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This paper is concerned with the uniqueness, existence, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in…
We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this…
We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition…
We consider core-periphery structured graphs, which are graphs with a group of densely and sparsely connected nodes, respectively, referred to as core and periphery nodes. The so-called core score of a node is related to the likelihood of…
Understanding the structure of real data is paramount in advancing modern deep-learning methodologies. Natural data such as images are believed to be composed of features organized in a hierarchical and combinatorial manner, which neural…
The influence of the microstructure of a polycrystalline material on its macroscopic deformation response is still one of the major problems in materials engineering. For materials characterized by elastic-plastic deformation responses,…
We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for…
In this paper, we study the evolution of smooth, closed planar curves under a fourth order biharmonic flow with an external forcing term. Such flows arise naturally in the theory of biharmonic maps and geometric variational problems…
The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a…
In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are…
This paper discusses the effect of sequential conflict resolution maneuvers of an infinite aircraft flow through a finite control volume. Aircraft flow models are utilized to simulate traffic flows and determine stability. Pseudo-random…
We study the Fisher model describing natural selection in a population with a diploid structure of a genome by differential- geometric methods. For the selection dynamics we introduce an affine connection which is shown to be the…
Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible…
We derive an equation of motion for interest-rate yield curves by applying a minimum Fisher information variational approach to the implied probability density. By construction, solutions to the equation of motion recover observed bond…
In this work, we consider the solid-state dewetting of an axisymmetric thin film on a curved-surface substrate, with the assumption that the substrate morphology is also axisymmetric. Under the assumptions of axisymmetry, the surface…
In this work, the author developed a multiple scattering model for heterogeneous elastic continua with strong property fluctuation and obtained the exact solution to the dispersion equation derived from the Dyson equation under the…
We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fr\'echet smooth manifolds…
The goal of this survey article is to explain and elucidate the affine structure of recent models appearing in the rough volatility literature, and show how it leads to exponential-affine transform formulas.
In this paper, we study a curve flow which preserves the anisotropic length of the evolving curve, and show that for any convex closed initial curve, the flow exists for all time and the evolving curve converges to a homothety of the…
Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly…