the importance of mathematics in the development of...

The Importance of Mathematics in the Development of Science and

Technology

BİLİM ve TEKNOLOJİNİN GELİŞİMİNDE MATEMATİĞİN ÖNEMİ

BİLİM ve TEKNOLOJİNİN GELİŞİMİNDE MATEMATİĞİN ÖNEMİ

Prof.Dr.Mustafa BAYRAM

Yıldız Teknik Üniversitesi

Kimya-Metalurji Fakültesi

Matematik Mühendisliği Bölümü

İstanbul

Essence and Role of Mathematics

Mathematics is an autonomous intellectual discipline, one of the clearest exponents of the creative power of the human mind. On the other hand, it plays a fundamental role in modern Science, has a strong influence on it and it has been influenced by it in an essential way.


There are two different Matehmatics:

A first dimension of Mathematics is in fact the PURE aspect, Mathematics as an art in its own right, a game that is played in our minds.

A second dimension of Mathematics is in fact the APPLICATION aspect.


Mathematicsc is an art that expresses

beauty in the form of axioms, theorems andlogical or numerical relations;

Applied role of Mathematics is more essential. In fact:

Mathematics has played a fundamental role in theformulation of modern Science since the very beginning; a scientific theory is a theory that has an adequatemathematical model;


The Mathematics that can be applied today covers all the fields of the mathematical science and not only some special topics; it concerns Mathematics of all levels of difficulty and not only simple results and arguments;

The capabilities of scientific computation have made numerical simulation an indispensable tool in the design and control of industrial processes.


In this presentation we will deal with this aspectwhereby Mathematics is language in which thepages of Science are written. Thanks to it therehas been a development of the combinationScience-Technology .

In deed, the daily practice of the physical sciencesand engineering hides huge amounts of highermathematics.


Moreover, the very concepts on which theirtheories are based are essentially mathematicalconcepts.

In the last decades we have seen the trend towards mathematization reach other disciplines,

like Economics, particularly the financialmarket, branches of Chemistry, Biologyand Medicine, and even the social sciences.


Mathematics should permit to assimilate the data and to understand the phenomena.

In the hands of the engineer, it is the tool that makes possible to buid a numerical or qualitative model whose analysis allows to make decisions and design artifacts in an efficients and reliable way.


Applied mathematics cover the classical areas like mathematical physics and mathematical methods for engineering, but it has today broader contours with the advent of scientific computation and numerical simulation.

Modeling, computational simulation and data analysis are essential tools in modern science and industry.

Galile’s and Newton’s Heirs(Galile ve Newton’un Mirascıları)

Two great historical figures fixed the key role of Mathematics ın the moments in which modern Science was being born.

Galileo formulated it,

Newton demonstrated it.


Leonardo da Vinci guessed the role of Mathematics in Science.

As is well known, modern Science appeared in Europe at the end of the Renaissance. It is not based upon Mathematics alone.


The fundamental pillar of the building in germ was aptly formulated by the English philosopher and politician Francis Bacon circa 1620 and consists of the experimental method.

Nature becomes the preferential object of philosophical investigation, we should learn to read and to understand it, and eventually to control it; observation is the means for comprehension and expertiment is the test of our predictions.


The sciences were formed around this method, first Physics, then Biology, Geology, Chemistry and so on.

Galileo was of course a committed defender of the experimental method, to which he contributed his famous astronomical and mechanical observations.


Isaac NEWTON (1642-1727), who shows the incontestable success of Galileo’s proposal as applied to mechanics. He attacks the basic problems debated during the century .

F ma


Universal attraction formula

2/F Gmm r

The Century of Reason and Lights

During the following three centuries, a part of that ocean has been filled with truth, science and mathematics, Science and Technology, the basis of the Industrial Revolution, have advanced with theories, reasoning and experiments.


As a consequence, the society of the XXth century has changed more radically with respect to the XVIIth century than anything that had happened in several thousand years before, since the onset of the great agricultural civilizations. The comfort of house, transportation and communications, and the healty of the present-day citizen rest upon technical bases completely unknown to the people of the XVIIth century.


Starting with G.W.Leibniz, a great philosopher and Newton’s rival in the famous and a bit sad “dispute of the Calculus”, a series of brilliant mathematicians, like Bernoulli family, Euler, D’Alembert, …….exploited the potential of the new Calculus and formulated mathematically all types of mechanical problems: shooting problems, problems concerning the fall of bodies, the motion of fluids, mechanical vibrations, minimization,…..


Toward the year 1738 Johann and Daniel Bernoulli establish the theoretical science of Hydrodynamics on the idealized basis of the so-callled perfect fluids.

The study is continued by Euler, who writes the famous equations(1755)


This equation is

0, 0u

u u p ut


Whose analytical solution turns out to be intractable at the time.

The XIXth Century, the Great Century of Sciene

The contribution of the XIXth century to Mathematics,both pure and applied,is surprising by its novelty, by its richness and multiplicity of topis, and by its very unexpetedness.

Let us begin our review with Physis.


•ELETRICITY and MAGNETISM: From Mihael Faraday toJ.C.Maxwell,experiments and partial laws over a road that counts the names of Gauss, Ampeere, Biot, Savart,Lenz,...till we arrive at the system of partial diferential equations that relates the electric and magnetic felds(1863), the work of James Clerk MAXWELL. Maxwell's equations are one of the major achievements of Mathematics in the 19th century.


THE REAL FLUIDS: The Navier-Stokes equations desribe real fluids and they govern the behavior of atmospheric phenomena (climate, Meteorology, Hydrology, the future Aeronautis).


THERMODYNAMCIS: which studies the exhange of heat, aquires solid mathematical foundations with James Joule, Saadi Carnot, J.R.Mayer,...It has strong infuene on the calculus with partial derivatives and the concept of exact diferential. This theory inludes the famous Second Law of Thermodynamics (law of entropy growth in the universe), a fundamental law in sciene.


Finally, let us mention STATISTICAL MECHANISC, assoiated to the names of L.Boltzmann and Gibbs, who carved a branch of Mathematial Physis on the basis of the calculus of probabilities, a disipline that had remained very much at the margin of this scientific adventure.

The XXth Century, a Century of Wonders

A mainfeature that stands out is the progressive mathematization of other scienes,which makes them appear as new horizons for Applied Mathematics.

THE THEORY OF RELATIVITY. Albert EINSTEIN,


The fundamental formula

E=mc2

about the mathematical equivalence of mass and energy


QUANTUM MECHANICS:Quantum

mechanics often describes and predicts the movement and behavior of particles at the atomic and subatomic levels. These include particles such as atoms, electrons, protons, and photons. According to quantum mechanics, the behavior and movement of particles at these microscopic levels are counter-intuitive, radically differing from anything observed in everyday life


AERONAUTIS: Aeronautics is the study of the science of flight. Aeronautics is the method of designing an airplane or other flying machine.

THE CALCULUS of PROBABILITIES: As a

mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data.


DETERMINISTIC CHAOS: Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions.

http://en.wikipedia.org/wiki/Applied_mathematics

http://en.wikipedia.org/wiki/Applied_mathematics

http://en.wikipedia.org/wiki/Physics

http://en.wikipedia.org/wiki/Economics

http://en.wikipedia.org/wiki/Biology

http://en.wikipedia.org/wiki/Philosophy

http://en.wikipedia.org/wiki/Dynamical_system_(definition)


NEW CONEPTS of SOLUTION in DIFFERENTIAL EQUATIONS(1930)

Engineering and Mathematics in the last revolution of the Century. Computers and Computational Mathematics


The Computational Word, a new word for Mathematics.

The computer word is changing little by little the daily life of the citizen: bankig transactions, electronic mail, ticket reservations, .....


Mathematical Modeling, Mathematical and Numerical Analysis, Simulation, Vizualization, control.

Compuational Biology, Computaional Fluid Dynamic. Weather prediction, Astrophysics, mining, indusrial engineering, ……………….


A view has emerged where Computational Science is now the third leg of the scientific method together with Theory and Experiment, and this view is nowadays strongly practiced in Physics, Chemistry and Engineering.

Trends at the beginning of the XXIth century

Mathematisc in the Sciences,

Industry,

Management and

Business


Celestial Mechanics. Problems of aerospatial sciene. Stability and chaos in dynamical systems. Strange attrators. Mechanics of solids and fuids in zero gravity.

Theory of fuids .Appliation to meteorology and climatology


Aeronautis. Hydrodynamical problems, supersonic and transonic fight. Shock waves and hyperbolic equations.

Modern Physis.The Mathematics of the atomic world and of elementary partiles. Group theory, renormalization and Galule theories, supersymmetry, YangMills equations, instantons, dilatons, branes,...


Astrophysis. General relativity, stella rmodels Mathematics of plasma physis, magnetohydrodynamis. Kinetic equations (Boltzmann,Landau,Fokker Plank, Vlasov,...).

Geosienes. Problems of resources and mining.Environmental Problems(PD equations)


Materials Science. Modeling and simulation of composite materials, magnetic material, polymers, glass, and paper.

(Linear and nonlinear elasticity, calculus of variations. Homogenization theory.)


Nanotechnology. Integrated optics, optical networks.(wigner equation)

Indusrial Engineering. Steel industry, blast furnaces, car industry.

Communication. Telecommunication and optical networks. (Maxwell equations and Fourier heat theory)


Management. (Discrete mathematics, graph theory, combinatorics.)

Computer Sciences. Mathematical logic, algorithmics, computational complexity, parallezization………………..


Control, Optimal control, robust control, nonlinear control, Predictive control, ……….

Automation and Robotics. Algebraic geometry and computation.

Information theory. Coding of messages, error-correcting codes….


Statistics in Science. Industry, Government and Business. Estimation and hypothesis testing, design of experiments. Reliabilitiy, survival analysis.

Chemistry. Quantum Chemistry: simulation of atomic and molecular structures through fundamental equations.


Optimization Theory and Mathematical Programming. Integer Programming: Facets, Subadditivity, and Duality.Nonlinear programming, convex programming. Iterative Methods.Industrial Design Optimization. Numerical methods, partial differential equations, calculus of variations, combinatorics, linear algebra.


Problems of optimal transportation. Problems of traffic (with continuous and discrete modeling). Network planning. Traffic in the Web.

Economy. Financial mathematics (option pricing,derivative trading, risk management,…) unites stochastic differential equations partial differential equations and free boundary problems.Models for the global economy.


Biology: Mathematical Ecology, Epidemiology,

Biometrics, Bio-informatics. Mathematics of Genetics, Computational phylogenetics. Nucleic Acid structure and function. Molecular evolution. Proteomics. Regulatory and developmental pathway inference. DNA computation. Sequence alignment, fuzzy reasoning. Mathematical modeling in biopolymerization.


Medicine: interaction fluid-structure as a model for the blood flow. Modeling and simulation of the function of other organs: brain, lungs and liver. Self-organization and fractal geometries. Computational asistence of surgery. Pharmacokinetics, tumor growth modeling. Computational neuroscience. The Mathematics of infectious diseases and epidemic spreading. Artificial organs, immune system modeling.


Medical imaging methods. Tomography: computerized tomography, 3D image reconstruction. Fourier and Radon Transforms, inverse problems.


Though Computational Mathematics (as different from Computer Science) permeates all fields of application, it deserves a mention in itself: numerical methods and codes; efficient algorithms; approximation, (a priori and a posteriori) error estimates, adaptive methods and adaptive models, multigrid and domain decomposition, multiscale analysis, numerics of random processes,…


On the other hand, Mathematical Modeling in its different variants (deterministic, continuous, discrete, …) leads to the problems of Model Validation and the techniques of obtaining and elaborating data on which validation is based


(see Statistics above), as well as the quite important (and debated) concept of hierarchy of models, a progressive way of approaching “reality” that is nowadays recognized and embedded into the toolkit of the applied scientist (the old idealists with their eternal truth will revolve in their graves; or will they not?).

METABOLIC CONTROL

ANALYsis

Prof.Dr.Mustafa BAYRAMYıldız Teknik University

Istanbu

Abstract

it is important to the establishment of mathematical models for kinetic analysis of a biochemical system. Metabolic control analysis allows one to quantify the behaviour of a metabolic pathway in steady state in terms of dimensionless coefficients. From the definition of metabolite and flux control coefficients and elasticities we are able to derive symbolic forms of these parameters, in terms of conventional kinetic parameters. At the simplest level we are able to substitute values of these kinetic parameters, to yield values for the metabolic control coefficients. Since we are substituting into symbolic equations we can always quarantee the conservation relationships hold. The basic relationships are the summation and connectivity theorems. The ability to define the control coefficient equations in matrix form not only allows easy solution by numerical inversion but also opens up the possibility of obtaining the algebraic solutions by symbolic manipulation of the matrix. However for matrixes longer than rank 4 or 5, this latter possibility, if done by hand, becomes very tedious and is prone to error. The solution to this problem is to develop computer software to automatically carry out this procedure.

Metabolism

The chemical changes that take place in a

cell or an organism that produce energy and

basic materials needed for important life

processes.

Anabolism-Synthesis of complex

molecules from simpler ones (e.g amino-acid

synthesis)

Catabolism-Breaking down of complex

molecules to simpler, smaller molecules (e.g.

glycolysis)

Metabolic Networks

Reactions:

Pathways:

Networks:

A B

A B C D

A B C E

D

Metabolic Network –

Definitions (I)

A B C E

D

Reaction Intermediate

Substrate Product

Active reaction

Inactive reaction

Metabolic Network –

Definitions (II)

A B C E

D

Exchange flux

SystemBoundary

Internal flux

Flux – The production or consumption of

mass per unit area per unit time.

Metabolic Network Analysis:The Stoichiometric Matrix

Let us consider following enzymes

kinetic systems

1. X1

2.

*

3.

5.

4.

+

X1 X6+ X2

+X2 2X5 X3

X3+X4 X5

+X5 X6X4

v1

v2

v3

v4

v5 *


Where , (i=1,2,…,6) are concentration metabolites. We define the metabolite concentration vector as:

1

2

3

4

5

6

x

x

xX

x

x

x

ixX

Metabolic Network Analysis:The Stoichiometric Matrix In order to construct the model, we first write

the stoichiometric reaction scheme that describes how the mrtabolites combine [reder, 1988].

It will be convenient to associate to the reaction scheme the matrix of rows and columns

constructed as follows: the column of represents the reaction , and we write in this column at row . For example, for metabolite and reaction 1,

N m r

Nj

j

i1x

( , ) (1,1).i j N N


1

1

1

, if the reaction produces molecules of .

, if the reaction consumes molecules of .

0, if the reaction neither produces nor consumes of .

ij

j

j

j

x

N x

x


1. X1

2.

*

3.

5.

4.

+

X1 X6+ X2

+X2 2X5 X3

X3+X4 X5

+X5 X6X4

v1

v2

v3

v4

v5 *

For the system


The stoichiometric is:

10010

11200

11000

01100

00110

00011

N

Metabolic Network Analysis: Rate Vector

We assume that the rate of change of the concentration of metabolite is the sum of the rreaction rates, each weighted by the corresponding stoichiometric coefficient of metabolites.

Let denotes the rate of the reaction J, The rate vector of the above system can be written as:

Tvvvvv 54321v

jv

Definition 1.

For each metabolite Xi :

where Sij is the stoichiometric coefficient of the reactant Xi in the reaction j with the flux vj .

Negative if Xi is a substrate, positive if Xi is a product.

[ ]iij j

j

d X dS v

dt dt

xNv

Metabolic Network Analysis:Rate Vector

For the system 1. X1

2.

*

3.

5.

4.

+

X1 X6+ X2

+X2 2X5 X3

X3+X4 X5

+X5 X6X4

v1

v2

v3

v4

v5 *

Metabolic Network Analysis:Rate Vector

We write

5

4

3

2

1

6

5

4

3

2

1

10010

11200

11000

01100

00110

00011

v

v

v

v

v

x

x

x

x

x

x

dt

d

Definition 2.(Mathematical Model)

A mathematical model of the biochemical kinetics is written as following differential equations:

where vj is a function of concentransion of metabolite xi (the number of xi is m) and external parameters z (the number of z is p).

);(:)( zxdt

dz Nv

x

Definition 3.

At Steady-state

0Nv

Example: Stoichiometric Matrix

A B C

D

v1 v2

v3

b1 v6

v5

Eb4

v4

b3

b2

v7

1 1

1 4 2 3

Matrix Notation

2 5 6 2

3 5 4 7 3

6 7 4

Stoichiometric MatrixBalance Equations

1 0 0 0 0 0 0: 0

1 1 1 1 0 0 0: 0

= 0 1 0 0 1 1 0: 0

0 0 1 1 1 0 00: 0

0 0 0 0 0 1 1: 0

v b

v v v v

v v v b

v v v v b

v v b

A

BN

CNv=

D

E

1 0 0 0

0 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Exchange Fluxes Internal fluxes

Metabolic Control Analysis

Çok enzimli bir metabolik sistemde yer alan enzimlerin kinetik özellikleri bu sistemin özelliklerinin belirlenmesinde etkilidir. Ancak sistemi oluşturan enzimlerden herhangi birinin konsantrasyonunda yapılacak küçük bir değişimin sistemin hızını nasıl etkileyeceği, enzimlere ait kinetik çalışmalar ile açıklanamaz.

Metabolik Kontrol Analizi-Tanım

Metabolik kontrol analizi, çok enzimli bir metabolik sistemde, sistem hızının ve metabolik konsantrasyonlarının bu sistemde yer alan enzimler tarafından nasıl kontrol edildiğini inceleme metodudur.

Metabolik Kontrol Analizi-Biraz Tarihçe

Metabolik kontrol analizine ait ilk teori

Kacser ile Burn [8] ve Heinrich ile

Rapoport [6, 7] tarafından ileri sürülmüştür. Bunların dışında gerek biyokimyacılar gerekse matematikçiler tarafından bu konuda yapılmış bir çok çalışma vardır

Metabolik Kontrol Katsayıları

Metabolik kontrol katsayıları;

Sistem hızı(Flux) kontrol katsayıları,

Konsantrasyon kontrol katsayıları ve

Esneklik katsayıları olmak üzere

üç kısımdan oluşur.

Sistem Hızı Kontrol Katsayıları

Çok enzimli bir sistemde, herhangi bir enzimin aktivitesindeki değişiklikler, o enzime ait kinetik parametrelerden herhangi birinde yapılacak bir değişikle veya enzim konsantrasyonunda yapılacak bir değişiklikle gerçekleştirilir.

Çok enzimli bir sistemde, enzimlerden birinin konsantrasyonlarında yapılacak küçük değişiklik, bu enzimin katalizlediği reaksiyonun hızında da değişikliğe neden olacaktır. Bu ise, sistemin hızında bir değişiklik meydana getirir. Bu değişiklikler, sistem hızı kontrol kantsayıları ile tanımlanır.


Bir metabolik sistem için sistem hızı

de i. enzimin konsantrasyonu olmak üzere, sistem hızı kontrol katsayısı

olarak tanımlanır.

/ ln

/ lni

v

E

i i i

v v vC

E E E

viE


Çok enzimli lineer bir sistemde, sistem hızı kontrol katsayısı 0 ile 1 arasında değişen değerler alır. Lineer olmayan sistemlerde bu değerler daha küçük veya daha büyük olabilirler.

Sistemdeki herhangi bir enzim için SHKK’nın 0olması bu enzimin sistem hızı üzerinde etkisi olmadığı, SHKK’nın 1 olması ise bu enzimin sistem hızını tek başına tümden kontrol ettiği anlamına gelir.


Lineer olmayan bir sistemde SHKK’nın sıfırdan küçük olması enzim aktivitesindeki artışın sistem hızında bir düşüşe neden olacağını gösterir.

Konsantrasyon Kontrol Katsayıları -Tanım

Konsantrasyon kontrol katsayısı (KKK), çok enzimli bir sistemdeki herhangi bir enzimin aktivitesinde yapılacak değişimin, sistemdeki ara metabolit konsantrasyonlarına etkisini gösterir.

Konsantrasyon Kontrol Katsayıları

KKK’nın pozitif değeri enzimin aktivitesindeki artışla birlikte metabolit seviyesinde de bir artış olduğunu gösterirken,

negatif KKK değeri, enzimin aktivitesindeki artışla metabolit seviyesinde bir düşüşolduğu anlamına gelir.

Konsantrasyon Kontrol Katsayıları - Tanım

bir metabolik sistem için S bir ara metabolit ve de i. enzimin konsantrasyonu olmak üzere konsantrasyon kontrol katsayısı,

olarak tanımlanır.

iE

/ ln

/ lni

S

E

i i i

S S SC

E E E

Esneklik Katsayıları

Çok enzimli bir sistemde, her bir enzimin katalizlediği reaksiyon, diğer kısmından ayrı olarak incelenebilir. Bu şekilde elde edilen bilgiler sadece o enzimin katalizlediği reaksiyona ait olduğu için lokal özellikler olarak adlandırılır.

Bir enzim için esneklik katsayısı (EK), sistemdeki substratlardan veya ürünlerden herhangi birindeki küçük değişimin o enzimin katalizlediği reaksiyonun hızına etkisinin bir ölçüsüdür.


Bir metabolik sistemde i. reaksiyonun hızı ve S de bu reaksiyonun ürün veya substratlarından biri olmak üzere esneklik katsayısı,

şeklinde tanımlanır.

iv

/ ln

/ lniv i i i

S

v v v

S S S


SHKK ve KKK bütün bir sistem ile ilgili olduğu için sistem özelliği olarak isimlendirilir.

EK ise tek bir reaksiyonu ilgilendirdiği için lokal bir özelliktir.

Toplam Teoremleri

Çok enzimli bir metabolik sisteme ait SHKK’ları toplam teoremlerini sağlamak zorundadır.

Teorem 1: n-tane enzim içeren bir sistem için sistem hızı, de i. enzimin konsantrasyonu olmak üzere

dir [8].

viE

1

1i

nv

E

i

C

Toplam Teoremleri

Teorem 2: n-tane enzim içeren bir sistem için herhangi bir ara metabolite ve de i.enzimin konsantrasyonu olmak üzere

eşitliği sağlanır [4, 6, 7].

jSiE

1

0j

i

nS

E

i

C

Konnektivite Bağıntıları

SHKK ve KKK bir sistem özelliği olup sistemi oluşturan tüm enzimlere ve parametrelere bağlıdır.

Esneklik katsayısı ise sistemdeki tek bir enzim ile ilgili lokal bir özelliktir.

Konnektivite bağıntıları, esneklik katsayıları ile kontrol katsayıları arasında ilişki kuran bağıntılardır


n-tane enzim içeren dallanmasız çok enzimli bir sistemde tane bilinmeyen vardır. Toplam teoremleri ile bir tanesi, SHKK ile ilgili diğer n-1 tanesi de KKK ile ilgili olmak üzere n-tane bağıntı tanımlamaktadır. O halde böyle bir sistem için tane konnektivite bağıntısı vardır.

2n

2n n


SHKK ile EK lar arasındaki ilişkiyi gösteren bağıntı ilk kez Kacser ve Burns [3, 4, 8] tarafından bulunmuş olup bir iç metabolit olmak üzere


jS

31 2

1 2 3 0

j j j

vv vv v v

E S E S E SC C C L


KKK ile EK’lar arasındaki ilişkiyi gösteren bağıntı ilk kez Wasterhoff ve Chen [4, 18] tarafından ileri sürülmüş olup ve birer ara metabolit olmak üzere


jS kS

31 2

1 2 3

1 eğer ise

0 eğer ise k k k

j j j

S S S vv v

E S E S E S

k jC C C

k j

L

Applications Let us consider the following two enzymes

system: aspartate aminotransferase (AAT) and malate dehydrogenase(MDH), that is,

1

2

v

AAT

v

MDH

aspartate ketoglutarate glutamate oxaloacetate

oxaloacetate NADH malate NAD

Applications

The system given above contain two substrates and two producs and the system at steady state, That is,

Where is overall flux.

1

2

0

0

v v

v v

v

ApplicationsFlux control coefficients for the given

system:

and

To calculate the above flux control coefficients we have taken the begining value of the experimental date.

0.993544v

AAT

AAT vC

v AAT

0.006456v

MDH

MDH vC

v MDH

Applications

Sum of all the flux control coefficients of a reaction system is

Summation theorem is provided.

0.993544 0.006456 1.0v v

AAT MDHC C

Applications

Metabolic control coefficients (Concentration control coefficients):

and

0.468897oaa

AAT

AAT oaaC

oaa AAT

0.468897oaa

MDH

MDH oaaC

oaa MDH

Applications

Sum of metabolic control coefficients

0.468897 0.468897 0oaa oaa

AAT MDHC C

Applications

Finally, Elasticity control coefficients,

and

1 1

1

0.0137677v

oaa

oaa v

v oaa

2 2

2

2.11890v

oaa

oaa v

v oaa

Application

Connectivity theorem:

The connectivity theorem is provided.

1 2 0.00000v vv v

AAT oaa MDH oaaC C

the importance of mathematics in the development of...

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