# Dejenie Alemayehu LakewHampton University | HIU · Department of Mathematics

Dejenie Alemayehu Lakew

Ph.D.

Professor(Asst.) Mathematics Department, Hampton University

## About

31

Publications

101,389

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29

Citations

Citations since 2016

Introduction

Dejenie Alemayehu Lakew (PhD) is a mathematician with rich experience in teaching, research and publications. His dictum is that mathematics is both useful and beautiful, it is a study of abstract structures of logic, reason and imagination.
He taught undergraduate & graduate courses in his long teaching career, author, co-author of articles in peer reviewed journals, and reviewer/editor.
Areas of research: Analysis - hypercomplex analysis, functional analysis, operator theory, PDEs, ODEs.

Additional affiliations

Education

August 1996 - July 2000

**University of Arkansas**

Field of study

- Mathematics (Clifford analysis and PDEs)

September 1987 - June 1988

September 1980 - July 1984

## Publications

Publications (31)

In article [1], the initial value problems (IVPs) studied were the ones that have solutions that are algebraic. In this article, initial value problems of difference equations of first and second order whose solutions are transcendental sequences, not algebraic are considered. Again, the discrete Laplace transform method and its inverse transform a...

In this short research article, we establish new results of initial value problems of discrete differential equations. The particular difference equations we consider at the moment are the ones with solutions that are generated as algebraic sequences, sequences of polynomials or of rational expressions. These results are obtained by a transform met...

In this short article we show solutions of first, second and higherorder PDEs develop as sums of functions that are orthogonal, the parts that evolve from the trace values of solutions on the boundary and the ones that evolve from values of the differential equations in the interior of the domain.That is, for the first order Cauchy BVP : Du=f in /O...

We investigate more initial value problems of difference equations of first and second order whose solutions are transcendental sequences using the method of the discrete Laplace transform and its inverse.

Mathematics is ubiquitous, not peculiar to humans but in nature, it is the lingua franca and modes vivendi of nature and the paper describes this phenomena.

The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space W^{1, 2}(Ω) as W^{1, 2}(Ω) = A^{2,2}(Ω)⊕D²(W₀^{3, 2}(Ω)) and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space W^{1,2}(Ω)⊖(W₀...

The main theme of this paper is to construct Clifford analytic-complete function systems in the general-ized Bergman spaces: BpCln:=kerD∩LpCln, and B(p;2)Cln:=kerD^2∩LpCln. These systems are used to approximate null solutions of elliptic partial differential equations of the Dirac /D and Laplace /Delta operators over an unbounded domains /Omega in...

The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space W^{1,2}(Ω) as W^{1,2}(Ω)=A^{2,2}(Ω)⊕D²(W₀^{3,2}(Ω)) and look at some properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space W^{1,2}(Ω)⊖(W₀^{1,2}(Ω))^{...

We establish new and different kinds of proofs of properties that arise due
to the orthogonal decomposition of the Hilbert space, including projections,
over the unit interval of one dimension. We also see angles between functions,
particularly between those which are non zero constant multiples of each other
and between functions from the kernel s...

We obtain further results of the discrete Laplace transform of transcendental sequences and solve initial value problems with a mix of algebraic and transcendental solutions.

In this short article we show an orthogonal decomposition of the Hilbert space as ℒ²(Ω)=A²(Ω)⊕(d/(dx))(W₀^{1,2}(Ω)), define orthogonal projections and see some of its properties. We display some decomposition of elementary functions as corollaries.

In this article we define the spherical π operator over domains in the (n-1)- unit sphere S^{n-1} of Rⁿ and develop new and analogous results. We introduce a spherical Dirac operator Γ_{α}: = Γ_{ω}+α for α in ℂ and Γ_{ω} = ω ^ D_{ω}, the anti-symmetric Grassmanian product of ω and D_{ω} = ∑_{j=1}ⁿe_{j}∂_{w_{j}}. We use a Gugenbauer polynomial Ψ_{α}...

In this article we develop few of the analogous theoretical results of
Clifford analysis over Orlicz-Sobolev spaces and study mapping properties of
the Dirac operator and the Teodorescu transform over these function spaces. We
also get analogous decomposition results of Clifford valued Orlicz spaces and
the generalized Orlicz-Sobolev spaces.

In this short note, we study an ordinary differential operator of infinite order defined by: ∑_{n=0}^{∞}((Dⁿ)/(n!))=:e^{D} where D:=(d/(dx))

We present norm estimates for solutions of first and second order elliptic
BVPs of the Dirac operator considered over a bounded and smooth domain of the
n-dimension Euclidean space. The solutions whose norms to be estimated are in
some Sobolev spaces and the boundary conditions as traces of solutions and
their derivatives are in some Slobodeckij sp...

In this short article, we study different problems described as initial value
problems of discrete differential equations and develop a transform method
called the sigma transform, a discrete version of the continuous Laplace
transform to generate solutions as rational functions of integers to these
initial value problems. Particularly we look how...

The main aim of this article is to study the hypercomplex π-operator over
${\mathbb{C}^{n+1}}$
via real, compact, n + 1-dimensional manifolds called domain manifolds. We introduce an intrinsic Dirac operator for such types of domain manifolds and define an intrinsic π-operator, study its mapping properties and introduce a Clifford–Beltrami equati...

In this paper we study singular integral operators which are hyper or weak over Lipschitz or Holder spaces and over weghted Sobolev spaces defined on unbounded domains in the standard $n$-D space $R^n$ for $n>0$. The $\pi$-operator in this case is one of the hyper integral operators which has been studied extensively than other hyper singular integ...

In this article, we define the spherical $\pi$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac operator $\Gamma_\alpha$ as an $\alpha$- shift of of $\Gamma_omega$, where $\Gamma_omega$ is the negative of the w...

In this short note, we present few results on the use of the discrete Laplace transform in solving first and second order initial value problems of discrete differential equations.

We introduce mollifiers in Clifford analysis setting and construct a sequence of $\C^{\infinity}$-functions that approximate a $\gamma$-regular function and a solution to a non homogeneous BVP of an in homogeneous Dirac like operator in certain Sobolev spaces over bounded domains whose boundary is not that wild. One can extend the smooth functions...

In this paper, we construct $\gamma$-regular $Cl_n$-minimal function systems in the generalized Bergman space : $W^{2,k}_\gamma(\Omega,Cl_n) which are $\gamma$-regular, Clifford valued over $\Omega$ that are used to approximate null solutions of the in-homogeneous Dirac operator in the best way.

For Ω a sufficiently smooth unbounded domain in ℝn we develop a decomposition result for the Sobolev space
. We also use modified Cauchy-Green type kernels to construct Clifford analytic-complete function systems in the generalized Bergman space
, where D
l is the l-th iterate of the Dirac operator, l is a positive integer less than n and n/(n − l...

The main theme of this paper is to construct Clifford analytic-complete function systems in the generalized Bergman spaces: BpCln(Ω):=kerD(Ω)∩LpCln(Ω), and Bp,2Cln(Ω):=ker▵(Ω)∩LpCln(Ω). These systems are used to approximate null solutions of elliptic partial differential equations of the Dirac and Laplace operators over an unbounded domain Ω in ℝn.

By using a sophisticated maximum principle, three over determined problems for elliptic equations in two variables are discussed.

By using a sophisticated maximum principle, three over determined problems for elliptic equations in two variables are discussed

In this article we define the spherical ��,Sn 1 operator over domains in the (n 1)D unit sphere Sn 1 of Rn and develop new and analogous results. We introduce a spherical Dirac operator � := !+�, where � 2 C and ! = ! ^ D! , the anti-symmetric Grassmanian product of ! with D! = n X i=1 ei @ @!i. We use a Gegenbauer polynomial �n�(! �) as a Cauchy k...

## Questions

Questions (37)

Consider the two:

*. " A weak divergence CDC method for biharmonic equation on triangular and tetrahedral meshes " by ..... and

*. " Chlodosky type - Bernstein Stancu operator of Korvokin -type approximation theorems .. " by ...

Both are titles of articles written for mathematics journals. The first clearly indicates that it is a mathematics paper, while the second, barely with list of names of individuals in sequence.

We all know that mathematics of man is as old as mankind and that of nature is as old as nature itself. Mathematicians study and write, extrapolate and expand the kingdom of mathematical knowledge freely with no taking possession of any kind. But a recent phenomena is unique, in which a mathematics article is barely understood to be mathematics, instead chains of names of humans. The above two examples describe what I am discussing. Just imagine, all those results in mathematics since antiquity, written in the name of those who established them (which many of them will be unknown), then we are in real estate business, and the very meaning of mathematics and its universality, ubiquities-ness lose its flavor.

Let us keep mathematics - mathematical.

## Projects

Projects (4)

Extend the location concept from material point to area, and scrutinize the ramifications.
Connect to Tti and first order gravity concepts.

To prove that solutions of first, second and higher order PDEs evolve as sums of functions that are orthogonal with respect to the inner product of a Sobolev space W^{k,2}(Ω) for k= 1, 2, 3,... The components are the parts that evolve from the trace values of solutions on the boundary ∂Ω of the domain and the ones that evolve from values of the differential equations in the interior of the domain Ω. I use the symbol ⊎ to denote an orthogonal sum of functions that are from orthogonal sum ⊕ of function subspaces.

The them of this project is to open a forum of acknowledgement to our former academic advisers who initiate, acclimate and guide us in to the frontiers of our fields of research and be reasons to our achievements. I share an article about my doctoral research adviser Prof. John Ryan of the University of Arkansas, his academic works, his doctoral students and his scholarly achievements in the field of Clifford Analysis. The writer of the article is Prof. Wolfgang Sprößig, Department of Mathematics and Informatics, Institute of Applied Analysis, TU Bergakademie Freiberg, Germany, a lead mathematician in Clifford analysis and applications. Acknowledge your former academic advisers by sharing their names and academic achievements here in this forum, their human dimensions as well if need be.