Volume 76 , issue 1-2 ( 2024 ) | back |

75 YEARS OF "MATEMATIČKI VESNIK" | 1$-$3 |

BI-CLOSING WORDS | 5$-$14 |

**Abstract**

We will show that factor codes that have a bi-closing word are bi-closing a.e. and have a degree. Moreover, a closing code with a bi-closing word into an irreducible shift space is constant-to-one. Then we study which properties may be invariant by codes that have a bi-closing word under factoring and extension. Moreover, we give some conditions for a bi-closing word and show that every closing open code in an irreducible shift space has a bi-closing word.

**Keywords:** Shift of finite type; sofic; synchronized; coded; closing; hyperbolic.

**MSC:** 37B10

COUPON COLLECTOR PROBLEM WITH PENALTY COUPON | 15$-$28 |

**Abstract**

In this paper we consider a generalization of the coupon collector problem where we assume that the set of available coupons consists of standard coupons and an additional penalty coupon, which does not belong to the collection and interferes with collecting standard coupons. Applying Markov chain approach the following problem is solved: how many coupons (on average) one has to purchase in order to complete a collection without interference or to collect $n$ more penalty coupons than standard coupons. Also, we obtain additional results related to the distribution of the waiting time until the collection is sampled without interference or until $n$ more penalty coupons than standard coupons is sampled.

**Keywords:** Coupon collector problem; penalty coupon; waiting time; Markov chain; transition probability matrix; random walk.

**MSC:** 60C05

ON THE GENERALIZED DISTANCE EIGENVALUES OF GRAPHS | 29$-$42 |

**Abstract**

For a simple connected graph $G$, the generalized distance matrix $D_{\alpha}(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. The largest eigenvalue of $D_{\alpha}(G)$ is called the generalized distance spectral radius or $D_{\alpha}$-spectral radius of $G$. In this paper, we obtain some upper bounds for the generalized distance spectral radius in terms of various graph parameters associated with the structure of graph $G$, and characterize the extremal graphs attaining these bounds. We determine the graphs with minimal generalized distance spectral radius among the trees with given diameter $d$ and among all unicyclic graphs with given girth. We also obtain the generalized distance spectrum of the square of the cycle and the square of the hypercube of dimension $n$. We show that the square of the hypercube of dimension $n$ has three distinct generalized distance eigenvalues.

**Keywords:** Generalized distance matrix (spectrum); spectral radius; hypercube; unicyclic graph.

**MSC:** 05C50, 05C12, 15A18

SOME NEW OBSERVATIONS ON $w$-DISTANCE AND $F$-CONTRACTIONS | 43$-$55 |

**Abstract**

The aim of this paper is to present some new observations about $w$-distance (in the sense of O. Kada, T. Suzuki, W. Takahashi, \emph{Nonconvex minimization theorems and fixed point theorems in complete metric spaces}, Math.\ Japonica \textbf{44}, 2 (1996), 381--391) and $F$-contractions (in the sense of D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces}, Fixed Point Theory Appl., \textbf{2012}:94 (2012)). Both concepts have been examined separately a lot, but there have been few attempts to connect them. This article is a step in filling this gap. Besides, some comments and improvements of results in the existing literature are presented.

**Keywords:** $w$-distance; $wt$-distance; $F$-contraction; $b$-metric space; metric-like space.

**MSC:** 47H10, 54H25

ON THE LOCAL CONTROLLABILITY FOR OPTIMAL CONTROL PROBLEMS | 56$-$65 |

**Abstract**

We consider an optimal control problem on the fixed interval of time with the right endpoint constraint. We introduce the concept of controllability for this problem. The main result of the paper states that if for the optimal control problem the Pontryagin maximum principle fails on the given admissible process then this process satisfies the controllability condition.

**Keywords:** Optimal control; controllability; Pontryagin's maximum principle.

**MSC:** 49K15

AN EFFICIENT SOLUTION APPROACH TO THE $p$-NEXT CENTER PROBLEM | 66$-$83 |

**Abstract**

An extension of the $p$-center problem, called the $p$-next center problem, is considered in this paper. In practice, it has been shown that centers can close suddenly due to a problem (accident, staff shortage, technical problem, etc.). In this case, customers should proceed to the backup center - the one closest to the closed center. Both the $p$-center problem and the $p$-next center problem are NP-hard, so approximation methods are suitable for solving them. In this paper, an efficient solution approach based on Skewed Variable Neighborhood Search (SVNS) is proposed for the $p$-next center problem. The performance of the proposed SVNS method is evaluated on a set of pmed instances with up to 900 nodes. The obtained computational results are presented and compared with the best results from the literature, confirming the efficiency and stability of the proposed method in solving the $p$-next center problem.

**Keywords:** Combinatorial optimization; $p$-next center problem; variable neighborhood search; fast interchange heuristic.

**MSC:** 90C27, 90C59, 90B80

UNCERTAINTY PRINCIPLES ASSOCIATED WITH THE SHORT TIME QUATERNION COUPLED FRACTIONAL FOURIER TRANSFORM | 84$-$104 |

**Abstract**

In this paper, we extend the coupled fractional Fourier transform of complex valued functions to that of the quaternion valued functions on $\mathbb{R}^4$ and call it the quaternion coupled fractional Fourier transform (QCFrFT). We obtain the sharp Hausdorff-Young inequality for QCFrFT and obtain the associated Rényi uncertainty principle. We also define the short time quaternion coupled fractional Fourier transform (STQCFrFT) and explore its important properties followed by the Lieb's and entropy uncertainty principles.

**Keywords:** Quaternion coupled fractional Fourier transform; short time quaternion coupled fractional Fourier transform; Lieb's uncertainty principle.

**MSC:** 11R52, 42B10, 42A05

GEODESICS OF RIEMANNIAN COMPLEX HYPERBOLIC PLANE | 105$-$117 |

**Abstract**

The complex hyperbolic plane is a symmetric space of negative sectional curvature; hence, it has the structure of a 4-dimensional connected solvable real Lie group with a left-invariant metric. We consider all non-isometric left-invariant Riemannian metrics on this group, denoted by ${\mathcal {CH}}^2$, and search for real geodesics corresponding to them. Using Euler-Arnold equations, one can translate the second-order differential equations of the geodesics on the group into the first-order equations on its Lie algebra. In the Kähler case we solve these equations on the Lie algebra of ${\mathcal {CH}}^2$, i.e. we explicitly find curves on algebra corresponding to the geodesics of the standard Einstein metric. Numerical solutions are used to visualize geodesic lines and geodesic spheres of various left-invariant Riemannian metrics.

**Keywords:** Complex hyperbolic plane; left-invariant metric; Euler-Arnold equations; geodesic lines; geodesic spheres.

**MSC:** 53C22, 22E60

DENSE BALL PACKINGS BY TUBE MANIFOLDS AS NEW MODELS FOR HYPERBOLIC CRYSTALLOGRAPHY | 118$-$135 |

**Abstract**

We intend to continue our previous papers on dense ball packing hyperbolic space $\mathbf{H}^3$ by equal balls, but here with centres belonging to different orbits of the fundamental group $\boldsymbol{Cw}(2z, 3 \le z \in \mathbb{N}$, odd number), of our new series of {\it tube or cobweb manifolds} $Cw = \mathbf{H}^3/\boldsymbol{Cw}$ with $z$-rotational symmetry. As we know, $\boldsymbol{Cw}$ is a fixed-point-free isometry group, acting on $\mathbf{H}^3$ discontinuously with appropriate tricky fundamental domain $Cw$, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every $\boldsymbol{Cw}(2z)$ is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group $\boldsymbol{W}(u; v; w = u)$ that is a complete Coxeter orthoscheme reflection group, extended by the half-turn $\boldsymbol{h}$ $(0 \leftrightarrow 3, 1 \leftrightarrow 2)$ of the complete orthoscheme $A_0A_1A_2A_3 \sim b_0b_1b_2b_3$ (see Figure 1). The vertices $A_0$ and $A_3$ are outer points of the (Beltrami-Cayley-Klein) B-C-K model of $\mathbf{H}^3$, as $1/u + 1/v \le 1/2$ is required, $3 \le u = w, v$ for the above orthoscheme parameters. For the above simple manifold-construction we specify $u = v = w = 2z$. Then the polar planes $a_0$ and $a_3$ of $A_0$ and $A_3$, respectively, make complete with reflections $\boldsymbol{a}_0$ and $\boldsymbol{a}_3$ the Coxeter reflection group, where the other reflections are denoted by $\boldsymbol{b}^0$, $\boldsymbol{b}^1$, $\boldsymbol{b}^2$, $\boldsymbol{b}^3$ in the sides of the orthoscheme $b^0b^1b^2b^3$. The situation is described first in Figure 1 of the half trunc-orthoscheme $W$ and its usual extended Coxeter diagram, moreover, by the scalar product matrix $(b^{ij}) = (\langle \boldsymbol{b}^i, \boldsymbol{b}^j \rangle)$ in formula (1) and its inverse $(A_{jk}) = (\langle \boldsymbol{A}_j, \boldsymbol{A}_k \rangle)$ in (3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme $W$, then its ball packings, densities, then those of the manifolds $\boldsymbol{Cw}(2z)$. As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general $W(u;v;w=u)$ as well, summarized at the end of our paper.

**Keywords:** Infinite series of hyperbolic space forms; cobweb or tube manifold derived by an extended complete Coxeter orthoscheme reflection group; ball packing by group orbits; optimal dense packing; hyperbolic crystallography.

**MSC:** 57M07, 57M60, 52C17

EMBEDDINGS TO RECTILINEAR SPACE AND GROMOV--HAUSDORFF DISTANCES | 136$-$148 |

**Abstract**

We show that the problem whether a given finite metric space can be embedded into $m$-dimensional rectilinear space can be reformulated in terms of the Gromov--Hausdorff distance between some special finite metric spaces.

**Keywords:** Gromov--Hausdorff distance; rectilinear space; isometric embeddings; finite metric spaces; normed spaces.

**MSC:** 46B85, 51F99, 53C23

$(\mathbb{ C}^{\ast})^{k}$-action on $\mathbb{ C} P^{N}$ | 149$-$158 |

**Keywords:** Algebraic torus actions; complex projective spaces; Grassmann manifolds; moment map.

**MSC:** 14M15, 14M25, 57S12, 53D20