A unified class of k-uniformly convex functions defined by the Dziok–Srivastava linear operator

The main purpose of this paper is to introduce a new class of functions which are analytic in the open unit disk . We obtain various results including characterization, coefficient estimates, and distortion and covering theorems, for functions belonging to the class .     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,

in the unit ball Ω of with Dirichlet boundary conditions, in the subcritical case. More precisely, we study the set of initial values in C₀(Ω) for which the resulting solution of (NLH) is global. We obtain very precise information about a specific two-dimensional slice of , which (necessarily) contains sign-changing initial values. As a consequence of our study, we show that is not convex. This contrasts with the case of nonnegative initial values, where the analogous set is known to be convex.     

The simplicity of certain groups of subanalytic diffeomorphisms

Let M be a connected real analytic manifold. We denote by , 1r<∞, the group of subanalytic C^r diffeomorphisms of M which are isotopic to the identity via a compactly supported subanalytic C^r isotopy. We show that satisfies Epstein's axioms. This implies that the commutator subgroup of is simple. Moreover, we show that the commutator subgroup of is dense in . As a corollary we obtain that is topologically simple.     

When and how an error yields a Dirichlet form

We consider a random variable Y and approximations Y_n, , defined on the same probability space with values in the same measurable space as Y. We are interested in situations where the approximations Y_n allow to define a Dirichlet form in the space where is the law of Y. Our approach consists in studying both biases and variances. The article attempts to propose a general theoretical framework. It is illustrated by several examples.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Poincaré's theorem for the modular group of real Riemann surfaces

Let Mod_g denote the modular group of (closed and orientable) surfaces S of genus g. Each element [h]Mod_g induces a symplectic automorphism H([h]) of . Poincaré showed that is an epimorphism. A real Riemann surface is a Riemann surface S together with an anticonformal involution σ. Let (S,σ) be a real Riemann surface, be the group of orientation preserving homeomorphisms of S such that h○σ=σ○h and be the subgroup of consisting of those isotopic to the identity by an isotopy in . The group plays the role of the modular group in the theory of real Riemann surfaces. In this work we describe the image by H of . Such image depends on the topological type of the involution σ.     

Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

Sequential definitions of compactness

A subset F of a topological space is sequentially compact if any sequence of points in F has a convergent subsequence whose limit is in F. We say that a subset F of a topological group X is G-sequentially compact if any sequence of points in F has a convergent subsequence such that where G is an additive function from a subgroup of the group of all sequences of points in X. We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G-sequential compactness. Sequential compactness is a special case of this generalization when G=lim.     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

Biharmonic hypersurfaces in Sasakian space forms

We consider the Boothby–Wang fibration of a strictly regular Sasakian space form N and find the characterization of biharmonic Hopf cylinders over submanifolds of . Then, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.     

Integral-type operators from iterated logarithmic Bloch spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball of and the radial derivative of h. In this paper we study the boundedness and compactness of the following integral operator:, from iterated logarithmic Bloch spaces to Zygmund-type spaces.     

Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

Let be a time scale such that . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation with boundary conditions , where φ_p(s)=|s|^p-2s with p>1, is continuous for i=1,2,…,m-1 and nonincreasing if . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential and difference equations . As an application, an example is given to illustrate our result.     

Norms of some operators on the Bergman and the Hardy space in the unit polydisk and the unit ball

Let H(X) be the class of all holomorphic functions on the set and uH(X). We calculate operator norms of the multiplication operators M_u(f)=uf, on the weighted Bergman space , as well as on the Hardy space H^p(X), where X is the unit polydisk or the unit ball in . We also calculate the norm of the weighted composition operator from the weighted Bergman space , and the Hardy space , to a weighted-type space on the unit polydisk.     

Region of variability for close-to-convex functions-II

For a complex number α with let be the class of analytic functions f in the unit disk with f(0)=0 satisfying in , for some convex univalent function in . For any fixed , and we shall determine the region of variability V(z₀,α,λ) for f(z₀) when f ranges over the class

In the final section we graphically illustrate the region of variability for several sets of parameters z₀ and α.     

Integral-type operators from Bloch-type spaces to Zygmund-type spaces

Let denote the space of all holomorphic functions on the unit ball . This paper investigates the following integral-type operator with symbol

where is the radial derivative of g. The boundedness and compactness of the operator T_g from Bloch-type spaces to Zygmund-type spaces are studied.     

Globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients

A new approach will be used to obtained the existence and uniqueness of positive almost periodic solution of the following nonlinear functional difference equation:

Δx(n)=-a(n)x(n)+f(n,x(n-τ(n))).