Volume 66 , issue 3 ( 2014 ) | back |

The (co)shape and (co)homological properties of continuous maps | 235$-$247 |

**Abstract**

The purpose of this paper is to investigate continuous maps from the standpoint of geometric topology and algebraic topology. Using a direct system approach and an inverse system approach of continuous maps, we study the (co)shape and (co)homological properties of continuous maps. Applications of the obtained results include the constructions of long exact sequences of continuous maps for the (co)homology pro-groups and (co)homology inj-groups, spectral Čech (co)homology groups and spectral singular (co)homology groups.

**Keywords:** Inverse system; direct system; pro-category; inj-category; shape category; coshape category; pro-group; inj-group;
Čech (co)homology group; singular (co)homology group; long exact sequence of map.

**MSC:** 54C56, 55N05, 55U40

On certain subclass of analytic functions defined by convolution | 248$-$264 |

**Abstract**

In this paper we use the principle of subordination between analytic functions and the convolution to introduce the class $\tilde S(f,g;A,B;\alpha,\beta)$. We obtain coefficient inequalities, distortion theorems, extreme points, radii of close to convexity, starlikeness and convexity for this class and modified Hadamard product of several functions belonging to it. Also, we investigate several distortion inequalities involving fractional calculus. Finally, we obtain integral means for functions belonging to this class.

**Keywords:** Univalent functions; subordination; Hadamard product; fractional calculus operators; integral means.

**MSC:** 30C45

On countably nearly paracompact spaces | 265$-$273 |

**Abstract**

The idea of countable near paracompactness of a topological space, as a natural offshoot of the well known concept of near paracompactness, is introduced and investigated in this article. In the process, the notion of semi nearly normal spaces is initiated. Apart from its characterizations, semi near normality is used for the investigation of countably nearly paracompact spaces.

**Keywords:** Near paracompactness; countable near paracompactness; semi nearly normal space; regular open set; $\delta$-open set.

**MSC:** 54D20, 54D99

Global existence and boundedness of solutions for a general activator-inhibitor model | 274$-$282 |

**Abstract**

The purpose of this paper is to prove global existence in time of solutions for a class of multi-component reaction diffusion systems called multiple Gierer-Meinhardt type. The system describes, following Gierer-Meinhardt's scheme, ``m'' substances in interaction. The nonlinearities present a difficulty since they are fractions. We prove the global existence by using a series of Lyapunov functionals.

**Keywords:** Reaction-diffusion system; global existence; Lyapunov functional.

**MSC:** 35K45, 35K57, 35K45

On some new mixed modular equations involving Ramanujan's theta-functions | 283$-$293 |

**Abstract**

In his second notebook, Ramanujan recorded altogether 23 $P$--$Q$ modular equations involving his theta functions. In this paper, we establish several new mixed modular equations involving Ramanujan's theta-functions $\varphi$ and $\psi$ which are akin to those recorded in his notebook.

**Keywords:** Modular equations; theta-functions.

**MSC:** 33D10, 11A55, 11F27

Some results on local spectral theory of composition operators on $l^p$ spaces | 294$-$300 |

**Abstract**

In this paper, we give a condition under which a bounded linear operator on a complex Banach space has Single Valued Extension Property (SVEP) but does not have decomposition property~$(\delta)$. We also discuss the analytic core, decomposability and SVEP of composition operators $C_\phi$ on $l^p$ $(1\leq p<ınfty)$ spaces. In particular, we prove that if $\phi$ is onto but not one-one then $C_\phi$ is not decomposable but has SVEP. Further, it is shown that if $\phi$ is one-one but not onto then $C_\phi$ does not have SVEP.

**Keywords:** Analytic core; composition operator; decomposability; decomposition property $(\delta)$; single valued extension property.

**MSC:** 47A10, 47A11, 47B33, 47B40

Trigonometric polynomial rings and their factorization properties | 301$-$314 |

**Abstract**

Consider the rings $S$ and $S^{\prime }$, of real and complex trigonometric polynomials over the field ${Q}$ and its algebraic extension ${Q}(i)$ respectively. Then $S$ is an FFD, whereas $S^{\prime}$ is a Euclidean domain. We discuss irreducible elements of $S$ and $S^{\prime}$, and prove a few results on the trigonometric polynomial rings $T$ and $T^{\prime}$ introduced by G. Picavet and M. Picavet in [Trigonometric polynomial rings, Commutative ring theory, Lecture notes on Pure Appl. Math., Marcel Dekker, Vol. 231 (2003), 419--433]. We consider several examples and discuss the trigonometric polynomials in terms of irreducibles (atoms), to study the construction of these polynomials from irreducibles, which gives a geometric view of this study.

**Keywords:** trigonometric polynomial, HFD, irreducible, wave.

**MSC:** 13A05, 13B30, 12D05

A note on modular curves and fundamental units of negative norm | 315$-$316 |

**Abstract**

We re-prove the fact that the fundamental unit of the ring of integers of a real quadratic number field is of negative norm whenever the discriminant is a prime number congruent to $1 \mod 4$.

**Keywords:** Hilbert modular surface, fundamental unit.

**MSC:** 11F41, 11R11

New version of property $(az)$ | 317$-$322 |

**Abstract**

In this paper we define new spectral properties $(h)$, $(gh)$, $(ah)$ and $(gah)$ as a continuation of [H. Zariouh, Property (gz) for bounded linear operators, Mat. Vesnik, 65 (2013), 94--103], which are variants to the properties $(z)$, $(gz)$ and $(az)$ introduced in the mentioned paper. The purpose of the paper is to study the relationship between these properties and other Weyl-type theorems.

**Keywords:** Property $(h)$, property $(ah)$, Weyl-type theorems.

**MSC:** 47A53, 47A10, 47A11

Compositions of Saigo fractional integral operators with generalized Voigt function | 323$-$332 |

**Abstract**

The principal object of this paper is to provide the composition of Saigo fractional integral operators with different forms of Voigt functions. An alternative explicit representation of the generalized Voigt function in terms of Laplace integral transform is shown and the relations between the left-sided and the right-sided Saigo fractional integral operators are established with the $_1F_1$-transform and the Whittaker transform, respectively. Many interesting results are deduced in terms of some relatively more familiar hypergeometric functions in one and two variables.

**Keywords:** Fractional operators, Voigt function, Whittaker transform, $_1{F}_1$-transform, Bessel function, Kampé de Fériet's function.

**MSC:** 26A33, 44A15, 33C15, 33C55

On linear maps approximately preserving the approximate point spectrum or the surjectivity spectrum | 333$-$342 |

**Abstract**

Let $X$ and $Y$ be superreflexive complex Banach spaces and let $\Cal{L}(X)$ and $\Cal{L}(Y)$ be the Banach algebras of all bounded linear operators on $X$ and $Y$, respectively. We describe a linear map $\phi:\Cal{L}(X)\to\Cal{L}(Y)$ that almost preserves the approximate point spectrum or the surjectivity spectrum. Furthermore, in the case where $X=Y$ is a separable complex Hilbert space, we show that such a map is a small perturbation of an automorphism or an anti-automorphism.

**Keywords:** Surjectivity spectrum; pseudo surjectivity spectrum; approximate point spectrum;
pseudo approximate point spectrum; approximately multiplicative map.

**MSC:** 47B48, 47A10, 46H05