Towards a geometrical interpretation of rainbow geometries

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.

Download Full-text

Algorithms for Stochastic Games with Geometrical Interpretation

Management Science ◽

10.1287/mnsc.15.7.399 ◽

1969 ◽

Vol 15 (7) ◽

pp. 399-415 ◽

Cited By ~ 41

Author(s):

M. A. Pollatschek ◽

B. Avi-Itzhak

Keyword(s):

Stochastic Games ◽

Geometrical Interpretation

Download Full-text

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

International Journal of Modern Physics A ◽

10.1142/s0217751x05025127 ◽

2005 ◽

Vol 20 (20n21) ◽

pp. 4797-4819 ◽

Cited By ~ 3

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.

Download Full-text

III. Octonions

Proceedings of the Royal Society of London ◽

10.1098/rspl.1895.0071 ◽

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.

Download Full-text