Fujio Kako

Solving Multivariate Algebraic Equation by Hensel Construction

Given a multivariate polynomial F(x, y, ...,z), this paper deals with calculating the roots ofF w.r.t.x in terms of formal power series or fractional-power series iny, ...,z. If the problem is regular, i.e. the expansion point is not a singular point of a root, then the calculation is easy, and the irregular case is considered in this paper. We extend the generalized Hensel construction slightly so that it can be applied to the irregular case. This extension allows us to calculate the roots of bivariate polynomial F(x, y) in terms of Puiseux series iny. For multivariate polynomial F(x, y, ...,z), we consider expanding the roots into fractional-power series w.r.t. the total-degree ofy, ...,z, and the roots are expressed in terms of the roots of much simpler polynomials.

Interaction of Ion-Acoustic Solitons in Multi-Dimensional Space.II

Numerical computations are made to study the collision process between two cylindrical or spherical solitons. The soliton resonance is found to play an important role in collision processes between two curved solitons as well as between two plane solitons

Computing floating-point gröbner bases stably

Computing floating-point gröbner bases stably.

Floating-Point Gröbner Basis Computation with Ill-conditionedness Estimation

Computation of Grobner bases of polynomial systems with coefficients of floating-point numbers has been a serious problem in computer algebra for many years; the computation often becomes very unstable and people did not know how to remove the instability. Recently, the present authors clarified the origin of instability and presented a method to remove the instability. Unfortunately, the method is very time-consuming and not practical. In this paper, we first investigate the instability much more deeply than in the previous paper, then we give a theoretical analysis of the term cancellation which causes loss of accuracy in various cases. On the basis of this analysis, we propose a practical method for computing Grobner bases with coefficients of floating-point numbers. The method utilizes multiple precision floating-point numbers, and it removes the drawbacks of the previous method almost completely. Furthermore, we present a practical method of estimating the ill-conditionedness of the input system.

Propagation of Solitons in a Dissipative Nonlinear Transmission Line

The perturbation theory for the inverse scattering transform applies to study a dissipative nonlinear transmission line, which is equivalent to the Toda lattice in the limit of vanishing dissipations. The time evolution of a one-soliton solution is calculated. It is found that the weak dissipations lead to change in the soliton parameters, the amplitude and the velocity, the creation of small solitons and the formation of a tail behind the initial soliton.

Term cancellations in computing floating-point Gröbner bases

We discuss the term cancellation which makes the floating-point Grobner basis computation unstable, and show that error accumulation is never negligible in our previous method. Then, we present a new method, which removes accumulated errors as far as possible by reducing matrices constructed from coefficient vectors by the Gaussian elimination. The method manifests amounts of term cancellations caused by the existence of approximate linearly dependent relations among input polynomials.

Inverse Method Applied to the Solution of Nonlinear Network Equations Describing a Volterra System

Equations for an irreversible nonlinear network, which is equivalent to Volterras competition equations, are solved by the inverse scattering method. The duality satisfied by the equations play an essential role not only in formulating the inverse method but also in finding solutions. N -soliton solutions obtained are in agreement with Hirota and Satsumas. The relation to N -soliton solutions of the Toda lattice is given explicitly.

Computational Construction of W-graphs of Heeke algebras H(q, n) for nup to 15

We construct by computer all W-graphs corresponding to irreducible representations of Heeke algebras H(q, n) for n up to 15, using a modification of a method proposed by lascoux and Schutzenberger (which fails for n > 13).

Anomalies in the diffraction of ion acoustic solitons

Abstract The diffraction of linear ion acoustic pulses and nonlinear ion acoustic solitons is examined experimentally. The results are compared with a numerical simulation and it is found that the soliton experiment exhibits anomalous features in addition to the numerically predicted behavior.

Numerical Analysis of the Buneman Instability in a Magnetic Field

The Buneman instability in a magnetic field is studied numerically using the fluid equations. It is shown that the instability due to the anomalous Doppler effect appears in addition to the Buneman instability. The detailed properties of these two instabilities are obtained by solving the dispersion equation.