arXiv preprints from January 1, 2026 through July 21, 2026 — 07:47:16 EST
The Kuramoto model has shaped our understanding of synchronization in complex systems, yet its phase-only formulation neglects amplitude dynamics that are intrinsic to many oscillatory networks. In this work, we revisit Kuramoto-type synchronization through networks of Stuart--Landau oscillators, which arise as the universal normal...
A proof will be presented that the existence of a non-trivial $Σ_1$-elementary embedding $j: V_{λ+3} \prec V_{λ+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the literature, notably including \cite{Goldberg2020}, \cite{Schlutzenberg2020}, and \cite{Woodin2010}, the latter...
This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. A correspondence between missing parts and the gaps of...
Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In...
We characterize the nuclearity of Toeplitz operators $T_μ: F_α^p \to F_α^q$ with Borel measure symbols for $1\leq p,q\leq \infty$. For positive measures $μ$ and $q\leq p$, we provide necessary and sufficient conditions in terms of the Berezin transform and establish a rigidity property for nuclearity across this range. In the case...
This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-Kähler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy and interpolating sesqui-energy functionals, restricted to the space of vector fields. Explicit...
A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order...
I argue that the Hodge structure on a Euclidean Clifford algebra $Cl(n)$ provides a way to generalise Kähler structure to higher dimensions, in the sense that the paired variables are now associated with $k-$ and $(n-k)-$ dimensional subspaces rather than with vectors. This puts a phase space structure on Clifford algebras, and so...
We mainly use the d-invariant surgery formula established by Wu and Yang \cite{wu2025surgerieslensspacestype} to study the distance one surgeries along a homologically essential knot between lens spaces of the form $L(p,1)$ and $L(q,2)$ where $p,q$ are odd integers.
In this paper, a two-step inertial Tseng extragradient method involving self-adaptive and Armijo-like step sizes is introduced for solving variational inequalities with a quasimonotone cost function in the setting of a real Hilbert space. Weak convergence of the sequence generated by the proposed algorithm is proved without assuming...
We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between...
We derive an Ito-type change-of-variables formula for Volterra Gaussian processes (including fractional Brownian motion with any Hurst parameter), based on the operator factorization framework. The Ito correction is expressed as a Stieltjes integral against the energy function Gamma^X(t) := ||Pi DX_t||_H^2, which equals E[X_t^2] for...
Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being...
This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically closed, the analogue of the Main Theorem of the representation theory of Galois orders by V....
We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon urn. For...
We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a...
In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a...
We consider general parameter to solution maps $θ\mapsto \mathcal G(θ)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing...
The convergence of the Conjugate Gradient method is subject to a locality limitation which imposes a lower bound on the number of iterations required before a qualitatively accurate approximation can be obtained. This limitation originates from the restricted transport of information in the graph induced by the sparsity pattern of the...
Let $X$ be a smooth projective variety over an arbitrary field $k$ of characteristic zero. We explore infinitesimal deformations of the Chow group $CH^{p}(X)$ via its formal completion $\widehat{CH}^{p}$, a functor defined on the category of local augmented Artinian $k$-algebras. Under a natural vanishing condition on Hodge cohomology...
A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the...
We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \fracπ{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $α:= \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of...
This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single element. A probabilistic argument using the Haar measure on the absolute Galois group of a number...
We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we identify a further class of global minimizers, all of which have constant Laplacian. Recent work...
We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\geq 0$. This leads us to conjecture what are the only possible obstructions to $|a^m-b^n|$ taking on infinitely many distinct...