We show that every language accepted by a nondeterministic auxiliary pushdown automaton in polynomial time (that is, every language in SAC 1 = Log(CFL)) can be accepted by a symmetric auxiliary pushdown automaton in polynomial time. Keywords-Symmetric Computation, Auxiliary Pushdown Automata, LogCFL, Reversible Computation

Abstract Given an elliptic curve E and a finite subgroup G, Vélu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of IG(P ) and the first elementary symmetric polynomial on the abscissas of th...

We report on some initial results of a brute-force search for determining the maximum correlation between degree-d polynomials modulo p and the n-bit mod q function. For various settings of the parameters n, d, p, and q, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previouslyanalyzed settings of parameters, where non-symmetric polynomial...

In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F → F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, pa...

The cover polynomial C(D) = C(D;x, y) of a digraph D is a twovariable polynomial whose coefficients are determined by the number of vertex coverings of D by directed paths and cycles. Just as for the Tutte polynomial for undirected graphs (cf. [11, 16]), various properties of D can be read off from the values of C(D;x, y). For example, for an n-vertex digraph D, C(D; 1, 0) is the number of Hami...

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some q-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite cou...

In this paper we prove the Dipper–James conjecture that the centre of the Iwahori–Hecke algebra of type A is the set of symmetric polynomials in the Jucys–Murphy operators. © 2006 Elsevier Inc. All rights reserved.

It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear group. We generalize this result showing that the abelianization of the algebra of the symmetric tensors of fixed order over a free associative algebra is isomo...

We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials Pn(z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments Mq(Pn) of even order q for q 32. We were also able to check a conjecture on the asymptotic behavior of Mq(Pn), namely Mq(Pn) ∼ Cq2, where Cq = 2...

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1, x2, . . .]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard...