Papers

We study finite braces whose additive and multiplicative groups are both cyclic. We reinterpret the classification of these braces from the perspective of regular subgroups of permutation groups, and identify which of them are biskew braces.

The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\geq 2$, put \[ D(k)=\gcd_{2\leq q\leq k+1}\binom{qk}{k}, \qquad n=k+1. \] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \[ D(k)=1 \quad\Longleftrightarrow\quad \frac{n}{P}>P. \] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.

We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses. We further prove sharp rates of convergence with an explicit bias-variance decomposition governed by a novel complexity measure. We show that the variance is independent of misspecification, while the bias depends on a source condition parameter known in the learning literature. For tensor product Sobolev spaces we obtain new rates that connect to spaces of functions with dominating mixed smoothness, substantially extending existing results and explaining why these estimators circumvent the curse of dimensionality. Our methodology, combining elements from both functional analysis and empirical process theory, allows for an asymptotic linearisation of the objective function that avoids both closed-form solutions and global Lipschitz assumptions, and may be of independent interest. The estimators are implemented in C++ and theory is supported by numerical experiments.

We study ergodic optimization problems for one-dimensional diffusions with resetting boundaries, namely diffusions that are instantaneously reset according to prescribed distributions upon reaching designated boundary points. For controlled diffusions with resetting boundaries, we consider optimization of the ergodic cost over a broad class of admissible real-time controls. We derive a Hamilton–Jacobi–Bellman equation with non-local boundary conditions and establish a verification theorem for optimal stationary Markov controls. In the case of one resetting and one reflecting boundary, we also prove an existence result under weak integrability assumptions. We then study renewal diffusions. Using their regenerative structure, we derive explicit formulas for the ergodic cost. These formulas are applied to optimization problems over families of drift coefficients, yielding explicit characterizations of optimal drift parameters in several examples. Finally, we study an infinite-penalty regime in which boundary hitting is prohibited and characterize the corresponding optimal ergodic cost.

We introduce the externally supported independence number $\alpha_{\rm es}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $\iota(G)$. We show that computing $\alpha_{\rm es}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $\alpha_{\rm es}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely describe the extreme graph classes attaining these bounds.

We investigate the approximation of continuous Fourier transforms by trigonometric sampling polynomials and their efficient evaluation by the nonequispaced fast Fourier transform (NFFT). While the NFFT is traditionally used for the evaluation of trigonometric polynomials, we show that it can also serve as an effective computational tool for the approximation of Fourier transform values. Building on ideas of M.~Ehler, K.~Gröchenig, and A.~Klotz \cite{EhGrKl24}, we derive explicit $\ell_\infty$ uniform error bounds between the Fourier transform and suitable sampling polynomials. The resulting estimates quantify the influence of the sampling width and truncation parameter and provide rigorous accuracy guarantees on entire frequency intervals. In contrast to previous analyses focusing mainly on discrete or $L_2$-type errors, our results yield uniform approximation bounds that are directly relevant for practical computations. Moreover, we prove a matching lower bound which shows that the derived rate in the truncation parameter is sharp. The derived theory leads to a simple algorithmic framework: first approximate the Fourier transform by a trigonometric sampling polynomial and then evaluate this polynomial efficiently by the NFFT. Numerical experiments confirm the theoretical convergence rates and demonstrate that accurate approximations of continuous Fourier transforms can be obtained with moderate computational effort.

Predicting the behavior of dynamical systems (DS) beyond the dynamical and parameter regimes observed in training is a pivotal and essentially unresolved problem in scientific ML. It is central to any good scientific theory, which we expect to be able to make predictions about regimes not covered by currently available data. Recent hierarchical and hyper-network guided approaches for DS reconstruction (DSR) enable training on many DS simultaneously, and revealed that extracted latent features are often related to crucial control parameters of the underlying DS that varied across the training corpus. However, true out-of-domain forecasting abilities of these models, e.g., across tipping points, remain limited, and fine-tuning, or even full model retraining, on time series from the new dynamical regime is usually required. Here, we mathematically analyze the root of these limitations in previous model formulations and identify three core shortcomings rooted in a mismatch between structural assumptions of the reconstruction model and typical properties of physical systems. We propose a combination of remedies for these shortcomings, most importantly feature splitting, and furthermore derive a closed-form bound on the reliable extrapolation range. We demonstrate empirically that our techniques allow for accurate zero-shot prediction into new dynamical regimes, outside the observed training regime, as, e.g., encountered across tipping points.

We study a spatial version of the Smoluchowski coagulation equation in which particles carry both an integer mass and a spatial location, and coagulate according to a mass-space product kernel. After coagulation, the resulting particle inherits the sum of the parent masses and a single spatial location. We provide a recursive formula for the solution and show that, while the first local mass moments are conserved up to gelation time at each location, the second local mass moments evolve according to a closed system of differential equations. When the spatial part of the kernel is of Mercer type, we further show that this system reduces to a replicator equation under a time rescaling and normalisation. We then use the fact that the replicator equation admits monotone functionals to quantify how spatial heterogeneity influences the rate of gelation and derive bounds on the gelation time. We also discuss the implications of these results for several important classes of Mercer kernels, including radial, diffusion, and translation-invariant kernels.

Let $d\ge 1$ and $\varphi: B_d\to\mathbb{D}$ be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb{C}^d$ and $\mathbb{D} = B_1$. Let $b: \mathbb{D} \to \mathbb{D}$ be a holomorphic function and $\mathcal {H}(b)$ denote the corresponding de Branges-Rovnyak space. We show that compactness of the composition operator $C_\varphi$ from $\mathcal{H}(b)$ to the Hardy space $H^2(B_d)$ is related to natural restrictions on the Nevanlinna counting functions of the slice-functions $\varphi_\zeta$, $\zeta\in \partial B_d$.

McKean-Vlasov-type stochastic differential equations (SDEs) are characterized by coefficients depending on both the state and the law of the solution. In this work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric $p$-rough path lift of its solution, with $p\in(2,3)$. After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean-Vlasov SDEs. Finally, we consider the associated particle system and propagation of chaos is established.

The broadcasting problem on trees concerns the transmission of binary information along the edges of a tree. In the formulation studied here, each vertex carries one of two colours, and along every edge the colour is transmitted with error probability $q$. The central question is whether the root colour can be reconstructed from the observed colours of all vertices in the tree, but crucially, without knowing which vertex is the root. We study the problem of reconstructing the root colour in a broadcasting process on random trees, focusing on the majority rule estimator. We introduce a novel and adaptable martingale approach for analysing the colour imbalance. This method provides a simpler and more powerful alternative to existing techniques. Our martingale approach unifies the analysis of the different tree models and leads to linear upper bounds for the error probability of the majority rule estimator in each case. This shows that, for a broad class of random tree models, the error probability of the majority rule admits a linear upper bound in $q$.

Rota-Baxter groups and skew braces are closely related algebraic structures, both providing set-theoretical solutions to the Yang-Baxter equation. In this paper, we extend these structures to the setting of affine schemes. First, we introduce affine Rota-Baxter groups and, by leveraging the duality between affine groups and Hopf algebras via their coordinate rings, prove the equivalence between affine Rota-Baxter groups and co-Rota-Baxter Hopf algebras. Next, we show affine Rota-Baxter groups can naturally give rise to the affine skew braces defined by Angiono, Galindo, and Vendramin. Conversely, any affine skew brace can be embedded into an affine Rota-Baxter group. By linking these to the relationship between affine skew braces and Hopf co-braces, we give new connections between co-Rota-Baxter Hopf algebras and Hopf co-braces. Finally, we propose the study of solutions to the Yang-Baxter equation within the framework of affine schemes, demonstrating that affine skew braces naturally give rise to such solutions.

Determining the conditions under which the direct product of graphs $G$ and $H$ satisfies $\mathrm{Aut}(G\times H)=\mathrm{Aut}(G)\times\mathrm{Aut}(H)$ has been a problem of considerable interest since Sabidussi's classic work in the 1950s. We call such a pair $(G,H)$ stable, and unstable otherwise. Although much progress has been made for graph pairs, the general digraph case has remained completely open. In this paper, we initiate the study of the stability of digraph pairs, and then focus on the stability of a single digraph $G$. This is defined as the stability of the pair $(G,K_2)$ and has been studied extensively when $G$ is undirected. We establish a necessary and sufficient condition for a connected digraph to be unstable, and use it to derive four sufficient conditions for circulant digraphs to be unstable. Moreover, we prove the nonexistence of nontrivially unstable finite arc-transitive circulant digraphs and nontrivially unstable Cayley digraphs of abelian groups of odd order.

We recently proved that if their initial velocity and magnetic field both are the sum of a classical 3D-part and some 2D-part (i.-e. depending only on the horizontal space variables), the solutions of the 3D-rotating magnetohydrodynamic (MHD) system converge (when the Rossby number goes to zero) towards those of a 2D-MHD system with six components and an additional 3D magnetic field transported by the 2D limit velocity. We also provided explicit convergence rates for large ill-prepared initial data. In this article, thanks to new estimates, we improve the convergence rates and allow larger initial data. Finally these improvements are adapted to the rotating fluids system.

We introduce and study compatible Lie conformal bialgebras as conformal counterparts of compatible Lie bialgebras. Such a structure consists of two compatible Lie conformal brackets and two compatible conformal cobrackets on the same $\C[\partial]$-module, and every simultaneous linear combination of them is again a Lie conformal bialgebra. We develop representations and matched pairs for compatible Lie conformal algebras, introduce compatible Lie conformal coalgebras, and establish their duality through the conformal dual in the finite case. For $\C[\p]$-modules that are free of finite rank, we prove the equivalence among compatible Lie conformal bialgebras, standard compatible conformal Manin triples and matched pairs. In the coboundary case, we characterize the tensors $r$ that determine compatible Lie conformal bialgebras. The characterization requires the symmetric part of $r$ to be invariant with respect to both brackets and three conformal Yang–Baxter conditions to hold. The first two conditions correspond to the two brackets separately, whereas the third is the compatible conformal Yang–Baxter condition. For comparison, we introduce the compatible conformal classical Yang–Baxter equation (CYBE) and show that each of its solutions satisfies these three conditions, while the converse fails. One explicit example shows that the first two conditions do not imply the third. Another shows that all three conformal Yang-Baxter conditions may hold even though $r$ is not a solution of compatible conformal CYBE.

We give a transfer principle for the fluctuations of linear statistics of finite particle systems around Lebesgue measure: if for a smooth kernel the variance decays polynomially with some exponent a compared to independent (non-interacting) particles, then the number variance over balls centred at almost every point decays with exponent min(1,a) times a log term if a=1, over a possibly reduced range of scales for non-periodic systems. We apply this principle to eigenvalues of random N*N Girko matrices, leveraging results of Cipolloni et al., and obtain the optimal perimeter-like number variance, on the microscopic and some mesoscopic scales range, after local averaging. We also apply the results to Coulomb gases, by transferring the results of Serfaty: we prove that 2D-Coulomb gases are 2-hyperuniform, i.e. they have surface order number variance, which is optimal, and that 3D Coulomb gases are 1-hyperuniform. In 2D, it allows to prove finite Coulomb energy and Wasserstein distance to Lebesgue measure.

It is well known that for every graded ideal $I$, the numbers of minimal generators $\mu(I^k)$ of its powers of $I$ are eventually increasing. However, its initial behavior can be surprisingly flexible. We prove that any prescribed finite pattern of increases, decreases, and equalities can occur among the first differences $\mu(I^{k+1})-\mu(I^k)$ of a suitable monomial ideal $I$ in $K[x,y]$. This provides a broad positive answer to a previously posed sign-realization problem and unifies several known constructions exhibiting unusual behavior of $\mu(I^k)$. Moreover, as a consequence, analogous sign-realization results are obtained for the index of reducibility of powers of $\mathfrak m$-primary ideals in two-dimensional regular local rings.

Applying machine learning techniques to solving long-standing mathematical conjectures can be particularly challenging due to their extreme reward sparsity. As an illustrative example, we consider Kalai's algebraic Hirsch conjecture and recast the construction of its counterexamples as a sparse-reward reinforcement learning problem on graphs. We propose a constrained options-based HRL framework with an equivariant graph neural network policy, which allows us to learn useful temporal abstractions for this task. We evaluate our approach over a wide range of degrees and demonstrate that it consistently outperforms classical RL algorithms as well as greedy search. By exploiting the hierarchical structure of the problem, we effectively provide a first-of-its-kind application of HRL to a problem in commutative algebra.

We relate the arithmetic of Bianchi groups to automorphism groups of Picard-rank-four $K3$ surfaces. Let $K$ be an imaginary quadratic field with ring of integers $\mathcal O_K$, and let $S_K=\operatorname{Herm}_2(\mathcal O_K)$ be the rank-four lattice of $2\times2$ Hermitian matrices over $\mathcal O_K$, equipped with the quadratic form $2\det$. For an odd integer $N\geq1$, we consider a very general $S_K(2N)$-polarized $K3$ surface $X_{K,2N}$. We prove that its automorphism group is commensurable with a level-$2N$ congruence subgroup of the Bianchi group. Furthermore, we also obtain exact realizations of congruence subgroups as full automorphism groups. Namely, if $K=\mathbb Q(i)$ or $K=\mathbb Q(\sqrt{-p})$, where $p$ is prime, then \[ \operatorname{Aut}(X_{K,2}) \cong P\Gamma_K(2). \] Thus, for every prime $p$, the projective principal congruence subgroup of level $2$ over $\mathcal O_{\mathbb Q(\sqrt{-p})}$ occurs as the full automorphism group of a Picard-rank-four $K3$ surface. At higher levels, the full automorphism group may be either the projective principal congruence subgroup or the strictly larger projective level subgroup $\operatorname{Bi}_K(2N)$, depending on the arithmetic of the primes dividing the level. We further explain these arithmetic groups geometrically. The surfaces $X_{K,2}$ arise as deformations of the Kummer surfaces $\operatorname{Km}(E_K\times E_K)$, yielding explicit double-cover models and genus-one fibrations. For $K=\mathbb Q(\sqrt{-2})$ and $K=\mathbb Q(\sqrt{-7})$, the automorphism group is generated by Mordell–Weil translations associated with genus-one fibrations coming from cusps, together with the covering involution. For $K=\mathbb Q(i)$ and $K=\mathbb Q(\sqrt{-3})$, we construct complete-intersection models in products of projective spaces and show that their automorphism groups are generated by deck involutions.

This paper studies the convolution operator on weighted Hahn sequence spaces. The boundedness and compactness of these operators, together with the multiplier algebras of the weighted Hahn space and its dual, are investigated. A complete characterization of the spectrum and fine spectrum is obtained, with illustrative examples. The introduction of the weighted framework leads to the emergence of new multiplier and spectral properties.

The co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is defined to be a graph with the set of all non-trivial proper subgroups of $G$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. The deleted co-maximal subgroup graph of $G$, denoted by $\Gamma^*(G)$, is defined as the graph obtained by removing the isolated vertices from $\Gamma(G)$. In this paper, we prove that for any finite group $G$, $\Gamma^*(G)$ is connected. Furthermore, we show that $\Gamma^*(G)$ either contains a cycle or is a star. When $\Gamma^*(G)$ contains a cycle, its girth is either $3$ or $4$. Finally, we classify all finite groups $G$ for which $\Gamma^*(G)$ is a star.

The aim of the paper is to compare in detail several notions of $BV$ space of functions of bounded variation in metric measure spaces $({\rm X},\mathsf{d},\mathfrak{m})$. Informally, they can be grouped in two classes, either by a relaxation procedure starting from a class of nicer functions (and with different notions of pseudo-gradient in the relaxation procedure) or by requiring good behaviour along a rich class of absolutely continuous curves. In the second approach, richness can be understood according to the notion of approximation modulus of [O. Martio, Adv. Calc. Var., 9 (2016)] or according to the notion of test plan introduced in [L. Ambrosio, N. Gigli, and G. Savaré, Invent. Math., 195 (2014)]. Extending [L. Ambrosio and S. Di Marino, J. Funct. Anal., 266 (2014)], we prove that all these approaches are isometrically equivalent in any locally complete metric measure space.

Quadratic programs (QPs) with sparsity constraint are generally NP-hard, and their efficient global solution depends crucially on tractable tight convex relaxations. In this paper, we propose a sparsity-cone semidefinite programming (SC-SDP) relaxation for sparse (indefinite) QPs. Unlike standard SDP liftings, such as the SDP–RLT relaxation, which involve a $(2n+1)$-dimensional semidefinite matrix, the proposed SC-SDP formulation uses only a $(n+1)$-dimensional matrix together with a single sparsity-cone constraint $\mathcal{K}$ to handle the relaxation of the $\ell_0$-norm constraint. We prove that SC-SDP is equivalent in strength to the SDP–RLT relaxation. We further study the sparsity cone $\mathcal{K}$, deriving structural characterizations and showing that projection onto $\mathcal{K}$ can be computed efficiently via a one-dimensional subproblem. Building on the dual of SC-SDP, we derive explicit presolving mechanisms, including a dual-fixing rule for individual variables, a screening-cut rule for excluding larger support patterns, and a dual-refinement step for improving presolving certificates. To solve the resulting relaxation SC-SDP efficiently, we develop a two-phase Riemannian-based augmented Lagrangian method and exploits the structured projection subproblems. Numerical experiments on several classes of sparse QPs show that SC-SDP preserves the bound quality of SDP–RLT while offering substantial computational advantages and practically effective presolving capabilities.

We analyze a generalized incompressible model proposed by Ohkitani [19]. This model is based on the observation that the two-dimensional Burgers' equation can be related to the incompressible Navier-Stokes equations by rotating the velocity gradient by 90 degrees. We present several results with initial data in $H^{3}$. First of all, we examine the inviscid model and show the existence and uniqueness of a local-in-time solution that blows up in finite time if the initial vorticity contains a negative part. In the presence of viscosity, we show the existence of a unique global-in-time solution without requiring a sign condition on the initial vorticity, establish the long-time behavior of the difference between two solutions, and derive temporal decay rates for the velocity field when the initial vorticity is non-positive.

Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $\delta$ and $\tau$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial-twuality polynomials to set systems, yielding partial-$\bullet$ polynomials with $\bullet\in\{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast\}.$ For partial-$\ast$ polynomials, Zhao and Yan proved that this polynomial is either even, odd, or both even-interpolating and odd-interpolating for every binary delta-matroid. In this paper, we extend this interpolation property to all the remaining nontrivial partial-twualities of binary delta-matroids. Consequently, for every binary delta-matroid and every $\bullet\in\{\ast,\times,\ast \times ,\times\ast ,\ast \times \ast \}$, the partial-$\bullet$ polynomial is either even, odd, or both even-interpolating and odd-interpolating. We also provide examples to show that the binary assumption is essential.

Over a qcqs scheme $S$, we analyze the birational localization $L_{\mathrm{bir}}\mathcal{H}^{\mathbb{A}^1}(S)$ of the motivic $\infty$-category $\mathcal{H}^{\mathbb{A}^1}(S)$. We establish that the associated localization functor $L_{bir}$ commutes with the bar construction, thereby preserving connectivity over fields. When the field is perfect, we show that a sheaf of groups is birational exactly when it is strongly $\mathbb{A}^1$-invariant and has trivial $\mathbb{G}_m$-contraction. For connected motivic spaces over such fields, this yields a canonical equivalence between $L_{bir}$ and the $S^{2,1}$-nullification functor $L^{2,1}$. For a general field $k$, we identify $\pi_0^b(X)$ with $\pi_0^{b\mathbb{A}^1}(X)$ for proper $k$-schemes and show that $\mathbb{A}^1$-connectedness is equivalent to birational connectedness for (ind-)proper $k$-schemes. This also confirms that $\pi_0^{b\mathbb{A}^1}(-)$ and $\mathbb{H}_0^{\mathbb{A}^1}(-)$ are stable birational invariants of smooth proper $k$-schemes, satisfying the expected universal property for invariants valued in birational sheaves (of sets and abelian groups respectively). Finally, we show that etale birational equivalences to $S$ are precisely the dense open immersions into $S$.

Let $P_3$ denote the three-dimensional paraboloid over a finite field of prime order in which $-1$ is not a square. We prove that the Fourier extension operator associated with $P_3$ maps $L^2$ to $L^r$ for $r>\frac{176}{51}=3.45098\ldots$. The argument combines the author's bilinear approach to the problem with point-line incidence estimates. We also prove that the extension operator associated with the paraboloid $P_6$ in six dimensions maps $L^2$ to $L^{8/3}$. This was previously known up to but not including the endpoint, and is the sharp $L^2$ estimate in six dimensions. Finally we observe that the endpoint restriction conjecture for $P_3$ in finite fields implies that the integer lattice points on the $3$-d Euclidean paraboloid are a $\Lambda(3)$ set.

We propose a scaled gap-function method for variational inequalities, mixed variational inequalities, and equilibrium problems over a closed convex set. The method drives a gap function to zero by combining a variable-metric scaled projection step with a modified non-monotone Armijo line search. The construction rests on a structural identity: the scaled projection step is exactly the maximizer that defines the Fukushima regularized gap, so the search direction is simultaneously the algorithmic step and the gap-defining direction. For variational and mixed variational inequalities with a strongly monotone, Lipschitz operator we establish global convergence and an R-linear rate, under a fixed or a controlled-change variable metric. The rate follows from the strong-monotonicity contraction and the resulting explicit gap error bound. For equilibrium problems we obtain global convergence and a gap error bound (Mastroeni's gap). Numerical experiments on controlled problems confirm the convergence guarantees and the predicted rate, and compare the method with standard extragradient and gap-descent methods. The combination of variable-metric scaling, a modified non-monotone line search, and gap-function descent appears to be new for these problem classes.

We study analytic paths of ergodic measures under quantitative constraints. For uniformly hyperbolic systems, we construct one-parameter families of ergodic measures whose prescribed Birkhoff averages vary affinely, whose metric entropies vary analytically, and whose endpoint entropy values are realized exactly. Along these paths, the integral of every Hölder continuous observable depends analytically on the parameter. In the mixing case, the measures may be chosen to be Bernoulli. We also prove a dimension counterpart for average conformal hyperbolic sets: the Hausdorff dimensions of the measures vary analytically along the path. Finally, for a class of partially hyperbolic diffeomorphisms with one-dimensional center, we construct analytic paths of ergodic measures whose center Lyapunov exponents are prescribed linearly and whose entropies vary nalytically.

We consider Neumann boundary value problems for the $p$-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Using spherical symmetrization under appropriate constraints, we derive Talenti-type comparison results in Lorentz spaces. We further show that, in contrast to the Robin case, the Neumann setting admits weaker constraints, which yields stronger comparison principles.