Fundamental hoop-algebras

In this paper, we investigate some results on hoop algebras and hyper hoop-algebras. We construct a hoop and a hyper hoop on any countable set. Then using the notion of the fundamental relation we define the fundamental hoop and we show that any hoop is a fundamental hoop and then we construct a fundamental hoop on any non-empty countable set

An Overview of Topological and Fuzzy Topological Hypergroupoids

On a hypergroup, one can define a topology such that the hyperoperation is pseudocontinuous or continuous.This concepts can be extend to the fuzzy case and a connection between the classical and the fuzzy (pseudo)continuous hyperoperations can be given.This paper, that is his an overview of results received by S. Hoskova-Mayerova with coauthors  I. Cristea , M. Tahere and  B. Davaz, gives examples of topological hypergroupoids and show that there is no relation (in general) between pseudotopological and strongly pseudotopological hypergroupoids. In particular, it shows a topological hypergroupoid that does not depend on the pseudocontinuity nor on strongly pseudocontinuity of the hyperoperation

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

A systems biology approach to multi-scale modelling and analysis of planar cell polarity in drosophila melanogaster wing

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Systems biology aims to describe and understand biology at a global scale where biological systems function as a result of complex mechanisms that happen at several scales. Modelling and simulation are computational tools that are invaluable for description, understanding and prediction these mechanisms in a quantitative and integrative way. Thus multi-scale methods that couple the design, simulation and analysis of models spanning several spatial and temporal scales is becoming a new emerging focus of systems biology. This thesis uses an exemplar – Planar cell polarity (PCP) signalling – to illustrate a generic approach to model biological systems at different spatial scales, using the new concept of Hierarchically Coloured Petri Nets (HCPN). PCP signalling refers to the coordinated polarisation of cells within the plane of various epithelial tissues to generate sub-cellular asymmetry along an axis orthogonal to their apical-basal axes. This polarisation is required for many developmental events in both vertebrates and non-vertebrates. Defects in PCP in vertebrates are responsible for developmental abnormalities in multiple tissues including the neural tube, the kidney and the inner ear. In Drosophila wing, PCP is seen in the parallel orientation of hairs that protrude from each of the approximately 30,000 epithelial cells to robustly point toward the wing tip. This work applies HCPN to model a tissue comprising multiple cells hexagonally packed in a honeycomb formation in order to describe the phenomenon of Planar Cell Polarity (PCP) in Drosophila wing. HCPN facilitate the construction of mathematically tractable, compact and parameterised large-scale models. Different levels of abstraction that can be used in order to simplify such a complex system are first illustrated. The PCP system is first represented at an abstract level without modelling details of the cell. Each cell is then sub-divided into seven virtual compartments with adjacent cells being coupled via the formation of intercellular complexes. A more detailed model is later developed, describing the intra- and inter-cellular signalling mechanisms involved in PCP signalling. The initial model is for a wild-type organism, and then a family of related models, permitting different hypotheses to be explored regarding the mechanisms underlying PCP, are constructed. Among them, the largest model consists of 800 cells which when unfolded yields 164,000 places (each of which is described by an ordinary differential equation). This thesis illustrates the power and validity of the approach by showing how the models can be easily adapted to describe well-documented genetic mutations in the Drosophila wing using the proposed approach including clustering and model checking over time series of primary and secondary data, which can be employed to analyse and check such multi-scale models similar to the case of PCP. The HCPN models support the interpretation of biological observations reported in literature and are able to make sensible predictions. As HCPN model multi-scale systems in a compact, parameterised and scalable way, this modelling approach can be applied to other large-scale or multi-scale systems.This study was funded by Brunel University

Advanced Concepts in Particle and Field Theory

Uniting the usually distinct areas of particle physics and quantum field theory, gravity and general relativity, this expansive and comprehensive textbook of fundamental and theoretical physics describes the quest to consolidate the elementary particles that are the basic building blocks of nature. Designed for advanced undergraduates and graduate students and abounding in worked examples and detailed derivations, as well as historical anecdotes and philosophical and methodological perspectives, this textbook provides students with a unified understanding of all matter at the fundamental level. Topics range from gauge principles, particle decay and scattering cross-sections, the Higgs mechanism and mass generation, to spacetime geometries and supersymmetry. By combining historically separate areas of study and presenting them in a logically consistent manner, students will appreciate the underlying similarities and conceptual connections across these fields. This title, first published in 2015, has been reissued as an Open Access publication

Theory and Metatheory of Atemporal Primacy

This thesis-on macro-ontology, physics, logic, and metalogical principles presents the findings, results, theorems, and metatheory that correct long-standing defects and deficiencies of current standard model (SM) physics and cosmology. It eliminates artificial SM anomalies, paradoxes, logical fallacies, absurdities, and conflicts with reality (and the findings of plasma physics, astronomy, ontology, epistemics, etc.). New theorems and metatheorems eliminate the illogic maintaining distortions of post-Einsteinian physics, its wildly speculative conjectures, and shibboleths (its unrealistic assumptions commonly accepted as facts). In critiques of misperceptions, misconceptions and misinterpretations of observations and data, this work exposes the common failure to recognize the severe deficiencies of theory, science, ontology, and metamathematics. New results prove that chronic failure to differentiate realities from theorems, hypotheses, conjectures, opinions, and beliefs prevents progress to better physics and STEM education. Therefore, this work also enables resolution of the long-standing logical incongruency of physics, psychology, cognitive science, and philosophy, especially ontology. So, the facts and proofs presented here can enable a new era of discovery, creativity, and technological progress