Volume 68 , issue 1 ( 2016 ) | back |

On starrable lattices | 1$-$11 |

**Abstract**

A starrable lattice is one with a cancellative semigroup structure satisfying $(x\vee y)(x\wedge y)=xy$. If the cancellative semigroup is a group, then we say that the lattice is fully starrable. In this paper, it is proved that distributivity is a strict generalization of starrability. We also show that a lattice $(X,\le)$ is distributive if and only if there is an abelian group $(G,+)$ and an injection $f:X\to G$ such that $f(x)+f(y)=f(x\vee y)+f(x\wedge y)$ for all $x,yın X$, while it is fully starrable if and only if there is an abelian group $(G,+)$ and a bijection $f:X\to G$ such that $f(x)+f(y)=f(x\vee y)+f(x\wedge y)$, for all $x,yın X$.

**Keywords:** Lattice; distributive lattice; starrable lattice

**MSC:** 06B99

Existence of a positive solution for a third-order three point boundary value problem | 12$-$25 |

**Abstract**

By applying the Krasnoselskii fixed point theorem in cones and the fixed point index theory, we study the existence of positive solutions of the non linear third-order three point boundary value problem $u'''(t)+a(t)f(t,u(t))=0$, $tın(0,1)$; $u'(0)=u'(1)=\alpha u(\eta)$, $u(0)=\beta u(\eta)$, where $\alpha$, $\beta$ and $\eta$ are constants with $\alphaın[0,\frac{1}{\eta})$, and $0<\eta<1$. The results obtained here generalize the work of Torres [Positive solution for a third-order three point boundary value problem, Electronic J. Diff. Equ. 2013 (2013), 147, 1--11].

**Keywords:** Third-order differential equations; three point boundary value problem; Krasnoselski fixed point in a cone; fixed point index theory

**MSC:** 34B10

Codes over hyperrings | 26$-$38 |

**Abstract**

Hyperrings are essentially rings, with approximately modified axioms in which addition or multiplication is a hyperoperation. In this paper, we focus on an important subclass of codes with additional structure called linear codes. Indeed, we introduce the notion of linear codes on finite hyperrings and we present a construction technique of cyclic codes over finite hyperrings. Since polynomial hyperrings are one of the main tools in our study, we analyze them too.

**Keywords:** hyperring; hyperring of polynomials; hyperideal; linear code; cyclic code

**MSC:** 20N20; 16Y99, 94B05

Some remarks on sequence selection properties using ideals | 39$-$44 |

**Abstract**

In this paper we follow the line of recent works of Das and his co-authors where certain results on open covers and selection principles were studied by using the notion of ideals and ideal convergence, which automatically extend similar classical results (where finite sets are used). Here we further introduce the notions of $ıc$-Sequence Selection Property ($ıc$-SSP), $ıc$-Monotonic Sequence Selection Property ($ıc$-MSSP) of $C_p(X)$ which extend the notions of Sequence Selection Property and Monotonic Sequence Selection Property of $C_p(X)$ respectively. We then make certain observations on these new types of SSP in terms of $\Cal{I}\tx{-}\gamma$-covers.

**Keywords:** Ideal convergence; $ıc$-SSP; $ıc$-MSSP; $ıga$-cover; $ıc$-Hurewicz property; $\SONE(ıGa,ıGa)$-space; $ıc\text{-}\gamma\gamma_{co}$-space

**MSC:** 54G99; 03E05, 54C30, 40G15

Integral inequalities of Jensen type for $\lambda$-convex functions | 45$-$57 |

**Abstract**

Some integral inequalities of Jensen type for $\lambda $-convex functions defined on real intervals are given.

**Keywords:** Convex function; discrete inequality; $\lambda$-convex function; Jensen type inequality

**MSC:** 26D15; 25D10

Directed proper connection of graphs | 58$-$65 |

**Abstract**

An edge-colored directed graph is called properly connected if, between every pair of vertices, there is a properly colored directed path. We study some conditions on directed graphs which guarantee the existence of a coloring that is properly connected. We also study conditions on a colored directed graph which guarantee that the coloring is properly connected.

**Keywords:** directed graph; edge coloring; proper connection

**MSC:** 05C15; 05C20

Construction of cospectral regular graphs | 66$-$76 |

**Abstract**

Graphs $G$ and $H$ are called cospectral if they have the same characteristic polynomial, equivalently, if they have the same eigenvalues considering multiplicities. In this article we introduce a construction to produce pairs of cospectral regular graphs. We also investigate conditions under which the graphs are integral.

**Keywords:** Eigenvalue; cospectral graphs; adjacency matrix; integral graphs

**MSC:** 05C50