scholarly journals On two linear vector spaces associated with a vector in an Ln

1940 ◽  
Vol 6 (3) ◽  
pp. 172-175
Author(s):  
Yung-Chow Wong
Keyword(s):  
Vector Field ◽  
Vector Spaces ◽  
Connected Space ◽  
Covariant Differentiation ◽  
Linear Vector ◽  
The Matrix ◽  
Image Position ◽  
Affinely Connected Space

Let υλ be a vector (i.e. a vector field) in an affinely connected space Ln, ∇k the symbol of covariant differentiation, and r the rank of the matrix ‖∇k υλ‖ then there exist two sets of n − r independent vectors and which satisfy respectively the equations

Characterization of integral curves of a linear vector field in Lorentz 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650073
Author(s):  
Tunahan Turhan ◽  
Nihat Ayyıldız
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Symmetric Matrix ◽  
Order System ◽  
Linear Vector ◽  
Integral Curves ◽  
The Matrix ◽  
Vector Formula

We propose a detail study of integral curves or flow lines of a linear vector field in Lorentz [Formula: see text]-space. We construct the matrix [Formula: see text] depending on the causal characters of the vector [Formula: see text] by analyzing the non-zero solutions of the equation [Formula: see text], [Formula: see text] in such a space, where [Formula: see text] is the skew-symmetric matrix corresponding to the linear map [Formula: see text]. Considering the structure of a linear vector field, we obtain the linear first-order system of differential equations. The solutions of this system of equations give rise to integral curves of linear vector fields from which we provide a classification of such curves.

scholarly journals On a type of Sasakian space

1972 ◽  
Vol 13 (4) ◽  
pp. 508-510 ◽  
Author(s):  
M. C. Chaki ◽  
D. Ghosh
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Positive Definite ◽  
Killing Vector Field ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Image Position ◽  
Conformally Recurrent

A Sasakian space [1]Mn (n = 2m + 1) is a Riemannian n-space with a positive definite metric tensor gij and a unit Killing vector field η which satisfies where the comma denotes covariant differentiation with respect to the metic tensor. In a recent paper [2] M. C. Chaki and A. N. Roy Chowdhury studied conformally recurrent spaces of second order, or briefly conformally 2-recurrent spaces, that is, non-flat Riemannian spaces Vn (n > 3) defined by where is the conformal curvature tensor: and alm is a tensor not identically zero.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Can the Number of D Electrons in Transition Metals be Measured from White Lines?

1997 ◽  
Vol 3 (S2) ◽  
pp. 957-958 ◽  
Author(s):  
P. Rez
Keyword(s):  
Transition Metals ◽  
Energy Loss ◽  
Electron Energy ◽  
Transition Elements ◽  
White Line ◽  
Line Intensities ◽  
Continuum Region ◽  
The Matrix ◽  
Image Position ◽  
Electron Energy Loss

Sharp peaks at threshold are a prominent feature of the L23 electron energy loss edges of both first and second row transition elements. Their intensity decreases monotonically as the atomic number increases across the period. It would therefore seem likely that the number of d electrons at a transition metal atom site and any variation with alloying could be measured from the L23 electron energy loss spectrum. Pearson measured the white line intensities for a series of both 3d and 4d transition metals. He normalised the white line intensity to the intensity in a continuum region 50eV wide starting 50eV above threshold. When this normalised intensity was plotted against the number of d electrons assumed for each elements he obtained a monotonie but non linear variation. The energy loss spectrum is given bywhich is a product of p<,the density of d states, and the matrix elements for transitions between 2p and d states.

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

A Purity Criterion for Pairs of Linear Transformations

1974 ◽  
Vol 26 (3) ◽  
pp. 734-745 ◽  
Author(s):  
Uri Fixman ◽  
Frank A. Zorzitto
Keyword(s):  
Perturbation Methods ◽  
Eigenvalue Problems ◽  
Vector Spaces ◽  
Infinite Dimension ◽  
Linear Transformations ◽  
Complex Vector ◽  
Algebraic Investigation ◽  
Image Position

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

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