多元整系数多项式因式分解的一种理论和算法

本文将ｔ（ｔ是大于２的整数）元整系数多项式看成为系数为ｔ－２元整系数多项式的二元多项式，建立了多元整系数多项式因式分解的一种新理论，进而得到了分解多元整系数多项式的一个有力的算法。     

A class of polynomial variable metric algorithms for linear optimization

In the paper, the behaviour of interior point algorithms is analyzed by using a variable metric method approach. A class of polynomial variable metric algorithms is given achieving O ((n/β)L) iterations for solving a canonical form linear optimization problem with respect to a wide class of Riemannian metrics, wheren is the number of dimensions and β a fixed value. It is shown that the vector fields of several interior point algorithms for linear optimization is the negative Riemannian gradient vector field of a linear a potential or a logarithmic barrier function for suitable Riemannian metrics. Research Partially supported by the Hungarian National Research Foundation, Grant Nos. OTKA-T016413 and OTKA-2116.     

Polynomial expansion of basis sets: Alternatives to fully optimized exponents

Alternatives based on polynomial expansions of gaussian basis set exponents are introduced and evaluated. The formulas presented here outperform methods based upon the even-tempered formula or combinations of it. They closely match the performance of other methods based upon larger polynomial expansions of the logarithm of the exponents using the same or one less parameter per orbital angular symmetry.     

Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields

We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.     

An application of the generalized alternating polynomials to the numerical solution of Fredholm integral equations

Generalized alternating polynomials have been introduced by the author earlier. In the present paper their indirect analogue is constructed for numerical solution of the Fredholm linear integral equations. Although the proposed method is a particular case of the general projection scheme, its valuable feature is the presence of a sequence of parameters, which, for sufficiently smooth kernels and inhomogeneous terms, serves as an indicator of the quality of approximation.     

A computer program for searching the best model for describing different experimental systems

An algorithm for searching the best polynomial analytical function for describing different experimental systems is presented. It is based

1. (1)on the generation of all possible analytical functions of a given order, with a given number of terms and with a given number of independent variables, and

2. (2)on the calculation of the parameters of all selected functions using the linear regression method.

To show the ability of the program two different examples are given:

1. (1) searching the best univariate polynomial model, and

2. (2) modelling of the stability of a two-component mixture as a function of three factors.

The polynomial reconstruction problem: The first 50 years

The problem of reconstructing the characteristic polynomial of a graph of order at least 3 from the collection of characteristic polynomials of its vertex-deleted subgraphs was posed by Cvetkovi? in 1973 as a spectral counter part to the well-known Ulam's reconstruction conjecture. Over the last 50 years, this problem has received notable attention, many positive results have been obtained, but in the general case the problem is still unresolved. In particular, no counter example is found in literature. In this expository paper we survey classical and some more recent results concerning the polynomial reconstruction problem, discuss some related problems, variations and generalizations.     

A fitting algorithm for real coefficient polynomial rooting

A new algorithm for computing all roots of polynomials with real coefficients is introduced. The principle behind the new algorithm is a fitting of the convolution of two subsequences onto a given polynomial coefficient sequence. This concept is used in the initial stage of the algorithm for a recursive slicing of a given polynomial into degree-2 subpolynomials from which initial root estimates are computed in closed form. This concept is further used in a post-fitting stage where the initial root estimates are refined to high numerical accuracy. A reduction of absolute root errors by a factor of 100 compared to the famous Companion matrix eigenvalue method based on the unsymmetric QR algorithm is not uncommon. Detailed computer experiments validate our claims.