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In this paper, the authors have proposed a new approach for generating multi-value sequence by utilizing primitive element, trace function, and k-th power residue symbol over odd characteristic field. Let p be an odd prime and Fp be an odd characteristic prime field, m be the degree of the primitive polynomial f(x), and k be a prime factor of p-1. In details, the procedure for generating multi-value sequence is as follows: primitive polynomial f(x) generates maximum length vector sequence, then trace function Tr(.) maps an element of extension field Fpm to an element of prime field Fp, next non-zero scalar A 2 Fp is added to the trace value, and finally k-th power residue symbol is utilized to map the scalars into (k + 1) values multi-value sequence. In this method, a certain mapping function Mk(.) is utilized during the autocorrelation calculation. Hence, our proposed multi-value sequence has some parameters such as p, m, k, and A. This paper discusses the period, autocorrelation, and cross-correlation properties of proposed multi-value sequence based on some experimental results.

International audience; We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is O_B(n^4 D^4 + n^2 D^4 τ), where n is the number of variables, D equals the bilinear Bezout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.

Let p be an odd prime number as the characteristic of finite field, this paper introduces a pseudo random binary sequence generated by a primitive polynomial of degree 2 over Fp, trace function, and Legendre symbol. Then, its period and autocorrelation are introduced with a small example.

The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah-Jones problem after \cite{AJ}) is to determine a (preferably optimal) integer $n_D$ such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension $n_D$, where $D$ denotes a tuple of integers called the "degree" of the algebraic maps and $n_D\to\infty$ as $D\to\infty$. In this paper we investigate this problem in the case when $X$ is a real projective space and $Y$ is a smooth compact toric variety.

By analogy to gauge field theory on principal bundles (Chap. 8), we develop SUSY gauge theory as theory of graded principal connections on principal graded bundles.

A primitive completely normal element for the extension E / F of Galois fields is a generator of the multiplicative group of E, which simultaneously is normal in E / K for every intermediate field K of E / F. Letting $$F=\mathbb {F}_q$$F=Fq und $$E=\mathbb {F}_{q^n}$$E=Fqn, we denote by $$P_n(q)$$Pn(q) the number of primitive elements of E, and by $$PCN_n(q)$$PCNn(q) the number of primitive completely normal elements for E / F. We prove that $$PCN_n(q) >0$$PCNn(q)>0 provided that $$\ln (2) q \ge (t(n)-1)\cdot (\ln (2)+n\ln (q))$$ln(2)q?(t(n)-1)·(ln(2)+nln(q)), where $$t(n):=\sum _{d|n}d$$t(n):=?d|nd denotes the sum of all positive divisors of n and $$\ln $$ln is the natural logarithm. Based on this estimate, we are able to prove that, for every fixed n, the quotient $$PCN_n(q)/P_n(q)$$PCNn(q)/Pn(q) converges to 1 as q goes to infinity. We also achieve effective versions of the first result: $$PCN_n(q)>0$$PCNn(q)>0 whenever $$q\ge n^{7/2}$$q?n7/2 and $$n\ge 7$$n?7, or when $$q\ge n^3$$q?n3 and $$n\ge 37$$n?37. The paper is self-contained and the proofs rely only on elementary combinatorics and number theory, as well as some basic calculus.

Let [Formula: see text] be a free Lie algebra of finite rank [Formula: see text] [Formula: see text]. We give another proof of the following criterion which is proven by Mikhalev and Zolotykh, using the idea of [Formula: see text]-primitivity: An endomorphism of [Formula: see text] preserving primitivity of elements is an automorphism.

Let E′ be a set of representatives of the cyclotomic cosets modulo 2 m + 1 for which each coset has the maximal size n. Let \(f_{a_{r}}\) be the function defined on \(\mathbb{F}_{2^{n}}\) by