Elements of The Theory of Computation

This book covers the modular invariant theory of finite groups, the case when the characteristic of the field divides the order of the group. It explains a theory that is more complicated than the study of the classical non-modular case, and it describes many open questions.

Quantum computers hold the promise of solving problems that would otherwise be intractable with conventional computers. Some prototypes of the simplest elements needed to build a quantum computer have already been implemented in the laboratory. The efforts now concentrate on combining these elements into scalable systems. In addition, alternative routes to creating large scale quantum computers are continuously being developed.
This volume gives a cross-section of recent achievements in both the theory and the practical realization of quantum computing devices. Samuel L. Braunstein (Reader, University of Wales, Bangor, and editor of the book "Quantum Computing - Where do we want to go tomorrow") and Hoi-Kwong Lo (Chief Scientist, MagiQ Technologies, Inc., NY) invited experts across many disciplines involved in the development of quantum computers to review their proposals in a manner accessible to the non-expert.

Prepare for Advanced Computer Science Courses. Learn how to program and think like a Computer Scientist.

This book presents an enticing introduction to tropical circuits and their use as a rigorous mathematical model for dynamic programming (DP), which is one of the most fundamental algorithmic paradigms for solving combinatorial, discrete optimization problems.

Economics of Information Security applies economics not to generate breakthroughs in theoretical economics, but rather breakthroughs in understanding the problems of security. Security, privacy and trusted computing are examined distinctly, using the tools of economics, and as elements of a larger dynamic system.

This monograph provides a self-contained presentation of the foundations of finite fields, including a detailed treatment of their algebraic closures. It also covers important advanced topics which are not yet found in textbooks: the primitive normal basis theorem, the existence of primitive elements in affine hyperplanes, and the Niederreiter method for factoring polynomials over finite fields.

The only single-source——now completely updated and revised——to offer a unified treatment of the theory, methodology, and applications of the EM algorithm
omplete with updates that capture developments from the past decade, The EM Algorithm and Extensions, Second Edition successfully provides a basic understanding of the EM algorithm by describing its inception, implementation, and applicability in numerous statistical contexts. In conjunction with the fundamentals of the topic, the authors discuss convergence issues and computation of standard errors, and, in addition, unveil many parallels and connections between the EM algorithm and Markov chain Monte Carlo algorithms. Thorough discussions on the complexities and drawbacks that arise from the basic EM algorithm, such as slow convergence and lack of an in-built procedure to compute the covariance matrix of parameter estimates, are also presented.

Introduction I. General remarks . 1 II. Notations . . 5 III. Lie algebras: some basics . 8 Chapter 1 Operator calculus and Appell systems I. Boson calculus . 17 II. Holomorphic canonical calculus . . 18 III. Canonical Appell systems . . 23 Chapter 2 Representations of Lie groups I. Coordinates on Lie groups . . . 28 II. Dual representations. 29 III. Matrix elements. . . 37 IV. Induced representations and homogeneous spaces . . 40 General Appell systems Chapter 3 I. Convolution and stochastic processes . 44 II. Stochastic processes on Lie groups . . 46 III. Appell systems on Lie groups. . 49 Chapter 4 Canonical systems in several variables I. Homogeneous spaces and Cartan decompositions . . 54 II. Induced representation and coherent states . 62 III. Orthogonal polynomials in several variables . 68 Chapter 5 Algebras with discrete spectrum I. Calculus on groups: review of the theory .. 83 II. Finite-difference algebra . 85 III. q-HW algebra and basic hypergeometric functions .. 89 IV. su2 and Krawtchouk polynomials .. . 93 V. e2 and Lommel polynomials . . 101 Chapter 6 Nilpotent and solvable algebras I. Heisenberg algebras . .. 113 II. Type-H Lie algebras .. 118 Vll III. Upper-triangular matrices .. . 125 IV. Affine and Euclidean algebras . . 127 Chapter 7 Hermitian symmetric spaces I. Basic structures . 131 II. Space of rectangular matrices . . 133 III. Space of skew-symmetric matrices . . 136 IV. Space of symmetric matrices . 143 Chapter 8 Properties of matrix elements I. Addition formulas .. . 147 II. Recurrences . .. 148 III. Quotient representations and summation formulas .. 149 Chapter 9 Symbolic computations I. Computing the pi-matrices . . 153 II. Adjoint group .. 154 III. Recursive computation of matrix elements .