Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators of rank and order two.

On an Extremal Problem concerning Bernstein Operators

* The second author is supported by the Alexander-von-Humboldt Foundation. He is on leave from: Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China.

Self Evolving Character Recognition Using Genetic Operators

In this paper, a novel approach for character recognition has been presented with the help of genetic operators which have evolved from biological genetics and help us to achieve highly accurate results. A genetic algorithm approach has been described in which the biological haploid chromosomes have been implemented using a single row bit pattern of 315 values which have been operated upon by various genetic operators. A set of characters are taken as an initial population from which various new generations of characters are generated with the help of selection, crossover and mutation. Variations of population of characters are evolved from which the fittest solution is found by subjecting the various populations to a new fitness function developed. The methodology works and reduces the dissimilarity coefficient found by the fitness function between the character to be recognized and members of the populations and on reaching threshold limit of the error found from dissimilarity, it recognizes the character. As the new population is being generated from the older population, traits are passed on from one generation to another. We present a methodology with the help of which we are able to achieve highly efficient character recognition.

Weighted Theorems on Fractional Integrals in the Generalized Hölder Spaces via Indices mω and Mω

Mathematics Subject Classification: 26A16, 26A33, 46E15.

Fractional Powers of Almost Non-Negative Operators

Mathematics Subject Classification: Primary 47A60, 47D06.

Rough Maximal Oscillatory Singular Integral Operators

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25

An Lp − Lq - Version of Morgan's Theorem Associated with Partial Differential Operators

2000 Mathematics Subject Classification: 42B10, 43A32.

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35

On Integral Means for Fractional Calculus Operators of Multivalent Functions

2000 Mathematics Subject Classification: Primary 30C45, Secondary 26A33, 30C80

Commutants of the Dunkl Operators in C(R)

2000 Mathematics Subject Classification: 44A35; 42A75; 47A16, 47L10, 47L80

Generalized Sobolev Spaces of Exponential Type Associated with the Dunkl Operators

2000 Mathematics Subject Classification: 35E45

q-Heat Operator and q-Poisson’s Operator

2000 Mathematics Subject Classification: 33D15, 33D90, 39A13

Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30

Inclusion Properties for Certain Classes of Analytic Functions Involving a Family of Fractional Integral Operators

2000 Math. Subject Classification: 30C45

Inversion Formulas for the q-Riemann-Liouville and q-Weyl Transforms Using Wavelets

Mathematics Subject Classification: 42A38, 42C40, 33D15, 33D60