ESAN - MAGAZINE
ESAN - MAGAZINE is a platform for members of ESAN and its affiliates to disseminate expository articles from education, science, technology, mathematics, engineering, social matters and economics that are friendly to professionals and non-professional members. The platform is intended to create interdisciplinary engagements in which professionals from different fields can share skills and established knowledge to create and enhance knowledge to provide solutions to problems.
Sunday, August 25, 2013
ESAN - MAGAZINE: The Ubiquitousness of Mathematics
ESAN - MAGAZINE: The Ubiquitousness of Mathematics: By Dejenie Alemayehu Lakew We say mathematics is a ubiquitous activity performed by nature at best and by we humans. The...
Saturday, August 24, 2013
ESAN - MAGAZINE: Economic Collapse – A Black Hole Syndrome
ESAN - MAGAZINE: Economic Collapse – A Black Hole Syndrome: Economic Collapse –A Black Hole Syndrome By Dejenie Alemayehu Lakew Mathematics justifies that greed and ex...
Economic Collapse – A Black Hole Syndrome
Economic Collapse –A Black Hole Syndrome
By
Dejenie Alemayehu Lakew
By
Dejenie Alemayehu Lakew
Mathematics justifies that
greed and extreme selfishness: causes economic collapse of societies.
Any system that is established from
relations between different groups in the system and the forces created
there off, continues to exist if the relations remain fair and forces that are
created from the relations remain on balance and valid to all parts. When one
of the forces dominate to the extent of diminishing the strength or eliminating powers
of others, then only a force of pulling towards the dominant group remains and
that leads to the collapse of the system.
I have few examples to validate this fact.
Example 1. Black hole -
galactic giant star that loses it’s force of balance against it’s own gravity.
The only force left that acts on it, is
its own gravity and the forces of pulling outward that emanate from itself and from
its surroundings that keep the balance of forces, in order the structure ( star) to exist, disappear due to its immense weight. The star
then collapses to either a mini size star or ultimately goes to its demise to a formation of a dark matter
of infinite density called a black hole where nothing escapes from it, and everything around sucked and devoured including light. Therefore the shining star
of a galaxy disappears from its starhood existence due to imbalance of forces
that holds it.
Example 2. Preys – predators in an ecosystem. We create an
ecosystem of the following things: grass, rabbits and foxes. Rabbits eat grass
and foxes consume rabbits and therefore rabbits are preys to foxes and foxes
are predators on rabbits. We can write mathematical models that studies this
system and consider different scenarios Their continuous co-existence is guaranteed only if they keep their desires in balance and greed in control. That
is, the rabbits should not eat all the grasses available and run out of food
and there by endanger their existence and threatens the existence of the foxes as well
since foxes consume rabbits. Similarly if the foxes get extremely greedy and
selfish and eat all the rabbits in certain time interval, then that is again a recipe for destruction of foxes as they will not have any thing to eat after
sometime as no rabbits left to live, reproduce and multiply. Therefore, greed
and extreme selfishness of the predators lead to the destruction of the vibrant
ecosystem of the two living species.
A mathematical model called Lotka-
Volterra of differential equations studies such relations and some other
similar systems in which species compete for resources, living areas, etc. or
others cooperate and live together creating a vibrant system that works for all
involved. But in the prey – predator model, the extreme case scenarios are that when foxes
are increasing in huge amounts and eat more rabbits, then the rabbits number decreases considerably to the point of disappearance. But there is always a perfect
condition in which the two species live together indefinitely, by keeping
selfish desires in check and live for ever or be greedy and extreme selfish but
disappear together for ever.
Example
3. Economic systems - systems formed from fiscal relations between
different sectors of a human society. In such systems, there are groups called
consumers that purchase services products and utilities and companies that
produces them, and the the fiscal/financial
relations created between them should indeed be healthy, honest, fair, ethical
and above all humane so that the economic relations formed and the forces of
the financial transactions created, stay on balance and remain valid for all
participants so that the economic structure created exists indefinitely.
Here we can look at two kinds of relations that are prevalent in such
systems:
(I) Type I : Relations within
companies or intra companies in which companies compete to get more markets and more customers
– their relations can be considered as competition to annihilate, in which the
existence of one is a treat to the growth and welfare of the other and
therefore working hard and playing any tricks to eliminate
the other is the motto of the game. Here is what capital theorists call it monopoly comes
to play.
(II) Type II : Relations between the
populus or majority consumers and companies – this relation is similar to the
prey - predator model in which companies play as predators while the majority
consumers as preys – they need each other but for wrong goals, from each sides
perspective. However, the economic relations created between these two groups,
consumers and companies, should again be healthy, fair, truthful or honest and
above all ethical and humane. If companies develop extreme greed and selfish
trends, losing sights of connecting to their consumers as their humane benefit
partners, and completely disregarding the difficulty of financial resources and
fiscals troubles of working people, taking an imaginable and unreasonable
amount of profits from these consumers, then a black hole syndrome will be
created between these two partners of the economy, which eventually lead to the
collapse of the economic system they created.
Case in point: The demise of real estate in America – real estate companies
and banks related to real estate business that offer loans for home buyers, were blinded by extreme profit making to an
unbearable height, wiping out the financial capabilities of their customers to
the point of being unable to pay their
timely bills, resulted in the abandoning
of homes by huge numbers and led to the complete collapse of the real estate business itself and bankings associated with it. The actual mathematical
justification here is very trivial. If a company uses a profit maximizing
function in which the inputs are from variables that are available based on the
input-response systems of the state of the economy, then trivially, one or more
of the inputs were put falsified, such as the capability of the consumers to sustain paying bills, while their income was dwindling by the day. Thus, the
calculations lack honesty, not being
truthful and therefore violates few of the fundamental rules of credit – trust
and the ethics of reporting facts
truthfully.
This again is a an example of a system that loses its stability by loosing
the forces of balance and fairness that
form the system and thereby creates its own destruction.
Lesson learnt: Although we do not have control over things that exist
outside of our power, such as stars changed to black holes, but we can avoid catastrophes on things we humans created to serve our selves and can have a
control over and make them function properly as needed in a robust way and make
them exist indefinitely. This is possible by keeping the forces that form the
system in balance and making human transactions ethical, fair, honest and
trustworthy.
Conclusion: In relational
existence, such as an economy, or other social matters, the innate behavior of species to damage self
to the welfare of others called altruism/selfless
is a very remote possible antithesis of
what is termed as selfishness/extreme
greed - benefiting self on the welfare of others. But between the two extremes,
selfless and selfishness, there is a golden mean – virtues of cooperation to the welfare of all and
even in some sense of positive competition
for betterment and growth, as long as the games are played by ethics and correct
rules, following principles of trust, honesty and responsibility, so that the
forces involved remain operational, valid and on balance so that all parties involved
remain partners of the process/system
and the system continues to exist indefinitely – with no black hole syndrome.
References:
[1] Rees M.
J., Volonteri M. Massive black holes: Formation and Evolution (2007).
[2] Dennis G.
Zill, A First Course in Differential Equations with Modeling Applications, 9th
ed.
[3] Aristotle,
Ethics (1976).
[4] Fehr E.,
Fischbacher U., The Nature of Human Altruism, Nature 425 (2003).
[5] Axelrod R.
and Hamilton W D., The Evolution of Cooperation, Science 1981.
Saturday, August 10, 2013
The Ubiquitousness of Mathematics
By
Dejenie Alemayehu Lakew
We say mathematics is a ubiquitous activity performed by
nature at best and by we humans. The mathematics of humans as an endeavor of
human intellect is a systematic study of space, quantity, numbers, change and
patterns or structures that either exist naturally or constructed abstractly
using the principles of logic and deductions largely aided by imagination. We
humans create models to study, navigate and discover the intricacies of the mathematics
of nature - the most common phenomena in which the universe is ruled under.
Nature is the greatest mathematician of all which does mathematics the best –
as mathematics is the working language of nature and of the greater universe
that is observable or otherwise. In this part of my exposition, I will write
the abundance of mathematics that is ubiquitous in nature.
I present few of the mathematics nature perfectly does and
speaks to us:
(1) The display
of sophisticated and intricate wonders, beauty and symmetries that are abundant
in nature, such as fractals and chaos. There is a branch of philosophy called
aesthetics that studies beauty and nature.
(2) Particular
suited elliptic paths planets and other galactic objects follow to rotate
around a central object such as the sun.
(3) The
formation and ultimate death of stars from a purely mathematical and physical perspective.
(4) The fascinating
natural process how a conception develops and the timing it requires to come
out of a mother’s womb.
(5) Shading of
their leaves trees do in cold tropics, to hibernate and protect themselves from
severe cold weather and the time they start to blossom when spring comes. The
hibernation mechanism is done partly by reducing the size of their parts
exposed to the outside environment in order to reduce the diffusion of cold in
to them – fascinating mathematics.
(6) The
periodicity of natural phenomena we see every year or season, that exist indefinitely
but in a bounded domain of either temporal or spatial. Periodicity in general is
one of nature’s way of displaying it’s work of mathematics.
(7) The sizing of petals or leave surfaces
by plants based on where they grow (arid
or wet and rainy zone ) to control evapo – transpiration. Here, we observe a sophisticated and extraordinary
mathematics performance of a resource management type in which a tree in a very
arid zone minimizes the size of its petals in an optimal way to:
* control evaporation, as the rate
at which water evaporates out is
directly proportional to the surface area of the petal, and at the same time
* enable the tree to track
enough amount of sun light in order to process its food.
These are few examples from the many
perfect mathematical performances of nature.
We humans try to understand
how nature does mathematics, by creating abstract models that imitate nature
and prove and justify the truth of things in nature. Things naturally work and function in an optimal way with
minimal errors and a small change in
parameters that govern a phenomena will create a huge change on the result
- which shows how nature is stable in a larger or what we call
global perspective but at the same time chaotic
locally. We see the chaotic part of
nature by looking at the effects of a
very minute change in the DNA results in a huge difference in creatures -- for instance we humans and chimpanzees have
almost similar DNA sequencing with a very minute differences, but that very
small difference creates that huge species difference.
Therefore as our mathematical activities, we represent
quantities, axiomatize, hypothesize/make conjectures and theorize through
mathematical expressions of symbols, variables and assumed to be properties, to
prove and validate what we assumed is naturally true and valid. For instance we
hypothesize and validate empirically that when a ball is rolling over a frictionless
inclined surface, the distance the ball covers is directly proportional to the
square on the time it takes to move from one point to the next lower point.
Next, I will discuss about a particular path of moving from
one point to the next lower point which expedites time. For curiosity, which path do you think
provides the shortest time in moving from one point to the next on a vertical
plane which lies below but not on a vertical line? You may think the one which is the shortest
segment or straight line segment that
connects the two points has the shortest time, but that is not true. There is a
longer path from the shortest path that will provide the shortest time – it defies
common sense but true.
Such paths are needed
to be followed to win in sports such as board skating and skiing. Every four
years at Summer Olympics, athletes of skiing compete in a mountain side that is
full of ice – called skiing. The game is to reach to the destination point down
the hill with the shortest time. Assume all the participants of the game have
same velocity, then one can ask, will there be a possibility of the existence
of one person with the shortest time? The answer is yes. Here is how.
Before I provide the
answer, let me say something about the history behind this path or curve of
shortest time called brachistochrone.
In 1696, a mathematician named Johann
Bernoulli challenged mathematicians of Europe by posing a problem called the brachistochrone problem. The problem was, given two points P and Q in a vertical plane in which both are not in a
vertical line but Q is below P. If a body is moving frictionless by only its own gravity along a path that connects both P and Q,
which path will be the one with the least time ? As I said, the shortest segment will not
provide the shortest time, but it is a curve called the brachistochrone – Greek
word, which is a concatenation of:
brachistos – shortest and chronos
– time. The answer was given by several mathematicians of the time, such as Isaac
Newton, Jacob Bernoulli (brother of Johann Bernoulli), Gottfried Leibniz, etc. Literally, the curve is a segment of a cycloid - a suspended cable on two
poles. Therefore the body should follow a brachistochrone,
the path with shortest time from P to Q.
Therefore, athletes who
compete for Summer Olympic of skiing, the one to be a winner, has to go from point to the next lower point making zigzag like movements but following a path of
a brachistochrone between consecutive
points, until he/she riches the destination point. The one who almost makes
such paths on the way down, although difficult to get those paths perfectly and
continuously, will be the winner. But
because they also have different speeds, the combination of their varying speeds
and the paths they follow enable one to be a winner.
Natural examples who use such paths - paths of shortest time
to pick their prey from below are seagulls or birds.
Seagulls or birds in general are one of the most fascinating
creatures of nature - the flights, swifts, turns, dives and rises they make.
Their flight mechanisms inspire humans the ambition to fly and hence a source
of research for applied mathematicians and engineers alike for designing planes
and their wings in regard to air dynamics and gravity to create levitation.
Besides their fascinating acrobatic flights and perfect flawless moves they
make, seagulls or birds also do amazing mathematics of differential geometry and
physics. When they move from above to pick a prey they see on the ground or
inside a sea or sea shore, the path they chose is not the straight segment from
their position to the prey, but the path with the shortest time to reach to the
prey – the brachistochrone. By choosing
such a perfect mathematically proven path, a path of shortest time, birds and
seagulls pick their prey swiftly and quickly – a fascinating natural act of
doing mathematics.
Therefore,
(8) Brachistochrone – the optimal nature’s
curve/path of smallest time.
References:
[1]. Courant R. and
Robinson H., What is mathematics? An elementary Approach to Ideas and Methods,
2nd ed. Oxford, England: Oxford University Press, 1996.
[2]. Wens D., The Penguin
Dictionary of Curves and Interesting Geometry, London, Penguin, p. 46, 1991.
[3]. Haws L. and Kiser
T., Exploring the Brachistochrone Problem, American Math. Monthly, 102,
328-336,1995.
[4]. Gardner M., The
Sixth Book of Mathematical Games from Scientific America, Chicago, IL:
University of Chicago Press, pp. 130-131,1984.
[5]. Mandelbrot Benoit,
Fractal Geometry of Nature, McMillan 1983.
[6]. Russ John, Fractal
Surfaces, Springer, ISBN 978-0-306-44702-0.
Wednesday, August 7, 2013
Establishment of ESAN - MAGAZINE
Re: Establishment of ESAN - MAGAZINE
As I have indicated to you in my previous post the necessity of a platform for ESAN members and its affiliates to disseminate expository articles on education, science, technology,
mathematics and issues that are socio-political, economic or cultural in content, to initiate a culture of publications.
Pursuant to that communication, I have established ESAN - MAGAZINE.
Articles considered for the magazine should have at least one or more of the following features:
* Motivational
* Inspiring
* Educational
* Fact supported
* Fibers of cohesion to the Ethiopian society.
I look forward hearing from you and call upon you to be actively participate in article production and make ESAN and its affiliate magazine a lively destination.
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