Volume 47, Issue 2

It is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

In this note we prove that if two implication algebras have isomorphic monoids of endomorphisms then they are isomorphic.

One of the pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Γ 0 ⋊ Γ 1 and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid Γ 1 and prove that it is isomorphic to the convolutive algebra of the groupoid Γ 0 ⋊ Γ 1 . We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.

We prove that the solution of the cyclic initial value problem u’ k = 1/2 - u k /(u k+1 + u k+2 ) (k Z/nZ), u(0)= x is convergent to an equilibrium μ (x) (1,…,1), and study the properties of the function x → μ(x) and its relation to Shapiro’s inequality.

On account of analytical importance or application in certain problems in radiation physics and nuclear technology, several interesting families of elliptic-type integrals were recently studied by many authors. The aim and objective of present paper is to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. In addition, to deriving known and various new elliptic-type integrals and their generalizations, these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.

In this paper, we study local asymptotic stability, global character and periodic nature of solutions of the system of rational difference equations given by x n +1= , y n =, n=0, 1,…, where the parameters a; b; c; d; e; f ∊ (0; ∞), and with initial conditions x 0 ; y 0 ∊ (0; ∞). Some numerical examples are given to illustrate our results.

In the present paper, we investigate the convergence and the approximation order of some Durrmeyer type operators in exponential weighted space. Furthermore, we obtain the Voronovskaya type theorem for these operators.

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev’s crossed product by an endomorphism, and is related to Exel’s interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.

In this paper, we obtain a couple of weak convergence results for nonself nearly asymptotically nonexpansive mappings. Our first result is for the Banach spaces satisfying Opial condition and the second for those whose dual satisfies the Kadec-Klee property.

In this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.

In this paper, common fixed point for four maps using an implicit contractive condition in a complete metric space is proved. Some periodic point results for such mappings are also obtained. These results extend and generalize several comparable results in the current literature.

In this paper, we study the stability of extended Weyl and Browdertype theorems for orthogonal direct sum S⊕T, where S and T are bounded linear operators acting on Banach space. Two counterexamples shows that property (ab), in general, is not preserved under direct sum. Nonetheless, and under the assumptions that Π 0 α (T) ⊂σ α (S) and Π 0 α (S) ⊂σ α (T), we characterize preservation of property (ab) under direct sum S⊕T. Furthermore, we show that if S and T satisfy generalized a-Browder’s theorem, then S⊕T satisfies generalized a-Browder’s theorem if and only if σ SBF - + (S ⊕T) = σ SBF - + (S) ∪ σSBF - + (T), which improves a recent result of [13] by removing certain extra assumptions.

We characterize minimal P γ-open sets in topological spaces. We show that any nonempty subset of a minimal P γ-open set is pre P-open. As an application of a theory of minimal Pγ-open sets, we obtain a sufficient condition for a P-locally finite space to be a pre P γ-Hausdorff space

A new class of functions called ‘R δ -supercontinuous functions’ is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity which already exist in the literature is elaborated. The class of Rδ-supercontinuous functions (Math. Bohem., to appear) properly contains the class of R z -supercontinuous functions which in its turn properly contains the class of R cl - supercontinuous functions (Demonstratio Math. 46(1) (2013), 229-244) and so includes all R cl -supercontinuous (≡clopen continuous) functions (Applied Gen. Topol. 8(2) (2007), 293-300; Indian J. Pure Appl. Math. 14(6) (1983), 767-772) and is properly contained in the class of R-supercontinuous functions (Demonstratio Math. 43(3) (2010), 703-723).

We prove two generalizations: the first to Das and Naik’s theorem for a pair of compatible maps without continuity; and the next as an extension of our first result to three self-maps on a metric space X without compatibility, under a stronger contraction type inequality and restricting the completeness of X to its subspace. The latter is a significant generalization of a recent result of Pant et al.

A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds, which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the viceversa can be done as well. In a physical language, this would correspond to the existence of a canonical connection. In this paper, we show an example of an algebra which always possesses such a connection.

It is known that every Sikorski space with the countably generated differential structure is smoothly real-compact. It means that every homomorphism from its differential structure, which forms a ring of smooth real-valued functions into the ring of real numbers, is an evaluation. This result is sharp: there is a non-smoothly real-compact Sikorski space with the differential structure which is not countably generated. We provide an easy example demonstrating this. By modifying this example we are able to show a certain shortcoming of the generator embedding, comparing to the canonical embedding, for Sikorski spaces. Finally, we note that a homomorphism from the ring of smooth functions of a Sikorski space into the ring of real numbers is an evaluation if and only if it is continuous.

Birth-and-death models are now a common mathematical tool to describe branching patterns observed in real-world phylogenetic trees. Liggett and Schinazi (2009) is one such example. The authors propose a simple birth-and-death model that is compatible with phylogenetic trees of both influenza and HIV, depending on the birth rate parameter. An interesting special case of this model is the critical case where the birth rate equals the death rate. This is a non-trivial situation and to study its asymptotic behaviour we employed the Laplace transform. With this, we correct the proof of Liggett and Schinazi (2009) in the critical case.

In this paper, a new two-step iterative method for solving symmetric positive semidefinite linear system of equations is presented. A sufficient condition for the semiconvergence of the method is also given. Some numerical experiments are presented to show the efficiency of the proposed method.

The aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy-Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.