The first general integrability criterion is due to Riemann. He observed that a necessary and sufficient condition for a function f(x) on [a, b] to be integrable in his sense is that for each ε > 0 there exists a δ > 0 so that

where {x0, x1, …, xn} is a partition of [a, b] with diameter less than δ (and ω(f, I) denotes the oscillation of f on the interval I).

Let f be a 2π-periodic function defined on [0, 2π]. We set

where suprema have been taken with respect to all partitions P:a = t0<t1<t2<…<tn<=b of any segment [a, b] contained in [0, 2π]. We call the pth total variation of f on [a, b]. If we denote pth total variaiton of f on [0, 2π] by Vp(f), then we can define Wiener's class simply by

The permanent of an n ✕ n matrix A = (aij) is the matrix function

1

where the summation is over all permutations in the symmetric group, Sn. An n ✕ n matrix A is a circulant if there are scalars a1 …, an such that

2

where P is the n ✕ n permutation matrix corresponding to the cycle (12 … n) in Sn.

Recall that an automorphism ϕ of an algebra A is called regular if, for all x∊A, ϕ(x) = x implies x = 0. For various types of algebras it is known that the existence of a regular automorphism implies that the algebra is radical. Here, for associative algebras, we relax the restrictions on the fixed points so that all automorphisms of commutative algebras immediately satisfy the hypothesis on the fixed points.

This paper is concerned with the infinite geometric products

and their generalizations to higher dimensions. Some new expressions and identities are derived for these products by using stochastic theory. The function is tabulated for p = 0(0.01)1.

Let Sn denote the permutation group on {1, 2, 3, …, n}. Let k ≥ 1 and n ≥ 1 be integers. For σ ∊ Sn we define ([1]) mk, n: Sn → Z by

Let pn ∊ Sn be the reverse permutation: that is,

In 1957, M. A. Subhankulov established the following identity

where ; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).

In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.

Let n and k be positive integers, k≥3. Denote by ϕ(n, k) the least positive integer such that if F is any family of more than ϕ(n, k) sets, each set with n elements, then some k members of F have pairwise the same intersection. In this paper we obtain a new asymptotic upper bound for ϕ(n, k), k fixed, n approaching infinity.

Dans un récent article [1], T. M. Apostol obtient une identité liant et , où f est une fonction complètement multiplicative, |f(n)| ≤ l, et p est un nombre premier. Il utilise cette identité pour prouver que

pour tout nombre premier p. Par la suite, T. M. Apostol obtenait d'autres identités [2].

It is shown that R is a biregular near-ring if and only if it is isomorphic with the near-ring of sections of a sheaf of reduced near-rings over a Boolean space. Also, some ideal properties of a biregular near-ring are proved. These are considered as generalizations of some works of R. Pierce on biregular rings.

In 1927, Jackson [5] obtained a transformation connecting a

where N is any integer, with a

viz.,

1

where | q | > l and |qγ-α-βN| > l.

Modules with finite spanning dimension were defined by P. Fleury [3] in an attempt to dualize the concept of Goldie dimension. In this note we study these modules in some detail, obtain an improved structure theorem for them and also extend the work done in [2] and [3]. Projective modules with finite spanning dimension turn out to be local or artinian.

It is well known (see [5]) that the maximal right quotient ring of a ring R is (von Neumann) regular if and only if R is (right) non-singular (every large right ideal is dense). In [8] it was shown that for a semigroup S, the regularity of Q(S), the maximal right quotient semigroup [7], is independent of the non-singularity of S. Nevertheless, right non-singular semigroups form an important class of semigroups.

Let be a finite measure space, and let T be a contraction in real Lp(X). (i.e. T is linear and ||T||≤1). It is said that the Dominated Ergodic Theorem holds for T, if there exists a constant cp such that, if M(T)f(x) = supn 1/n then ||M(T)f||p ≤ cp ||f||p for every f in Lp.

This paper presents variants of the Holder inequality for integrals of functions (as well as for sums of real numbers) and its inverses. In these contexts, all possible transliterations and some extensions to more than two functions are also mentioned.

As a contribution to various investigations [1-11] about packing of convex bodies with certain conditions imposed on the number of neighbours of each body, V. Chvátal [12] recently proved the following theorem: If in a packing of translates of a square each square has at least six neighbours then the density of the packing is at least 11/15.

One of the nicest theorems of elementary number theory is the two-square theorem of Fermat.

A prime p is expressible as the sum of 2 squares if and only if -1 is a quadratic residue mod p.

One of the interesting mathematical problems is whether the system of four Diophantine equations

1

2

has a solution in x, y, z, I, m, n, w. To this day the problem has not been shown to be impossible, nor has it been solved.

‖ A ‖ désignera le cardinal de A. Soient T une théorie complète du premier ordre dans un langage dénombrable, M un modèle de T, X un sous-ensemble de M; soit T(X) l'ensemble des formules closes à paramètres dans X satisfaites par M; on appelle n-type de T sur X un ensemble consistant avec T(X) et maximal de formules à paramètres dans X à n variables libres.

In [3] an attempt was made at proving the following result:

(A) An N-module M over a d.g. near-ring is injective if and only if for each right ideal u of N and each N-homomorphism f:u→M there exists an element mϵM with f(a) = ma for all aϵu.

In this note we present two examples. The first is a counterexample to (A) and the second illustrates one point at which the attempt made in [3] fails.