Volume 67 , issue 1 ( 2015 ) | back |

Constructions of $(m,n)$-hyperrings | 1$-$16 |

**Abstract**

In this paper, the class of $(m,n)$-hyperrings is introduced and several properties are found and examples are presented. $(m,n)$-hyperrings are a generalization of hyperrings. We define the fundamental relation $\Gamma^*$ on an $(m,n)$-hyperring $R$ such that $R/\Gamma^*$ is the smallest $(m,n)$-ring, and then some related properties are investigated.

**Keywords:** $(m,n)$-hyperring; $n$-ary operation; $n$-ary hyperoperation; $n$-ary hypergroup; fundamental equivalence relation.

**MSC:** 16Y99

Power mean inequality of generalized trigonometric functions | 17$-$25 |

**Abstract**

The author here studies the convexity and concavity properties of the generalized $p$-trigonometric functions in the sense of P. Lindqvist with respect to the Power Mean.

**Keywords:** Eigenfunctions $\sin_p$; Power Mean; generalized trigonometric functions.

**MSC:** 33C99, 33B99

Fixed points of a pair of locally contractive mappings in ordered partial metric spaces | 26$-$38 |

**Abstract**

Common fixed point results for mappings satisfying locally contractive conditions on a closed ball in a $0$-complete ordered partial metric space have been established. The notion of dominated mappings of Economics, Finance, Trade and Industry has also been applied to approximate the unique solution to non-linear functional equations. Our results improve some well-known, primary and conventional results.

**Keywords:** Common fixed point; Banach mapping; closed ball; dominated mapping, $0$-complete partial metric space.

**MSC:** 47H10, 46S40, 54H25

Remote filters and discretely generated spaces | 39$-$51 |

**Abstract**

Alas, Junqueira and Wilson asked whether there is a discretely generated locally compact space whose one point compactification is not discretely generated and gave a consistent example using CH. Their construction uses a remote filter in $\omega\times{}^{\omega}2$ with a base of order type $\omega_1$ when ordered modulo compact subsets. In this paper we study the existence and preservation (under forcing extension) of similar types of filters, mainly using small uncountable cardinals. With these results we show that the CH example can be constructed in more general situations.

**Keywords:** Discretely generated space; one-point compactification; remote point; small uncountable cardinals.

**MSC:** 54A25, 54D40, 54D80, 54A35, 03E17

A uniqueness result for the Fourier transform of measures on the paraboloid | 52$-$55 |

**Abstract**

A finite measure supported by a paraboloid of revolution $\Sigma$ in $\Bbb R^3$ and absolutely continuous with respect to the natural measure on $\Sigma$ is entirely determined by the restriction of its Fourier transform to a plane if and only if this plane is normal to the axis of $\Sigma$.

**Keywords:** Heisenberg uniqueness; Fourier transform; measure; paraboloid

**MSC:** 42B10, 46F12

On the polar derivative of a polynomial | 56$-$60 |

**Keywords:** Polynomials; inequalities in the complex domain; polar derivative; Bernstein's inequality.

**MSC:** 30A10, 30C10, 30E10

Ding projective modules with respect to a semidualizing module | 61$-$72 |

**Abstract**

In this paper, for a fixed semidualizing module $C$, we introduce the notion of $D_C$-projective modules which are the special setting of $G_C$-projective modules introduced by White [D. White, Gorenstein projective dimension with respect to a semidualizing module, J. Commut. Algebra 2(1) (2010) 111--137]. Then we investigate the properties of $D_C$-projective modules and dimensions, in particular, we give descriptions of the finite $D_C$-projective dimensions.

**Keywords:** semidualizing; $D_C$-projective module; $C$-projective module.

**MSC:** 13B02, 13D05

Some remarks on paramedial semigroups | 73$-$77 |

**Abstract**

Semigroups satisfying some type of generalized commutativity were considered in quite a number of papers. S. Lajos, A. Nagy and M. Yamada dealed with externally commutative semigroups. N. Stevanović and P. V. Protić in [Structure of weakly externally commutative semigroups, Algebra Colloq. 13:3 (2006) 441-446], introduced the notion of weakly externally commutative semigroup and gave a structural description for some subclasses of this class of semigroups. In this paper we consider a class which is a generalization of the class of externally commutative semigroups.

**Keywords:** (Weakly) externally commutative semigroups; congruence; semilattice decomposition.

**MSC:** 20N02