Security is one of the topmost concerns of any computing model and cloud computing is no exception. Consumers or enterprises moving into cloud need to feel secure about their computing facilities and more importantly about the data they are disclosing to service providers. This is a critical choice for enterprises accustomed to safeguarding data sitting in their own centers in the traditional way of computing. Cloud computing promotes the concept of working on proprietary data and applications outside their jurisdiction.

It is often said that, ‘cloud computing is an excellent idea, but cloud security is not good’. The common perception is that the cloud services are not inherently secure. Cloud computing creates scope for scores of possibilities although like any other technology, there are some risks associated with it which can be overcome if understood correctly.

A cloud computing environment can be formed in different ways and there can be several ‘cloud formations’. Each form is not suitable for every kind of business operations. Consumers must first understand which among these forms which is best-suited for their purpose. This little judgment can make the security of cloud computing even better than that of traditional in-house computing environments. This chapter focusses on these important aspects.

THE SECURITY CONCERN IN CLOUD

Traditional computing systems used to create a security boundary by placing firewalls at the gateways through which the network used to communicate with the outer world. Firewall blocks unwanted traffic trying to access the network it protects and thus only authenticated accesses are allowed into the system. Thus, malicious accesses get blocked at firewalls in the traditional data centers in order to keep the system protected from outside threats.

But this strategy only makes sense when all applications and data reside within one network of security perimeter. Traditional data centers allow perimeterized (i.e. within organization's own network boundary or perimeter) access to computing resources. But, the de-perimeterization (to open-up the interaction with outer network) and erosion of trust boundaries that was happening in enterprise applications, have been amplified and accelerated by cloud computing.

The algebra of operators in the space of functions is of considerable importance in quantum mechanics. The function space of interest in quantum mechanics is the one in which the functions are square integrable in the sense described below. In this chapter, we study the algebra of operators in the space of square integrable functions.

Space of Square Integrable Functions

Consider the space of complex-valued functions of a real variable x (a ≤ x ≤ b) in which the scalar product between the functions g(x) and f (x), denoted by (g(x), f (x)), is defined by

As a consequence of the definition of scalar product given above,

is finite. A function f (x) for which the integral in the equation above is finite is said to be square integrable. It may be verified that the definition of the scalar product given above satisfies all the axioms of scalar product, namely,

In particular, as a consequence of the axioms above, follows the Schwarz inequality (see (2.8)) which, in the present case, assumes the form

An important consequence of this inequality is the result derived in Ex. 4.1 establishing that any linear combination of square integrable functions is also square integrable, which asserts that the space of square integrable functions forms a vector space.

With appropriate capacity planning in cloud computing, the large pool of virtualized resources creates an illusion of an infinite source of computing resources. In the preceding chapters, it has been discussed how applications often need to scale to manage the varying workload and how resources are dynamically provisioned in order to meet the needs of scaling in cloud computing. It has also been discussed that horizontal scaling is more prepared for large scale scaling in distributed environment like cloud.

Upon obtaining provisioned or de-provisioned dynamic resources, the horizontal scaling provides elasticity to cloud computing. But, this flexibility comes with lots of implementation complexity. Load balancing is the mechanism through which these features are implemented. Apart from these, as there are multiple resources available to serve a particular type of service request, and in a distributed computing environment it becomes necessary to distribute proportionately those service requests among the available resources so that none of them becomes overloaded and degrades the performance of the entire system. It is the run-time logic or algorithm of distributing the service requests that has to be capable of evenly distributing the workload. This chapter discusses different types of load balancing techniques along with their pros and cons. The possible parameters for consideration while working out load balancing algorithms have also been discussed. A properly-designed workload distribution and balancing architecture reduces resource underutilization, improves the performance of a system and also brings down the cost of computation.

LOAD BALANCING

Load balancing in distributed computing systems is an essential technique to distribute processing and communication activities evenly across the resources in the network so that no single computing resource gets overloaded. Among different exercises, load balancing is especially important for applications which often deal with a large and unpredictable numbers of service requests. Load balancing is a technique that distributes load evenly among multiple computing resources such as processor, memory etc.

Through load balancing, the incoming service requests are distributed among available computing resources. An efficient load balancing mechanism improves average resource utilization rate and therefore enhances the overall performance of a system.

This chapter identifies the class of continuously varying one-dimensional potentials for which the time-independent Schrödinger equation can be solved analytically exactly by reducing it to the hypergeometric or the confluent hypergeometric form by means of the transformation of the dependent and the independent variables. Most of the widely studied exactly solvable problems in quantum mechanics are special cases of the potentials in the said classes. The problem of finding analytically exact solution of the Schrödinger equation by reducing it to the hypergeometric or the confluent hypergeometric form has been addressed in different ways, for example, in [58–61].

To that end, we start with the one-dimensional Schrödinger equation (9.1) for the wave function of a particle of mass m and energy E in the potential V(x),

where the ‘prime’ denotes differentiation with respect to the independent variable.We consider first the case when the functions f (z) and u(z) are independent of “ and for which (11.4) assumes the form of either the (i) hypergeometric or (ii) the con_uent hypergeometric equation.We will subsequently consider the class of potentials for which (11.4) is reducible to the con_uent hypergeometric form under energy-dependent transformation.

Potentials for which Schrödinger Equation is Reducible to Hypergeometric Form

By assuming f (z) and u(z) to be independent of, in this section we find the form of the potential U(z) for which (11.4) assumes the form of the hypergeometric equation.

As discussed in the Appendix B, a second-order ordinary differential equation is reducible to the hypergeometric equation if it has three regular singularities, including the singularity, if any, at infinity and no other singularity.

The easy access to high-performance computing resources in cloud computing has not only made the process-intensive activities smarter but the data-intensive computing activities also have taken center stage. The nature of data have radically changed with this revolutionary utility service; hence their processing and storage requirements vary. Large data-sets are generated and produced everyday are sent for processing in the high-performance computing environments.

Like the traditional storages, users can store and access multimedia files of various formats like text, image, audio and video in cloud also, but the storage requirements have been altered for efficient processing of the large data-sets which are produced in cloud every hour. The traditional enterprise level files and data storage systems were not sufficient to satisfy all of the data-intensive and high-performance computing requirements.

Efficient file handling to support parallel and distributed operations of large data-sets needed an entirely new file system format. Hence, researchers and computing vendors have come up with suitable storage solutions to achieve optimal performance in cloud like high-performance environment. This chapter focuses on all of these advancements made to fulfill those requirements.

REQUIREMENTS OF DATA-INTENSIVE COMPUTING

Data-intensive computing presents a challenge to computing systems in terms of delivering high-performance. Large volume complex data-sets cannot be processed centrally in a single node and require partitioning and distribution over multiple processing nodes. Thus, data-intensive computing is I/O-bound and requires rapid movements of data in large numbers. This requires appropriate management of data in transaction. Data modelling, partitioning, node assignment and accumulation are some of the critical parts of this computing. Consumers’ requirements and technical aspects related to the storage facility in high-performance computing environment are different from the traditional storage system in many ways. Traditional enterprise storage systems are no more sufficient to tackle those issues.

Scalability and high-performance distributed data processing are complementary to each other. Distribution of large data-set among as-many-nodes as required for processing promotes scalability of the application. Suitable file systems are required to support this distribution and scaling proficiently. It has been observed that data-intensive computing often involves process-intensive computing too. Complex data-sets present challenges before the computing system.

The future often arrives silently. In the digital world, cloud computing which emerged silently, has already made a significant place of its own and has shown enough potential to be the future of computing. The advantages it offers are plenty, and this is what makes cloud computing revolutionary.

Books on this evolving subject started appearing in the market towards the end of the last decade. A handful of books have been published so far, and they provide useful resource material. While a few among them have been written in order to discuss conceptual aspects of the subject, others have been written from research or application perspective. As the research or application oriented books serve specific purposes only, they are not appropriate for general readers of the subject.

Cloud computing is not a single technology; rather, it is a combination of multiple technologies. Hence, it is necessary to understand all those individual technologies in order to understand cloud computing as a whole. Many books have been written with their focus on some particular aspects of cloud computing, and thus, they have not been able to cover all the essential topics of cloud computing with equal importance. This book is written from conceptual perspective, for the sake of new learners of the subject, and it tries to fill the void for a book covering all the fundamentals in detail, with simplified explanations in a learner friendly manner.

The author felt the need for such a book while teaching the subject to the undergraduate engineering students, as well as software professionals who wanted to develop a basic understanding of the subject. It is either inadequate coverage or a complex approach of most books available in the market, that makes it difficult for new learners to develop a clear conception about the various aspects of these technologies, and they therefore do not satisfy the fundamental queries of students. Thus, it is a lack of books covering all the fundamentals in detail which encouraged the author to write this book.

The author is aware of the fact that computing and IT industry are fast merging into a singular cloud computing environment, and in the near future there will be a requirement for a large pool of workforce with sound insight into this subject.

Some Mathematical Formulas

This chapter discusses the quantum mechanical formulation in the position and the momentum representations. The position representation provides a natural means of constructing the quantum analog of a classical mechanical problem. The momentum representation emerges as the Fourier transform of the position representation and often provides a convenient means of studying the time evolution of the states.

Position Representation

For the sake of simplicity we consider first a particle constrained to move on the real axis. Recall that, according to the second postulate of quantum mechanics, to every classical dynamical variable there corresponds a unique hermitian operator. Accordingly, let the hermitian operator corresponding to classical position x be denoted by. Let be an eigenstate of the position operator corresponding to the eigenvalue x, i.e.,

where the second equation above is due to the hermiticity of. A state

in the state space of the particle is then represented by the function

called the wave function. By virtue of the axioms of the scalar product,. The representation of states and operators in the basis of the eigenstates fjxig of O x is called the position representation. As is expected of a self-adjoint operator, the eigenstates fjxig of O x constitute complete orthonormal set. The completeness relation reads

The orthonormality relation reads Hence, in terms of its position representative x(x), any state can be represented as

Now, we know from the fourth postulate that is the probability density which, on invoking (8.2), implies that