#### Version-1 (Nov-Dec 2017)

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**Abstract: **The objective of the study was to examine the effects of small-group learning and whole-class discussion on students' mathematics achievement. The study also sought to determine the small-group learning and whole-class discussion load that affects student mathematic achievement. In order to conduct the study an approach of action research was employed on one section of First Year Mathematics Students of Mettu University. First observation of students' activity and their perception towards mathematics small-group learning and whole-class discussion was made. Students' perceptions of small-group learning and whole-class discussion gave valuable information to improve the approaches of small-group learning and whole-class discussion so as to improve student achievement.........

**Keywords:** Improving, small-group learning, whole-class discussion, Students' achievement

[1]. Schmidt, W.H.; McKnight, C.C.; Raizen, S.A. 1997. A splintered vision: an investigation of U.S. science and mathematics education. Dordrecht, Netherlands, Kluwer Academic Publishers.

[2]. Secada, W.G. 1992. Race, ethnicity, social class, language, and achievement in mathematics. Grouws, D.A., ed. Handbook of research on mathematics teaching and learning, p. 623–60. New York,

[3]. Macmillan. Skemp, R.R. 1978. Relational understanding and instrumental understanding. Arithmetic teacher (Reston, VA), vol. 26, p. 9–15. 45

[4]. Slavin, R.E. 1990. Student team learning in mathematics. In: Davidson, N., ed. Cooperative learning in math: a handbook for teachers, p. 69–102. Reading,

[5]. MA, Addison-Wesley. 1995. Cooperative learning: theory, research, and practice. 2nd edition. Boston, Allyn & Bacon.

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**Abstract: **In this paper we shall study fuzzy transportation problem, and we introduce an approach for solving
a wide range of such problem by using a method which apply it for ranking of the fuzzy numbers. Some of the
quantities in a fuzzy transportation problem may be fuzzy or crisp quantities. In many fuzzy decision problems,
the quantities are represented in terms of fuzzy numbers may be triangular or trapezoidal. Thus, some fuzzy
numbers are not directly comparable. First, we transform the fuzzy quantities as the cost, coefficients, supply
and demands, into crisp quantities by using Centorid ranking method [4] and then by using the VAM algorithm
to solve and obtain the solution of the problem .

**Keywords:** Fuzzy set, Fuzzy transportation problem, Trapezoidal Fuzzy number, Ranking Technique.

[1] Hictchcock FL. The distribution of a product from several sources to numerous localities. Journal of Mathematical Physics. 1941; 224230.

[2] Zadeh LA. Fuzzy sets, information and control. 1965; 8: 338353

[3] Bellman RE, Zadeh LA. Decision making in a fuzzy environment. Management Sci. 1970; 17: 141164.

[4] Zimmermann HJ. Fuzzy set theory and its applications. Fourth Edition ISBN 0792374355; 1934.

[5] Zimmermann HJ. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems. 1978; 4555.

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**Abstract: **We consider Bianchi type III string cosmological model in the presence of viscous fluid and . To
solve the Eainstein's field equations for Bianchi type III space time, time has been obtained by assuming the
condition the coefficient of the viscosity is proportional to the expansion scalar, , expansion scalar is
proportional to shear scalar, and cosmological constant is proportional to the Hubble parameter
. The physical and geometrical behaviour of cosmological model are discussed.

**Keywords:** Bulk viscosity, cosmological constant , Bianchi-III space time, expansion scalar, shear, Hubble
parameter.

[1] Nightingale J.P. (1973); J. Astrophys 185, 105.

[2] Pavon D; Bafaluy J. Nad Jou D. (1991); class quantum gravity, 8, 357.

[3] Maartens R. (1995); class quantum Gravity, 12, 455.

[4] Kalyani D. and Singh G.P. (1997); In new direction in relativity and cosmology.

[5] Singh T.; Beesham A. and Mbokazi W.S. (1998); Gen. relativity, Granit 30, 537.

[6] Ng, Y. J. (1992); Int. J. Mod Phys. D1, 145.

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**Abstract: **The generalized difference Riesz sequence space 𝑟𝑞(ℳ,Δ𝑛𝑚,𝑢,𝑝,𝑠) of non absolute type was recently introduced and studied by some authors. This paper is devoted to characterize the classes (𝑟𝑞 ℳ,Δ𝑛𝑚,𝑢,𝑝,𝑠 ,ℓ∞), 𝑟𝑞 ℳ,Δ𝑛𝑚,𝑢,𝑝,𝑠 ,𝑐 𝑎𝑛𝑑 (𝑟𝑞 ℳ,Δ𝑛𝑚,𝑢,𝑝,𝑠 ,𝑐0) of infinite matrices and characterize a basic theorem where ℓ∞,𝑐 𝑎𝑛𝑑 𝑐0 denotes respectively the space of bounded sequences, space of all convergent sequences and space of all sequences converging to zero.

**Keywords:** Riesz sequence space, sequence space of non absolute type, paranormed sequence spaces and Matrix transformations.

[1] O. Toeplitz, Uberallegemeine Lineare mittelbidungen, Prace Math., 22, 1991, 113-119.

[2] I. J. Maddox, Element of functional analysis, The University Press, Cambridge 1988.

[3] G. M, Petersen, Regular matrix transformations, Mcgraw-Hill, London, 1966.

[4] N. A. Sheikh and A. H. Ganie, A new paranormed sequence space and some matrix transformations, Acta Math. Acad. Paedago. Nyregy., 28 (1), 2012, 47-58.

[5] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (2), 1981, 169-176.

[6] M. Et and R. Çolak, On some generalized sequence spaces, Soochow. J. Math., 21, 1995, 377-386.

[7] B. Altay and F. Bașar, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Mathematical Journal, 55 (1), 2003, 136-147.

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**Abstract: **This study was aimed to analyze probability task completion of elementary school students. In this
qualitative research, data was collected by giving probability tasks (probability of an event and probability
comparison) and interview. The result showed that students used a numerator strategy in solving probability of
an event task, and they used different strategy in solving probability comparison. Boy with high math ability
used strategy by considering set with less non target event, boy with low math ability used strategy by
considering set with more target event, girl with high math ability used strategy by considering set with greater
difference in favor of target event, and girl with low math ability used strategy by considering...........

**Keywords:** Probabilistic Thinking, Elementary Student, Probability Task, Gender

[1] L. S. Hirsch & A. M. O'Donnell. Representativeness in statistical reasoning: Identifying and assessing misconceptions. Journal of

Statistics Education, 9(2), 2001, 61-82.

[2] T. Kvatinsky & R. Even. Framework for teacher knowledge and understanding about probability. In Proceedings of the Sixth

International Conference on Teaching Statistics (CD), Cape Town, South Africa: International Statistical Institute, 2002.

[3] T. HodnikČadež & M. Škrbec. (2011).Understanding the concepts in probability of pre-school and early school children. Eurasia

Journal of Mathematics, Science & Technology Education, 7(4), 2011, 263-279.

[4] J. Way. Chance connections. Paper presented at the mathematical association of victoria. 2008. Retrieved from

http://www.mav.vic.edu.au/files/conferences/2008/Way/WayJ2008.doc

[5] F. M. Taylor. Why Teach Probability in the Elementary Classroom. Louisiana Association of Teachers Mathematics Journal, 2(1),

2011

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**Abstract: **This paper is an introduction for solving Delay Differential Equations (DDEs) using market
equilibrium. By market we mean the conditions under which producers sell and consumers buy a certain
commodity. The term market is used when only one commodity is being bought and sold and the word multimarket
is used when more than one commodity is involved. The price, demand and supply of any one good affects the
prices etc. of any other goods and vice-versa. DDEs discussed in this paper are linear. The most fundamental

Functional Differential Equations(FDEs) is the linear first order Delay Differential Equations (DDEs).Both
DDEs and FDEs are used as modeling tools in models in Economics............

**Keywords:** Delay differential equations, Demand function, Equivalent systems, Market equilibrium, Method of
characteristics, Method of steps, Stability of the equilibrium, Supply function.

[1] ASl, F.M. and Ulsoy, A.G., Analysis of a system of Linear Delay Differential Equations, Journal of Dynamic Systems, Measurement

and Control,2003, V.125, PP215-223.

[2] Bender, A.E., and Neuwirth, L.P. Traffic Flow : Laplace Transforms, American Mathematical Monthly, Vol.80, (1973), 417-423.

[3] Driver, R.D., Sasser, D.W and Slater, M.L., The Equationy'(t)=ay(t)+by(t-𝜏) with 'small' Delay, American Mathematical Monthly,

Vol.80, 1973, pp 990-995.

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**Abstract: **This article presents a bound estimation for two functions and consequently puts forwards several inequalities with their proofs. The new inequalities are helpful in estimating the bounds of certain functions and their better proofs are still calling more mathematical skills.

**Keywords:** Analytic solution, bound estimation, inequality

[1] M J Cloud, byron C Drachman, Inequalities-WithApplicationsToEngineering, Springer, 1998

[2] Herman J, Šimša J, Kučera R. Algebraic Inequalities, Springer, 2000

[3] B G Pachpatte, Mathematical inequalities, North-Holland mathematical library, 2005

[4] Vasile Cirtoaje, Algebraic Inequalities - Old And New methods, Gil Publishing House,2006

[5] Vo Quoc Ba Can, Cosmin Pohotata, Old and New Inequlaities,Gil Publishing House,2008

[6] Vasile Cirtoaje,Vo Quoc Ba Can & Tran Quoc Anh, Inequalities with Beautiful Solutions,Gil Publishing House,2009.

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Paper Type |
: | Research Paper |

Title |
: | Zero Average Method to Finding an Optimal Solution of Fuzzy Transportation Problems |

Country |
: | India. |

Authors |
: | A. Edward Samuel || P.Raja |

: | 10.9790/5728-1306015663 |

**Abstract: **In this paper, a proposed method, namely, Zero Average Method (ZAM) is used for solving fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of the transportation cost only but there is no uncertainty about the demand and supply of the product. In this proposed method transportation cost are represented by generalized fuzzy numbers. To illustrate the proposed method a numerical example is solved and the obtained results are comparing with the results of existing methods. The proposed method is very easy to understand and apply on real life transportation problems for the decision makers......

**Keywords:** Fuzzy Transportation Problem (FTP); Generalized Trapezoidal Fuzzy Number (GTrFN); Ranking function; Zero Average Method (ZAM).

[1]. Amarpreet Kaur, Amit Kumar, A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers, Applied soft computing, 12(3) (2012), 1201-1213.

[2]. S. Chanas, W. Kolodziejckzy, A.A. Machaj, A fuzzy approach to the transportation problem, Fuzzy Sets and Systems, 13 (1984), 211-221.

[3]. S. Chanas, D. Kuchta, A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems, 82(1996), 299-305.

[4]. A. Charnes, W.W. Cooper, The stepping-stone method for explaining linear programming calculation in transportation problem, Management Science, l (1954), 49-69.

[5]. W.W. Charnes, W.W. Cooper and A. Henderson, An introduction to linear programming Willey, New York, 1953..