Clear and concise introduction to matrices with elegant proofs; Of interest to scientists from many disciplines; Gives many interesting applications to different parts of mathematics, such as algebra, analysis and complexity theory; Contains 160 exercises, half of them on advanced material; Includes at least one advanced result per chapter

CAPITALIZE ON YOUR BRIGHT IDEAS WITH LEGAL DESIGN THINKING Organizations today innovate to survive in a competitive, complex, and interconnected business world. They co-create with others outside their own organization to succeed. But capturing the value of these bright ideas separately is often very complicated. The Innovation Matrixsimplifies and structures innovation management. Start looking at innovation and Intellectual Property (IP) from a pragmatic perspective. This book will take you through three key steps (THINK-STRATEGIZE-ACT) to build an innovation and IP strategy. You will learn to use IP--which is at the core of every innovation--as a tool to define your strategy and manage the innovation process. It includes many examples, templates, and checklists to help put theory into practice. Test yourself with case studies, and use our hands-on Tools (legal design thinking) for your own projects and cases.

This book includes papers on recent research carried out in the field of metal-matrix composites (MMCs). Processing, microstructure, and mechanical properties of MMCs and unreinforced matrix alloys will be covered with a focus on aluminum, titanium, nickel, and copper MMCs. Those involved in the research of MMCs and unreinforced alloys, particularly in aerospace, space, and automotive materials research, will find this volume indispensible.

In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical fixed points. These special type of fixed points or special realized limit cycles are always guaranteed as we use only MOD matrices as operators with its entries from modulo integers. However this sort of results are NP hard problems if we use reals or complex numbers.

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling.