Fractal mathematical over extended finite fields Fp[x]/(f(x))

In this paper, we have defined an algorithm for the construction of iterative operations, based on dimensional projections and correspondence between the properties of extended fields, with respect to modular reduction. For a field with product operations R(x) ⊗ D(x), over finite fields, GF[(pm)n−k]. With Gp[x]/(g(f(x)), whence the coefficient of the g(x) is replaced after a modular reduction operation, with characteristic p. Thus, the reduced coefficients of the generating polynomial of G contain embedded the modular reduction and thus simplify operations that contain basic finite fields. The algorithm describes the process of construction of the GF multiplier, it can start at any stage of LFSR; it is shift the sequence of operation, from this point on, thanks to the concurrent adaptation, to optimize the energy consumption of the GF iterative multiplier circuit, we can claim that this method is more efficient. From this, it was realized the mathematical formalization of the characteristics of the iterative operations on the extended finite fields has been developed, we are applying a algorithm several times over the coefficients in the smaller field and then in the extended field, concurrent form.

A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.

A General Threshold Signature Scheme Based on Elliptic Curve

Based on Elliptic Curve cryptosystem, a threshold signature scheme characterized by (k,l) joint verification for (t,n) signature is put forward. After being signed by a signer company employing (t, n) threshold signature scheme, the informationmis transmitted to a particular verifier company, and then the signature is verified through the cooperation ofkones from the verifier company withlmembers, so as to realize a directional transmission between different companies. Finally, the application examples of the company encryption communication system, the generating polynomial of company private key and public key were given. The security of this scheme is based on Shamir threshold scheme and Elliptic Curve system, and due to the advantages of Elliptic Curve, the scheme enjoys wider application in practice.

A Bivariate Matrix Padé-Type Method of the 2-D Filters

In this paper,according to the special generating polynomial, a class of bivariate matrix Padé-type approximation (BMPTA) is given by introducing a bivariate matrix-valued linear functional on the scalar polynomial space.An application in state-space realization of the 2-D filters is also given in the end.

Computing Invariants for Hybrid Systems

This paper address the problem of generating invariants of hybrid systems. We present a new approach, for generating polynomial inequality invariants of hybrid systems through solving semi-algebraic systems and quantifier elimination. From the preliminary experiment results, we demonstrate the feasibility of our approach.

Relative equilibria of point vortices and the fundamental theorem of algebra

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.

Polytool: Polynomial interpretations as a basis for termination analysis of logic programs

Our goal is to study the feasibility of porting termination analysis techniques developed for one programming paradigm to another paradigm. In this paper, we show how to adapt termination analysis techniques based on polynomial interpretations—very well known in the context of term rewrite systems—to obtain new (nontransformational) termination analysis techniques for definite logic programs (LPs). This leads to an approach that can be seen as a direct generalization of the traditional techniques in termination analysis of LPs, where linear norms and level mappings are used. Our extension generalizes these to arbitrary polynomials. We extend a number of standard concepts and results on termination analysis to the context of polynomial interpretations. We also propose a constraint-based approach for automatically generating polynomial interpretations that satisfy the termination conditions. Based on this approach, we implemented a new tool, called Polytool, for automatic termination analysis of LPs.

The Eulerian distribution on centrosymmetric involutions

Combinatorics International audience We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in S(n) satisfying the property sigma(i) + sigma(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.