Algorithmic Methods in Queues and in the Exploration of Point Processes

This is a review of methodology for the algorithmic study of some useful models in point process and queueing theory, as discussed in three lectures at the Summer Institute at Sozopol, Bulgaria. We provide references to sources where the extensive details of this work are found. For future investigation, some open problems and new methodological approaches are proposed.

Investigating the Relations of Attacks, Methods and Objects in Regard to Information Security in Network TCP/IP Environment

Possibilities for investigations of 43 varieties of file formats (objects), joined in 10 groups; 89 information attacks, joined in 33 groups and 73 methods of compression, joined in 10 groups are described in the paper. Experimental, expert, possible and real relations between attacks’ groups, method’ groups and objects’ groups are determined by means of matrix transformations and the respective maximum and potential sets are defined. At the end assessments and conclusions for future investigation are proposed.

An Iterative Method for the Matrix Principal n-th Root

In this paper we give an iterative method to compute the principal n-th root and the principal inverse n-th root of a given matrix. As we shall show this method is locally convergent. This method is analyzed and its numerical stability is investigated.

Distance Matrix Approach to Content Image Retrieval

As the volume of image data and the need of using it in various applications is growing significantly in the last days it brings a necessity of retrieval efficiency and effectiveness. Unfortunately, existing indexing methods are not applicable to a wide range of problem-oriented fields due to their operating time limitations and strong dependency on the traditional descriptors extracted from the image. To meet higher requirements, a novel distance-based indexing method for region-based image retrieval has been proposed and investigated. The method creates premises for considering embedded partitions of images to carry out the search with different refinement or roughening level and so to seek the image meaningful content.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

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

A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations

2000 Mathematics Subject Classification: 65H10.