The purpose of this note is to record a few formulas relating the indices of various lattices and sublattices all arising from the decomposition of a euclidean space E into three mutually orthogonal subspaces E = E0 ⊥ E1 ⊥ E2 which are rational with respect to a given lattice Λ ⊂ E. In the case that Λ is unimodular these formulas simplify to give very simple identities between various intersection lattices.

The relation between the subclass X of balanced one-sided ideals, which contains all two-sided ideals, and weakly prime one-sided ideals is considered. If a member P of X is prime, then it is prime in its idealizer I (P). Furthermore, if the ring is left and right noetherain, then I (P) modulo P is a prime Goldie ring.

The following Theorem is proved: Let R be a semi-prime ring in which either (xy)n − xnyn or (xy)n − ynxn is central, for all x,y in R where n > 1 is a fixed integer. Then R is commutative.

In terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

Characterizations of efficiency and proper efficiency are given for classes of multiple objective fractional optimization problems. These results are then applied to the case of multiple objective fractional linear problems. A dual problem is given for the multiple objective fractional problem and it is shown that for a properly efficient primal solution the dual solution is also properly efficient.

We prove the existence of nontrivial periodic solutions of some nonlinear ordinary differential equations with time-dependent coefficients using variational methods.

A closed subset c of [0,∞]×[−∞,∞] is called a barrier if

(i) (∞,x) ∈ C, x,

(ii) (t, ±∞) ∈ C, t,

(iii) (t, x) ∈ C implies (s, x) ∈ C, S ≥ t.

Given a Brownian motion (B (t)) Starting at the origin and a barrier C, let τ(C) be inf{t: (t, B (t)) ∈ C}. A random variable x (or a distribution F) is called achievable if there exists a barrier C so that B (τ(C)) is distributed as x (F). In this paper we shall show that if x is bounded above or below with finite mean or if x has zero mean and E (|x| log+ |x|) < ∞ then x is achievable. This result gives a partial answer to a problem raised by Loynes [7].

In this paper we extend Kurose's structure theorem to characterize a closed derivation in the algebra of continuous functions on the unit square, under the conditions that the range is the whole algebra and the kernel is the set of all functions depend only on the second variable, as a partial derivative with respect to signed measures on the unit square.

It is shown that if the ring A is left Artinian and L1 and L2 are left ideals of AL1 is nilpotent modulo L2 if L1 is nil modulo L2.

In this note we prove the uniqueness of a projection onto a given subspace with strictly contractive complement. We also show that, if one completely contractive projection is invariant under another, then the two commute.

Recently, Ky Fan extended his will known lemma (which is an extension of the classical theorem of Knaster, Kuratowski and Mazurkiewica) to the noncompact case. Using this result, another interesting lemma of Fan is generalized in this paper. As applications of our theorem, we obtain a generalizationof Browder's variational inequality and derive Fan's other recent results directly from our theorem. Also, in this paper, we give a slight extension recent results of K. K. Tan, which themselves are generalizations of many well-known results on minimax and variational inequalities.

For 0 < p < 1, let Sp denote the class of functions f (z) which are meromorphic and univalent in the unit disk U, with the normalisations f (0) = 0, f′(0) = 1 and f (p) = ∞, and let Sp (a) denote subclass of Sp consisting of those functions in Sp whose residue at the pole in equal to a. In this paper, we determine, for values of the residue a in a certain disk Δp, the greatest possible outer area over all functions in the class Sp (a). We also determine additional information concerning extremal function if the reside a dose not lie in Δp.

If S is a (multiplicative) Semigroup and a ∈ S, the binary operation ∘ defined on the set S by x ∘ y = x a y is associative and the resulting semigroup is called a variant of S. We study the congruence a defined on S by saying that two elements are α-related if and only if they determine the same variant of S. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involving D-equivalence).

In this paper we study the additive group structure of subdirectly irreducible rings and their hearts. We give and example of a torsion-free, non-reduced abelian group which is not the underlying additive group of any associative subdirectly irreducible ring. It is a counterexample to a theorem in Feigelstock's book “Additive Groups of Rings.”

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.

We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:

If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

Necessary and sufficient conditions are given for certain classes of eventually regular semigroups to the group-bound or even periodic.

It is shown that a non-discrete locally compact Hausdorff group has each of its proper closed subgroups finite (respectively, discrete) if and only if it is topologically isomorphic to the circle group (respectively, the circle group or the group of real numbers).