Steven Duplij – författare

The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to "generalize" it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the "concrete way" and provide consequent "polyadization" of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang-Baxter equation, find its constant solutions, and introduce polyadic tensor categories.

Suitable for university students of advanced level algebra courses and mathematical physics courses.

Key Features

Provides a general, unified approach Widens readers perspective of the possibilities to develop standard algebraic structures Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations Presents applications of "polyadization" approach to concrete algebraic structures

This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers.

In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way.

Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.

Key Features

Provides new mathematical methods for quantum computing

Presents material in a widely accessible way

Contains methods for unconventional computing where there is computational complexity

Provides methods to increase speed and efficiency

A presentation of outstanding achievements and ideas, of both eastern and western scientists, both mathematicians and physicists. Their presentations of recent work on quantum field theory, supergravity, M-theory, black holes and quantum gravity, together with research into noncommutative geometry, Hopf algebras, representation theory, categories and quantum groups, take the reader to the forefront of the latest developments. Other topics covered include supergravity and branes, supersymmetric quantum mechanics and superparticles, (super) black holes, superalgebra representations, and SUSY GUT phenomenology. Essential reading for workers in the modern methods of theoretical and mathematical physics.

This collection presents original mathematical ideas and unusual constructions in theoretical physics. They were developed locally at Kharkov University, Ukraine, and now can be presented to Western readers. The subjects are broad and cover an array of ideas, from regular generalization of supermanifolds and category theory, coalgebras, the Yang-Baxter equation and Hopf algebras, to nontrivial aspects of nilpotent mechanics, fermionic gravity and multigravity, the constraintless Hamiltonian approach to singular theories, and the application of polyadic structures to representation theory and quantum groups. The book can attract researchers and post-graduate students who are interested in new developments and methods in modern theoretical physics and related directions in mathematics.