Abstract

In this paper, we give a proof for Sehgal-Guseman theorem of fixed point in rectangular $b$-metric spaces. Our result is supported with a suitable example. As a corollary of our results, we obtain fixed point results of contraction mappings in $b$-metric spaces.

Keywords: Fixed points, rectangular $b$-metric space.

MSC: 47H10

Abstract

In earlier works we have investigated the densest packings and the thinnest coverings by congruent hyperballs based on the regular prism tilings in $n$-dimensional hyperbolic space $\HYN$ ($ 3 \le n \in \mathbb{N})$. In this paper we study a large class of hyperball packings in $\HYP$ that can be derived from truncated tetrahedron tilings. In order to get an upper bound for the density of the above hyperball packings, it is sufficient to determine this density upper bound locally, e.g. in truncated tetrahedra. Thus we prove that if the truncated tetrahedron is regular, then the density of the densest packing is $\approx 0.86338$. This is larger than the Böröczky-Florian density upper bound for balls and horoballs. Our locally optimal hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$, but we describe a hyperball packing construction, by the regular truncated tetrahedron tiling under the extended Coxeter group $[3, 3, 7]$ with maximal density $\approx 0.82251$. Moreover, we show that the densest known hyperball packing, related to the regular $p$-gonal prism tilings, can be realized by a regular truncated tetrahedron tiling as well.

Keywords: Hyperbolic geometry; hyperball packings; packing density; Coxeter tilings.

MSC: 52C17, 52C22, 52B15

Abstract

In this note we construct a commutative diagram in filtered Lagrangian Floer homology that involves a product of certain Lagrangian submanifolds. As a corollary, we prove the Künneth formula for Lagrangian Floer homology. We also prove that the Künneth formula for Lagrangian Floer homology lifts through a Lagrangian type of Piunikhin-Salamon-Schwarz map to the Künneth formula for Morse homology.

Keywords: Filtered Lagrangian Floer homology; Künneth formula; PSS isomorphism.

MSC: 53D40, 55U25, 53D12

Abstract

Based on the recent important result of S. S. Chang, H. W. Joseph Lee, C. K. Chan, L. Wang, L.J. Qin [Appl. Math. Comput. 219 (2013) 10416--10424], we study in this article the split feasibility problem for a countable family of multi-valued $\kappa-$strictly pseudo-contractive mappings and total asymptotically strict pseudo-contractive mappings in infinite dimensional Hilbert spaces. The main results presented in this paper improve and extend the aforementioned result.

Keywords: Split feasibilty problem; multi- valued mappings; pseudocontractive mapping; total asymptotically strict pseudocontractive mapping; Hilbert space.

MSC: 47H09, 47H10, 49J20, 49J40

Abstract

In 1999, De Smet et al.\ conjectured the generalized Wintgen inequality for submanifolds in real space forms. This conjecture is also known as the DDVV conjecture and it was proved by Ge and Tang. Recently, Mihai established such inequality for Lagrangian submanifold in complex space forms. In this paper, we obtain the generalized Wintgen inequality for bi-slant submanifolds in locally conformal Kaehler space forms. Further, we discuss the particular cases of this inequality i.e.\ for semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, invariant submanifolds and anti-invariant submanifolds in the same ambient space.

Keywords: Bi-slant submanifold; locally conformal Kaehler space forms; Wintgen inequality.

MSC: 53B05, 53B20, 53C40

Abstract

A space $X$ is {\it weakly star countable} if for each open cover $U$ of $X$ there exists a countable subset $F$ of $X$ such that $\overline{\bigcup_{x\in F}St(x, U)}=X$. In this paper, we investigate topological properties of weakly star countable spaces.

Keywords: Star countable; weakly star countable.

MSC: 54D20, 54A35

Abstract

The inclusions of classes of functions with bounded partial $\Lambda$-variation into the classes of functions with bounded total harmonic variation are established. The result is applied to the problem of convergence of rectangular partial sums for multiple Fourier series.

Keywords: Generalized variation; multiple Fourier series.

MSC: 42B05, 26B30

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The degree of a vertex $a\in V(G)$ is denoted by $d_{G}(a)$. The general sum-connectivity index of $G$ is defined as $\chi_{\alpha}(G)=\sum_{ab\in E(G)}(d_{G}(a)+d_{G}(b))^{\alpha}$, where $\alpha$ is a real number. In this paper, we compute exact formulae for general sum-connectivity index of several graph operations. These operations include tensor product, union of graphs, splices and links of graphs and Hajós construction of graphs. Moreover, we also compute exact formulae for general sum-connectivity index of some graph operations for positive integral values of $\alpha$. These operations include cartesian product, strong product, composition, join, disjunction and symmetric difference of graphs.

Keywords: General sum-connectivity index; graph operations.

MSC: 05C07