Volume 58, Issue 5

A class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ1 and ℱ2 the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ1 and ℱ2 are partially ordered by the class-theoretical inclusion. We prove that ℱ1 satisfies the infinite distributivity law and that ℱ2 is isomorphic to a principal ideal of ℱ1.

In the paper, there are proved some properties of the asymptotic density using the permutations.

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations $$ \bar x $$ = φ(x), ȳ = ȳ($$ \bar x $$) = L(x) + y(x), $$ \bar z $$ = $$ \bar z $$($$ \bar x $$) = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed.

In this paper, we obtain the necessary and sufficient conditions for having the maximum principle and existence of positive solutions for some cooperative systems involving Schrödinger operators defined on unbounded domains. Then, we deduce the existence of solutions for semi-linear systems. Finally we discuss the generalized maximum principle (gf q-positivity) for non cooperative systems.

Oscillatory and asymptotic behaviour of solutions of a class of fourth order nonlinear neutral difference equations of the form $$ \Delta ^2 (r(n)\Delta ^2 (y(n) + p(n)y(n - m))) + q(n)G(y(n - k)) = 0 $$ and $$ (E) \Delta ^2 (r(n)\Delta ^2 (y(n) + p(n)y(n - m))) + q(n)G(y(n - k)) = f(n) $$ are studied under the assumption $$ \sum\limits_{n = 0}^\infty {\tfrac{n} {{r(n)}}} $$ < ∞, for different ranges of p(n). Sufficient conditions are obtained for the existence of positive bounded solutions of (E).

The idea of I-convergence was introduced by Kostyrko et al (2001) and also independently by Nuray and Ruckle (2000) (who called it generalized statistical convergence) as a generalization of statistical convergence (Fast (1951), Schoenberg(1959)). For the last few years, study of these convergences of sequences has become one of the most active areas of research in classical Analysis. In 2003 Muresaleen and Edely introduced the concept of statistical convergence of double sequences. In this paper we consider the notions of I and I*-convergence of double sequences in real line as well as in general metric spaces. We primarily study the inter-relationship between these two types of convergence and then investigate the category and porosity position of bounded I and I*-convergent double sequences.

In this article we study different properties of convergent, null and bounded sequence spaces of fuzzy real numbers defined by an Orlicz function, like completeness, solidness, symmetricity, convergence free etc. We prove some inclusion results, too.

We show that a 2-homogeneous polynomial on the complex Banach space c 0 l 2i) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c 0(l 2i).

In the paper the unknown distribution function is approximated with a known distribution function by means of Taylor expansion. For this approximation a new matrix operation — matrix integral — is introduced and studied in [PIHLAK, M.: Matrix integral, Linear Algebra Appl. 388 (2004), 315–325]. The approximation is applied in the bivariate case when the unknown distribution function is approximated with normal distribution function. An example on simulated data is also given.