Related papers: Purely arithmetic PDE's over a p-adic field I: del…
It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system…
We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic…
We study natural D-modules on the moduli stack of elliptic curves over a field of characteristic zero. We use this to produce an algebro-geometric version of the algebra of higher depth mock modular forms, studied from a Conformal Field…
In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…
PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…
We introduce a new family of discontinuous Galerkin (DG) finite element schemes for the discretization of first order systems of hyperbolic partial differential equations (PDE) on unstructured simplex meshes in two and three space…
The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations,…
Solving partial differential equations (PDEs) can be prohibitively expensive using traditional numerical methods. Deep learning-based surrogate models typically specialize in a single PDE with fixed parameters. We present a framework for…
We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical…
Let $p$ be a prime number, $V$ a discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…