Solvability of systems of polynomial congruences modulo a large prime

We consider the following polynomial congruences problem: given a prime p, and a set of polynomials of total degree at most d, solve the system for solution(s) in . We give a randomized algorithm for the decision version of this problem. For a fixed number of variables, the sequential version of the algorithm has expected time complexity polynomial in d, m and log p; the parallel version has expected time complexity polylogarithmic in d, m and p, using a number of processors, polynomial in d, m and log p. The only point which prevents the algorithm from being deterministic is the lack of a deterministic polynomial time algorithm for factoring univariate polynomials over . Received: April 9, 1997.     

Adaptive finite-time control of uncertain nonlinear systems with the powers of odd rational numbers

This paper studies adaptive finite-time control problem of a class of nonlinear systems with dynamic and parametric uncertainties. The power of systems in this paper is any positive odd rational number but not necessary equal to or greater than one. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterise the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, adding a power integrator, adaptive control and FTISS methods, an adaptive state feedback controller is designed to drive the states to the origin in finite time while maintaining all the closed-loop signals bounded.     

The systems of primitive roots the degree and rank of prime numbers

Using the notions of the rank and the degree of a prime p≠2 relative to a residue class g(modp), where g is a positive integer with (g,p)= 1, we study the systems of primitive roots. This study leads to the understanding of the behavior of the discrete periodical signals whose values are the digits of the inverses of integers expressed in an arithmetic system with base g.     

On Hadamard powers of polynomials

We introduce the concept of the Hadamard power of a polynomial formed by real powers of its coefficients. We show that the Hadamard power of a Hurwitz polynomial remains Hurwitz.     

Trace representation and linear complexity of binary sequences derived from Fermat quotients

We describe the trace representations of two families of binary sequences derived from the Fermat quotients modulo an odd prime p （one is the binary threshold sequences and the other is the Legendre Fermat quotient sequences） by determining the defining pairs of all binary characteristic sequences of cosets, which coincide with the sets of pre-images modulo p2 of each fixed value of Fermat quotients. From the defining pairs, we can obtain an earlier result of linear complexity for the binary threshold sequences and a new result of linear complexity for the Legendre Fermat quotient sequences under the assumption of 2p-1≠ 1 mod p2.     

On the arithmetic complexity of matrix Kronecker powers

In this paper we study the arithmetic complexity of computing the pth Kronecker power of an n × n matrix. We first analyze a straightforward inductive computation which requires an asymptotic average of p multiplications and p – 1 additions per computed output. We then apply efficient methods for matrix multiplication to obtain an algorithm that achieves the optimal rate of one multiplication per output at the expense of increasing the number of additions, and an algorithm that requires O(log p) multiplications and O(log²p) additions per output.     

On a special class of primitive words

When representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its Watson-Crick complement θ(u), where θ denotes the Watson-Crick complementarity function. Thus, an expression which involves only a word u and its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called pseudo-primitive words relative to θ or simply θ-primitive words, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique θ-primitive root of a given word, and we give some constraints forcing two distinct words to share their θ-primitive root. Also, we present an extension of the well-known Fine and Wilf theorem, for which we give an optimal bound.     

About a property of a class of linear time-varying systems derived from the controllability and observability matrices

This correspondence deals with an application of a theorem of Silverman and Meadows which specifies conditions forntime-variablen-dimensional vectors to be linearly dependent. From this theorem conditions are derived which allow one to verify whether a linear time-varying system admits a realization in which either the state matrixAand the state-input matrixB, or the state matrixAand the state-output matrixC, are constant.     

On an integral of powers of hypergeometric or trigonometric functions

Expressions for the numerical evaluation of the integrals(1), (2) for p∈N, λ∈R are given, which generalize recent results obtained by Fettis using a different method. In this context, observations on some of the formulas given in [4] of the Bibliography are made and an extension is suggested.     

On the stabilizing property of SIORHC

It is shown that the stabilizing property of SIORHC (stabilizing I/O receding horizon control) holds for general stabilizable discrete-time linear plants irrespective of the condition that the plant has no poles at the origin.     

On some entropy functionals derived from Rényi information divergence

We consider the maximum entropy problems associated with Rényi Q-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized.The Rényi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure Q(x) and recover in a limit case some well-known entropies.