iosr-Jm   Volume-17 ~ Issue-2 ~ Mar. – Apr. 2021

Abstract: The notion of g\tilde\alpha-closed sets in a topological spaces are introduced by R.Devi et. al. [2]. In this paper, we introduced the concept of slightly g\tilde\alpha -continuous functions and study the basic properties and preservation theorems of this function.

Keywords: clopen set, gα-continuous map, slightly gα-continuous map. AMS(2000) Subject classification : 54A05, 54D05 54D10, 54D45.

[1]. M. Caldas and S. Jafari, On g-US spaces, Universitatea Din Bacau Stud II SI Cercetari Stiintifice (Mathematica), No. 14 (2004), 13-20.
[2]. R. Devi, A. Selvakumar and S.Jafari, On gα-closed sets in Topological spaces, Asia Mathematika, Vol. 3, No. 3 (2019), 16-22.
[3]. Erdal Ekici and Miguel Caldas, Slightly γ-continuous Functions, Bol. Soc. Oaran. Mat., (35) Vol. 22, No. 2 (2004), 63-74.
[4]. R.C. Jain, The role of regularly open sets in general topology, Ph.D Thesis, Meerut University, Institute of Advanced Studies, Meerut, India (1980).
[5]. O. Njastad, On some classes of nearly open sets, Pacif. J. Math., 15 (1965), 961-970..

Abstract: First, the 3x+1 problem is transformed into the decision of the terms valued 1 of the all-odd 3x+1 sequences. Then, the property of the all-odd 3x+1 sequences is used to obtain the equations with equal terms and the sufficient and necessary conditions for the all-odd 3x+1 sequences to have equal terms. Then, the uniqueness of the characteristic solutions of the equations with equal terms is proved. At last, based on the above results the 3x+1 problem is proved to be true

Keywords: the 3x+1 problem; the all–odd 3x+1 sequences; equations with equal terms; characteristic solutions MSC2010: 11A99

[1]. Jeffrey C. Lagarias, editor, The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010
[2]. M. R. Feix and J. L. Rouet, The (3x+1)/2 Problem and its generalization: a stochastic approach, Proceedings of international Conference on the Collatz Problem and Related Topics, August 5-6, 1999, Katholische Universitat Eichstatt, Germany
[3]. M. Chamberland, A Dynamical Systems Approach to the 3x+1 Problem, Proceedings of international Conference on the Collatz Problem and Related Topics, August 5-6, 1999, Katholische Universitat Eichstatt, Germany
[4]. E. Belaga, Reflecting on the 3x+1 Mystery, Proceedings of international Conference on the Collatz Problem and Related Topics, August 5-6, 1999, Katholische Universitat Eichstatt, Germany

Abstract: The design of experiments (DoEs) have much recent interest and this is likely to grow as more and more simulation models are used to carry out research. A good experimental design should have at least two important properties namely projective property (non-collapsing) and Space-filling (design points should be evenly spread over the entire design space) property. Any Latin Hyper-cube design (LHD) is inherently preserve projective property. In consequence, in sense of space-filling, Optimal LHDs is required for good DoEs. In this study, we consider maximin LHDs obtained by Iterated Local search (ILS) heuristic approach in which inter-site distances are measured in Euclidean distance measure. We have compared the performance and effectiveness of ILS approach with some well-known approaches available in the literature regarding maximin LHDs in Euclidean distance measure.......

Keywords: Design of experiments, Audze-Eglais LHDs, Iterated Local search, Optimal criteria

[1]. Applegate D., W. Cook and A. Rohe, 1999, "Chained Lin-Kernighan for large traveling salesman problems", Technical Report No. 99887, Forschungsinstitut f
.. u r Diskrete Mathematik, University of Bonn, Germany.
[2]. Bates S. J., Sienz J. and Langley D.S., 2003, "Formulation of the Audze-Eglais Uniform Latin Hypercube design of experiments",
Advanced in Engineering Software, Vol. 34, Issue 8, pp. 493-506.
[3]. Baum, E. B., 1986(b), "Iterated descent: A better algorithm for local search in combinatorial optimization problems", Technical
report, Caltech, Pasadena, CA Manuscript.
[4]. Baum, E. B.,1986(a), "Towards practical "neural" computation for combinatorial optimization problems", In J. Denker, editor,
Neural Networks for Computing, pp. 53–64, AIP conference proceedings.
[5]. Baxter, J., 1981, "Local optima avoidance in depot location", Journal of the Operational Research Society, Vol. 32, pp. 815 –819

Abstract: Problems in both physical and social sciences are generally described by mathematical models in multi-dimensional variable spaces. Then their solutions require skills and a good understanding of mathematical operations. When you ask most professionals and students to define ZERO and ONE or UNITY, they are puzzled and can't really answer it. They were taught in the elementary school that multiplication of numbers is like a recurrent addition, but when you ask them to explain why the product of two negative numbers is positive, they are again at a loss. So, in this paper we first define a set of unified elementary mathematical operations consisting of eight steps, which include definitions.....

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Abstract: The quotient coarse category Qcrs, equipped with a small amount of extra structure, is a Baues cofibration category [12], [13]. In this article we show that the pointed quotient coarse category PQcrs is also a Baues cofibration category and use the coarse cofibration category machinery to define controlled and coarse homotopy groups, compute these groups for coarse spheres, and define relative coarse homotopy. For the last we show that any two classes in the coarse category PQcrs, that are strongly coarsely homotopic relative to , will be relatively coarsely homotopic..

Keywords: Baues cofibration Category, The quotient coarse category, relative coarse homotpy

[1]. H. J. Baues. Algebraic Homotopy, volume 15 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,
[2]. H. J. Baues. Combinatorial Foundation of Homology and Homotopy. Springer Monographs in Mathematics. Springer-Verlag. Berlin, (1999). Applications to spaces, diagrams, transformation groups, compactifications, differential algebras, algebraic theories, simplicial objects, and resolutions.
[3]. A. N. Dranishnikov. Asymptotic Topology. Uspekhi Mat. Nauk, 55(6(336)......

Abstract: This work is about Group Divisible Designs (GDDs) of block size four on three groups of different sizes n1 = 4, n2 = n and n3 = n + 1 where n ≥ 4. We first establish necessary conditions for the existence of the GDD using relationships between the parameters of the GDD and then prove that these conditions are sufficient for several families of GDDs..

Keywords: Group Divisible Designs, Blocks, Parameters

[1]. C. J. Colbourn and J. H Dinitz. Handbook of combinatorial de signs CRC Press, 2006.
[2]. A. Chaiyasena, S. P. Hurd, N. Punnim and D. G Sarvate. Group divisible designs with two association classes. Journal of Combinatorial Mathematics and Combinatorial Computing (82) (2012), (179).
[3]. S. P. Hurd and D. G. Sarvate. Group divisible designs with two association classes and with groups of sizes 1, 1, and n. Journal of Combinatorial Mathematics and Combinatorial Computing (75) (2010), (209).
[4]. W. Lapchinda, N. Punnim and N. Pabhapote. GDDs with Two Associate Classes and with Three Groups of Sizes 1, n n and λ1 < λ2. Thailand-Japan Joint Conference on Computational Geometry and Graphs, (2012), (101-109).
[5]. W. Lapchinda, N. Punnim and N. Pabhapote. GDDs with two associate classes and with three groups of sizes 1, n and n. Australasian J. Combinatorics, (58) (2014), (292-303).

Abstract: In this paper we suggest an extended axiomatic approach to probability so as to cover the notion of conditional probability - erstwhile requiring a separate definition.

Keywords Axioms for Probability Function, Conditional Probability

[1]. P. Billingsley. Probability and Measure. Wiley Series in Probability and Statistics. Wiley, 2012.
[2]. W. Feller. An Introduction to Probability Theory and Its Applications: Volume I. Number v. 1 in Wiley series in probability and mathematical statistics. John Wiley & sons., 1968.
[3]. A. Stuart and K. Ord. Kendall's Advanced Theory of Statistics: Volume 1: Distribution Theory. Number v. 1; v. 1994 in Kendall's Advanced Theory of Statistics. Wiley, 2009

Abstract: Background: Vekua equation is an areolar equation from a complex function, which cannot be solved in general case. Its origin is from a practice problem from the theory of elasticity.
Results: In the paper zeros of the solutions of some special cases of Vekua equation are considered and the results are formulated in theorems.

Key Word: Areolar derivative, Areolar equation, Vekua equation, nonhomogeneous linear differential equation

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упругости, 1909
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Университета, 1965
[4]. S. Brsakoska, Operator differential equations from the aspect of the generalized analytic functions, MSc thesis, Skopje, 2006
[5]. S. Brsakoska, Some contributions in the thory of Vekua equation, PhD thesis, Skopje, 2011.