Fourth-rank tensors of the thirty-two crystal classes: multiplication tables

1984 ◽  
Vol 391 (1800) ◽  
pp. 149-179 ◽  
Keyword(s):  
Associative Algebra ◽  
Point Of View ◽  
Complex Numbers ◽  
Hypercomplex Numbers ◽  
Algebraic Point ◽  
Multiplication Tables

The fourth-rank tensors that embody the elastic or other properties of crystalline anisotropic substances can be partitioned into a number of sets in order that each shall acknowledge the symmetry of one or more of the crystal classes and moreover shall make up a closed linear associative algebra of hypercomplex numbers for the purpose of calculating the sums, products and inverses of its constituent tensors, to which end coordinate invariant expressions of the tensors are adopted. The calculations are simplified immensely, and ensuing physical analyses are well prepared for, once the structure of every algebra is unravelled completely in terms of a number of separate subalgebras isomorphic to familiar algebras such as the binary one of the complex numbers, the quaternary one of the 2x2 matrices and the octonary one of the complex quaternions. The fourth-rank tensors do not seem to have been submitted previously to the present algebraic point of view, and nor do those of any other rank: a parallel, but less intricate, development can be provided for the second-rank ones.

scholarly journals CONFLICTS AND FAIR TESTING

2006 ◽  
Vol 17 (04) ◽  
pp. 797-813 ◽  
Author(s):  
ROBI MALIK ◽  
DAVID STREADER ◽  
STEVE REEVES
Keyword(s):  
Discrete Event Systems ◽  
Process Algebra ◽  
Discrete Event ◽  
Denotational Semantics ◽  
Point Of View ◽  
Algebraic Point ◽  
Event Systems ◽  
Common Task ◽  
Testing Theory ◽  
Process Algebraic

This paper studies conflicts from a process-algebraic point of view and shows how they are related to the testing theory of fair testing. Conflicts have been introduced in the context of discrete event systems, where two concurrent systems are said to be in conflict if they can get trapped in a situation where they are waiting or running endlessly, forever unable to complete their common task. In order to analyse complex discrete event systems, conflict-preserving notions of refinement and equivalence are needed. This paper characterises an appropriate refinement, called the conflict preorder, and provides a denotational semantics for it. Its relationship to other known process preorders is explored, and it is shown to generalise the fair testing preorder in process-algebra for reasoning about conflicts in discrete event systems.

scholarly journals Localizability of unicycle mobiles robots: An algebraic point of view

2012 ◽  
Author(s):  
Hugues Sert ◽  
Wilfrid Perruquetti ◽  
Annemarie Kokosy ◽  
Xin Jin ◽  
Jorge Palos
Keyword(s):  
Point Of View ◽  
Algebraic Point

MODELS OF HOPFIELD-TYPE QUATERNION NEURAL NETWORKS AND THEIR ENERGY FUNCTIONS

2005 ◽  
Vol 15 (01n02) ◽  
pp. 129-135 ◽  
Author(s):  
MITSUO YOSHIDA ◽  
YASUAKI KUROE ◽  
TAKEHIRO MORI
Keyword(s):  
Neural Networks ◽  
Information Processing ◽  
Energy Function ◽  
Point Of View ◽  
Complex Numbers ◽  
Energy Functions ◽  
Fully Connected ◽  
Complex Valued

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

Multidimensional Numbers and the Genomatrices of Hydrogen Bonds

2010 ◽  
pp. 193-206
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Hydrogen Bonds ◽  
Genetic Code ◽  
Structural Features ◽  
Hyperbolic Type ◽  
Complex Numbers ◽  
Hypercomplex Numbers ◽  
Molecular Systems ◽  
Natural Systems ◽  
Double Numbers ◽  
Mathematical Properties

This chapter returns to the kind of numeric genetic matrices, which were considered in Chapter 4-6. This kind of genomatrices is not connected with the degeneracy of the genetic code directly, but it is related to some other structural features of the genetic code systems. The connection of the Kronecker families of such genomatrices with special categories of hypercomplex numbers and with their algebras is demonstrated. Hypercomplex numbers of these two categories are named “matrions of a hyperbolic type” and “matrions of a circular type.” These hypercomplex numbers are a generalization of complex numbers and double numbers. Mathematical properties of these additional categories of algebras are presented. A possible meaning and possible applications of these hypercomplex numbers are discussed. The investigation of these hyperbolic numbers in their connection with the parameters of molecular systems of the genetic code can be considered as a continuation of the Pythagorean approach to understanding natural systems.

MAGNETIC GROUP ON THE PSEUDOSPHERE

1992 ◽  
Vol 06 (21) ◽  
pp. 3525-3537 ◽  
Author(s):  
V. BARONE ◽  
V. PENNA ◽  
P. SODANO
Keyword(s):  
Magnetic Field ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Constant Magnetic Field ◽  
Coherent States ◽  
Compact Operators ◽  
Point Of View ◽  
Algebraic Point ◽  
Planar Limit ◽  
Discrete Part

The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.

scholarly journals Connectivity of pseudomanifold graphs from an algebraic point of view

2015 ◽  
Vol 353 (12) ◽  
pp. 1061-1065 ◽  
Author(s):  
Karim A. Adiprasito ◽  
Afshin Goodarzi ◽  
Matteo Varbaro
Keyword(s):  
Point Of View ◽  
Algebraic Point

A BERGMAN KERNEL IN ℝn, HOLOMORPHIC FUNCTIONS, AND RELATED NON-ASSOCIATIVE ALGEBRAS

2007 ◽  
Vol 18 (10) ◽  
pp. 1113-1132
Author(s):  
ZBIGNIEW BŁOCKI
Keyword(s):  
Dirichlet Problem ◽  
Bergman Kernel ◽  
Holomorphic Functions ◽  
Point Of View ◽  
Complex Numbers ◽  
Associative Algebras ◽  
Natural Way

For domains in ℝn we construct the Bergman kernel on the diagonal using solutions of the Dirichlet problem. Starting from this, in a natural way we obtain an algebra 𝔸n of dimension n(n - 1)/2 + 1 over ℝ and a class of holomorphic functions valued in 𝔸n. Of course 𝔸2 is the field of complex numbers, and it turns out that 𝔸3 is the algebra of quaternions, whereas for n ≥ 4, 𝔸n is non-associative. Holomorphic functions can be written as f + ω, where f is a (real-valued) function and ω a differential 2-form such that d* ω = df and dω = 0. We investigate the main properties of the obtained objects, especially from the analytic point of view.

A lattice of extension rings for a commutative ring

1978 ◽  
Vol 84 (2) ◽  
pp. 225-234 ◽  
Author(s):  
D. Kirby ◽  
M. R. Adranghi
Keyword(s):  
Point Of View ◽  
Local Rings ◽  
Algebraic Point ◽  
Multiple Points ◽  
One Dimensional ◽  
Maximal Ideals ◽  
Blowing Up ◽  
Abstract Case ◽  
Algebroid Curve

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

Freeness in classes without equality

1999 ◽  
Vol 64 (3) ◽  
pp. 1159-1194 ◽  
Author(s):  
Raimon Elgueta
Keyword(s):  
Structural Properties ◽  
Universal Algebra ◽  
Reduction Process ◽  
Point Of View ◽  
Basic Notion ◽  
Algebraic Point ◽  
Generic Problem

AbstractThis paper is a continuation of [27], where we provide the background and the basic tools for studying the structural properties of classes of models over languages without equality. In the context of such languages, it is natural to make distinction between two kinds of classes, the so-called abstruct classes, which correspond to those closed under isomorphic copies in the presence of equality, and the reduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described in [27] is to investigate under what conditions this reduction process does not alter the metatheory of a class.Here we focus our attention on a concrete aspect of this generic problem that we import from universal algebra, namely the existence and description of free models. As in [27], we can find here again the basic notion of protoalgebraicity, which was originally introduced in [7] as the weakest condition to guarantee that the reduction process behaves reasonably well from an algebraic point of view. Our concern, however, takes us to handle a further notion, that of semialgebraicity, which corresponds to the notion of equivalential logic of [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process.

scholarly journals PARAGRASSMANN ANALYSIS AND QUANTUM GROUPS

1992 ◽  
Vol 07 (23) ◽  
pp. 2129-2141 ◽  
Author(s):  
A. T. FILIPPOV ◽  
A. P. ISAEV ◽  
A. B. KURDIKOV
Keyword(s):  
Differential Operator ◽  
Quantum Groups ◽  
Natural Generalization ◽  
Point Of View ◽  
Algebraic Point ◽  
Root Of Unity ◽  
Deformation Parameters ◽  
One And Many

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being root of unity are established.

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