80 research outputs found
A Non-Archimedean Wave Equation
Let K be a non-Archimedean local field with the normalized absolute value
. It is shown that a ``plane wave'' , where f is a Bruhat-Schwartz complex-valued test function on K,
, ,
satisfies, for any f, a certain homogeneous pseudo-differential equation, an
analog of the classical wave equation. A theory of the Cauchy problem for this
equation is developed.Comment: 17 pages; the final version, to appear in Pacif. J. Mat
Non-Archimedean Group Algebras with Baer Reductions
Within the concept of a non-Archimedean operator algebra with the Baer
reduction (A. N. Kochubei, On some classes of non-Archimedean operator
algebras, Contemporary Math. 596 (2013), 133--148), we consider algebras of
operators on Banach spaces over non-Archimedean fields generated by regular
representations of discrete groups.Comment: Final version, to appear in Algebras and Representation Theor
Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic
We consider a class of partial differential equations with Carlitz
derivatives over a local field of positive characteristic, for which an analog
of the Cauchy problem is well-posed. Equations of such type correspond to
quasi-holonomic modules over the ring of differential operators with Carlitz
derivatives. The above class of equations includes some equations of
hypergeometric type. Building on the work of Thakur, we develop his notion of
the hypergeometric function of the first kind (whose parameters belonged
initially to ) in such a way that it becomes fully an object of the
function field arithmetic, with the variable, parameters and values from the
field of positive characteristic
Fractional-Hyperbolic Systems
We describe a class of evolution systems of linear partial differential
equations with the Caputo-Dzhrbashyan fractional derivative of order in the time variable and the first order derivatives in spatial
variables , which can be considered as a fractional analogue
of the class of hyperbolic systems. For such systems, we construct a
fundamental solution of the Cauchy problem having exponential decay outside the
fractional light cone .Comment: Final version. available at http://link.springer.com/journal/1354
Dwork-Carlitz Exponential and Overconvergence for Additive Functions in Positive Characteristic
We study overconvergence phenomena for -linear functions on a
function field over a finite field . In particular, an analog of
the Dwork exponential is introduced
Analysis and Probability over Infinite Extensions of a Local Field
We consider an infinite extension of a local field of zero characteristic
which is a union of an increasing sequence of finite extensions. is
equipped with an inductive limit topology; its conjugate is a
completion of with respect to a topology given by certain explicitly
written seminorms. We construct and study a Gaussian measure, a Fourier
transform, a fractional differentiation operator and a cadlag Markov process on
. If we deal with Galois extensions then all these objects are
Galois-invariant.Comment: 24 pages, LaTex; to appear in Potential Analysi
Analysis and Probability over Infinite Extensions of a Local Field, II: A Multiplicative Theory
Let be a projective limit, with respect to the renormalized norm
mappings, of the groups of principal units corresponding to a strictly
increasing sequence of finite separable totally and tamely ramified Galois
extensions of a local field. We study the structure of the dual group ,
introduce and investigate a fractional differentiation operator on , and the
corresponding L\'evy process. Part I: Potential Anal., 10 (1999), 305-325.Comment: 11 pages, LaTe
Polylogarithms and a Zeta Function for Finite Places of a Function Field
We introduce and study new versions of polylogarithms and a zeta function on
a completion of at a finite place. The construction is based
on the use of the Carlitz differential equations for -linear
functions.Comment: 15 pages, LaTeX-2
General Fractional Calculus, Evolution Equations, and Renewal Processes
We develop a kind of fractional calculus and theory of relaxation and
diffusion equations associated with operators in the time variable, of the form
where
is a nonnegative locally integrable function. Our results are based on the
theory of complete Bernstein functions. The solution of the Cauchy problem for
the relaxation equation , , proved to be (under some
conditions upon ) continuous on and completely monotone,
appears in the description by Meerschaert, Nane, and Vellaisamy of the process
as a renewal process. Here is the Poisson process of intensity
, is an inverse subordinator.Comment: To appear in Integral Equations and Operator Theor
Differential Equations for F_q-Linear Functions
We study certain classes of equations for -linear functions, which are
the natural function field counterparts of linear ordinary differential
equations. It is shown that, in contrast to both classical and -adic cases,
formal power series solutions have positive radii of convergence near a
singular point of an equation. Algebraic properties of the ring of -linear
differential operators are also studied.Comment: 17 pages, LaTex-2
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