Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property

1989 ◽  
Vol 19 (11) ◽  
pp. 1299-1314 ◽  
Author(s):  
Christian Schindler
Keyword(s):  
Geometrical Interpretation ◽  
Lebesgue Decomposition ◽  
Decomposition Property

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

Geometrical interpretation of the multi-point flux approximation L-method

2009 ◽  
Vol 60 (11) ◽  
pp. 1173-1199 ◽  
Author(s):  
Yufei Cao ◽  
Rainer Helmig ◽  
Barbara I. Wohlmuth
Keyword(s):  
Geometrical Interpretation ◽  
Flux Approximation

The deformation multiplicity of a map germ with respect to a Boardman symbol

2001 ◽  
Vol 131 (5) ◽  
pp. 1003-1022 ◽  
Author(s):  
C. Bivià-Ausina ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Geometrical Interpretation ◽  
Algebraic Multiplicity ◽  
Homogeneous Map

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

Algorithms for Stochastic Games with Geometrical Interpretation

1969 ◽  
Vol 15 (7) ◽  
pp. 399-415 ◽  
Author(s):  
M. A. Pollatschek ◽  
B. Avi-Itzhak
Keyword(s):  
Stochastic Games ◽  
Geometrical Interpretation

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

scholarly journals III. Octonions

1896 ◽  
Vol 59 (353-358) ◽  
pp. 169-181
Keyword(s):  
Linear Function ◽  
Geometrical Interpretation

Octonions is a name adopted for various reasons in place of Clifford’s Bi- quaternions . Formal quaternions are symbols which formally obey all the laws of the quaternion symbols , q (quaternion), x (scalar), ρ (vector) ϕ (linear function in both its ordinary meanings), ϕ' (conjugate of ϕ ), i, j, k, ζ , K q , S q , T q , U q , V q . Octonions are in this sense formal quaternions. Each octonion symbol, however, requires for its specification just double the number of scalars required for the corresponding quaternion symbol. Thus, of every quaternion formula involving the above symbols there is a geometrical interpretation more general than the ordinary quaternion one, an octonion interpretation.

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