The Journal of Mathematics

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The Journal of Mathematics, Impact Index: 3.58, ISJN: 8352-7329, Imprint: Photon, Category: Peer Reviewed Indexed International Journal


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The Problem of a Griffith Crack at the Interface of Two Bonded Dissimilar MicropolarElastic Half-planes 


Rajendra Kumar Tripathi 

Pioneer Mahila Mahavidayalya Lakhperabagh, Barabanki, India 


Keywords: Fredholm Integral Equations, Legendre Polynomial, Dual Integral Equation, Riemann Hilbert Problem 


Abstract 

This paper deals with a two dimensional crack problem i.e. (Griffith Crack Problem) located at the interface of two half-planes and symmetrically situated with respect to y-axis which is perpendicular to the interface is considered. These two half-planes y > 0 and y < 0 are occupied by elastic materials with the constants where is the modulus of rigidity and is the coefficient of heat conduction with the relation =3-4 (i = 1,2) where is the poison ratio. The crack is given by y=0, aj< |x| <bj (j=1,2,3,…n). The upper and lower surface of the crack are subjected to a prescribed pressure p(x), boundary and continuity conditions are applied to the inside and outside of the surface of the crack. The constant internal pressure is applied to the surfaces of the crack. The analysis is carried out by a system of simultaneous dual internal equations which are further reduced to the system of Riemann-Hilbert problem (Green and Lindsay, 1972). Solution of the equations is obtained by using Fourier sine and cosine transforms. Expressions for evaluating the stress intensity factors at the tip of crack are derived. 


Citation 

RK Tripathi, 2019. The Problem of a Griffith Crack at the Interface of Two Bonded Dissimilar MicropolarElastic Halfplanes. The Journal of Mathematics. Photon 109, 180-190 


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Photon Ignitor: ISJN83527329D896225062019




Comparisions of stimultaneous shrinkage and variable selection methods: the lasso, and elastic net regression method 


Suleiman S. a* Buba A.b, Usman U.a 


a Department of Mathematics, Usmanu Danfodiyo University, Sokoto 

b Department of Mathematics, Federal University, Birni-Kebbi 


Keywords: 

Ordinary least squares, multicollinearity, Simultaneous shrinkage and variable selection.


Abstract 

Ordinary Least Squares (OLS) often does poorly in both prediction and interpretation. Penalization techniques have been proposed to improve OLS. This study evaluates the performance of the OLS and some shrinkage methods namely the LASSO and Elastic-Net for remedying multicollinearity, with these methods compared under varying collinearity levels and sample sizes based on criteria such as bias, efficiency and prediction. The result from the simulations shows that the estimates of the LASSO and E-Net are more efficient than those of OLS only when the sample size is small in the presence of very high multicollinearity (ρ≥0.99). The result also indicates that regardless of the level of multicollinearity, OLS remains the least biased and is also most accurate in terms of prediction. It was found out that the shrinkage approaches reduce the problems of collinearity, but does not ultimately solve them, therefore, at small sample sizes (≤500), one may consider the shrinkage methods to reduce the effect of multicollinearity. However, when the sample size is very large, the performance of OLS remains the best even in the presence of multicollinearity. 


Citation: 

Suleiman S. Buba A., Usman U., 2017. Comparisions of stimultaneous shrinkage and variable selection methods: the lasso, and elastic net regression method. The Journal of Mathematics. Photon 108 168-179 


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Photon Ignitor: ISJN83527329D874506122017



Relation Between Lucas Sequence and Generalized Fibonacci-Like Sequence 


Arfat Ahmad Wani*a, G. P. S. Rathoreb, Kiran Sisodiyaa 


a School of Studies in Mathematics, Vikram University Ujjain, India 

b Department of Statistics, Horticulture College Mandsaur, India 


Keywords

Fibonacci Sequence, Lucas Sequence, Generalization of Fibonacci-Like sequence and Binet’s Formula. Msc 2010 classifications: 11B39, 11B37, 11B39 



Citation: 

Wani. A.A.*, Rathore G.P.S., Sisodiya K., 2017. Relation Between Lucas Sequence and Generalized Fibonacci-Like Sequence. The Journal of Mathematics, Photon 108, 163-167 



Abstract

In this paper we present the distinct relations between Lucas sequence and the generalized Fibonacci-Like sequence that is we obtain some identities on the common factors of Lucas sequence and the generalized Fibonacci-Like sequence. Moreover the proof of these identities can be obtained by Binet’s formulae.















This article reports on how “base complement additions (BCA)” was used to help students, in grades 4 and 5, do compound subtraction (subtraction that involves renaming). The research addresses the question of how BCA compares to decomposition (D) as a strategy for teaching and learning compound subtraction, in terms of arithmetic-skill development. The data collection covered nine weeks; and pre-intervention, intervention and post-intervention design was used and computational tests and interviews were two main evaluation instruments used for the study. Participants’ understanding the use of BCA helped them convert compound subtraction to simple subtraction. Findings revealed that participants prefer using base complement additions to the decomposition (borrowing) strategy for BCA is easier for them to use. Suggestions for further research on BCA and D were made. 









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