Bent functions are important cryptographic Boolean functions. In order to enumerate eight-variable bent functions, we solve the following three key problems. Firstly, under the action of AGL(7, 2), we almost completely classify R(4, 7)/R(2, 7). Secondly, we construct all seven-variable plateaued functions from the orbits of R(4, 7)/R(2, 7). Thirdly, we present a fast algorithm to expand plateau...

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (198...

In this paper we present an overview of the implications of our previously derived results for positive definite kernels on the general theory of positive definite functions. We begin by exploring the consequences of a set of differential inequalities on the global behaviour of a smooth positive definite function of one real variable. Then we propose a natural extension of this study to the com...

axioms for a space on which we can define integration. Furthermore, the set of such functions is an algebra containing 1 and which separates points, so is dense in C(Ω) by the Stone-Weierstrass theorem. Let us call the space of such functions Cfin(Ω). If we define an integral I on Cfin(Ω) then, by the Stone-Weierstrass theorem it extends to C(Ω) and therefore, by the Riesz representation theore...

The sum. difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propos...

We give sign-reversing involution proofs of a pair of two variable Pfaffian identities. Applications to symmetric function theory are given, including identities relating Pfaffians and Schur functions. As a corollary we are able to compute the plethysm p2 ° skn •

Macdonald defined in [M1] a remarkable class of symmetric polynomials Pλ(x; q, t) which depend on two parameters and interpolate between many families of classical symmetric polynomials. For example Pλ(x; t) = Pλ(x; 0, t) are the Hall-Littlewood polynomials which themselves specialize for t = 0 to Schur functions sλ. Also Jack polynomials arise by taking q = t and letting t tend to 1. The Hall-...

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the deri...

In this paper we develop with considerable details a theory of multivector functions of a p-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., the A-directional derivative (where A is a p-vector) and the generalized concepts of curl, divergence and gradie...