Low-dimensional lattices. III. Perfect forms

1988 ◽  
Vol 418 (1854) ◽  
pp. 43-80 ◽  
Keyword(s):  
Quadratic Forms ◽  
Automorphism Groups ◽  
Low Dimensional ◽  
Minimal Vectors ◽  
Lattice Packings

The perfect lattices or quadratic forms in dimensions n ≼ 7 found by Barnes, Stacey and others are studied, and their automorphism groups, orbits of minimal vectors and eutactic coefficients are determined. It is shown that just 30 of the 33 known seven-dimensional perfect lattices are extreme. A simple and self-contained proof is given of the classification of perfect lattices in dimensions n ≼ 4 and hence of the determination of the densest lattice packings in these dimensions.

Low-dimensional lattices. II. Subgroups of GL(n, ℤ)

1988 ◽  
Vol 419 (1856) ◽  
pp. 29-68 ◽  
Keyword(s):  
Quadratic Forms ◽  
Automorphism Groups ◽  
Geometric Description ◽  
Low Dimensional ◽  
Minimal Vectors

The maximal finite irreducible groups of n x n integers for n = 4, 5, . . . , 9, 11, 13, 17, 19, 23 were determined by Dade, Ryskov, Bülow, Plesken & Pohst and Plesken, as the automorphism groups of certain quadratic forms. This paper presents a geometric description of the corresponding n -dimensional lattices, and gives coordinates which display their symmetries and minimal vectors. Some very interesting lattices appear.

scholarly journals Petrographic structures and Hardy – Weinberg equilibrium

2020 ◽  
Vol 242 ◽  
pp. 133
Author(s):  
Yury VOYTEKHOVSKY ◽  
Alena ZAKHAROVA
Keyword(s):  
Quadratic Forms ◽  
Dimensional Space ◽  
Complete Classification ◽  
Algebraic Expression ◽  
Diagonal Form ◽  
Hardy Weinberg Equilibrium ◽  
Algebraic Forms ◽  
Canonical Diagonal Form

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data.

scholarly journals Ternary quadratic types

1948 ◽  
Vol 241 (836) ◽  
pp. 399-456 ◽  
Keyword(s):  
Quadratic Forms ◽  
Complete System ◽  
Ternary Quadratic Forms ◽  
Irreducible System

The first part of this paper deals with the determination of the complete system of concomitants of five or fewer ternary quadratic forms. In the second part of the paper it is shown that this system is irreducible, and that from it may be deduced the irreducible system of ternary quadratic types, thus giving a classification of the irreducible concomitants of any number of ternary quadratics.

scholarly journals AUTOMORPHISM GROUPS OF QUANDLES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250008 ◽  
Author(s):  
MOHAMED ELHAMDADI ◽  
JENNIFER MACQUARRIE ◽  
RICARDO RESTREPO
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Affine Group ◽  
Low Dimensional Topology ◽  
Inner Automorphism ◽  
Mathematical Sciences ◽  
Low Dimensional

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [B. Ho and S. Nelson, Matrices and finite quandles, Homology Homotopy Appl.7(1) (2005) 197–208.], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [S. Carter, S. Kamada and M. Saito, Surfaces in 4-Space, Encyclopaedia of Mathematical Sciences, Vol. 142, Low-Dimensional Topology, III (Springer-Verlag, Berlin, 2004)]. Since computations of automorphisms of quandles relate to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

THE ESSENCE, THE NEED FOR EVALUATION AND TYPES OF ECONOMIC POTENTIAL OF ENTERPRISES

2017 ◽  
Author(s):  
Olexandr Yemelyanov ◽  
Keyword(s):  
Economic Activity ◽  
Business Strategies ◽  
Enterprise Management ◽  
Value Assessment ◽  
Economic Potential ◽  
Enterprise Value ◽  
Development Assessment ◽  
State Of The Environment

The formation of business strategies of enterprises should be based on a preliminary assessment of their current and future economic opportunities. Such an assessment is to establish the value of the total economic potential of enterprises and its individual varieties. The purpose of this study was to clarify the essence of the economic potential of enterprises, justify the need for its evaluation and selection of its types. The main approaches to interpreting the terms "potential" and "enterprise potential" are identified. These include resource, result, resource-result, resource-target, and result-target concepts. It is established that the potential of any object, including the enterprise, can be interpreted as a set of its external functional properties, which this object shows or can show in a certain state of the environment in which it is located. Accordingly, the assessment of the potential of an object should be based on the identification and determination of its external properties, taking into account the environment in which the object is located. The main situations in which there is a need for information about certain components of the economic potential of the enterprise are identified, and the types of this potential and the consumers of the information about their level, corresponding to these situations, are determined. In particular, such situations include management of production and sales, management of financial and economic results of the enterprise, management of enterprise development, assessment of the company's need for various types of resources, assessment of enterprise value, assessment of current and future impact of the enterprise on the economy of the country (region), etc. The features of classification of types of enterprise potential existing in the scientific literature are supplemented by the following ones: by the main types of economic activity, by the dynamics of changes in the economic and production system of the enterprise, by consequences for the subject of enterprise potential assessment, by the stages of economic activity, enterprise potential, etc. The obtained results make it possible to improve the understanding of the complex patterns that underlie the formation of the economic potential of enterprises.

State Failure and the Political Violence Phenomenon: A Comparative Analysis of Iraq and Syria Cases

2016 ◽  
Vol 4 (2) ◽  
pp. 170 ◽  
Author(s):  
K. Eylem Özkaya Lassalle
Keyword(s):  
Cold War ◽  
Political Violence ◽  
State Failure ◽  
The Political ◽  
Collective Violence ◽  
Short Run ◽  
Definition Of ◽  
The Cold War

The concept of failed state came to the fore with the end of the Cold War, the collapse of the USSR and the disintegration of Yugoslavia. Political violence is central in these discussions on the definition of the concept or the determination of its dimensions (indicators). Specifically, the level of political violence, the type of political violence and intensity of political violence has been broached in the literature. An effective classification of political violence can lead us to a better understanding of state failure phenomenon. By using Tilly’s classification of collective violence which is based on extent of coordination among violent actors and salience of short-run damage, the role played by political violence in state failure can be understood clearly. In order to do this, two recent cases, Iraq and Syria will be examined.

Method of determination of the text direction on the image with the use of convolutional neural network

2020 ◽  
pp. 96-99
Author(s):  
P.L. Nikolaev
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
Deep Neural Network ◽  
Binary Classification ◽  
Synthetic Data ◽  
Real Data ◽  
Method Of Determination ◽  
Classification Of Images

This article deals with method of binary classification of images with small text on them Classification is based on the fact that the text can have 2 directions – it can be positioned horizontally and read from left to right or it can be turned 180 degrees so the image must be rotated to read the sign. This type of text can be found on the covers of a variety of books, so in case of recognizing the covers, it is necessary first to determine the direction of the text before we will directly recognize it. The article suggests the development of a deep neural network for determination of the text position in the context of book covers recognizing. The results of training and testing of a convolutional neural network on synthetic data as well as the examples of the network functioning on the real data are presented.

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