transitive digraphs that are not directed cycles. The question has been reduced by Giudici and Xia to the case where \(G\) is almost simple. This work succeeds our 2025 paper (Yin and Chen), which addressed \({}^3\!D_4(q)\), \(G_2(q)\), \({}^2\!F_4(q)'\), \(F_4(q)\), \(E_6(q)\), and \({}^2\!E_6(q)\). Together with Chen, Giudici, and Praeger's work on \({}^2\!B_2(q)\) and \({}^2\!G_2(q)\), it answers the question for all exceptional groups.

We discuss the connections between tautological systems and the local cohomology of cones over homogeneous spaces. We study a derived version of tautological systems, related to the Chevalley–Eilenberg complex, and show that in many cases it underlies a complex of mixed Hodge modules.

Let $S_g$ be a closed orientable surface and let $\Psi\colon \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb{Z})$ be the representation induced by the action on first homology. We investigate the characteristic polynomials of integral symplectic matrices arising from mapping classes of algebraically finite type and give a complete characterization in the cyclotomic case: for $n\geq 3$, the polynomial $\varphi_n(x)$ is realized by a mapping class of algebraically finite type if and only if $n$ has at most two distinct prime divisors. Consequently, if $n$ is square-free and has at least three distinct prime divisors, then every mapping class with characteristic polynomial $\varphi_n(x)$ is pseudo-Anosov. This gives a cyclotomic complement to the Casson–Bleiler homological criterion and yields a complete criterion for a symplectic polynomial to be realized only by pseudo-Anosov mapping classes.

This paper proposes a quasi-BP decoding scheme for BCH codes that preserves the parallelizable structure of belief propagation while exploiting code automorphisms and optimized redundant parity-check matrices. To eliminate the computationally expensive $\tanh$ and $\tanh^{-1}$ functions in check node updates, we further introduce a neural-network-based variant that replaces them with a lightweight convolutional neural network trained under a triple-constraint loss function enforcing non-negativity and order consistency. Simulation results for three BCH codes demonstrate that quasi-BP decoding achieves competitive frame error rate performance, with a gap within 0.25 decibels compared with belief propagation decoding of an LDPC code of similar blocklength. The neural-network-based variant incurs negligible performance loss while enabling stable deployment with arithmetic operations on hardware accelerators. Concatenation with an ordered statistics decoding variant further bridges the gap to the maximum-likelihood bound. Hence, the proposed schemes offer a viable path toward high-throughput, low-latency decoding of BCH codes in next-generation communication systems.

We have embedded the category of smooth real $n$-manifolds faithfully into the category of associative varieties, \cite{S2603}. Many constructions on smooth manifolds can then be generalized to associative varieties. Here we will generalize the well known interpretation of connections on manifolds to connections on associative varieties. This eventually proves the existence of geodesics in an associative variety.

We consider the structure of the spine of the supersingular $\ell$-isogeny graph for one of the cases which arXiv:2502.03613 was not able to fully describe, $\ell = 2$ and $p = 71, 119\pmod{120}$. We find the distance, eccentricity, and diameter functions, of the components of the spine without the non-trivial edge not defined over $\mathbb{F}_p$. Using these functions, we find the mean diameter of the spine and show how this value distinguishes the different structures of the spine. Thus, allowing us to use explicit computations to provide heuristics on the behavior of the spine's structure as $p$ varies.

Contemporary machine learning is founded almost entirely on Archimedean mathematics: data are embedded in real vector spaces, similarity is measured by Euclidean-type metrics, learning is formulated through real-valued loss functions, and optimization is driven by differential or gradient-based methods. Recent work has shown that p-adic and ultrametric methods can enrich neural architectures, hierarchical representation, classification, and information processing. Rather than constructing a p-adic analogue of existing neural networks, we advocate a more fundamental viewpoint. We propose that a genuinely non-Archimedean theory of learning should be derived intrinsically from the topology, geometry, algebra, and measure structure of ultrametric spaces. To this end, we formulate the Principle of Intrinsic Non-Archimedean Learning and introduce an axiomatic framework in which information states are organized by nested ultrametric balls, learning is represented by hierarchical refinement and redistribution of information, and scale is determined by valuation depth. In this framework, neurons, layers, activation functions, gradients, and backpropagation are not assumed to be primitive concepts but possible computational realizations of a deeper mathematical theory. We prove a basic structural result showing that, on \(\Zp\) refinement depth coincides with common-prefix depth and that the finite quotients \(\mathbb Z/p^N \mathbb Z \) preserve the rooted ball hierarchy through level N. Consequently, hierarchy is not an auxiliary structure to be learned but is already encoded in the intrinsic geometry of the underlying information space

As a continuation of our previous work, where bifurcations of periodic and antiperiodic orbits near an equilibrium were studied for autonomous differential delay systems with one or two delays, this paper investigates bifurcations of periodic and antiperiodic solutions for five classes of non-autonomous potential delay differential systems with multiple delays. By reformulating these systems as generalized Hamiltonian systems on Euclidean spaces, we apply the Hamiltonian bifurcation theory recently developed by the author to establish alternative bifurcation results of both Fadell–Rabinowitz and Rabinowitz type.

We investigate bifurcations of periodic and antiperiodic solutions near equilibria for four classes of potential differential-delay equations involving one or two delays. By reformulating each system as a (generalized) Hamiltonian system, and applying the Hamiltonian bifurcation theory recently developed by the author we obtain bifurcation results of both Fadell–Rabinowitz and Rabinowitz type. The analysis relies on the computation of Maslov–type indices for fundamental solutions of the associated linear Hamiltonian systems.

Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(\delta_\pi\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(\delta_\pi\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.

Projection-based dynamical systems and projection neural networks offer a continuous-time approach to solving variational inequalities by driving the state toward its projection onto the feasible set. However, most existing models are built on a fixed Euclidean or constant metric, which can lead to poor conditioning and slow convergence when the operator or feasible set geometry is highly anisotropic. This paper introduces a state-dependent metric projection dynamical system, referred to as a state-dependent scaled projection neural network (SD-SPNN), in which the geometry of the projection operator evolves smoothly with the state. The dynamics generalize classical projected dynamical systems and projection neural networks by embedding continuous-time preconditioning directly into the flow. Under standard monotonicity assumptions on the operator and mild regularity conditions on the metric, we establish existence of solutions, an exact equilibrium-solution correspondence with the underlying variational inequality, and Lyapunov-based stability properties. The framework unifies Euclidean, constant-metric, and state-dependent projection neural network within a single continuous-time model. Numerical experiments illustrate how state-dependent metrics reshape the transient geometry of the projected dynamics while converging to the same equilibrium.

We prove that étale and de Rham cohomology algebras of a smooth proper rigid-analytic space over a finite extension of $\mathbf{Q}_p$ are formal if the rigid-analytic space satisfies the weight-monodromy conjecture. This is achieved by showing that the underlying $E_\infty$-algebra of a monodromy-pure $E_\infty$-algebra in Weil–Deligne representations is formal. We give examples of smooth proper rigid-analytic surfaces whose cohomology algebras are not formal.

Interpreting optimizers as gradient-flow discretizations has motivated applying higher-order Runge-Kutta (RK) integrators to neural networks. We build a representative Adam variant (Bogacki-Shampine 3(2) RK pair, FSAL reuse, local-error step control) and evaluate it under a strict compute-matched protocol giving every method the same gradient-evaluation budget - an accounting this literature rarely enforces. Under it the RK variant loses to plain Adam on training loss in both minibatch and full-batch (RK's best-case) training. Instrumenting it shows the "adaptivity" is illusory: normalized error stays far below tolerance, the step size pins at its growth cap from step one (98-100 percent of steps), and no rtol x hmax x h0 setting makes it act; tolerances spanning 100x give bit-identical trajectories. The method is exactly fixed-step Adam with an averaged gradient at 3-4x cost. Repairing it (true reject branch; error on the applied map) reverses the full-batch result - about 40x lower training loss than tuned Adam - and a fixed-step control isolates adaptivity (an emergent warmup-and-growth schedule) as the mechanism. But the gain is fragile to the initial step size and does not reach test accuracy. A pre-registered follow-up rules out the obvious explanations: deeper minimization does not overfit, and an explicit temperature knob only hurts - leaving a trajectory effect, the controller selecting a minimum generalizing 1.3-3.4 points below first-order descent at equal depth. An n=10 study confirms one secondary effect: gradient averaging is a genuine implicit regularizer, beating lr-matched Adam and AdamW on 10/10 seeds - yet RMSprop and NAdam match or beat it at a third the per-step cost. Higher-order adaptive integration buys deeper deterministic minimization and a small regularization effect, but nothing a cheaper, well-tuned first-order baseline does not already provide.

The Generalized Riemann Hypothesis (GRH) has long defined the expected bounds for the smallest prime in an arithmetic progression and the least quadratic non-residue. However, this hypothesis primarily addresses the horizontal location of non-trivial zeros. In this paper, we show that incorporating the vertical spacing–or pair-correlation–of these zeros allows us to surpass these classical bounds. By combining these two zero-distribution perspectives, we establish sharper estimates for both problems under GRH and specific pair-correlation hypotheses, thereby providing a new link between pair-correlation phenomena for Dirichlet L-functions and these two classical problems.

We prove fundamental space lower bounds for exact random sampling using an entropy source of i.i.d. uniform bits. A classic result from information theory shows that generating $n$ discrete random variables $X_1, \dots, X_n$ requires at least $H(X_1, \dots, X_n)$ input random bits on average, where $H$ is the Shannon entropy function. How much space must a random sampling algorithm use in order to approach this information-theoretically optimal entropy bound? We prove that any random sampling algorithm that is exact for arbitrary discrete target distributions and consumes at most $H(X_1,\ldots,X_n)+\varepsilon n+o(n)$ input bits in expectation for every output process must use $\Omega(\log(1/\varepsilon))$ bits of space. In fact, i.i.d. sampling from the single distribution $\mathrm{Bernoulli}(1/3)$ already forces at least $(1/{5.116201}-o(1))\log(1/\varepsilon)$ bits of space. If the sampler handles a family of infinitely many Bernoulli distributions, we show a sharper bound of at least $\log(1/\varepsilon)$ bits of space. We also prove lower bounds for general i.i.d. sampling: for almost every distribution on $k$ outcomes, the space is at least $(1/(k+1)-o(1))\log(1/\varepsilon)$ bits. The proof technique is based on a graph-theoretic analysis of the amount of information that any algorithm can store in its state. Finite state spaces force short cycles around the state-transition graph, and the loss around such cycles reduces to Diophantine lower bounds on fractional parts of integer combinations of log-probabilities. To the best of our knowledge, these results comprise the first known space lower bounds for entropy-efficient random sampling.

We show, as a proof of concept, that least-squares very weak formulations of elliptic problems can be effectively discretized by neural networks possessing low regularity, provided the test functions are drawn from appropriately smooth spaces. Apart from the immediate computational benefit of avoiding automatic differentiation, this approach, evaluated across various neural network spaces, demonstrates good performance even in challenging contexts, such as singular solutions and high dimensional settings. Particular attention is paid to trial functions based on step activations and one bit quantized linear functions, which are amenable to efficient hardware-oriented implementations.

The PDHG algorithm provides a new capability for solving difficult linear programming (LP) problems: the ability to find low-accuracy solutions quickly. While such solutions may not be applicable in all applications of LP, one possible use is to accelerate heuristics for finding feasible solutions to Mixed-Integer Programming (MIP) problems, where these approximate LP solutions can hopefully guide a heuristic towards accurate and high-quality MIP solutions. While a monolithic heuristic that exploits PDHG solutions would be useful, we pose a broader question here: could we replace the default LP solver in a modern MIP solver with PDHG to accelerate most (or all) of its existing heuristics? We find that doing so does have some obvious drawbacks, preventing us from using several powerful MIP techniques, but it leaves most others unaffected, ultimately resulting in a heuristic that often finds high-quality, high-accuracy solutions faster than current state-of-the-art strategies.

Working over uncountable algebraically closed fields, we extend the theorems of Kollár-Lieblich-Olsson-Sawin on reconstructing varieties from their Zariski topological spaces. In particular, we adapt their results to arbitrary quasi-projective varieties in arbitrary characteristic, and thus we give positive answers to each of the relevant `speculations' made by the original authors in our setting. Our proofs use techniques from model theory: in particular, we employ a general model-theoretic setting for algebro-geometric reconstruction problems, known as the `Zilber trichotomy for ACF-relics'.

Maximal quantum leakage (MQL) is a worst-case information leakage measure that quantifies an adversary's inference advantage gained from accessing quantum encoding of classical data with arbitrary measurements. While MQL admits an exact characterization for a given ensemble of quantum states, its robustness to implementation imperfections has not been systematically studied. In this paper, we analyze the sensitivity of maximal quantum leakage under perturbations of the quantum encoding. We establish a continuity bound in terms of the trace distance between ideal and perturbed quantum states, and show, via an example, that this bound is attainable. We further derive fidelity-based and relative-entropy-based sufficient conditions for bounding the variation of maximal quantum leakage, and illustrate numerically that these conditions can be loose.

A set $S \subseteq V$ is called a semitotal dominating set of $G=(V,E)$ if every vertex in $V \setminus S$ is adjacent to at least one vertex in $S$, and every vertex in $S$ is within distance 2 of another vertex in $S$. The corresponding decision problem is NP-complete even for unit disk graphs. In this paper, we present a 5-factor approximation algorithm for the Minimum Semitotal Domination problem on unit disk graphs in the graph-based input model. The algorithm processes the layers of a Breadth-First-Search tree and constructs a maximal independent set whose vertices satisfy the semitotal condition. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(n + m)$ time, and hence in $O(n^2)$ time in the worst case. This improves the previously known 5.75-approximation algorithm with $O(n^3)$ running time.

We study the sample covariance error of centered Gaussians. A remarkable breakthrough [66] established the correct error scaling order and explicitly revealed the critical role of both the effective rank and the true covariance spectrum. In this work, we move beyond scaling characterizations and determine the precise limiting value of the error's spectral norm. To do so, we develop a generic framework based on Random Duality Theory (RDT). Within this framework, we first determine closed-form, explicit RDT-based upper bounds. We then establish complementary lower bounds by introducing a novel bilinear-quadratic RDT lower-bounding mechanism. By combining this mechanism with a two-replica systems bounding strategy, we show that our lower and upper bounds match in large-dimensional contexts. Our theoretical results are supplemented with numerical evaluations and simulations, demonstrating an excellent agreement already for problem sizes on the order of thousands.

Reverse-mode automatic differentiation is commonly given a denotational account in which each source type has a single cotangent type. Variant types obstruct this simply typed representation because the valid cotangent space depends on the branch selected at run time. Existing correctness results therefore use primal-indexed families of cotangent spaces, whose natural internal language is dependently typed. We show that the same dependency can be represented in an ordinary nondependent target. The cotangent fibres of each source type are embedded in a common ambient type, and a primal-indexed idempotent selects the valid fibre. Semantically, this amounts to passing from the constant-family model to its Karoubi completion. For a category $\mathcal C$ and a regular infinite cardinal $\kappa$, we prove that the constant-family inclusion extends to an equivalence $\mathrm{Kar}(\mathrm{Copow}*\kappa(\mathcal C)) \simeq \mathrm{Fam}*\kappa(\mathcal C)$ precisely when $\mathcal C$ is Cauchy complete and every $\kappa$-small family admits a common retract host. We also construct the resulting coproducts explicitly. Applying this theorem, we obtain a bicartesian closed semantics for reverse-mode automatic differentiation with variants using only ordinary target types, projectors, and backpropagators. Splitting the generated idempotents recovers the established dependent semantics. Thus dependent cotangent families and simply typed ambient cotangents equipped with projectors are equivalent presentations of the same denotational transformation.

We study the intrinsic right $q$-radial derivative at integral superdimension $M=m-2n$, using an independent formal parameter $Q$ before the specialisation $Q=q^M$. The exterior-blade action carries the parity factor $(-1)^{|I|}$, and the corresponding Fischer theory uses right multiplication. We establish finite-support compatibility, faithful coordinate realisations in a sufficient finite-block range, and localised Green and determinant-localised Fischer decompositions. For one vector, we determine the kernel and cokernel explicitly, including the additional singular monomial and missing image exactly when $M\in-2\mathbb N_0$. In degree zero the right Fischer operator is diagonal on auxiliary exterior blades, with eigenvalue $[M-p]_q+p$ at nonempty support rank $p$. For positive integral $M$, resonances in $0<q<1$ occur exactly at ranks $p>M$: each rank has one simple resonance, these values increase strictly with $p$, and the corresponding kernels and singular right-monogenic elements are explicit. The two-vector Fischer determinant is nonzero for positive $M$ and $0<q<1$.

We consider the problem of group testing, in which one seeks to identify a subset of defective items of size $k$ from a larger set of $n$ items based on pooled tests. We introduce a selectable threshold model, in which each test has an associated threshold that can be chosen, such that the test outcome is 1 if and only if the number of defectives in the test is no smaller than that threshold. In settings with a large or unbounded maximum threshold, we establish conditions under which high-probability recovery can be attained with a rate (i.e., the asymptotic ratio of $\log_2{n \choose k}$ to the number of tests) approaching its maximum possible value of 1. Moreover, in the case of a fixed maximum threshold, we establish an achievable number of tests using simple and computationally efficient decoding methods, and a converse that holds under suitable regularity conditions on the test design, with the two coinciding in the dense limit (i.e., $\theta$ approaching one in the scaling $k = \Theta(n^{\theta})$).