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"A discussion of developments in the field of bifurcation theory, with emphasis on symmetry breaking and its interrelationship with singularity theory. The notions of universal solutions, symmetry breaking, and unfolding of singularities are discussed in detail. The book not only reviews recent mathematical developments but also provides a stimulus for further research in the field."

"This textbook is perfect for a math course for non-math majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas. It includes many topics commonly found in sampler courses, like Platonic solids, Euler’s formula, irrational numbers, countable sets, permutations, and a proof of the pythagorean theorem. All of these topics serve a single compelling goal, understanding the mathematical patterns underlying the symmetry that we observe in the physical world around us."

"Mathematical symmetry and chaos come together to form striking, beautiful color images throughout this impressive work, which addresses how the dynamics of complexity can produce familiar universal patterns.The book, a richly illustrated blend of mathematics and art, was widely hailed in publications as diverse as the New York Review of Books, Scientific American, and Science when first published in 1992. This much-anticipated second edition features many new illustrations and addresses the progress made in the mathematics and science underlying symmetric chaos in recent years; for example, the classifications of attractor symmetries and methods for determining the symmetries of higher dimensional analogues of images in the book. In particular, the concept of patterns on average and their occurrence in the Faraday fluid dynamics experiment is described in a revised introductory chapter."

"We impose the Higgs mechanism to break the Grand Unified Theory based on SU(6) symmetry, SU(6) GUT. We investigate and search for the Higgs multiplets which are approriate to generate masses for both fermions and gauge bosons. We have found that the most minimal Higgs multiplets are < 15 >,< 20 > and < 21 > to realize three steps of symmetry breaking in SU(6) GUT down to the standar SU(2) × U(1) model. However within this minimal SU(6) GUT, we can reproduce only some fermions spectra as N4e, νe,N6e, Ne and d quarks, although we have deployed the most general forms of Higgs multiplets. On the other hand, the present approach have succeeded in generating the reguler gauge bosons, W±, Z0 and Aµ."

"This monograph explores nonoscillation and existence of positive solutions for functional differential equations and describes their applications to maximum principles, boundary value problems and stability of these equations. In view of this objective the volume considers a wide class of equations including, scalar equations and systems of different types, equations with variable types of delays and equations with variable deviations of the argument. Each chapter includes an introduction and preliminaries, thus making it complete. Appendices at the end of the book cover reference material."

"The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior)."

"Generalized solutions of operator equations and extreme elements, presents recently obtained results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The presented results offer new methods of identifying these solutions and studying their properties. These new methods involve the application of a priori estimations and a general topological approach to construct generalized solutions of linear and nonlinear operator equations. "