This paper focuses on some main distinctions that are necessary to analyze mathematical knowledge from a learning point of view and explain how many students come up against difficulties at each level of the curriculum. From a learning point of view, visualization cannot be used as an immediate and obvious support for understanding, even though it is the only relevant cognitive modality in math...

In the present paper certain operational formulae involving RiemannLiouville and Kober fractional q-integral operators for an analytic function are derived. The usefulness of the main results are exhibited by considering some examples which also yield q-extensions of some known results for ordinary hypergeometric functions of one and more variables.

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. For q = 1 it is known that these hypergeometric functions are related to zonal spherical polynomials for GL(N,C)/U(N) symmetric space. We show that multivariate hypergeometric functions are tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of...

We establish a large n complete asymptotic expansion for q-Laguerre polynomials and a complete asymptotic expansion for a q-Bessel function of large argument. These expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems. We also give a new derivation of an asymptotic formula due to Littlewood (1907).

This work addresses the unification of some basic functions and thresholds used in nonparametric estimation of signals by shrinkage in the wavelet domain. The Soft and Hard thresholding functions are presented as degenerate smooth sigmoid based shrinkage functions. The shrinkage achieved by this new family of sigmoid based functions is then shown to be equivalent to a regularisation of wavelet ...

It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are fuzzy value fuzzy set, i...

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato [Phys. Rev. E 57, 495 (1998)] introduced a stochastic optimization techniq...

An essential component of the human diet, L-tryptophan is critical in a number of metabolic functions and has been widely used in numerous research and clinical trials. This review provides a brief overview of the role of L-tryptophan in protein synthesis and a number of other metabolic functions. With emphasis on L-tryptophan's role in synthesis of brain serotonin, details are provided on the ...

Geometric moment invariants (GMIs) have been widely used as basic tool in shape analysis and information retrieval. Their structure and characteristics determine efficiency and effectiveness. Two fundamental building blocks or generating functions (GFs) for invariants are discovered, which are dot product and vector product of point vectors in Euclidean space. The primitive invariants (PIs) can...