Hyper Symmetries

Fivefold symmetry is common in flowers, fruits, molecules, logos, and buildings, but it is a forbidden symmetry in the world of crystals. A few years ago, the so-called quasicrystals were discovered displaying fivefold symmetry, and it caused a minirevolution in crystallography. There has been increased awareness of fivefold symmetry in all domains of human interest ever since. The present book brings together authors and ideas on a common theme from mathematics, the sciences, design, and anthropology to history, literature, and the arts. Its 29 chapters are an offering by scientists and humanists from 13 countries to a broad readership of professionals and laypersons about fivefold symmetry and the areas that are being bridged by this unique concept.

This is the third volume in the Single Monad Model of the Cosmos series. The second volume introduced the Duality of Time Theory, which provided elegant solutions to many persisting problems in physics and cosmology, including super-symmetry and matter-antimatter asymmetry. In addition to uniting the principles of Relativity and Quantum theories, this theory can also explain the psychical and spiritual domains; all based on the same discrete complex-time geometry. Super-symmetry, and quantum gravity, are realized only with the two complementary physical and psychical worlds, while the spiritual realm is governed by hyper-symmetry, which mirrors the previous two levels together, and all these three realms mirror the ultimate level of absolute oneness that describes the symmetry of the divine presence of God and His Beautiful Names and Attributes. This "ULTIMATE SYMMETRY" is a modern scientific account of the same ancient mystical, and greatly controversial, theory of the "Oneness of Being" that is often misinterpreted in terms of "pantheism", but it is indeed the concluding gnostic knowledge of God and creation. Otherwise, how can we understand the origin of the cosmos, with both or either of its corporeal and incorporeal realms, without referring to its Originator! In the literal sense, ultimate or perfect symmetry may seem to be trivial, because it means that all possible transformations in such a symmetric system are invariant. The system we are talking about here is the whole Universe that we are watching and experiencing its immense and sometimes shattering changes every moment of time. Yet many great philosophers, such as Parmenides and Ibn al-Arabi, maintained their firm belief that reality is unchanging One and existence is timeless and uniform, while all apparent changes are mere illusions induced by or in our sensory faculties. Nevertheless, since we are living inside it, this illusion is as good as reality for us. Therefore, we still need to explain how the Universe is being formulated. Only when are able to transcend beyond the current chest of time, we shall discover that we were living a dream, and we shall be able to see the whole Universe as unchanging symmetry. The Single Monad Model and the resulting Duality of Time Theory provide the link between this apparent dynamic multiplicity of creation and the ultimate metaphysical oneness. In fact, the complex-time geometry concludes that we are imagining the reality because we are observing it from a genuinely imaginary time dimension. Since the ultimate reality is One, we cannot view it from outside, because there is none! Thus, as we quoted in the Introduction, in the Book of Theophanies, Ibn al-Arabi ascribes to God as saying: Listen, O My beloved! I am the conclusive entity of the World. I am the center of the circle (of existence) and its circumference. I am its simple point and its compound whole. I am the Word descending between heaven and earth. I have created perceptions for you only to perceive Me. If you then perceive Me, you perceive yourself. But don't ever crave to perceive Me through yourself! It is through My Eyes that you see Me and see yourself. But through your own eyes you can never see Me! This Theophany of Perfection summarizes the Ultimate Symmetry between the single point and the encompassing space. It also summarizes the instantaneous process of creation, or re-creation, which is breaking this symmetry into the two arrows of time, that produce particles and anti-particles, and then restoring it through each subsequent annihilation. This reunion is also the fundamental cause of motion, which is formulated as the Principle of Love that leads to the stationary action that is the initial assumption of most physics theories including Relativity and Quantum Field theories.

The quest for the unification of fundamental interactions has become the most challenging frontier of sciences in the 21st century. This book presents a detailed analysis and systematic investigation of the foundations of the hyperunified field theory (HUFT) in light of the path integral formulation with the least action principle. Alternative to other unification theories, the starting point of HUFT is initiated from a simple notion that the universe is made of the fundamental building block which is always moving and obeys the basic rule. Such a rule is delved into in this book by proposing the maximum locally entangled-qubits motion principle together with the scaling and gauge invariance principle. These two basic guiding principles are demonstrated to lay the foundations of HUFT, which enable enables us to discuss a series of long-standing fundamental questions, such as: why does the fundamental building block of nature appear as an entangled qubit-spinor field? what brings about the fundamental symmetry of nature? how does the inhomogeneous hyperspin gauge symmetry govern all basic forces? what is the nature of gravity and space-time? how can the space-time dimension and qubit-spinor field be categorized? why do we live in a universe with only four-dimensional space-time? why are there more than one family of leptons and quarks? how does the early universe evolve to be inflationary? what is the nature of dark matter and dark energy?Foundations of the Hyperunified Field Theory will be of great interest to graduate and senior undergraduate students, junior and senior researchers in theoretical physics, quantum field theory, particle physics, gravitational theory, cosmology, as well as mathematical physics and general physics.

This book is a collection of 12 innovative research papers in the field of hypercompositional algebra, 7 of them being more theoretically oriented, with the other 5 presenting strong applicative aspects in engineering, control theory, artificial intelligence, and graph theory. Hypercompositional algebra is now a well-established branch of abstract algebra dealing with structures endowed with multi-valued operations, also called hyperoperations, having a set as the result of the interrelation between two elements of the support set. The theoretical papers in this book are principally related to three main topics: (semi)hypergroups, hyperfields, and BCK-algebra. Heidari and Cristea present a natural generalization of breakable semigroups, defining the breakable semihypergroups where every non-empty subset is a subsemihypergroup. Using the fundamental relation β on a hypergroup, some new properties of the β-classes are obtained by De Salvo et al., who introduced and investigated the notion of height of a β-class. Based on the properties of a cyclic hypergroup of particular matrices, Krehlik and Vyroubalova describe the symmetry of lower and upper approximations in certain rough sets connected with this hypergroup. These results suggest an application to the study of detection sensors. In the framework of hyperrings and hyperfields theory, a new line of research has been developed regarding hyperhomographies on Krasner hyperfields, with interesting applications in cryptography (Vahedi et al.) and new fuzzy weak hyperideals were defined in Hv-rings by using the concept of fuzzy multiset (Al Tahan et al.), for which some algebraic properties were obtained. Two articles are dedicated to the study of BCK-algebras. Bordbar et al. present the properties of the relative annihilator in lower BCK-semilattices, whereas several types of intuitionistic fuzzy soft ideals in hyper BCK-algebras were defined and studied by Xin et al. Increasing numbers of researchers are interested in the applicative aspects of algebraic hypercompositional structures. For example, new properties related with symmetric relations are emphasized by Chvalina and Smetana for the structures and hyperstructures of artificial neurons. Novak et al. present a mathematical model based on elements of algebraic hyperstructure theory, used in the context of underwater wireless sensor networks. A construction of granular structures using m-polar fuzzy hypergraphs and level hypergraphs is illustrated in Luqman et al. using examples from a real-life problem. In the last paper in this book, Akram et al. discuss some properties related to edge regularity for q-rung picture fuzzy graphs.

The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, and the relationships among NET- semihypergroups, regular semihypergroups, NET-hypergroups and regular hypergroups are systematically are investigated. Moreover, pure NET-semihypergroup and pure NET-hypergroup are investigated, and a strucuture theorem of commutative pure NET-semihypergroup is established. Finally, a new notion of weak commutative NET-semihypergroup is proposed, some important examples are obtained by software MATLAB, and the following important result is proved: every pure and weak commutative NET-semihypergroup is a disjoint union of some regular hypergroups which are its subhypergroups.

This book provides an upto date information on metric, connection and curva ture symmetries used in geometry and physics. More specifically, we present the characterizations and classifications of Riemannian and Lorentzian manifolds (in particular, the spacetimes of general relativity) admitting metric (i.e., Killing, ho mothetic and conformal), connection (i.e., affine conformal and projective) and curvature symmetries. Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of a comprehensive collection of the works of a very large number of researchers on all the above mentioned symmetries. (b) We have aimed at bringing together the researchers interested in differential geometry and the mathematical physics of general relativity by giving an invariant as well as the index form of the main formulas and results. (c) Attempt has been made to support several main mathematical results by citing physical example(s) as applied to general relativity. (d) Overall the presentation is self contained, fairly accessible and in some special cases supported by an extensive list of cited references. (e) The material covered should stimulate future research on symmetries. Chapters 1 and 2 contain most of the prerequisites for reading the rest of the book. We present the language of semi-Euclidean spaces, manifolds, their tensor calculus; geometry of null curves, non-degenerate and degenerate (light like) hypersurfaces. All this is described in invariant as well as the index form.

Volume 4 of the 5-volume Quantum Nanochemistry covers quantum (physical) chemical theory of solids and orderability and addresses the electronic order problems in the solid state viewed as a huge molecule in special quantum states, including also the bondonic treatment of the graphene nano-ribbons, along basic crystallographic principles, from geometrical-, to chemical- to physical- (x-ray) crystallography with featured examples, and energetic correlating symmetry discussion on orderability in nanochemical compounds.

Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. Dynamical Symmetry introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in research journals. Though Lie groups play a key role in elementary particle physics, their connection with differential equations is more often exploited in applied mathematics and engineering. Dynamical Symmetry bridges this gap in a novel manner designed to help readers establish new connections in their own areas of interest. Emphasis is placed on applications to physics and chemistry. Applications to many of the other sciences illustrate both general principles and the ubiquitousness of dynamical symmetries.