Showing papers in "Open Mathematics in 2010"
TL;DR: In this article, the authors extend the study of fractal operator theory for multivalued operators on complete b-metric spaces to the case of complete or compact metric spaces.
Abstract: Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.
191 citations
TL;DR: In this paper, the authors introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemanian manifold and give necessary and sufficient conditions for a Langrangian submersion to be totally geodesic.
Abstract: We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold of such submersions.
138 citations
TL;DR: In this paper, asymptotic properties of the third-order neutral differential equation were studied. But the objective of this paper was not to study the convergence of non-oscillatory solutions of (E) to zero, but to establish sufficient conditions that all nonoscillatorial solutions converge to zero or all oscillatory solutions are oscillatory.
Abstract: The objective of this paper is to study asymptotic properties of the third-order neutral differential equation $$
\left[ {a\left( t \right)\left( {\left[ {x\left( t \right) + p\left( t \right)x\left( {\sigma \left( t \right)} \right)} \right]^{\prime \prime } } \right)^\gamma } \right]^\prime + q\left( t \right)f\left( {x\left[ {\tau \left( t \right)} \right]} \right) = 0, t \geqslant t_0 . \left( E \right)
$$
. We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.
41 citations
TL;DR: In this paper, the authors show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix and the normal curvature ellipse.
Abstract: In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of two geometric figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We construct a family of surfaces with flat normal connection.
40 citations
TL;DR: A new proof is given showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree and how the result can be used to prove the inherent ambiguity of languages of infinite trees.
Abstract: We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.
35 citations
TL;DR: In this paper, the authors introduce and investigate weighted statistical approximation properties of a q-analogue of the Baskakov and Baskak-Kantorovich operators, using a weighted modulus of smoothness, and give some estimations for error in the case 0 < q < 1.
Abstract: In the present paper we introduce and investigate weighted statistical approximation properties of a q-analogue of the Baskakov and Baskakov-Kantorovich operators. By using a weighted modulus of smoothness, we give some direct estimations for error in the case 0 < q < 1.
35 citations
TL;DR: In this article, the Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces defined by the norm of norm of the norm.
Abstract: The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces \( \hat H_s^r \left( \mathbb{R} \right) \) defined by the norm
$$ \left\| {v_0 } \right\|_{\hat H_s^r \left( \mathbb{R} \right)} : = \left\| {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\|_{L_\xi ^{r'} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1} {2}} , \frac{1} {r} + \frac{1} {{r'}} = 1 $$
.
34 citations
TL;DR: In this paper, the authors established direct estimates for the q-Baskakov operator using the second order Ditzian-Totik modulus of smoothness, and defined and studied the limit q-baskakov operators.
Abstract: We establish direct estimates for the q-Baskakov operator introduced by Aral and Gupta in [2], using the second order Ditzian-Totik modulus of smoothness. Furthermore, we define and study the limit q-Baskakov operator.
32 citations
TL;DR: In this paper, the q-analogue of the Pochhammer k-symbol introduced by Diaz and Teruel has been studied in combinatorial and probabilistic ways.
Abstract: We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Diaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.
26 citations
TL;DR: In this article, it was shown that there is also an equivalence of the singular derived categories of the fibers f−1(0) and g−1 (0) of a Calabi-Yau hypersurface in a weighted projective space.
Abstract: Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → \( \mathbb{A}^1 \), g:Y → \( \mathbb{A}^1 \). Assuming that there exists a complex of sheaves on X × \( \mathbb{A}^1 \)Y which induces an equivalence of Db(X) and Db(Y), we show that there is also an equivalence of the singular derived categories of the fibers f−1(0) and g−1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.
22 citations
TL;DR: In this paper, the authors introduced a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of almost-paralellionic hermitian manifolds.
Abstract: In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kahler manifolds.
TL;DR: In this article, a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six is presented, which is based on Galois descent.
Abstract: We present a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. We start with cubic surfaces in the hexahedral form of L. Cremona and Th. Reye. For these, we develop an explicit version of Galois descent.
TL;DR: In this article, the authors considered a fourth order eigenvalue problem with a spectral parameter both in the equation and in the boundary condition, and derived asymptotic formulae for eigenvalues and eigenfunctions.
Abstract: We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in Lp(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.
TL;DR: In this article, a generalization in q-calculus of Stancu operators is presented, where the modulus of continuity and Lipschitz type maximal function are considered.
Abstract: This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.
TL;DR: In this paper, it was shown that for any connected graph G of order n ≥ 3 and δ(G) ≥ 2, the maximum number of edges in a matching in G can be characterized with sdγghazi t�� (G)=2γ�γ� (G)−1.
Abstract: A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ
t
(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt
(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sdγ
t
(G) ≤ 2γ
t
(G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sdγ
t
(G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sdγ
t
(G)=2γ
t
(G)−1.
TL;DR: In this paper, a model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered, and the existence and uniqueness theorem of the classic solution is proved.
Abstract: A model of coupled parabolic and ordinary differential equations for a heterogeneous catalytic reaction is considered and the existence and uniqueness theorem of the classic solution is proved.
TL;DR: In this paper, the authors characterised Green's relations and ideals in E(X, σ) under composition of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.
Abstract: Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α
−1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.
TL;DR: In this article, the q-Szasz-Durrmeyer operators were introduced and a local approximation result for continuous functions in terms of moduli of continuity was given.
Abstract: In the present paper, we introduce the q-Szasz-Durrmeyer operators and justify a local approximation result for continuous functions in terms of moduli of continuity. We also discuss a Voronovskaya type result for the q-Szasz-Durrmeyer operators.
TL;DR: In this paper, it was shown that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, where e is unit element of A, then T is an algebra isomorphism.
Abstract: Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces MA and MB, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: MB → MA and a closed and open subset K of MB such that
$$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$
for all a ∈ A, where e is unit element of A. If, in addition, \( \widehat{T\left( e \right)} = 1 \) and \( \widehat{T\left( {ie} \right)} = i \) on MB, then T is an algebra isomorphism.
TL;DR: In this article, the authors initiated the study of signed k-domination in digraphs and presented some sharp lower bounds for γkS(D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number.
Abstract: Let k ≥ 1 be an integer, and let D = (V; A) be a finite simple digraph, for which dD− ≥ k − 1 for all v ɛ V. A function f: V → {−1; 1} is called a signed k-dominating function (SkDF) if f(N−[v]) ≥ k for each vertex v ɛ V. The weight w(f) of f is defined by \( \sum
olimits_{v \in V} {f(v)} \). The signed k-domination number for a digraph D is γkS(D) = min {w(f|f) is an SkDF of D. In this paper, we initiate the study of signed k-domination in digraphs. In particular, we present some sharp lower bounds for γkS(D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs and digraphs.
TL;DR: In this article, a generalization of Kannan's fixed point theorem is presented, in order to observe the condition of kannan mappings more deeply, and prove a generalisation of k-means.
Abstract: In order to observe the condition of Kannan mappings more deeply, we prove a generalization of Kannan’s fixed point theorem.
TL;DR: In this article, the k-isogeny class of an abelian variety with commutative endomorphism algebra over a finite field k is uniquely determined by a Weil polynomial fA without multiple roots.
Abstract: Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial fA without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of fA(1 − t).
TL;DR: In this paper, the structure of minimal normal subgroups of totally inert groups was investigated and it was shown that infinite locally graded simple groups cannot be totally inert, even if every subgroup of the group is inert.
Abstract: A subgroup H of a group G is inert if |H: H ∩ Hg| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.
TL;DR: In this paper, a characterisation of α1-matrices and α2matrices is given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.
Abstract: This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.
TL;DR: In this article, it was shown that the moduli space of anti-self-dual connections on a (4,4)-manifold is also hypercomplex, and admits a strong HKT metric.
Abstract: A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkahler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.
TL;DR: In this article, necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field are given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups.
Abstract: This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.
TL;DR: Some problems involving the classical Hardy function are discussed in this paper, where the odd moments of Z(t) and the distribution of its positive and negative values are discussed, as well as its distribution of odd moments.
Abstract: Some problems involving the classical Hardy function $$
Z\left( t \right) = \zeta \left( {\frac{1}
{2} + it} \right)\left( {\chi \left( {\frac{1}
{2} + it} \right)} \right)^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-
ulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right)
$$
, are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
TL;DR: In this paper, the realizability of groups of order 64 as Galois groups over arbitrary fields was examined and necessary and sufficient conditions were provided for 134 of the 200 noncyclic groups that are not direct products of smaller groups.
Abstract: This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.
TL;DR: In this paper, the concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters in the form of pre-BCK algebras.
Abstract: The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework We thus recapture theorems already known in the literature, as well as new ones As a by-product we introduce the class of pre-BCK algebras
TL;DR: It is proved that the isomorphism relation for ω-tree-automatic boolean algebras is not determined by the axiomatic system ZFC and is neither a Σ21- set nor a Π21-set.
Abstract: An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set.