This paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.

The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.

Let $\chi $ be a locally compact metric space and let $\mathcal{A}$ be any of the Lipschitz algebras $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, or $\text{lip}_{\alpha }^{0}\,\chi $. In this paper, we show, as a consequence of rather more general results on Banach algebras, that $\mathcal{A}$ is $C$-character amenable if and only if $\chi $ is uniformly discrete.

This note proves another special case of a conjecture of U. Oberst. Oberst considered [1], for any small category , the X abelian category of abelian group-valued functors on , and the X functor Colim: which takes each diagram to its colimit. The question is, when is Colim exact? For its relationships, see [1], It is a sufficient condition that each component of is upward filtered. Oberst conjectured that it is also necessary, and proved this under some conditions. He mentioned particularly the case that is a monoid, i.e. a category with one object. We shall verify the conjecture in that case.

We show that if the family $\mathcal{O}$ of orbits of all vector fields on a subcartesian space $P$ is locally finite and each orbit in $\mathcal{O}$ is locally closed, then $\mathcal{O}$ defines a smooth Whitney A stratification of $P$. We also show that the stratification by orbit type of the space of orbits $M/G$ of a proper action of a Lie group $G$ on a smooth manifold $M$ is given by orbits of the family of all vector fields on $M/G$.

Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

We study the representations of extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ where $q$ is $N$-th primitive root of unity ($({{\mathbb{C}}_{q}}$ is the quantum torus in two variables). We first prove that $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ for a suitable number of copies is a quotient of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Thus any finite dimensional irreducible module for $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$lifts to a representation of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Conversely, we prove that any finite dimensional irreducible module for $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ comes from above. We then construct modules for the extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)\oplus \mathbb{C}{{d}_{1}}\oplus \mathbb{C}{{d}_{2}}$ which is integrable and has finite dimensional weight spaces.

A class of weakly coupled systems of semilinear elliptic partial differential equations is considered in an exterior domain in ℝN, N > 3. Necessary and sufficient conditions are given for the existence of a positive solution (componentwise) with the asymptotic decay u(x) = O(|x|2-N) as |x| —> ∞. Additional results concern the existence and structure of positive solutions u with finite energy in a neighbourhood of infinity.

We are concerned with the nth-order linear differential equation

1

where the coefficients are continuous. Aliev [1, 2] showed, in papers unavailable to the author that for n = 4

(see Definition 2). Theorems 1 and 5 give respectively nth-order generalizations of these two results.

Let p > l, r≠1, and let f(x) be a non-negative function defined in [0, ∞). The following inequality is due to G. H. Hardy [5, Ch. IX]:

1.1

where according as r>1 or r < l.

A Room design of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square of side 2n - 1, so that each of the (2n - 1)2 cells of the array is either empty or contains just two distinct objects; each of the 2n objects occurs just once in each row and in each column; and each (unordered) pair of objects occurs in just one cell.

We define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that

(1)

(2)

In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined by

where the summation is over a square-reduced residue system (mod r).

B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and

(3)

As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

Let G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

Let X and Y be metrizable compact spaces and μ and v be nonzero continuous measures on X and Y, respectively. Then there is no bounded operator from the space of bimeasures BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1 (μ X v); in particular, if X and Fare nondiscrete locally compact groups, then there is no bounded projection from BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1(X X Y).

Consider a life testing experiment in which (Xl, X2,…,Xn) are such that (n—1) of them are distributed as f(x, σ)=(l/σ)e-x/σ, x ≥ σ, > 0 and one of them is distributed as f(x, σ/α), 0 < α< l. A priori each Xi has probability 1/n of being a spurious observation distributed as f (x,σ/α). For such an experiment Kale and Sinha [2] showed that if ur denotes the probability that X(r), the rth component of the order statistic, corresponds to the spurious observation, then u1 < u2 < … < un.

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.

Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.

Nous considérons dans cette note une classe d′équations différentielles fonctionnelles linéaires dont les coefficients sont analytiques; des résultats sur l′ordre des solutions et leur comportement asymptotique seront obtenus.

Il s'agit d'une généralisation à des équations du deuxiéme ordre d'une propriété d'équations d'ordre un [2]: la démonstration du théoréme principal demeure valable dans ce dernier cas.

Completely positive maps defined by an irreducible correspondence between two von Neumann algebras M and N are introduced. We give results about their structure and characterize, among them, those which are extreme points in the convex set of all unital completely positive maps from M to N. As particular cases we obtain known results of M. D. Choi [4] on completely positive maps between complex matrices and of J. A. Mingo [8] on inner completely positive maps.

Higher dimensional mixed norm type inequalities involving certain integral operators are characterized in terms of the corresponding lower dimensional inequalities.

Let ${{a}_{1}},...,{{a}_{9}}$ be nonzero integers and $n$ any integer. Suppose that ${{a}_{1}}+\cdot \cdot \cdot +{{a}_{9}}\,\equiv \,n\,\left( \bmod \,2 \right)$ and $\left( {{a}_{i}},\,{{a}_{j}} \right)\,=\,1$ for $1\,\le \,i\,<\,j\,\le \,9$. In this paper we prove the following:

(i) If ${{a}_{j}}$ are not all of the same sign, then the cubic equation ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ has prime solutions satisfying ${{p}_{j}}\,\ll \,{{\left| n \right|}^{{1}/{3}\;}}\,+\,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{14+\varepsilon }}$.

(ii) If all ${{a}_{j}}$ are positive and $n\,\gg \,\max {{\left\{ \left| {{a}_{j}} \right| \right\}}^{43+\varepsilon }}$, then ${{a}_{1}}p_{1}^{3}\,+\cdot \cdot \cdot +\,{{a}_{9}}\,p_{9}^{3}\,=\,n$ is solvable in primes ${{p}_{j}}$.

These results are an extension of the linear and quadratic relative problems.