iosr-Jm   Volume-14 ~ Issue-5 ~ Sep-Oct 2018

Abstract: In this paper we deal with the fuzzy optimization of the mean number of customers and the mean waiting time of a customer in the queue in a preemptive priority discipline with two priority classes where the preemptive units do not return to service but are lost. Poisson arrival, Exponential service time, single server and infinite waiting line are assumed. Fuzzyfying the parameters in the mean number of customers and the mean waiting time of a customer in the queue, optimization is obtained using statistical technique. A Numerical example for fuzzy optimization is illustrated.

Keywords: Reliability Optimization, DMAIC, down time, Equipment availability, optimal maintenance cost, PRCM,

[1]. Baron.O, Scheller-Wolf.A and Wang.J. M/M/c Queue with Two Priority Classes, Journal of Operation Research. 2015;63(3):733-
749.
[2]. Bailey.N. On queuing processes with bulk service, Journal of the Royal Statistical Society,Series B,.1954;16(1): 80-87.
[3]. Bohm.W.Lattice path counting and the theory of queues, Journal of Statistical Planning and Inference.2010:140(8); 2168 -2183.
[4]. Barry.J.Y. A priority queueing problem, Opre.Res., 1956;4:385-386.
[5]. Champernowne.D.G,.An elementary method of solution of the queueing problem with a single server and constant parameters,
Journal of the Royal Statistical Society. Series B (Methodological), 1956:18(1); 125 - 128..

Abstract: As a generalization of hesitant fuzzy implicative filters of hoops, we introduce the concept of (M,N)- hesitant fuzzy implicative filters. The relationships between (M,N)-hesitant fuzzy implicative filters and implicative filters are discussed by using the notion of (M, N)- -level sets of hesitant fuzzy sets. Several conditions for a (M,N)-hesitant fuzzy filter to be a (M,N)-hesitant fuzzy implicative filter are presented. Moreover, some characterizations of (M,N)-hesitant fuzzy implicative filters in regular hoops are derived.

Keywords: hoop, (M,N)-hesitant fuzzy implicative filter,hesitant fuzzy set

[1]. G Georgescu, L Leustean, and V Preoteasa, Pseudo-hoops, Journal of Multiple Valued logic and Soft Computing, 11, 2005, 153-184.
[2]. Y Yang, L Lu, and Q Wang, Pseudo valuation on hoop-algebra respect to filters, International Journal of Mathematics and
Statistics, 2 (11), 2017, 12-19.
[3]. M Wang, X Xin, and J Wang, Implicative pseudo valuations on hoops. Chinese Quarterly Journal of Mathematics, 33 (1), 2018,
51-60.
[4]. M Kondo, Some types of filters in hoops, International Symposium on Multiple-Valued Logic, 47, 2011, 50-53.
[5]. C Luo, X Xin, and P He, n-fold (positive) implicative filters of hoops, Italian Journal of Pure & Applied Mathematics, 38, 2017,
631-642..

Abstract: The three-dimensional Zabolotskaya--Khokhlov equation (ZK) is a nonlinear second order partial differentialmodel for sound waves propagation. In this work, some ordinary differential reductions of this uation have been givenby two-step using of the Lie point symmetry group method. Then, some new exact similarity solutions of the ZK equationhave been obtained by solvingthe reduced ODEs.The solutions can be used to clarifying the propagation of a boundedtwo-dimensional acoustic beam in nonlinear medias.

Keywords: Zabolotskaya--Khokhlov equation (ZK), Symmetry algebra, Lie point symmetry group, Optimal systemof sub-algebras, Reduction of equation, Similarity solution

[1]. N.S. Bakhvalov, Y.M. Zhileikin, E. A. Zabolotskaya, Nonlinear theory of sound beams, Am. Inst. Phys, NewYork, 1987.
[2]. A. V. Bocharov, A. M.Verbovetsky, Symmetriesand Conservation Lawsfor Differential Equations of MathematicalPhysics,Author address: DiffeoUtopia.
[3]. A.R. Chowdhury, M. Nasker, Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions, J.Phys. A: Math. Gen. 19: 1775--1782, 1986.
[4]. P.A. Clarkson, M.D Kruskal, New similarity reductions of the boussinesq equation, J. Math. Phys. 30: 2201--2213. 1989.
[5]. Y.N. Grigoriev, .N.H. Ibragimov, .V.F. Kovalev, S.V. Meleshko, Symmetries ofIntegro-DifferentialEquations, Springer, Dordrecht, 2010.

Abstract: In this paper the concept of statistical Bourbaki-Cauchy double sequence is introduced. Some inclusion relations between Bourbaki-Cauchy double sequence and statistical Bourbaki-Cauchy double sequence is also explored. We shall also give some essential analogous definitions of Bourbaki completeness and Bourbaki boundedness for double sequence in metric spaces which are characterized in terms of functions and preserved statistical Bourbaki-Cauchy double sequence.

Keywords: Bourbaki-Cauchy double sequence, Statistical Bourbaki-Cauchy double sequence and Statistical Bourbaki-Cauchy bounded.
2010 Mathematics subject classification: Primary 40F05, 40J05, 40G05

[1]. J. Christopher, The asymptotic density of some k-dimensional sets, Amer. Math. Monthly 63(1956), 399 – 401.
[2]. 2.Cakalli, H. (2008). Sequential defiitions of compactness. Appl. Math. Lett. 21(6), 594-598.
[3]. Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicum, 2 (3–4), 241–244.
[4]. Fridy, J. A., & Orhan, C. (1993). Lacunary statistical summability. Journal of Mathematical Analysis and Applications, 173, 497–504.
[5]. Ilkhan, M. & Kara, E. E. (2018). A new type of statistical Cauchy sequence and its relation to Bourbaki completeness. Cogent Mathematics & Statistics, 5, 1-9.

Abstract: In this paper, consider the controllability of a class of fractional dynamical systems with control delay. Necessary and sufficient conditions for the controllability of fractional linear systems with control delay are obtained. The results obtained in this paper are important for the study of controllability of nonlinear fractional dynamical systems with control delay. An example is also provided to illustrate the main results.

Keywords: Controllability, Delay, Fractional dynamical system, Mittag-Leffler matrix function

[1]. Vijayakumar S. Muni, Venkatesan, Govindaraj and Raju K. George, Controllability of fractional order semilinear systems with a
delay in control, Indian Journal of Mathematics, 60(2), 2018, 311-335.
[2]. R. Joice Nirmala, K. Balachandran, L. Rodríguez and J. J. Trujillo, Controllability of nonlinear fractional delay dynamical systems,
Reports on Mathematical Physics, 77, 2016, 87-104.
[3]. Binbin He, Huacheng Zhou and Chunhai Kou, The controllability of fractional damped dynamical systems with control delay,
Commun Nonlinear Sci Numer Simulat, 32, 2016, 190-198.
[4]. Annamalai Anguraj, Subramaniam Kanjanadevi, Non-instantaneous impulsive fractional neutral differential equations with state
dependent delay, Progress in Fractional Differentiation and Applications, 3(3), 2017, 207-218.

Abstract: In this paper, we will solve three portfolio models with singular covariance matrix. These portfolio models include Mean-Variance Portfolio Model, Value-at-Risk portfolio Model, and Conditional Value-at-Risk portfolio Model. By studying and calculating, we fond: the effective boundary of these three types of portfolio models must be the effective boundary of their maximal linearly independent groups or their maximal linearly independent groups and risk-free assets

Keywords: Singular Covariance Matrix; Maximum Linear Unrelated Group; Effective borders

[1] R.Tyrrell Rockafellar, Stanislav Uryasev. Optimization of conditional value-at-risk.
[2] HaixiangYao, Zhongfei Li, Xun Li, Yan Zeng.Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset.Journal of Industrial and Management Optimization,2017, 13(3):1273-1290.(SSCI , SCI)
[3] Miao Zhang, Ping Chen,Haixiang Yao(Corresponding author). Mean-variance portfolio selection with only risky assets under regime
switching.Economic Modelling, 2017, 62: 35–42. (SSCI)
[4] Haixiang Yao,Zhongfei Li, Duan Li.Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable
liability.European Journal of Operational Research, 2016, 252(3): 837–851.(SSCI , SCI)
[5] Haixiang Yao, Zhongfei Li, Yongzeng Lai. Mean-CVaR portfolio selection: A nonparametric estimation framework. Computers& Operations Research, 2013, 40: 1014-1022. (SCI,SSCI)

Abstract: In this article, we study the existence and stability results for second order neutral stochastic functional differential equations driven by fractional Brownian motion. Our method of investigating the stability of solutions is based on successive approximation approach and Lipschitz conditions being imposed

Keywords: Second – order system, non Lipschitzian conditions, fractional Brownian motion, mild solution

[1]. Arthi,JH.Park and HY.Jung Exponential stability for second-order neutral stochastic differential equations with impulse. INt.J.Control 88 (6), (2015), 1300-1309.
[2]. P.H Bezandry, Existence of almost periodic solutions for semi linear stochastic evolution equations driven by fractional Brownian motion, Eltronic Journal of Differential Equations 2012(2012) 1-21.
[3]. B.Boufoussiand S.Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics and Probability Letters, 82 (2012), 1549-1558.
[4]. T.Caraballo, J.Real and T.Taniguchi, The exponential stability of neutral Stochastic delaypartial differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 295-313
[5]. T. Caraballo, M.J Garrido – Atienza, and T. Taniguchi, The existence and exponential behaviour of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinearnal. 74, pp. 3671 – 3684, (2011)