Volume 63, Issue 5

State MV-algebras were introduced by Flaminio and Montagna as MV-algebras with internal states. Di Nola and Dvurečenskij presented the notion of state-morphism MV-algebra which is a stronger variation of a state MV-algebra. Rachůnek and Šalounová introduced state GMV-algebras (pseudo-MV algebras) and state-morphism GMV-algebras, while the state BL-algebras and state-morphism BL-algebras were defined by Ciungu, Dvurečenskij and Hyčko. Recently, Dvurečenskij, Rachůnek and Šalounová presented state Rℓ-monoids and state-morphism Rℓ-monoids. In this paper we study these concepts for more general fuzzy structures, namely pseudo-hoops and we present state pseudo-hoops and state-morphism pseudo-hoops.

Characterizations of fuzzy filters in a BE-algebra are established. Conditions for a fuzzy set to be a fuzzy filter are given. For a fuzzy set µ the least fuzzy filter containing µ is constructed. The homomorphic properties of fuzzy filters of a BE-algebra are provided. Finally, characterizations of Noetherian BE-algebras and Artinian BE-algebras via fuzzy filters are obtained.

We introduce the notion of gi-algebra as a generalization of dual BCK-algebra, and define the notions of strong, commutative and transitive gi-algebra, and then we show that an interval ↑l = {a ∈ P | l ≤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation ∼D on a transitive gi-algebra P and show that the quotient set P/∼D is a gi-algebra and a dual BCK-algebra.

Jordens and Sturm investigated the link between closure systems on sets and closure systems on partitions. We extend that study to the wider framework of partial partitions, and highlight better the relation between these two families of closure systems. Then we consider the construction of a closure operator on partial partitions by the iterated application a set operator to the blocks of a partial partition; the resulting closure system fits into our framework.

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ℓ-groups. We mention the relationship of clean unital ℓ-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ℓ-groups with distinguished weak order unit.

In 2006, P. J. Cameron and J. Nešetřil introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finitely generated substructures of the structure extends to an endomorphism of the structure. In several recent papers homomorphism-homogeneous objects in some well-known classes of algebras have been investigated (e.g. lattices and semilattices), while finite homomorphism-homogeneous groups were described in 1979 under the name of finite quasiinjective groups. In this paper we characterize homomorphism-homogenous monounary algebras of arbitrary cardinalities.

For an m × n matrix B = (b ij)m×n with nonnegative entries b ij, let B(k, l) denote the set of all k × l submatrices of B. For each A ∈ B(k, l), let a A and g A denote the arithmetic mean and geometric mean of elements of A respectively. It is proved that if k is an integer in ($$\tfrac{m} {2}$$,m] and l is an integer in ($$\tfrac{n} {2}$$, n] respectively, then $$\left( {\prod\limits_{A \in B\left( {k,l} \right)} {a_A } } \right)^{\tfrac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}} \geqslant \frac{1} {{\left( {_k^m } \right)\left( {_l^n } \right)}}\left( {\sum\limits_{A \in B\left( {k,l} \right)} {g_A } } \right),$$ with equality if and only if b ij is a constant for every i, j.

We generalize and improve several inequalities of the Čebyšev-Grüss-type using least concave majorants of the moduli of continuity of the functions involved. Our focus is on normalized positive linear functionals. We discuss a problem posed by the two Gavreas and also give the solution of a stronger one. In a section about the non-multiplicativity of positive linear operators it is demonstrated that the previous use of second moments is not quite the right choice. This is documented in the case of the classical Hermite-Fejér and de La Vallée Poussin convolution operators.

In the present paper we estimate a Voronovskaja type quantitative estimate for a certain type of complex Durrmeyer polynomials, which is different from those studied previously in the literature. Such estimation is in terms of analytic functions in the compact disks. In this way, we present the evidence of overconvergence phenomenon for this type of Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane. In the end, we mention certain applications.

The associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given.

In this paper we establish a set of necessary and sufficient conditions in order that |C, 0|k ⇒ |R, p n|s and |R, p n|k ⇒ |C, 0|s for the case 1 < k ≤ s < ∞. As a corollary, we obtain that a crucial assumption of [BOR, H.: A new result on the high indices theorem, Analysis 29 (2009), 403–405] is omitted and that the other one is not only sufficient but also necessary for his consequence to hold.

In the present paper we construct q-Szász operators that preserve the third test function e 2. Rate of global convergence is obtained in the frame of weighted spaces. Furthermore, we obtain a Voronovskaja type theorem for these operators.

The purpose of this paper is to establish some new approximation theorems of common fixed points for a countable family of total asymptotically quasi-nonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. [J. Math. Anal. Appl. 333 (2007), 128–141], [J. Math. Anal. Appl. 326 (2007), 960–973], [Internat. J. Math. Math. Sci. 2009, Article ID 615107, 17 pp.] and others.

In this paper we study the local convergence of the method $$0 \in f\left( {p,x_k } \right) + A\left( {x_{k + 1} - x_k } \right) + F\left( {x_{k + 1} } \right),$$ in order to find the solution of the generalized equation $$find x \in X such that 0 \in f\left( {p,x} \right) + F\left( x \right).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $$\nabla _x f\left( {\bar p,\bar x} \right)$$, we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.

In this paper, we mainly consider the first and second arc length variational problems on the exponential statistical manifold, and give the variational formulae.

Let A be the class of analytic functions in the unit disk $$\mathbb{D}$$ with the normalization f(0) = f′(0) − 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $$\mathbb{D}$$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $$\mathbb{D}$$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.

The present paper deals with continuous extreme-like selections for the Vietoris hyperspace of countably compact spaces. Several new results and applications are established, along with some known results which are obtained under minimal hypotheses. The paper contains also a number of examples clarifying the role of countable compactness.

We consider a random subgraph G n(p) of a finite graph family G n = (V n, E n) formed by retaining each edge of G n independently with probability p. We show that if G n is an expander graph with vertices of bounded degree, then for any c n ∈ (0, 1) satisfying $$c_n \gg {1 \mathord{\left/ {\vphantom {1 {\sqrt {\ln n} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\ln n} }}$$ and $$\mathop {\lim \sup }\limits_{n \to \infty } c_n < 1$$, the property that the random subgraph contains a giant component of order c n|V n| has a sharp threshold.

We establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator.