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.

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 The Integrated Photometry of Globular Clusters in the Vilnius Photometric System

1980 ◽  
Vol 85 ◽  
pp. 483-484
Author(s):  
K. Zdanavičius ◽  
V. Straižys
Keyword(s):  
Globular Clusters ◽  
Color Index ◽  
Photometric System ◽  
Intrinsic Values ◽  
Average Color ◽  
Color Excess ◽  
Free Parameters ◽  
Range Of Variation

Thirty-three globular clusters of our Galaxy were observed with the filters of the Vilnius photometric system UPXYZVS with 3450, 3740, 4050, 4660, 5160, 5440, 6550 å filters (Straižys 1977). For the classification of clusters in metallicities the reddening-free parameters QUYV QPYV and QXYV can be used. In Figure 1 these Q parameters, having a range of variation of the order of 0.4, are plotted against metallicity values from Kukarkin (1974). The parameter QPYZ has an even larger range of variation (of the order of 0.6). For determination of color excesses of clusters every color index can be used if its intrinsic values for a given metallicity defined by quantities Q are known (Figure 2). Average color excesses determined from the diagrams QUYV, (Y-V)o; QPYV, (Y-V)o and QXYV (Y-V)o and transformed to EB-V in Figure 3 are compared with color excesses from Kukarkin (1974). To summarize, the Vilnius system presents a number of metallicity sensitive, reddening-free parameters which can be used for [Fe/H] and color-excess determinations of globular clusters.

IMPLEMENTATION OF SUPPORT VECTOR MACHINES FOR CLASSIFICATION OF CLINICAL DATASETS

2010 ◽  
Vol 07 (04) ◽  
pp. 347-356
Author(s):  
E. SIVASANKAR ◽  
R. S. RAJESH
Keyword(s):  
Feature Extraction ◽  
Dimensional Space ◽  
Principal Component ◽  
Kernel Functions ◽  
Support Vector ◽  
Optimal Hyperplane ◽  
Higher Dimensional ◽  
Vector Machines ◽  
Map Data

In this paper, Principal Component Analysis is used for feature extraction, and a statistical learning based Support Vector Machine is designed for functional classification of clinical data. Appendicitis data collected from BHEL Hospital, Trichy is taken and classified under three classes. Feature extraction transforms the data in the high-dimensional space to a space of fewer dimensions. The classification is done by constructing an optimal hyperplane that separates the members from the nonmembers of the class. For linearly nonseparable data, Kernel functions are used to map data to a higher dimensional space and there the optimal hyperplane is found. This paper works with different SVMs based on radial basis and polynomial kernels, and their performances are compared.

scholarly journals Adinkra (in)equivalence from Coxeter group representations: A case study

2014 ◽  
Vol 29 (06) ◽  
pp. 1450029 ◽  
Author(s):  
Isaac Chappell ◽  
S. James Gates ◽  
T. Hübsch
Keyword(s):  
Dimensional Space ◽  
Solution Space ◽  
Group Representations ◽  
Dimensional Solution ◽  
Signed Permutation ◽  
Higher Dimensional ◽  
Dimensional Space Time ◽  
Definition Of

Using a Mathematica TM code, we present a straightforward numerical analysis of the 384-dimensional solution space of signed permutation 4×4 matrices, which in sets of four, provide representations of the 𝒢ℛ(4, 4) algebra, closely related to the 𝒩 = 1 (simple) supersymmetry algebra in four-dimensional space–time. Following after ideas discussed in previous papers about automorphisms and classification of adinkras and corresponding supermultiplets, we make a new and alternative proposal to use equivalence classes of the (unsigned) permutation group S4 to define distinct representations of higher-dimensional spin bundles within the context of adinkras. For this purpose, the definition of a dual operator akin to the well-known Hodge star is found to partition the space of these 𝒢ℛ(4, 4) representations into three suggestive classes.

scholarly journals Computing the Discrete Compactness of Orthogonal Pseudo-Polytopes via Their D-EVM Representation

2010 ◽  
Vol 2010 ◽  
pp. 1-28 ◽  
Author(s):  
Ricardo Pérez-Aguila
Keyword(s):  
Dimensional Space ◽  
Efficient Algorithms ◽  
Video Sequences ◽  
Higher Dimensional ◽  
Discrete Compactness ◽  
Representation Scheme ◽  
2D And 3D ◽  
Orthogonal Polytopes

This work is devoted to present a methodology for the computation of Discrete Compactness in -dimensional orthogonal pseudo-polytopes. The proposed procedures take in account compactness' definitions originally presented for the 2D and 3D cases and extend them directly for considering the D case. There are introduced efficient algorithms for computing discrete compactness which are based on an orthogonal polytopes representation scheme known as the Extreme Vertices Model in the -Dimensional Space (D-EVM). It will be shown the potential of the application of Discrete Compactness in higher-dimensional contexts by applying it, through EVM-based algorithms, in the classification of video sequences, associated to the monitoring of a volcano's activity, which are expressed as 4D orthogonal polytopes in the space-color-time geometry.

Learning Classification in the Olfactory System of Insects

2004 ◽  
Vol 16 (8) ◽  
pp. 1601-1640 ◽  
Author(s):  
Ramón Huerta ◽  
Thomas Nowotny ◽  
Marta García-Sanchez ◽  
H. D. I. Abarbanel ◽  
M. I. Rabinovich
Keyword(s):  
Olfactory System ◽  
Mushroom Body ◽  
Hebbian Learning ◽  
Dimensional Space ◽  
Control Mechanisms ◽  
Kenyon Cells ◽  
Higher Dimensional ◽  
Odor Classification ◽  
Range Of Values

We propose a theoretical framework for odor classification in the olfactory system of insects. The classification task is accomplished in two steps. The first is a transformation from the antennal lobe to the intrinsic Kenyon cells in the mushroom body. This transformation into a higher-dimensional space is an injective function and can be implemented without any type of learning at the synaptic connections. In the second step, the encoded odors in the intrinsic Kenyon cells are linearly classified in the mushroom body lobes. The neurons that perform this linear classification are equivalent to hyperplanes whose connections are tuned by local Hebbian learning and by competition due to mutual inhibition. We calculate the range of values of activity and size fo the network required to achieve efficient classification within this scheme in insect olfaction. We are able to demonstrate that biologically plausible control mechanisms can accomplish efficient classification of odors.

scholarly journals Classification of 5-dimentional subalgebras for 6-dimentional nilpotent Lie algebras

2020 ◽  
Vol 56 (2) ◽  
pp. 175-188
Author(s):  
U. L. Shtukar
Keyword(s):  
Lie Algebras ◽  
Standard Method ◽  
Classical Problem ◽  
Vector Spaces ◽  
Nilpotent Lie Algebras ◽  
New Approach ◽  
Zero Characteristic ◽  
Higher Dimensional

In this paper, we consider the classical problem of the classification of subalgebras of small dimensional Lie algebras. We found all 5-dimentional subalgebras of 6-dimentional nilpotent Lie algebras under the field with the zero characteristic. As is known, up to isomorphism all 6-dimensional nilpotent Lie algebras (their number is 32) were received by V. V. Morosov. However, the standard method based on the Campbell – Hausdorf formula is not effective for the determination of subalgebras of Lie 5- or higher dimensional algebras. In our research, we use a new approach to the solution of the problem of the determination of 5-dimensional subalgeras of indicated 6-dimensional nilpotent Lie algerbas, namely, the application of canonical bases for subspaces of vector spaces.

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