# International Journal of Pure Mathematics

**E-**ISSN: 2313-0571

Volume 4, 2017

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of NAUN Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

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**Volume 4, 2017**

Title of the Paper: **Metalogic in the Context of Logical Varieties**

Authors: **M. Burgin**

Pages: **64-78**

Abstract: The development of logics brought forth a multiplicity of various logical systems. This situation demands building sound foundations and common formalism for all these systems. Construction of logical foundation is done in metalogic. In the same way as metamathematics studies formalized mathematical theories, metalogic studies theories in logic, or logics and logical calculi. The discipline of logic has been developed with the aim to model and study human thinking and reasoning. A more realistic understanding relates logic only to reasoning but extends logical applications from humans to computers. Reasoning is a mental and verbal activity. Any activity is based on actions and operations organized by procedures and algorithms. That is why procedures and algorithms are basic for the development of different logics, their study and application. In this work, we study structures used in logic with the aim to reflect primary goals of logic as a discipline. Logical structures are stratified forming three levels: logical languages on the first level, logical calculi on the second level and logical varieties and prevarieties on the third level. Here only syntactic structures of logics, namely, deductive logical calculi, varieties, prevarieties and corresponding languages are considered. Semantic and pragmatic structures are studied elsewhere.

Title of the Paper: **Reliability Functions Based on Fuzzy Probability Densities**

Authors: **Reinhard Viertl, Owat Sunanta**

Pages: **59-63**

Abstract: In general, real life time data are imprecise numbers, also called fuzzy. This kind of imprecision is a result of measurement and/or observation, including time observation, which is a characteristic of interest in the area of reliability. Therefore, estimation methods for reliability characteristics have to be adapted to such fuzzy life time data for realistic results. In this contribution, definitions of fuzzy numbers and related foundations are explained. Then, a generalized reliability function is defined based on fuzzy probability densities. The generalized function is generated by fuzzy stochastic information and, hence, a fuzzy valued function describing the adapted life time distribution.

Title of the Paper: **Semi-Symmetric Metric Connection on Doubly Warped Product Manifolds**

Authors: **Sibel Sular**

Pages: **52-58**

Abstract: We find relations between Levi-Civita connection and semi-symmetric metric connection of a doubly warped product manifold M=f B×b F. We also obtain some results of Einstein doubly warped product manifolds with respect to a semi-symmetric metric connection.

Title of the Paper: **Construction of a Pre-Contrast Function on a Deformed Exponential Family**

Authors: **Hiroshi Matsuzoe**

Pages: **48-51**

Abstract: A pre-contrast function is a direction dependent distance like function. Such a function naturally arises in the theory of non-conservative statistical inference or in quantum information geometry. On the other hand, a deformed exponential family is a set of probability distributions which contains long tailed probability distributions. Such probability distributions play important roles in the study of complex systems or in anomalous statistical physics. For a long tailed probability distribution, the standard expectation does not exist in general. Hence the notion of escort expectation has been introduced. In this paper, we apply an escort expectations to pre-contrast functions. That is, we construct a pre-contrast function on a deformed exponential family by means of an escort expectation. After giving preliminaries of geometry of contrast functions and pre-contrast functions, we review foundations of anomalous statistics. In particular, we study the notion of escort expectations. Then we construct a pre-contrast function on a set of non-exponential type probability density functions.

Title of the Paper: **Doubly Warped Product Manifolds with Respect to a Semi-Symmetric Non-Metric Connection**

Authors: **Sibel Sular**

Pages: **41-47**

Abstract: The main goal of the present paper is to get new equalities with respect to a special connection titled ‘’Semi-symmetric non-metric connection’’. Whoever wants to study manifol or submanifold theory with this type of special connection, our new equalities can be used in the future studies. In this study, we calculate Koszul formulas on doubly warped product manifolds endowed with the semi-symmetric non-metric connection and we give relations between Levi-Civita connection and semi-symmetric non-metric connection of a doubly warped product manifold M=f B×b F. We also get some results of Einstein doubly warped product manifolds with a semi-symmetric non-metric connection.

Title of the Paper: **Horocycle Trajectories and their Limit-Strings on a Complex Hyperbolic Space**

Authors: **Toshiaki Adachi**

Pages: **37-40**

Abstract: We take trajectory-harps for a Kahler magnetic field of strength √|c| on a complex hyperbolic space CHn© of constant holomorphic sectional curvature c. We show that distance functions between their limit-strings and their arch-trajectories are not bounded though their limit points in the ideal boundary are the same.

Title of the Paper: **Prime Geodesic Theorem for Compact Even-Dimensional Locally Symmetric Spaces of Real Rank One**

Authors: **Muharem Avdispahic, Dzenan Gusic**

Pages: **26-36**

Abstract: We improve the error term in DeGeorge’s prime geodesic theorem for compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature.

Title of the Paper: **Homogeneous Hopf Hypersurfaces in a Complex Hyperbolic Space and Extrinsic Shapes of their Trajectories**

Authors: **Tuya Bao, Toshiaki Adachi**

Pages: **22-25**

Abstract: We study homogeneous real hypersurfaces in a complex hyperbolic space whose characteristic vectors are principle. We characterize some of them by investigating properties of extrinsic shapes of some curves on these hypersurfaces which are associated with their characteristic tensor fields.

Title of the Paper: **RCD: An R Package for Estimating Robust Copula Dependence**

Authors: **Yi Li, Adam Ding**

Pages: **17-21**

Abstract: The robust copula dependence (RCD) [1, 2] is recently introduced as an equitable dependence measure: it measures the dependence according to the strength of association regardless of the functional shape, treating linear and nonlinear relationships among the data equitably. It is useful to detect nonlinear relationships in data exploration. We introduce a new R package rcd for implementing the estimation of RCD using two methods: the kernel density estimation (KDE) and the k-nearest-neighbour (KNN) density estimation, with the latter one has smaller computational complexity in highdimensional settings. The parallel programming with the Rcpp and RcppParallel packages is used to further speed up the estimators. The numerical performance of different estimators are evaluated with numerical experiments. The usage of functions in the rcd package is illustrated with numerical examples.

Title of the Paper: **Application of the Fast Automatic Differentiation to the Computation of the Gradient of the Energy of Atoms’ System with Respect to Tersoff Parameters**

Authors: **Alla Albu, Vladimir Zubov**

Pages: **12-16**

Abstract: Gradient optimization methods are often used to tackle problems of computer modeling of materials’ crystal structures. This raises the need to determine the exact value of the gradient of the Tersoff potential using specific parameters of this potential for the modeled substance. Based on the technique of the Fast Automatic Differentiation the formulas that allow the calculation of the exact value of the above-mentioned gradient were derived.

Title of the Paper: **Correctness of the Initial-Boundary Value Problem and Discrete Analogs for One Nonlinear Parabolic Integro-Differential Equation**

Authors: **Temur Jangveladze, Zurab Kiguradze, Maia Kratsashvili**

Pages: **7-11**

Abstract: First type initial-boundary value problem for one nonlinear parabolic integro-differential equation is considered. This model is based on Maxwell system describing the process of the penetration of a magnetic field into a substance. Semi-discrete and finite difference schemes are studied. Attention is paid to the investigation not only power type that already were studied but more wide cases of nonlinearities. Existence, uniqueness and long-time behavior of solutions are fixed too.

Title of the Paper: **Investigation and Approximate Solution of Two Systems of Nonlinear Partial Differential Equations**

Authors: **Temur Jangveladze**

Pages: **1-6**

Abstract: Two systems of nonlinear partial differential equations are considered. Both systems are obtained at mathematical modeling of process of electromagnetic field penetration in the substance. In the quasistationary approximation, this process, taking into account of Joule law is described by nonlinear well known system of Maxwell equations. Taking into account heat conductivity of the medium and again the Joule law, the different type nonlinear system of partial differential equations is obtained. Investigation and approximate solution of the initial-boundary value problems are studied for these type models. Linear stability of the stationary solution is studied. Blow-up is fixed. Special attention is paid to construction of discrete analogs, corresponding to one-dimensional models as well as to construction, analysis and computer realization of decomposition algorithms with respect to physical processes for the second system. Averaged additive semi-discrete models, finite difference schemes are constructed and theorems of convergence are given.