We compute the rst N coeecients of the reciprocal r(x) of a given polynomial p(x), (r(x)p(x) = 1 mod x N , p(0) 6 = 0), by using, under the PRAM arithmetic models , O(h log N) time-steps and O((N=h)(1 + 2 ?h log (h) N)) processors, for any h, h = 1; 2; : : : ; log N, provided that O(log m) steps and m processors suuce to perform DFT on m points and that log (0)

Cooperating animals frequently show closely coordinated behaviours organized by a continuous flow of information between interacting partners. Such real-time coaction is not captured by the iterated prisoner's dilemma and other discrete-time reciprocal cooperation games, which inherently feature a delay in information exchange. Here, we study the evolution of cooperation when individuals can dy...

Let \(\gamma_{-1}\) be the absolutely continuous measure on \(\mathbf{R}^n\) whose density is reciprocal of a Gaussian function. further \(\mathscr{A}\) natural self-adjoint Laplacian \(L^2(\gamma_{-1})\). In this paper, we prove that Riesz transforms associated with order one or two are weak type \((1,1)\), but those higher not.

Using the Stickelberger-Swan theorem, the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field will be hereby characterized. It will be shown that in the case of binary fields such a characterization can be presented in terms of the exponents of the monomials of the self-reciprocal polynomial.

Let Hm(z) be a sequence of polynomials whose generating function ∑∞ m=0 Hm(z)t m is the reciprocal of a bivariate polynomial D(t, z). We show that in the three cases D(t, z) = 1 + B(z)t + A(z)t, D(t, z) = 1 + B(z)t + A(z)t and D(t, z) = 1 + B(z)t + A(z)t, where A(z) and B(z) are any polynomials in z with complex coefficients, the roots of Hm(z) lie on a portion of a real algebraic curve whose e...

Many applications give rise to structured, in particular T-palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P (λ)x = 0, where P (λ) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P (λ) due to the palindromicity are preserved. I...

Many applications give rise to structured, in particular T-palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P (λ)x = 0, where P (λ) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P (λ) due to the palindromicity are preserved. I...

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, . . . , mr&minus1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are co...