Intrinsic Functions on Semi-Simple Algebras

1967 ◽  
Vol 19 ◽  
pp. 590-598 ◽  
Author(s):  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Ground Field ◽  
The Real ◽  
Simple Algebras

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on a linear associative algebra , with identity, over the real field R or the complex field C. In this paper we shall consider a semi-simple algebra = ⊕ … ⊕ over R or C with simple components . Let G be the group of all automorphisms or anti-automorphisms of which leave the ground field elementwise invariant, and let H be the subgroup of G such that Ω = (i = 1, 2, … , t) for each Ω in H.

Classes of Functions on Algebras

1966 ◽  
Vol 18 ◽  
pp. 139-146 ◽  
Author(s):  
C. G. Cullen ◽  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Finite Dimensional

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.

Intrinsic Functions on Matrices of Real Quaternions

1963 ◽  
Vol 15 ◽  
pp. 456-466 ◽  
Author(s):  
C. G. Cullen
Keyword(s):  
Division Algebra ◽  
Matrix Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Real Quaternions ◽  
Algebra Over A Field

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

scholarly journals The algebra of parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part

2015 ◽  
Vol 159 (2) ◽  
pp. 219-237 ◽  
Author(s):  
CHARLES BOUBEL
Keyword(s):  
Riemannian Manifold ◽  
Associative Algebra ◽  
Ricci Curvature ◽  
Holonomy Group ◽  
Simple Algebra ◽  
Riemannian Metric ◽  
Generic Type ◽  
Simple Part ◽  
Different Types ◽  
Simple Algebras

AbstractOn a (pseudo-)Riemannian manifold (${\mathcal M}$, g), some fields of endomorphisms i.e. sections of End(T${\mathcal M}$) may be parallel for g. They form an associative algebra $\mathfrak e$, which is also the commutant of the holonomy group of g. As any associative algebra, $\mathfrak e$ is the sum of its radical and of a semi-simple algebra $\mathfrak s$. Here we study $\mathfrak s$: it may be of eight different types, including the generic type $\mathfrak s$ = ${\mathbb R}$ Id, and the Kähler and hyperkähler types $\mathfrak s$ ≃ ${\mathbb C}$ and $\mathfrak s$ ≃ ${\mathbb H}$. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrize it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.

Normed Right Alternative Algebras Over the Reals

1972 ◽  
Vol 24 (6) ◽  
pp. 1183-1186 ◽  
Author(s):  
José I. Nieto
Keyword(s):  
Division Algebra ◽  
Real Field ◽  
Division Algebras ◽  
Classical Theorem ◽  
Complex Field ◽  
The Real ◽  
Alternative Algebras ◽  
Finite Dimensional

One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.

Order Characterization of the Complex Field

1978 ◽  
Vol 21 (3) ◽  
pp. 313-318
Author(s):  
Lino Gutierrez Novoa
Keyword(s):  
Upper Bound ◽  
Ordered Set ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Ordered Field ◽  
The One ◽  
Definition Of ◽  
Real Number Field

It is well known that the real number field can be characterized as an ordered field satisfied the “least upper bound” property.Using the idea of n -ordered set, introduced in [3], and generalizing the notion of l.u.b. in a suitable way, it is possible to give a similar categorical definition of the complex field.With these extended meanings, the main theorem of this paper (Theorem 7 in the text) is stated almost identically to the one for the real field. Any directly two-ordered field, in which the "supremum property" holds, is isomorphic to the complex field.

scholarly journals The characterization of generalized Jordan centralizers on algebras

2017 ◽  
Vol 35 (3) ◽  
pp. 225 ◽  
Author(s):  
Quanyuan Chen ◽  
Xiaochun Fang ◽  
Changjing Li
Keyword(s):  
Continuous Mapping ◽  
Additive Mapping ◽  
Real Field ◽  
Complex Field ◽  
Von Neumann ◽  
The Real

In this paper, it is shown that if $\mathcal {A}$ is a CSL subalgebra of a von Neumann algebr and $\phi$ is a continuous mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})-(m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I))\in \mathbb{F}I $ for any $A\in \mathcal {A}$, where $\mathbb{F}$ is the real field or the complex field, then $\phi$ is a centralizer. It is also shown that if $\phi$ is an additive mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})=m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I) $for any $A\in\mathcal{A}$, then $\phi$ is a centralizer.

Stochastic simulation of the spatio-temporal field of the average daily heat index in Southern Russia

2020 ◽  
Vol 82 ◽  
pp. 149-160
Author(s):  
N Kargapolova
Keyword(s):  
Heat Waves ◽  
Numerical Models ◽  
Heat Index ◽  
Real Field ◽  
Gaussian Field ◽  
The Real ◽  
Southern Russia ◽  
Weather Stations ◽  
Spatio Temporal ◽  
Temporal Field

Numerical models of the heat index time series and spatio-temporal fields can be used for a variety of purposes, from the study of the dynamics of heat waves to projections of the influence of future climate on humans. To conduct these studies one must have efficient numerical models that successfully reproduce key features of the real weather processes. In this study, 2 numerical stochastic models of the spatio-temporal non-Gaussian field of the average daily heat index (ADHI) are considered. The field is simulated on an irregular grid determined by the location of weather stations. The first model is based on the method of the inverse distribution function. The second model is constructed using the normalization method. Real data collected at weather stations located in southern Russia are used to both determine the input parameters and to verify the proposed models. It is shown that the first model reproduces the properties of the real field of the ADHI more precisely compared to the second one, but the numerical implementation of the first model is significantly more time consuming. In the future, it is intended to transform the models presented to a numerical model of the conditional spatio-temporal field of the ADHI defined on a dense spatio-temporal grid and to use the model constructed for the stochastic forecasting of the heat index.

scholarly journals Role Playing In the Field Of Education

2018 ◽  
pp. 1-13
Author(s):  
Yuan Lo
Keyword(s):  
Real Life ◽  
Role Playing ◽  
Real Field ◽  
Real Situation ◽  
The Real ◽  
Artificial Environment ◽  
Life Education

The character and status are presented together. Others have to play the role. The real situation is to be presented in a simple way. It can be understood how to adapt yourself to the real field. The role of the actress is to be revealed. Students get real-life education in the artificial environment. Performances of speech and expression are improved.

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