JUNE 2019| VOL.1 |ISSUE 1
JOURNAL OF MATHEMATICAL SCIENCES & COMPUTATIONAL MATHEMATICS
A Publication of
Society for Makers, Artists, Researchers and Technologists,
USA 6408 Elizabeth Avenue SE, Auburn, Washington 98092.
www.jmscm.thesmartsociety.org 2 JMSCM, Vol 1, Issue 1, 2019
Editorial and Administrative Information
From Managing Editor’s Desk
“Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency” - ‘Descartes’
The Journal of Mathematical Sciences & Computational Mathematics (JMSCM) is a peer reviewed, international journal which promptly publishes original research papers, reviews and technical notes in the field of Pure and Applied Mathematics and Computing. It focuses on theories and applications on mathematical and computational methods with their developments and applications in Engineering, Technology, Finance, Fluid and Solid Mechanics, Life Sciences and Statistics.
The spectrum of the Journal is wide and available to exchange and enrich the learning experience among different strata from all corners of the society. The contributions will demonstrate interaction between various disciplines such as mathematical sciences, engineering sciences, numerical and computational sciences, physical and life sciences, management sciences and technological innovations. The Journal aims to become a recognized forum attracting motivated researchers from both the academic and industrial communities. It is devoted towards advancement of knowledge and innovative ideas in the present world.
It is the first issue of the Journal and we expect to bring it to the zenith through the cooperation of my friends and colleagues.
We sincerely believe that the quarterly issues of the Journal will receive patronage from our esteemed readers and contributors throughout the globe.
Valuable suggestions from any corner for further improvement of the Journal are most welcome and highly appreciated.
Our thanks are due to the SMART SOCIETY, USA in bringing out the Journal in an orderly manner.
Saktipada Nanda
Biswadip Basu Mallik
Journal of Mathematical Sciences & Computational Mathematics
3 JMSCM, Vol 1, Issue 1, 2019 Editorial Board and Editorial Advisory Board members:
Editor in Chief:
Prof. Daniele Ritelli Fellow of American Mathematical Society Associate Professor, Department of Statistical Sciences "Paolo Fortunati", University of Bologna, Bologna Italy. Email:[email protected]
Board Members:
Ahmad Alzaghal Assistant Professor, Department of Mathematics, Farmingdale State College (SUNY), Whitman Hall, Room 180 G, T. 631-794-6516, 2350 Broadhollow Road, Farmingdale, NY 11735. Email:[email protected] Dr. Mohammad Reza Safaei Research Associate, Department of Civil and Environmental Engineering, Florida International University , 1501 Lakeside DriveMiami, Florida , USA Email:[email protected] Douglas Thomasey Assistant Professor, Department of Mathematics, University of Lynchburg , 1501 Lakeside Drive , Lynchburg, VA 24501-3113 Email:[email protected] Vinodh Kumar Chellamuthu Assistant Professor, Department of Mathematics, Dixie State University, 225 South University Avenue, St. George, UT 84770 Phone: 435-879-4256 Buna Sambandham Assistant Professor, Department of Mathematics,
Journal of Mathematical Sciences & Computational Mathematics
4 JMSCM, Vol 1, Issue 1, 2019
Dixie State University, 225 South University Avenue, St. George, UT 84770 Phone: (435) 652-7762. Dr. Marjan Goodarzi Senior Researcher, Department of Mechanical Engineering, Lamar University, Texas, USA Email: [email protected] Ashley McSwain Assistant Professor, Department of Mathematics, Salt Lake Community College, 4600 South Redwood Road, Salt Lake City, UT 84123 Email: [email protected] Garth Butcher Assistant Professor, Department of Mathematics, Salt Lake Community College, 4600 South Redwood Road, Salt Lake City, UT 84123 Email: [email protected] Prof. Dr. Abdon Atangana Professor, Department of Mathematics, University of the Free State, Bloemfontein, South Africa. Email: [email protected] Prof. Belov Aleksei Yakovlevich Professor, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel Email: [email protected] Prof.Mohammad Mehdi Rashidi Professor of Mechanical Engineering Tongji University, Shanghai, China. Email: [email protected] Prof. Poom Kumam Professor, Department of Mathematics, KMUTT Fixed point Research Laboratory,
Journal of Mathematical Sciences & Computational Mathematics
5 JMSCM, Vol 1, Issue 1, 2019
Faculty of Science, King Mongkut’s University of Technology Thonburi(KMUTT). Bangkok 10140, Thailand Email: [email protected] Prof. Dr. Abdalah Rababah Professor, Department of Mathematical Science, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE. Email: [email protected] Prof. Sheng- Lung Peng Professor, Department of Computer Science and Information Engineering, National Dong Hwa University, Taiwan Email: [email protected] Prof. Dr. Dimitar Kolev Department of Fundamental Sciences (PBZN), Academy of Bulgarian Ministry Interior, Sofia, Bulgaria. Email: [email protected] Prof.Mohammad Jaradat Professor , Department of Mathematics, Qatar University, Qatar. Email: [email protected] Prof. Fahd Jarad Professor Department of Mathematics, Cankaya University, Ankara, Turkey. Email: [email protected] Prof. Bilel Krichen Professor, Department of Mathematics, Faculty of Sciences of Sfax,, Tunisia. Email: [email protected] Prof. Chuan-Ming Liu Professor, Dept. of Computer Science and Information Engineering , National Taipei University of Technology (Taipei Tech) , Taipei 106, TAIWAN Email:[email protected]
Journal of Mathematical Sciences & Computational Mathematics
6 JMSCM, Vol 1, Issue 1, 2019
Prof. Alfaisal A. Hasan Professor , Department of Basic and Applied Sciences, Arab Academy for Science, Technology & Maritime Transport, Egypt Email: [email protected] Prof. Fevzi Erdogan Professor Department of Econometrics, YuzuncuYil University, Van-65080, Turkey. Email: [email protected] Prof. Dr. Eric Flores Medrano Professor, Faculty of Physical Mathematical Sciences, Benemérita Autonomous University of Puebla, Puebla, Mexico, Email:[email protected] Prof. Dr. Ayse Dilek Maden Professor, Department of Mathematics, Faculty of Science, Selcuk University,Konya, Turkey, Email: [email protected] Prof. Ahmet Yildiz Professor , Department of Mathematics, Inonu University, Malatya, Turkey Inonu University, Malatya, Turkey Email: [email protected] Prof.Cristiane De Mello Professor , Department of Mathematics, Federal University of the State of Rio de Janeiro (UNIRIO), Av Pasteur, 458 - Urca, Rio de Janeiro, Brazil Email: [email protected] Prof. Dr.Zakia Hammouch Department of Mathematics, Faculty of Sciences and Techniques, Moulay Ismail University Errachidia,
Journal of Mathematical Sciences & Computational Mathematics
7 JMSCM, Vol 1, Issue 1, 2019
Morocco. Email: [email protected] Prof. Omar Abu Arqub Professor of Applied Mathematics, The University of Jordan. Email: [email protected] Prof. Meng-Chwan Tan Associate Professor , Department of Mathematical Physics, National University of Singapore, Singapore Email: [email protected] Prof. Ilhame Amirali Associate Professor, Department of Mathematics, Faculty of Arts & Sciences, Duzce University, Duzce-81620, Turkey Email: [email protected] Prof. Nei Carlos Dos Santos Rocha Associate Professor , Institute of Mathematics-IM-UFRJ, Federal University of Rio de Janeiro- UFRJ, Brazil Email: [email protected] Prof. Hanaa Hachimi Associate Professor in Applied Mathematics & Computer Science, National School of Applied Sciences ENSA, IbnTofail University, PO Box 242, Kenitra14000, Morocco. Email: [email protected] Prof. Dr. M.B. Ghaemi Associate Professor of Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, 1684613114, Tehran, Iran. Email: [email protected] Prof. Marcel David Pochulu Associate Professor, Pedagogical Academic Institute of Basic and Applied Sciences, National University of Villa Maria, Villa Maria, Argentina, Email: [email protected]
Journal of Mathematical Sciences & Computational Mathematics
8 JMSCM, Vol 1, Issue 1, 2019
Prof. Amr Abd Al-Rahman Youssef Ahmed Assistant Professor in Applied Mathematics, (Quantum Information focused on Heisenberg models), High Institute for Engineering and Technology at El-Behira, Egypt. Email: [email protected]
Prof. Wutiphol Sintunavarat Assistant Professor, Thammasat University, Thailand Email: [email protected]
Prof. Umit Yildirim Assistant Professor of Mathematics, Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa University, 60100 Tokat, Turkey Email: [email protected]
Prof. Ebenezer Bonyah Senior Lecturer, Department of Mathematics Education, University of Education Winneba (Kumasi Campus), Kumasi, Ghana. Email: [email protected]
Dr. Godwin Amechi Okeke Senior Lecturer, Department of Mathematics, School of Physical Sciences, Federal University of Technology, Owerri, PMB 1526 Owerri, Imo State, Nigeria. Email: [email protected]
Dr. Mehmet Giyas Sakar Department of Mathematics Faculty of Science Van Yuzuncu Yil University Van-65080, TURKEY. Email: [email protected]
Journal of Mathematical Sciences & Computational Mathematics
9 JMSCM, Vol 1, Issue 1, 2019
Patron
Prof. (Dr) Satyajit Chakrabarti Director Institute of Engineering & Management.
Managing Editor
Prof. (Dr) Saktipada Nanda Professor Department of Electronics & Communication Engineering, Institute of Engineering & Management.
Development Content Editors
Prof. Biswadip Basu Mallik Assistant Professor Department of Basic Science & Humanities, Institute of Engineering & Management. Prof. (Dr) Krishanu Deyasi Associate Professor Department of Basic Science & Humanities, Institute of Engineering & Management. Prof. Santanu Das Assistant Professor Department of Basic Science & Humanities, Institute of Engineering & Management.
Contact Us:
Society for Makers, Artist, Researchers and Technologists. 6408 Elizabeth Avenue SE, Auburn, WA 98092, USA.
Email: [email protected]
Phone- 1-425-605-0775
Journal of Mathematical Sciences & Computational Mathematics
10 JMSCM, Vol 1, Issue 1, 2019
PAGE No. CONTENT
11 A mathematical model to study the effect of porous parameter on blood flow through an atherosclerotic arterial segment having slip velocity
Sibashis Nanda, Sayudh Ghosh & Ronit Chaudhury 21 Artificial neural network
Mohammad Ehteshamullah 26 Union of Brain Computer Interface and Internet of Things: An Integrated Platform to Enhance Cognitive Interaction in Real- time
Aritra Mukherji & Nirmalya Ganguli 31 Pi, revisited
A. K. Chatterjee
35 A novel approach for selecting the potentially best performing team in IPL
Sannoy Mitra, Tiyash Patra & Raima Ghosh
41 Field Aided Semiconductor Superlattices, the Einstein Relation and All That
J. Pal, M. Debbarma, N. Debbarma, Paulami Basu Mallik and K. P. Ghatak
Journal of Mathematical Sciences & Computational Mathematics
11 JMSCM, Vol 1, Issue 1, 2019
A MATHEMATICAL MODEL TO STUDY THE EFFECT OF POROUS PARAMETER ON BLOOD FLOW THROUGH AN ATHEROSCLEROTIC ARTERIAL SEGMENT HAVING SLIP VELOCITY
1Sibashis Nanda, 2Sayudh Ghosh & 3Ronit Chaudhury
1Founder and CEO, LEARNIKX EDUCATION,Sydney, Australia E-mail: [email protected]
2Graduate Teaching Assistant Department of Mechanical Engineering Colorado State University, Fort Collins, Colorado, USA Email: [email protected]
3Department of Mechanical Engineering Institute of Engineering & Management Salt Lake Electronics Complex, Kolkata-700091. India Email: [email protected] Abstract
This theoretical investigation focusses on blood flow through a multiple stenosed human artery under porous effects. A mathematical model is developed for estimating the effect of porous parameter on blood flow taking Harschel-Bulkley fluid model (to account for the presence of erythrocytes in plasma) and artery as circular tube with an axially non-symmetric but radially symmetric mild stenosis. The mathematical expression for the geometry of the artery with stenoses is given by the polynomial function model. The velocity slip condition is also given due weightage in the investigation. It is necessary to study the blood flow through such type of stenosis to improve the arterial system. An extensive quantitative analysis is carried out by performing large scale numerical computations of the measurable flow variables having more physiological significance. The variations of velocity profile, volumetric flow rate and pressure gradient with porous parameter are calculated numerically by developing computer codes. Their graphical representations with appropriate scientific discussions are presented at the end of the paper.
Key words: stenoses, stenotic geometry, stenotic height, H-B fluid model, Darcy’s number, fluid index parameter, slip velocity, flow flux
Journal of Mathematical Sciences & Computational Mathematics
12 JMSCM, Vol 1, Issue 1, 2019
INTRODUCTION
Among all the fatal diseases of the human body, circulatory disorders are still a major cause of morbidity or death. A systematic study on the rheological and hemodynamic properties of the streaming blood with the mechanical behavior of blood vessel walls could play a significant role in the basic understanding, diagnosis and treatment of many cardiovascular, cerebrovascular and arterial diseases. Arteries and veins are narrowed by abnormal and unusual deposition of cholesterols, fats, plaques and other suspended matters in their inner lining leading to the cardiovascular disease – atherosclerosis (medically termed stenosis). This narrowing of body passage, tube or orifice is a frequently occurring phenomenon in human artery. One of its most serious consequences is the increasing resistance to blood flow bringing about significant alterations in pressure distribution, wall shear stress and the flow resistance. The tragedy of aging is that plaques build up within narrow arteries and this phenomenon makes them stiffer (less elastic) that restricts (sometimes blocks) the regular flow of blood in the human physiological system. This type of constriction causes insufficient flow of blood in heart and may lead to stroke, ischemia and heart attack.
A wide variety of analytical as well as experimental studies on blood flow through the arterial segments having a single or multiple stenosis were carried out by several investigators applying different blood models (Newtonian or non-Newtonian) and various geometry of the stenosis (polynomial, smooth cosine or exponential). Experimental observations confirmed that blood is predominantly a suspension of erythrocytes (red cells) in plasma and may be better represented by non-Newtonian model when it flows through narrow arteries at low shear rate, particularly in diseased state. A good number of mathematicians, physicists and medical professionals have contributed their experimental and theoretical research works on cause and remedies of restriction of blood flow in the constricted artery of the human body.
It is a fact that all the phenomena in the real world show non-linear attitude. Therefore, for evaluating a phenomenon in our non-real world, the scientists should make their efforts to develop a mathematical model that exhibits the physical system or phenomena’s nearly exact behavior. So, for any investigation on blood flow in stenotic region, the rheological and physical nature of the blood together with the geometry of the stenosis were modeled mathematically by the researchers.
In the recent past, quite a good number of analytical as well as experimental investigations were performed by several researchers to explore the effect of arterial constriction on the flow characteristics of blood applying different fluid models for blood and geometry of the stenosis. It is accepted that blood may be fairly closely represented by Harschel-Bulkley (H-B) model-a non- Newtonian blood model at low shear rates when flowing through a tube of diameter 0.095 mm or less. In order to understand the effect of stenosis on blood flow through and beyond the narrowed
Journal of Mathematical Sciences & Computational Mathematics
13 JMSCM, Vol 1, Issue 1, 2019 segment of the artery many investigations have been undertaken both experimentally as well as analytically.
Sankar and Hemlatha (2007) presented an analytical study on the pulsatile flow of blood through catherized artery by modeling blood as Harschel-Bulkley fluid and the catheter and artery as rigid coaxial cylinders. The output of the study established decrease of velocity and flow rate and increase of wall shear stress and longitudinal impedance for the increase in value of yield stress with other parameters fixed. The pulsatile flow of blood through mild stenosed artery was investigated by Sankar and Lee (2009) considering the HB model of blood. They observed that the plug core radius, pressure drop and wall shear stress increase with the increase of yield stress and the stenosis height.
It has been observed that in case of some arterial diseases, a porous structure is formed in the lumen of an artery by fatty substances – cholesterol as well as due to blood clots. Walls are porous and generally Darcy’s law is used to investigate the problem analytically. Krishna et. al. (2012) investigated analytically the effects of various parameters like Darcy number (for porous medium), slip parameter and radius on velocity and frictional force and obtained results for volumetric flow rate and frictional forces. Singh (2012) presented a mathematical analysis on the effect of stenosis shape parameter and height on the flow resistance. The study reveals that flow resistance decreases as shape parameter increases and it shows increasing trend with the increase of stenosis height and length. In the analysis, he adopted two phase macroscopic model for blood and the polynomial model to represent the geometry of the stenosis. Shah (2013) developed a mathematical model for the analysis of blood flow through diseased blood vessels under the influence of porous parameter. It was observed that the wall shear stress increases with the increase of porous parameter, stenosis size and length. On the other hand, the wall shear stress decreases as the stenosis shape parameter increases. The Harschel-Bulkley fluid model for blood with flow model represented by Navier- Stokes and the continuity equations were adopted in the study. In the analysis, the traditional no slip boundary condition was employed. But a number of theoretical and experimental studies on blood flow have suggested the likely presence of slip (a velocity discontinuity) at the flow boundaries (or in their immediate neighborhood) [Biswas (2000)]. Thus in blood flow modeling, consideration of a velocity slip at the stenosed vessel wall will be quite relevant. Hematocrit-a representation of the oxygen carrying capacity of the blood (its normal values for an adult male is 40-54% and for an adult female is 36-46%) was not involved in the investigation.
Nanda et. al. (2017) developed a mathematical model for studying blood flow through an elastic artery with the consideration of slip velocity at the inner wall of the artery. The study reveals considerable alterations in flow characteristics due to the presence of elastic property of blood vessel wall and the presence of velocity slip at the wall.
It gives us an opportunity to develop a mathematical model for estimating the effect of porous parameter on blood flow taking Harschel-Bulkley fluid model and artery as circular tube with an axially non-symmetric but radially symmetric mild stenosis. The velocity slip condition is also
Journal of Mathematical Sciences & Computational Mathematics
14 JMSCM, Vol 1, Issue 1, 2019 given due weightage in the investigation. It is necessary to study the blood flow through such type of stenosis to improve the arterial system. An extensive quantitative analysis is carried out by performing large scale numerical computations of the measurable flow variables having more physiological significance. The variations of velocity profile, volumetric flow rate and pressure gradient with porous parameter are calculated numerically by developing computer codes. Their graphical representations with appropriate scientific discussions are presented at the end of the paper.
MATHEMATICAL FORMULATION OF THE PROBLEM
Let us consider the axisymmetric flow of blood through a a uniform circular artery with an axially non-symmetric but radially symmetric mild stenosis specified at the position shown in Fig. 1.
The geometry of the stenosis assumed to be manifested in the arterial segment is given by m1 m R(z) 1 AL0 (z d) (z d) ,d z d L0 ______(1) R0 1 ,otherwise where
R(z) : Radius of the tube with stenosis R0 : Radius of the tube without stenosis
L : Length of the artery L0 : Length of the stenosis
: Max. height of the stenosis in the lumen ( R0 ) m : Stenosis shape parameter (m 2) mm /(m1) d : Location of the stenosis in the artery and A . R L m m 1 0 0
METHOD OF SOLUTION
The constitutive equation in one dimensional form for Herschel-Bulkley fluid in terms of the axial velocity of blood (u) with the shearing stress , is given by
Journal of Mathematical Sciences & Computational Mathematics
15 JMSCM, Vol 1, Issue 1, 2019
1 n du 0 , 0 f ( ) (2) dr 0, 0
r dp Rc dp du where , measure of yield stress 0 , the strain rate e 2 dz 2 dz dr : viscosity coefficient of blood n : flow behavior index r : radius of artery and p : pressure gradient [The values of 'n'for blood flow problems are generally taken to lie between 0.9 and 1.1]
(Basu Mallik & Nanda S. [9])
The Darcy’s equation governing the flow of blood through a porous media is given by p Q k. (3) where k :porous parameter, viscosity coefficient of blood
The present mathematical model is devoted to estimate the effect of porous parameter on blood (treated as an incompressible non-Newtonian fluid) flow. The governing Navier- Stokes equation in cylindrical polar co-ordinates is given by u u u 1 p 2u 1 u 2u v u (4) 2 2 t r z z r r r z v v v 1 p 2v 1 v v 2v v u (5) t r z r r 2 r r r 2 z 2 1 rv u The continuity equation is 0 (6) r r z
BOUNDARY CONDITIONS:
The slip (a velocity discontinuity) at the flow boundaries gives the following boundary conditions:
u us (slip velocity) at r R0 and u 0 at r R(z) (7)
ANALYTICAL SOLUTION: Applying the governing equation of motion for steady incompressible blood flow with pressure gradient through the mid stenosis in an artery is reduced to the following form:
1 p r P r , where, P, P being a constant. Integrating with respect to r, P r r z 2
Journal of Mathematical Sciences & Computational Mathematics
16 JMSCM, Vol 1, Issue 1, 2019 The equation of velocity gradient can be obtained as:
1/푛 푑푣 푃 1/푛 = ( ) (푟 − 푟 ) 푑푟 2µ 푝
Now, integrating the above equation by considering slip velocity us we will have,
푃 1/푛 푛 푛+1 푛+1 푉 = 푈 + ( ) × {(푟 − 푟 ) 푛 − (푅 − 푟 ) 푛 } 푆 2µ 푛 + 1 푝 푝
푅 Now the total flow flux can be calculated via the following equation: 푄 = 2휋푟 × 푉 푑푟 ∫0
3푛+1 3푛+1 2푛+1 2푛+1 ( ) ( ) ( ) ( ) 1⁄ 푛 푛 푛 푛 2 푃 푛 푛 (푅−푟푝) (−푟푝) (푅−푟푝) (−푟푝) Or, 푄 = 휋푅 푈푆 + 2휋 ( ) × [ 3푛+1 − 3푛+1 + 푟푝 { 2푛+1 − 2푛+1 } − 2µ 푛+1 ( ) ( ) ( ) ( ) 푛 푛 푛 푛
푛+1 푛 (푅 − 푟푝) × 푅] and the pressure P from the above equation can be calculated as:
푃 푛 푛 + 1 2 푛 2µ × ( ) × [푄 − 휋푅 푈푆] = 푛 3푛+1 3푛+1 2푛+1 2푛+1 푛 (푅 − 푟 ) 푛 (−푟 ) 푛 (푅 − 푟 ) 푛 (−푟 ) 푛 푛+1 ( )푛 푝 푝 푝 푝 푛 2휋 [ 3푛 + 1 − 3푛 + 1 + 푟푝 { 2푛 + 1 − 2푛 + 1 } − (푅 − 푟푝) × 푅] ( 푛 ) ( 푛 ) ( 푛 ) ( 푛 ) NUMERICAL RESULTS AND DISCUSSION:
In order to have an estimate of the qualitative and quantitative effects of the physical and rheological parameters involved in the analysis, it is necessary to quantify them. The values of different material constants and other parameters have been taken from standard literatures. P 550, 0.004, us 0.05, p 0.001, 0.2, L0 0.5, d 0.25, z 0.75 Computer R0 codes are developed and the graphs are plotted using MATLAB 8.5.
Fig. 2: Variation of flow velocity with radial distance
Journal of Mathematical Sciences & Computational Mathematics
17 JMSCM, Vol 1, Issue 1, 2019
The variation of flow velocity (v) with radial distance (r) for n (fluid index parameter) = 0.95, 1.00, 1.05 is exhibited in Fig. 2. In the normal stage, the flow velocity should be minimum in the vicinity of the flow boundary but the trend may be justified due to the presence of stenosis in the artererial segment. The tendency is similar for = 0.95, 1.00, 1.05.
Fig. 3: Variation of flow velocity with axial distance, taking =0.2 R0 Fig. 3 illustrates the variation of flow velocity with axial distance for = 0.95, 1.00, 1.05 taking
(stenotic height) = 0.2. The velocity is proportional to r that is the presence of stenosis does not influence the longitudinal velocity.
Fig. 4: Variation of flow velocity with axial distance, =0.2, 0.3, 0.4, 0.5, 0.6
The variation of flow velocity with radial distance for (fluid index parameter) = 0.95, 1.00, 1.05 is demonastrated in Fig. 4. The flow velocity sharply declines in the stenotic region and converge for the values of the under consideration.
Journal of Mathematical Sciences & Computational Mathematics
18 JMSCM, Vol 1, Issue 1, 2019
Fig. 5: Variation of flow flux with stenotic height
Taking n (fluid index parameter) = 0.95 and 1.05, the variation of flow flux with stenotic height is exhibited in Fig. 5. The Flow flux shows an increasing tendency for increase in the values of stenotic height for both values of n.
Fig. 6: Variation of flow flux with axial distance
Taking slip velocity 0.0 (no slip), 0.025 and 0.05, the variation of flow flux with axial distance is demonastrated in the above figure (6). The flow flux is minimum for minimum slip velocity. So, velocity discontinuity has an important role in the calculation of flow flux.
P Fig. 7: Flow resistance vs. Stenotic height Q
The variation of flow resistance with stenotic height for n 0.95,1.05are presented.
Journal of Mathematical Sciences & Computational Mathematics
19 JMSCM, Vol 1, Issue 1, 2019
P Fig. 8: Flow resistance vs. Stenotic height Q
The variation of flow resistance with stenotic height for us 0.0,0.025,0.05are examined.
CONCLUSION
It is now established that lifesaving medicines applied for treatment of many fatal diseases like cancer, diabetes, heart failure damage some other organs /cells of the human body. If the location of the stenosis is detected without any surgery, then appropriate medicine may be sent to the affected area/cells with the help of Nano technology. This may open a new dimension in the treatment of some fatal diseases and more and more theoretical as well as experimental research in this field are suggested to overcome the restrictions imposed on the analysis. Also, the outcome of this analysis may be shared for further investigation in this emerging field and for providing primary support to the deceased where adequate medical/surgical support is not available.
REFERENCES:
1.Charm, S.E. and Kurland, G.S. –“Blood Rheology in Cardiovascular Fluid Dynamics, Academic Press, London, 1965 2.Astrita, G. and Marrucci, G.; Principles of Non-Newtonian Fluid Mechanics, McGRaw –Hill, New York, USA (1974). 3. Liepsch, D. W.; “Flow in tubes and arteries – A comparison”. Biorheology, 23, 395-433 (1986) 4.Ku, D. N., Blood flow in arteries, Ann. Rev. Fluid Mech, 29, 399-434, (1997) 5. McDonald, D.A., Blood flow in arteries, Edward Arnold Publishers, London (1974) 6. Sankar, D.S. and K. Hemlatha; “Pulsatile flow of Harschel-Bulkley fluid through catherized arteries-a mathematical model”. Applied Mathematical Modeling, 31, 1497-1517 (2007)
7. Sankar, D.S. and Lee,U.: “Math. Modeling of pulsatile flow of non-Newtonian fluid in stenosed arteries”. Communication in Non-linear Sci. Numer. Simul., 14, 2971-2981 (2009)
8. Nanda, S., Basu Mallik, B., Das S.; “Study on the effect of non-Newtonian nature of blood flowing through an elastic artery with slip condition”: J. of Chem., Bio., and Phy. Sc.: Sec. C, 7(4), 934-942
Journal of Mathematical Sciences & Computational Mathematics
20 JMSCM, Vol 1, Issue 1, 2019
9. Basu Mallik, B. and Nanda, S.: A Math. Anal. Of Blood Flow through Stenosed Arteries: A non- Newtonian Model, IEM Int. J. of Mgt. & Tech., 2(1), 59 – 63 (2012)
10. Shah, S. R.; “A math. Model. for the anal. Of blood flow through diseased blood vessels under the influence of porous parameter: J. Biosc. Tech., 4(6), 2013, 534-541.
11. Verma, N. K., Mishra, S, Siddique, S.U., Gupta, R.S. ; Effect of slip velocity on blood flow through a catherized artery, Applied Mathematics, 2, 764-770 (2011).
12. Kumar, S. “Study of blood flow using Power law and Harschel-Bulkley non-Newtonian fluid model through elastic artery”, Proceedings of ICFM, 229-235 (2015).
13. Ponalgusamy, R; “Blood flow through an artery with mild stenosis: A Two-layered model, Different shapes of stenosis and slip velocity at the wall”, Journal of Applied Sciences, 7, 1071-1077 (2007).
14. Nanda, S., Basu Mallik, B., Das S.; Effect of slip velocity and stenosis shape on blood flow through an atherosclerotic arterial segment using Bingham-Plastic fluid model. Int. Journal of Appl. Mathematics and Mechanics, 5(1), (2016).
15. Singh, Bijendra, Joshi, Padma & Joshi, B.K.; “Blood flow through an artery having radially non- symmetric mild stenosis”, Applied Mathematical Sciences, 4(22), 1065-1072 (2010).
16. Singh, A.K. “Effects of shape parameter and length of stenosis on blood flow through improved generalized artery with multiple stenosis”, Advances in Appl. Mathematical Biosc., 3(1), 41-48 (2012).
Journal of Mathematical Sciences & Computational Mathematics
21 JMSCM, Vol 1, Issue 1, 2019
ARTIFICIAL NEURAL NETWORK
Mohammad Ehteshamullah
Department of Electronics and Communication Engineering, Institute of Engineering & Management, India
E-mail: [email protected]
Keywords: Artificial Neural Network (ANN), Neurons
Abstract
Artificial Neural Networking or ANN is the way by which computers can mimic the biological nervous system by building a huge number of simulated neurons, which are joined together in a number of ways to form networks. A neural network has multiple processors instead of just one central processor and thus is very task efficient. They cannot be made to perform a specific task because they learn via experience, which is unlike any other computer processor. They follow certain learning laws while receiving feedback from the environment. This paper gives an overview of Artificial Neural Network, its working and training.
Introduction
We view computers as an advance piece of technology capable of doing large amount of computations at a very low amount of time. They have helped us complete tasks which would have been impossible without them. However, certain tasks like identification of faces, people’s accents or recognising emotions, humans are still far superior than even the fastest and the most sophisticated super computer in the planet. For an example, when we meet an old friend, our brain tends to recognise their face immediately. A computer which has been programmed to recognise faces would run a scan, trying to match the person’s face with all the faces stored into its data. Such process might take from minutes to even days. This could be partly because the algorithm that computers use to learn new things or adapt to certain environment is much different to that of a human. To change that, the concept of developing Artificial Neural Network came into existence in the early 1950s. The ANN concentrates of developing a huge number of simulated neurons, connected in a non-linear and parallel way to complete task. Some of the advantages of an ANN is –
1. Non-linearity An Artificial Neural Network contains an enormous connection of simulated neuron, unlike computers today which are connected linearly. 2. Adaptivity: Neural Network can adapt the free parameters to the changes in the surrounding environment. 3. Evidential Response: ANN can make computers give decisions with a measure of confidence
Journal of Mathematical Sciences & Computational Mathematics
22 JMSCM, Vol 1, Issue 1, 2019 4. Fault Tolerance: Artificial Neural Network follows what is called ‘Graceful Degradation’. Basically, this means that the computer allows certain tolerance to some degradation in the software of in the simulated neurons. Thus, Artificial Neural Networks take a different approach to solve a particular problem. Today, neural networks discussions are occurring everywhere. Their promise seems very bright as nature itself is the proof that this kind of thing works.
Construction
A neural Network can be constructed in a computer by having a deeper look on how information is transferred through a human brain. Certain information is first received through our sensory organs, which are then transferred through neurons to our brain, which performs certain calculations in regards to what response we want, and then sends that particular response. This exact process is imitated by the Artificial Neural Network. Neural network builds a huge number of simulated neurons, which are joined together in a number of ways to form networks. These simulated neurons are arranged in the form of layers, which works just like our nerves. These layers are connected through nodes which contain activation function. When one layer receives certain information, it passes on to the next layer. The layer might itself do the computation or pass it along. The following diagram shows how these connections are made:
There is a total of 3 layers, the input layer imitates the sensory organs, which receives data from the user. The hidden layer is where the learning mechanism is put to use. The hidden layer could be singular, or multi-layered. A neural network consisting of multiple hidden layers can also be called as deep learning neural network.
Working
Scientists McCulloch and Pitts developed a network model for an ANN. This network model consisted of several inputs (x1, x2, x3…., xn) with each input’s effect being decided by the amount of weight associated with it (wk1, wk2, wk3…., wkn). The input is then multiplied with its corresponding weight to give the synaptic weights (uk). A bias is associated along with the input to fit the prediction. These synaptic weights are then summed up along with the bias to give in the
Journal of Mathematical Sciences & Computational Mathematics
23 JMSCM, Vol 1, Issue 1, 2019 transfer function to give a net input (vk). vk is then passed through an activation function, which contains a certain threshold, and only allows certain limited number of output. The output is thus obtained. All these process is represented in the diagram below.
The threshold is a particular graph that we feed into the activation function. The graph can be a unit step function graph, where if vk is any positive integer, the output will be 1, otherwise 0.
Threshold function can also be a signum function, sigmoid function, or a Rectified Linear Unit function or RELU function.
Let’s take an example to better understand the model. Consider that we have already trained an ANN on how to decide whether an individual is hospitalised or not. Based on the inputs such as the individual’s age, gender, their location and income. Older people have a much higher chance of being hospitalised than younger men, and statistically men have a higher chance of being hospitalised than women. Our ANN can be trained to find the correlation between the distance of the hospital to the individual’s location, and higher income means being able to afford the bills of the hospital, thus higher chance of being in the hospital. The threshold function is a unit step function. If the net input (vk) is positive, then the individual is hospitalised, and if it is negative, then the individual is not hospitalised. The hidden layer does the main computations to decide the output. The weight of each particular input and the bias is developed by the program itself, based on how it has been trained.
Training is a vital step of making an ANN after constructing the model. A human child cannot learn everything on its own. It is the job of the parents and the teachers to constantly guide and supervise the child to explain them how to perform certain tasks, what things are morally right or wrong. But their learning is not just limited to the knowledge acquired by their supervisors. There are certain things which an individual learns on its own during its growth. In a very similar way, the ANN is trained. There are three ways by which an ANN can be trained:
1. Supervised Training: In a supervised training, we have a large housing of datas that we feed into the ANN. Thus, the ANN knows what type of inputs will be fed to it and what output needs to be derived. Thus, the input data is run through multiple programs in the hidden layers in such a way to derive the desired output. This is called forward propagation.
Journal of Mathematical Sciences & Computational Mathematics
24 JMSCM, Vol 1, Issue 1, 2019 In each complete step an error is found, which the computer recognises. We try to minimise the weights of the neuron for those that are contributing more to the error, and this happens while we go back to the neurons. This is called backward propagation. The computer repeats these steps, slowing the amount of adjustment made to the weights at every step, adjusting the weights in such a way that the error is minimised, until the desired output is found. Process of reduction of error can take a lot of time and power, so we had to develop a smart way to do these steps. A process called gradient descend is used to tackle with the problem of error reduction. 2. Unsupervised Training: In an unsupervised training process, only the inputs of the network has been provided, while the computer adjusts the weights on its own to give the desired output. The learning process is similar to that of the supervised training, but the decision whether the output is relevant or not is decided by the computer itself. This training allows the computer to learn new things on its own and adapt to certain changes without being told to do so. This is called “Artificial Intelligence”. 3. Semi-supervised Training: In a semi-Supervised training process, the output is unknown, but the network provides the feedback whether the output is correct or incorrect. Applications
An ANN is going to be beneficial to the human society in many ways, some of which are listed below:
1. Forecasting of weathers and stock markets. 2. Biometric securities such as facial recognition, fingerprint recognition or iris recognition. 3. Development of Artificial Intelligences among computers and making of robots. 4. Development of self-driving cars and regulation of traffic. 5. Diagnosis of certain undetectable diseases. 6. Recognition of handwriting and certain other calligraphy. Conclusion
Neural networking is the future of computers. Its applications have already been vastly adopted in our day to day life such as in electronics like smartphones and smart TVs, to development of “Artificial Intelligence (AI)”. With introduction of NPUs for dedicated AI tasks, our computer processor does not need to rely upon its CPU for every ANN tasks. The computer’s dependency on us humans would lessen as they would learn how to adapt and change from time to time and reduce errors, create self-organisations and perform real time operation. With active study going on upon how to improve upon its learning law and create better ones, we must expect computers to soon outsmart us.
Acknowledgement
I would like to express my gratitude my parents and my sister for being supportive of my research and guiding me through each step. Special thanks to Dr. Ankur Bhattacharjee for guiding me with important notification and details regarding the event.
Journal of Mathematical Sciences & Computational Mathematics
25 JMSCM, Vol 1, Issue 1, 2019
References
[1] Lippmann, R.P., 1987. An introduction to computing with neural nets. IEEE Accost. Speech Signal Process.
[2] Ms. Sonali B. Maind, Ms. Priyanka Wankar, 2014. Research Paper on Artificial Neural Network
Journal of Mathematical Sciences & Computational Mathematics
26 JMSCM, Vol 1, Issue 1, 2019
Union of Brain Computer Interface and Internet of Things: An Integrated Platform to Enhance Cognitive Interaction in Real-time 1Aritra Mukherji & 2Nirmalya Ganguli
1Department of Electronics and Communication Engineering, Institute of Engineering & Management, Kolkata, India. 2Department of Electrical Engineering, Institute of Engineering & Management, Kolkata, India.
Abstract
Brain Computer Interface (BCI) is a platform which receives brain signals, measures and analyses them, providing a pathway for the human brain to interact with external utilities in real-time. It is entirely independent of the normal output of peripheral nerves and muscles. On the other hand, with the exposure of Internet of Things, the concept of connectivity of devices has evolved. The number of connected devices is expected to grow phenomenally across multiple industries, thereby boosting productivity and efficiency in coming years. This paper elaborates the procedure of developing a system merging Brain Computer interface and internet of things, the possible applications of human-thing cognitive interactivity and the challenges we face while working with it.
Key words: Brain-Computer Interface, Internet of Things, cognitive, interactivity.
INTRODUCTION
According to the statistical records, in the last decade around 50 million devices have been connected to the Internet. The number of interconnected devices exceeded our population back in 2008. With the revolution of Internet of Things, people can interact and control a number of devices through a wide range of applications available on our smartphones, laptops, etc. The global worth of IoT devices is projected at $6.2 trillion by 2025. On the other hand, the popularity of Brain-Controlled Interface is growing in recent years which allows establishment of a direct communication pathway between the human brain and any external device in real-time. In recent years, the world has seen the ability of BCI to decode the thinking capabilities of man and using the relevant information obtained to control external devices. Some common examples being, mind-controlled robotic hand (underneuro-prosthetics), mind-controlled wheelchair and other IoT based appliances. To establish human-cognitive interactivity, BCI proves to be advantageous as it has the inherent privacy setup. This is because brain activity is invisible in nature. Another advantage of BCI is that the received data is real-time in nature as a person just need to think about the interaction instead of performing any physical task.
While working on BCI platform we face a number of challenges as it is time consuming and not completely efficient. The environmental factors are also a major factor for decreased efficiency.
Journal of Mathematical Sciences & Computational Mathematics
27 JMSCM, Vol 1, Issue 1, 2019 In this paper we are highlighting the working principle of the integrated platform of BCI, IoT and the problems we face while working with BCI, the possible applications and the current works on human-thing cognitive interaction.
The Integrated System: Working Principle Internet of Things is a platform of interrelated electronic devices, machineries, objects etc, which are having a unique identification and are capable of interacting and exchanging data via Internet without any human-human interaction, thereby executing the practical applications with utmost efficiency in real-time.
The standpoint of integration of Brain Computer Interface with the Internet of Things, the correlation of BCI and IoT is given below:
Fig 1: Correlation among BCI and IOT SENSORS/DEVICES Brain cells communicate with each other by transmission of small electrical signals. Under normal conditions, to perform a particular task, these signals are sent to the muscles via the central nervous system. Brain Computer Interface is a platform which creates an alternative pathway to interact with external utilities in real-time without normal output of peripheral nerves and muscles. For execution of the following process, the initial requirement is collection of brain signals. A number of technologies can be used to collect these signals, the most common being Electroencephalography (EEG). There are other popular technologies like Magneto-
Journal of Mathematical Sciences & Computational Mathematics
28 JMSCM, Vol 1, Issue 1, 2019 encephalography (MEG) and Functional Near-Infrared Spectroscopy (fNIR) for the collection of brain signals. As they involve the use of expensive machineries, they are not suitable for household application. The most common way of collection of brain signals is by placing electrodes on the scalp of a person, or by implanting electrodes inside the brain by surgical method which is commonly termed as implantable BCI. These electrodes sense the small electrical signals and amplify them to detect the active regions of the brain.
CONNECTIVITY
Connectivity is an important segment of Internet of Things. There are a number of communication portals available for execution of the same, for example, WiFi, Bluetooth, Satellite, Cellular, RFID, NFC, LPWAN and Ethernet. The sensors can be connected to any of these communication portals based on the desired range and thereby facilitate to send the received signal data from the brain to the cloud. For the organizations it will be difficult to store the massive amount of data produced by the devices in the coming years, as it involves a huge cost. Cloud is a platform which is designed to store the amount of data received from the sensors and process them for real-time applications. Selecting the most suitable connectivity portal is essential because an erroneous choice may lead to poor performance or increased cost which is not desirable. The technical, commercial and ecosystem requirements are to be analysed before selecting the suitable connectivity portal.
DATA PROCESSING
This is the core of the integrated system of BCI and IoT. In 2012, a team of IEEE members proposed a Deep Learning Framework. The cloud consists of this Deep Learning Framework to process the received data from the sensors. The processing of the data is a tri-level process.
• Data Replication and Shuffling Data replication allows copying of application database to a secondary clone database by capturing the small changes in data occurring from time to time. It ensures availability, increased parallelism, and security. The Data replication is not instantaneous as it involves increased cost. So it captures the data when the system is not in use. This causes a time delay, which is termed as latency. Now, EEG signals are arranged as one- dimensional vectors in practice. To use this data efficiently, we need to replicate it to a higher dimensional space. The massive amount of replicated data is then shuffled to improve the predictive performance and machine learning model quality.
• Selective Attention Mechanism As different categories of data signify different characteristics of the brain, it is difficult to specify the fragments with utmost relevant information. Selective Attention Mechanism is designed to select the relevant information received from the brain signals and preferably process it while suppressing the redundant data. This increases the efficiency of the integrated system.
Journal of Mathematical Sciences & Computational Mathematics
29 JMSCM, Vol 1, Issue 1, 2019 • Weighted Average Spatial LSTM Classifier Recurrent Neural Networks (RNN) is a class of neural networks which is expert in learning crucial information from sequential data. A section of RNN, which is applicable for practical use is, Long Short-Term Memory (LSTM). Now, in Weighted Average Spatial LSTM Classifier, the LSTM output is obtained by averaging the last two weighted outputs to enhance the stability of the neural network as it is continuously fluctuating.
USER INTERFACE
User Interface deals with the act of sending the relevant information obtained to the user end for executing the practical applications. The several ways by which a user can interact with the system are by receiving automatic notification, monitoring information proactively and control the system remotely. A significant example of Brain-Controlled Interface with Internet of Things being Smart Living Environmental Auto-adjustment Control System (BSLEACS). It monitors the mental state of the user and adapts the surrounding accordingly. Also the integration of smart house and healthcare is a recent approach of this model.
Problems we encounter while working in this platform
• BCI is primarily based on the emotional state of the user.
• Environmental factor is also a prime factor which affects BCI efficiency. • It is incapable of detecting the complex signal control of the brain. So it is not suitable for practical purpose as it is not sufficiently efficient.
• Data pre-processing is time consuming and dependent upon the skill of the person. • The most suitable area of the brain for decoding the signals is yet to be found.
CONCLUSION
This paper primarily deals with the working principle of BCI and IoT integrated field for human- thing cognitive interaction. Initially, we discussed about the suitable brain wave sensor for home application based on the cost effectiveness. Later, we gave a brief idea on the connectivity suitable for various conditions based on the range of control. We highlighted the data processing segment which was recently proposed. It uses a deep learning framework. The processed data was then sent to the user end for executing the IoT enabled appliances. We also elaborated the problems we face while working in this platform, thereby suggesting the fields which require improvement. The eradication of the mentioned problems will increase the efficiency of this platform, making practical use of this system possible.
Journal of Mathematical Sciences & Computational Mathematics
30 JMSCM, Vol 1, Issue 1, 2019 REFERENCES:
[1]“Internet of Things Meets Brain-Computer Interface: A Unified Deep Learning Framework for Enabling Human-Thing Cognitive Interactivity” by Xiang Zhang, Lina Yao, Shuai Zhang, Salil Kanhere, Michael Sheng, Yunhao Liu.
[2]”How IoT works?” from www.leverege.com
[3]”The Pros and Cons of IoT Connectivity Options for Asset Tracking” from www.leverege.com
[4]”What is IoT Cloud (Salesforce IoT Cloud)?” from SearchSalesforce.com
[5]”What is Data Replication?” from www.intricity.com.
Journal of Mathematical Sciences & Computational Mathematics
31 JMSCM, Vol 1, Issue 1, 2019
PI, REVISITED
A. K. Chatterjee
Department of Basic Science and Humanities, Institute of Engineering & Management, D1 Management House, Salt Lake, Sector V, Kolkata-700091, West Bengal, India
Abstract
An attempt is made to show how an important mathematical constant (namely ), in use for ages, is evaluated computers are used, no doubt, but what is the philosophy behind the program created to carry out the evaluation (to any specified degree of accuracy). The discussion focuses on what exactly is (or are) the cornerstone(s) of such programs.
Key words: Pi, Gregory’s Series, Maclaurin’s series, Accuracy
We take a look into how the well-known constant “ π ” is evaluate to some specified degree of accuracy. Any student in middle school first encounter π , just after the concept of a circle is introduced. A very natural question forms in the minds of these young learners: What is the area of a circle? Is there any formula that comes in handy? Teacher provide the answer by starting that the area of a circle is r2 . And it is at this stage that is introduced as being a constant ratio 22 that of the circumference of ANY circle to its diameter, the FIRST value assigned being . (we 7 355 later learnt that is also a good value to use for , in calculations. 113
As a student moves through the middle school level does not bother him anymore, and he is content with being able to solve elementary problems like what linear distance a circular wheel traverses when it rotates a specified number of times, its radius being given. There is a certain change in scenario when the student steps into the IXth standard and is introduced to the three systems of measuring angle in trigonometry.
When the circular system is discussed the student encounters conversion relation like π radians = 180 degrees
The student gets a little more serious about the non-recurring, non-terminating value of just after completing high school, and before stepping into graduate level courses or during its progress. The series of James Gregory, on which the evaluation of depends, creeps in later on and is stated below:
Journal of Mathematical Sciences & Computational Mathematics
32 JMSCM, Vol 1, Issue 1, 2019
tan3 tan 5 tan 7 tan - - ....ad inf ( ) 3 5 7 4 4
xxx357 (This may equivalent cast into the form to tan...... -1 xx infinity.) 357
The series unlike the Maclaurin series for tan the agreement , express the argument in terms of its trigonometric tangent i.e. tan . If be replaced by and replaced by 1, we get 4 π 1 1 1 =1- + - +...... 4 3 5 7
This series opens the gates to writing a suitable program to obtain the value of , to any desired degree of accuracy (more the number of terms taken from the series on the right, more accurate is the result).
This series has an inherent difficulty in that it converges very slowly. This is an indirect way of saying that (literally) thousands of terms have to be taken to give a value of up to (say) eight or ten places of decimals. It is known for example, that about 80,000 to 1, 00,000 terms are needed for a value correct to four decimal places.
There is an avenue to overcome this difficulty, by using certain ALLIED SERIES, the most common being that due to Euler:
11 π tan+tan=tan(1)=-1-1-1 234
In other words
11 π =4(tan-1 + tan -1 ) 23 11 = 4[tan-1 ] 4[tan 1 ] 23 11 11 ()()35 ()()35 11 4[ 22 ....] 4[ 33 ....] 2 3 5 3 3 5
It is easy to see that both the series in brackets because successive terms get smaller and smaller with appreciable rapidity.
One very rapidly converging series for is that due to Machin (stated in 1706).
Journal of Mathematical Sciences & Computational Mathematics
33 JMSCM, Vol 1, Issue 1, 2019 The series goes thus:
11 4 tantan11 45239
It is evident that the Gregory series for the second term on the right converges with formidable speed.
Formulas similar to the above are captured in the generalized Machin-like formulae, shown below:
N 1 an ctan()0 cn 4 n1 bn
here an and bn are positive integer such that an < b n & c n is a signed non-zero integer and c0 is a positive integer.
William Rutherford [1-3] used the formula
111 4 tantantan111 457099
It may be observed that this is just a variation of the Machin’s formula stated earlier, the two signed 1 terms in the end conveniently combining to tan-1 .In other words 239 111 tan--1-1-1 tantan 7099239
Another very simple and important relation that enables to evaluate π is: r= 1 π2 =, r=1 r2 6 which can be written as r= 1 6= π2 r=1 r2
Any program to evaluate from this identify simply exploits the rule: take the sum of the squares of the reciprocals of the natural numbers upto and including N, a pre-specified natural number (generally large, say about 1000 or more). Multiply the sum by 6, and take the square root of the result to obtain the value of . As N is made larger and larger, the successive values of that emerge are with greater and greater degrees of accuracy.
Journal of Mathematical Sciences & Computational Mathematics
34 JMSCM, Vol 1, Issue 1, 2019 References
[1]Apostol, Tom M., Program Guide and Workbook to accompany the videotape on "The Story of ", Project Mathematics! California Institue of Technology, 1989.
[2] Beckmann, Petr, A History of , St. Martin's Press, New York, 1971.
[3] Salikhov, V. Kh, Journal of Usp. Mat. Nauk Vol. 63, 163-164, (2008)
Journal of Mathematical Sciences & Computational Mathematics
35 JMSCM, Vol 1, Issue 1, 2019
A novel approach for selecting the potentially best performing team in IPL
1Sannoy Mitra, 2Tiyash Patra and 3Raima Ghosh
1Electronics and Communication Engineering, Institute of Engineering & Management, India
(Email: [email protected])
2 Information Technology, Institute of Engineering & Management, India
(E-mail: [email protected])
3Electronics and Communication Engineering, Institute of Engineering & Management, India
(E-mail: [email protected])
Keywords: Cricket analytics; Indian Premier League; Performance quantification
Abstract
In this paper, we propose to design a framework to analyze what should be the playing eleven of a particular squad. The framework designed uses data mining to find out the player statistics and hence helps to choose the best possible playing eleven on the basis of the pitch. This is done by creating a rank system on the basis of strike rate, economy rate, average runs, wickets taken of each player. Then we analyze their performance with the help of the designed framework and find out the best playing eleven. Henceforth, the paper provides a method to find out the best possible team when pitch conditions are provided by the user.
Introduction The art of prediction and its corresponding results ages back to the ancient times when prophecies were made. Similarly, cricket has its roots deep into history and is often touted as the gentleman's game. With the passage of time, like every other sport, cricket has also evolved dramatically in every dimension. The 5-day test match has given way to the 50 over One Day International and currently the most prevalent being the T-20 format.
Though the sport previously only catered to international matches, in current years we have seen the growth of cricket premier leagues where franchisees are owned by the aristocrats and businessmen. The most popular T-20 League in the world is currently the Indian Premier League (IPL).
Journal of Mathematical Sciences & Computational Mathematics
36 JMSCM, Vol 1, Issue 1, 2019 The league featured 10 teams at maximum till date, each representing a certain state. The players were drafted into the team via auction. Hence auction plays a very important role in these leagues all over the world. There are multiple franchises bidding for a marquee player so that the team can get the best of the world in their squad.
The Indian Premier League also has some rules. Only 4 overseas players are allowed in the playing 11 of every match. Hence while bidding for the player or deciding the playing 11, a lot of the squad selection is dependent on the playing conditions and the type of player the team needs at that situation. Due to the limitation of overseas players in the squad, a lot of domestic talent from India gets a chance to hog the limelight.
Our paper deals with this concept and caters to the following interests:
1. It analyses the player statistics of all the players who have played in T-20 leagues all over the world.
2. Further, it helps to select the best possible playing 11 from a given squad of 20 according to the field conditions.
Proposed framework:
The following flowchart and the corresponding elaboration is our suggested framework for selecting the best possible playing eleven from a squad of 20 players.
Flowchart:
A. Data Source
B. Cleaning
C. Data segregation
D. Performance quantification
E. Quantification of team potential
F. Output
Journal of Mathematical Sciences & Computational Mathematics
37 JMSCM, Vol 1, Issue 1, 2019
Detailed Procedure:
A. 1. DATA COLLECTION FROM WEBSITES: ESPN CRICINFO,WIKIPEDIA 2. STORING THEM IN DATABASE
B. 1. GO THROUGH AND CHECK DATA 2. REMOVE INCONSISTENCIES OR ERRORS 3. GET COMPLETE DATASET READY
C. DATA SEGREGATION TO BATSMAN, BOWLERS, WICKET KEEPERS AND ALL-ROUNDERS 1. FROM BATTING AVERAGE 2. FROM ECONOMY 3. FROM WICKETS TAKEN
D. PERFORMANCE CLASSIFICATION BASED ON PLAYER ROLE: 1.
FOR FOR BOWLER FOR ALL- BATSMEN ROUNDERS
0.7* STRIKE 0.6*ECONOMY RATE 0.25*BATTING RATE AVERAGE 0.25* STRIKE RATE
0.3* 0.4* WICKETS 0.25*ECONOMY BATTING RATE AVERAGE 0.25* WICKETS TAKEN PER MATCH
Journal of Mathematical Sciences & Computational Mathematics
38 JMSCM, Vol 1, Issue 1, 2019 2. *RANKING ON CONDITIONS PROVIDED (FOREIGN/DOMESTIC) E. QUANTIFICATION OF THE TEAM POTENTIAL 1. INPUT OF PITCH TYPE AND 20 NAMES 2. SELECTION OF PLAYING 11 a. PITCH TYPE & FORMAT OF TEAM b. SELECTION OF PLAYERS OUT OF 20 BASED ON RANK
CALCULATION OF ANALYTICS (RANKING SYSTEM) STRIKE RATE: AVERAGE RUNS: ECONOMY RATE(RPO) 0-90: 1 0-10: 1 >12: 1 90-130: 2 10-25: 2 7-12: 2 130-200: 3 25-35: 3 4-7: 3 >200: 4 >35 :4 0-4: 4
FOR BATSMAN: SCORE= (0.7* SRR+ 0.3*ARR)/ 4
FOR BOWLER: SCORE= (0.6*ERR + 0.4*WT)/ 6.4
FOR ALL ROUNDER SCORE= (0.25* SRR + 0.25* ARR + 0.25*ERR + 0.25* WT)/5.5 FOR WICKETKEEPER: SCORE= W/ M where, SRR= STRIKE RATE RANK ARR= AVERAGE RUN RANK
Journal of Mathematical Sciences & Computational Mathematics
39 JMSCM, Vol 1, Issue 1, 2019 ERR= ECONOMY RATE RANK WT= WICKETS TAKEN PER MATCH W= NUMBER OF WICKETS M= TOTAL MATCHES PLAYED
Playing 11 format according to pitch:
Dusty Green Flat
1) Foreign 1) Foreign 1) foreign ● Batsmen-2 ● Opener-1 ● Batsman-1 ● Spinner-1 ● Wicketkeeper ● Spinner-1 ● All-rounder-1 -1 ● Allrounder-1 ● Pacer-1 ● All-rounder-1 2) Indian 2) Indian ● N.Batsmen-2 ● N.Batsman-1 ● N.Bowler-2 2) Indian ● N.Pacer-1 ● D.Bowler-1 ● N.Batsman-1 ● D.Batsman-1 ● D.Wicketkee ● N.All- ● D.Pacer-1 per-1 rounder-1 ● Wicketkeeper-1 ● Uncapped all- ● N.Pacer-1 ● Uncapped batsman- rounder-1 ● D.Batsmen-2 1 ● D.Spinner-1 ● Pacing all-rounder-1 ● D.Pacer-1
3) Foreign/Indian spinner-1
Conclusion The paper proposes a novel framework which when executed shall help any IPL team strategize every match. This model can be further enhanced implementing machine learning algorithms, which will make work for the support staff easier as we shall take into consideration the statistic of players from every possible twenty20 leagues happening in and around the world before computing the best possible playing eleven.
Acknowledgment We would heartily like to thank our mentor, Prof. Amit Kr. Das (Department of Computer Science Engineering, IEM) for his tremendous support so that we could execute our vision to the best of our means. It would have been impossible for us to have completed this framework design had he Journal of Mathematical Sciences & Computational Mathematics
40 JMSCM, Vol 1, Issue 1, 2019 not guided us in the right way and given the necessary resources.
References:
[1] Jayanth, Anthony, Abhilasha, Shaik, Srinivasa: ‘A team recommendation system and outcome prediction for the game of cricket’, Journal of Sports Analytics 4, 2018
[2] Perera, Davis, Swartz: ‘Optimal lineups in Twenty20 cricket’, Journal of Statistical Computation and Simulation
[3] Khan, Biswas, Kabir: ‘A quantitative approach to influential factors in One Day International Cricket: Analysis based on Bangladesh’, Journal of Sports Analytics
[4] Davis, Perera, Swartz: ‘Player Evaluation in Twenty20 cricket’, Journal of Sports Analytics 1(2015)
Journal of Mathematical Sciences & Computational Mathematics
41 JMSCM, Vol 1, Issue 1, 2019
Field Aided Semiconductor Superlattices, the Einstein Relation and All That
1J. Pal, 2M. Debbarma, 3N. Debbarma, 4Paulami Basu Mallik and 5K. P. Ghatak
1Department of Physics, Meghnad Saha Institute of Technology, Nazirabad,
P.O. Uchepota, Anandapur, Kolkata-700150, India
2Department of Physics, Women’s College, Agartala, Tripura- 799001, India
3Department of Computer Science and Engineering,National Institute of Technology, Agartala, Tripura-799055, India
4Chartered Engineer, Department of Electronics & Communication Engineering, The Institution of Engineers (India), 8, Gokhale Road, Kolakta-700020. India
5Department of Computer Science and Engineering, University of Engineering and Management and Institute of Engineering and Management, Kolkata and Jaipur, India
ABSTRACT
In this paper we study the Einstein relation for the diffusivity mobility ratio (DMR) under magnetic quantization in III-V, II-VI, IV-VI and HgTe/CdTe SLs with graded interfaces by formulating the appropriate electron statistics. We have also investigated the DMR in III-V, II-VI, IV-VI and HgTe/CdTe effective mass SLs in the presence of quantizing magnetic field respectively. The DMRs in quantum wire
GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe SLs and the corresponding effective mass SLs have further been studied. It appears that the DMR oscillates both with inverse quantizing magnetic field and electron concentration for GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe superlattices with graded interfaces. The DMR decreases with increasing film thickness and decreasing electron concentration for the said superlattices under 2D quantization of wave vector space.
Keywords: Einstein Relation, Semiconductor Superlattices, Magnetic Quantization, Quantum Wire Superlattices
Introduction It is well known that Keldysh [1] first suggested the fundamental concept of a superlattice (SL), although it was successfully experimental realized by Esaki and Tsu [2]. The importance of SLs in the field of nanoelectronics have already been described in [3-5]. The most extensively studied III-V SL is the one consisting of alternate layers of GaAs and Ga1-xAlxAs owing to the relative ease of fabrication. The GaAs layers forms quantum wells and Ga1-xAlxAs form potential barriers. The III-V SL’s are attractive for the realization of high speed electronic and optoelectronic devices [6]. In addition to SLs with usual structure, SLs with more complex structures such as II-VI [7], IV-VI [8] and HgTe/CdTe [9] SL’s have also been proposed. The IV-VI SLs exhibit quite different properties as compared to the III-V SL due to the peculiar band structure of the constituent Journal of Mathematical Sciences & Computational Mathematics
42 JMSCM, Vol 1, Issue 1, 2019 materials [10]. The epitaxial growth of II-VI SL is a relatively recent development and the primary motivation for studying the mentioned SLs made of materials with the large band gap is in their potential for optoelectronic operation in the blue [10]. HgTe/CdTe SL’s have raised a great deal of attention since 1979, when as a promising new materials for long wavelength infrared detectors and other electro-optical applications [11]. Interest in Hg-based SL’s has been further increased as new properties with potential device applications were revealed [11, 12]. These features arise from the unique zero band gap material HgTe [13] and the direct band gap semiconductor CdTe which can be described by the three band mode of Kane [14]. The combination of the aforementioned materials with specified dispersion relation makes HgTe/CdTe SL very attractive, especially because of the possibility to tailor the material properties for various applications by varying the energy band constants of the SLs. In addition to it, for effective mass SLs, the electronic subbands appear continually in real space [15]. We note that all the aforementioned SLs have been proposed with the assumption that the interfaces between the layers are sharply defined, of zero thickness, i.e., devoid of any interface effects. The SL potential distribution may be then considered as a one dimensional array of rectangular potential wells. The aforementioned advanced experimental techniques may produce SLs with physical interfaces between the two materials crystallo graphically abrupt; adjoining their interface will change at least on an atomic scale. As the potential form changes from a well (barrier) to a barrier (well), an intermediate potential region exists for the electrons. The influence of finite thickness of the interfaces on the electron dispersion law is very important, since; the electron energy spectrum governs the electron transport in SLs. In recent years there has been considerable work in studying the Einstein relation (a very important transport quantity for modern nano-devices) under different physical conditions [19-68]. In this paper, we shall study the Einstein relation for the diffusivity –mobility-ratio (DMR)under magnetic quantization in III-V, II-VI, IV-VI and HgTe/CdTe SLs with graded interfaces in sections 2.1 to 2.4. From sections 2.5 to 2.8, we shall investigate the DMR in III-V, II-VI, IV-VI and HgTe/CdTe effective mass SLs in the presence of quantizing magnetic field respectively. In sections 2.9 to 2.16, we shall investigate the DMR in the aforementioned SLs in the presence of two dimensional size quantizations. In section 3, the doping and magnetic field dependences of the DMRs have been studied by taking GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe SLs and the corresponding effective mass SLs as examples. In the same section, we have also studied the doping and thickness dependences of the DMR for the said quantum wire SLs. The section 3 contains the result and discussions as appropriate for this paper.
2 Theoretical Background 2.1Einstein relation under magnetic quantization in III-V superlattices with graded interfaces The energy spectrum of the conduction electrons in bulk specimens of the constituent materials of III-V SLs whose energy band structures are defined by three band model of Kane can be written as
Journal of Mathematical Sciences & Computational Mathematics
43 JMSCM, Vol 1, Issue 1, 2019
2 2 * ( k ) (2mi ) EG(E, Egi , i ) (1)
2 Egi i E Egi i E Egi 3 i = 1,2,…, G(E, Egi , i ) . 2 Egi Egi i E Egi i 3
* is Planck constant, iskm electron wave ve ctor, is i the effective electron mass at the edge of the conduction band, isEE the electron energy,is thegii band gap and is the spin orbit splitting constant.
Therefore, the dispersion law of the electrons of III-V SLs with graded interfaces can be expressed, following Jiang and Lin [16], as 1 cos,L kE k 2 os2 where Lab000 is the period length, a0 and b0 are the widths of the barrier and the well respectively, Ek, 2cosh Ek , cos Ek , Ek , sinh Ek , sin Ek , s s s s s s 2 K2 E, k K E, k 1 s 2 s 0-3KEk 2 ,s cosh Ek , s sin Ek , s 3 KEk 1 , s sinh Ek , s cos Ek , s K E,, k K E k 21ss 22 2K E , k - K E , k cosh E , k cos E , k 0 1s 2 s s s
33 5 KE ,k 5 K E , k 1 2 ss 1 34K21 E , ks K E , k s sinh E , k s sin E , k s 12 K E,, k K E k 12ss
K12 E,, kss K E k , Ek, s , E,, kKEss ka 100 , o is the interface K21 E,, kss K E k 1/ 2 2mE* width, E,, kKE kb , 2 2 , EVE' , V ss 200 K1 E,(,,) ks 2 G E V o 2 2 k s 0 0 is the potential barrier encountered by the electron VEE , α 1/ E , 0 gg21 i gi 1/ 2 2mE* 1 2 and k2 k 2 k 2 . K21 E,( 1 kGss , Ek , )2 s x y
In the presence of a quantizing magnetic field B along z-direction, the simplified magneto- dispersion relation can be, written as=
Journal of Mathematical Sciences & Computational Mathematics
44 JMSCM, Vol 1, Issue 1, 2019
1/2 112 eB 2 knz ELn , 0 3 Lo 2 1 where (,)[cos{(,)}]nEnE -12 , n is the Landau quantum number, 2
n,2cosh,cos,,sinh, En En En En E
2 K n, E 1 sinn , E 02 3 K n , E cosh n , E K n, E 2 2 K n, E 2 sinn , E 3 K1 n , E sinh n , E cos n , E K n, E 1 22 2K n , E K n , E cosh n , E cos n , E 0 1 2
33 5K n , E 5 K n , E 1 12 34K21 n , E K n , E sinh n , E sin n , E 12K n , E K n , E 21
K12 n,, E K n E n,,,,- E n E K1 n E a 0 0 K21 n,, E K n E 1/2 * 2mE2 2 eB1 K1 n,(-,,) E 2 G E Vo 2 2 n 2 1/2 * 2mE1 2 eB1 Considering n,,,(,,) E K2 n Es b 0 0 and K 2 n E 2 G E 1 1 n 2 only the lowest miniband, since in an actual SL only the lowest miniband is significantly populated at low temperatures, where the quantum effects become prominent, the electron concentration ( n0
e Bg nmax )can be written as, nTn ETn Ev ,, 091922 FSLFSL L0 n0 (4)
where gv is the valley degeneracy,
1/ 2 s 2 eB1 2 T910 n,(, EEFSLFSL ) - nnL , Tn9291 EL,,, rFSLFSL T n E 2 r1 2r 2r L( r ) 2 k T 1 212 r 2r , r is the set of real positive integers whose upper B 2r EFSL limit is s, (2r) is the Zeta function of order 2r ,kB is the Boltzmann constant and T is the temperature. The DMR can be expressed as
Journal of Mathematical Sciences & Computational Mathematics
45 JMSCM, Vol 1, Issue 1, 2019
D 1 n0 1 n0[ ] (5) The eE FSL use of equations (4) and (5) leads to the expression of the DMR in this case as
nmax TnETnE ,, D 1 9192 FSLFSL n0 (6) e nmax TnETnE9192 ,,FSLFSL n0
2.2 Einstein relation under magnetic quantization in II-VI superlattices with graded interfaces The energy spectrum of the conduction electrons of the constituent materials of II-VI SLs are given by [17] 22 22 ks kz Ek **0 s 7 22mm,1,1 and 22k * EG E,,(8) Eg 22 2m2
* * where m,1 and m ,1 are the transverse and longitudinal effective electron masses respectively at the edge of the conduction band for the first material and 0 is the splitting of the two spin-states by the spin –orbit coupling and the crystalline field .
Journal of Mathematical Sciences & Computational Mathematics
46 JMSCM, Vol 1, Issue 1, 2019 The electron dispersion law in II-VI SLs with graded interfaces can be expressed as 1 cos,L kE k 9 os2 1 wheer ,
111111(E , kEssssss ) kE 2cosh{ k ( , )}cos{ ( , )}( ,E )sinh{ kE k ( , )}sin{ (Ek , )} 22 {K34 ( E , kKss )}{ E k ( , )} 04113[(-3 ( , ))cosh{K E ( kEssss , kE )}sin{ kK E ( k , )} (3 ( , )) K43( E , kKss )( E , k ) sinh{11 (E , kEss )}cos{ k ( , )}] 22 03411[2({K ( E , kKssss ) -{ E kE ( kE , k)} )cosh{ ( , )}cos{ ( , )} 3 3 1 5{K3 ( E , ks ) 5{K4 ( E , ks )} [ 34K4311 ( E , kssss ) K ( E , kE )]sinh{ kE k ( , )}sin{ ( , )}]] 12( KE4 ,kKss )( E , k ) 3
K34( E , kKss )( E , k ) 11300(E , kEsss ) kK [], E ( k , a )( , )[], K43( E , kKss )( E , k ) * 2mE2 2 1/2 140032(EkK ,ssso )( 2 Ek , )[], bK ( EkGE , ) [(, V , ) 2 ks ] and
* 22 2m ,1 ks 1/2 K40 E,[[]](10) kEkss 2* 2m,1
In the presence of a quantizing magnetic field B along z-direction, the simplified magneto- dispersion relation can be, written as
22112 eB knzo EL2 n1 , 11 L0 2 where, 2 -1 1 11n,cos, En E 2
Journal of Mathematical Sciences & Computational Mathematics
47 JMSCM, Vol 1, Issue 1, 2019
n,2cosh,cos,, En En En En sinh, E 11111 2 K n, E 3 sin,3,cosh,1041n EK n En E K n, E 4 2 K n, E 4 sin,3,sinh,cos,1311n EK n En En E K n, E 3 2 2,,cosh,cos,K n EK n En En E 03411
3 3 5,5K n EK n , E 1 34 34,,sinh,sin,K4311 n E K n En En E 12,K n E K n, E 4 3
K34 n,, EK n E 1130nEnEK,,,,-,,,, 0 1400 nE anEK nE b K43 n,, EK n E 1/2 2m* 2 eB1 2 K30 n,, 2 EE , 2G E2 Vn 2 and 1/2 1/2 2m* eB 112 eB ,1 K4 n, EE 2*nn0 m 22 ,1 The electron concentration in this case can be expressed as e Bg nmax nT n ETv n E ,, 12 093942 FSLFSL 2 L0 n0 where, 1/2 s 2 eB1 2 T93109493 n,,-and, EnFSLFSLFSLFSL E nLT n EL r T n E , 2 r1 The use of equations (12) and (5) leads to the expression of the DMR as
nmax T n,, E T n E D 1 93FSL 94 FSL n0 13 e nmax T93 n,, EFSL T 94 n E FSL n0 2.3Einstein relation under magnetic quantization in IV-VI superlattices with graded interfaces
Journal of Mathematical Sciences & Computational Mathematics
48 JMSCM, Vol 1, Issue 1, 2019 The E-k dispersion relation of the conduction electrons of the constituent materials of the IV-VI SLs can be expressed [18] as
2 1/ 2 EE 222222 ggii (14) Ea kbisizisizisiy kc kd ke kf k 22
2 2 2 2 2 2 where, a , b , cP 2 , dP 2 , e and f . i i ii, ii, i i 2m,i 2m ,i 2m,i 2m ,i
The electron dispersion law in IV-VI SLs with graded interfaces can be expressed as
Journal of Mathematical Sciences & Computational Mathematics
49 JMSCM, Vol 1, Issue 1, 2019
1 cos,L kE k 15 os2 2 where, E,2cosh,cos,, kE kE sinh,sin, kE kE kE k 222222 ssssss 22 K56 E,, kKss E k 06225[(-3, )cosh,sin,(3,)K E kEssss kE kK E k K65 E,, kKss E k sinh,cos,]22E kEss k
22 2,-,cosh,K E kK E kE k cos, Ek 0562 ss ss 2
33 5,5,K E kK E k 1 56ss 34,,sinh,sin,K6522 E kssss K E kE kE k 12,,K E kK E k 65ss
K56 E,, kKss E k 22500EkEkK,,,,,,,sssss 2600 Ek aEkK Ek b K65 E,, kKss E k 1/2 1/2 1 222 K61121 E, kx ,, kEH yx y H k kH 3141511E EH kx,,, kH yx yiii k kHb f and 2 1 f 22 i H213i k xyigi,22,, kHE b i d igii f Ee ii x yi f a b k kH 2 ii 22 bfii 1 H k,44448, kHbdbfEfEk 222 22 k efb af 41i x yiiiiigi gx y iii iiii 2 22 22 22 22 2 H5i kx,48444 kHk yix yi ka i1 b i e ii fb ii ef ix ak yi i ke g f i E b i 44444444444efdef EabEabdabfEbeEbcbE22222222 af eEf cf E a iiiii gii giiiiiiiiiiiii gi i gi ii g ii i gi ii g i E2 b 222 d 2 f g 2E b dE b22 fd2 f E gi iii i giii i ig i ii i g and
1/2 1/2 2 K50320 E, kx ,,,, 42520 kE yx V yx 1222 yx HE y V H k kH k kE V H H kk The simplified magneto-dispersion relation in this case can be written as
1/2 112 eB k n, E L2 n 16 zo2 L0 2 where,
Journal of Mathematical Sciences & Computational Mathematics
50 JMSCM, Vol 1, Issue 1, 2019
2 1 n,cos,,,2cosh, En En En-1 En En cos,, En E sinh, , 2222222 2 22 K56 n,, EK n E sin,[(3,206225n )cosh,EK n En sin,(3,) En EK n E K65 n,, EK n E sinh,22 cos,n En E
2 K n, E 6 sin,3,sinh,252n EK n En En Ecos, 2 K n, E 5 2 2,,cosh,K n E cos, K n En En E 05622
33 5,5,K n EK n E 1 56 34,,sinh,K6522 n E K sin, n En En E 12,,K n EK n E 65
K n,, E K n E 1/2 1/2 56 2 2250n,,,,- En 0 E ,, 611 K n 21 E aK n EEH H nE H 314151EH n H n K65 n,, E K n E 1 22 H1i b i f i
2 1 2 eB 1 f and , i , HnHE21iig iii bdf gi22 iiEe i fa bn H3i 2 ii 2 22 bfii
1 2 eB1 H nHb db44448 f222 Ef Ene f b a f 41iii ii i gi gi i ii i ii 2 2 1 22e Be B 11 22 22 2 H51iii nHna i i ii48444 b ii e ii fb i ef g ane i f E b 22 i 44444444444efdef EabEabdabfEbeEbc22222222 bE af eEf cf E a iiiii gii giiiiiiiiiiiii gi i gi ii g ii i gi ii g i Eb2 2 d 2 fg 2 2 2 Ebd 2 Ebf 2 2 dfE and gii i ii gii i gii i iig i 1/ 2 2 1/ 2 KnE , EVH EVHnHn EVHHn . 5 0 32 0 42 52 0 12 22
The electron concentration in this case can be expressed as
e Bg nmax nT n ETv n E,, 095962 FSLFSL (17) L0 n0
Journal of Mathematical Sciences & Computational Mathematics
51 JMSCM, Vol 1, Issue 1, 2019
1/ 2 2 eB1 2 where, TnEnEnL9520 ,,- FSLFSL and 2 s TnELrTnE9695 ,,FSLFSL . r1
The use of equations (17) and (5) leads to the expression of the DMR as
nmax T n,, E T n E D 1 95FSL 96 FSL n0 n (18) e max T95 n,, EFSL T 96 n E FSL n0
2.4 Einstein relation under magnetic quantization in HgTe/CdTe superlattices with graded interfaces
The dispersion relation of the conduction electrons of the constituent materials of HgTe/CdTe SLs can be expressed [13] as
22 3 ek2 k (19) E * 2128m1 sc
22k (20) * EGE E122 g , 2m2 The electron energy dispersion law in HgTe/CdTe SL is given by
1 cos,L kEk (21) os2 3 where, E,2cosh,cos,,sinh,sin, kE kE kE kE kE k 333333 ssssss
72 K E,, k K E k 78 ss 0-3KEk 8 ,s cosh 3 Ek , s sin 3 EkKEk , s 3 7 , s sinh 3 Ek , s cos 3 Ek , s K E,, k K E k 87ss
22 2K E , k - K E , k cosh E , k cos E , k 0 7s 8 s 3 s 3 s
33 5K E ,5 kK E k , 1 87ss 34K7833 E ,, kssss K sinh E kE kE , k sin , 12,,K E kK E k 78ss
K78 E,, kss K E k 3 Ek, s , 3E,, kss K 7 E k a 0 0 K87 E,, kss K E k
Journal of Mathematical Sciences & Computational Mathematics
52 JMSCM, Vol 1, Issue 1, 2019
1/ 2 2 B2224 AE B B AE 3 e 2 EkKEkb,, , 0 0 0 2 , B , A and 3800 ss K8 E,, kx k y 2 k s 0 * 2A 128 2m sc 1 1/ 2 2mE* 2 2 . KEkGEVEk702 ,(,,)sgs 2 2
The magneto dispersion relation in this case can be expressed as
1/2 2 eB 112 knzo ELn3 , 22 L 2 o
2 -1 1 where, 33nEnE,cos, , 2 n,2cosh,cos,,sinh, En En En En E 33333
2 Kn E, 7 sin,3,cosh,3083n EK n En E K n, E 8
2 K n, E 8 sin,3,sinh,cos,3733n EK n En En E K n, E 7
22 2,,cosh,cos,K n EK n En En E 07833
33 5,5,K n EK n E 1 78 34,,sinh,sin,K8733 n E K n En En E 12,,K n EK n E 87 Kn EKn,, E 78 , n,,- EKn Ea , n,, E K n E b , 3 nE, 3700 3 8 0 0 Kn87 EKn,, E 1/ 2 B222 AE B B 4 AE 2 e B 1 0 0 0 , and K8 n, E 2 n 22A 1/ 2 2mE* 2 eB 1 2 . K7 n,,, E 2 G E V 0 2 2 n 2
The electron concentration in this case can be expressed as
e Bg nmax nv T n,, E T n E 02 97 FSL 98 FSL (23) L0 n0
Journal of Mathematical Sciences & Computational Mathematics
53 JMSCM, Vol 1, Issue 1, 2019
1/ 2 2 eB1 2 where, TnEnEnL9730 ,,- FSLFSL and 2 s TnELrTnE9897 ,,FSLFSL . r1
The use of equations (23) and (5) leads to the expression of the DMR as
nmax TnETnE ,, D 1 9798 FSLFSL n0 n (24) e max TnETnE9798 ,,FSLFSL n0
2.5 Einstein relation under magnetic quantization in III-V effective mass superlattices
Following Sasaki [15], the electron dispersion law in III-V effective mass superlattices (EMSLs) can be written as
1 2 kfE212 kkk cos,, (25) xyz 2 L0 in which, , 222 , f Ek,yz ,cos,,cos,, kaaC 10 EkbD 10 120 10 EkaaC 1 EkbD Ek kkk yz 2 1 2 1/ 2 1 1/ 2 mm** mm** 22, 22, a1 14 a2 14** mm** mm 11 11 1/ 2 1/ 2 2mE* 2mE* CE kG,,, E Ek 1 2 and DE kG,,, E Ek 2 2 . 11 2 g1 12 2 g2
In the presence of an external magnetic field along x-direction, the simplified magneto dispersion law in this case can be written as
2 knx E4 , (26)
112 2 eB in which, n,cos, Ef n En1 , 4 2 L0 2 fnE , a1 cos aCnE 0 1 , bDnE 0 1 , a 2 cos aCnE 0 1 , bDnE 0 1 ,
Journal of Mathematical Sciences & Computational Mathematics
54 JMSCM, Vol 1, Issue 1, 2019
1/ 2 2mE* 2 eB1 Cn EGEEn,,, 1 and 11 2 g1 2 1/ 2 2mE* 2 eB1 Dn EGEEn,,, 2 . 12 2 g2 2
The electron concentration in this case can be expressed as
e Bg nmax nTnETnEv ,, (27) 0999102 FSLFSL n0
s 1/ 2 where, T99 n,, EFSL 4 n E FSL and T910 n,, EFSL L r T 99 n E FSL . r1
The use of equations (27) and (5) leads to the expression of the DMR as
nmax Tn ETn,, E D 1 99910 FSLFSL n0 (28) e nmax Tn99910 ETn,,FSLFSL E n0
2.6 Einstein relation under magnetic quantization in II-VI effective mass superlattices
Following Sasaki [15], the electron dispersion law in II-VI EMSLs can be written as 1 2 kfE212 kkk cos,, (29) zxys 2 1 L0 in which, , 222 , fEkk1 ,x , y a 302 cos aCEk , s bDEk 02 , s a 402 cos aCEk , s bDEk 02 , s kkksxy 1 2 1/ 2 1 2 1/ 2 ** ** mm mm 22, 22, a 14 a4 14 3 mm** mm** ,1,1 ,1 ,1 1/ 2 1/ 2 1/ 2 * 22 * 2m ,1 k 2m 2 C E, kEk s and D E,,, kEG E Ek 2 . 20ss2* 22sgs 2 2 2m,1
Under magnetic quantization along z-direction, the simplified magneto dispersion law can be expressed as
2 knz E5 , (30)
Journal of Mathematical Sciences & Computational Mathematics
55 JMSCM, Vol 1, Issue 1, 2019
112 2 eB in which, nEfnEn,cos, 1 , 51 2 L0 2 fn13020240202 Eaa,cos,,cos,, Cn Eb Dn Eaa Cn Eb Dn E
1/ 2 1/ 2 1/ 2 2m* e Be B 112 ,1 and CE20 kEnn, 2* m 22 ,1 1/ 2 2m* 2 eB1 Dn EEGEEn,,, 2 . 22 2 g2 2
The electron concentration in this case can be expressed as
e Bg nmax nTnETnEv ,, 09119122 FSLFSL (31) 2 n0
s 1/ 2 where, TnEnE9115 ,,FSLFSL and Tn912911 EL ,, rTnFSLFSL E . r1
Thus, using equation (41) and (1.11) leads to the expression of the DMR as
nmax Tn ETn ,, E D 1 911912 FSLFSL n0 n (32) e max Tn911912 ETn ,,FSLFSL E n0
2.7 Einstein relation under magnetic quantization in IV-VI effective mass superlattices
Following Sasaki [15], the electron dispersion law in IV-VI, EMSLs can be written as 1 2 k2cos 1 f E , k , k k 2 (33) x2 2 y z L0 in which, fEkk, ,cos, a ,, ,cos, aCEkk ,, , bDEkk a aCEkk bDEkk , 25 y 0 zy 30 zy 36 zy zy 0 z30 3
2 1/ 2 1 ** mm22 a5 14 mm** 11
2 1/ 2 *2 2 2 2 2 2 2 2 mi a i aCaeE iiiigig eE EaCeE gi iig 2 CeE iig 2 EaC gii 2 eaE iig ab22 i i i i i i i ii
1/ 2 1/ 2 2 C3 E,,,,, ky k z EH 1121 H k y k z E H 3141 EH k y k z H 51 k y k z ,
Journal of Mathematical Sciences & Computational Mathematics
56 JMSCM, Vol 1, Issue 1, 2019
1/ 2 1/ 2 2 DE31222324252 kkEHHkkE,,,,,yzyzyzyz HEHkkHkk an
2 1/ 2 1 mm** 22. a6 14 mm** 11
Thus, in the presence of a quantizing magnetic field along x-direction, the simplified magneto dispersion law in this case can be written as
2 k nx E 6 , (34) in 112 2 eB which, n, E cos1 f n , E n , 62 2 L0 2 fn25030360303 Eaa,cos,,cos,, Cn Eb Dn Eaa Cn Eb Dn E
1/ 2 1/ 2 Cn EEHHnE, HEHnHn 2 31121314151
1/ 2 1/ 2 D n, EEHHnE HEHnHn 2 . 31222324252
The electron concentration in this case can be expressed as
e Bg nmax nTn ETn Ev ,, 09139142 FSLFSL (35) n0
s 1/ 2 where, Tn9136 En ,, EFSLFSL and Tn914913 EL ,, rTnFSLFSL E . r1
Thus, using equation (35) and (5) leads to the expression of the DMR as
nmax T n,, E T n E D 1 913FSL 914 FSL n0 n (36) e max T913 n,, EFSL T 914 n E FSL n0
2.8 Einstein relation under magnetic quantization in HgTe/CdTe effective mass superlattices
Following Sasaki [15], the electron dispersion law in HgTe/CdTe EMSLs can be written as
1 2 kf212 E k kkcos, , (37) xy z 2 3 L0
Journal of Mathematical Sciences & Computational Mathematics
57 JMSCM, Vol 1, Issue 1, 2019 in which, , fEk3 , a 704 cos aCEk , bDEk 04 , a 804 cos aCEk , bDEk 04 , 2 1 2 1/ 2 1 1/ 2 mm** mm** 22, 22, a1 14 a2 14** mm** mm 11 11 1/ 2 1/ 2 B2224 AE B B AE 2mE* 0 0 0 2 and 2 2 . C4 E, k 2 k D42 E,,, k 2 G E Eg k 2A 2
In the presence of an external magnetic field along x-direction, the simplified magneto dispersion law in this case can be written as
2 kx 6 n, E (38)
112 2 eB in which, n,cos, Efn En 1 , 63 2 L0 2 fn37040480404 Eaa,cos,,cos,, Cn Eb Dn Eaa Cn Eb Dn E 1/ 2 BAEBBAEe22242 B 1 000 and Cn4 En, 2 22A 1/ 2 2mE* 2 eB1 D n,,, E2 G E E n . 42 2 g2 2
The electron concentration in this case can be expressed as
e Bg nmax nTn ETn Ev ,, 09159162 FSLFSL (39) n0
s 1/ 2 where, Tn9156 En ,, EFSLFSL and Tn916915 EL ,, rTnFSLFSL E . r1
The use of equations (39) and (5) leads to the expression of the DMR as
nmax Tn ETn ,, E D 1 915916 FSLFSL n0 n (40) e max Tn915916 ETn ,,FSLFSL E n0
2.9 Einstein relation in III-V quantum wire superlattices with graded interfaces The electron dispersion law in III-V quantum wire superlattices (QWSLs), can be written following equation (2) as
Journal of Mathematical Sciences & Computational Mathematics
58 JMSCM, Vol 1, Issue 1, 2019
1 knnEnn2 ,,, (41) zxyxy 2 8 L0 2 2 -1 1 2 where 88nnEnnExyxy,,cos,, , nnndndxyxxyy, , 2 nx ( 1,2 ,3 ,. . . ) and ny ( 1,2,3,...) are the size quantum numbers along the x- and y- direction respectively and dx and dy are the nanothickness along the respective directions, nnE, , 2cosh nnE , , cos nnE , , nnE , , sinh nnE , , 8 x y 8 x y 8 x y 8 x y 8 x y
2 KnnE9 xy,, sin,,3,,cosh,,nnEKnnEnnE 80108 xyxyxy KnnE10 xy,, 2 KnnE10 xy,, sin,,3,,sinh,,cos,,nnEKnnEnnEnnE 8988 xyxyxyxy KnnE9 xy,, 2 0910882,K ,, nnEKx ,cosh, yx yx ,cos, yx nnEnnEnnE y , 33 5K n , n , E 5 K n , n , E 1 9 x y 10 x y 34KnnEKnnE10 x , y , 9 x , y , sinh 8 nnE x , y , sin 8 nnE x , y , 12 K n,,,, n E K n n E 10 x y 9 x y KnnEKnnE,,,, 910 xyxy , nnEKnnEa,,,,- , 8 nnExy,, 8900 xyxy KnnEKnnE,,,, 109 xyxy 1/ 2 2mE* 2 , nnEKnnEb,,,, and Kn9022 nEGxyxy,,(-,,), E Vn n 2 81000 xyxy
* 1/ 2 2mE1 Kn1011 nEGxyxy,,(En ,,), n2 . Considering only the lowest miniband, since in an actual SL only the lowest miniband is significantly populated at low temperatures, where the quantum effects become prominent, the relation between the 1D electron concentration ( n1D ) and the Fermi energy in the present case can be written as,
nn 2g xymax max nT n nv ET n n E , ,, , (42) 1917918Dx y FQWSLx y FQWSL nnxy11
Journal of Mathematical Sciences & Computational Mathematics
59 JMSCM, Vol 1, Issue 1, 2019
1/ 2 where, TnnEnnEnn,,,,-, and 9178 xyFQWSLxyFQWSLxy s TnnELrTnnE,,,, . 918917 xyFQWSLxyFQWSL r1
The use of equations (42) and (5) leads to the expression of the DMR in this case as
nn xymaxmax TnnETnnE,,,, 917918 xyFQWSLxyFQWSL D 1 nn11 xy nn (43) e xymaxmax TnnETnnE917918 xyFQWSLxyFQWSL,,,, nnxy11
2.10 Einstein relation in II-VI quantum wire superlattices with graded interfaces
The electron dispersion law in II-VI QWSLs, can be written as 1 knnEnn2 ,,, (44) zxyxy 2 9 L0 2 -1 1 where, 99nnEnnExyxy,,cos,, , 2 nn, ,2cosh, Enn Enn ,cos,Enn ,, , sinh,Enn , E 99999 x yx yx yx yx y 2 K11 nxy,, n E sin, ,3,n n ,cosh, EK , n n En n E 90129 x yx yx y K12 nxy,, n E 2 K12 nxy,, n E sin, ,3,nnEK ,sinh, ,cos, nnEnnEnnE , 91199 x yx yx yx y K11 nxy,, n E 2 01112992,K ,, nnEK ,cosh,x yx yx ,cos, yx nnEnnEnnE y , 33 5K n , n , E 5 K n , n , E 1 11 x y 12 x y 34KnnEKnnE12 x , y , 11 x , y , sinh 9 nnE x , y , sin 9 nnE x , y , 12 K n,,,, n E K n n E 12 x y 11 x y K n,,,, n E K n n E 11 x y 12 x y , n,,,,- n E K n n E a , 9 nxy,, n E 9 x y 11 x y 0 0 K n,,,, n E K n n E 12 x y 11 x y
9nx,,,, n y E K 12 n x n y E b 0 0 ,
1/ 2 2m* 2 K11 nx,,,,, n y E 2 E G E V 0 2 2 n x n y
Journal of Mathematical Sciences & Computational Mathematics
60 JMSCM, Vol 1, Issue 1, 2019
1/ 2 * 2m 2 1/ 2 and ,1 . KnnEEnnnn120 xyxyxy,,,, 2* 2m ,1
The electron concentration in this case can be expressed as
nn 2g xymas max nv T n,,,, n E T n n E (45) 1D 919 x y FQWSL 920 x y FQWSL nnxy11
1/ 2 where, TnnEnnEnn,,,,-, and 9199 xyFQWSLxyFQWSLxy s TnnELrTnnE,,,, . 920919 xyFQWSLxyFQWSL r1
The use of equations (45) and (5) leads to the expression of the DMR as nn xymas max Tn n,,,, ETn n E 919920 xyFQWSLxyFQWSL D 1 nn11 xy nn (46) e xymas max Tn919920 nxyFQWSLxyFQWSL,,,, ETn n E nnxy11
2.11 Einstein relation in IV-VI quantum wire superlattices with graded interfaces The electron dispersion law in IV-VI QWSLs can be written as
1 knnEnn2 ,,, (47) zxyxy 2 10 L0 2 -1 1 where, 10nx, n y , E cos 10 n x , n y , E , 2 nn, ,2cosh, Enn Enn ,cos,Enn ,, , sinh,Enn , E 1010101010 x yx yx yx yx y 2 K11 nxy,, n E sinn , n , E 3 K n , n , E cosh n , n , E 10 x y 0 12 x y 10 x y K12 nxy,, n E 2 K12 nxy,, n E sinnnEKnnE , , 3 , , sinh nnE , , cos nnE , , 10 x y 11 x y 10 x y 10 x y K11 nxy,, n E 2 02,K 11121010 ,, nnE ,cosh,x yx yx , yxK cos, y nnEnnEnnE ,
Journal of Mathematical Sciences & Computational Mathematics
61 JMSCM, Vol 1, Issue 1, 2019
33 5,,5,,Kn n EKn n E 1 1112 xyxy 34,,,,sinh,,sin,,Kn12111010 nxyxyxyxy EKn n En n En n E 12 Kn n,,,, EKn n E 1211 xyxy KnnEKnnE,,,, 1112 xyxy , nnEKnnEa,,,,- 10 nnExy,, 101100 xyxy KnnEKnnE,,,, 1211 xyxy
1/ 2 1/ 2 2 KnnEEHHnnE121121314151 xyxyxyxy,,,,, HEHnnHnn , 1 HnnHEbdf21ixyigiiigiii,22, ixy Ee fa bnn , ii 1 HnnHb,44448, db f EfEnne 222 f ba f , 41ixyiiii igigxyi i ii i ii 2 22 221 2 HnnHnna51ixyixyi ,4,844,4 i i be iiiiixyi fb ig ef i anne f E b i 44444efd efE2 abE 2 abd abfE 444444 beE 2 bc 2 bEa 2 feE 2 fc 2 fEa 2 iii iigi iig i iii iiig i iig i ii igi i iig i ii igi i EbdfgEb222222 dEb fdf E 222 and giiiigiigiiiigiiii
1/ 2 1/ 2 2 K110320 nnEEVx, yx 42520 ,,,, yx 1222yx y HEVH nnH nnEVHH nn .
The electron concentration in this case can be expressed as
nn 2g xymax max nTn n ETnv n E,,,, (48) 1921922DxyFQWSLxyFQWSL nnxy11
1/ 2 where, Tn n,,,,-,, En n En n E and 92110 xyFQWSLxyFQWSLxyFQWSL s TnnEL,,,, rTnnE . 922921 xyFQWSLxyFQWSL r1
The use of equations (48) and (5) leads to the expression of the DMR as
nn xymax max T n,,,, n E T n n E 921 x y FQWSL 922 x y FQWSL D 1 nn11 xy nn (49) e xymax max T921 nx,,,, n y E FQWSL T 922 n x n y E FQWSL nnxy11
2.12 Einstein relation in HgTe/CdTe quantum wire superlattices with graded interfaces The electron dispersion law in HgTe/CdTe QWSLs can be written as
Journal of Mathematical Sciences & Computational Mathematics
62 JMSCM, Vol 1, Issue 1, 2019
1/ 2 1 knnEnnE,,,, (50) zxyxy 2 11 L0
2 1 where, -1 , 1111nnEnnExyxy,,cos,, 2 nnE, , 2cosh nnE , , cos nnE , , nnE , , sinh nnE , , 11 x y 11 x y 11 x y 11 x y 11 x y 2 K13 nxy,, n E sinn , n , E 3 K n , n , E cosh n , n , E 11 x y 0 14 x y 11 x y K14 nxy,, n E 2 KnnE14 xy,, sin,,3,,sinh,,cos,, nnEKnnEnnEnnE 11131111 xyxyxyxy KnnE13 xy,, 22 0131411112,,,,cosh,,cos,,KnnEKnnEnnEnnE xyxyxyxy 33 5,K ,5, n , n EK n n E 1 1314 x yx y 34,K14131111 ,, nnEKx ,sinh, yx yx , yx nnEnnEnnE sin, y , 12 K n, n ,, EK , n n E 1413 x yx y KnnEKnnE,,,, 1314 xyxy , nnEKnnEa,,,,- , 11 nnExy,, 111300 xyxy KnnEKnnE,,,, 1413 xyxy
111400nnEKnnEbxyxy,,,, ,
1/ 2 BAEBBAE2224 000 , and Kn14 nxyxy,,, En n 2 2A 1/ 2 2mE* Kn n,,,,, EG E Vn n2 . 13022 xyxy 2
The electron concentration in this case can be expressed as
n 2g nmaxy max nv T n,,,, n E T n n E (51) 1D 923 x y FQWSL 924 x y FQWSL nnxy11
1/ 2 where, T n,,,,-, n E n n E n n and 923 x y FQWSL 11 x y FQWSL x y
s T n,,,, n E L r T n n E . 924 x y FQWSL 923 x y FQWSL r1
Journal of Mathematical Sciences & Computational Mathematics
63 JMSCM, Vol 1, Issue 1, 2019
The use of equations (51) and (5) leads to the expression of the DMR as n nmaxmaxy TnnETnnE,,,, 923924 xyFQWSLxyFQWSL D 1 nn11 xy (52) n n e maxmaxy TnnETnnE923924 xyFQWSLxyFQWSL,,,, nnxy11
2.13 Einstein relation in III-V effective mass quantum wire superlattices The electron dispersion law in III-V, effective mass quantum wire superlattices (EMQWSLs) can be written as knnE2 ,, (53) xyz 12
2 2 1 2 ny n in which, 1 , nn, z 1212nnEfnnEnnyzyzyz,,cos,,, 2 yz L dd 0 yz f nnE, ,cos, aaCnnE ,, ,cos, ,, , bDnnEaaCnnE bDnnE 1270 y 50 zy 580 zy 50 zy 5 zy z 1/ 2 2mE* Cn n,,,,, EG E En n1 and 51 yzgyz 2 1 1/ 2 2mE* Dn n,,,,, EG E En n2 . 51 yzgyz 2 1
The electron concentration in this case can be expressed as nn 2g yzmax max nTn n ETnv n E,,,, (54) 1925926DyzFQWSLyzFQWSL nnyz11
1/ 2 where, TnnEnnE,,,, and 92512 yzFQWSLyzFQWSL
s T n, n ,, EL , r T n n E . 926925 y z FQWSLy z FQWSL r1
The use of equations (54) and (5) leads to the expression of the DMR as nn yzmax max T n,,,, n E T n n E 925 y z FQWSL 926 y z FQWSL D 1 nn11 yz nn (55) e yzmax max T925 ny,,,, n z E FQWSL T 926 n y n z E FQWSL nnyz11
Journal of Mathematical Sciences & Computational Mathematics
64 JMSCM, Vol 1, Issue 1, 2019 2.14 Einstein relation in II-VI effective mass quantum wire superlattices The dispersion law in II-VI, EMQWSLs can be written as knnE2 ,, (56) zxy 13
1 2 in which, nnEfnnEnn,,cos,,, 1 , 1313 xyxyxy 2 L0 fnnEaa,,cos,,,,cos,,,, CnnEb DnnEaa CnnEb DnnE 1390 xyxyxyxyxy 606100 606
1/ 2 * 1/ 2 2 1/ 2 2m ,1 CnnEEnnnn60 xyxyxy,,,, 2* 2m ,1
1/ 2 2m* and DnnEEGEEnn,,,,, 2 . 62 xygxy 2 2
The electron concentration in this case can be expressed as
nn 2g xymax max nTn n ETnv n E,,,, (57) 1927928DxyFQWSLxyFQWSL nnxy11
1/ 2 where, TnnEnnE,,,, and 92713 xyFQWSLxyFQWSL s TnnEL,,,, rTnnE . 928927 xyFQWSLxyFQWSL r1
Thus, using equation (57) and (511) leads to the expression of the DMR as
nn xymax max T n,,,, n E T n n E 927 x y FQWSL 928 x y FQWSL D 1 nn11 xy nn (58) e xymax max T927 nx,,,, n y E FQWSL T 928 n x n y E FQWSL nnxy11 2.15 Einstein relation in IV-VI effective mass quantum wire superlattices The dispersion law in IV-VI, EMQWSLs can be written as k2 n,, n E (59) x14 y z
Journal of Mathematical Sciences & Computational Mathematics
65 JMSCM, Vol 1, Issue 1, 2019
1 2 in which, nnEfnnEnn,,cos,,, 1 1414 yzyzyz 2 L0 fnnEaa,,cos,,,,cos,,,, CnnEb DnnEaa CnnEb DnnE 1490 yzyzyzyzyz 707100 707
1/ 2 1/ 2 2 CnnEEHHnnE71121314151 yzyzyzyz,,,,, HEHnnHnn
1/ 2 1/ 2 2 DnnEEHHnnE71222324252 yzyzyzyz,,,,, HEHnnHnn .
The electron concentration in this case can be expressed as nn 2g yzmaxmax nTnnETnnEv ,,,, (60) 1929930DyzFQWSLyzFQWSL nnyz11
1/ 2 where, TnnEnnE,,,, and 92914 yzFQWSLyzFQWSL s TnnEL,,,, rTnnE . 930929 yzFQWSLyzFQWSL r1
Thus, using equation (60) and (5) leads to the expression of the DMR as nn yzmax max Tn n,,,, ETn n E 929930 yzFQWSLyzFQWSL D 1 nn11 yz nn (61) e yzmax max Tn929930 nyzFQWSLyzFQWSL,,,, ETn n E nnyz11
2.16 Einstein relation in HgTe/CdTe effective mass quantum wire superlattices The dispersion law in HgTe/CdTe, EMQWSLs can be written as knnE2 ,, (62) xyz 15
1 2 in which, n, n ,cos, Ef n ,, n En1 n , 1515 y zy zy z 2 L0 fnnE, , a cos aCnnE , , bDnnE , , a cos aCnnE , , bDnnE , , 15 y z 11 0 8 y z 0 8 y z 12 0 8 y z 0 8 y z 1/ 2 BAE2224 B BAE 00 0 and C8 ny, n zy ,, En z n2 2A 1/ 2 2mE* D n,,,,, n E2 G E E n n . 82 y z 2 g2 y z
Journal of Mathematical Sciences & Computational Mathematics
66 JMSCM, Vol 1, Issue 1, 2019 The electron concentration in this case can be expressed as nn 2g yzmaxmax nTnnETnnEv ,,,, (63) 1931932DyzFQWSLyzFQWSL nnyz11
1/ 2 where, TnnEnnE,,,, and 93115 yzFQWSLyzFQWSL
s TnnELrTnnE,,,, . 932931 yzFQWSLyzFQWSL r1
The use of equations (63) and (5) leads to the expression of the DMR as nn yzmaxmax TnnETnnE,,,, 931932 yzFQWSLyzFQWSL D 1 nn11 yz nn (64) e yzmaxmax TnnETnnE931932 yzFQWSLyzFQWSL,,,, nnyz11
Results and discussion
Using the appropriate equations, we have plotted the DMR in figure 1 as a function of inverse quantizing magnetic field for GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe superlattices with graded interfaces as shown by curves (a), (b), (c) and (d) respectively. Using the same equations, we have plotted the DMR as function of electron concentration for GaAs/Ga1- xAlxAs, CdS/CdTe superlattices, as shown by curves (a) and (b), where as, in figure 3, the same has been drawn for PbTe/PbSnTe and HgTe/CdTe superlattices respectively as shown by curves (c) and (d) respectively. We have plotted the DMR in figure 4 and as a function of inverse quantizing magnetic field for the aforementioned effective mass superlattices respectively. Using the same equations, we have plotted the DMR as function of electron concentration for GaAs/Ga1- xAlxAs, CdS/CdTe effective mass superlattices, as shown by curves (a) and (b) in figure 5, where as the same has been drawn for PbTe/PbSnTe and HgTe/CdTe effective mass superlattices respectively as shown by curves (c) and (d) in figure 6 respectively. It appears from the said figures that the DMR oscillates both with 1/B and n0 due to SdH effect, although the rates of variations are totally band structures dependent for all types of superlattices as considered here. We have plotted the normalized 1D DMR as function of film thickness for GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe quantum wire superlattices with graded interfaces as shown by curves (a), (b), (c) and (d) of figure 7. In figure 8, we have plotted all cases of figure 7 as function of electron concentration per unit length. It appears from both the figures 7 and 8 that the DMR increases with decreasing film thickness and increasing electron concentration per unit length respectively, although the numerical values are totally band structure dependent. We further plot the normalized 1D DMR in quantum wire effective mass superlattices of the aforementioned materials as functions of film thickness and electron concentration per unit length as shown by
Journal of Mathematical Sciences & Computational Mathematics
67 JMSCM, Vol 1, Issue 1, 2019 curves (a), (b), (c) and (d) in figures 9 and 10 respectively. It appears that the DMR in this case decreases with film thickness and increases with electron concentration for all the cases.
Conclusion
In this paper an attempt is made to study the Einstein relation for the diffusivity mobility ratio (DMR) under magnetic quantization in III-V, II-VI, IV-VI and HgTe/CdTe SLs with graded interfaces by formulating the appropriate electron statistics. We have also investigated the DMR in III-V, II-VI, IV-VI and HgTe/CdTe effective mass SLs in the presence of quantizing magnetic field respectively. The DMRs in quantum wire GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe SLs and the corresponding effective mass SLs have further been studied. It appears that the DMR oscillates both with inverse quantizing magnetic field and electron concentration for GaAs/Ga1-xAlxAs, CdS/CdTe, PbTe/PbSnTe and HgTe/CdTe superlattices with graded interfaces. The DMR decreases with increasing film thickness and decreasing electron concentration for the said superlattices under 2D quantization of wave vector space.
References
1. Keldysh. L.V., 1962, Journal of Soviet Physics Solid State, 4, pp 1658 2. Esaki. L. and Tsu. R., 1970, IBM Journal of Research and Develop., 14, pp 61 3. Tsu. R., 2005, Superlattices to nanoelectronics, Elsevier; Ivchenko. E.L. and Pikus. G.,1995, Superlattices and other heterostructures, Springer-Berlin; Bastard. G., 1990, Wave mechanics applied to heterostructures, Editions de Physique, Les Ulis, France 4. Bose. P. K., 1997, In Handbook of Theory of optical process in semiconductors, bulk and microstructures, Oxford University Press 5. Ghatak. K. P. and Mitra. B., 1992, ,International Journal of Electronics, 72, pp 541-552 6. Vaidyanathan. K. V., Jullens. R. A., Anderson. C. L. and Dunlap. H. L., 1983, Journal of Solid State Electron., 26, pp 717-723 7. Wilson. B. A., 1988, IEEE Journal of Quantum Electron., 24, pp 1763 8. Krichbaum. M., Kocevar. P., Pascher. H. and Bauer. G., 1988, IEEE, Journal of Quantum Electron., 24, pp 717 9. Schulman. J. N. and McGill. T. C., 1979, Journal of Applied Physics Letters., 34, pp 663 10. Kinoshita. H., Sakashita. T. and Fajiyasu. H., 1981, Journal of Applied Physics, 52, pp 2869 11. Ghenin. L., Mani. R. G., Anderson. J. R. and Cheung. J. T., 1989, Journal of Physical Review B, 39, pp 1419 12. Hoffman. C. A., Mayer. J. R., Bartoli. F. J., Han. J. W., Cook. J. W., Schetzina. J. F. and Schubman. J. M., 1989, Journal of Physical Review B., 39, pp 5208 13. Yakovlev. V. A., 1979, Properties of Zero Gap Semiconductors, Journal of Sov. Phys. Semicon., 13, pp 692 14. Kane. E. O., 1957, Journal of Physics and Chemistry of Solids, 1, pp 249 15. Sasaki. H., 1984, Journal of Physical Review B, 30, pp 7016 16. Jiang. H. X. and Lin. J. Y., 1987, Journal of Applied Physics, 61,pp 624 17. Hopfield. J. J., 1960 , Journal of Physics and Chemistry of Solids, , 15, pp 97
Journal of Mathematical Sciences & Computational Mathematics
68 JMSCM, Vol 1, Issue 1, 2019
18. Foley. G. M. T. and Langenberg. P. N., 1977 , Journal of Phys. Rev. B, 15B, pp 4850 19. Kroemer H., 1978, Journal of IEEE Trans, Electron Devices, 25, pp 850
20. Einstein A.,1905, Journal of Ann der Physik, 17, pp 549
21. Wagner. C.,1933, Journal of Physik Chemistry, B21, pp 24
22. Lade R. W., 1965, Journal of Proceeding IEEE, 52,pp 743
23. Landsberg P. T. , 1952, Journal of Proceedings of the Royal Society of London. Series A, , 213, pp226
24. Landsberg P. T., 1981, European Journal of Physics, 2, pp213
25. Wang C. H. and Neugroschel A.,1990 , IEEE Journal of Electron. Dev. Lett., 11, pp 576.
26. Y. Leu I. and Neugroschel A.,1993, IEEE Journal of Trans. Electron. Dev., 40, pp1872
27. Stengel. F.,Mohammad S. N. and Morkoç. H.,1996, Journal of Applied Physics, 80, pp 3031
28. Pan H. J., C. Wang W., B. Thai K., Cheng C.C., Yu K.H., W. Lin K.,- Wu C. Z. and Liu W. C.,2000, Journal of Semiconductor Science and Technology, 15,pp 1101.
29. Mohammad S. N. 2004, Journal of Applied Physics 95, pp 4856
30. K. Arora, 2002, Journal of Applied Physics Lett. 80, pp3763
31. Mohammad S. N. 2004, Journal of Applied Physics 95, pp 7940
32. Mohammad S. N., 2004, Journal of Philos. Mag., 84, pp 2559
33. Mohammad S. N. ,2005, Journal of Applied Physics, 97, pp 063703
34. Suzue K., Mohammad S. N. , F. Fan Z., Kim W., Aktas O., E. Botchkarev A. and Morkoç H.,1996, Journal of Applied Physics 80, pp 4467
35. Das. P. K. , Ghatak. KP, 2019, Journal of nanoscience and nanotechnology ,19, pp 2909-2912
36. Fan Z.,Mohammad S. N. , Kim W., Aktas O.,E. Botchkarev A., Suzue K. and Morkoç H.,1996, Journal of Electronic Materials, 25,pp 1703
37. Lu C., Chen H., Lv X., Xia X. and Mohammad S. N. 2002, Journal of Applied Physics, 91, pp 9216 38. Dmitriev V. G. and . Markin Yu V, 2002, Journal of Semiconductors 34, pp 931 39. Ghatak. K.P., Chakrabarti. S, Chatterjee. B, 2018, Journal of Materials Focus, 7, 361-362
40. G. Park , D., Tao M., Li D.,E. Botchkarev A., Fan Z.,Mohammad S. N. and Morkoç H.,1996, Journal of Vacuum Science & Technology B,14,pp 2674
41. Chen. Z.,G. Park D.,Mohammad S. N. and Morkoç H.,1996, Journal of Applied Physics Lett. 69, pp230
Journal of Mathematical Sciences & Computational Mathematics
69 JMSCM, Vol 1, Issue 1, 2019
42. Landsberg P. T.,1984, Journal of Applied Physics 56, pp 1696
43. Mohammad.S. N and T. H. Abidi S.,1987, Journal of Applied Physics 61, pp 4909
44. Landsberg P. T,and A. Hope S., 1977, Journal of Solid State Electronics, 20, 421 45. Ghatak. K.P., Chakrabarti. S, Chatterjee. B, Das. P. K., Dutta. P, Halder. A, 2018, Journal of Materials Focus, 7, 390-404
46. SL Singh, SB Singh, KP Ghatak, 2018, Journal of nanoscience and nanotechnology, 18, 2856-2874
47. Ghatak. K.P., Mitra. M, Paul. R, Chakrabarti. S, 2017, Journal of Computational and Theoretical Nanoscience, 14, pp 2138-2229
48. Nag. B. R.and Chakravarti A. N.,1975, Journal of Solid State Electron., 18, pp 109
49. Chatterjee. B, Debbarma. N, Mitra. M, Datta. T, Ghatak. K.P., 2017, Journal of Nanoscience and Nanotechnology, 17, 3352-3364
50. Nag B. R.and Chakravarti A. N.,1981, Journal of Physica Status Solidi (a), 67, pp K113
51. Chakravarti A. N.and P. Parui D.,1972, Journal of Phys. Letts, 40A, pp 113
52. Choudhury. S., De D., Mukherjee S., Neogi A., Sinha A., Pal M., K. Biswas S., Pahari S., Bhattacharya S. and Ghatak K.P.,2008, Journal of Computation and Theoretical Nanoscience, 5, pp 375 53. Bouchaud J. P. and Georges A.,1996, Journal of Physics Report, 195, pp127
54.Marshak A. H., 1987, Journal of Solid State Electron., 30, pp 1089 55. Chakravarti A. N., Dhar A., Ghosh K. K. and Ghosh S.,1981,Journal of Applied Physics, A26, pp 165- 169 56.Nag.B.R.,1980, Electron Transport in Compound Semiconductor, Springer-Verlag, Germany
57. Ghatak K.P., Ghoshal A. and Biswas S.N.,1993, Journal of Nouvo Cimento, 15D, pp 39-58
58. Ghatak K.P.and Bhattacharyya D.1994, Journal of Physics Letters A , 184, pp 366-369
59. Ghatak K.P.and Bhattacharyya. D., 1995, Journal of Physica Scripta, 52, pp 343
60. Ghatak K.P., Nag B and Bhattacharyya D., 1995, Journal of Low Temp. Phys. 14, pp 1
61. Ghatak K.P.. and Mondal M.,1987, Journal of Thin Solid Films, 148, pp 219
62. Ghatak K.P.K. Choudhury A., Ghosh S. andN. Chakravarti A.,1980, Journal of Applied Physics, 23, pp 241-244
64. Ghatak K.P..,1991, Influence of Band Structure on Some Quantum Processes inTetragonal Semiconductors, D. Eng. Thesis, Jadavpur University, Kolkata, India.
Journal of Mathematical Sciences & Computational Mathematics
70 JMSCM, Vol 1, Issue 1, 2019
65. Ghatak K.P., Chattropadhyay N. and Mondal M.,1987, Journal of Applied Physics A, 44, pp 305-312
66. Biswas S.N. and Ghatak K.P.,1987, Handbook of Proceedings of the Society of Photo-optical and Instrumentation Engineers (SPIE), Quantum Well and Superlattice Physics, USA, Vol. 792, , pp 239
67. Mondal M. and Ghatak K.P.,1987, Journal of Physics C, Solid State Physics, 20, p 1671
68. Ghatak K.P. and Mondal M. 1992, Journal of Applied Physics 70, p 1277
Journal of Mathematical Sciences & Computational Mathematics
71 JMSCM, Vol 1, Issue 1, 2019 FIGURES
Fig 1 The plot of the DMR as a function of inverse quantizing magnetic field for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe superlattices with graded interfaces.
Journal of Mathematical Sciences & Computational Mathematics
72 JMSCM, Vol 1, Issue 1, 2019
Fig 2 The plot of the DMR as a function of electron concentration for (a) GaAs/Ga1-xAlxAs and (b) CdS/CdTe superlattices with graded interfaces.
Journal of Mathematical Sciences & Computational Mathematics
73 JMSCM, Vol 1, Issue 1, 2019
Fig 3 The plot of the DMR as a function of electron concentration for (c) PbTe/PbSnTe and (d) HgTe/CdTe superlattices with graded interfaces.
Journal of Mathematical Sciences & Computational Mathematics
74 JMSCM, Vol 1, Issue 1, 2019
Fig 4 The plot of the DMR as a function of inverse quantizing magnetic field for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe effective mass superlattices.
Journal of Mathematical Sciences & Computational Mathematics
75 JMSCM, Vol 1, Issue 1, 2019
Fig 5 The plot of the DMR as a function of electron concentration for (a) GaAs/Ga1-xAlxAs and (b) CdS/CdTe effective mass superlattices.
Journal of Mathematical Sciences & Computational Mathematics
76 JMSCM, Vol 1, Issue 1, 2019
Fig 6 The plot of the DMR as a function of electron concentration for (c) PbTe/PbSnTe and (d) HgTe/CdTe effective mass superlattices.
Journal of Mathematical Sciences & Computational Mathematics
77 JMSCM, Vol 1, Issue 1, 2019
Fig 7 The plot of the 1D DMR as a function of film thickness for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe superlattices with graded interfaces.
Journal of Mathematical Sciences & Computational Mathematics
78 JMSCM, Vol 1, Issue 1, 2019
Fig 8 The plot of the 1D DMR as a function of surface electron concentration for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe superlattices with graded interfaces.
Journal of Mathematical Sciences & Computational Mathematics
79 JMSCM, Vol 1, Issue 1, 2019
Fig 9 The plot of the 1D DMR as a function of film thickness for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe effective mass superlattices.
Journal of Mathematical Sciences & Computational Mathematics
80 JMSCM, Vol 1, Issue 1, 2019
Fig 10 The plot of the 1D DMR as a function of surface electron concentration for (a) GaAs/Ga1-xAlxAs (b) CdS/CdTe (c) PbTe/PbSnTe and (d) HgTe/CdTe effective mass superlattices.
Journal of Mathematical Sciences & Computational Mathematics