Monte Carlo Algorithm for Solving Integral Equations with Polynomial Non-Linearity. Parallel Implementation

An iterative Monte Carlo algorithm for evaluating linear functionals of the solution of integral equations with polynomial non-linearity is proposed and studied. The method uses a simulation of branching stochastic processes. It is proved that the mathematical expectation of the introduced random variable is equal to a linear functional of the solution. The algorithm uses the so-called almost optimal density function. Numerical examples are considered. Parallel implementation of the algorithm is also realized using the package ATHAPASCAN as an environment for parallel realization.The computational results demonstrate high parallel efficiency of the presented algorithm and give a good solution when almost optimal density function is used as a transition density.

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)

Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively, with n > m, three new, and easy to understand methods — along with the more efficient variants of the last two of them — are presented for the computation of their subresultant polynomial remainder sequence (prs). All three methods evaluate a single determinant (subresultant) of an appropriate sub-matrix of sylvester1, Sylvester’s widely known and used matrix of 1840 of dimension (m + n) × (m + n), in order to compute the correct sign of each polynomial in the sequence and — except for the second method — to force its coefficients to become subresultants. Of interest is the fact that only the first method uses pseudo remainders. The second method uses regular remainders and performs operations in Q[x], whereas the third one triangularizes sylvester2, Sylvester’s little known and hardly ever used matrix of 1853 of dimension 2n × 2n. All methods mentioned in this paper (along with their supporting functions) have been implemented in Sympy and can be downloaded from the link http://inf-server.inf.uth.gr/~akritas/publications/subresultants.py

The Legendre Formula in Clifford Analysis

2000 Mathematics Subject Classification: 30A05, 33E05, 30G30, 30G35, 33E20.

Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group

2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.