Volume 70 , issue 3 ( 2018 ) | back |

On the $L^p$ boundedness of semiclassical Fourier integral operators | 189$-$203 |

**Abstract**

In this paper, we investigate the $L^p$-boundedness of semiclassical Fourier integral operators defined by symbols $a(x,\xi)$ which behave in the spatial variable $x$ like $L^p$ functions and are smooth in the $\xi$ variable.

**Keywords:** $h$-Fourier integral operators; symbol and phase; $L^p$-boundedness.

**MSC:** 35S30

A fixed point theorem for mappings with a contractive iterate in rectangular $b$-metric spaces | 204$-$210 |

**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

Hyperball packings in hyperbolic $3$-space | 211$-$221 |

**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

Filtered Lagrangian Floer homology of product manifolds | 222$-$232 |

**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

Split feasibility problem for countable family of multi-valued nonlinear mappings | 233$-$242 |

**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

Generalized wintgen inequality for bi-slant submanifolds in locally conformal Kaehler space forms | 243$-$249 |

**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

Remarks on weakly star countable spaces | 250$-$256 |

**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

On the classes of functions of bounded partial and total $\Lambda$-variation | 257$-$266 |

**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

Exact formulae of general sum-connectivity index for some graph operations | 267$-$282 |

**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 Haj\'{o}s 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