AdigraphC is called a zero divisor if there exist non-isomorphic digraphs A and B for which A×C ∼= B ×C , where the operation is the direct product. In other words,C being a zero divisormeans that cancellation property A×C ∼= B×C ⇒ A ∼= B fails. Lovász proved that C is a zero divisor if and only if it admits a homomorphism into a disjoint union of directed cycles of prime lengths.Thus any digra...

Let p be a prime and let φ ∈ Zp[x1, x2, . . . , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a φ-zero sequence if φ(T ) = 0; a sequence in Zp is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Define g(φ, Zp) to be the smallest integer l such that every sequence in Zp of length l contains a φ-zero seque...

Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentially nilpotent over F with a special emphasis on the case that F = Zp is a finite field. A necessary condition on F is observed for...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...

We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that (x; q; a) ...

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with non-constant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T ·Z[T ]. For any N > 1 invertible in K and finite subgroup T ⊂ E(K) of order N , we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic r...

We define minimal atomic complexes and irreducible complexes, and we prove that they are the same. The irreducible complexes admit homological characterizations that make them easy to recognize. These concepts apply both to spaces and to spectra. On the spectrum level, our characterizations allow us to show that such familiar spectra as ko, eo2, and BoP at the prime 2, all BP 〈n〉 at any prime p...

Let Fq be a finite field with q elements, ψ a non-zero element of Fq, and n an integer ≥ 3 prime to q. The aim of this article is to show that the zeta function of the projective variety over Fq defined by Xψ : x n 1 + · · · + x n n − nψx1 . . . xn = 0 has, when n is prime and Xψ is non singular (i.e. when ψ 6= 1), an explicit decomposition in factors coming from affine varieties of odd dimensi...

Policies that support entrepreneurship are keys to the generation of new business. In Brazil, seed capital, installation of technology parks, programs and zero interest financing, economic subsidy as Program First Innovative Company (PRIME) are examples of incentive policies. For the implementation of PRIME, in particular the Brazilian Innovation Agency (FINEP) decentralized operationalization ...