Geometrical Illustrations of Algebraic Identities

Geometrical analysis of the algebraic cluster model

Three-body clusters are studied in the algebraic cluster model. Particular attention is paid to the case of three identical particles. It is shown in a geometrical analysis that the harmonic oscillator, the deformed oscillator and the oblate symmetric top are contained as special solutions. An application of the oblate top limit to the nucleus 12 C suggests that the low-lying spectrum can be de...

ILLUSTRATIONS FROM GEOMETRY OF THE THEORY OF ALGEBRAIC QUATERNIONS By

Algebraic geometrical methods for hierarchical learning machines

Hierarchical learning machines such as layered perceptrons, radial basis functions, Gaussian mixtures are non-identifiable learning machines, whose Fisher information matrices are not positive definite. This fact shows that conventional statistical asymptotic theory cannot be applied to neural network learning theory, for example either the Bayesian a posteriori probability distribution does no...

Algebraic-Geometrical Formulation of Two-Dimensional Quantum Gravity

We find a volume form on moduli space of double-punctured Riemann surfaces whose integral satisfies the Painlevé I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivati...

Cocycle Deformations of Algebraic Identities and R-matrices

For an arbitrary identity L = R between compositions of maps L and R on tensors of vector spaces V , a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit examples with calculations. Relations to the Kauffman bracket and knot in...