Ryad A Ghanam - Academia.edu (original) (raw)

Papers by Ryad A Ghanam

Optimal sampling regimes for estimating predator-prey dynamics

Research in Mathematics

arXiv (Cornell University), Apr 4, 2021

In a recent paper, a new method was proposed to find the common invariant subspaces of a set of m... more In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper invstigates the more general problem of putting a set of matrices into block triangular or block-diagonal form simultaneously. Based on common invariant subspaces, two algorithms for simultaneous block triangularization and block diagonalization of sets of matrices are presented. As an alternate approach for simultaneous block diagonalization of sets of matrices by an invertible matrix, a new algorithm is developed based on the generalized eigen vectors of a commuting matrix. Moreover, a new characterization for the simultaneous block diagonalization by an invertible matrix is provided. The algorithms are applied to concrete examples using the symbolic manipulation system Maple.

Extracta Mathematicae

We obtain minimal dimension matrix representations for each decomposable fivedimensional Lie alge... more We obtain minimal dimension matrix representations for each decomposable fivedimensional Lie algebra over R and justify in each case that they are minimal.

Annali di Matematica Pura ed Applicata (1923 -)

A problem that is frequently encountered in a variety of mathematical contexts is to find the com... more A problem that is frequently encountered in a variety of mathematical contexts is to find the common invariant subspaces of a single or of a set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Plücker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of a totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple. Our main motivation lies in Lie symmetry, where the invariant subspaces of the adjoint representations for the Lie symmetry algebra of a differential equation must be known explicitly and comprehensively in order to determine a...

Arab Journal of Mathematical Sciences

PurposeThis study aims to find all subalgebras up to conjugacy in the real simple Lie algebra so(... more PurposeThis study aims to find all subalgebras up to conjugacy in the real simple Lie algebra so(3,1).Design/methodology/approachThe authors use Lie Algebra techniques to find all inequivalent subalgebras of so(3,1) in all dimensions.FindingsThe authors find all subalgebras up to conjugacy in the real simple Lie algebra so(3,1).Originality/valueThis paper is an original research idea. It will be a main reference for many applications such as solving partial differential equations. If so(3,1) is part of the symmetry Lie algebra, then the subalgebras listed in this paper will be used to reduce the order of the partial differential equation (PDE) and produce non-equivalent solutions.

Cornell University - arXiv, Dec 20, 2016

We consider a non-linear Timoshenko system of partial differential equations (PDEs) with friction... more We consider a non-linear Timoshenko system of partial differential equations (PDEs) with frictional damping term in rotation angle. The nonlinearity is due to the arbitrary dependence on the rotation moment. A Lie symmetry group classification of the arbitrary function of rotation moment is presented. Optimal system of one-dimensional subalgebras of the non-linear damped Timoshenko system are derived for all the non-linear cases. All possible invariant variables of the optimal systems for the three non-linear cases are presented. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided.

Cornell University - arXiv, 2015

We obtain minimal dimension matrix representations for each indecomposable fivedimensional Lie al... more We obtain minimal dimension matrix representations for each indecomposable fivedimensional Lie algebra over R and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.

Cogent Mathematics & Statistics, 2020

For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of ... more For each of the six-dimensional indecomposable nilpotent Lie algebras, the geodesic equations of the associated canonical Lie group connection are given. In each case, a basis for the associated Lie algebra of symmetries is constructed and analyzed.

This paper is concerned with finding minimal dimension linear representations for six-dimensional... more This paper is concerned with finding minimal dimension linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. It is known that all such Lie algebras can be represented in gl(6, R). After discussing the classification of the 24 such Lie algebras, it is shown that only one algebra can be represented in gl(4, R). A Theorem is then presented that shows that 13 of the algebras can be represented in gl(5, R). The special case of filiform Lie algebras is considered, of which there are five, and it is shown that each of them can be represented in gl(6, R) and not gl(5, R). Of the remaining five algebras, four of them can be represented minimally in gl(5, R). That leaves one difficult case that is treated in detail in an Appendix.

For each of the two and three-dimensional indecomposable Lie algebras the geodesic equations of t... more For each of the two and three-dimensional indecomposable Lie algebras the geodesic equations of the associated canonical Lie group connection are given. In each case a basis for the associated Lie algebra of symmetries is constructed and the corresponding Lie brackets are written down.

Computational and Mathematical Methods, 2020

This article points out some of the shortcomings of a paper by Ceballos et al. In particular, rep... more This article points out some of the shortcomings of a paper by Ceballos et al. In particular, representations of the indecomposable six-dimensional complex nilpotent Lie algebras are corrected. In addition, the class of representations is extended so as to include all the indecomposable six-dimensional real nilpotent Lie algebras. A test case is studied in detail in which it is shown that there is no representation of one of the six-dimensional real nilpotent Lie algebras using 5 × 5 matrices. This test case points up some of the severe difficulties that one will encounter when implementing an algorithm such as the one developed by Ceballos et al. The case of the seven-dimensional complex nilpotent Lie algebras is briefly touched upon and several representations are corrected. K E Y W O R D S complex and real nilpotent Lie algebra, minimal dimension matrix representation, six and seven dimensions 1 INTRODUCTION We refer to a recent paper by Ceballos et al.. 1 The article introduces a routine for constructing matrix representations of minimal dimensional for a nilpotent Lie algebra starting from the Lie brackets. It is applied to the problem of finding representations of complex nilpotent Lie algebra in dimensions six and seven. In this note, we should like to add a few clarifying comments. Up to isomorphism, there are 20 complex nilpotent Lie algebra of dimensions six. As far as we can see, 11 of the 20 representations given in Reference 1 appear to be incorrect and here we give corrected versions. The main problem in implementing the algorithm given in Reference 1, resides in the fact that the authors have not ensured that, for the solutions they found, the resulting matrices are linearly independent; nor is checking for linear independence an easy process to implement. In addition, one must verify that generators found in the solution do produce the Lie algebra in question. There are a number of other interesting related issues. One such question is to find representations of real nilpotent Lie algebras. As far we are concerned, workers in the field are mainly concerned with real Lie algebras and real representations. We take up the issue of the classification of the real nilpotent Lie algebras of dimension six in Section 5. It turns out that there is just one such algebra for which the first step in the algorithm presented in Reference 1 does not accurately predict the dimension of the minimal representation. This algebra is 6.24 = −1 in Reference 2. Its relation to the listing used in Reference 1 will be explained in Section 6. The interest here consists in examining 6.24 = −1 as a case study in which it is shown that actually there is no 5 × 5 representation. The complexity of the argument should give one pause for thought in establishing the existence or nonexistence of a particular size representation. An outline of this article is as follows. In Section 2, we consider the Lie algebra n of n × n strictly upper triangular matrices. In Section 3, we mention Hochschild's extension of Ado's theorem for the existence of nil-representations of

Symmetry, 2019

In this investigation, we present symmetry algebras of the canonical geodesic equations of the in... more In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

Tamkang Journal of Mathematics, 2006

We study the main properties of the generalized neutral orthogonal group O(2, 2) and its Lie alge... more We study the main properties of the generalized neutral orthogonal group O(2, 2) and its Lie algebra o(2, 2). We also give an explicit isomorphism between the Lie algebras su(1, 1) ⊕ su(1, 1) and o(2, 2). We use this isomorphism to classify the subalgebras of o(2, 2).

Journal of Symbolic Computation, 2017

Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the sam... more Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.

Arabian Journal of Mathematics, 2017

We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions fiv... more We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.

Journal of Mathematics, 2016

All the simple and then semisimple subalgebras of gl(4,R) are found. Each such semisimple subalge... more All the simple and then semisimple subalgebras of gl(4,R) are found. Each such semisimple subalgebra acts by commutator on gl(4,R). In each case the invariant subspaces are found and the results are used to determine all possible subalgebras of gl(4,R) that are not solvable.

Journal of Physical Mathematics, 2016

This paper is concerned with finding linear representations for seven-dimensional real, indecompo... more This paper is concerned with finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras. We consider the first 39 algebras presented in Gong's classification which was based on the upper central series dimensions. For each algebra, we give a corresponding matrix Lie group, a representation of the Lie algebra in terms of left-invariant vector field and left-invariant one forms.

Communications in Nonlinear Science and Numerical Simulation, 2016

Highlights  Presents explicit and implicit formulations for anomalous diffusion in heterogeneous... more Highlights  Presents explicit and implicit formulations for anomalous diffusion in heterogeneous porous media for single phase flow  Formulates the wellbore model based on the modified Darcy's law.  Presents examples to study the effect of memory parameter on pressure.

Journal of Geometry and Physics, 2015

A constructive version of the Frobenius integrability theorem-that can be programmed effectively-... more A constructive version of the Frobenius integrability theorem-that can be programmed effectively-is given. This is used in computing invariants of groups of low ranks and recover examples from a recent paper of Boyko, Patera and Popoyvich [BPP].

A note on bi-invariant metrics on Lie groups

Let G be a connected real Lie group. A general procedure for constructing bi-invariant pseudo- Ri... more Let G be a connected real Lie group. A general procedure for constructing bi-invariant pseudo- Riemannian structures on G using the canonical symmetric connection is described. It is applied to two 4-dimensional solvable Lie groups G, giving in each case a 2-parameter family of bi-invariant pseudo-Riemannian structures.