The enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

Many — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

The purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

Let X be a random variable whose frequency function is

1.1

Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ${{\mathbb{R}}^{n}}$.

Let K0 and K1 be equational classes of algebras of the same type. The smallest equational class K containing K0 and K1 is the join of K0 and K1; in notation, K = K0 ∨ K1. The direct product K0 × K1 is the class of all algebras α which are isomorphic to an algebra of the form a0 × a1, a0 ∈ K1. Naturally, K0 × K1 ⊆ K0 ∨ K1, Our first theorem states a very simple condition under which K0 × K1 = K0 ∨ K1, and an additional condition under which the representation α ∨ a0 × a1 unique.

The convolution sum $\sum{_{m<n/16}\,\sigma (m)\sigma (n\,}-16m)$ is evaluated for all $n\,\in \,\mathbb{N}$. This evaluation is used to determine the number of representations of $n$ by the quadratic form $x_{1}^{2}\,+\,x_{2}^{2}\,+\,x_{3}^{2}\,+\,x_{4}^{2}\,+\,4x_{5}^{2}\,+\,4x_{6}^{2}\,+\,4x_{7}^{2}\,+\,4x_{8}^{2}$.

In this paper the author prove that standard modules of classical groups whose Langlands quotients are generic are irreducible. This establishes a conjecture of Casselman and Shahidi for this important class of groups.

We obtain a representation for the supercritical Dawson-Watanabe superprocessin terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems.

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

We prove the "folklore" results that both the attractor A and invariant measure μ of an N-map Iterated Function System (IFS) vary continuously with variations in the contractive IFS maps as well as the probabilities. This represents a generalization of Barnsley's result showing the continuity of attractors with respect to variations of a parameter appearing in the IFS maps. Some applications are presented, including approximations of attractors and invariant measures of nonlinear IFS, as well as some novel approximations of Julia sets for quadratic complex maps.

Fieldhouse observed that any finitely presented left R-module P is projective with respect to pure exact sequences, i.e.

can always be completed to a commutative diagram when the sequence is pure exact. A left R-module A is absolutely pure if it is a pure submodule of every module which contains it.

In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the $\text{S}$-curvature. We show some relationships among the flag curvature, the $\text{S}$-curvature, and the new non-Riemannian quantity.

For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.

G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

A base a pseudoprime is an integer n such that

1

A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a Carmichael number, then n is the product of k(>2) distinct primes P1,P2,P3, … Pk, and Pi-1|n-1(i=1, 2, 3, …, k).

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.

Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,

(1)

Equality holds only if x(t) = Kt for some constant K.

In this paper some cardinal inequalities for Urysohn spaces are established. In particular the following two theorems are proved:

(i)If where [A]θ denotes the θ-closed hull of A, i.e., the smallest θ-closed subset of X containing A;

(ii), where aL(X, X) is the smallest cardinal number m such that for every open cover of X there is a subfamily for which

We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.