We use spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.

A computational method to obtain simplest normal forms, which we refer as hypernormal forms, for vector fields having a linear degeneracy corresponding to a double zero eigenvalue with geometric multiplicity one, is presented. The procedure of simplifying the classical normal forms requires some hypothesis on the nonlinear terms of the vector field. Once the hypothesis are established, differen...

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the form pq or p2, there are ...

The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study algebraic and geometric notions related to bilinear forms (especially the concept of orthogonality in all its manifestations: orthogonal vectors, orthogonal subspaces, and orthogonal bases). Section 1 defines a bilinear form on a vector space a...

In recent years, there has been greater attempt to find numerical solutions of differential equations using wavelet's methods. The following method is based on vector forms of Haar-wavelet functions. In this paper, we will introduce one dimensional Haar-wavelet functions and the Haar-wavelet operational matrices of the fractional order integration. Also the Haar-wavelet operational matrices of ...

Both vector 1-forms (tensor elds of type (1,1)) and Lie brackets map (tangent) vectors to the same. The former do so in a manner that is linear over the ring of functions; the latter are bilinear, but only over the constants. By composing one Lie bracket and m vector 1-forms (we consider 2 m 4) in all possible ways many maps of vector elds are obtained, bilinear over the constants. We study lin...

An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and in CR geometry. 1.1. Introduction. Vector-valued forms play a key role in the study of higher codimensional geometries. For example, they occur naturally in t...

The purpose of this paper is to introduce differential forms to readers without a background in differential geometry. With the presumption that the reader is experienced in advanced calculus, we provide motivation for learning some of the most important concepts in differential geometry by re-observing the theorems of vector calculus.