We present a survey of results on the problem coloring 3-chromatic graphs with the least number of colors in polynomial time. We emphasize on the semi-definite programming based breakthroughs that have occurred towards this problem in the recent past.

We consider a monic polynomial of even degree with symbolic coefficients. We give a method for obtaining an expression in the coefficients (regarded as parameters) that is a lower bound on the value of the polynomial, or in other words a lower bound on the minimum of the polynomial. The main advantage of accepting a bound on the minimum, in contrast to an expression for the exact minimum, is th...

The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjectur...

In this paper we suggest a hybrid method for computing the smallest eigenvalue of a symmetric and positive definite Toeplitz matrix which takes advantage of two types of methods, Newton’s method for the characteristic polynomial and projection methods based on rational interpolation of the secular equation.

In this paper we describe the retrieval of some real fish properties to establish them on a virtual fish. These include swimming motions. We introduce a special function which extracts a definite description of a fish. We use polynomial interpolation and spline to estimate fish body changes (moves), then reproduce such on a display.

We describe an added-mass partitioned (AMP) algorithm for solving fluid-structure interaction (FSI) problems involving inviscid compressible fluids interacting with nonlinear solids that undergo large rotations and displacements. The computational approach is a mixed Eulerian-Lagrangian scheme that makes use of deforming composite grids (DCG) to treat large changes in the geometry in an accurat...

Positive definite matrices structured by orthogonal polynomial systems allow a Cholesky type decomposition of their inverse matrices in O(n2) steps. The algorithm presented in this paper uses the three-term recursion coefficients and the mixed moments of the involved polynomials.

We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to extend the renormalization theory of unimodal maps to multimodal maps.

Let R be a principal ideal domain with quotient field F . An R-lattice is a free R-module of finite rank spanning an inner product space over F . The classification problem asks for a reasonably effective set of criteria to determine when two given R-lattices are isometric; that is, when there is an inner-product preserving isomorphism carrying one lattice onto the other. In this paper R is the...

Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν . Our implementation uses the split-radix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log 2 n real arithmetic operations without explicit us...