Papers

Let \(H\) be a finite hypergraph with rank $r$ and minimum degree \(d\). In the Maker–Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of \(H\), with Maker moving first, and Breaker seeks to maximize the minimum degree $\delta(H_{\mathrm B})$ of his spanning subhypergraph $H_{\mathrm B}$. We prove that, for every fixed \(r\ge2\) and all sufficiently large \(d\), Breaker has a deterministic strategy satisfying \[ \delta(H_{\mathrm B})\ge \frac{d}{r+1}-\sqrt{d\log d}. \] For graphs, this improves the classical universal lower bound \(d/4\) to \(d/3-\sqrt{d\log d}\). The proof introduces a virtual balancing process, encodes local imbalance by a multiplicative risk, and keeps the resulting risk vector in the Shearer region throughout the game.

We explore a new growth condition for Lipschitz functions around local minimizers, recently proposed in the context of Goldstein-subgradient-based algorithms and their behavior in practice. On generic examples, these algorithms are often observed to converge approximately linearly. We focus on objectives that are piecewise twice continuously differentiable. In that case, the new growth condition holds when the objective is, in addition, strongly convex. In the nonconvex case, we prove that quadratic growth in conjunction with a regularity property for the active gradients suffices. Computational experiments illustrate the importance of the assumptions.

The edge-of-stability (EoS) phenomenon of full-batch Adam has been widely observed, yet its underlying dynamical mechanism remains poorly understood. In this paper, we identify Adam's second-moment adaptation as a negative-feedback mechanism that drives the dynamics toward the stability boundary. We characterize this mechanism through the *active curvature*, namely, the preconditioned curvature along the preconditioned gradient direction, and establish rigorous characterizations in progressively richer settings: rank-one quadratics with momentum, diagonal quadratics, on which the active curvature separates from the sharpness, and general objectives. Importantly, the mechanism predicts *gradient reversal* of full-batch Adam near the edge: consecutive gradients repeatedly point in nearly opposite directions, as we observe across fully connected networks, ResNets, ViTs, LSTMs, GPT-2 medium, and Adam-family optimizers. Consistent with this picture, averaging iterates suppresses these fast oscillations and produces smoother and lower loss curves. Together, these results provide an important first step towards fully understanding the dynamical behavior of Adam's EoS through active curvature and gradient reversal.

The randomly pivoted Cholesky algorithm is one of the leading methods for computing a low-rank approximation to a large positive-semidefinite matrix. However, while it consistently achieves accuracy comparable to or better than competing methods of its type in experiments, its theoretical analysis lags somewhat behind other methods. This paper closes this gap, proving that randomly pivoted Cholesky produces an approximation with expected error within a $1+\varepsilon$ factor of the optimal rank-$r$ approximation in $\mathcal{O}(r/\varepsilon + r\sqrt{\log r})$ steps. This result nearly matches the optimal complexity $\Theta(r/\varepsilon)$ for any low-rank approximation method based on a partial Cholesky decomposition (also known as a column Nyström approximation). The paper also presents bounds on the randomly pivoted Cholesky trace and spectral-norm errors that hold with high probability. The mathematical argument is largely due to GPT 5.6-Sol (Pro), with some refinements by the author.

Understanding the features captured by the hidden layers of neural networks is a fundamental challenge in machine learning, despite their widespread success across various classification problems. In this work, we propose an algebraic framework for examining neural networks that model classification problems. Certain results, such as the correspondence between the neural network and neural ideals, algorithms for computing the neural ideals, and a stabilization theorem that enables approximation of the neural ideals, are first established. As an application to the framework, we present algorithms to identify and interpret the features captured by each hidden-layer neuron. Along with these theoretical developments, the practical performance has been demonstrated on the MNIST digit dataset, and the results highlight the pivotal role of neural ideals as a mathematical and computational tool for analyzing the features captured by neural networks. Further, we develop an interactive software that builds on the presented framework to visualize the features captured by each neuron. This tool is available at https://github.com/yvs1967/neural-network-representation-explorer

Let \(d\ge2\) be fixed and let \(G_n\) be finite abelian groups of odd orders \(N_n\to\infty\). We determine the centered diameter-\(d\) critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing \(M_n=(N_n-1)/2\), we prove that the normalized first distance-\(d\) coverage times satisfy \sum_{[x]\in(G_n\setminus\{0\})/\{\pm1\}} \delta_{\frac{N_n^{d-1}}{d!}\tau_{n,[x]}^d-\log M_n} \xrightarrow{d} \PPP(e^{-z} $\,dz). Consequently, the number of antipodal defects in the critical window converges in total variation to a Poisson law, the diameter transition has the Gumbel profile \(e^{-e^{-c}}\), and the diameter hitting time has Gumbel fluctuations. In the original generator-density parametrization this yields the sharp fixed-\(d\) threshold constant \(d!/2^d\) throughout the odd-order abelian class. For \(d=2\), we additionally obtain an exact path–cycle decomposition of the target representation graphs.

We study reducibility of the characteristic polynomials of a class of finite Jacobi pencils. We classify the diagonal equality patterns for which generic connected couplings yield reducibility and use this classification to prove that for $n\ge4$, the connected reducible locus has codimension at least two in the connected parameter space. For $n=2m+1$, we also prove that for every fixed connected coupling vector, the locus of diagonal potentials that admit a constant branch has codimension $m$.

We develop a theory of universal decoding built on the pairwise error probability (PEP) primitive of a companion paper. The PEP and its error spectrum are defined for an arbitrary decoding metric, and while raw metric values across a family share no common scale, the PEP supplies one. Universal decoding – decoding well simultaneously against a whole family of metrics or channels – becomes the problem of merging the per-metric spectra into a single rule, which we construct from a clipped inverse-PEP statistic. The construction rests on a Kraft-type inequality for decoding, valid for any input prior: per output, every metric canonicalizes into a conditional probability assignment on the codewords, and the merge is the normalized-maximum-likelihood envelope of the induced family. We prove the rule is random-coding universal (it loses only a vanishing rate relative to the best metric in the family, against every channel), show it is an asymptotic minimax/equalizer rule, and derandomize it: over a subexponential channel family a single deterministic code inherits the guarantee. For discrete memoryless channels the construction reduces to types and recovers the maximum-mutual-information decoder, its tilted variant for non-uniform memoryless input, and finite-state universal decoders. It extends to decoding with an erasure option (deterministically, uniformly over erasure margins) and, via a discretization of separable metric families, to continuous alphabets: for AWGN with deterministic interference it attains the matched-ML exponent up to an explicit typical-set cap, and for ISI channels the best exponent in a family of equalize-and-decode rules, under explicit assumptions on the equalizers and the channel spectrum and the same cap. Throughout, universality is a corollary of the PEP analysis rather than a separate theory.

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n) = (\mathbb{Z}/r\mathbb{Z})^n \rtimes S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a product of resultants of smaller, partition-indexed subsystems. Combining this with the classical resultant–discriminant relation, we show that the discriminant of a $G(r,n)$-invariant homogeneous polynomial splits explicitly into a product of resultants of smaller subsystems, considerably easier to compute; we illustrate both decompositions with worked examples.

We consider the Cauchy problem for semilinear classical damped wave equations with general nonnegative nondecreasing nonlinearities, without imposing a modulus-of-continuity structure or convexity. A Dini-type integral condition expressed directly in terms of the nonlinearity gives small data global in-time existence and extends the nonlinear framework of Ebert–Girardi–Reissig [Math. Ann. 378 (2020), 1311–1326]. When this Dini-type integral diverges and the effective total mass of the initial data is positive, convergence of the Keller–Osserman tail integral is necessary and sufficient for finite-time blow-up of the maximal mild Sobolev solution, particularly, divergence of the tail integral gives global grow-up. The resulting integral criteria distinguish the contributions of the nonlinearity near zero and at infinity in governing small data global in-time existence with diffusion decay, global grow-up, and finite-time blow-up.

We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range $0<\gamma<2$. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.

For $\mathcal H:\mathcal B^\alpha\to\mathcal B^\alpha$ with $1<\alpha<2$, we obtain an exact norm representation. In particular, the norm is given by an explicit Gamma-function formula for $1<\alpha\leq3/2$, whereas for $3/2<\alpha<2$ it strictly exceeds the corresponding Gamma-function expression. For $\mathcal H:\mathcal B^\alpha\to\mathcal B^\alpha_{\log}$, we determine the exact norm: it is equal to $3/2$ for $1<\alpha\leq4/3$ and is given by an explicit Gamma-function formula for $4/3<\alpha<2$. We also determine the exact norm of $\mathcal H:H^1\to H^1_{\log}$ and show that it is equal to $\pi$. Finally, for $0<\alpha<1$, we determine the exact norm of $\mathcal H:H^\infty_\alpha\to\mathcal B^{\alpha+1}$, which is given by an explicit Gamma-function formula.

We study how spherical $2$-designs arising from finite group orbits can be completed to spherical $4$-designs by adjoining further orbits, allowing weights in the general theory. For an irreducible real orthogonal $G$-module $W$ with $\mathbb D=\operatorname{End}_G(W)\in\{\mathbb R,\mathbb C,\mathbb H\}$, we consider the multiplicity-two representation $W\oplus W$ and retain the failure of the $2$-design equation $M^*M=\frac12 I_2$ as a quadratic defect. When the invariant quartics are determined by the Hermitian Gram matrix, the fourth-moment problem reduces to a mean and covariance condition on these defects. This yields a sharp lower bound for the total weight of the correction orbits; at equality, their normalized defects form a weighted spherical $2$-design in the associated defect space. The quartic condition holds for the multiqubit Clifford groups in every dimension $r\geq 1$, giving an unbounded-dimensional family with a fixed three-dimensional defect space; among equality cases using the minimum number of correction orbits, the defect geometry is always a regular tetrahedron. As a complementary unweighted example, we construct a $378$-point $W(E_6)$-invariant spherical $4$-design in $S^{11}$ and prove that it is sharp among unweighted invariant completions containing a spherical $2$-design orbit.

For normal matrices A and B, the classical Hoffman-Wielandt theorem bounds the optimal matching distance between their spectra by the Frobenius norm of A-B. We prove the sharper estimate d(\sigma(A),\sigma(B))^2 \le \sum_{k=1}^{\lfloor (n+1)/2 \rfloor} s_k(A-B)^2, where s_k(M) are the singular values of M, involving only the first \lfloor (n+1)/2 \rfloor singular values of the difference. Consequently, d(\sigma(A),\sigma(B)) \le \sqrt{\lfloor (n+1)/2 \rfloor}\,\|A-B\|, which improves the classical dimension-free bound for all 3 \le n \le 16. The proof combines a min-max duality for optimal matching distances with spectral subspace overlaps and the monotonicity of singular values under rectangular compressions.

Examples and counterexamples define the boundaries of mathematical propositions, test foundational conjectures, and resolve open structural problems in abstract algebra. In loop theory, an Osborn loop is a crucial generalization of a Moufang loop satisfying a specialized variable-sandwiching identity. While small finite non-associative examples are known, constructing large examples of proper Osborn loops (loops that are neither conjugacy closed nor Moufang) remains a computational bottleneck. In this paper, we establish an efficient framework for generating large proper Osborn loops by taking the direct product of non-associative conjugacy closed (CC) loops and Moufang loops. We outline the boundary constraints necessary to prevent structural collapse into sub-varieties and provide explicit examples up to order 2025 validated via the GAP package LOOPS.

In 1984, Roland Häggkvist posed the problem of constructing Hamiltonian graphs of order $n$ with large minimum degree and no $(n-2)$-cycle. He remarked that he did not know of such a graph with minimum degree at least three. We solve this problem by proving the following two results. (1) For every integer $d\ge 3$ and every integer $n\ge 15d-14,$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-2)$-cycle. (2) For every integer $d\ge 3,$ every positive integer $k,$ and every integer $n\ge (k+1)[(d-1)(k+3)+1],$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-s)$-cycle for any $s\in\{1,2,\dots,k\}.$ The proofs are constructive. We also pose several open problems.

Using determinantal divisors of integral matrices, that is, greatest common divisors of minors of fixed order, we introduce the notion of type of a homomorphism $f$ of complex tori, which is similar to the type of a polarization. We show that the type is invariant under composition with isomorphisms, and that it completely describes the kernel of $f$ as a group. More precisely, the quotient of the kernel by its connected component containing 0 is a product of cyclic groups whose orders are determined by the type of $f$. Since the type can be computed from any rational representation of $f$, this gives an effective way to determine the kernel of a homomorphism. As a consequence, we compute the classic invariants degree and exponent of $f$, when $f$ has finite kernel. When $f$ is an isogeny, we also compute the type of its inverse isogeny from that of $f$. Finally, we compare the type of a polarization to the type of its associated isogeny.

An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.

Let $R(A,X)$ denote the largest ideal of a primitive axial algebra $(A,X)$ that contains no axis from the specified generating set $X$, and let $J(A)$ be the intersection of the maximal ideals of $A$. Mamontov, Shpectorov, and Zhelyabin asked whether $R(A,X)=J(A)$ always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis $a,b$ and multiplication $a^2=a$, $ab=2b$, and $b^2=b$ is a primitive axial algebra for an explicit fusion law and the generating set $X={a,b}$. Its complete ideal lattice is $0<\mathbb{F}b<A$, whence $R(A,X)=0$ and $J(A)=\mathbb{F}b$; in particular, the primitive axis $b$ lies in $J(A)$. Over $\mathbb{C}$, this axial presentation is equivalent to the previously classified $D(-1),{e_2,a_6}$ presentation with fusion law $F_{D3}$. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.

In this paper, we establish several higher-dimensional generalizations of refined Bohr-type inequalities for bounded holomorphic functions mapping into the unit polydisk $\mathbb{P}\Delta(0;1_n)$ in $\mathbb{C}^n$. First, we formulate multidimensional analogues of sharp Bohr-type inequalities originally established by Liu et al. [Bull. Sci. Math. 173 (2021) 103054], incorporating both squared coefficient terms and area functional components. Second, we provide improved inequalities for a recent multidimensional extension by Ahamed et al. [Complex Anal. Oper. Theory 20(6) (2026), 142] by introducing an analogous term corresponding to the area functional. Finally, we extend a refined Bohr-type inequality involving the term $\vert{}f(z)-a_0\vert{}$ to the setting of several complex variables. All the results are shown to be sharp.

In this short note we show that certain definition from the work of Kanigowski and Ravotti can be relaxed. More specifically, we show that the definition of good special flows, that is, flows exhibiting a form of mechanism of mixing, shearing of Birkhoff sums can be relaxed. Specifically, we show that the disintegration corresponding to almost partitions, which are uniformly sheared along the direction of the flow, of the base can depend on $k$-tuple of times. This note serves as a complementary material to an article on mixing of higher order in flows of the author (details in the note), where we use the new definition of good special flows to show that flows of Fayad on $n$-torus, $n\geq 3$, are mixing of any order.

Turing completeness of a programming language or system characterizes its expressive power; and the strong Church-Turing hypo-/thesis refines such from qualitative to polynomial-time equivalence. Exact Real Computation (ERC) is a novel numerical programming language paradigm: for the imperative processing of continuous data as entities appearing as exact, i.e. devoid of rounding errors [doi:10.1007/978-3-662-44199-2_107]. ERC has been designed [doi:10.46298/lmcs-20(2:17)2024] as convenient and practical alternative, namely proven qualitatively equivalent, to the Turing machines originally underlying Computable Analysis [doi:10.1007/978-3-642-56999-9,doi:10.1007/978-1-4684-6802-1]. The present work quantitatively strengthens this qualitative Turing-completeness: We assign bit-costs to ERC's operational primitives (including partial/multivalued tests) in such a way that any real function incurring polynomial cost becomes Turing-computable in polynomial time, and vice versa. Runtime measurements on implementations in the iRRAM C++ library confirm our theoretical performance predictions.

In this article, we survey some fundamental results and recent advances in the design and analysis of Boolean bent functions from the perspective of their relation to the completed Maiorana-McFarland class.

This paper introduces new characterizations, decomposition theorems, and stability results for convex sequences. We show that a sequence is convex precisely when its epigraph satisfies a midpoint convexity condition, thereby connecting discrete and geometric notions of convexity. A decomposition result proves that any sequence can be written as the difference of two convex sequences, with generalizations to higher-order convexity. We construct nontrivial convex minorants for bounded-below sequences and establish a Hyers-Ulam-type stability theorem showing that any approximately convex sequence can be uniformly approximated by a genuine convex sequence without significantly altering its values. Finally, for a concave sequence, we characterize those subsequences that are convex in it by providing slope inequalities and monotone auxiliary sequences. We explore the interplay among convexity, subadditivity, and periodically indexed subsequences.

We compute several new Laplace convolution integrals involving exponential functions, error functions, and parabolic cylinder functions. We extend our results using formulas for repeated integration and differentiation. We apply these convolution integrals to heat transfer and linear viscoelastic problems. As a by-product, we obtain new integral representations for the error function and the complementary error function.

In this paper, we study planar virtual knotoids, which generalize virtual knots and knotoids both. We consider the problem of whether two given planar virtual knotoids can be transformed into each other via a sequence of local moves and what is the shortest sequence length. By introducing $r$-covering of planar virtual knotoids, we determine the homotopy relationship between different planar virtual knotoids, and obtain lower bounds of the Gordian distance between them. On the basis of these results, we demonstrate calculations of the exact Gordian distance of several given pairs of homotopic planar virtual knotoids. Furthermore, we investigate $\Delta$-moves and virtual region crossing changes for planar virtual knotoids, and derive lower bounds of the Gordian distance with respect to both moves via $r$-coverings.

We investigate finite non-degenerate set-theoretic solutions to the Yang–Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour — apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ — they can be made arbitrarily close to $1$.

The notion of a base of computability $\mathscr{C}$ in a category $\mathscr{C}$ was introduced as a tool to generate computability models, in the sense of Longley and Normann, from categories. In this paper we introduce the category $\mathsf{CatBaseComp}$ of categories with a base of computability, and we show that $\mathsf{CatBaseComp}$ has all pie limits. We prove that a Grothendieck fibration lifts a base of computability in the base category to a base of computability in the total category of the fibration, and conversely, a pullback-preserving Grothendieck fibration maps a base of computability in the total category to a base of computability in the base category of the fibration. Connecting $\mathsf{CatBaseComp}$ with the semantics of dependent type theory, we show that $\mathsf{CatBaseComp}$ is a type-category, or a (fam, $\Sigma$)-category with a terminal object. Moreover, we prove that CatBaseComp is a (2-fam, $\Sigma$)-category, a 2-categorical generalisation of a (fam, $\Sigma$)-category. Finally, we describe the canonical (2-dep, $\Sigma$)-structure of CatBaseComp, i.e., the canonical dependent arrows of CatBaseComp that are compatible with its (2-fam, $\Sigma$)-structure.

We study polylogarithmic Chabauty–Kim loci for $S$-integral points on $\mathbb{P}^1\smallsetminus\{0,1,\infty\}$ over number fields. We compare motivic and étale Selmer schemes, describe Galois actions on Selmer schemes and period rings, and give an explicit formula for the localisation map on the polylogarithmic Selmer scheme. For imaginary and real quadratic fields, we derive equations and determine several polylogarithmic and full Chabauty–Kim loci and show that Kim's Conjecture holds in several new cases. In other cases, we show that the polylogarithmic Chabauty–Kim method is insufficient to cut out precisely the $S$-integral points, even when combined with $S_3$-symmetrisation, due to additional points arising from $p$-adic roots of unity. As a further application, we give a Chabauty–Kim theoretic proof of the $S$-Selmer Section Conjecture for imaginary quadratic fields when $S$ is empty or $S$ consists of a single prime not fixed by complex conjugation.

We introduce group-valued coloured interchange processes, a class of continuous-time random walks on wreath products generated by transpositions and arbitrary symmetric base-group transitions. We give a complete representation-theoretic characterisation of their spectral gaps by decomposing the associated quasi-regular representation and determining precisely which irreducible representations are required. For abelian base groups, we realise these representations on multislices and identify their Laplacians with discrete Schrödinger operators. We then classify the minimal families of irreducible representations that determine the spectral gap for every choice of transition rates.