Information theory and dimensionality of space

We present an information-theoretic approach to the optimal representation of the intrinsic dimensionality of data and show it is a noninteger. Since optimality is accepted as a physical principle, this provides a theoretical explanation for why noninteger dimensions are useful in many branches of physics, where they have been introduced based on experimental considerations. Noninteger dimensions correlate with lesser density as in the Hausdorff dimension and this can have measurable effects. We use the lower density of noninteger dimension to resolve the problem of two different values of the Hubble constant obtained using different methods.

CONSCIOUSNESS AND SELF-CONSCIOUSNESS IN PSICHOANALYTICAL, PHENOMENOLOGICAL AND EXISTENTIAL DISCOURSE

This article is focused on the analysis of the problem of deep contexts of irrational reflection in the studies of great thinkers of the twentieth century and in the further research. The author analyses specifics of conceptualization of self-knowledge and structuring of psychics as well as the features of methodology of cognition and self-consciousness in the context of definition of values and priorities for a development of the person. The study includes a review of socio-historical determinants of the human psychics and behavior which leads to a conclusion that the human psychics does not depend on any limitations, such as the three-dimensionality of space and time. It does not obey the laws of physics and has a certain superpersonal layer in its structure. Additionally, the author deep dives in the main concepts and problems of irrational reflection, demonstrates the examples of their application in situational contexts and summarizes theoretical interrelations and the most important conceptual discussions. The theoretical significance of the problem of cognition and self-consciousness is determined by the central role of man as a person in society and history. The analysis shows that the methodology of cognition and self-consciousness should be based on the principles of axiological disengagement, unity of logical and historical aspects, as well as on the coherence of social theory and practice.

Dimensionality, symmetry and the Inverse Square Law

Kant suggested that Newton's Inverse Square Law (ISL) determines the dimensions of space to be three. Much has been written in the philosophical literature about Kant's suggestion, including specific arguments attempting to link the ISL to three-dimensionality. In this article, we explore one such argument and demonstrate that it fails to support the link Kant purports to make between the ISL and the three-dimensionality of space. At best, the link that can be made is between the ISL and symmetry.

Introduction

Chapter 1 introduces various essential ideas on second-order phase transitions and the theoretical challenges that accompany them. Furthermore, it focuses on important issues, for example, correlation length, correlation functions, scaling laws and behaviour, energy minimization, entropy maximization, dimensionality of space and order parameters and critical exponents, etc. It introduces and also devotes a short discussion to the Ising model and its most significant developments during the years of its study, as well as a short background about Ising himself. The chapter also contains two appendices that summarize all relevant results of ensembles of classical statistical mechanics and quantum statistical mechanics.

DIMENSÕES FÍSICAS, GEOMETRIA, PERSPECTIVA, FILOSOFIA E ARTE

A ideia de um espaço tridimensional começou a se formar no século XV. Antes disso, em um mundo dominado pelo aristotelismo, o espaço era vinculado à superfície e não ao volume. Foi através das artes que essa realidade começou a mudar. A perspectiva racional, definida aqui como um recurso gráfico que utiliza o efeito visual de linhas convergentes para criar a ilusão de tridimensionalidade do espaço e das formas representadas sobre uma superfície plana de um papel ou tela, nascida a partir de uma retomada da geometria euclidiana, entrou em cena para ficar no século XVI. Entre os muitos nomes que poderiam ser citados, destacaram-se o matemático e filósofo John Dee, o arquiteto e designer Filippo Brunelleschi, e o pintor e matemático Pierro della Francesca. Através da combinação das ideias e realizações desses três atores é possível entender uma época de transição entre o antigo e o moderno, em termos de ciência e arte.PALAVRAS-CHAVE: Espaço tridimencional. Geometria. Perspectiva racional. Filosofia e arte. ABSTRACTThe idea of a three-dimensional space began to form in the fifteenth century. Before that, in a world dominated by Aristotelianism, space was bound to the surface and not to the volume. It was through the arts that this reality began to change. The rational perspective, defined here as a graphic resource that uses the visual effect of converging lines to create the illusion of three-dimensionality of space and forms represented on a flat surface of a paper or canvas, born from a resumption of Euclidean geometry, came into the scene to stay in the sixteenth century. Among the many names that could be cited were the mathematician and philosopher John Dee, the architect and designer Filippo Brunelleschi, and the painter and mathematician Pierro della Francesca. By combining the ideas and achievements of these three actors it is possible to understand a time of transition between the old and the modern, in terms of science and art.KEYWORDS: Three-dimensional space. Geometry. Rational Perspective. Philosophy and Art.

The Fractal Calculus for Fractal Materials

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

Conditions of Emergence

Chapter 1 describes the disparate conditions for the emergence of higher-dimensioned space as a cultural object. It gives an account of Immanuel Kant’s original work on space, and particularly his thoughts on the dimensionality of space, considering this formulation ‘foundational’ for the nineteenth-century novel. Reading scholarly discussion in British periodicals it identifies the persistent use of analogy as a rhetorical device for explaining the ideas of dimensionality. It identifies, too, the fact that geometry itself is a model of the more abstract form that is space, alerting us to a structural shift between domains early in the life cycle of the fourth dimension, as it leaves geometry—a domain of pure thought—to enter space, a phenomenon of the physical world. It also considers Henry More’s notion of ‘spissitude’, an earlier iteration of the fourth dimension.