Volume 71 , issue 1-2 ( 2019 ) | back |

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

Analytic solution of fractional advection dispersion equation with decay for contaminant transport in porous media | 5$-$15 |

**Abstract**

Advection and dispersion are the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. The aim of this paper is to find concentration of contaminant in flowing groundwater using fractional advection dispersion equation with decay involving Hilfer derivative with respect to time. Time fractional advection-dispersion equation describe particle's motion with memory in time. The solution of time fractional advection dispersion equation with decay is obtained in terms of Mittag-Leffler function and Green function. The effect of the decay is to reduce mass and concentration of the solution, which is a function of time and space variable.

**Keywords:** Time fractional advection dispersion equation; decay rate coefficient; Fourier transform; Laplace transform; Hilfer derivative; Mittag-Leffler function.

**MSC:** 34A08, 26A33

The existence of homogeneous geodesics in special homogeneous Finsler spaces | 16$-$22 |

**Abstract**

A well known result by O. Kowalski and J. Szenthe says that any homogeneous Riemannian manifold admits a homogeneous geodesic through any point. This was proved by the algebraic method using the reductive decomposition of the Lie algebra of the isometry group. In previous papers by the author, the existence of a homogeneous geodesic in any homogeneous pseudo-Riemannian manifold and also in any homogeneous affine manifold was proved. In this setting, a new method based on affine Killing vector fields was developed. Using this method, it was further proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic and any homogeneous Finsler space of odd dimension admits a homogeneous geodesic. In the present paper, the affine method is further refined for Finsler spaces and it is proved that any homogeneous Berwald space or homogeneous reversible Finsler space admits a homogeneous geodesic through any point.

**Keywords:** Homogeneous Finsler space; homogeneous geodesic.

**MSC:** 53C22, 53C60, 53C30

On fixed points in the context of $\boldsymbol b$-metric spaces | 23$-$30 |

**Abstract**

In this paper, we obtain sufficient conditions for the existence of common fixed points in the framework of ordered $b$-metric spaces. Our results generalize some recent results in the literature. Also, to illustrate the usability of the results we give an adequate example in which $b$-metric is not continuous.

**Keywords:** Fixed point; $C$-class function; dominating mappings; $b$-metric space.

**MSC:** 47H10, 54H25

$LM$-fuzzy metric spaces and convergence | 31$-$44 |

**Abstract**

We study a category of lattice-valued metric spaces that contains many categories of lattice-valued metric spaces studied before. Furthermore, introducing suitable convergence structures, we are able to show that the category of these lattice-valued metric spaces is a coreflective subcategory of the category of lattice-valued convergence spaces and we even characterize lattice-valued metric spaces by convergence.

**Keywords:** $LM$-fuzzy metric space; $L$-convergence tower space; $M$-valued distance distribution function.

**MSC:** 54A20, 54A40, 54E35, 54E70

Applications of PDEs to the study of affine surface geometry | 45$-$62 |

**Abstract**

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}\rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde\nabla)$ be another affine structure on $M$ which is strongly projectively flat. We show that $\mathcal{Q}(\mathcal{M})=\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\nabla=\tilde\nabla$ and that $\mathcal{Q}(\mathcal{M})$ is linearly equivalent to $\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\mathcal{M}$ is linearly equivalent to $\tilde{\mathcal{M}}$. We use these observations to classify the flat Type $\mathcal{A}$ connections up to linear equivalence, to classify the Type $\mathcal{A}$ connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type $\mathcal{A}$ connections where the Ricci tensor is non-degenerate up to affine equivalence.

**Keywords:** Type $\mathcal{A}$ affine surface; quasi-Einstein equation; affine Killing vector field; locally homogeneous affine surface.

**MSC:** 53C21

Spectral approximation of a strain-limiting nonlinear elastic model | 63$-$89 |

**Abstract**

We construct a numerical algorithm for the approximate solution of a nonlinear elastic limiting strain model based on the Fourier spectral method. The existence and uniqueness of the numerical solution are proved. Assuming that the weak solution to the boundary-value problem possesses suitable Sobolev regularity, the sequence of numerical solutions is shown to converge to the weak solution of the problem at an optimal rate. The numerical method represents a finite-dimensional system of nonlinear equations. An iterative method is proposed for the approximate solution of this system of equations and is shown to converge, at a linear rate, to the unique solution of the numerical method. The theoretical results are illustrated by numerical experiments.

**Keywords:** Spectral method; convergence; nonlinear elasticity; limiting strain models.

**MSC:** 65N35, 74B20

On rational Pontryagin homology ring of the based loop space on a four-manifold | 90$-$103 |

**Abstract**

In this paper we consider the based loop space $\Omega M$ on a simply connected manifold $M$. We first prove, only by means of the rational homotopy theory, that the rational homotopy type of $\Omega M$ is determined by the second Betti number $b_{2}(M)$. We further consider the problem of computation of the rational Pontryagin homology ring $H_{*}(\Omega M)$ when $b_{2}(M)\leq 3$. We prove that $H_{*}(\Omega M)$ is up to degree $5$ generated by the elements of degree $1$ for $b_{2}(M)=3$.

**Keywords:** Rational Pontryagin homology; based loop space; four-manifolds.

**MSC:** 55P35, 55P62, 57N13

Note on combinatorial structure of self-dual simplicial complexes | 104$-$122 |

**Abstract**

Simplicial complexes $K$, in relation to their Alexander dual $\widehat{K}$ , can be classified as self-dual ($K=\widehat{K}$), sub-dual ($K\subseteq \widehat{K}$), super-dual ($K\supseteq \widehat{K}$), or transcendent (neither sub-dual nor super-dual). We explore a connection between sub-dual and self-dual complexes providing a new insight into combinatorial structure of self-dual complexes. The {\em root operator} associates with each self-dual complex $K$ a sub-dual complex $\surd K$ on a smaller number of vertices. We study the operation of {\em minimal restructuring} of self-dual complexes and the properties of the associated {\em neighborhood graph}, defined on the set of all self-dual complexes. Some of the operations and relations, introduced in the paper, were originally developed as a tool for computer-based experiments and enumeration of self-dual complexes.

**Keywords:** Alexander dual; self-dual complexes; triangulations; combinatorial classification.

**MSC:** 55M05, 05A15, 05E45, 55U10

Isometry group of Gromov-Hausdorff space | 123$-$154 |

**Abstract**

The present paper is devoted to investigation of the isometry group of the Gromov-Hausdorff space, i.e., the metric space of compact metric spaces considered up to isometry and endowed with the Gromov-Hausdorff metric. The main goal is to present a complete proof of the following result by G. Lowther (2015): the isometry group of the Gromov-Hausdorff space is trivial.

**Keywords:** Gromov-Hausdorff space; Gromov-Hausdorff distance; compact metric spaces; isometry groups; Metric Geometry.

**MSC:** 53C23, 54E45, 51F99

Mikha\uılichenko group of matrices over skew-fields | 155$-$161 |

**Keywords:** Group of matrices; generalized matrix multiplication; skew-field.

**MSC:** 20G99, 20H25, 15A30

Moment relations of record values from a Peng-Yan extended Weibull distribution | 162$-$168 |

**Abstract**

We derive new recurrence relations between the single and double moments of record values from a Peng and Yan extension of the Weibull distribution. Also, we provide a series representation of a single record moment.

**Keywords:** Recurrence relations; moments; records.

**MSC:** 62G30, 62G99

Approximate Bahadur efficiency of Henze-Meintanis exponentiality tests with comparison | 169$-$179 |

**Abstract**

In this paper, we present a class of tests proposed by Henze and Meintanis which is derived from the empirical characteristic function, and determine the asymptotic Bahadur efficiencies for two tests from the class. We compare those tests in Bahadur sense with the likelihood ratio tests and some other recent tests.

**Keywords:** Asymptotic efficiency; V-statistics; empirical characteristic function.

**MSC:** 60G10, 62G20

Intuitionistic unprovability | 180$-$189 |

**Abstract**

In 1952, S.C. Kleene introduced a Gentzen-type system $G3$ which is designed to be suitable for showing that the given sequents (and consequently the corresponding formulae) are unprovable in the intuitionistic logic. We show that some classes of predicate formulae are unprovable in the intuitionistic predicate calculus, using the system $G3$ and some properties of sequents that remain invariant throughout derivations in this system. The unprovability of certain formulae obtained by Kleene follows from our results as a corollary.

**Keywords:** Sequent calculus; intuitionistic logic; unprovability.

**MSC:** 03B20, 03B22, 03F03, 03F05