Journal of Mathematical Sciences & Computational Mathematics (JMSCM)
(ISSN Number (Online) - 2644-3368)
(ISSN Number (Print) - 2688-8300)


Volume1 Issue4:


ELECTRON ENERGY SPECTRA IN QUANTIZED STRUCTURES

J. Pal1,P. K. Das2,M. Mitra3 and K. P. Ghatak2,4

1Department of Physics, Meghnad Saha Institute of Technology, Nazirabad, P.O. Uchepota, Anandapur, Kolkata-700150, India
2Department of Basic Science and Humanities, Institute of Engineering & Management, D1, Management House Salt Lake, Sector V Kolkata -700091, India
3Department of Applied Physics, University College of Science and Technology, 92 APC Road, Kolkata-700009, India
4University of Engineering Management, University Area, Plot, Street Number 03, Action Area III, B/5, Newtown, Kolkata, West Bengal 700160
Page Number: 374-438

In this paper we study the Electron Energy Spectra (EES) in quantum wells (QWs) of heavily doped (HD) non-linear optical, III-V, II-VI, IV-VI and stressed Kane type compounds by formulating HDEES in each case respectively considering all the specialties of the energy band constants of the said materials . It is noted that the complex EES in many cases in HDS, instead of real one, occurs from the existence of the essential poles in the corresponding EES in the absence of band tails. The EES in QWs is Quantized 2D closed surfaces. As a collateral study we have also investigated the effective mass (EM), Density-of-states (DOS) function and the electron statistics in this context for the purpose of comprehensive understanding. The EM exists in the forbidden zone, which is impossible without the effect of band tailing. In the absence of band tails, the EM in the band gap of semiconductors is infinity. Besides, depending on the type of unperturbed carrier energy spectrum, the new forbidden zone will appear within the normal energy band gap for HDS. Under certain limiting conditions all the results for all the models get simplified the wellknown results of an isotropic parabolic energy bands which exhibit the mathematical compatibility of our present generalized analysis.
Keywords:Electron Energy Spectra, Heavy Doping, Quantum Wells, Effective Mass, Density-of-States Function (DOS)


DOI: doi.org/10.15864/jmscm.1401

ON AN INEQUALITY IN L q VERSION

Barchand Chanam

Department of Mathematics National Institute of Technology Manipur,Langol, Imphal-795004, Manipur, India Email:[email protected]
Page Number: 439-450

If f(z) = nΣ v=0 c vzv is a polynomial of degree n which does not vanish in |z|=1, then for 0|z|= (ρ) |ƒ(z)|<= n (ρ+t)n-1 ‾ (t+r) n {1-t(t-ρ(n|c0|-t|c1)|)n ‾ (t2+ ρ2)n|c0|+ 2t2ρc1(ρ- r ‾ t+ρ)(t+r ‾ t+ρ)n-1} max|z|-r|ƒ(z)| In this paper, we prove an interesting inequality, which not only extends the above result to Lq version as a particular case, but also gives some interesting known results as corollaries.
Keywords: Polynomial,Lq Inequalities, Maximum Modulus.


DOI: doi.org/10.15864/jmscm.1402

RIEMANN AND DIOPHANTINE’S CONTRIBUTION IN THE FIELD OF NUMBER THEORY

Pratik Kumar 1,Vikas Kumar2, Md Saad Ahmed3, Rajnish Tiwari4 and Md Saquib5

1,2 Department of Computer Science Engineering Institute of Engineering & Management EP Block, Sector V, Salt Lake, Kolkata-700091. India
3,5Department of Electronic and Communication Engineering Institute of Engineering & Management EP Block, Sector V, Salt Lake, Kolkata-700091. India
4Department of Mechanical Engineering Institute of Engineering & Management EP Block, Sector V, Salt Lake, Kolkata-700091. India Corresponding Author Email: [email protected]
Page Number: 451-455
There have been several fascinating applications of Number Theory in key cryptography. Key cryptography enables many technologies we take for granted, such as the ability to make secure online transactions. The purpose of this survey paper is to highlight certain important such applications. Prime numbers constitute an interesting and challenging area of research in number theory. Diophantine equations form the central part of number theory. An equation requiring integral solutions is called a Diophantine equation. In the first part of this paper, some major contribution in number theory using prime number theorem is discussed and some of the problems which still remains unsolved are covered. In the second part some of the theorems and functions are also discussed such as Diophantine Equation, Goldbach conjecture,Fermats Theorem, Riemann zeta function and his hypothesis that still remain unproved to this day . The Chinese hypothesis is a special case of Fermat’s little theorem. As proved later in the west, the Chinese hypothesis is only half right . From the data of this study we conclude that number theory is used in computer network and applications in cryptography.We came to know about the purpose of Diophantine equation , Squarefree natural number,Zeta function,Fermat’s theorem and Chinese hypothesis which is a special case for Fermat’s theorem.
Keywords: Chinese hypothesis, Diophentine equation, Fermat’s theorem, Goldbach conjecture, squarefree natural number.


DOI: doi.org/10.15864/jmscm.1403

SUMMATION FORMULAE INVOLVING BASIC HYPERGEOMETRIC AND TRUNCATED BASIC HYPERGEOMETRIC FUNCTIONS

Harwinder Kaur1 & A.K.Shrivastav2

1Department of Mathematics RIMT University, Mandi Gobindgarh ,Fathegarh Sahib, Punjab (India). Email: [email protected]
2Department of Mathematics RIMT University, Mandi Gobindgarh, Fathegarh Sahib, Punjab (India). Email: [email protected]
Page Number: 456-461

In this paper, making use of concept splitting the limits of summation identities and Bailey’s lemma, an attempt has been to establish the transformations involving Bailey’s lemma and Truncated Basic Hypergeometric Series.
Keywords: Bailey’s lemma, q-series, Basic Hypergeometric functions, Truncated Basic Hypergeometric functions.


DOI: doi.org/10.15864/jmscm.1404

F (R) GRAVITY WITH K-ESSENCE SCALING RELATION AND COSMIC ACCELERATION

Somnath Mukherjee1, Debashis Gangopadhyay 2

1Department of Physics, Dharsa Mihirlal Khan Institution [H.S], G.I.P Colony, Howrah-711112, India. Kalpana Chawla Centre for Space and Nano Sciences (KCCSNS), Kolkata700010, India Email: [email protected]
2Department of Physics, Sister Nivedita University, DG 1/2, Action Area 1, Newtown Kolkata-700156, India
Page Number: 462-483

A modified gravity theory with f (R) = R2 coupled to a dark energy lagrangian L = −V (φ)F (X) , X = ∇μφ∇μφ, gives plausible cosmological scenarios when the modified Friedman equations are solved subject to the scaling relation X( dF )2= Ca(t)−6. This relation is already known to be valid, for constant potential V (φ), when L is coupled to Einstein gravity. φ is the k-essence scalar field and a(t) is the scale factor. The various scenarios are: (1) Radiation dominated Ricci flat universe with deceleration parameter Q = 1. The solution for φ is an inflaton field for small times. (2) Q is always negative and we have accelerated expansion of the universe right from the beginning of time and φ is an inflaton for small times. (3)The deceleration parameter Q = −5, i.e., we have an accelerated expansion of the universe. φ is an inflaton for small times. (4)A generalisation to f (R) = Rn shows that whenever n > 1.780 or n < −0.280 , Q will be negative and we will have accelerated expansion of the universe. At small times φ is again an inflaton.The results remind us of other physical phenomena where existence of scaling relations signal some sort of universality for theories with different microscopic lagrangians. Here this is seen in the case of Einstein gravity and modified gravity theories.
Keywords:Lagrangian, Hubble parameter, pressure, dark energy, ac- celeration.


DOI: doi.org/10.15864/jmscm.1405

ANALYSIS OF SYSTEM RELIABILITY OF AN ILLUMINATED DARK ROOM BY TIT2FNS FUZZY DATA WITH SOME OF ITS ARITHMETIC OPERATIONS

1Rahul Kar,2A.K. Shaw

1Department of Mathematics, Springdale high school, Kalyani, India Email: [email protected]
2 Department of Mathematics, Regent Education and Research Foundation, Kolkata Email: [email protected]
Page Number: 484-495

According to modern view uncertainty is considered essential to science and technology, it is not only the unavoidable plague but also it has impact a great utility. Generally, fuzzy sets are used to analyse fuzzy system reliability. To analyse the fuzzy system reliability, the reliability of each component of the system is considered as a Triangular intuitionistic Type 2 fuzzy number (TIT2FN). At first, TIT2FN and their arithmetic operations are introduced. Expressions for computing the fuzzy reliability of a series system and a parallel system following TIT2FN have been described. Here, an imprecise reliability model of an electric network model of a dark room is taken. To compute the imprecise reliability of the above said system, the reliability of each component of the systems is represented by TIT2FN. A corresponding numerical example is presented.
Keywords:Triangular intuitionistic Type 2 fuzzy number (TIT2FN), System reliability, Parallel system, Series system.


DOI: doi.org/10.15864/jmscm.1406

RECREATIONAL MATHEMATICS

Part One

Arun Kumar Chatterjee

Department of Basic Science and Humanities, Institute of Engineering & Management, D1 Management House Salt Lake, Sector-V, Kolkata-700091, India

Page Number: 496-501

Over centuries mathematicians have generated a wealth of rigorous and high level mathematics that is the armoury of pure mathematicians. But there is an interesting segment of mathematics that can justifiably be consigned to a different realm, which is the world of recreational mathematics. In this paper we will visit a few interesting areas of this fascinating domain.
Keywords: Rigorous, armoury, realm, recreational.


DOI: doi.org/10.15864/jmscm.1407

COMPARISON OF NUMERICAL METHODS IN PREDICTING THE GROWTH OF NORMAL AGRICULTURAL ASSETS

Godspower C. Abanum

Department of Mathematics/Statistics, Ignatius Ajuru University of Education, Port Harcourt, Nigeria Corresponding Author: [email protected]

Page Number: 502-523

In this paper, I considered the comparison of numerical methods in predicting the biodiversity gain due to the variation of α1 and α2 together on biodiversity scenario. However, when the model parameter values α1 and α2 are increase, the normal agricultural variable also changes. By comparing the patterns of growth in these two interacting normal agricultural data, we have finite instance of biodiversity due to the application of four numerical methods such as ODE45, ODE23, ODE23tb and ODE15s. We have found the numerical prediction upon using these four numerical methods which are similar and robust, hence we have considered ODE45 numerical simulation to be computationally more efficient than the other three methods. The novel result we have obtained in this study have not been seen elsewhere.
Keywords: ODE45, ODE23, ODE23tb, ODE15s, Normal Agriculture


DOI: doi.org/10.15864/jmscm.1408

A REVIEW ON CRYPTOCURRENCY: THE VOLATILITY OF BITCOINS

1Vikash Gupta,2 Shrey Rungta 3Vishnu Soni

1,2,3Institute of Engineering & Management Salt Lake Electronics Complex, Kolkata-700091. India Email: [email protected]
Page Number: 524-530

This study seeks to analyze conceptual, innovative, marketing and quantitative aspects of Bitcoin (BTC) and how these are reflected in the volatility of its return. After describing basic concepts of digital currencies and BTC, an electronic currency created in 2009, we contextualize BTC as a financial innovation. A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution .We propose a solution to the double-spending problem using a peer-to-peer network. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they'll generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone In my contribution I want to analyses this currency-system from a sociocy bernetic point of view .After presenting the basic mechanisms of Bitcoin money creation, the value regulation etc. We want to focus the basic processes of self-organization in this high-complex social system.
Keywords:Bitcoin, Innovation.


DOI: doi.org/10.15864/jmscm.1409

MATHEMATICAL MAT SCORE PREDICTION & POST-MAT ACADEMIC PERFORMANCE ANALYSIS ON MANAGEMENT STUDENTS

1Sk. Samim Ferdows, 2Joydeep Dey, 3Akash Nag, 4Sk. Sahin Ferdows, 5Sunil Karforma

1Department of Mathematics M.U.C. Women’s College, Burdwan, West Bengal, India Email: [email protected]
2,3Department of Computer Science M.U.C. Women’s College, Burdwan, West Bengal, India Email: [email protected], [email protected]
4Student of Hemsheela Model School Durgapur, Paschim Bardhaman, West Bengal, India Email: [email protected]
5Professor & Head, Department of Computer Science The University of Burdwan, Burdwan, West Bengal, India Email: suni[email protected]
Page Number: 531-542

In India, the Management Aptitude Test (MAT) score is taken as the primary indicator of a student’s ability to be enrolled in the management courses. It has been observed in many cases during the group discussion and personal interview phases for MBA course admission, that candidate with poor MAT scores do well than those with higher scores. Hence, in that context, this research was carried out in order to assess the validity of MAT scores in predicting future success and also to measure the nature and strength of association between MAT scores and academic achievements. We also look at the correlation between students’ academic performance in their higher-secondary & under-graduate levels, and their MAT scores.
Keywords: Management Aptitude Test (MAT) score, academic achievement, management courses


DOI: doi.org/10.15864/jmscm.1410