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 Petrographic structures: Khibiny ijolites and urtites

Vestnik MGTU ◽  
2021 ◽  
Vol 24 (2) ◽  
pp. 160-167
Author(s):  
Yuri Leonidovich Voytekhovsky ◽  
Alena Alexandrovna Zakharova
Keyword(s):  
Crystalline Rocks ◽  
Diagonal Form ◽  
Standard Description ◽  
Khibiny Massif ◽  
Constitutional Principle ◽  
Hardy Weinberg Equilibrium ◽  
Definition Of ◽  
Canonical Diagonal Form ◽  
Strict Definition

In addition to the standard description of the structures and textures of crystalline rocks the mathematical approaches have been proposed based on a rigorous determination of the petrographic structure through the probabilities of binary intergrain contacts. In general, the petrographic structure is defined as an invariant aspect of rock organization, algebraically expressed by the canonical diagonal form of the symmetric Pij matrix and geometrically visualized by structural indicatrices - surfaces of the 2nd order. The agreed nomenclature of possible petrographic structures for an n-mineral rock is simple: the symbol Snm means that there are exactly m positive numbers in the canonical diagonal form of the Pij matrix. New types of barycentric diagrams have been proposed. To describe the massive texture, the concept of Hardy - Weinberg equilibrium has been proposed. This boundary classifies barycentric diagrams into areas within which canonical types of Рij matrices and topological types of structural indicatrices are preserved. The change in the organization of the rock within a type is quantitative, the transition from one type to another means structural restructuring. The methods are used to describe ijolites and urtites of the Khibiny massif, the Kola Peninsula. In the modern taxonomy of rocks, the boundaries between them are mostly conditional and are drawn according to the contents of rock-forming minerals, for example, between ijolites and urtites - according to the contents of nepheline and pyroxene. The strict definition of the petrographic structure proposed by the authors makes it possible to introduce into petrography the constitutional principle (structure + composition), which is successfully acting in mineralogy.

Progress in mineralogy

1977 ◽  
Vol 286 (1336) ◽  
pp. 235-237
Keyword(s):  
Earth Science ◽  
Complete Classification ◽  
Applied Physics ◽  
X Ray Diffraction ◽  
Full Effect ◽  
Spin Resonance ◽  
Great Progress ◽  
The Impact

Since the fiftieth anniversary of the Mineralogical Society in 1926, there has been an epoch of great progress, resulting from the impact of applied physics and from the generally widened horizons of Earth science. Description of the morphology of crystals by means of the goniometer, determination of the optics of minerals in transmitted light and of their chemistry by wet methods had already been carried to an advanced stage, but in the eight years up to 1934 the full effect of the application of X-ray diffraction to crystallography by von Laue, W. H. & W. L. Bragg, Jackson, Maugin, Pauling, W. H. Taylor, Warren, West and Wyckoff was felt, leading to a virtually complete classification of minerals on the basis of atomic structure (Bragg 1937). This has stood the test of time for all minerals save chrysotile, and has been fundamental to most other developments in mineralogy. Active fields in structure analysis today include the basis of ordering over octahedral and tetrahedral sites in silicates, and the factors controlling bond-lengths and angles; nuclear magnetic resonance (Bloch 1946; Purcell 1946) and electron spin resonance (Zavoisky 1945) are contributory techniques.

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 On Universal Binary Hermitian Forms

Integers ◽  
2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Scott Duke Kominers
Keyword(s):  
Quadratic Forms ◽  
Complete Classification ◽  
Hermitian Forms

AbstractEarnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying upon Ramanujan's list of universal quadratic forms and the Bhargava–Hanke 290-Theorem. Our methods bypass the

scholarly journals SIMPLIFIED NUMERICAL FORM OF UNIVERSAL FINITE TYPE INVARIANT OF GAUSS WORDS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350037
Author(s):  
TOMONORI FUKUNAGA ◽  
TAKAYUKI YAMAGUCHI ◽  
TAKAAKI YAMANOI
Keyword(s):  
Normal Form ◽  
Computer Program ◽  
Finite Type ◽  
Group Structure ◽  
Complete Classification ◽  
Finite Type Invariants ◽  
Partial Classification ◽  
Type Invariant

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

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.

scholarly journals Symmetry, topology and the maximum number of mutually pairwise-touching infinite cylinders: configuration classification

2017 ◽  
Vol 4 (1) ◽  
pp. 160729
Author(s):  
Peter V. Pikhitsa ◽  
Stanislaw Pikhitsa
Keyword(s):  
Cross Sections ◽  
Adjacency Matrix ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Classification ◽  
Three Dimensions ◽  
Theory Of Graphs ◽  
Three Dimensional Space

We provide a complete classification of possible configurations of mutually pairwise-touching infinite cylinders in Euclidian three-dimensional space. It turns out that there is a maximum number of such cylinders possible in three dimensions independently of the shape of the cylinder cross-sections. We give the explanation of the uniqueness of the non-trivial configuration of seven equal mutually touching round infinite cylinders found earlier. Some results obtained for the chirality matrix, which is equivalent to the Seidel adjacency matrix, may be found useful for the theory of graphs.

The decomposition of crystal families

1980 ◽  
Vol 88 (2) ◽  
pp. 245-263 ◽  
Author(s):  
J. D. Jarratt
Keyword(s):  
Dimensional Space ◽  
Canonical Decomposition ◽  
Space Groups ◽  
Higher Dimensional ◽  
Result Theorem ◽  
Free Parameters

In the course of their enumeration of all 4-dimensional space groups, Brown, Bülow, Neubüser, Wondratschek and Zassenhaus have introduced concepts appropriate to the study of n-dimensional crystallography for n ≥ 4 (see (2), (3)). One such concept is that of a crystal family. Families seem to be particularly useful as a framework within which to study higher dimensional crystallography, primarily because they determine a classification of all the standard crystallographic objects and overcome the traditional confusion over crystal systems (see (2); pp. 16–17, (9)). In this paper, techniques are developed for the determination of all rationally decomposable families in a given dimension from the indecomposable families of lower dimensions. These techniques place emphasis on three geometric invariants of families: the decomposition pattern; the canonical decomposition pattern; and the number of free parameters. This, it is felt, further reinforces their position as fundamental objects. The key result (Theorem 5.4) is:a family uniquely determines, and is uniquely determined by, the constituent families in the canonical decomposition.

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.

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