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## Publications

Publications (48)

Purpose
This study aims to find all subalgebras up to conjugacy in the real simple Lie algebra s o ( 3,1 ) .
Design/methodology/approach
The authors use Lie Algebra techniques to find all inequivalent subalgebras of s o ( 3,1 ) in all dimensions.
Findings
The authors find all subalgebras up to conjugacy in the real simple Lie algebra s o ( 3,1 )...

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 formu...

The Covid-19 outbreak of 2020 has required many governments to develop mathematical-statistical models of the outbreak for policy and planning purposes. This work provides a tutorial on building a compartmental model using Susceptibles, Exposed, Infected, Recovered, Deaths and Vaccinated (SEIRDV) status through time. A Bayesian Framework is utilize...

This paper studies the canonical symmetric connection ∇ associated to any Lie group G. The salient properties of ∇ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to ∇ in the special case where the Lie algebra 𝔤 of G, has a codimension one abelian nilradica...

This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codim...

Ecologists are interested in modeling the population growth of species in various ecosystems. Specifically, logistic growth arises as a common model for population growth. Studying such growth can assist environmental managers in making better decisions when collecting data. Traditionally, ecological data is recorded on a regular time frequency and...

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 dia...

During the current COVID-19 pandemic, decision makers are tasked with implementing and evaluating strategies for both treatment and disease prevention. In order to make effective decisions, they need to simultaneously monitor various attributes of the pandemic such as transmission rate and infection rate for disease prevention, recovery rate which...

This work shows for the first time the viability of using the Bayesian paradigm for both estimation and hypothesis testing when applied to fractional differential equations. Two distinct fractional differential equation models were explored using simulated data sets to determine the performance of the Bayesian inferential methods across values of α...

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 st...

Ecologists are interested in modeling the population growth of species in various ecosystems. Studying population dynamics can assist environmental managers in making better decisions for the environment. Traditionally, the sampling of species and tracking of populations have been recorded on a regular time frequency. However, sampling can be an ex...

The Covid-19 outbreak of 2020 has required many governments to develop mathematical-statistical models of the outbreak for policy and planning purposes. This work provides a tutorial on building a compartmental model using Susceptibles, Exposed, Infected, Recovered and Deaths status through time. A Bayesian Framework is utilized to perform both par...

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.

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...

Symmetries of the canonical geodesic equations of indecomposable nilpotent Lie groups of dimension five are constructed. For each case, the associated system of geodesics is provided. In addition, a basis for the associated Lie algebra of symmetries as well as the corresponding non-zero Lie brackets are listed and classified.

The infinitesimal algebra of Lie symmetries of the Eikonal equation is shown to be isomorphic to o(n+1,2) when there are n independent variables. An explicit basis is found that is aligned with the standard basis coming from the standard matrix representation of o(n+1,2) thereby making it possible to read off inequivalent one-dimensional symmetry v...

We obtain minimal dimension matrix representations for each decomposable five-dimensional Lie algebra over R and justify in each case that they are minimal.

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 t...

The extraction of natural gas from the earth has been shown to be governed by differential equations concerning flow through a porous material. Recently, models such as fractional differential equations have been developed to model this phenomenon. One key issue with these models is estimating the fraction of the differential equation. Traditional...

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

We consider a nonlinear Timoshenko system of partial differential equations (PDEs) with a 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. An optimal system of one-dimensional subalgebr...

A nonlinear transport model for single-phase gas through tight rocks, is combined with a fractional calculus method, to produce a new time-fractional advection-diffusion transport model for the pressure field, í µí± = í µí±(í µí±¥, í µí±¡) in the flow of gas through tight porous reservoirs. Solutions for different fractional order, 0 <∝< 1, and f...

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...

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.

A diffusion problem involving a time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are...

In reservoir engineering, an oil reservoir is commonly modeled using Darcys diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in...

We obtain minimal dimension matrix representations for each indecomposable five-dimensional 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.

A constructive version of the Frobenius integrability theorem – that can be pro-grammed e↵ectively – 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].
( to appear in Journal of Geometry and Physics)

It is known how to find minimal dimension matrix representa-tions for four-dimensional complex Lie algebras. The method depends on constructing left symmetric structures. In this note it is explained how to obtain the representations directly and also how to extend the results to real Lie algebras. Two different bases for the four-dimensional Lie a...

We consider a homogeneous fractional diffusion prob-lem in an infinite reservoir sometimes called a "modified" diffusion equation. The equation involves a (nonlocal in time) memory term in the form of a time fractional derivative (of the Laplacian). For the sake of reducing the computational domain to a bounded one we establish appropriate "artific...

In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer...

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain ma...

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain ma...

In order to characterize the systems of second-order ODEs which admit a regular Lagrangian function, Noether symmetries for the geodesic equations of the canonical linear connection on Lie groups of dimension three or less are obtained, so the characterization of these geodesic equations through their Noether’s symmetries Lie Algebras is investigat...

We construct new examples of four-dimensional Einstein metrics with neutral signature and two-dimensional holonomy Lie algebra.

A restricted version of the inverse problem of Lagrangian dynamics for the canonical linear connection on a Lie group is studied. Specifically for solvable Lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo-Riemannian metric on the corresponding linear Lie group are found. Of the 19 such metrics fo...

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).

This paper is concerned with finding linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. After surveying the literature a preliminary classification is made based on the lower central series. Thereafter, a finer classification is based on Lie algebra cohomology and the dimension of the automorphism group. Finally...

This paper gives a comprehensive analysis of the inverse problem of Lagrangian dynamics for the geodesic equations of the canon-ical linear connection on Lie groups of dimension four. Starting from the Lie algebra, in every case a faithful four-dimensional representation of the algebra is given as well as one in terms of vector fields and a represe...

Einstein neutral metrics in dimension four are constructed. They provide examples of holonomy type A16.

Two spacetime metrics are constructed that provide examples that were
conjectured to exist in a recent paper by Hall and Lonie. The first metric
is an Einstein space that has a null recurrent vector field. The second
metric is conformally flat and also has a null recurrent vector field.

The Lie algebra isomorphism between su(1,1)×su(1,1) and o(2,2) is used to obtain a list of subalgebras of the latter. The resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the Lie algebra of the holonomy group of a neutral metric in four dimensions. The conclusions, taken in conjunction with previously...

Two spacetime metrics are constructed that provide examples that were conjectured to exist in a recent paper by Hall and Lonie. The first metric is an Einstein space that has a null recurrent vector field. The second metric is conformally flat and also has a null recurrent vector field.

Typescript. Thesis (Ph. D.)--University of Toledo. "A dissertation [submitted] as partial fulfillment of the requirements of the Doctor of Philosophy degree in Mathematics." Includes bibliographical references (leaves 58-59).

## Projects

Projects (4)

1. Develope algorithm for constructing invariant subspaces of all dimensions of a finite set of matrices.
2. Provide Maple code to find all the invariant subspaces of all dimensions for a finite set of matrices.
3. Develope algorithm for simultaneous block triangularization and block diagonalization of a finite set of matrices.

Estimate the parameter, the fraction of a fractional differential equation, while adequately quantifying the uncertainties associated with error and prediction.

Finding a matrix Lie group for finite low-dimensional Lie algebras