Related papers: Evolution Equations and Functions of Hypergeometri…
We introduce a natural method of computing antiderivatives of a large class of functions which stems from the observation that the series expansion of an antiderivative differs from the series expansion of the corresponding integrand by…
In the number theory in positive characteristic, there are analogues of some special values introduced by Carlitz, Carlitz gamma values and Carlitz zeta values for instance. Each of them is further developed to arithmetic gamma values and…
In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in…
We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of…
We investigate the abstract Cauchy problem for a quasilinear parabolic equation in a Banach space of the form \( du_t -L_t(u_t)u_t dt = N_t(u_t)dt + F(u_t)\cdot d\mathbf X_t \), where \( \mathbf X\) is a \( \gamma\)-H\"older rough path for…
Classes of an equivalence relation on a module V over a supertropical semiring, called rays, carry the underlaying structure of "supertropical trigonometry" and thereby a version of convex geometry which is compatible with quasilinearity.…
We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the…
In this series of papers we aim to provide a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the…
I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D…
We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…
We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation…
The abstract elliptic and parabolic equations on exterior domain are considered. The equations have top-order variable coefficients. The separability properties of boundary value problems for elliptic equation and well-posedness of the…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely…
In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We…
We present some results and open problems related to expansions of the field of real numbers by hypergeometric and related functions focussing on definability and model completeness questions. In particular, we prove the strong model…
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…
In this paper, we establish a Bloch-type growth theorem for generalized Bloch-type spaces and discuss relationships between Dirichlet-type spaces and Hardy-type spaces on certain classes of complex-valued functions. Then we present some…
Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which…
Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in…