Mohammad Ashraf

The theory of differential identities in associative rings and algebras is the basis of this monograph. Informally, an identical relation involving arbitrary elements in the underlying rings (or algebras) along with the unknown differential function is called a differential identity in a ring (or algebra). Invariant theory, non-commutative geometry, mathematical physics, and the theory of rings and algebras are just a few of the fields where this abstract theory has proved to be an effective instrument for solving a wide range of challenging issues, and as the twenty-first century has arrived, the theory of differential identities has found enormous applications in resolving a number of unresolved problems in the theory of rings. This volume summarizes the findings and approaches that have significantly advanced the field during the previous three decades. The first chapter provides a brief introduction to the topic. The following three chapters cover the various kinds of derivations in rings and algebras as well as the interactions between the structure of some classes of rings with involution and the behavior of the underlying derivations, generalized derivations, skew derivations, and b-generalized derivations, as well as their corresponding properties. Chapter 5 explores the characterization of several kinds of higher derivable mappings and the structure of Lie and Jordan-type higher derivations. Although the book contains numerous applications of the conclusions presented in these chapters, the last chapter mostly focuses on the application of derivations. This research monograph is useful for researchers working in the area of differential identities in rings and algebras. It provides a comprehensive and authoritative account of research findings.

This book provides a comprehensive knowledge of linear algebra for graduate and undergraduate courses. As a self-contained text, it aims at covering all important areas of the subject, including algebraic structures, matrices and systems of linear equations, vector spaces, linear transformations, dual and inner product spaces, canonical, bilinear, quadratic, sesquilinear, Hermitian forms of operators and tensor products of vector spaces with their algebras. The last three chapters focus on empowering readers to pursue interdisciplinary applications of linear algebra in numerical methods, analytical geometry and in solving linear system of differential equations. A rich collection of examples and exercises are present at the end of each section to enhance the conceptual understanding of readers. Basic knowledge of various notions, such as sets, relations, mappings, etc., has been pre-assumed.

This book provides a comprehensive knowledge of linear algebra for graduate and undergraduate courses. As a self-contained text, it aims at covering all important areas of the subject, including algebraic structures, matrices and systems of linear equations, vector spaces, linear transformations, dual and inner product spaces, canonical, bilinear, quadratic, sesquilinear, Hermitian forms of operators and tensor products of vector spaces with their algebras. The last three chapters focus on empowering readers to pursue interdisciplinary applications of linear algebra in numerical methods, analytical geometry and in solving linear system of differential equations. A rich collection of examples and exercises are present at the end of each section to enhance the conceptual understanding of readers. Basic knowledge of various notions, such as sets, relations, mappings, etc., has been pre-assumed.

This book clarifies some misunderstandings about money by tying the concept of money to the goods and services sector of the economy. In addition, it demystifies the process of money creation on the part of central banks. The phenomenon of money is ubiquitous; it has been around for tens of thousands of years, if not longer. Indeed, no modern economy could function without money. For many, however, the concept of money remains elusive. Worse still, misinformation abounds, which leaves the uninitiated vulnerable to fraud. This lack of understanding has serious policy implications as well. When policymakers lack a firm grasp of the concept, policy is likely to be flawed and its effects are likely to be detrimental to the body politic. After providing a brief history of money, the author details the role of money in the division of labor and specialization, in economic growth, and in an interconnected world. Throughout the book, he points out the pitfalls of fallacious thinking. In recent policy debates, such thinking has led to proposals ranging from the re-institution of the gold standard to supplying limitless money as suggested by Modern Monetary Theory.