Volume 66, Issue 6

A mutual relationship between MV-algebras and coupled semirings as established by L. P. Belluce, A. Di Nola, A. R. Ferraioli and B. Gerla is extended to lattice effect algebras and so-called characterizing triples. We show that this correspondence is in fact one-to-one and hence every lattice effect algebra can be considered as an ordered triple consisting of two semiring-like structures and an antitone involution which is an isomorphism between these structures.

We study the periods of Morgan-Voyce sequences modulo an integer m and the periods of Morgan-Voyce sequences over elliptic curves.

Let R be a ring and I is a nonzero ideal of R . A mapping F : R → R is called a multiplicative generalized derivation if there exists a mapping g : R → R such that F ( xy ) = F ( x ) y + xg ( y ), for all x , y ∈ R . In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: F ([ x , y ]) = 0, F ( xoy ) = 0, F ([ x , y ]) = ± [ x , y ], F ( xoy ) = ±( xoy ), F ([ x , y ]) = ±( xoy ), F ( xoy ) = ±[ x , y ], F ([ x , y ]) = ±[ F ( x ), y ], F ( xoy ) = ±( F ( x ) oy ), F ( xy ) ± xy ∈ Z , F ( xy ) ± yx ∈ Z , F ( xy ) ±[ x , y ] ∈ Z , F ( xy ) ±( xoy ) ∈ Z , for all x , y ∈ I .

We establish conditions guaranteeing the existence of (generalized) regularly varying solutions to n th order linear differential and q -difference equations. The proofs are based mainly on classical Poincaré’s and Perron’s theorems and certain transformations. In some special cases, our results reduce to the existing ones, thus actually we offer an alternative approach to some parts of the asymptotic theory of differential equations made in the framework of regular variation. For higher order cases, our statements are essentially new. Another important feature of this paper is that it reveals connections among various results and somehow revises some of them.

A linear operator D 𝒫 k f ( z ) : S → S is introduced, where S denotes the class of univalent analytic functions in open unit disk and 𝒫 k is an arbitrary monotonically increasing function in k . We study a new class 𝒯 𝒫 k ( α 1 , α 2 , α 3 , β ) of analytic functions related to D 𝒫 k f ( z ). The main object of this paper is to give the radius equation between the class 𝒯 𝒫 k ( α 1 , α 2 , α 3 , β ) and the close-to-convex class of order α (0 ≤ α < 1). Also, we apply the extremal theorem to maximize | f (𝒳) ( z )| over 𝒯 𝒫 k ( α 1 , α 2 , α 3 , β ), where 𝒳 = {0, 1, 2, …, k }, all the results are sharp.

In this paper, we research the existence of solutions for a class of Riemann-Liouville fractional evolution inclusions with nonconvex right hand side. Our main results obtain the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. In the end, we give an example to illustrate the applications of the abstract results.

Linear time-varying Volterra difference equations are considered. By a novel approach, we get some new explicit criteria for global exponential stability. Some examples are given to illustrate the obtained results. To the best of our knowledge, the obtained results are new.

The results corresponding to some theorems of S. Lal [Appl. Math. Comput. 209 (2009), 346–350] and the results of the authors [In: Banach Center Publ. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, pp. 339–351] are shown. The same degrees of pointwise approximation as in mentioned papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated.

Let 𝔄 and 𝔅 be two Banach algebras and let θ be 𝔄 nonzero character on 𝔅 . In this paper, we deal with the θ -Lau product 𝔄 × θ 𝔅 that was first introduced by A. T. Lau for certain Banach algebras known as Lau algebras and recently by M. S. Monfared for all Banach algebras; we study pseudo-amenability, pseudo-contractibility and character pseudo-amenability of 𝔄 × θ 𝔅 and their relations with 𝔄 and 𝔅 .

Let G be a semi direct product G = ℝ d ⋊ ℝ k . On G we consider a class of second order left-invariant differential operators of the form Lα=∑j=1de2λj(a)∂xj2+∑j=1k(∂aj2−2αj∂aj),${{\cal L}_\alpha } = \sum\limits_{j = 1}^d {\mkern 1mu} e{{\mkern 1mu} ^{2{\lambda _j}(a)}}\partial _{{x_j}}^2 + \sum\limits_{j = 1}^k {(\partial _{{a_j}}^2 - 2{\alpha _j}{\partial _{{a_j}}})},$ where a ∈ ℝ k and λ 1 ,..., λ d ∈ (ℝ k )*. It is known that bounded 𝓛 α -harmonic functions on G are precisely the “Poisson integrals” of L ∞(ℝ d ) against the Poisson kernel ν α which is a smooth function on ℝ d . We prove an upper bound of ν α and its derivatives.

Using a family of ordered cones, we define a suitable order to investigate projective limit topologies for cones. Under the strict separation property, the projective limit cone topology is proved to be equivalent to some of the X -topologies embedded on a subcone of the product cone in its topology. Also, we discuss the cones of convex subsets in projective limits; in particular, we show that a cone of convex subsets of the projective limit cone in its topology carries the projective limit of the cones of convex subsets in its components.

We characterize Banach spaces X with spaces with property ( wL ), i.e. spaces with the property that every L -subset of X * is weakly precompact. We prove that a Banach space X has property ( wL ) if and only if for any Banach space Y , any completely continuous operator T : X → Y has weakly precompact adjoint if and only if any completely continuous operator T : X → ℓ ∞ has weakly precompact adjoint. We prove that if E is a Banach space and F is a reflexive subspace of E * such that ⊥ F has property ( wL ), then E has property ( wL ). We show that a space E has property RDP * (resp. the DPrcP ) if and only if any closed separable subspace of E has property RDP * (resp. the DPrcP ). We also show that G has property ( wL ) if under some conditions K w * ( E *, F ) contains the dual of G .

We study quadratic integral equations in Orlicz spaces on the interval [ a , b ]. We relate growth conditions with appropriate spaces of solutions and then we investigate different cases. Our approach allows to consider discontinuous solutions as well as to unify the results for both quadratic and classical integral equations. Many different types of growth conditions for investigated functions are considered.

The main aim of this paper is to present some theorems in order to guarantee existence and uniqueness of best proximity points under rational contractivity conditions using very general test functions. To illustrate the variety of possible test functions, we include some examples of pairs of functions which are included in innovative papers published in the last years. As a consequence, we prove that our results unify and extend some recent results in this field.

We introduce and investigate a new concept of almost Weyl structures on null hypersufaces in indefinite Kenmotsu manifolds, tangent to the structure vector field. We show that, under some conditions, there are foliations on almost Weyl null hypersufaces which are Einstein-Weyl with scalar curvature of constant sign.

The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher-order infinitesimal symmetries which provide a simplifying “linear approximation” of general groups of higher-order symmetries. The classical Lie’s approach is appropriately adapted.

We study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a strongly separately continuous mapping to be continuous on a product of an arbitrary family of topological spaces. Moreover, we characterize the discontinuity point set of strongly separately continuous function defined on a subset of countable product of finite-dimensional normed spaces.

In this paper we study the homogeneous Wick differential equation associated to the quantum white noise (𝚀𝚆𝙽) Euler operator ΔEg,Q$\Delta _E^{g, Q}$ acting on generalized operators. ΔEg,Q$\Delta _E^{g, Q}$ is defined as sum of the extension of the 𝚀𝚆𝙽-Gross Laplacian and the 𝚀𝚆𝙽-conservation operator. It is shown that the operator ΔEg,Q$\Delta _E^{g, Q}$ has a representation in terms of the 𝚀𝚆𝙽-derivatives {Dc−,Dc+:c∈N}$\{ D_c^ -, D_c^ +: \;c \in N\} $. The poisson equation is worked out as a non homogeneous Wick differential equation associated to ΔEg,Q$\Delta _E^{g, Q}$.

Errors-in-variables (EIV) models have been studied for many years. Some rules for treating them in the multivariate case are given in the paper from several viewpoints. An evaluation of an influence of inaccurate entries of a design matrix on the estimator of models parameters is studied. The problem of linearization of the multivariate EIV model is studied as well.

In this work, we study codes over the ring 𝓡 k, m = 𝔽 2 [ u , v ]/ 〈 u k , v m , uv − vu 〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F2km$\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

Let R be a commutative ring with Z ( R ) the set of zero-divisors and U ( R ) the set of unit elements of R . The total graph of R , denoted by T (Γ( R )), is the (undirected) graph with all elements of R as vertices, and for distinct x , y ∈ R , the vertices x and y are adjacent if and only if x + y ∈ Z ( R ). We study the domination number of T (Γ( R )). It is shown that if R = Z ( R ) ∪ U ( R ), then the domination number of T (∪( R )) is finite provided R has a maximal ideal of finite index. Moreover, if R=∏ i=1nFi$R = \prod\limits_{i = 1}^n {{F_i}} $, where F i is a field for each 1 ≤ i ≤ n and t = | F 1 | ≤ | F 2 | ≤ ··· ≤ | F n |, then the domination number of T (Γ( R )) is equal to t - 1 provided t = | F i | for every 1 ≤ i ≤ n , and is equal to t otherwise. Finally, for an R -module M it is shown that the total domination number of the total graph of the idealization (Nagata extension) R (+) M is equal to the domination number of the total graph of R provided M is a torsion free R -module or R = Z ( R ) ∪ U ( R ).