Related papers: Introducing an Analysis in Finite Fields
This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
Ultrafunctions are a particular class of functions defined on a Non Archimedean field E. They have been introduced and studied in some previous works. In this paper we develop the notion of fine ultrafunctions which improves the older…
A new transform over finite fields, the finite field Hartley transform (FFHT), was recently introduced and a number of promising applications on the design of efficient multiple access systems and multilevel spread spectrum sequences were…
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear…
All squigonometric functions admit derivatives that can be expressed as polynomials of the squine and cosquine. We introduce a general framework that allows us to determine these polynomials recursively. We also provide an explicit formula…
The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…
We study cyclic finite Galois extensions of the rational function field of the projective line P^{1}(F_q) over a finite field F_q with q elements defined by considering quotient curves by finite subgroups of the projective linear group…
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.
We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).
We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are…
This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\bK$, one looks for a smoothness basis for $\bK^*$…
Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of…
The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same…
We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…
From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries…
Recently, a convergent series employing a non-Gaussian initial approximation was constructed and shown to be an effective computational tool for the finite size lattice models with a polynomial interaction. Here we numerically examine the…
In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes…