Bachelier (1900), Einstein (1905) and Smoluchowski (1915) provided a theory of the peculiar erratic motion of small particles suspended in a liquid, first described in 1826 by the English botanist Brown. In a series of papers beginning in 1920 Wiener undertook a mathematical analysis of Brownian motion.

It is well known that

1

where An denotes the number of "up-down" or alternating permutations

2

of 1, 2, …, n. The numbers A2n and A2n+1 are known as the secant and tangent numbers respectively and A2n = (—l)"E2n , where En is the Euler number.

In this paper we construct the hitting time distributions for stochastic processes Xk , taking on values amongst the integers 0, 1, …, d -1 for which has a smooth polynomial density with respect to the Lebesgue measure on [0,1].

An arithmetic function f is said to be multiplicative if f(mn) = f(m)f(n), whenever (m, n) = 1 and f(1) = 1. The Dirichlet convolution of two arithmetic functions f and g, denoted by f • g, is defined by f • g(n) = Σd|n f(d)g(n/d). Let w(n) denote the product of the distinct prime factors of n, with w(l) = 1. R.

Abstract. Let denote the graph (k times) where is the strong product of the two graphs G and H. In this paper we prove the conjecture of J. Zaks [3]: For every connected graph G with at least two vertices there exists an integer k = k(G) for which the graph is hamiltonian.

In [A1] is defined a class of Markov operators on C(X) (X compact T 2), called Generalized Averaging Operators (g.a.o.) which yield an easy solution to the following problem: given a fixed Markov operator T, find necessary and sufficient conditions on any other Markov operator R for the relation ker T ⊂ker R to hold. The main application of this is to inclusion relations between matrix summability methods.

Dans cet article, nous établirons un théorème sur la représentation de certaines séries de Dirichlet (ceci généralise un résultat de [3]). Avec ce nouveau résultat nous donnerons seulement quelques applications qui évidemment pourraient être augmentées de beaucoup. Par exemple, nous donnerons une expression asymptotique pour

1

The main result in this note is the following

Theorem: Let G be a finite solvable group. There exists a permutation σ of the set G such that {g • σ(g); g∈G} = G if and only if the Sylow 2-subgroup of G is non-cyclic or trivial

In the study of particular categories of integral domains, it has proved useful to know which overrings of the domains of interest lie within the category, and indeed whether all such overrings do. (Recall: an overring of R is a ring T with R ⊆ T ⊆ quotient field of R.) Two classes of domains classically studied in this setting are Prüfer domains and one-dimensional Noetherian domains. Since both of these classes are contained in the category of coherent domains, it is natural to investigate this category in this setting.

In [2], Herstein proves the following result:

Theorem. Let R be a prime ring, d≠0 a derivation of R such that d(x) d(y) = d(y) d(x) for all x, y ∈ R. Then, if char r≠2, R is commutative, and if char R = 2, R is commutative or an order in a simple algebra which is 4-dimensional over its center.

Let {ank} be a matrix defined by

1

and n taking only non-negative integer values.

Let f(x) ∈ L [0, 2π] and be periodic with period 2π outside this interval. Let the Fourier series associated with the function f(x) be given by

and let

where s is a constant.

A surface of order three F in the real projective three-space P 3 is met by every line, not in F, in at most three points.

In the present paper, we determine the existence and examine the distribution of elliptic, parabolic and hyperbolic points; that is, the differentiable points of F which do not lie on any line contained in F.

Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M2 be the maximum of the absolute values of the r × r minors of the augmented matrix (A | B). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M2. (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then .

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximation

and the extremal transformations U whenever they exist.

Without appealing to the Cauchy theorem or its corollaries, it is proved that the real and imaginary parts of a non-constant complex-valued analytic function of several complex variables are functionally independent. This unifies and generalizes some results sporadically treated in standard treatises on function theory.

An upper bound for P[Σ Xi≥tσ, Σ Yi ≥ tσ], where (Xi, Yi ), i = 1, 2, …, n are bounded independent random variables, was given by Mullen (1973). An improvement to the bound is possible without further assumptions.

Let U be a bounded open subset of the complex plane. By a well known result of A. M. Davie, C(bU) is the uniformly-closed linear span of A(U) and the powers (z-zi )-n , n = 1, 2, 3, … with zi a point in each component of U. We show that if A(U) is a Dirichlet algebra and bU is of infinite length, then one power of (z - zi ) is superfluous.

We extend the following result to several variables: For any sequence {Nj }, we have C{Nj } = C{Mj }, with {Mj } logarithmically convex i.e.