(ISSN Number (Online) - 2644-3368)

(ISSN Number (Print) - 2688-8300)

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.

If f(z) =

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.

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.

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.

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.

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.

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.

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.

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.