Volume 44, Issue 2

Given a mapping f from a finite set X n on itself, we explore the finite topology τ f defined by the stable subsets E of X n ; i.e., the subsets E of X n which satisfy f ( E ) ⊆ E . Necessary and sufficient conditions are found in order that a finite topology τ may correspond to the topology τ f for a certain mapping f . A number of results relating properties of topology τ f with properties of the mapping f itself are also exposed.

The dual category with respect to the category of differential groups is defined and investigated. The objects of this category are algebras, called Hopf–Sikorski (H-S) algebras, the axioms of which combine the axioms of Sikorski’s algebras with modified axiomas of Hopf algebras. Morphisms of this category are structural mappings corresponding to Hopf algebras that are smooth in the sense of Sikorski. As an example, we discuss the H-S algebra of the Lorentz group.

A method is presented for finding the sign and defect of a linear functional L when it operates on an n -convex function.

The object of the present paper is to investigate the coefficients estimates, distortion properties, the radii of starlikeness and convexity, subordination theorems, partial sums and integral mean inequalities for classes of functions with two fixed points. Some remarks depicting consequences of the main results are also mentioned.

In this paper we define a general integral operator Bazilevič and we obtain sufficient conditions for univalence of this integral operator.

In this paper, we further investigate the class of functions ϕ * ( n, p, λ, α ) which are analytic in the open unit disk △ = { z : | z | < 1}, and involve the combinations of the representations of p-valently starlike and convex functions. We obtain several generalized results on the modified-Hadamard product of the class ϕ * ( n, p, λ, α ), which extend the corresponding results obtained by Altintas et al. [Computers Math. Applic. 30(2), 9-16, (1995)]; Darwish, Aouf [Mathematical and Computer Modelling(2007), doi:10.1016/j.mcm.2007.08.016]. Futher, by fixing the second coefficient of functions in ϕ * ( n, p, λ, α ), we introduce the special class ϕc*(n,p,λ,α)$\phi _c^* (n,p,\lambda ,\alpha )$ and discuss the extreme points.

In this paper, we study Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis 3 . We characterize timelike biharmonic Legendre curves in terms of their curvature and torsion. Moreover, we obtain the position vectors of timelike biharmonic Legendre curves and we construct parametric equations of Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis 3 .

We find four q -analogues of general reduction formulas from Buschman and Srivastava together with some special cases, e.g. q -analogues of reduction formulas for Appell- and Kampé de Fériet functions. A proper q -analogue of the notation △( l ; λ ) by MacRobert, Meijer and Srivastava is given, and the definition of q -hypergeometric series is generalized accordingly.

The existence of positive periodic solutions for a class of second order impulsive differential equations is studied. By using fixed point theorem in cone, new existence results of positive periodic solutions are obtained without assuming the existence of positive periodic solutions of the corresponding continuous equation.

In this paper we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problem x(Δm)(t)=f(t,x(t)),     t∈T,x(0)=0,xΔ(0)=η1,…,x(Δ(m-1))(0)=ηm-1,    η1,…ηm-1∈E,$$\matrix{\hfill {x^{(\Delta m)} (t)} & { = f(t,x(t)),\;\;\;\;\;t \in T,} \hfill \cr \hfill {x(0)} & { = 0,} \hfill \cr \hfill {x^\Delta (0)} & { = \eta _1 , \ldots ,x^{(\Delta (m - 1))} (0) = \eta _{m - 1} ,\;\;\;\;\eta _1 , \ldots \eta _{m - 1} \in E,} \hfill \cr } $$ where x (Δ m ) denotes a m th order Δ - derivative, T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence ( a n ) in T and a n → ∞), E — a Banach space and f is a continuous function or satisfies Carathéodory’s conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result. As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

Let X , Y be p-homogeneous, complete F-spaces, r > 0, δ > 0. A function f : X → Y is an ( r, δ )-isometry, if |‖f(x)-f(y)‖r-‖x-y‖r|≤δ   for x,y∈X.$$\left| {\left\| {f(x) - f(y)} \right\|^r - \left\| {x - y} \right\|^r } \right| \le \delta \;\;\;{\rm{for}}\;x,y \in X.$$ We consider the following problem: does for each ε > 0, r > 0 there exist a δ > 0 such that for each surjective ( r, δ )- isometry f there exists an isometry U : X → Y such that sup x ∈ X ‖ f ( x ) − U ( x ) ‖ ≤ ε .

A fixed point theorem for three mappings on a metric space into itself is proved. This result extends the results obtained in [1] from two mappings to three mappings, and after that, a generalization for an arbitrary number of mappings is obtained. As corollaries of these results we obtain the extending of Theorems of Nešić, Rhoades, Chatterjea, Rus and Kannan for an arbitrary number of mappings.

Let φ and ψ be holomorphic maps on the open unit disk 𝔻 such that φ (𝔻) ⊂ 𝔻 and H (𝔻) be the space of holomorphic functions on 𝔻. For a non-negative integer n, define a linear operator Dφ,ψn$D_{\varphi ,\psi }^n $ as Dφ,ψnf=ψ⋅(f(n)οφ)$D_{\varphi ,\psi }^n f = \psi \cdot \left( {f^{(n)} \circ \varphi } \right)$, f ∈ H (𝔻). In this paper, we characterize boundedness and compactness of Dφ,ψn$D_{\varphi ,\psi }^n $ on the Bergman space 𝒜 2 . We also compute the upper and lower bounds of essential norm of this operator on the Bergman space.

In this paper, we consider a generalized integration operator Ig,φ(n)f(z)=∫0zf(n)(φ(ζ))g(ζ)dζ$$I_{g,\varphi }^{(n)} f(z) = \int\limits_0^z {f^{(n)} (\varphi (\zeta ))g(\zeta )d\zeta } $$ induced by holomorphic maps g and φ of the open unit disk 𝔻, where φ (𝔻) ⊂ 𝔻 and n is a positive integer. We characterize boundedness and compactness of Ig,φ(n)$I_{g,\varphi }^{(n)} $ from Bloch type spaces to weighted BMOA spaces by using logarithmic Carleson measure characterization of the weighted BMOA spaces.

In this paper we introduce the notion of semi-invariant submanifolds of a Lorentzian almost contact manifold. We study their principal characteristics and the particular cases in which the manifold is a Lorentzian Sasakian manifold or a Lorentzian Sasakian space form.

The notion of almost cl-supercontinuity (≡ almost clopen continuity) of functions (Acta Math. Hungar. 107 (2005), 193–206; Applied Gen. Topology 10 (1) (2009), 1–12) is extended to the realm of multifunctions. Basic properties of upper (lower) almost cl-supercontinuous multifunctions are studied and their place in the hierarchy of strong variants of continuity of multifunctions is discussed. Examples are included to reflect upon the distinctiveness of upper (lower) almost cl-supercontinuity of multifunctions from that of other strong variants of continuity of multifunctions which already exist in the literature.

We discuss properties of density-type topologies 𝒯 ψ connected with condition (Δ 2 ) similar to the condition considered in the theory of Orlicz spaces.

We find a way to calculate the term with lowest exponent in the Kauffman bracket polynomial of a closed 3-string braid. This leads us to a new proof of the faithfulness of the Temperley-Lieb representation of the 3-string braid group B 3 . In particular we will prove that for any link with braid index 3 either the coefficient of the lowest degree term or the coefficient of the highest degree term of the Jones polynomial is equal to ±1.