arXiv preprints from January 1, 2026 through September 8, 2026 — 23:33:59 EST
We study the distribution of $2$-torsion in class groups and narrow class groups of cubic fields and cubic orders subject to prescribed shape conditions. The \emph{shape} of a cubic order in a number field is a natural geometric invariant taking values in the modular surface $\mathbb{H}/\operatorname{GL}_2(\mathbb{Z})$. Fix a subset...
We introduce Minary, a computational framework designed as a candidate for the first formally provable autopoietic primitive. Minary represents interacting probabilistic events as multi-dimensional vectors and combines them via linear superposition rather than multiplicative scalar operations, thereby preserving uncertainty and...
This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently...
Let $F: T^{1,0}M\rightarrow[0,+\infty)$ be a strongly convex complex Finsler metric on a complex manifold $M$ and $\pmb{J}$ the canonical complex structure on the complex manifold $T^{1,0}M$. We give a geometric characterization of strongly convex Kähler-Berwald metrics. In particular, we prove that $\pmb{J}$ is horizontally parallel...
Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central...
We consider classical/quantum correspondence in Lindblad evolution with jump operators for which the corresponding Fokker--Planck equation is subelliptic. This allows us to consider the physical model proposed by Zurek and Paz, and to extend some of the recent mathematical results of Hernandez, Ranard and Riedel, Galkowski and...
Fractional Burgers equations pose substantial challenges for classical numerical methods due to the combined effects of nonlocality and shock-forming nonlinear dynamics. In particular, linear approximation frameworks-such as spectral, finite-difference, or discontinuous Galerkin methods-often suffer from Gibbs-type oscillations or...
This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight error bounds on the quality of the approximations are obtained.
It is well-known that an associative algebra shares the same growth and Gelfand-Kirillov dimension (GK-dimension) as its associated monomial algebra with respect to a degree-lexicographic order. This article mainly investigates the relationship between the GK-dimension of an algebra and that of its associated monomial algebra with...
We develop the specification and orbit-decomposition approach to equilibrium states for parabolic rational maps of the Riemann Sphere. Our result extends the well-known results on uniqueness of equilibrium states in this setting, notably the results of Denker, Przytycki and Urbański. We extend the class of potentials from Hölder to...
This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning over this class as a structured infinite-dimensional regression problem with multiscale sparsity....
For a graph $G$ on $n$ vertices, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which has been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar graph $G$. We show that the version of this problem for multigraphs (allowing parallel edges) is...
This paper presents a computational framework for modeling wave propagation in geometrically linear elastic materials characterized by algebraically nonlinear constitutive relations. We derive a specific form of the nonlinear wave equation in which the nonlinearity explicitly appears in the time-derivative terms that govern the...
In this paper, we investigate Liouville-type theorems for stationary solutions to the shear thickening fluid equations in a slab. We show that the axisymmetric solution must be trivial if its local $L^\infty$-norm grows mildly as the radius $R$ grows. Also, a bounded general solution $u$ must be trivial if $ru^r$ is bounded. The proof...
Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on...
We consider random linear unbounded operators on a Banach space $\mathcal{X}$. For example, such random operators may be random quantum channels. The Law of Large Numbers is known when $\mathcal{X}$ is a Hilbert space, in the form of the usual Law of Large Numbers for random operators, and in some other particular cases. Instead of...
We study the fluctuation behavior of individual eigenvalues of kernel matrices arising from dense graphon-based random graphs. Under minimal integrability and boundedness assumptions on the graphon, we establish distributional limits for simple, well-separated eigenvalues of the associated integral operator. A sharp probabilistic...
In the Material Point Method (MPM), accurately imposing Neumann-type thermal boundary conditions, particularly convective heat flux boundaries, remains a significant challenge due to the inherent nonconformity between complex evolving material boundaries and the fixed background grid. This paper introduces a novel Virtual Heat Flux...
By their nature it is difficult to differentiate chaotic dynamical systems through measurement. In recent years, work has begun on using methods of Topological Data Analysis (TDA) to qualitatively type dynamical data by approximating the topology of the underlying attracting set. This comes with the additional challenges of high...
The explicit constraint force method (ECFM) was recently introduced as a novel formulation of the physics-informed solution reconstruction problem, and was subsequently extended to inverse problems. In both solution reconstruction and inverse problems, model parameters are estimated with the help of measurement data. In practice,...
Half-space models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit rich boundary phenomena that alter the asymptotic behavior familiar from their full-space counterparts. A distinguishing feature of these systems is the presence of a boundary parameter that governs a transition between subcritical, critical, and...
We compute numerically the $L^2$ Marcinkiewicz-Zygmund constants of cubature rules, with a special attention to their role in polynomial approximation by orthogonal bases. We test some relevant rules on domains such as the interval, the square, the disk, the triangle, the cube and the sphere. The approximation power of the...
In this study, we introduce the theory of what we call Hecke vector-forms. A Hecke vector-form can be viewed as a vector function representation of some quasiautomorphic form that transforms like an automorphic form on an arbitrarily chosen Hecke triangle group. In other words, because quasiautomorphic forms have complicated...
In this paper we study the Lyapunov spectrum rigidity for random walks of expanding maps on unit circle $\mathbb{S}^1$ and Anosov diffeomorphisms on $d$-torus $\mathbb{T}^d$. Let $ν$ be a probability supported on the set of expanding maps on $\mathbb{S}^1$ or a neighborhood of a generic Anosov automorphisms on $\mathbb{T}^d$. If the...
We prove the Kato square root estimate for second-order divergence form elliptic operators $-div(A\nabla)$ on a bounded, locally uniform domain $D \subseteq \mathbb{R}^n$, for accretive coefficients $A \in L^\infty(D; \mathbb{C}^n)$, under the Robin boundary condition $ν\cdot A\nabla u + bu = 0$ for a (possibly unbounded) boundary...