Recently,Williams expressed all coefficients of one hundred and twenty-six eta quotients in terms of σ(n), σ(n/2), σ(n/3) and σ(n/6), and Yao, Xia and Jin, expressed only even coefficients of one hundred and four eta quotients in terms of σ3(n), σ3(n/2), σ3(n/3) and σ3(n/6). The Fourier series expansions of a class of eta quotients in terms of σk-1(n), σk-1(n/2), σk-1(n/3) and σk-1(n/6) for k = 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 have been expressed by the author. The Fourier series expansions of a class of eta quotients in M_{2 }(Γ_{0}, χ) in terms of σ(n), σ(n/2), σ(n/3) and σ(n/6) has been found by Alaca and the Fourier series expansions of a class of eta quotients in M_{4} (Γ_{0}, χ) in terms of σ3(n), σ3(n/2), σ3(n/3) and σ3(n/6) has been determined by the author. Here, we will determine the coefficients of the Fourier series expansions of a class of eta quotients in M_{6} (Γ_{0}, χ) in terms of σ5(n), σ5(n/2), σ5(n/3), σ5(n/6) and Fourier coefficients of the eight eta quotients.

The aim of this paper is to introduce the concepts of fuzzy translation to fuzzy associative ideals in BCK/BCI-algebras. Also, the notion of fuzzy extensions and fuzzy multiplications of fuzzy associative ideals with several related properties are investigated. The relationships between fuzzy translations, fuzzy extensions and fuzzy multiplications of fuzzy ideals are investigated.

In this paper, we prove fixed point theorem of a self mapping in non-normal cone hexagonal metric spaces. Our result extend and improve some recent results of Azam et al., [Banach contraction principle on cone rectangular metric spaces, Applicable Analysis and Discrete Mathematics, 3 (2), 236 - 241, 2009], Rashwan and Saleh [Some Fixed Point Theorems in Cone Rectangular Metric Spaces, Mathematica Aeterna, 2 (6): 573 - 587, 2012], Garg and Agarwal, [Banach Contraction Principle on Cone Pentagonal Metric Space, J. Adv. Studies Topol., 3 (1), 12 - 18, 2012], Garg, [Banach Contraction Principle on Cone Hexagonal Metric Space, Ultra Scientist, 26 (1), 97 - 103, 2014], and others.

In this article, we suggest a new approach while solving two phase simplex method. The method sometimes involves less iteration than in the Simplex Method or at the most an equal number because the method attempts to replace more than one basic variable simultaneously. While dealing with Two Phase Simplex Method a new method [1,2,3,4,5,6] (Quick Simplex Method) can be applied in Phase I and also in Phase II.

This has been illustrated by giving the solution of solving Two Phase Simplex Method problems. It is also shown that either the iterations required are the same or less but iterations required are never more than those of the Simplex Method.

This paper introduces a new heuristic function with high efficiency for an optimum solving of the 8-puzzle, this one being the double of the Chebyshev distance. The comparative study is realized among this new heuristic (Chebyshev heuristic), the Hamming heuristic and the Manhattan heuristic using A* algorithm implemented in Java. The Chebyshev heuristic function is more informed than Hamming and Manhattan heuristics. This paper also presents the necessary stages in object oriented development of an interactive software dedicated to simulate the A* search algorithm for 8-puzzle. The modeling of the software is achieved through specific UML diagrams representing the phases of analysis, design and implementation, the system thus being described in a clear and practical manner. In order to confirm the obtained theoretical results which show that Chebyshev heuristic is more efficient, two performance criteria were used: space complexity and time complexity. The space complexity was measured by the number of generated nodes from the search tree, by the number of the expanded nodes and by the effective branching factor. The time complexity was measured by the running time. From the experimental results obtained by using the Chebyshev heuristic, improvements were observed regarding space and time complexity of A* algorithm versus Hamming and Manhattan heuristics.

In the present work, extenics approach is used for modeling a problem of software testing. The approach is useful for generation of the test cases through the sequence diagram designed by the use of Unified Modeling Language. Authors created formalized sequence diagram by use of n-dimensional matter element property which is converted into message flow graph. Parsed data is generated through Extended Markup Language (XML) from the sequence diagram and converted the dual of message flow graph.

A traversal algorithm is designed and implemented on the dual of message flow graph for generation of valid test cases. Further test cases are minimized which satisfies path coverage criteria. The entire work is implemented on object oriented Java programming language through a case study of departure activity of aircraft.

We examine the spectral geometry of η-Einstein S-manifolds and compute the the spectral coefficients for S-space form and obtain the results analogue to Patodi’s results for Riemannian manifolds and J. Park results for η−Einstein Sasakian manifolds. We show that how an η−Einstein S-manifold and S-space forms are spectrally determined.