We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is p...

The Jacobi algorithm for eigenvalue calculation of symmetric matrices can be performed with a CORDIC algorithm as its basic module. Recently, a simpliied Jacobi algorithm, by employing approximate rotations based on CORDIC rotations, was proposed. It fully exploits the binary data structure and reduces the overall computational cost signiicantly. In this paper an error analysis of the approxima...

Jacobi sets have been identified as significant in multi-field topological analysis, but are defined in the domain of the data rather than in the Reeb Space. This distinction is significant, as exploiting multi-field topology actually depends on the projection of the Jacobi set into the Reeb Space, and the details of its internal structure. We therefore introduce the Jacobi Structure of a Reeb ...

We analyze the relationship between the covering of the Jacobi group and the squeezed states. We attach some nonclassical states to the Jacobi group. The matrix elements of the Jacobi group are presented.

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new...

In this paper we give extensions of the mass formula for biweight enumerators and the Jacobi weight enumerators of binary self-dual codes and binary doubly even self-dual codes. For binary doubly even self-dual codes, our formula is expressed in terms of the root system E8 embedded in C4 for biweight enumerators, while the root system D4 is employed for Jacobi weight enumerators. For self-dual ...

Among all lattice reduction algorithms, the LLL algorithm is the first and perhaps the most famous polynomial time algorithm, and it is widely used in many applications. In 2012, S. Qiao [24] introduced another algorithm, the Jacobi method, for lattice basis reduction. S. Qiao and Z. Tian [25] improved the Jacobi method further to be polynomial time but only produces a Quasi-Reduced basis. In t...

We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [12, 17, 19], to all Delannoy numbers and certain Jacobi polynomials. Another specialization provides a weighted lattice path enumeration model for shifte...

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 ...

Figure 1. Convergence factors for various discretizations of the Laplacian (Case 1). The lines show smoothing factors of optimally weighted point Jacobi (solid line), line Jacobi (dashes), and unweighted point RB (alternating dashes). The symbols correspond to numerical calculations using V(1,0) cycles with point Jacobi (), line Jacobi (+), and point RB (2), and also two-level (1,0) cycles usin...