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...

In this work, we propose an asymptotic preserving scheme for a non-linear kinetic reactiontransport equation, in the regime of sharp interface. With a non-linear reaction term of KPP-type, a phenomenon of front propagation has been proved in [9]. This behavior can be highlighted by considering a suitable hyperbolic limit of the kinetic equation, using a Hopf-Cole transform. It has been proved i...

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...

The spectral properties of Jacobi and periodic Jacobi matrices are analyzed and algorithms for the construction of Jacobi and periodic Jacobi matrices with prescribed spectra are presented. Numerical evidence demonstrates that these algorithms are of practical utility. These algorithms have been used in studies of the periodic Toda lattice, and might also be used in studies of inverse eigenvalu...

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative...

We develop a discrete analogue of the Hamilton–Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton–Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton– Jacobi equation is discrete only in time, and is shown to recover the Hamilton–Jacobi equation in the continuous-time...

The relation between two Morse functions defined on a common domain can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces have shown to be useful in various applications. Unfortunately, in practice functions often contain noise and...

This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...