M MATRIX

A brief but concise review of methods to generate P-matrices (i.e., matrices having positive principal minors) is provided and motivated by open problems on P-matrices and the desire to develop and test efficient methods for the detection of P-matrices. Also discussed are operations that leave the class of P-matrices invariant. Some new results and extensions of results regarding P-matrices are included.

Diffusion-type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M-matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M-matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M-matrix for rectangular elements and can be an M-matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.

In this article a new preconditioner from class of (I+S)-type based on the Modified Accelerated
Overrelaxation(MAOR) iterative method has been introduced and convergence properties of the proposed
method have been analyzed and compared with the some other preconditioners. Moreover, comparisons
between different splittings are derived. Numerical example is also given to illustrate our results.

The linear systems with M and H-matrices as coefficient matrix many often arise in science and engineering. In this article we consider some preconditioners of [12] for solving a linear system with the above coefficient matrix. Also we use preconditioned AOR method for the solution of the systems. The convergence theorems for the preconditioners also presented. A numerical example is given to test the methods.

Diffusion-type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M-matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M-matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M-matrix for rectangular elements and can be an M-matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra \MA closed under transposition. The associative algebra \MA is irreducible iff its commutant \MA^{\prime} coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of {sp}(\infty,{\bb R}) corresponding to the field {{\bb R}} of reals, of u(∞, ∞) associated with the field {{\bb C}} of complex numbers, and of so*(4∞) related to the algebra {{\bb H}} of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and U(N,{{\bb H}})=Sp(2N) , respectively. Lecture at the workshops 'Lie Theory and Its Applications in Physics', 18-24 June 2007, Varna, Bulgaria; 'Infinite- Dimensional Algebras and Quantum Integrable Systems', 23-27 July, 2007, Faro, Portugal; and 'Supersymmetries and Quantum Symmetries', 30 July-4 August, 2007, Dubna, Russia.