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...

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

    

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, 9^th 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.

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, 2^nd 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.

Establishment of ESAN - MAGAZINE

To : Members of ESAN and its affiliates,

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.