A Project Proposal for Funding by

Department of Mathematics Addis Ababa

University, ETHIOPIA Submitted to

International Science Program (ISP), Uppsala

University, Sweden August , 2017

Applicant (Research Group Leader)

Tilahun Abebaw Kebede (PhD)

Department of Mathematics,

Addis Ababa University,

P.O.Box 1176, Addis Ababa, Ethiopia

Tel. -251 11 123 9461(Office); +251 911 063510 (Mobile)

E-mail: tilahunabebaw@yahoo.com OR tilahun.abebaw@aau.edu.et

Project Title: Capacity Building in Researchand Graduate Education in Mathematics inEthiopia

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I. Proposed Research Project

1.1 Purpose and Significance of the Project

The population of Ethiopia is projected close to 100 Million and the government of Ethiopia is undertaking

a major expansion in higher learning institutions to produce skilled manpower for various sectors of the

economy in the country. In view of this, eleven new public Universities have been established in the last few

years alone in various regions of the country, thereby bringing the total number of public universities in the

country to 41.

To relieve these new universities from shortage of staff and to strengthen the existing ones there is a high

demand and an extensive pressure on Addis Ababa University to expand its graduate programs. To meet

this demand at the national level and play its role in this ambitious development plan, the university has

reminded all its units to revise their curricula and enroll larger number of graduate students in both MSc

and PhD programs. In this regard mathematics stands out as one of the vital components in the program of

the university. Therefore, cognizant of the gravity of the problem the Department of Mathematics responded

by expanding its graduate programme. It has started a Sandwich PhD program (with Stockholm University

which was Financed by ISP) and an in-house PhD programme since 2010 and training a good number of

candidates in these programmes.

However, to implement the PhD program successfully there is a felt need to formulate research agenda and

help students to be part of the research by the senior academic staff of the Department. Since the available

expertise in the department is very limited, it is necessary to network the programmes of the Department

with regional and international level researchers in Mathematics. By developing research agenda we can

form collaborations with experts in various fields in Mathematics. In the areas where the Department lacks

expertise we need to formulate a sandwich type PhD training in collaboration with some willing professors

from Swedish Universities. In the past we have successfully completed the PhD training of four students

in sandwich scheme with Stockholm University in the areas of Algebra (three) and Analysis (one, female).

We hope that our current plan will also produce trained manpower in Applied Mathematics, where we

have already formed collaborations with Linkoping University and Chalmers University of Technology. One

staff member of our Department has got admission at Linkoping University to pursue a Sandwich PhD in

Optimization.

In this project we have formulated four different components of research agenda (in each component there

are different research problems, where different expertise from our Department and other collaborating

Universities will participate) in which Masters and PhD level students will participate. These are in the

areas of (see the details of each specific research proposal below):

1. Algebra and Analytic Number Theory

2. Partial Differential Equations, Functional Analysis, and Complex Analysis

3. Combinatorics & Graph Theory

4. Optimization and Applied Mathematics

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The research topics selected are thought to be interdisciplinary in nature and we tried to involve applied

and pure research topics as much as possible. If carried out as planned, the results of the research topics

proposed will contribute to the advancement of science in general and some of them can be implemented to

solve specific and critical societal problems in Ethiopia. The plan in the project is to recruit more female

candidates and increase the participation of females in both Masters and PhD programs and these can be

motivated by giving support from the project fund.

1.2 Background

The Department of Mathematics, Addis Ababa University is the oldest department in the country and has

been training students in Mathematics at BSc, MSc and recently at PhD levels. Currently, the Department

has 35 full-time academic staff of which only 2 are associate professors, 15 are assistant professors(five of

them are graduates of the PhD program of the Department) and all others are lecturers with M.Sc. degree

in Mathematics looking for further study at PhD level.

The Department has started offering Masters programme in Mathematics since 1984 and a sandwich type

PhD programme since 2001. Currently, due to highest demand from the government and the commitment

to expand graduate programs on part of the university, the Department has undergone a major curriculum

reform and established an in-house PhD program on top of the existing large number of enrollment for the

Masters programme. To overcome the shortage of expertise in various research areas in mathematics, the

department has established a network with several universities and research institutions abroad. Through

these cooperative networks several professors and researchers have shown interest and commitment to assist

in the new PhD programme.

1.3 Overall and Specific Objectives of the Project

The general objective of the project is to expand the contribution of research and postgraduate education in

Mathematical Sciences both at national and at the global level by Building the Capacity of the Department

of Mathematics, Addis Ababa University, Ethiopia. This can be realized through the achievement of the

following specific objectives:

• To increase the number and quality of Mathematics instructors in Ethiopian Universities by helping 5

to 8 PhD candidates to graduate every year.

• To increase research output of the Department by assisting academic staff members and PhD students

to publish 5 to 10 of their research works in reputable international journals and conference proceedings

per year.

• To establish new research areas in Applied streams at the Department, which are relevant in solving

practical problems in the country.

1.4 Strategy and Plan

Each of the five areas of specific research agenda that are proposed as components of this project will be

developed further and streamlined to admit PhD students in each of the areas. Collaborating researchers

will be invited to give short term courses relevant to research topics and start the problems with local senior

staff as well as the PhD students (as it has been the case since 2005). Moreover, some topics which fit the

level of MSc students will be prepared so that Masters students can work on. The financial supports to

graduate students given by the government is too small (less than 400 USD per month) and the cost of living

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in Addis Ababa is getting increasingly expensive and difficult to afford with the aforementioned salary. Due

to these, the students prefer to look for graduate study elsewhere either in Europe or America whereas those

who already joined the graduate programme of the Department usually take much of their time in looking

for additional part-time jobs to finance their own accommodation and their families.

Because of this compelling fact graduate students have very limited time to focus on research work. In order

to circumvent this problem, we are planning to support some capable students, using merit based criteria

(as we have been doing it in our previous grant period, it was very successful). Female students will be given

priority. Masters programme students can select the thesis option of the programme and the PhD students

focus on research and complete their dissertation within the given time table (3 to 4 years).

1.5 Expected Financing

The Department has applied and short listed for funding from Simon’s Foundation for the next five years,

from 2018-2022. From local sources the government supports the minimum living cost of the students that

are sponsored by state and covers their tuition fee. Moreover, the university will finance the transport and

local costs of some of the visiting scientists who will participate in the in-house PhD programme of the

Department.

1.6 Expected Outcome, Impact and Dissemination

Through the support that we get from ISP (if this proposal is accepted and approved) the Department

will train 10 Masters students in thesis options every year. Currently, most students finish their masters

program through additional course work and a project option as it might not take longer time as doing thesis.

The Department started an in-house PhD program in 2010, around 17 students have graduated from the

program and they are working in different Universities in Ethiopia, some of them have got different positions

in different places, to mention but a few;

1. Addisalem Abathun, PhD graduate of our Department in July 2016 has been appointed as the Graduate

Program Coordinator at the Department of Mathematics, Addis Ababa University.

2. Sebsibew Atikaw, PhD graduate of our Department in July 2016 has been appointed as the Commis-

sioner for Science and Technology commission for Amhara Regional State by President of Amhara

Regional State.

3. Sebsibe Teferi, PhD graduate of our Department in July 2016 has been appointed as the Associate

Dean for College of Natural Sciences, Kotebe Metropolitan University.

4. Ketsela Hailu, PhD graduate of our Department in 2015 has been appointed as the Head of Registrar

of Addis Ababa Science and Technology University.

Currently there are over 30 students in the program. If supported by ISP, each of the research components

will enroll at least 2 additional PhD students and we expect from the earlier enrollment at least 8 of them

to finalize their PhD dissertations by the end of the first year of the project (2018).

Moreover, the proposed research topics will produce at least 15 publications in reputable scientific journals

by the end of 2020 and at least 1 working report by each component groups.

At least one scientist from abroad in each area of the components proposed will visit the Department and

at least 1 senior staff and 1 PhD student will participate in a regional workshop or international conferences

and present their research results from each component per year (details of how to do will be outlined).

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1.7. Collaboration with Other Scientists

Research collaborations has been already established with Shiferaw Berhanu (Prof., Temple University, USA),

Torbjrn Larsson (Prof. , Linkoping University ,Sweden), Ann-Brith Strmberg (Prof. Chalmers University

of Technology, Sweden), Elizabeth Wulcan (Asso. Prof. , Chalmers University of Technology, Sweden),

Michael Patriksson (Prof. Chalmers University of Technology, Sweden) Minassie Ephrem (Assoc. Prof.,

Costal Carolina University, USA), Marco Gaviano and Giovani Poru (Prof., Cagliari University, Italy),

Ahmed Mohammed (Prof., Ball State University, USA) and Habtu Zegeye (Prof., Botswana International

University of Science and Technology, Botswana) for the area of Analysis and Differential Equations, with

Abdulkadir Hassen (Prof., Rowan University, USA), Rolf Kallstram (Prof., University of Gavle, Sweden)

and Rikard Bogvad (Prof., University of Stockholm, Sweden) for the field of Algebra and Number Theory,

with Melkamu Zeleke (Prof., William Paterson University, USA), Akalu Tefera (Prof. Grand Valley State

University, USA ) for Combinatorics and Graph Theory, with Stephan Dempe (Prof., Technical University of

Bergakademie, Freiberg, Germany) and Montaz Ali (Prof., Witwatersrand University, South Africa) for the

area of Optimization Theory and its Applications, and Senelani D. Hove-Musekwa (Assoc. Prof. National

University of Science and Technology, Bulawayo, Zimbabwe), Aziz Ouhinou (Researcher at AIMS, University

of Stellenbosch, South Africa), and Rachid Ouifki (Researcher at SACEMA, University of Stellenbosch, South

Africa) for Mathematical Epidemiology stream.

1.8 Postgraduate Students

Most of the graduate students who are enrolled in the program are financed by the government of Ethiopia

even though their bursaries are very small (as mentioned in Section 4) and the amount allocated for their re-

search work is extremely inadequate. The masters programme is for 18 months whereas the PhD programme

is for 3 to 4 years, if students allocate their fulltime for the study. However, due to increasing prices on basic

goods and low salaries, most of them do not finish within the intended time. Specially the masters students

do not take the thesis option as doing research may prolong their study time. Therefore, subsidizing their

studies with ISP funding will help the masters students participate in the thesis option of the programme,

thereby helping them participate in the research work of the Department. Moreover, the subsidy help the

PhD students finish their study within the intended time and encourage them to come up with a quality

output.

The support for selected masters students will last for 6 to 10 months of their study time whereas the

financial support for PhD students will be for 3 years and we are planning to support between 10 and 15

each year. In addition to this, the sandwich PhD students may require additional funding for their travel

and stay in Sweden.

Budget Request

To finance the above mentioned activities and to carry out the accompanied research proposals, it requires

about SEK 3,632,400. The details of this is given in the application form.

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II. Proposed Research Problems in

Different Areas of Mathematics

The following will provide a brief description of the research projects that students will undertake under

the various groups of researchers in the previously stated areas of Mathematics under the supervision of the

expertise in the areas.

The research problems are given in the following areas.

1. Algebra and Analytic Number Theory

2. Partial Differential Equations, Functional Analysis, and Complex Analysis

3. Combinatorics & Graph Theory

4. Optimization and Applied Mathematics

The details of each component is given below.

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Component I: Algebra and Analytic

Number Theory

1.1 Computational Commutative Algebra

Persons involved:

1. Mohammed Tesemma (Spelman College, USA)

2. Berhanu Bekele (Addis Ababa University)

The theory of Grobner bases is one of the active areas of research in computational commutative algebra

and algebraic geometry. As Grobner bases is for ideals of a polynomial ring a parallel theory was developed

for subalgebras of polynomial and Laurent polynomial rings by Robbiano and Sweedler in [7]. This analogue

theory is called SAGBI bases. The term SAGBI is an acronym for “Subalgebra Analogue to Grobner Bases

for Ideals”. Unlike Ideals it is known that not every subalgebra of a multivariable polynomial ring is finitely

generated. This problem directly translates to the fact that not every subalgebra have finite SAGBI bases.

This challenge identifying which subalgebras have finite SAGBI bases is noted early on in the paper by [7].

They also give interesting examples like the subalgebra k[x, xy − y2, , xy2] which have a finite SAGBI basis

under one monomial order while it fails to admit such finite bases under another. This example warns us that

subalgebras are also dependent on the monomial order. It is worth mentioning that there are uncountably

many monomial orders by Robbiano [6].

A characterization of all subalgebras that have finite SAGBI bases is still a wide open and challenging

problem. Nevertheless there are several special cases where the subalgebra have finite SAGBI bases. Herre

is short list of some recent results for reference. [1, 2, 3, 4, 5, 7, 8, 9, 10].

Our plan in this project here is to study new families of subalgebras that have finite SAGBI bases or identify

others that doesnot. Currently we have two graduate students in the computational algebra group here at

Addis Ababa University, one of them is directly working on a finite SAGBI bases problem and another on

SAGBI bases related to Invariant theory. As we learn more we have identified more problems of varying

difficulty once our current students completed their project. We anticipate to admit additional graduate

student/s to our group who will do a similar project in SAGBI bases and its applications.

Phase of the training process:

After successful completion of the graduate algebra class a candidate for our project will take additional

topics courses on:

1. Introduction to commutative algebra

2. Grobner and SAGBI bases in Commutative algebra

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3. Introduction to invariant theory and connection to SAGBI bases

4. Additional topics courses as necessary

5. Continuous registration for research and seminars

Each student have at least one advisor and one coadvisor who follow up the progress of the candidate as

he/she go through the various stages of their graduate study.

1.2 Topologizing filters on f-rings of continuous real-valued func-

tions

Persons Involved:

Nega Arega (Addis Ababa University)

Introduction

The set FilRR of all right topologizing filters on a fixed but arbitrary ring R is both a complete lattice

under inclusion, and a monoid with respect to an order compatible, but in general noncommutative binary

operation :. It is known that the order dual [FilRR]du of FilRR is always left residuated, meaning, for each

pair F ,G ∈ FilRR there exists a smallest filter H ∈ FilRR such that H : G ⊇ F , but is not, in general, right

residuated (there exists a smallest filter H such that G : H ⊇ F). This thesis is an investigation of rings R

for which [FilRR]du is both left and right residuated. A sufficient, but not necessary, condition for two-sided

residuation to hold is that [FilRR]du is commutative, by which is meant, the monoid operation on FilRR is

commutative.

An f-ring is a lattice-ordered ring in which a∧ b = 0 and c ≥ 0 imply ca∧ b = ac∧ b = 0. It is a commutative

lattice ordered ring with unity denoted by 1. Every ordered ring is an f-ring, since, in an ordered ring,

a ∧ b = 0 implies either a = 0 or b = 0. Any abelian lattice-ordered group G can be made into an f-ring

by defining ab = 0 for all a, b ∈ G. The ideals of an f-ring are called l-ideals. My project for the coming

three years will focus topologizing filters of l-ieals of f-rings in general and, in particular, l-ideals of f-rings

of continuous real-valued functions C(X) on a completely regular hausdorff space X.

In this project C(X) is a ring of continuous real-valued functions on a completely regular Hausdorf regular

space X. All rings here are commutative rings with unity and Ideal we mean proper ideal.

The objective of this project is to:

1. Characterize topologizing filters on l-ideals of f-rings.

2. To investigate the residuation property of topologizing filters on l-ideals of f-rings.

3. We specialize (1) and (2) for the case of f-rings of continuous real-valued functions C(X).

Work plan during the project years

1. During the first year of the project period we will recruit two potential PhD candidates. I will arrange

together with the department to invite the main supervisor to come and offer the basic courses designed

for them in collaboration with me.

2. By the mid of the second year the PhD candidates will take comprehensive exam and in the mean time

we will give articles and prepare them to read and present in the way they understood and will get

feedback from the co-supervisor as well as from Algebra Research Group and their by prepare their

research proposal. We plan to defend their research proposal by the end of the second year.

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3. During the third year of the project period the PhD students will go to the abroad for three or more

months to the place where their main supervisor is found and do their research work and present their

work for him. The remaining time will be devoted to write their thesis and will complete their study

in their fourth year.

1.3 Research on Complex analysis and D-module theory

Persons involved:

1. Addisalem Abathun (Addis Ababa University)

2. Tilahun Abebaw (Addis Ababa University)

3. Sebsibew Atekaw (Gonder University)

4. Berhanu Bekele (Addis Ababa University),

5. Elizabeth Wulcan (Chalmers University of Technology, Sweden),

6. Rikard Bogvad (Stockholm University, Sweden)

7. Shiferaw Berhanu (Temple University, USA)

1.3.1 Bernstein-Sato Ideals

Bernstein-Sato ideals are interesting invariants of systems of differential equations for a variety of reasons

connected to geometry of complex varieties.

Building on Abebaw and Bogvad ([43, 44], see also Abebaw’s thesis at Addis Ababa University), Sebsibew

Atekaw (Gonder University) recently computed these ideals for a central line arrangement (see his 2016

thesis at Addis Ababa University and his licentiate at Stockholm University).

The main project would be to extend this to more examples (of a hyperplane arrangement type), and in

these examples explore the connection to geometry. The department at Addis Ababa has a strong connection

with the Singular people at Kaiserslautern (through the chair, Dr. Berhanu Bekele), with one members of

the group was already was involved in Sebsibew Atekaw’s thesis.

There are several natural concrete problems, some of which would be suitable for a graduate student. We

mention three:

1. How do the BS ideals behave with respect to restriction and deletion of arrangements?

2. Can one get an overview of low-dimensional behaviour? Is it possible to make the existing Singular

algorithm more efficient?

3. There is an equivalent formulation of [AB1], [AB2] in terms of perverse sheaves by Goncalvez (licentiate

thesis at Stockholm University, 2015). In general perverse sheaves have a quiver type description. Can

such a description be used to formulate a quiver type definition of BS-ideals, in the hope of simplifying

calculations of BS-ideals?

1.3.2 Euler-Poincare currents, zero-counting measures and hypergeometric func-

tions

There is a very large literature on the asymptotic behaviour of zero sets of sequences of polynomials, mainly

dealing with the case of one variable. Zero-sets can be understood in terms of positive measures or more

generally in higher dimensions in terms of positive currents (see Demailly’s book).

9

Addisalem Abathun ([41, 42]) calculates, in her 2016 Ph. D. thesis at AAU, local asymptotics for generalized

hypergeometric polynomials (in one variable), and she also makes (and proves for the Gauss hypergeometric

polynomials) a conjecture on the global behaviour in a degenerate case, studied by Duren, Driver et al

(see references in [41, 42]. This conjecture seems possibly within reach by inductive methods based on the

saddle point method (see [42]), and would be suitable for a graduate student advised by Abathun and her

collaborators.

The preceding project used measures to understand zero-sets. A natural generalization would be to study

asymptotic properties of certain hypergeometric systems of differential equations (in several variables), the

so called GKZ systems; then positive currents should be the appropriate apparatus to study zero-sets of

polynomial solutions. Bogvad has an ongoing project to generalize the use of piecewise harmonic functions

to plurisubharmonic functions, and apply this in particular to GKZ systems. Certain cases of asymptotic

behaviour in higher dimensions are easy to analyze (see [46]), and similar problems would be suitable for

Ph. D. students, perhaps mentored in a group.

1.4 Analytic Number Theory

Persons Involved:

1. Abdulkadir Hassen (Rowan University ,USA)

2. Hunduma Legesse Geleta (Addis Ababa University)

Introduction

Currently, three graduate students are working in the field of Analytic Number Theory ( Eyerus W/Yohannes,

Demessie Ergabus and Tewolde G/Egziabher).

Eyerus has been working on Fractional Hypergeometric Bernoulli Polynomials, which is a generalization

of Hypergeometric Bernoulli polynomials. She also study the connection of these family of polynomials

to the Hermite Polynomials. Classical Bernoulli numbers Bk and the Bernoulli polynomials Bk(x) are of

fundamental importance in several parts of Analysis and in the Calculus of Finite Differences and have

applications in various other fields such as Statistics, Numerical Analysis, Combinatorics, and so on. The

Hermite Polynomials are also play an important role in various fields and have interesting properties and ap-

plications. Some of the generalizations of the classical Bernoulli Numbers and Polynomials were investigated

by (Dichler, A. Hassen, H. Nguyen) and the references cited in each of these earlier works. The classical

Bernoulli Polynomials and their generalizations by hypergeometric Bernoulli polynomials are usually defined

respectively by means of generating functions.

Demessie Ergabus, just began research on ”Results Related to Hamburger’s Theorem.” He uses the

theory of automorphic integrals with period functions, specifically the log-polynomial sums on the Hecke

groups. He characterize the log-polynomial sums with different parameters.

The third student, Tewolde G/Egziabher has began working on ”Dirichlet Series with Functional Equa-

tions and Arithmetical Identities.” He also uses automorphic integrals and try to establish analogous equiv-

alence relations of the functional equation with automorphic integrals involving rational period function for

the Hecke group.

Below we describe the proposed problems:

Title: Dirichlet Series with Functional Equations and Arithmetical Identities.

John H.Hawkins and Marvin I.Knopp showed the equivalence of the functional equation to the modular

integral with rational periodic functions of weight 2k , k ∈ Z on the theta group Γθ. Sister Ann M.Heath

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derived the equivalence of the functional equation to two arithmetical identities developed in the Hawkins

and Knopp context, but with respect to the full modular group Γ(1).

The full Modular group, Γ(1), is the set of matrices M =

(a b

c d

)with a,b,c,d ∈ Z ad− bc = 1. Γ(1) may

be considered as group of linear fractional transformations acting on the Riemann sphere:

Mz =az + b

cz + d.

Thus, M, and -M can be identified. Γ(1) is generated by S and T where S(z) = z + 1 and T (z) = −1z and

T 2 = (ST )3 = I or as matrices by S =

(1 1

0 1

), T =

(0 −1

1 0

)Rational periodic function for modular group Γ(1) are rational functions q(z) which occur in functional

equation of the form

F (z + 1) = F (z).

z−2kF

(−1

z

)= F (z) + q(z).

wherek ∈ Z. F is a function of holomorphic in H which restricted in the growth at ı∞.This growth can be

written as the Fourier expansion of F(z) at ı∞. that is

F (z) =

∞∑n=0

ane2πinz,

where =z = y > yo > 0 and µ ∈ Z . The function F is called an entire Modular integral of weight 2k and

period q(z).Where

q(z) =∑Lr=1 Crf(z, 0) +

∑Pj=1

∑kr=1 Crjfr(z, αj).

fr(z, 0) = z−r − (−1)rz−2k+r, fr(z, αj) = (z − αj)− (−1)rα−rz−2k+r(z + 1

αj

)−r.

Φ(s) = (2π)−s

Γ(s)φ(s).

φ(s) is the Dirichlet series φ(s) =∑∞n=1

anns , with s = σ + it such that φ(s) converges absolutely for

σ > 1 + β.

Φ (2k − s)− i2kΦ (s) = Rk (s) . where

Rks (s) = i2kp∑j=1

k∑r=1

Crj

(−1

αj

)rB (s, r − s) (iαj)

s −B (2k − s, r − (2k − s))(i−sα2k−s

j

).

The functional equation

Φ(2k − s)− i2kΦ(s) = Rk(s)

is equivalence to

1

Γ(ρ+ 1)

∑n≤x

′an(x− n)ρ =

(−1)k

(2π)ρ

∞∑n=1

an

(xn

) ρ+2k2

Jρ+2k(4π√nx)

−p∑j=1

k∑r=1

Crj

(−1

α j

)r(iαj)

r(2π)rxr+ρ

Γ(r + ρ+ 1)

×φ(r, r + ρ+ 1;−iαj2πx)

+

p∑j=1

k∑r=1

Crj

(−1

α j

)r(i)−2k(2π)2kx2k+ρ

Γ(2k + ρ+ 1)

×φ(r, 2k + ρ+ 1;

−2πx

iαj

)−

L∑n=k

(−i)n(2π)nxn+ρ

Γ(n+ ρ+ 1)− i2k−n(2π)2k−nx2k−n+ρ

Γ(2k − n+ ρ+ 1)

.

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and

Φ(2k − s)− i2kΦ(s) = Rk(s)

is equivalent to (−1

s

d

ds

)ρ [1

s

∞∑n=1

ane−s√n

]

= i−2k26k+ρπ2k− 12 Γ(2k + ρ+

1

2)

∞∑n=1

an

(s2 + 16π2n)2k+ρ+ 12

+−2ρ√πs2ρ+1

P∑j=1

k∑r=1

crj

(− 1

αj

)r Γ(r + ρ+

1

2)Ψ(2)

(r,−ρ+

1

2;

s2

8πiαj

)

− αj2k

(iαjs

2

8π

)r−2k

Γ(2k + ρ+1

2)Ψ(1)

(r, r − 2k − ρ+

1

2;iαjs

2

8π

)+

2ρ√πs2ρ+1

L∑m=k

cm

(8πi

s2

)2k−m

Γ(2k −m+ ρ+1

2)

−(−8πi

s2

)mΓ(m+ ρ+

1

2),

our main goal is to establish analogous equivalence relations of the functional equation with automorphic

integrals involving rational period function for the Hecke group G(λ).

The Hecke group G(λ). is the group of linear fractional transformation generated by

T (z) = −1z and Sλ(z) = z + λ, for all z ∈ H and λ ∈ R+.

Outline of the Proposal for the Period 2018-2020

Year 2018:

We also have a plan to accept at least two students in the academic year 2017/2018 and these two students

will be on course work. The other three students complete some of the proposed problems and present the

results in local and international conferences.

Year 2019:

The two new students began seminar works and defend their proposal. The three students finalize their

work, write dissertations for defence and defend at Addis Ababa University.

Year 2020:

We will accept another two students and they will be on course work. The other two students complete some

of the proposed problems and present the results in local and international conferences.

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Component II: Partial Differential

Equations, Functional Analysis, and

Complex Analysis

2.1 ITERATIVE APPROXIMATION OF FIXED POINTS OF NON-

LINEAR OPERATORS AND THEIR APPLICATIONS

Persons Involved:

1. Mengistu Goa Sangago (Addis Ababa University)

2. Habtu Zegeye Hailu (Botswana International University of Science and Technology, Botswana)

3. Tadesse Bekeshie Gerbaba (AAU)

4. Sebsibe Teferi Woldeamanuel (Kotebe Metropolitan University, Ethiopia)

5. Tesfalem Hadush Meche (Jigjiga University, Ethiopia)

6. Wondimu Woldie Kassu (Wolaita Sodo University, Ethiopia)

INTRODUCTION

The study of nonlinear operators had its beginning about the start of the twentieth century with investiga-

tions into the existence properties of solutions to certain boundary value problems arising in ordinary and

partial differential equations.

In the last century, because of its applications in nonlinear analysis, optimization, economics, game theory,

and so on, the theory of variational and equilibrium problems with their relation to fixed point theory has

emerged as a rapidly growing area of research. In the recent past, many mathematicians devoted their

attention to study the existence of solutions of variational inequalities, equilibrium problems, split feasibility

problems and fixed point problems. The iterative methods for finding the approximate solutions of variational

problems, equilibrium problems, fixed point problems and split feasibility problems are also another key area

of interest due to its direct connection with applied sciences. Very recently, several iterative methods have

been investigated to solve the variational inequality problems, equilibrium problems and split feasibility

problems in such a way that the solutions of these problems are the fixed points of some nonlinear operators.

RESEARCH PROBLEMS

The earliest techniques, largely devised by Picard [174], involved the iteration of an integral operator to obtain

solutions to such problems. In 1922, these techniques of Picard were given precise abstract formulation by

13

Banach [175] and Cacciopoli [176] which is now generally called the Banach contraction mapping principle.

The contraction mapping principle involves the existence and uniqueness of solutions of differential equations

and is one of the most useful fixed point theorems. As a consequence of the importance of its applications,

fixed point theory has developed into an area of independent interest, for example, (i) variational inequality

(ii) random fixed points, (iii) equilibrium problems, etc.

The theory of variational Inequalities, which was introduced by Stampacchia [177], has emerged as an in-

teresting and fascinating branch of mathematical and engineering sciences. The ideas and techniques of

variational inequalities are being applied in structural analysis, economics, optimization, operations research

etc. It has been shown that variational inequalities provide the most natural, direct, simple, and efficient

framework for a general treatment of some problems arising in various fields of pure and applied sciences. In

recent years, there have been considerable activities in the development of numerical techniques including

projection methods, Wiener-Hopf equations, auxiliary principle, and descent framework for solving varia-

tional inequalities (see, e.g., Noor [178], Poliquin et al. [179], and the references therein).

Equilibrium problems which were introduced by Blum and Oettli [180], have had a great impact and in-

fluence in pure and applied sciences. It has been shown that numerous problems in physics, optimization

and economics reduce to finding a solution of an equilibrium problem. The equilibrium problem includes

and unifies a wide class of problems, such as variational inequality problems, complementarity problems,

Nash equilibrium problems, optimization problems, fixed point problems. One of the most important and

interesting topics in the theory of equilibrium is to develop algorithms for solving equilibrium problems and

their generalizations[see for example: Wang et al. [181], Meche et al.[182], Ansari et al.[183], etc].

In this project, we plan to study the problems of existence, uniqueness and regularity of solutions of varia-

tional inequality problems, generalized equilibrium problems and fixed point of random operators. Next, we

plan to develop efficient iterative methods for approximating these solutions and investigate applications of

such operators in Economics, physical sciences and Biomathematics. In addition, we investigate techniques

in analysis which support the study of mathematical modeling which improves the social and economic

problems of the society.

EXPECTED GRADUATES

Graduates from the program will be readily employable by academic units of Mathematics at institutions of

higher learning and also ready to undertake research in different areas of Mathematical Sciences. This will

strengthen the quality and quantity of science in Ethiopia. There is one PhD candidate on the pipline who

will complete his study in the 2017/2018 academic year. The research team in collaboration will supervise

10 MSc students and 6 PhD students in the research area.

EXPECTED RESEARCH OUTPUTS

The research team will publish six research articles in international reputable journals and ten conference

proceedings.

2.2 Problems in Partial Differential Equations

Persons Involved:

Ahmed Mohammed (Ball State University, USA)

The following proposals are meant to give a rough guideline for problems that will lead to at least two

dissertations in the area of partial differential equations. These proposals may lead to more questions that

14

may need deeper investigations. PhD candidates who choose to work on problems related to the proposal

will gain extensive research experience in an area that is of interest to a wide segment of researchers in

PDE. In addition to my current student from Hawassa University, I anticipate to supervise two or more

PhD students from Addis Ababa University with an expected total publication of at least four papers in

very good journals. My current student from Hawassa university is slated to finish in one year and we are

expecting two publications out of his dissertation.

I - Infinite Boundary Value Problem for fully nonlinear PDEs

For fixed constants 0 < λ ≤ Λ, let Aλ,Λ := A ∈ Sn : λIn ≤ A ≤ ΛIn, where Sn is the set of n × n real

symmetric matrices. By X ≤ Y for X,Y ∈ Sn we mean Y −X is a positive semidefinite matrix.

We recall the extremal Pucci operators:

P+λ,Λ(X) = Λ tr(X+)− λ tr(X−) = Λ

∑ei(X)>0

ei(X) + λ∑

ei(X)<0

ei(X),

P−λ,Λ(X) = λ tr(X+)− Λ tr(X−) = λ∑

ei(X)>0

ei(X) + Λ∑

ei(X)<0

ei(X),(1)

where X+ and X− are the positive and negative parts of X, respectively, and ei(X), i = 1, . . . , n, are the

eigenvalues of X, not necessarily distinct. Here, tr(B) denotes the trace of matrix B.

For (x, t, p,X) ∈ Ω× R× Rn × Sn, we set:

M+(t, p,X) := P+λ,Λ(X) + γ(x)|p|+ χ(x) t−, (2)

M−(t, p,X) := P−λ,Λ(X)− γ(x)|p| − χ(x) t+, (3)

where γ, χ ∈ C(Ω) are non-negative functions. Let H : Ω×R×Rn×Sn → R be a continuous function such

that for all x, y ∈ Ω, t, s ∈ R, p, q ∈ Rn and X,Y ∈ Sn the following hold:

(H-1) H(x, 0, 0, O) = 0, and

M−(t− s; q − p;Y −X) ≤ H(x, t, q, Y )−H(x, s, p,X) ≤ M+(t− s; q − p;Y −X).

(H-2) |H(x, t, p,X)−H(y, t, p,X)| ≤ K‖X‖|x− y|+ ω((1 + |p|)|x− y|)

for a constant Kγ0 and a function ω : R+ → R+ such that ω(0+) = 0.

We remark that condition (H-1) implies that H is uniformly elliptic; that is

λ tr(Y −X) ≤ H(x, t, p, Y )−H(x, t, p,X) ≤ Λ tr(Y −X) forX ≤ Y ;

and that H is non-increasing in t; that is H(x, t, p,X)−H(x, s, p,X) ≤ 0 for s ≤ t.One objective of this project is to study the existence and uniqueness of positive solutions to

H(D2u,Du, u, x) = f(x, u) in Ω, and u =∞ on ∂Ω (4)

under the structural conditions (H-1) and (H-2) on H.

In dealing with (4) we are guided by the recent work [66] when H has the form H(X, ξ, r, x) = tr(A(x)X) +

b(x)T ξ + c(x)r. Here b(x)T denotes the transpose of b(x) ∈ Rn. In investigating existence of solutions

to (4), the function γ and χ in (2) and (3) are allowed to be unbounded on Ω, but are controlled by the

requirement that γ(x) = o(dist(x, ∂Ω)) and χ(x) = o(dist2(x, ∂Ω)) as dist(x, ∂Ω) → 0. However, boundary

asymptotic estimates and uniqueness of solutions to (4) will require that γ and χ be bounded functions,

which without loss of generality, can be taken as positive constants. The project’s objective is to investigate

the structure on f(x, t) in order for problem (4) to admit a solution.

15

II - A Harnack inequality for the sub-Laplacian in Carnot groups

We illustrate the concept of Harnack inequality with non-negative harmonic functions on an open set Ω ⊂ Rn,

that is non-negative solutions of ∆u = 0 in Ω. It is well known that there is a constant C > 0 such that

maxB

u ≤ C minB

u for all non-negative harmonic functions u in Ω and all balls B contained in Ω. The important

point here is that C does not depend on the solution u. The question of whether such inequality holds for

nonnegative solutions of other PDEs has been of interest for a long time, and many important developments

have occurred over the years. The interest in such inequalities stems from the indispensable roles they play

in the investigation of the existence, a priori estimates and regularity of solutions of PDEs, as well as in the

understanding of other qualitative behaviors of such equations. The most recent and remarkable application

was demonstrated in Perelman’s use of a version of Harnack inequality for Ricci flow to settle the century-old

Poincare conjecture.

Let G be a Carnot group of step r ∈ N. That is, G is a connected and simply connected Lie group whose

Lie algebra g has a stratification

g = V1 ⊕ V2 ⊕ · · · ⊕ Vr

into direct sums of finite dimensional vector spaces Vj over R such that

[V1, Vi−1] = Vi, for i = 2, · · · , r and [V1, Vr] = 0.

Let X1, · · · , Xn be a set of vector fields that form a basis of V1. The degenerate elliptic PDE Lu =∑nj=1X

2j u is called a sub-Laplacian. A famous theorem due to Hormander asserts that this operator is

hypoelliptic. In other words, if Lu = f for some distributions u and f and f is smooth, then u is also

smooth. This property of the differential operator L makes it extremely useful in PDEs. Given an open and

bounded set Ω ⊂ G, one important problem is to investigate sufficient conditions on f : R+ → R+ in order

for Harnack inequality to hold for nonnegative solutions of

(2.1) Lu = f(u) in Ω.

A crucial step to understanding this is to get a global L∞ estimate of all non-negative solutions of (2.1).

One way this can be achieved is by studying the problem Lw = f(w) in B(ξ,R) and w = ∞ on ∂B(ξ,R)

where B(ξ,R) is the so-called Koranyi ball of radius R > 0 with center ξ. This already points to working

with a class of functions f that satisfy the so-called Keller-Osserman condition, a condition that forces f

to grow very fast at infinity. However, again guided by our recent work, (see [64] and [65]), we will require

a stronger condition on f and this is expected to lead to the desired Harnack inequality for non-negative

solutions of (2.1).

Carnot groups also provide a framework for the study of large solutions. As far as I am aware there has

not been a systematic study of such solutions in the context of Carnot groups. Therefore there is a trove of

problems that may need investigation in such settings.

2.3 Problems in Operator-Related Function Theory

Persons Involved:

1. Tadesse Bekeshie(Addis Ababa University)

2. Tesfa Mengestie (Western Norway university of applied sciences, Norway)

Summary of the project description:

This project deals with research that takes place in the interface between operator theory, complex analysis

and functional analysis. A celebrated example of the fruitful interaction between these two fields, opening

16

up a new area of research, was A. Beurling’s theorem [72] that gives a complete description of the invariant

subspaces of the shift operator on a separable Hilbert space by relating it to the operator of multiplication

by the independent variable z on the Hardy space H2. Moreover, the invariant subspace problem 1 reduces

to a special question about invariant subspaces of the operator of multiplication by z on the Bergman space,

as it was shown by Hedenmalm, Richter and Seip in 1996 [76].

Another relevant example is the Carleson measures problem on certain spaces of analytic functions. Such

problem reduces to the boundedness problem for the two weighted Hilbert transform [80]. Nazarov and

Volberg [80] used this and refuted the then Cohn’s [75] conjecture about Carleson measure for model spaces2

by finding a counterexample among Hilbert operators acting on weighted L2 spaces. These problems were

further studied by Y. Belov, T. Mengestie, and K. Seip in [71, 78] where complete interpolating sequences

are described in terms of boundedness of two weight discrete Hilbert transforms.

Another example illustrating the fruitful interplay between the three theories is the question of interpolating

sequences in model subspaces K2I of H2. It has been related to the invertibility problem for certain Toeplitz

operators when the sequence of points are close to the spectrum of the generating inner function I. In this

project we aim to continue studying problems in operator-related function theory as specifically described

below.

The team will mainly work on spectral and dynamical properties of Volterra-type integral operators, compo-

sition and weighted compositor operators, and the differential operator acting on generalized Fock spaces. In

contrast to the boundedness and compactness properties which are well studied, there are considerably fewer

investigation on other spectral and dynamical properties of the operators. The theory of the Volterra-type

integral operators began with a work A. Calderon on commutators of singular integral operators [73] in

harmonic analysis in 1965. The theory was introduced to the complex analysis society by Pommerenke [81]

in 1997 in his work on study of BMO functions.

For holomorphic functions f and g, the Volterra-type integral operator Vg and its companion Jg are defined

by

Vgf(z) =

∫ z

0

f(w)g′(w)dw and Jgf(z) =

∫ z

0

f ′(w)g(w)dw.

Applying integration by parts in any one of the above integrals gives the relation

Vgf + Jgf = Mgf − f(0)g(0),

where Mgf = gf is the multiplication operator of symbol g.

These operators arise naturally in the study of semigroups of composition operators on spaces of analytic

functions which in turn is applied to study properties of the resolvent operators of the semigroups [82]. Ques-

tions about the symbols g for which the operators Vg and Jg have bounded, compact, and other operator

theoretic properties have been a subject of high interest. The operators have especially attracted lots of in-

terest and have been extensively studied for many spaces following the works of Aleman and Siskakis [69, 70]

on Hardy and Bergman spaces. Analogous results on some general weighted Bergman spaces were shown

in [70] with Bloch and little Bloch spaces replacing BMOA and VMOA in the characterization of bound-

edness and compactness. For more information on the subject, we refer to [68, 83] and the references therein.

Fixed point approach to the invariant subspace problem:

Despite the simplicity of its statement, the invariant subspace problem remains one of the most famous and

elusive open questions in functional analysis. Our team will also study this problem thoroughly following

1The invariant subspace problem asks whether every bounded linear operator on a separable Hilbert space has a nontrivial

closed invariant subspace.2Cohn’s [75] conjecture in model spaces was that uniform embedding of the reproducing kernels characterizes the Carleson

measures for on model subspaces.

17

the fixed point approach. The general idea of the fixed point approach to the invariant subspace problem

is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an

operator on a Banach space. This idea was started by [77] where as the idea of using fixed point theorems

for multivalued maps in the search for invariant subspaces was first introduced by Androulakis. Lomonosov

gave a very short proof using the Schauder fixed point theorem that if the operator T on a Banach space

commutes with a non-zero compact operator, then T has a non-trivial invariant subspace.

2.4 Optimization and Eigenvalue Problems

Persons Involved:

1. Giovanni Porru (Cagliari University, Italy )

2. Tadesse Abdi (Addis Ababa University)

The project in this section will be carried out by Feyissa Kebede, a new PhD student who has recently

started working under the guidance of Professor Giovanni Porru and Dr. Tadesse Abdi.

The project concerns optimization problems related to the eigenvalue problem

−∆u = λg(x)u, u > 0 in Ω, u = 0 on ∂Ω,

where the weight g(x) is a function which is positive in a subset with a positive measure, and λ is the

principal eigenvalue, Ω ∈ RN is a bounded smooth domain. Here λ depends on g, so we write λg. We

assume g belongs to a class of rearrangements G. Two measurable functions f(x) and g(x) defined in Ω are

said to have the same rearrangement if

|x ∈ Ω : f(x) ≥ β| = |x ∈ Ω : g(x) ≥ β| ∀β ∈ R.

If g0(x) is a bounded function positive in a subset of positive measure, we denote by G = G(g0) the class of

its rearrangements. We shall consider the optimization problems

ming∈G

λg and maxg∈G

λg.

Results of existence, non-existence and uniqueness about these problems are already known, see [87] and

references therein. Thus we plan to investigate the properties of the maximizers and minimizers whenever

they exist.

In dimension N = 2, particularly interesting is the case when g(x) = χE − χF , where E and F are disjoint

subsets of Ω with fixed measures |E| = α > 0 and |F | = β ≥ 0, 0 < α + β < |Ω|. Biologically, such opti-

mization problems are motivated by the question of determining the most convenient spatial arrangement of

favorable and unfavorable resources for a species to survive or to decline. The question may have practical

importance in the context of reserve design or pest control. Assuming Ω has a special geometry, we plan to

find shape and location of E and F when g(x) = χE − χF corresponds to an optimal configuration.

Another interesting case is when g(x) = hχD + H(1 − χD), where h and H are positive constants with

h < H and |D| = β ∈ (0, |Ω|). The corresponding eigenvalue problem is related to the vibration of an elastic

membrane fixed at the boundary ∂Ω, built out of two different materials. The optimizers of λg correspond

to the distribution of the materials in Ω so to minimize or maximize the first mode of the vibration. We

shall investigate the following question: does a minimizer Dm or a maximizer DM inherit some symmetry

property of Ω? These phenomena are often referred as symmetry preserving or symmetry breaking.

By the same methods used to discuss the optimization of the eigenvalues λg, one can investigate the opti-

mization of the energy integral

Ig =

∫Ω

|∇ug|2dx,

18

where u = ug is the solution of the boundary value problem

∆u = g(x) in Ω, u = 0 on ∂Ω.

Similarly to the previous case one can investigate the optimization problems

ming∈G

Ig and maxg∈G

Ig.

Results of existence and uniqueness about these problems are well known, see [84, 85, 86]. Also in this case

we plan to find properties of the maximizers and minimizers whenever they exist.

Expected outcomes

Feyissa Kebede who is investigating this problem is expected to finish by 2020. At least two publications are

expected from the work in the dissertation. By the end of 2018 we expect to communicate the first article

for publication in a reputable Journal and the second paper before the defense.

2.5 Analysis of Boundary-Domain Integral Equations in 2D

Persons Involved:

1. Tsegaye Gedif Ayele (Addis Ababa University, Ethiopia)

2. Sergey E. Mikhailov (Brunel University, UK)

Background

Partial differential equations (PDEs) with variable coefficients often arise in mathematical modelling of

inhomogeneous media (e.g. functionally graded materials or materials with damage induced inhomogeneity)

in solid mechanics, electromagnetics, thermo-conductivity, fluid flows through porous media, and other areas

of physics and engineering.

Generally, explicit fundamental solutions are not available if the PDE coefficients are not constant, preventing

reduction of boundary value problems (BVPs) for such PDEs to explicit boundary integral equations (BIEs),

which could be effectively solved numerically. Nevertheless, for a rather wide class of variable-coefficient PDEs

it is possible to use instead an explicit parametrix (Levi function) associated with the fundamental solution of

the corresponding frozen-coefficient PDEs, and reduce BVPs for such PDEs to systems of boundary-domain

integral equations (BDIEs) for further numerical solution of the latter, see, e.g., [161, 166, 167, 169, 170, 171]

and references therein. The BDIE analysis is useful for the BDIE discretisation and numerical solution to

obtain by this way a numerical solution of the associated BVP.

Although the theory of BDIEs in 3D is well developed, cf. [167, 163, 161, 172], the BDIEs in 2D need a

special consideration due to their different equivalence properties. As a result, one needs to set conditions on

the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based

integral layer potentials and hence the unique solvability of BDIEs.

Problem Formulation

Let Ω be a domain in R2 bounded by a smooth curve ∂Ω, and let n(x) be the exterior unit normal vector

defined on ∂Ω. The set of all infinitely differentiable function on Ω with compact support is denoted

by D(Ω). The function space D′(Ω) consists of all continuous linear functionals over D(Ω). The space

Hs(R2), s ∈ R, denotes the Bessel potential space, and H−s(R2) is the dual space to Hs(R2). We define

19

Hs(Ω) = u ∈ D′(Ω) : u = U |Ω for some U ∈ Hs(R2). Note that H1(Ω) coincides with the Sobolev space

W 12 (Ω), with equivalent norms. The space Hs(Ω) is the closure of D(Ω) with respect to the norm of Hs(R2),

and for s ∈ (− 12 ,

12 ), the space Hs(Ω) can be identified with H(Ω), see e.g. [162].

We shall consider the scalar elliptic differential equation

Au(x) =

2∑i=1

∂

∂xi

[a(x)

∂u(x)

∂xi

]= f(x) in Ω, (5)

for a(x) ∈ C∞(R2), a(x) > c > 0, where u is unknown function and f is a given function in Ω.

Boundary-Domain Integral Equations for Dirichlet BVP associated to PDE (5) are analysed in [158] and for

Neumann BVP in [157]. Using an appropriate parametrix (Levi function), the corresponding problems are

reduced to some direct segregated systems of BDIEs. Due to the different equivalence properties of BDIEs

in 2D, authors set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility

of corresponding parametrix-based integral layer potentials as well as invertibility for the hypersingular

operator (the co-normal derivative of the double layer potential) and hence the unique solvability of BDIEs.

• Problem 1. Developing results in [157, 158] further, we plan to dirive and investigate BDIEs for BVPs

associated to PDE (5) in united as well as localised formulations both for bounded and unbounded

domains which are studied in [171, 172, 173] for 3D case.

The two-operator approach formulated in [168] for a certain non-linear problem employs a parametrix of

another (second) PDE, not related with the PDE in question, for reducing the BVP to a BDIE system.

Since the second PDE is rather arbitrary, one can always chose it in such a way, that its parametrix is known

explicitly. The simplest choice for the second PDE is the one with an explicit fundamental solution.

In [159, 160], to analyse the two-operator approach, one of its linear versions is applied to the mixed (Dirichlet-

Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE and reduce it to four different

BDIE systems.

• Problem 2. Extending the results both in [157, 158] and [159, 160], we plan to analyse the two-

operator versions of BDIEs in 2D.

In [165], segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and asso-

ciated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial

differential equation with a variable coefficient are formulated. The PDE right-hand sides belong to the

Sobolev space H1(Ω) or H1(Ω), when neither classical nor canonical co-normal derivatives are well defined.

Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non- uniqueness, and

as well as Fredholm property and invertibility of the BDIE operators are analysed in Sobolev (Bessel poten-

tial) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate

finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

• Problem 3. Based on results and conclusions of [165], a version of the two-operator approach differing

from the one considered in [159, 160] is to be applied for analysis of extended BDIEs for Dirichlet BVP

for a second-order scalar elliptic differential equation with variable coefficient and with right-hand side

from H−1(Ω).

Strategy and Plan

Currently 5 (Five) Ph.D candidates are working under the general theme: Analysis of Boundary-Domain

Integral Equations (BDIEs) and one awarded his PhD in 2016.

In this project in collaborations with partner universities, we plan to supervise 2 M.Sc and 2 PhD students.

20

Expected outcomes, impact and dissemination

Knowledge gained during the sponsorship period of International Science Programme (ISP) will be eval-

uated and disseminated through publications in journals and reports at conferences,through postgraduate

programmes in particular training and advising of M.Sc and PhD students.

2.6 Problems on systems of first order partial differential equations

Persons Involved:

Shiferaw Berhanu (Temple University, USA)

Jemal Yesuf, a new doctoral student under the guidance of S. Berhanu will investigate the local and mi-

crolocal regularity of solutions u of a system of first order partial differential equations with complex valued

coefficients. These first order partial differential equations arise from the study of involutive structures.

Consider a pair (M,V) where M is a manifold, and V is a subbundle of the complexified tangent bundle

CTM which is involutive, that is, the bracket of two sections of V is also a section of V. We will refer to

the pair (M,V) as an involutive structure. The involutive structure (M,V) is called locally integrable if

the orthogonal of V in CT ∗M is locally generated by exact forms. A function (or distribution) u is called a

solution of the involutive structure (M,V) if Lu = 0 for every smooth section L of V. When (M,V) is locally

integrable, the local and microlocal regularity of solutions u has been studied extensively during the past 30

years. In particular, when (M,V) is a CR manifold (that is, V and its conjugate V have satisfy V ∩V = 0),and the bundle V is locally integrable, the regularity of the solutions continues to attract the attention of

many researchers. It is well known that if the manifold M and the bundle V are real analytic and the pair

(M,V) is involutive, then it is locally integrable. However, when the pair is only C∞, then local integrability

may not hold. The general problem that Yusuf will investigate concerns conditions that guarantee regularity

of solutions for nonlocally integrable structures. The first specific problem he will study can be described as

follows: suppose (M,V) is an involutive structure. A submanifold X ⊆M is called maximally real if at each

p ∈ X, its complex tangent space satisfies CTXp⊕Vp = CTpM. Suppose Y ⊆M is another maximally real

submanifold and the point p ∈ X ∩Y . Suppose the covector ξ ∈ T ∗pM is a direction in the characteristic set

of the bundle V (which means that ξ annihilates V), and ιX : X → M, ιY : Y → M denote the inclusion

maps, then if (M,V) is locally integrable, for any solution u on a neighborhood of p, i∗X(ξ) /∈ WF(uX) if

and only if i∗Y (ξ) /∈ WF(uY ). Here uX , uY denote the restrictions of u to X and Y respectively, and WF

denotes the wavefront set which means the C∞ wavefront set when the pair (M,V) is smooth, and the real

analytic wavefront set when (M,V) is real analytic. Yusuf will explore whether this result holds when the

pair (M,V) is not locally integrable. In addition, in the locally integrable case, when (M,V) is in the Gevrey

call Gs, he will explore whether the preceding result is valid for the Gevrey wavefront set of the restrictions

uX and uY when the maximally real submanifolds X and Y are in some Gevrey class Gt, t related to s. We

note that in the important case when the manifoldM = Cn and it is equipped with the bundle generated by

the Cauchy Riemann operators ∂∂zj

, 1 ≤ j ≤ n, a submanifold X is maximally real if and only if X is totally

real of maximal dimension n. The solutions in this case are the holomorphic functions, and it is not hard to

see that such sets X are sets of uniqueness for holomorphic functions in the sense that if two holomorphic

functions u and v agree on X, then they agree on a neighborhood of X. A similar result holds for any

locally integrable structure (M,V). That is, if X ⊆ M is a maximally real submanifold, and u and v are

solutions that agree on X, then they agree on a neighborhood of X. This is another reason why maximally

real submanifolds are very important in the study of involutive structures.

Microlocal regularity of solutions for first order linear and non linear nonlinear equations has been investi-

gated by several researchers including Abraham - Berhanu, Adwan-Berhanu, Barostichi-Petronilho, Berhanu,

21

Baouendi-Treves, Hanges-Treves, Lerner-Morimoto-Xu, and many other researchers. A key tool that has

been employed in these works is the FBI transform (FBI for Fourier-Bros-Iagolintzer), a nonlinear transform

of the form

Fu(x, ξ) =

∫Rm

eiξ·(x−x′)−|ξ||x−x′|2u(x′) dx′, x, ξ ∈ Rm

and it has been used in numerous works to study the regularity (smoothness, analyticity, and Gevrey

regularity) of solutions of linear and nonlinear partial differential equations.

A more general version of this transform has been introduced by Berhanu and Hounie A simple example of

this generalized transform includes the class

Fu(x, ξ) =

∫Rm

eiξ·(x−x′)−|ξ|p(x−x′)u(x′) dx′, x, ξ ∈ Rm

where p(x) =∑|α|=2k aαx

α is a real-valued, positive homogeneous polynomial. In his Ph.D. dissertation

defended in June, 2016 (under S. Berhanu), Abraham Hailu proved that these more general nonlinear Fourier

transforms (with p(x) as above) characterize the Gevrey spaces. Hussein will investigate whether Gevrey

regularity can be characterized by more general FBI transforms, for example, taking p(x) to be a real

analytic or even in a Gevrey class and satisfying some positivity assumption. He will also explore examples

that demonstrate the flexibility of the more general class of transforms developed by Berhanu and Hounie.

Specifically, he will look for a class of complex vector fields where the regularity of the solutions can be

studied more easily using these more general transforms rather than the classical FBI transform with a

quadratic phase function.

Expected outcomes

By the end of 2020, Jemal Yusuf is expected to finish his doctoral studies with at least one publication in a

good journal. S. Berhanu is ready to advise one more new student in this area of mathematics.

2.7 The Method of Upper and Lower Solutions for Nonlinear Bound-

ary Value Problems

Persons Involved

Tadesse Abdi (Addis Ababa University)

Introduction

The notion of lower solution of differential equations traces its origin back to Picard’s iteration whereas the

method of lower and upper solutions is linked to G. S. Dragoni. The method of successive approximations

due to E. Picard is regarded as the first step and early work in the theory of lower and upper solutions. G.

S. Dragoni introduced the method of lower and upper solutions for nonlinear Dirichlet problem,u

′′= f(x, u, u

′)

u(a) = α

u(b) = β

with the assumption of continuity on f : [a, b]× Ω→ R, Ω ⊂ R2.

This method guarantees existence of a solution localized between a lower and upper solution by appeal-

ing to a monotone iterative scheme [29]. In his 1998 paper C. De Coster et al [22] proved existence and

22

localization of solution on a bounded domain Ω ⊂ RN for second order elliptic BVP for which the maximum

principle applies by assuming that f is a nonlinear Caratheodory function. Existence of solutions of Cauchy

problems by way of this method have been proved by Perron and extension of Perron’s result to system of

initial value problems was credited to Muller [31]. Monotone iterative schemes along with the method of

lower and upper solutions render an efficient technique to prove existence of solution of differential equations

of first and higher order. This method has been employed to probe various types of differential equations,

namely ODEs, PDEs and FDEs [29]. It has found wider application in the study of boundary value problems

of nonlinear elliptic and parabolic types long back [32] while its treatment for hyperbolic partial differential

equations is relatively recent until it is considered by V. Lakshmikanthan [29].

Problem Statement and Plan

We consider the Dirichlet problem −∆u = f(x, u,∇u) in Ω

u = 0 on ∂Ω

where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω. We study this problem assuming that a

lower solution η(x) ∈ C2(Ω) and an upper solution ζ(x) ∈ C2(Ω)) satisfying η(x) ≤ ζ(x) ∀x ∈ Ω is available.

A function η is a lower solution if

−∆η ≥ f(x, η,∇η)

and a function ζ is an upper solution if

−∆ζ ≤ f(x, ζ,∇ζ)

With these functions in place, we seek for appropriate conditions on f that ensures existence of positive

solution. With the information of existence of a solution u and its location η(x) ≤ u(x) ≤ ζ(x) we proceed

to propose an efficient iterative scheme for explicit construction of a stable classical solution.

There are various problems that are interesting for graduate students, namely, exploring existence of positive

solution under different regularity assumptions on f and with different boundary conditions. In particular

for a problem with gradient dependent nonlinearity on f one needs a priori bound on the derivative of the

unknown function and quadratic growth rate on f which can be realized by using Bernstein or Nagumo con-

dition. We try to workout on conditions that may weaken these growth conditions. In addition, investigation

of iterative schemes for problems with gradient dependent nonlinearity and for systems with singularity is

another direction of research.

23

Component III: Combinatorics &

Graph Theory

Persons Involved:

1. Samuel A. Fufa (Addis Ababa University)

2. Yirgalem Tsegaye (Addis Ababa University)

3. Melkamu Zeleke (William Paterson University, USA)

4. Akalu Tefera (Grand Valley State University, USA)

5. Zelealem B. Yilma (Carnegie Mellon University, Qatar )

Statement of the Project

Introduction

Combinatorics and Graph Theory are branches of modern mathematics that deal with the existence, enumer-

ation, analysis, and optimization of discrete structures and states. These two areas have become important

branches of mathematics since the middle of the twentieth century because of the major impact that com-

puters have had, and continue to have, in our society. Large-scale problems that were previously difficult to

solve are being solved with the advent of computers. But computers do not function independently; they

need to be programmed to carry out these functions. The basis for these computer programs often are

combinatorial algorithms or involve heavy use of results from graph theory. Analysis of these algorithms for

efficiency and effectiveness with regard to running time and storage requirements also requires knowledge of

array of tools in combinatorics and graph theory.

The overall objective the Combinatorics and Graph Theory Research Group is to introduce graduate students

interested in this area to open problems in the field and help them solve some of these problems and thereby

contribute to the advancement of mathematical research in Ethiopia. Over the next three to five years, the

group’s research will focus on:

1. The combinatorics of permutations,

2. The Riordan group and its applications in combinatorics, and

3. Problems in graph and hypergraph theory.

24

The Project

Combinatorics of Permutations

Let q = q1 · · · qk be a permutation in Sk (the symmetric group of order k), and let k ≤ n. We say that the

permutation p = p1p2 · · · pn ∈ Sn contains q as a pattern if there are k entries pi1 , pi2 , . . . , pik in p so that

i1 < i2 · · · < ik, and pia < pib if and only if qa < qb. Otherwise we say that p avoids q.

The study of permutation patterns began with Donald Knuth’s consideration of stack-sorting in 1968 [102].

Knuth showed that a permutation π can be sorted by a stack if and only if π avoids the pattern 231 ,

and that these permutations are enumerated by the Catalan numbers. Shortly thereafter, Robert Tarjan

investigated sorting networks in 1972 [116], while Vaughan Pratt showed in 1973 that the permutation π

can be sorted by a double-ended que if and only if for all k, π avoids patterns 5274 · · · (4k+ 1)(4k− 2)3(4k)1

and 5274 · · · (4k + 3)(4k)1(4k + 2)3 and every permutation that can be obtained from either of these by

interchanging the last two elements or the 1 and the 2 [106]. Because this collection of permutations is

infinite, it is not immediately clear how long it takes to decide if a permutation can be sorted by a double-

ended que. In 1984, Rosenstiehl and Tarjan presented a linear (in the length of π) time algorithm which

determines if π can be sorted by a double-ended que [107].

Another major influence on the early development of the study of permutation patterns came from enumer-

ative combinatorics, and focused on finding formulas for the number of permutations in Sn which avoid a

fixed permutation pattern q. Two patterns p and q are called Wilf-equivalent if Sn(p) = Sn(q) for every

positive integers n, where Sn(q) denotes the number of n-permutations that avoid the pattern q. Now,

define a relation ∼ on Sk by p ∼ q if and only if Sn(p) = Sn(q). Clearly ∼ is an equivalence relation and

an equivalence class determined by the relation ∼ is called a Wilf-class. The central theme in the theory

of pattern-avoiding permutations is to classify all patterns up to Wilf-equivalence [90]. Simion and Schmidt

showed using bijective methods that there is only one Wilf class in S3 [110]. After knowing the result for

patterns of length three, one may think that we will now return to patterns of length four, and then to the

general case, and obtain similar results. In fact, this is not the case and enumeration and classification of

permutations that avoid patterns of length greater or equal to 4 is not an obvious task and it is currently

an active area of research in enumerative combinatorics [90, 113, 114].

Our research group plans to focus on the following permutation combinatorics problems where graduate

students can be involved with minimal prerequisites:

1. Martin Klazar proved the famous Stanley-Wilf conjecture that for any permutation pattern q of length

k, Sn(q) ≤ cnq where cnq = 154k2(k2

k ) [101]. The constants that Klazar obtains for the Stanley-Wilf

conjecture are obviously large. For example, it can be shown using the fact that Sn(q) = 1n+1

(2nn

)for

patterns q of length k = 3 that Sn(q) ≤ 4n. Improving the upper-bound constant cq in the Stanley-Wilf

conjecture is an important open problem and we plan to have our graduate students, led by Professor

Akalu Tefera, tackle this problem for various subclasses of permutations of length k.

2. Another important problem in this area is calculating the Stanley-Wilf limit L(q) = limn→∞n√Sn(q).

For example, L(q) = 4 for patterns q of length k = 3. There are three Wilf classes for patterns q

of length k = 4. The Stanley-Wilf limit for two of these three classes are known and L(1234) = 9

and L(1342) = 8. However, L(1324) is still an open problem. Miklos Bona showed that Stanley-Wilf

limits are not always integers or even rational numbers by calculating L(12453) = 4 + 9√

2 [91]. It

is still an open problem as to whether the Wilf-Stanley limits are always algebraic. If so, can their

degree be more than two? Can their degree be arbitrarily large? Anders Claesson, Vit Jelinek, and

Einar Steingrimsson showed that L(q) ≤ 4l2 for a layered pattern of length l. They conjectured that

L(τ) ≤ 4l2 for any pattern of length l and their conjecture is still open [94].

25

3. Let A and P be matrices whose entries are either equal to 0 or 1, and let P be of size k × l. We say

that A contains P if A has a k × l submatrix Q so that if Pi,j = 1, then Qi,j = 1, for all i and j. If A

does not contain P, then we say that A avoids P. The Furedi-Hajnal conjecture states that if P is any

permutation matrix, and f(n, P ) is the maximum number of 1 entries that a P-avoidnig n×n matrix A

can have, then there exists a constant cP so that f(n, P ) ≤ cPn. Marcus and Tardos gave a spectacular

proof of this conjecture [104], and it was using this conjecture that Martin Klazar proved the famous

Stanley-Wilf conjecture stated above. Define P to be minimally nonlinear if f(n, P ) is nonlinear but

replacing any 1-entry in P with 0 yields a pattern P with f(n, P ) ≤ cn. As any 0-1 matrix with three

1-entries has a linear extremal function any pattern P containing four 1-entries with f(n, P ) nonlinear

is minimally nonlinear. These patterns are all known, but no other minimally nonlinear patterns are

known. Finding any minimally nonlinear patterns P with more than four 1-entries is another problem

our group will focus on.

4. The set of all permutations form a poset P with respect to pattern containment as defined above. The

Mobius function of an interval [x, y] in a poset P is defined recursively as follows: For all x, we set

µ(x, x) = 1 and

µ(x, y) = −∑

x≤z<y

µ(x, z).

Finding the Mobius function of intervals in the permutation poset is an active area of research and our

group, led by Dr. Samuel A. Fufa, will conduct investigation on evaluating this function for various

classes of permutations.

The first known result in this area is the formula obtained by Bruce Sagan and Vincent Vatter for

intervals of layered permutations [108]. Burstein et. al. obtained a formula for separable and de-

composable permutations in 2011 [93]. It is conjectured that the maximum value of µ(σ, π) for any

separable permutation π of length n ≥ 3 is obtained by a permutation of this form, for k that is roughly

n/2. For separable σ and τ , the value of |µ(σ, τ)| cannot exceed the number of occurrences of σ and

τ , but nothing similar is known for the general case.

It is not difficult to obtain families of intervals [σ, τ ] with µ(σ, τ) = 0. It is, however, an open problem

to characterize such intervals completely. As a matter of fact, it will be more interesting to characterize

those intervals for which |µ(σ, τ)| equals the number of occurrences of σ in π.

5. Another important aspect of the poset of permutations is the topology of the order complexes of its

intervals. There are some indications from the research done on this topic that large classes of intervals

in the permutation poset P have topology that can be described easily, and thus that the homology and

homotopy type of these intervals can be well understood. Gaining such understanding may well lead to

significant progress in answering questions about the Mobius function since the Mobius function of a

poset is equal to the reduced Euler characteristic of its order complex [117, 92]. It is very important to

find out whether order complexes arising from intervals of permutation posets are homotopy equivalent

to wedges of spheres of the same dimension.

Given an interval I = [σ, π] ∈ P, let 4(σ, π) be the order complex of the interior of the interval I. We

would like graduate students in our group to explore possible answers to questions such as:

(a) For which σ and π does 4(σ, π) have the homotopy type of a wedge of spheres?

(b) Let Γ be the subcomplex of 4(σ, π) induced by those elements τ of [σ, π] for which µ(σ, τ) 6= 0. Is

Γ a pure complex, that is, do all its maximal simplices, with respect to inclusion, have the same

dimension?

(c) If σ occurs precisely once in π, and µ(σ, π) = ±1, is 4(σ, π) homotopy equivalent to a sphere?

(d) For which σ and π is 4(σ, π) shellable?

26

The Riordan Group and Its Applications in Combinatorics

Let Fn be the collection of formal power series of the form

f(x) ∈ R[[x]]|f(x) = anxn + an+1x

n+1 + · · · , where an 6= 0.

The Riordan group R = F (1)0 × F (1)

1 is the set of pairs (g(x), f(x)) of formal power series g(x) ∈ F0, with

g(0) = 1, and f(x) ∈ F1, with f ′(0) = 1, equipped with the binary operation

(g, f) · (u, v) = (gu(f), v(f)).

The identity element is (1, x) and the inverse of (g, f) is given by

(g, f)−1

=

(1

g(f), f

).

To each element (g(x), f(x)) ∈ R, we associate a matrix M(g, f), with entries in the ring R, by defining its

(n, k)-th term to be

tn,k = [xn]g(x)f(x)k.

We call the matrix M(g, f) the Riordan array associated to the element (g(x), f(x)) ∈ R. The Fundamental

Theorem of Riordan Arrays states that if A(x) and B(x) are the generating functions of the column vectors

A = (a0, a1, a2, · · · )T and B = (b0, b1, b2, · · · )T ,

then

(g, f) ·A = B if and only if B(x) = g(x)A(f(x)).

Riordan Arrays gained attention after Lou Shapiro, Seyoum Getu and Wen-Jin Woan published their seminar

work on the Riordan Group in 1989 [109]. The central concept in their work is the development of a group

structure which unifies many themes in enumeration. One can use this concept to obtain remarkable results

in enumerative combinatorics [112]. For example, multiplying the Riordan Array D = (1, xC(x)2), where

C(x) is the Catalan generating function, with a periodic column vector

A = (0, 1, 0, 0,−1, 0, 1, 0, 0,−1, · · · )T ,

1 0 0 0 0 · · ·0 1 0 0 0 · · ·0 2 1 0 0 · · ·0 5 4 1 0 · · ·0 14 14 6 1 · · ·...

......

......

...

·

0

1

0

0

−1...

=

0

1

2

5

13...

,

we notice that the first few terms of the column vector on the right hand side of the above equation are

every other terms in the Fibonacci sequence. Since the generating function of the periodic column vector is

clearly A(x) = x−x4

1−x5 , we obtain, by applying the Fundamental Theorem of Riordan Arrays, the generating

function identity

x− x2

1− 3x+ x2=xC(x)

2 −(xC(x)

2)4

1−(xC(x)

2)5

which relates the terms of every other Fibonacci numbers to some subsets of ordered trees [99].

With the exception of some work done on alternative characterizations of Riordan arrays by Renzo Sprugnoli

et. al. [98, 103], the underlying theory of Riordan arrays is not well developed and the long term plan of our

27

research group is to develop detailed theory and provide an algebraic frame work that helps in discovering

and proving generating function identities such as the one given above.

One of the most important results in the study of the Riordan group and the associated Riordan arrays is

what is known as Roger’s sequence characterization of entries of the array [111]. More precisely, Roger’s

theorem states that an infinite lower triangular array M(g, f) = (tn,k)n,k∈N is a Riordan array if and only

if sequences A = a0 6= 0, a1, a2, · · · and Z = z0 6= 0, z1, z2, · · · exist such that for each n, k ∈ N ,

tn+1,k+1 =

∞∑j=0

ajtn,k+j

and

tn+1,0 =

∞∑j=0

zjtn,j .

Similar sequence characterizations of some of the well known subgroups of the Riordan group are ob-

tained recently [100]. For example, the Riordan array M(g, f) = (tn,k)n,k∈N is in the Bell subgroup

B = (g(x), f(x)) ∈ R|f(x) = xg(x) = (g(x), xg(x))|g(x) ∈ F (1)0 if and only if there is a sequence

C = c0, c1, c2, · · · such that

tn+1,k = tn,k−1 +∑j≥0

cjtn−j,k for n, k = 0, 1, 2, · · · .

Graduate students in our group will conduct research on obtaining similar characterizations for other sub-

groups of the Riordan group under the guidance of Professor Melkamu Zeleke.

Another important area of research involving the Riordan group is algebraic and combinatorial characteri-

zations of its involutions [95, 96]. Since a Riordan array that describes a combinatorial situation often has

all positive entries on and below the main diagonal and cannot itself have order 2, we define an element D

in the Riordan group to have pseudo-order 2 if DM has order 2, where M = (1,−x). Clearly, AMA−1 has

order 2 for any element A in the Riordan group. An element of pseudo-order 2 in the Riordan group is called

a pseudo involution. Students in our group will also explore some open questions in this area such as:

1. Can every element of order 2 in the Riordan group be written as AMA−1 for some element A in the

Riordan group?

2. If the element of order 2 or pseudo-order 2 has combinatorial significance can we find an A which has

a related combinatorial significance?

3. If D = (g(x), f(x)) has order 2, can we characterize g(x) in terms of f(x)?

Extremal Graph Theory

In graph theory, as in many fields of mathematics, one is often interested in determining the maxima or

minima of certain functions and identifying the points of optimality. One such result is the classical theorem

of Mantel that the number of edges in an n-vertex triangle-free graph is at most bn2/4c (with equality only

for the equitably partitioned complete bipartite graph). The generalization of this result by Paul Turan is

widely regarded as the landmark result that launched the area of extremal graph theory.

Theorem 1 (Turan’s Theorem). For r ≥ 2, let Tr(n) be the complete r-partite graph on n vertices where

each part has size either dn/re or bn/rc. Let tr(n) be the number of edges in Tr(n).

If G is a graph on n vertices which does not contain Kr+1 as a subgraph, then e(G) ≤ tr(n), with equality

holding only when G = Tr(n).

28

For this particular problem, Erdos and Simonovits proved a stability property; a Kr+1-free graph G with

e(G) ‘almost’ optimal has a structure ‘close’ to the unique extremal graph. Such stability results, while

interesting on their own right, have also been instrumental in proving various exact results.

An area of active interest is the theory of supersaturated graphs, where one studies the behavior of graphs

which exceed the Turan threshold, ex(n, F ), for some simple graph F . In particular, we are interested in

determining the minimum number of copies of F contained in such graphs. The origin of this area may

be traced back to an unpublished result of Rademacher in 1941; it has since become a rich subfield of

extremal graph theory. In the paper “Supersaturation problem for color-critical graphs”, Oleg Pikhurko and

Zelealem Belaineh Yilma, extend results of Erdos [123, 124, 125]; Lovasz and Simonovits [130, 131] ; and,

most recently, Mubayi [132], to show, for example, that

Theorem 2. If G is a graph on 2n vertices and n2 + q edges, where 1 ≤ q < n, then G contains at least

qn(n− 1)(n− 2) copies of C5.

In the paper, similar results, both exact and asymptotic, are obtained for color-critical graphs, those graphs

that contain an edge whose deletion reduces the chromatic number of the graph. The proofs follow a two

step process: first, stability is proved by applying theorems such as the Removal Lemma. Next, the results

are obtained by studying the limiting behaviors and relationships between various graph polynomials.

One reason for working with color-critical graphs is the fact that the corresponding extremal graphs are

Turan graphs. It is interesting to consider the supersaturation problem for graphs whose extremal structures

are not as well-understood. Two particularly interesting instances, and subjects for future work, are the

Petersen graph, whose extremal structures are known, and the 4-cycle, whose extremal structures are known

only for very few values. In fact, the following conjecture of Erdos and Simonovits [126] is still open:

Conjecure 1. If G has at least ex(n,C4) + 1 edges, then G contains at least 2 copies of C4.

Graph Colorings and Labelings

Coloring and labeling problems have a rich history in graph theory. There are now numerous variants that

are actively studied; a dynamic survey is maintained by Gallian [127]. One such problem is that of antimagic

labelings. An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges

to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of

labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling.

It has been conjectured by Hartsfield and Ringel [128] that all graphs except K2 are antimagic and various

methods have been employed to prove that certain classes of graphs are antimagic. One interesting result,

due to Hefetz, Saluz, and Tran [129], uses Alon’s Combinatorial Nullstellensatz [121] to prove that a graph

on pk (p an odd prime) vertices that admits a Cp-factor is antimagic.

The Combinatorial Nullstellensatz is an interesting tool which we believe would be amenable to proving that

other classes of graphs are antimagic. Already, some partial results have been obtained for a subset of graphs

which contain a vertex of large degree. One research goal of the group is to further investigate applications

of the Combinatorial Nullstellensatz to problems of graph labelings.

Packings and Coverings in Hypergraphs

For a hypergraph H, let ν(H) and τ(H) denote its packing number and covering number, respectively. It is

easily seen that τ(H) ≥ ν(H); however, there is no function of ν(H) that bounds τ(H) from above in general.

Indeed, much work has been done to find such bounding function for various classes of hypergraphs. One

29

well known result is Konig’s Theorem, which states that if H is a simple bipartite graph, then τ(H) = ν(H).

An interesting natural extension of Konig’s Theorem has been conjectured by Ryser:

Conjecure 2. If H is an r-uniform r-partite hypergraph, then τ(H) ≤ (r − 1)ν(H).

Even the seemingly simple case where ν(H) = 1 is open.

In a recent paper [122], Chepoi, Estellon and Vaxes consider the above problem, that is, bounding τ(H)

from above, where the hypergraph H is a ball-hypergraph of some planar graph G. We intend to study such

problems for different classes of hypergraphs.

Statement of Procedures

This proposal is a joint research project by Professors Melkamu Zeleke, Akalu Tefera, Zelealem Belaineh

Yilma, Yirgalem Tsegaye and Samuel Asefa Fufa. The research problems on the combinatorics of permu-

tations and Riordan arrays are based on the selected topics courses offered to graduate students at Addis

Ababa University by Professors Akalu Tefera and Melkamu Zeleke during their most recent visits to the

department. The motivation for the Mobius function problems is some work done on the Mobius function of

pointed integer and set partitions by Dr. Samuel A. Fufa during his doctoral studies at Addis Ababa Univer-

sity. The graph and hypergraph theory problems in this project are extensions of research work conducted

by Dr. Zelealem Belaineh Yilma as part of his doctoral and post doctoral work and students interested in

these problems will work under the guidance of Dr. Yirgalem Tsegaye.

The following is an outline of the project plan with estimates of timing.

• Fall 2017 - Spring 2018: The two current doctoral students, Mesfin Masre and Birhanu Gebreyes,

will complete their qualifying examination by October 2017 and start working on some of the open

problems outlined in this proposal.

• Fall 2018 - Spring 2019: Up to four additional doctoral students who are going to be admitted

during the 2017-2018 academic year will start working on the remaining problems.

• Fall 2019 - Spring 2020: Current doctoral students finalize results on their research work for

publication and prepare for thesis defense.

Significance of the project

The mathematical problems that will be investigated in this project are important questions in discrete

mathematics and any progress made will contribute to the advancement of mathematical research in Ethiopia.

The results of the project will also be interesting enough for publication in reputable mathematics journals

such as Journal of Discrete Mathematics, Annals of Combinatorics, Advances in Applied Mathematics,

Electronic Journal of Combinatorics, Bulletin of the Institute of Combinatorics and its Applications, Ars

Combinatoria, etc.

In addition, the problems proposed here have many interesting extensions for future study. Support to

successfully complete the proposed project will encourage the investigators to pursue various extensions of

the problems in this project in a timely manner and use some of the results to direct future student projects.

As a direct result, the research group and, in general, the mathematics department, will be able to increase

both the number of students it accepts into the Ph.D. program and the number of Masters students who

30

enroll in the research-track. As further motivation and support for students, as well as for the necessary

dissemination of results, the research group will organize bi-weekly seminars in combinatorics and graph

theory.

31

Component IV: Optimization and

Applied Mathematics

4.1 Traffic Flow Optimization

Berhanu Guta

4.1.1 Introduction

Transport is both a means and an outcome of urbanization and development. The fast growing economic

activities in Ethiopia has brought about many changes including, but not limited to, high rate of urbanization,

an increasing demand for transportation services, and a need for quality infrastructure. In response to these

demands and needs the government of Ethiopia has been building modern road networks throughout the

country, in particular in Addis Ababa. The total length and standard of road networks and the railways

that have been built in Addis Ababa in the last few years alone are evidences for this effort. According to

[?], improved accessibility in Ethiopia increases value added per worker. Furthermore, it has been shown

that, access to an all-whether road in rural Ethiopia reduced poverty by 6.9% and increased consumption by

16.3% [15]. Despite the important and positive roles that transportation facilities have on social, economic,

and political activities of the country, it can also generate negative impacts through road accidents and

traffic congestion if roads are not properly designed and traffic flow is not effectively managed. Moreover,

the construction and maintenance of transport infrastructure may demand either unnecessarily huge budget

or a compromise of quality if it has to be handled just in the traditional way without employing suitable

optimization techniques that optimize desirable measures of performances. It is apparent that nowadays

optimization technology is at the forefront of every modern scientific and technological innovation. Therefore,

in support of the development endeavors and in order to minimize the negative consequences of road facilities,

we would like to propose to conduct a scientific study and develop an innovative framework based on

optimization technology regarding traffic flow management. Briefly, the research is aimed at the study

of efficient and effective means of traffic flow management.

4.1.2. Background

Rapid urbanization and increasing demand for transportation burdens urban road infrastructures. When

growth in economic activities significantly outpaces the growth in transportation infrastructure, cities expe-

rience congestion to levels that make mobility difficult [139]. Road transport policies, however, should seek

to manage congestion on a cost-effective basis with the aim of reducing the negative impact that congestion

imposes upon road users. From a transportation professional’s perspective, congestion relates to an excess

of vehicles on a portion of roadway at a particular time, resulting in speeds that are slower, sometimes

much slower, than free-flow speeds [140]. There are two type of congestion: recurring and non-recurring.

Recurring congestion are caused by topographic and physical barriers to movement; the discontinuity of the

32

road network; design and operational deficiencies. Non-recurring congestion are created by incidents, surging

demand, inclement weather, work zones/street closures, and driver behavior [139]. Congestion results when

traffic demand approaches or exceeds the available capacity of the road system [141]. While this is only

a simple concept, it is not straightforward to determine what triggers congestion until effective congestion

measures are set.

Ethiopia is one of emerging economies in Africa. Fueled by the unprecedented economic growth, Ethiopia has

been undergoing rapid urbanization over the last few years. Consequently, one of the biggest challenges that

Ethiopia would face in the coming years is the effective management of urbanization, which is taking place

rapidly throughout the country and particularly in Addis Ababa. In the process of a shift from agriculture led

economy to a manufacturing, urban areas come to the centre stage of development process and experience a

population growth at an alarming rate. Along with its growing population, Addis Ababa city needs to address

transportation demands which require a comprehensive analysis of the current transportation condition and

developing a system that can forecast the future conditions. We firmly believe that any strategy that we put

in place to mitigate traffic congestion should take in to account the present traffic condition and be able to

forecast the state of the future traffic. This can only be achieved through the use of state-of-the-art models

that can forecast traffic with high accuracy. It is therefore desirable to develop techniques to estimate as

accurately as possibly what the forthcoming traffic conditions may be, so that correct and timely action can

be taken.

4.1.3 Problems/or Gaps in urban traffic flow management

Addis Ababas transportation sector needs to be able to keep pace with the rising demand and growing

economy. Despite the extensive activities of constructing and expanding road networks, there are observable

serious traffic congestions and a rising concern of road users in Addis Ababa city which is getting worse

compared to years ago. This is characterized by a common phenomenon that people are standing on a

queue of line waiting for a public transport and also vehicles making a queue of line on roads, especially

during peak hours of a day. Traffic congestion has a potential to influence investment decisions as it limits

accessibility of residents which results in lost opportunities for both public and business. In fact, it impacts

every aspect of our life including our economy, our social life, the quality of our air, safety, mobility, and

the fuel consumption. Fuel is the major imported commodity to the country with huge budget. Despite

this fact, traffic jams are causing millions (if not billions) of liters of fuels to be wasted as vehicles have to

run their engine while they are on a queue of traffic, and this is equated to a loss in millions of Birr at

national level. No need to mention the loss that can be caused by the accidents because of traffic congestion

and related issues. As it is mentioned earlier, construction of new road is a straight forward solution to the

problem, but it is not always feasible in practice because of space and budget limitation. Besides, because

of poor traffic forecast it may happen that newly built roads have very little effect in mitigating a traffic

congestion problem. To address the city’s transport problems, the government has been expanding road

infrastructures (such as building new railway and roads with more lanes) to increase the capacity of the

current road networks and also injected new public buses in to the system, but traffic congestion remain a

problem. This indicates that the construction of road infrastructure is no longer the only option to overcome

the problem of traffic congestion. There is a general feeling that, if this situation is not intervened as soon

as possible, then it might reach an irreversible stage where any attempt would barely produce any result.

Addis Ababa being the seat of government of the Federal Democratic Republic of Ethiopia and also home

to the African Union, the Economic Commission for Africa, diplomats and other international organization,

need to have a modern urban road and traffic management system. So, we believe that, it is very important

to act on the problem before it is too late. Therefore, in this study, we aim to evaluate the performance

of selected road networks in Addis Ababa, and plan to develop new approaches and technologies based on

optimization techniques that would mitigate the current traffic congestion problems and forecast the future

traffic conditions.

33

With regard to the public transport system in Addis Ababa, the passengers who use busses are usually

subjected to long waiting times at every bus station and unable to plan their travel time since there is no

working bus schedule (travel timetables) for the bus lines. Moreover the passengers who need to change

travel line from train to bus or from bus to train, where the two systems intersect, are subjected to another

long waiting times at bus/train stations since there is no matching travel timetables (schedules) between the

two transport systems. Therefore, it is highly needed to study the public transport system in Addis Ababa

and establish optimal strategies for bus and train transport systems that enhance the quality of their public

service.

4.1.4. Research questions

Route selection optimization in road design based on cost minimization requires comprehensive formulation

of cost and utility functions and the development of efficient solution algorithms. At the same time, traffic

congestion is one of the major problems in urban areas particularly in Addis Ababa. Therefore, we pose the

following research question with respect to the three themes to be investigated.

• What is the performance of the existing road networks in Addis Ababa in terms of their Level of Service

(LOS)?

• What are the triggering factors for the traffic congestion? And what is the extent of the congestion?

• What is the prevailing type of congestion? Recurring? Non-recurring? Or both?

• What are the capacities, operational inefficiencies and bottlenecks in the existing roadway system?

• What capacity enhancement is required (and how far) to manage the congestion without adding a new

road or civil structure?

• Is there any more efficient and effective design of roads at cross-points and roundabouts that helps to

reduce the traffic congestion?

• How do traffic lights operate? And what is a more efficient way of programming the traffic light (signal)

patterns that minimizes the waiting time of cars?

• Is there any efficient and effective way of coordination between different public transportation systems

that enhances their quality of services and also help to reduce congestion?

4.1.5. Objectives

4.1 . General objectives : Developing a framework for intelligent traffic flow management through the

optimization of traffic operations

4.2. Specific objectives

– Identifying the bottlenecks that can be removed through the use of optimization techniques with-

out a need to add any civil infrastructure

– Developing effective and efficient strategies that mitigate the traffic congestions in the city.

– Developing efficient and effective strategies for coordinating public transport

– Developing a software that produces optimal bus schedules and coordinate bus-train system

34

4.1.6. Research Methodology

4.1.6.1. Research outline We conduct this study in three phases. The first phase concerns the traffic data

generation method. So, we begin our study by devising a reliable and executable traffic data collection

method. We plan to use both manual and automatic data collection methods. We also use secondary

data that is readily available at Addis Ababa City Road Transport Agency (ACRTA) and Addis Ababa

traffic agency. Concurrently, we want to develop a methodology for handling big data that may arise

from the new traffic data collection techniques. Once we collect and store the data, the second phase

of the research deals with developing of relevant mathematical models that can fit the field data and

simulate the actual problems at hand. These models include mathematical models that deal with route

selection problem and traffic flow models and mathematical models that deal with traffic congestion

problem; and network models and others that deal with traffic forecast. We note that traffic congestion

is a multi-faceted problem so that, we plan to take different but coordinated solution approaches to

tackle the problem. This include, optimization of traffic signal patterns and timing, optimizing the

locations of traffic light installation, and programming the city bus routes thereby minimizing the

waiting time. Using the data, we continuously calibrate our models until the best models are obtained.

As part of this stage, we will develop solution methods and algorithms that can be implemented in to

computer software.

The final phase is devoted to writing computer scripts and developing computer software. Specifically,

we plan to produce the following tools: a tool that produces efficient city bus schedules, a tool that

coordinates efficient traffic light (signal) patterns, and a tool that forecasts traffic conditions. We also

plan to use commercial software and integrate them in to the video camera in a cost-effective manner

for data collection. The resulting tools need will be tested using the available data and be compared

with the existing practice.

4.1.6.2. Description of the research procedure

The procedure or steps that we will follow to conduct the research are mainly: collecting relevant

primary and secondary data, organizing analyzing the data, developing data base, developing relevant

mathematical models, solving the problems (the mathematical models) and simulations, analyzing the

results by comparing with the actual situations and make improvements, and develop relevant ap-

plication tools and software. In particular, in this part of the study we plan to work on congestion

management actions, in particular, strategies for minimizing the recurring congestion and enhancing

the quality of service for public transportation system. These strategies focus on enhancing roadway

capacity. Capacity can be enhanced in two ways: by restoring the lost (or wasted) capacity and by

adding new road capacity. The lost capacity can be restored by eliminating traffic operational ineffi-

ciencies, removing bottlenecks in the existing roadway system and utilizing the existing infrastructure

and facilities optimally. The second strategy (building new road) is beyond the reach of this study. So,

our approach is restoring the lost capacity.

However, we understand that these two strategies are sometimes inseparable. For instance, to improve

the operation of city buses we might need to have bus exchange terminals, or to maximize the capacity

of an existing road network we might need to have parking areas so that on-road parking be prohibited.

That means, besides looking for optimal ways of utilizing the existing facilities and system, we need

to look also for desirable new facilities and their optimal locations or arrangement.

Moreover, consecutive signalized intersections (traffic lights) have uncoordinated signals which result

in unnecessary waiting and accumulation of vehicles at each of the signalized intersections. In order to

identify optimal signal patterns and timing, we need to collect a data regarding the distance and ride

time between consecutive signalized intersections, traffic volume and the signal cycles and timing. In

this regard, one of the advantages video camera based traffic counting is that, this technology is useful

for intersection capacity analysis and signal timing.

35

This study begins by analyzing the current state of operation and coordination of public transport

system and thereby explores the possibilities for improvement using optimization technology. A survey

on public opinion will be conducted to learn more about what improvements are desired and also to

know if a fare increase is accepted in an exchange for improved service (such as minimized waiting time

and maximized frequency and reliability of the city bus). We will apply optimization techniques to the

bus scheduling, timetable scheduling, and bus route configuration with the objective of minimizing the

waiting time and the buss operation cost.

Therefore, by designing efficient public bus service system we can gain the following three advantages:

First, it will encourage private vehicle owners to use public buses (which in fact contributes to the

congestion relief strategy); Second, it increases the reliability and reduces the waiting times for pas-

sengers; Third it reduces the operational cost of the public bus and maximizes the revenue through

fare increment. Part of this design consists in determining an optimal bus route configuration and

stop locations based on travel demand survey that will be conducted at the beginning of the study.

Our basic premise in dealing with public transport for congestion relief is that, a rational traveler will

choose a route with least cost, minimum travel time, and reliability of travel time. For any road user

travel time, reliability of travel time, and travel costs are the most important elements. Congestion

affects exactly these travel features.

Concurrently, we will contact Addis Ababa Road Authority and Addis Ababa Traffic Agency to know

if there is any plan to improve or expand the use of traffic light at more intersections. This will be

very helpful for us because, if there is such a plan, then we will be working hand-in-hand so that the

desired change can be attained.

4.1.6.3. Data collection methodology

The ability to forecast traffic parameters such as flows and density in urban or rural area on a road

network is one of the most important approaches in managing future traffic congestion. The success

of such prediction depends, to a large extent, on the accuracy of traffic data and the model used for

prediction. We embark on our study by collecting traffic data that are necessary to analyze the per-

formance of roads and investigating the causes of traffic congestion. These data include traffic volume

and its growth trend, traffic density, vehicle speed, delay at intersections, and road capacity. Traffic

volume studies are conducted to determine the number, movements, and classifications of vehicles on

a road at a given location. These data are used to identify critical traffic flow time, to determine the

influence of large vehicles and pedestrian on the flow, and to document traffic volume trends. Since

we cannot cover every road network in this survey, we take a sample of the existing roads and collect

the respective data from them. Our plan to take road samples is as follows. We divide Addis Ababa

city in to a number of zones, called traffic zones. This approach, which is recognized in transportation

modeling, has an advantage for big data management and to consider different type of road segment

in the sampling process. For simplicity, we can use each sub-city as our traffic zones. We understand

that some of the roads cross the boundary of zones and in this situation we collect traffic data in the

respective zones. Then we identify congested roads in each zone. They are automatically added to the

sample. But for the remaining road networks we use a random sampling technique to get them in to

the sample space. Prior to the actual data collection, a preliminary survey of the road networks in each

sub-city will be carried out, and will be coded. The codes have two purposes; they are to be used for

the random sampling purpose and to easily document the data for the later uses. Once we identify the

sample road networks in each zone, we may want to distinguish them from one another based on some

characteristic, such as the number of lanes. This technique may insight the type of measure that one

should take to mitigate congestion. As we indicated in the previous section, we also plan to conduct

extensive data survey regarding the travel demand of customers, the public bus operation system, the

travel (ride) time of busses along their lines, the locations of bus stops and intersections of bus lines.

These data will be used to design optimal bus route configurations; stops locations along bus routes,

frequency of the service, and optimal schedules. In order to plan efficient public bus service, these

36

attributes have a consequential effect on the waiting time minimization, arrive-departure synchroniza-

tion, and minimization of bus operation cost. Because travel demand data are so crucial, we need the

cooperation of Anbessa City Bus Service Enterprise (ACBSE). We plan to use two strategies to collect

the data. First, in collaboration with ACBSE we take a sample of bus routes in each traffic zone, print

a new bus tickets which contain some travel information such as origin-destination of the passenger,

purpose of the travel, frequency of the travel in a day, etc. A person who sales the tickets also counts

the number of passenger getting off of the bus at each stop (also known as passenger alighting). We

do this at each stops including the start and end terminals for all buses using the route. The second

strategy is by using household survey and workplace survey in each zone. We prepare a diary of travel

in which to record relevant travels on a daily basis. The diary will contain information such as origin-

destination of the trip, departure time of the travel, departure location of the travel, purpose of travel,

frequency of travel, regularity of travel, length of travel time, travel cost, mode of travel, reliability of

the mode, comfort and safety of the mode, number and location of connections etc. This information

help us in estimating the demand along the sample routes by using demand predictive models and

data analytics. Moreover, these data are used to inform the decision makers about the future demand

based on the current demand pattern.

Moreover, in the phase 2 and phase 3 of the research, we will organize conferences/workshops on which

we (and other invited researchers) will present our results/findings regarding the problems at hand in

order to obtain further inputs from stakeholders and other experts.

4.6. Expected outputs and outcomes of the Project

4.1.7.1. Expected outputs

Based on the intended objectives, we expect the following developments and outputs from this research.

– Software packages

– New techniques for traffic data collection

– Travel demand database program

– Traffic management policy recommendations and guidelines

– Publications in scientific journals

– MSc theses and PhD dissertations

4.1.7.2. Expected outcomes

With the appropriate financial support and diligence in execution of the research plan, we expect to

get the following benefits and outcomes from the research.

– Getting reliable traffic data from the database program

– Improved traffic forecast

– Improved service quality of the public bus transport

– The ability to forecast travel demand

– Reduced traffic congestion problem

– Knowledge production, and refining the existing body of knowledge

– Open a research avenue towards the use of modern technologies for improving traffic flow in urban

areas.

– Personnel specialized in the area at MSc and PhD level

37

4.2 Investigation of Behavior Change and Resource Allocation (IBCRA):

for Infectious Diseases Management Programs

Persons Involved

Semu Mitiku Kassa (Addis Ababa University)

4.1.1 Problem Formulation

The population in Sab-Sahara Africa is suffering most from epidemics of infectious diseases as compared

to other part of the world. For example, globally over 36.7 million people are leaving with HIV/AIDS and

of these cases about 70.5% live in Sub-Saharan Africa. More than 1.8 million children under the age of

15 live with HIV/AIDS globally and among them more than 83 percent live in Sub-Sahara African region.

According to the World Bank report, a further 2.1 million people were infected in the year 2015 alone. Every

day over 5,700 new HIV infections occur worldwide. Out of these more than 95 percent occur in low- and

middle-income countries[150]. In general global HIV prevalence (proportion of people with HIV) is remaining

at the same level, although the global number of people with HIV is rising because of new infections and

longer survival times, and continuously growing global total population. In the absence of a curing medicine

and a working vaccination, the investment to control the HIV epidemics is mainly to reduce the total number

of new infections and the rate of progression to AIDS.

The other example is, Tuberculosis (TB) which is a chronic infectious disease mainly caused by Mycobac-

terium tuberculosis (MTB). Even though a pharmaceutical treatment for infected individuals has been in

place for long time the burden of TB is still high in our regions. In 2015, an estimated 10.4 million new

(incident) TB cases were reported worldwide and of which 1.0 million (10%) were children[151]. The effect

of HIV has exacerbated further the burden of TB. Moreover, multi-drug-resistant tuberculosis (MDR-TB)

is an increased global problem with most cases are caused due to noncompliance to the rules by the patients

during treatment of drug sensitive TB. Statistical data shows that in 2015, there were an estimated number

of 480,000 new multi-drug resistant TB cases and an additional 100,000 people with rifampicin-resistant TB

[152] who were also newly eligible for MDR-TB type treatments. Since multidrug-resistant tuberculosis is

expensive to treat and treating such patients for longer time is difficult, the emergence of resistant strains is

a headache for many of the developing countries.

Malaria is a mosquito-borne disease that currently affects over 100 countries worldwide; but the highest

incidence and mortality rates are reported in sub-Saharan Africa. The World Health Organization (WHO)

estimates that 660,000-971,000 people die every year from malaria and approximately 90% of the deaths

occur in children under five years of age. According to the latest estimates[4], between 2000 and 2015,

malaria case incidence was reduced by 41% and malaria mortality rates by 62%. At the beginning of 2016,

malaria was considered to be endemic in 91 countries and territories, down from 108 in 2000. Despite some

remarkable progress in reducing the burden of malaria due to the use of ITNs (Insecticide-Treated-Bed-Nets)

and other products in the last 15 years, malaria continues to have a devastating impact on peoples health

and livelihoods. Updated estimates indicate that 212 million cases occurred globally in 2015, leading to 429

000 deaths, most of which were in children aged under 5 years in Africa [153].

The above three endemic diseases represent the categories of sexually transmitted diseases, airborne diseases,

and vector-borne diseases, respectively. To manage such endemic diseases countries in the region have

been putting policies in place and allocating the meager resources they have. These policies and resource

allocation programs are usually assisted by mathematical models and analyses that simulate the possible

outcome of the control measures. However, most of the mathematical models that have been used so

far assume that the behavior of individuals towards reducing their risky engagements remains constant or

unchanged in the course of the outbreak. Realistically, the change in human behavior can significantly

influence the spread of infectious disease as has been witnessed, even recently during the influenza and

38

Ebola outbreaks. Therefore, including the effects of behavioral changes during an epidemic can possibly

give a better approximation to the reality. Moreover, such models can also guide how best to synchronize

the public health educational campaigns with other control efforts to yield the best possible outcome on the

disease management program. In addition, to allocate resources to medical intervention programs the criteria

used so far are the cost-effectiveness ratios of the various intervention types, as recommended by health

economics theory[154]. Actually this leads to a Give-it-all or nothing strategy which may not be suitable

for health management programs in developing countries. Moreover, to establish the Cost-Effectiveness

criteria one needs to collect adequate data on the population distribution and generate cost-effectiveness

data for individual and community-level interventions, which is usually more difficult to implement. Even in

the perfect information case, ‘Allocation by Cost-Effectiveness’ (ACE) does not allow for several important

factors such as, increasing or diminishing marginal returns to scale, mutual exclusivity of programs, and

interaction of program outcomes[155]], which are necessary to be taken into consideration.

4.2.2 Objectives of the Project

Therefore, the main objectives of this study are:

1. to formulate a mathematical model, which takes the change in human behaviour into consideration,

and to recommend the best possible and cost effective combination of disease management programs.

2. to propose an efficient solution method for the mathematical problems that arise from the applications

(for example, multilevel optimization problems with multiple followers).

3. to formulate a better resource allocation procedure at a country level, which can possibly measures

and controls the actual dynamics of the disease, instead of simply using the cost-effectiveness criteria.

4. Description of research approach

To investigate the actual effect of spontaneous behavioural changes in disease dynamics, we shall incorporate

the behavioural change dynamics into the specific disease model by considering either the theory of Diffusion

of Innovations or the Game theoretic approach of Imitation dynamics or both. Then using the tools from

dynamical systems we will analyse the models mathematically. Once the models with generic parameter

values are analysed we shall collect country level specific data about the diseases described above. It is

known that policies and programs for disease control are set depending on the social make-up of the society,

governance system of the country, and the economic conditions. Therefore, they differ from country to

country even if there is some communality. Due to this fact, we shall collect data from at least three

countries chosen from the Sab-Saharan Africa region and the models will be calibrated using this actual

data. Then after, numerical simulations as well as bifurcation analysis could be employed to come up with

possible predictions and to recommend on which attributes of the diseases should the authorities to focus

on for effective outcomes in the control of the diseases. Moreover, optimal control analysis will be used to

formulate the best combination strategy in the fight against the epidemics. Moreover, we may also investigate

the dependence of the parameters on the climate variables to analyze the evolution of the epidemics.

The study on the resource allocation procedure will use the actual disease dynamics and hierarchical or

multilevel programming techniques. To solve such mathematical programming problems, we will study in

depth the existing mathematical analysis and methods and we shall also propose new solution methods

especially for multilevel multifolower problems with various mathematical structures. But for application

purposes we shall formulate a heuristic type implementable algorithm. Using the resulting methods we will

generate a technical (or decision support) tool that helps to efficiently allocate resources for disease control

programs and that gives a way to monitor the use of resources generated at each level in the hierarchy. We

also expect that the same tool could be used to monitor the specific impact of the investment on the disease

dynamics.

39

4.2.3 Materials, Methods and Techniques

We shall formulate mathematical models that reflect the dynamics of the real problem and analyze them

mathematically and propose an efficient solution approach for a numerical implementation. To calibrate the

model and estimate parameters, we will collect macro-level data on countrys disease management programs,

from at least three countries in Sub-Saharan Africa region. Then we will test and validate the formulated

and mathematically analyzed models, simulate the results numerically and finally come up with specific

recommendations and procedures for effective disease management programs as well as an efficient, macro-

level resource allocation procedure for disease control programs.

40

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