Authors:John Baptist Gauci, Jean Paul Zerafa Abstract: The Erd$\H{o}$s-Faber-Lovász Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of $G$ can be properly coloured using $n$ colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining $n$-cliques. We here provide a quick and easy algorithm to colour the vertices of $G$ in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.
Authors:Preeti Gupta, Sunil Hans, Abdullah Mir Abstract: In this paper, we shall obtain the bounds for the derivative of a rational function in the supremum norm on the unit circle in both the directions by involving the moduli of all its zeros. The obtained results strengthen some recently proved results.
Authors:Khangembam Babina Devi, Kshetrimayum Krishnadas, Barchand Chanam Abstract: Let $p(z)$ be a polynomial of degree $n$ having no zero in $ z < k$, $k\leq 1$, then Govil [Proc. Nat. Acad. Sci., $\textbf{50}$, (1980), 50-52] proved
$\max\limits_{ z =1} p'(z) \leq \dfrac{n}{1+k^{n}}\max\limits_{ z =1} p(z) $,
provided $ p'(z) $ and $ q'(z) $ attain their maxima at the same point on the circle $ z =1$, where
Authors:Aziz Blali, Rachid Ameziane Hassani, Abdelkhalek El Amrani, Mouad El Beldi Abstract: In this paper, we introduce tensor product semigroups of operators on locally convex spaces. The basic properties are presented. We give multiple relations between the tensor product semigroups and its components. The generator of such semigroups is studied.
Authors:Antonio Leaci, Franco Tomarelli Abstract: We establish some notation and properties of the bilateral Riemann-Liouville fractional derivative $D^s.$ We introduce the associated Sobolev spaces of fractional order $s$, denoted by $W^{s,1}(a,b)$, and the Bounded Variation spaces of fractional order $s$, denoted by $BV^{s}(a,b)$: these spaces are studied with the aim of providing a suitable functional framework for fractional variational models in image analysis.
Authors:Alberto Cavicchioli, Fulvia Spaggiari Abstract: We completely recognize the topological structure of the ten compact euclidean space forms with special minimal tetrahedra, constructed by face pairings in nice papers of Molnár [8-9]. From these polyhedral descriptions we derive special presentations with two generators for the fundamental groups of the considered manifolds. Our proofs also show that such group presentations completely characterize the euclidean space forms among closed connected $3$-manifolds. The results have also didactical importance.