Parallelohedrons of Higher Dimension and Partition of N-Dimensional Spaces

2021 ◽  
pp. 198-239
Keyword(s):  
Hilbert Problem ◽  
Dimensional Space ◽  
Geometric Theory ◽  
Careful Analysis ◽  
Higher Dimension ◽  
Geometric Properties ◽  
New Approach ◽  
Modern Knowledge ◽  
Structure Of Matter

For more than 100 years in science, many researchers, when trying to solve Hilbert's 18th problem of constructing n-dimensional space, used the principles of the Delaunay geometric theory. In this book, as a result of a careful analysis of the work in this direction, it is shown that the principles of the Delaunay theory are erroneous. They do not take into account the features of figures of higher dimensionality, do not agree with modern advances in the physics of the structure of matter, and lead to erroneous results. A new approach to solving the 18th Hilbert problem is proposed, based on modern knowledge in the field of the structure of matter and the geometric properties of figures of higher dimension. The basis of the new approach to solving the 18th Hilbert problem is the theory developed by the author on polytopic prismahedrons.

scholarly journals On the Structure of Parallelohedrons of Higher Dimension

2019 ◽  
Vol 8 (2) ◽  
pp. 69-96 ◽  
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Hilbert Problem ◽  
Dimensional Space ◽  
Geometric Theory ◽  
Careful Analysis ◽  
Higher Dimension ◽  
Geometric Properties ◽  
New Approach ◽  
Modern Knowledge ◽  
Structure Of Matter

For more than 100 years in science, many researchers, when trying to solve Hilbert's 18th problem of constructing n-dimensional space, used the principles of the Delaunay geometric theory. In this paper, as a result of a careful analysis of the work in this direction, it is shown that the principles of the Delaunay theory are erroneous. They do not take into account the features of figures of higher dimensionality, do not agree with modern advances in the physics of the structure of matter, and lead to erroneous results. A new approach to solving the 18th Hilbert problem, based on modern knowledge in the field of the structure of matter and the geometric properties of figures of higher dimension, is proposed. The basis of the new approach to solving the 18th Hilbert problem is the theory developed by the author on polytopic prismahedrons.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

The alternative derivation procedure of equations of the law of the junction for the study of the p-n junction theory

2020 ◽  
pp. 002072092097813
Author(s):  
Algirdas Baskys
Keyword(s):  
Student Engagement ◽  
Student Interest ◽  
Student Survey ◽  
New Approach ◽  
Alternative Derivation ◽  
Student Activity ◽  
Comprehensive Knowledge ◽  
Modern Knowledge ◽  
The Law ◽  
Derivation Procedure

The methodology that increases the student interest and provides modern knowledge to the study of the p-n junction theory has been proposed. It is based on two methods: increasing the interest of students using a new approach in derivation of equations of the law of the junction that allows obtaining more comprehensive knowledge about the operation of the p-n junctions and engaging students using the storytelling technique. The reaction of students to the lecture, obtained statistics on the student activity and the results of the student survey show that the proposed methodology allows increasing the student engagement in the p-n junction theory lectures.

scholarly journals A Constraint Solver for Flexible Protein Model

2013 ◽  
Vol 48 ◽  
pp. 953-1000 ◽  
Author(s):  
F. Campeotto ◽  
A. Dal Palù ◽  
A. Dovier ◽  
F. Fioretto ◽  
E. Pontelli
Keyword(s):  
Structure Prediction ◽  
Dimensional Space ◽  
Protein Structures ◽  
Empirical Evaluation ◽  
Geometric Constraints ◽  
Conformational Space ◽  
Loop Modeling ◽  
Geometric Properties ◽  
Constraint Solver ◽  
Multi Body

This paper proposes the formalization and implementation of a novel class of constraints aimed at modeling problems related to placement of multi-body systems in the 3-dimensional space. Each multi-body is a system composed of body elements, connected by joint relationships and constrained by geometric properties. The emphasis of this investigation is the use of multi-body systems to model native conformations of protein structures---where each body represents an entity of the protein (e.g., an amino acid, a small peptide) and the geometric constraints are related to the spatial properties of the composing atoms. The paper explores the use of the proposed class of constraints to support a variety of different structural analysis of proteins, such as loop modeling and structure prediction. The declarative nature of a constraint-based encoding provides elaboration tolerance and the ability to make use of any additional knowledge in the analysis studies. The filtering capabilities of the proposed constraints also allow to control the number of representative solutions that are withdrawn from the conformational space of the protein, by means of criteria driven by uniform distribution sampling principles. In this scenario it is possible to select the desired degree of precision and/or number of solutions. The filtering component automatically excludes configurations that violate the spatial and geometric properties of the composing multi-body system. The paper illustrates the implementation of a constraint solver based on the multi-body perspective and its empirical evaluation on protein structure analysis problems.

A Geometric Theory for Formulation of Form, Profile and Orientation Tolerances: Problem Formulation

1998 ◽  
Author(s):  
J. B. Gou ◽  
Y. X. Chu ◽  
H. Wu ◽  
Z. X. Li
Keyword(s):  
Configuration Space ◽  
Minimization Problem ◽  
Geometric Theory ◽  
Problem Formulation ◽  
Euclidean Group ◽  
Geometric Properties ◽  
Symmetry Subgroup ◽  
Minimum Zone ◽  
Constrained Minimization Problem ◽  
Linear Programming Problems

Abstract This paper develops a geometric theory which unifies the formulation and evaluation of form (straightness, flatness, cylindricity and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. In the paper, based on an an important observation that a toleranced feature exhibits a symmetry subgroup G0 under the action of the Euclidean group, SE(3), we identify the configuration space of a toleranced (or a symmetric) feature with the homogeneous space SE(3)/G0 of the Euclidean group. Geometric properties of SE(3)/G0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/G0, with G0 being the symmetry subgroup of the underlying feature. We transform the non-differentiable minimization problem into a differentiable minimization problem over an extended configuration space. Using geometric properties of SE(3)/G0, we derive a sequence of linear programming problems whose solutions can be used to approximate the minimum zone solutions.

Space-Optimized Tree: A Connection+Enclosure Approach for the Visualization of Large Hierarchies

2003 ◽  
Vol 2 (1) ◽  
pp. 3-15 ◽  
Author(s):  
Quang Vinh Nguyen ◽  
Mao Lin Huang
Keyword(s):  
Hierarchical Structure ◽  
Dimensional Space ◽  
Tree Structure ◽  
Relational Data ◽  
Two Dimensional ◽  
Approach Space ◽  
Link Diagram ◽  
New Approach ◽  
Semantic Zooming ◽  
Tree Visualization

This paper describes a new approach, space-optimized tree, for the visualization and navigation of tree-structured relational data. This technique can be used especially for the display of very large hierarchies in a two-dimensional space. We discuss the advantages and limitations of current techniques of tree visualization. Our strategy is to optimize the drawing of trees in a geometrical plane and maximize the utilization of display space by allowing more nodes and links to be displayed at a limited screen resolution. Space-optimized tree is a connection+ enclosure visualization approach that recursively positions children of a subtree into polygon areas and still uses a node–link diagram to present the entire hierarchical structure. To be able to handle the navigation of large hierarchies, we use a new hybrid viewing technique that combines two viewing methods, the modified semantic zooming and a focus+ context technique. While the semantic zooming technique can enlarge a particular viewing area by filtering out the rest of tree structure from the visualization, the focus+context technique allows the user to interactively focus, view and browse the entire visual structure with a reasonable high-density display.

scholarly journals The Catenary Method as an Alternative to the Horizontal Directional Drilling Trajectory Design in 2D Space

Energies ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 1112 ◽  
Author(s):  
Rafał Wiśniowski ◽  
Krzysztof Skrzypaszek ◽  
Paweł Łopata ◽  
Grzegorz Orłowicz
Keyword(s):  
Dimensional Space ◽  
Trajectory Design ◽  
Directional Drilling ◽  
Horizontal Directional Drilling ◽  
Industrial Practice ◽  
New Approach ◽  
Pipeline Networks ◽  
Pulling Forces ◽  
Calculation Algorithms ◽  
Gas Consumption

An increase in demand for energy natural resources has stimulated the development of gas pipeline networks in Europe, as well as globally. Also, in Poland in recent years there has been a significant increase in natural gas consumption. Therefore, it is necessary to build new pipelines networks using dig and no–dig techniques. Horizontal Directional Drilling is one of the most popular trenchless technologies. The aim of this article is to present a new approach in the design of HDD trajectories in two–dimensional space (2D). A review of the trajectories used so far has been provided, offering calculation algorithms to determine their well path. Then, the original catenary method is proposed, taking into account natural deflection of casing pipes. Applicable formulas and computational algorithms have been given, together with a computational example which enables comparison of the classical design methodology with the new one. According to the authors, due to natural stress distribution, the catenary method allows the use of smaller pulling forces during installation and ensures longer pipeline life. Therefore, it should be used in industrial practice as an alternative to current designing methods.

scholarly journals A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system

Automatica ◽  
2008 ◽  
Vol 44 (11) ◽  
pp. 2834-2839 ◽  
Author(s):  
Domenico Prattichizzo ◽  
Lorenzo Ntogramatzidis ◽  
Giovanni Marro
Keyword(s):  
Hamiltonian System ◽  
Geometric Properties ◽  
New Approach

Dilatation properties of regular local rings

1959 ◽  
Vol 55 (1) ◽  
pp. 1-9
Author(s):  
D. G. Northcott
Keyword(s):  
Dimensional Space ◽  
Geometric Theory ◽  
Commutative Rings ◽  
Ideal Theory ◽  
Local Rings ◽  
Abstract Form ◽  
One Dimensional ◽  
Regular Local Rings ◽  
Dimensional Projective Space ◽  
The Ideal

The results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.

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