Volume 22, Issue 2

In this paper, we introduce and examine the concepts of I 2 ( u )-convergence of order (α,β) and uniformly strong p -Cesàro summability of order (α,β), of double sequences of complex (or real) numbers, where α,β$\alpha ,\beta $ are real numbers such that 0<α,β≤1${ 0&lt;\alpha ,\beta \le 1}$. Also, we establish the main results and prove some inclusion relations between uniformly strong p -Cesàro summability and I 2 ( u )-convergence of order (α,β).

In this paper, let X be a finite set, D be a complete X -semilattice of unions and Q={T1,T2,T3,T4,T5,T6,T7,T8}$Q=\lbrace T_1,T_2,T_3,T_4,T_5, T_6,T_7,T_8\rbrace $ be an X -subsemilattice of D where T1⊂T3⊂T5⊂T6⊂T8$T_1\subset T_3\subset T_5\subset T_6\subset T_8$, T1⊂T3⊂T5⊂T7⊂T8$T_1\subset T_3\subset T_5\subset T_7\subset T_8$, T2⊂T3⊂T5⊂T6⊂T8$T_2\subset T_3\subset T_5\subset T_6\subset T_8$, T2⊂T3⊂T5⊂T7⊂T8$T_2\subset T_3\subset T_5\subset T_7\subset T_8$, T2⊂T4⊂T5⊂T6⊂T8$T_2\subset T_4\subset T_5\subset T_6\subset T_8$, T2⊂T4⊂T5⊂T7⊂T8$T_2\subset T_4\subset T_5\subset T_7\subset T_8$, T2∖T1≠∅$T_2\setminus T_1\ne \emptyset $, T1∖T2≠∅$T_1\setminus T_2\ne \emptyset $, T4∖T3≠∅$T_4\setminus T_3\ne \emptyset $, T3∖T4≠∅$T_3\setminus T_4\ne \emptyset $, T6∖T7≠∅$T_6\setminus T_7\ne \emptyset $, T7∖T6≠∅$T_7\setminus T_6\ne \emptyset $, T2∪T1=T3$T_2\cup T_1=T_3$, T4∪T3=T5$T_4\cup T_3=T_5$, T6∪T7=T8$T_6\cup T_7=T_8$. Using the characteristic family of sets, the characteristic mapping and base sources of Q , we characterize the class whose elements are each isomorphic to Q . We generate some advanced formulas in order to calculate the number of regular elements α of B X ( D ) satisfying V(D,α)=Q$V(D,\alpha )=Q$, in an efficient way.

We develop the method of maximal likelihood for infinite-dimensional Hilbert spaces and prove several theorems about consistency and asymptotic normality.

In this paper, we introduce the definitions of shadowing and average shadowing properties for iterated function systems and give some examples characterizing these definitions. We prove that an iterated function system has the shadowing property if and only if the step skew product corresponding to the iterated function system has the shadowing property. Also, we study some notions such as transitivity, chain transitivity, chain mixing and mixing for iterated function systems.

In this paper, we prove a well-known theorem under weaker conditions. This theorem also includes some known and new results.

In this paper, we show how to extend Voronov's construction of L ∞ -structures by using the splitting of a graded Lie algebra into the direct vector space sum of two subalgebras, of which one is abelian, to just a graded Lie algebra inclusion without an algebraic complement. The construction uses certain Verma modules of the universal enveloping algebra of the graded Lie algebra, and it is fairly explicit in terms of convolution formulas. Voronov's result [J. Pure Appl. Algebra 202 (2005), 133–153] and Bandiera's result on nonabelian complements [http://arxiv.org/abs/1304.4097] occur as particular cases.

Some necessary and sufficient conditions for the existence of the ℍ-derivative of a quaternion function are given in terms of its differential.

The classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other p -step nilpotent Lie algebras are obtained by contraction of the rigid ones.

In this paper, an optimal control problem is considered for a system of fourth order hyperbolic equations with constant coefficients. The gradient of the functional is calculated and the necessary and sufficient conditions of optimality in the form of an integral inequality are derived.

Starting with an operator defined on the family of open sets in a topological Baire space 〈X,τ〉${\langle X, \tau \rangle }$ we can define a lower density operator on the family of sets having the Baire property with respect to the τ-topology. The investigation of the abstract density topologies generated by such operators is the main subject of this paper.

In this paper, we establish some new Hadamard-type inequalities using elementary well-known inequalities for functions whose absolute values of the second derivatives are α-, m -, (α, m )-logarithmically convex.

A set X in a real Hilbert space H is called an a t -set if every three-element subset of X forms either an acute-angled triangle or a right-angled triangle. The maximal cardinality of an a t -set in an infinite-dimensional H is found. Furthermore, the number of right angles in the unit cube [0,1]n${[0,1]^n}$ is calculated. As an application, a simple solution of a well-known problem is given, concerning the maximal cardinality of a strong a t -set in the Euclidean space ℝ n .

In this paper, we consider higher-order Bernoulli and poly-Bernoulli mixed type polynomials and give some interesting identities for the polynomials arising from umbral calculus.

This paper deals with the existence of solutions for a system of k equations x''=f(t,x,x'),${x^{\prime \prime }=f(t,x,x^{\prime }),}$ where f:[0,1]×ℝk×ℝk→ℝk${f\colon [0,1]\times \mathbb {R}^{k}\times \mathbb {R}^{k}\rightarrow \mathbb {R}^{k}}$ is a vector function, subject to various nonlocal boundary conditions. The method is to impose an a priori bound condition on f and apply the Leray–Schauder fixed point theorem. Our results extend some results in the references and also provide the information on the existence of stationary solutions for the heat equation with nonlinear gradient source terms.

In this paper, we introduce the notion of fp n -injective and fp n -flat modules, and investigate their properties. In particular, we prove that every left R -module has an fp n -injective cover (resp. preenvelope) and every right R -module has an fp n -flat cover (resp. preenvelope). We also study the exchange properties of fp n -injective and fp n -flat modules, as well as fp n -injective precovers (resp. preenvelopes) and fp n -flat precovers (resp. preenvelopes), under (almost) excellent extensions of rings.

In this paper, we prove the following result. Suppose that E is an arbitrary non-empty closed set of real numbers (mod2π)${(\operatorname{mod} 2\pi )}$. Then for any positive λ (0<λ<∞${0&lt;\lambda &lt;\infty }$), there exists an algebroid function (not meromorphic) w ( z ) of order λ in the unit disc, such that every point eiθ${e^{i\theta }}$, θ∈E${\theta \in E}$, on the unit circle is a Borel point of order λ of w ( z ) and w ( z ) has no other Borel point of order λ. This result generalizes the related results of a meromorphic function due to Lo Yang, Guanghou Zhang, Daochun Sun and Zongsheng Gao.